Optimal Stopping with General Risk Preferences∗ Jetlir Duraj† Abstract We characterize in terms of primitives on risk preferences the stopping and the continuation regions of optimal stopping of diffusions. We consider separately the case of a naive agent who is unaware of the possible time inconsistency and the case of a sophisticated agent who is fully aware of a possible time inconsistency in her behavior. We apply the general result to several well-known risk preference models. The applications suggest that specific models of probability weighting are the only risk preferences which can exhibit extreme behavior in the sense of naive agents always continuing with positive probability or sophisticated agents never starting. Keywords: Optimal Stopping, Non-Expected Utility, Naive Agent, Sophisticated Agent. JEL Classification: D01, D03, D81.

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Introduction

There is a vast literature in economics which applies optimal stopping problems to economic situations. Applications include models of gambling, investment, search behavior, but also models of experimentation and information acquisition. In a general stopping decision problem an agent is facing a sequence of lotteries over prizes and deciding at each moment in time whether to stop the sequence of lotteries and leave with the cumulative realizations of the lotteries till that moment in time, or to continue and face another prize lottery. For example, in a job search model the prizes could be a monetary compensation the agent receives for performing a task and lotteries model uncertain compensation in the next period. In a casino gambling setting lotteries are bets on monetary outcomes whose realizations will add or subtract to current wealth of the gambler. The main approach used in the literature postulates a static preference of the agent over prize lotteries and determines the continue/stop choice over time based on these preferences. Classical work uses Expected Utility as a model for the preferences over lotteries. Violations of the main behavioral implication of Expected Utility, the Independence axiom, are well-documented in the empirical and experimental literature and alternative non-Expected Utility models for decision making under risk have been developed in the literature. In Barberis (2012), Ebert, Strack (2015) and Ebert, Strack (2016) the authors ∗

I am especially grateful to Drew Fudenberg and Tomasz Strzalecki for continual support and advice. I am thankful to Krishna Dasaratha and Kevin He for making extensive comments on earlier drafts of the paper. I also thank Harry Di Pei, Jonathan Libgober, Eric Maskin and Matthew Rabin for many useful discussions. Any errors are mine. † [email protected]

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consider the implications of a specific non-Expected Utility model in optimal stopping problems, namely Cumulative Prospect Theory. This paper characterizes behavior in optimal stopping problems for general risk preferences which are required to satisfy only minimal technical properties. More precisely, we don’t impose any functional restrictions on risk preferences, besides that they are complete, transitive, continuous and that they additionally satisfy first-order-stochasticdominance-monotonicity (FOSD-monotonicity).1 If the risk preference of the agent in the optimal stopping problem is not Expected Utility, the agent will not be dynamically consistent. It follows that knowing the risk preference of the agent is not sufficient for the full description of her behavior in dynamic problems. Therefore, auxiliary rules are needed to fully specify dynamic behavior. Here we mostly focus on the two rules most used in the literature: naivet´ e and sophistication. The optimal stopping problem has two sides: the preference and the technology side. For the technology side of the problem, we assume that the sequence of wealth lotteries facing the agent comes from a diffusion.2 Diffusions have become popular in economic theory as a modeling tool of exogenous variation in the prices of assets or wealth. In particular, the diffusion assumption allows us to compare our results with related, more specialized work on optimal stopping with non-EU preferences, e.g. the CPT model used in Ebert, Strack (2015) and Ebert, Strack (2016), where the same diffusion assumption is made. The main contribution of this work is the full characterization of the continuation and stopping regions for general risk preferences for both naive and sophisticated agents as well as the application of these general results to a wide range of classes of well-known non-Expected Utility risk preferences. We focus first on the naive case. We give a full characterization of the continuation and stopping regions of a naive agent. This gives as a byproduct the full characterization of the starting/not starting decision of an agent who has commitment. A preference condition called weak Risk Aversion plays a central role in the characterization of naive behavior. Weak Risk Aversion says that an agent always weakly prefers to get the expected value of a lottery with probability one than face the lottery. We show that a sufficient and necessary condition for an agent to continue with positive probability when facing any ’Martingale’ diffusion is that the agent’s risk preferences violate weak Risk Aversion at the state of the diffusion. We then show how this result generalizes to arbitrary diffusions. For an arbitrary diffusion, continuation at the current state of the diffusion happens with positive probability if and only if the same agent but who rescales the prize space appropriately and so that starting point of the optimal stopping problem is mapped to zero, is weakly Risk Averse at zero. It turns out this full characterization of naive behavior, albeit seemingly convoluted, is amenable enough to allow for characterizations of naive behavior for risk preferences far beyond Expected Utility. Violations of weak Risk Aversion can be checked 1

A preference satisfies FOSD-monotonicity, if the agent always prefers a lottery to another lottery, if the former first order stochastically dominates the latter. 2 Diffusions have a rich structure in the sense that different stopping policies can induce a relatively large set of feasible lotteries, especially when compared to alternative discrete-time stochastic processes. One could think of the diffusion as an ad-hoc model of a ’deep’ financial market or of an ’advanced’ casino setting offering a wide variety of gambling/investment opportunities. Having a ’relaxed’ side of the technology side shifts focus to the preference side of the model.

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easily if one has a utility representation of the agent’s risk preferences and for all wellknown models of risk preferences there are utility specifications which allow the needed violations of weak Risk Aversion. Given the importance of weak Risk Aversion for the naive case, we also relate it in the online Appendix to the paper to the traditional concept of Risk Aversion, defined as aversion to mean-preserving spreads. Namely, we find the appropriate relaxation of Independence which when added to weak Risk Aversion becomes equivalent to Risk Aversion. We establish as well a similar connection between the negation of weak Risk Aversion and Risk Loving attitude, defined as preference for mean-preserving spreads. To analyze the sophisticated case in generality, we make use of the assumption of strict FOSD-monotonicity in addition to the basic axioms of rational and continuous preference. We model the sophisticated case as a game between selves, namely as a game between current selves and future selves. This is in general a difficult game to study as it has a continuum of players, besides the large and intractable strategy space of all possible stopping strategies. Consequently, we restrict the strategy space of the agent and the equilibrium concept of the game the sophisticated selves are assumed to play. We assume that the sophisticated agent can only use pure Markov stopping policies and we allow the agent to randomize among them at time zero only. Due to the continuity of the process, this is equivalent in our setting to restricting the agent’s policy choice to the set of simple threshold stopping times and randomizations among them at time zero: the sophisticated agent stops whenever the process leaves an interval of wealth levels and she can randomize at time zero among these intervals of wealth levels. The same setup, but without the mixing at time zero, is used in Ebert, Strack (2016) in the special case of CPT preferences. The mixing at time zero may be beneficial for an agent who has convex risk preferences. We also show that whenever the risk preference of the agent is quasi-convex the agent doesn’t randomize at time zero and so chooses among those simple threshold stopping times which she knows can be implemented by her future selves. An alternative solution concept to Markov Nash Equilibrium is the stronger concept of Markov Perfect Equilibrium. In general, there is no Markov subgame perfect equilibrium in pure strategies in this model. For tractability reasons we focus on Markov Nash equilibria in this paper. For the sophisticated case, the characterization of the continuation and stopping regions for a fixed diffusion depends on the local comparison of a technological constraint originating from the diffusion, called the win probability of the diffusion, and of a preference constraint which we call the calibration function of the risk preference. These can be explained as follows. For any two distinct prizes and an intermediate prize, the diffusion started at the intermediate prize and stopped whenever it reaches either the highest or the lowest of the three prizes defines a binary lottery with support on the lowest and highest of the three prizes. This lottery is identified uniquely with the probability of the highest prize. The latter probability varies as the intermediate prize varies, according to a function which we call the win probability for the two distinct prizes picked at the start. Moreover, for any two distinct prizes and an intermediate prize there exists due to our assumptions on the risk preference a unique lottery with support on the highest and lowest of the three prizes to which the agent is indifferent when the alternative is to get the intermediate prize for sure. This lottery is again identified uniquely with the probability it puts on the high prize. As the intermediate prize varies, this probability varies according 3

to a function we call the calibration function. The sophisticated agent is able to implement a strategy consisting of stopping whenever the current state of the diffusion is above some threshold or below some other, lower threshold if and only if the technological constraint is looser than the preference constraint: the win probability for the two thresholds has to be higher than the calibration function, and this for all intermediate prizes. Otherwise, there will always exist a self, identified with an intermediate prize within the interval bounded by the thresholds, who will disobey the continuation recommendation. This characterization helps us find an if and only if condition on preferences, called extreme sensitivity to risk, which uniquely characterizes sophisticated agents who never start any diffusion. Extreme sensitivity to risk is given whenever for all fixed pairs of high and low prize, the slope of the win probability function at the low prize is infinite or the slope of the win probability at the high prize is zero. It turns out that for most models of risk preference it is easy to find conditions under which extreme sensitivity to risk is not given and thus the sophisticated agent starts some diffusion. The only exception are certain models of probability weighting. We illustrate the general results for both the naive and the sophisticated case with different classes of risk preferences. These include well-known models of risk preferences which are either quasi-concave, quasi-convex or satisfy Betweenness. We show that our general characterization results allow one to find easy-to-verify sufficient conditions which ensure that any of naive or sophisticated agent stops or continues any particular diffusion. Our results also generalize and clarify the extent of the results in Ebert, Strack (2015) and Ebert, Strack (2016), who focus on Prospect Theory and who just as in this paper, use the diffusion assumption. Our results suggest that extreme behavior like the naive agent always continuing with positive probability and the sophisticated agent never starting irrespective of the diffusion faced can generically not be replicated outside of models of probability weighting. Moreover, within the class of probability weighting there exists specifications where neither of the extreme behaviors mentioned above occur. Finally, underlining the extreme character of the above mentioned ‘always continuing and/or never starting’ properties, we show that all other preference models we apply our characterizations results to, are similar to Expected utility in the sense that, even a naive risk loving agent will stop very ‘unfavorable’ diffusions and even a sophisticated risk averse agent will choose to start diffusions which are ‘favorable’ enough. In the online Appendix we also give foundations for the model we use in terms of a history-dependent collection of preference relations defined over the set of stopping times. We characterize uniquely Expected Utility by an axiom on dynamic behavior in this setting: Dynamic Consistency of Preferences. Dynamic Consistency of Preferences says that a strategy which is dominated by an alternative one, no matter the future continuation of the decision problem, cannot be preferred to the alternative one at the current moment of time. The characterization of Expected Utility through dynamic axioms like Dynamic Consistency of Preferences in finite dynamic choice problems is a classical result, which to the best of our knowledge, hasn’t been established in our setting of continuous time and with stopping times as objects of choice. Since the proof is elaborate and non-trivial in our setting, we include it in the online Appendix. The fact that the classical characterization of Expected Utility as the unique static preference compatible with Dynamic Consistency holds in our setting is valuable, as it 4

provides a theoretical justification in continuous time for the necessity of either of the naivet´e and sophistication axioms, whenever the agent’s static risk preference is not Expected Utility. The rest of the paper is organized as follows. The next subsection discusses related literature. Section 2 introduces formally the set up used in this paper and the related technical machinery and definitions. Section 3 is devoted to general results for the cases of naive and sophisticated agents. Section 4 contains applications to different classes of risk preferences illustrating the value of the general results. The last section concludes. The Appendices in the main body of the paper contain the proofs of the results in the main body of the paper and some technical details about the stochastic process used in this paper. The online Appendix contains an analysis of behavioral foundations of the general model used in this paper and characterizes which axioms of dynamic behavior have to be relaxed if the agent is not an Expected Utility-maximizer. It also contains additional applications of the general characterization results and a characterization of the relation between weak Risk Aversion and Risk Aversion.

1.1

Related Literature.

To the best of our knowledge, Karni, Safra (1990) is the first paper which considers an optimal stopping problem with general risk preferences. They look at a model of an agent without recall, who faces a finite stream of sampling opportunities from a known distribution. At each discrete moment in time the agent decides whether to draw another sample or to stop. The payoff of the agent when stopping is the realization of the last sample. They concentrate on the sophisticated case and solve for the optimal randomized strategy without recall (Markov policy). They find, that if the agent’s preferences are quasi-convex, the optimal stopping rule is deterministic and takes a simple threshold form. This is similar to both our result about the sophisticated agent with quasi-convex preferences not randomizing in period one as well as to our choice Markov strategies for the sophisticated case. Barberis (2012) considers a finite horizon gambling model, where an agent who has either commitment or is otherwise naive or sophisticated about her time-inconsistency, faces a finite stream of binary lotteries and has preferences of the Cumulative Prospect Theory (CPT) sort. He shows through examples and simulations that CPT preferences can explain a wide range of behavior, both in the naive and sophisticated case if the agent is facing zero-mean bets. Our examples and applications here show that this wide range of behavior is, at least in part, not a feature of CPT per se, but more generally of risk preferences. Namely for the Markovian, more general counterpart of zero-mean bets in continuous time, Martingale diffusions, we can easily characterize optimal stopping behavior through our general results. For example, we show that for a naive agent, the relevant risk preference primitive deciding continuation/stopping is weak Risk Aversion, a general property not only confined to CPT models. Barberis (2012) also emphasizes the fact that his CPT model and especially probability weighting, causes the agent to be dynamically inconsistent. Our result in the supplementary material formally proves that dynamic inconsistent behavior is not only not restricted to CPT, but is a general feature of non-Expected utility models in our setting. In Ebert, Strack (2015) the authors find sufficient conditions for the extreme result 5

of always continuing with positive probability any diffusion for the naive agent3 and in Ebert, Strack (2016) they find sufficient conditions for the extreme result of never starting any diffusion for the sophisticated agent. Our results help us characterize fully when in the case of a CPT sophisticated agent the never-starting result holds. Moreover, they complement the main result of Ebert, Strack (2015) by showing that other well-known risk preference models, be they quasi-concave or quasi-convex, don’t exhibit the extreme behavior in the naive case. He et al. (2016) also consider an agent with CPT preferences who faces an infinite sequence of fair bets and either has commitment or is naive about any possible dynamic inconsistency. In this setting they find tight characterizations in terms of the functional forms of CPT for the behavior of the agent under commitment. They also characterize naive behavior under certain CPT specifications. In particular, they find sufficient conditions when the naive agent never stops. Their analysis of the naive case is less complete than ours, which is understandable due to the discrete time stochastic process they use. Due to the different assumption on the stochastic process, their analysis can be considered complementary to the one in this paper, for the case of CPT preferences only. Finally, dynamic foundations for Expected Utility as a risk preference in finite lottery tree environments are well known in the literature (see Hammond (1988), Hammond (1989), Gul, Lantto (1990) and references mentioned therein). In this environment, Expected Utility is equivalent to two axioms of dynamic choice: Consistency and Dynamic Consistency of Preferences. We prove in the online appendix that a similar result holds in our continuous time diffusion setting. Besides serving as a formal motivation for auxiliary rules of dynamic behavior, such as naivet´e and sophistication, this also justifies the use of continuous time methods in our setting.

2

Set Up

An agent is equipped with a complete and transitive preference  over the space of Borel probability distributions over a prize space [w, b], i.e. a preference over ∆([w, b]).4 We denote by δx for x ∈ [w, b] the degenerate probability distribution which yields x with probability one. In text we will use interchangeably the name lottery for probability distributions from ∆([w, b]). Throughout we will assume the following about the static risk preference of the agent. Axiom 1 - Continuity: The sets {F ∈ ∆([w, b]) : F G} and {F ∈ ∆([w, b]) : F G} are closed in the topology of convergence in distribution. 3

Ebert, Strack (2015) allow only for pure strategies in their model and so interpret their result as the naive agent never stopping. Henderson et al. (2017) shows by examples that when mixed strategies are allowed in the set up of Ebert, Strack (2015) the naive agent may optimally choose to stop with positive probability, albeit less than one. In light of this result, the right interpretation of the main result in Ebert, Strack (2015) is that of a naive agent never stopping with probability one, or equivalently, of always continuing with positive probability. 4 Almost all of the preference models we apply our general results to are defined for lottery prizes in a bounded interval as [w, b]. Moreover, we consider this assumption as much more realistic than the alternative assumption of [w, +∞), which would allow for infinitely large amounts of wealth to be feasible.

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Axiom 2 - FOSD-monotonicity If F strictly FOSD-dominates G, then F G.5 It is a standard result, that under Axiom 1 the agent has a utility function V : ∆([w, b])→R which is continuous in the topology of convergence in distribution. Under Axiom 2 this utility function is increasing with respect to the FOSD-order. For the characterization of naive behavior we don’t need Axiom 2. We nevertheless impose it in all of the following analysis, as it is standard in the decision-theoretic analysis of risk preferences. The agent in our model faces a sequence of lotteries in continuous time, which are generated by a diffusion dXt = µ(Xt )dt + σ(Xt )dWt ,

X0 = y0 .

(1)

We call a pair (X, y0 ) consisting of a diffusion process and a starting point y0 ∈ [w, b] a stopping problem. Here the exogenous uncertainty is modeled with help of the Brownian motion (Wt )t∈R+ and the drift µ : [w, b]→R together with the volatility σ : [w, b]→(0, +∞) are assumed Lipschitz continuous.6 We also assume that the diffusion is stopped, once it leaves [w, b]: the lower bound is a limited liability constraint of the agent, while the upper bound excludes gambles with arbitrarily large prizes. The assumption that the variance coefficient σ is bounded away from 0 means in economic terms, that the uncertainty that the agent faces at each moment in time is always non-negligible, independently of the current state of the diffusion. We call diffusions satisfying these properties regular diffusions. One can think of regular diffusions as modeling the value of an asset. Implicitly, in applications of this model we are assuming away bubbles or that arbitrarily large positive or negative wealth is possible (limited liability) and that the uncertainty remains non-negligible throughout time. We consider the process Xt , t ≥ 0 as living in the Wiener space C([0, ∞), [w, b]) of continuous functions with image in [w, b] and adapted to the filtration of the Brownian motion, which we denote by F = (Ft )t≥0 . The diffusion model in (1) is more general than it seems. In particular, we can model costs of continuation into the drift of the diffusion as long as they are time independent.7 Discounting could also be easily introduced; for example if the diffusion in (1) is of the geometric Brownian motion sort, as long as it is assumed that the discounting factor is constant over time.8 If we would allow for timedependent discounting, this would introduce an additional source of dynamic inconsistency besides the one generated by the violation of Independence for the static risk preference. We avoid this important other source of dynamic inconsistency in this paper. 5

F FOSD-dominates G if for all x ∈ [w, b] we have F (x) ≤ G(x). This is an incomplete, transitive, reflexive binary relation on ∆([w, b]). The relation ‘strictly FOSD-dominates’ is the irreflexive part of the ‘FOSD-dominates’ relation. 6 Lipschitz continuity and that the variance coefficient is bounded away from zero are standard assumptions to ensure existence of unique, strong solutions of the stochastic differential equation (1). Lipschitz continuity can be relaxed without changing the results of this paper. See Appendix A for more on this and more generally about the justification of the regularity assumption we make on the diffusion. 7 For a given cost parameter c, the net drift including costs of continuation would be modified to (µ(Xt ) − c)dt. 8 E.g. if (1) reads dXt = µXt dt + σXt dWt , X0 = y0 > 0 then discounting flow payoffs by a constant discounting factor exp(−r) can be modeled as looking instead at the diffusion process dXt = (µ−r)Xt dt+ σXt dWt , X0 = y0 > 0.

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One way to think of the regular diffusion assumption is that of an ad hoc model of a ’deep’ financial market, which allows investors to replicate through stopping strategies payoffs of a very wide variety of assets. This set of strategies is not unrestricted though: one has to assume away strategies of the doubling type, which would allow an agent with arbitrarily negative starting wealth to achieve arbitrarily positive wealth levels. One possible such strategy is ’waiting for the best prize b’. The problem with this kind of strategy is that they are not uniformly integrable, i.e. it would take on average an infinite amount of time for the agent to implement it. Due to limited liability, waiting on average for an infinite amount of time is not feasible in our model. In all, the set of pure strategies available to the agent in this setting is identified with the set of uniformly integrable stopping times. For parts of the analysis we also assume an outside source of randomization in the form of a randomization device, independent of the Brownian motion driving the uncertainty of the prizes, which the agent can use to mix between possible pure strategies. Formally, this source can be identified with the measure space ([0, 1], B([0, 1]), λ)), where B([0, 1]) stands for the Borel sigma-Algebra of [0, 1] and λ stands for the Lebesgue measure. The following definition formally defines the types of strategies we are considering. Definition 1. (1) (Pure Strategies) A stopping time τ is a [0, ∞]−valued random variable, such that for all t ∈ [0, ∞), the event {τ ≤ t} is contained in Ft . (2) (Mixed Strategies) A randomized stopping time is a (B([0, 1]) × F)-progressively measurable9 function κ : [0, 1] × C([0, ∞))→[0, ∞] such that for every r ∈ [0, 1], κ(r, ω) is a stopping time. (3) A stopping time is called a simple threshold stopping time if it is given by τa,c = inf{t ≥ 0|Xt ≤ a or Xt ≥ c}, for some a < c, a, c ∈ [w, b]. Randomized stopping times are well-known in mathematical game theory.10 See for example Laraki, Solan (2005) who in their Definition 1 use the same concept of randomized stopping times as in part (2) of the above definition. The idea is that, independently of the realization of the path of the diffusion, a coin is thrown at the beginning that determines which stopping time the agent then decides to implement. Implicit in this definition is that the recommendation of the coin is binding and will be followed by the agent. Intuitively, with a simple threshold stopping time the agent wants to cap both losses and winnings. It is well-known that when  has an Expected Utility representation, the solution to the optimal stopping problem can always be taken to be of this form. They can be identified with pure Markov strategies of the agent, because, as is clear from the definition, the decision of whether to stop or not at a specific time t depends only on Xt and not on the path of the diffusion before time t. This is a stronger requirement than 9

See Appendix A for more details on this technical definition. Different papers use different names for different concepts of randomization with stopping times. See for example Shmaya, Solan (2014) who call our mixed strategies concept mixed stopping times and define another concept of randomization of stopping times and call it randomized stopping times. They show that under weak regularity conditions, the two concepts are equivalent for optimal stopping purposes in the sense that for a randomized stopping time of any of the two concepts there exists an appropriate stopping time of the alternative randomization concept so that the joint distribution of paths of the process and stopping time is the same. 10

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that for general stopping times (part (1) in Definition 1), where the decision to stop or continue at time t may depend on the whole path of the diffusion till time t. Finally, note that a simple threshold stopping time is a stopping time and the latter is a randomized stopping time where κ is a constant function in its first argument. Moreover, note that we are assuming that the diffusion is stopped as soon as it leaves the space of possible prizes [w, b]. This ensures uniform integrability for all types of strategies we consider. Therefore we call all the strategies satisfying the conditions of Definition 1, feasible strategies. In the literature one sometimes finds an alternative description of stopping times through the history dependent stopping policies they induce. Call a function s : ∪t≥0 Ft →∆({0, 1}) a stopping policy. Say that it is a pure stopping policy if its image is actually {0, 1}. Here 1 is interpreted as the agent continuing and 0 as the agent stopping. For each t ≥ 0 and event A ∈ Ft , s gives the probability the agent continues after history A. If the stopping policy is pure, then s says whether the agent continues or stops with probability one after history A. Markov stopping policies can be rewritten as functions s : [w, b]→∆({0, 1}) and they are pure when they can be rewritten as functions s : ∪t≥0 Ft →{0, 1}. Intuitively, the Markov property of the stopping policy is satisfied if the stopping decision at time t after any possible history depends only on the current value of the diffusion Xt . Formally, we say that s is a Markov stopping policy if for every t ≥ s ≥ 0, A ∈ Ft , B ∈ Fs such that the trace sigma-algebras A ∩ σ(Xt ), B ∩ σ(Xs ) are equal, it holds s(A) = s(B).11 We clarify in the next section that pure Markov stopping policies can be identified with simple threshold stopping times. Denote by FX the distribution induced by the random variable X on the measure space ([w, b], B([w, b])), where B([w, b]) is the Borel sigma-algebra of [w, b]. Then the distribution induced by a stopping time τ is FXτ . The distribution induced by a randomized stopping time κ is denoted again by FXκ and is given by Z 1 FXκ(r,·) (s)dr. (2) FXκ (s) = 0

The problem facing the naive agent and the agent with commitment is: sup

V (FXκ ).

κ randomized stopping time

For the sophisticated agent without commitment we restrict her strategy space at time zero to randomized stopping times κ, s.t. κ(r, ω) is a simple threshold stopping time. This is without loss of generality regarding the distributions FXκ induced. More on this in subsection 3.2. Our model assumes, that the primitive preference  is not defined over the whole path of the asset price, but that rather the agent only cares about the final outcome of the diffusion. A more general model could consider how risk preferences change as new information in the form of the realization of diffusion values arrives and would also consider welfare implications of such ‘news’. This kind of model is outside the scope of this paper. The observable behavior of the agent consists of a stopping region, where the agent stops with probability one and a continuation region, where the agent stops with positive 11

Note that σ(Xt ) is a sub-sigma-algebra of B([w, b]).

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probability. It may be optimal for the agent to choose a stopping strategy which prescribes stopping at the current wealth level with positive probability strictly less than one. As Henderson et al. (2017) show, this is the case if the agent is naive and her risk preference are some suitable specifications of Cumulative Prospect Theory (CPT). In the following Definition we formalize the concepts of stopping/continuation region. This definition is valid for all three agent types we are considering in this paper. Definition 2. (1) For a diffusion X and y0 ∈ [w, b] say that y0 is in the stopping region of X, if the agent stops with probability one in the stopping problem (X, y0 ). (2) For a diffusion X and y0 ∈ [w, b] say that y0 is in the continuation region of X, if y0 is not in the stopping region of X. There are three possible behavioral assumptions about the beliefs of the agent regarding her future behavior in the literature.12 First, the agent could have commitment. In this case, the agent’s choice maximizes her period-0 preferences and the optimal strategy of period-0 will be fully implemented. She cannot interfere with the implementation of the optimal strategy, once that has been chosen. Second, the agent could have no available commitment possibilities and also be naive about her Dynamic Inconsistency, i.e. she thinks at each period, that she will follow through with her decided plan, but then she (mostly) doesn’t. In particular, a naive agent thinks she will behave as an agent with commitment. If the preferences don’t satisfy Dynamic Consistency, this implies the agent holds irrational expectations of future behavior. Finally, the agent could again have no available commitment possibilities but is sophisticated about her Dynamic Inconsistency, i.e. she knows what future selves will actually do, when she is deciding about her current strategy. Sophistication requires the agent to exhibit a high degree of rationality, while not considering any normative revision to her preferences.13 Notice, that the behavior of an agent who has commitment can be fully characterized by the behavior of the (naive) agent at period zero, because the naive agent at period zero believes that she will act in the future as though she has commitment (but generally fails to do so). On the other hand, due to the Markov assumption on the prize process X, if one knows the commitment solution for all starting wealth levels, one can construct the naive solution by pasting together the different commitment solutions as the prize process evolves with time. Since the two cases are essentially analyzed by the same procedure, except for Theorem 1, we defer commenting in detail on the case of the agent with commitment in the following. Consequently, when we speak of a naive or sophisticated agent we always mean an agent without commitment possibilities. 12

See also Barberis (2012) for a related discussion. Note that a sophisticated agent is not necessarily dynamically consistent. But she is aware of the possible dynamic inconsistency and thus restricts her strategy space to strategies she knows, will not lead to behavior exhibiting violations of Dynamic Consistency. 13

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3

General Characterization of Optimal Stopping Behavior

We consider in turn the case of a naive agent and the case of a sophisticated agent. To avoid the nuisance of dealing with indifference cases we require the following convention: Convention: Unless otherwise stated, whenever the agent is indifferent between stopping and continuing, she will stop.14

3.1

The Naive Case and weak Risk Aversion

Risk aversion is usually defined as aversion to mean-preserving spreads.15 One implication of risk aversion is that the agent always prefers to get the mean of the lottery for certain rather than face the lottery. It is true that under the Expected Utility hypothesis the two concepts are equivalent. Definition 3. Say that the agent is weakly risk averse (wRA) at x ∈ (w, b) if for all lotteries F with E[F ] = x we have F δE[F ] . If instead there exists F with E[F ] = x and F δE[F ] , we say the agent is not weakly risk averse (not wRA) at x. Clearly, not wRA at some x implies that the agent is not risk averse. To characterize the solution to the optimal stopping problem of the naive (and also for the commitment case), we first characterize the set of lotteries achievable by stopping strategies. For a stopping problem (X, y0 ) define the scaling function   Z z Z x µ(t) dt dz. (3) S(x, y0 ) = exp −2 2 y0 y0 σ (t) It depends on the diffusion only through the normalized drift function z→ σµ(z) This 2 (z) . function will appear often in the characterizations that follow. As it turns out in the following results, it is always sufficient to know for each stopping problem (X, y0 ) the corresponding scaling function S(·, y0 ), rather than the full dynamic (1) to characterize optimal stopping behavior. Given a fixed diffusion X as in (1) say that a stopping time implements a distribution F ∈ ∆([w, b]) if the distribution of the random variable given by the stopped process Xτ is equal to F . The following Lemma gives the set of feasible distributions over final prizes the agent can achieve, if she starts at y0 . Proposition 1. 1. For a stopping problem (X, y0 ) with scaling function S, randomized stopping times can implement a distribution F if and only if it is contained in FX (y0 ) = {F ∈ ∆([w, b] : Ex∼F [S(x, y0 )] = 0}. FX (y0 ) is a convex, compact set (in the topology of convergence in distribution). 14

We could consider other tie-breaking rules, but exposition for some of our applications and results is easier under the ones chosen here. 15 See e.g. Definition 6.D.2 and Example 6.D.2 in Mas-Colell, Whinston, Green (1995).

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2. Every distribution in FX (y0 ) can be induced by an optimal stopping time (without randomization). 3. Binary distributions from FX (y0 ) can be induced by simple threshold stopping times. Moreover, the following are equivalent for both a naive and sophisticated agent in an optimal stopping problem. (a) The agent uses a pure Markovian stopping policy if she starts. (b) The agent picks a simple threshold stopping time if she starts. Parts 1) and 2) follow from results in Ankirchner et al. (2015). See Appendix A for details. FX (y0 ) is the technologically feasible set of distributions of the agent when she faces the stopping problem (X, y0 ). From now on, we will suppress the diffusion X from the notation and write F(y0 ) instead. It will be clear from the context which diffusion X underlies F(y0 ). Compactness of F(y0 ) implies that under our Continuity assumption on preferences there will always exist an optimal strategy/optimal distribution out of F(y0 ). For F ∈ ∆([w, b]) and S : [w, b]→R strictly increasing, denote by F ◦ S −1 the distribution function from ∆([S(w), S(b)]) given by (F ◦ S −1 )(z) = F (S −1 (z)) for all z ∈ [S(w), S(b)]. It is clear that the map that sends each F ∈ ∆([w, b]) to its corresponding F ◦ S −1 ∈ ∆([S(w), S(b)]) is bijective. As a last piece of convention, whenever S(·, y0 ) is the scaling function of a stopping problem (X, y0 ), and f is a real-valued function such that fˆ(·) = f ◦ S −1 (·, y0 ) is welldefined, we will write for this function shortly f ◦S −1 , with the understanding that (X, y0 ) is clear from the context. Now we can state the main Theorem for the naive case, which gives a complete characterization of naive behavior. Its usefulness will be clearer when we apply it in Section 4. Theorem 1. 1. The naive agent without commitment and the agent with commitment continue in the stopping problem (X, y0 ) if and only if the feasible set F(y0 ) contains a lottery which is strictly preferred to δy0 . 2-a) If the agent is not weakly risk averse at y0 ∈ (w, b), then she continues with positive probability any Martingale diffusion started at y0 . 2-b) If the agent is weakly risk averse at y0 , then she stops all Martingale diffusions started at y0 in finite time with probability one. Moreover, irrespective of any starting point y 6= y0 she will stop with positive probability any stopping problem (X, y), if X is a Martingale diffusion. 3. The agent with utility V stops a diffusion with scaling function S at y0 if and only if the agent, whose utility over lotteries G in ∆([S(w, y0 ), S(b, y0 )]) is given by VS (G) = V (G ◦ S), is weakly risk averse at 0. Part 1) is an obvious restatement of the problems of the naive agent and the agent with commitment. Part 2) and 3) characterize precisely the stopping region of the naive agent: 2) does this for Martingale diffusions, while 3) uses the result in 2) to extend the characterization to arbitrary diffusions. Intuitively, a stopped Martingale diffusion corresponds to lotteries which have the same expectation as the starting point of the diffusion. If an agent is weakly risk averse at y0 12

she always prefers the certain amount y0 equal to the expectation of a lottery induced by the Martingale diffusion, rather than facing the lottery. On the other hand, if she is not weakly risk averse at y0 there exists a lottery with expectation y0 she prefers to getting y0 for sure. This lottery can be induced through a pure stopping strategy and so the agent continues with positive probability. This is the content of part 2) of Theorem 1. Finally, a general diffusion, not necessarily a Martingale, is equivalent in probabilistic terms to a Martingale diffusion where the prize space [w, b] has been appropriately rescaled. Since the agent’s utility V depends on a stopping strategy only through the lottery induced by it, this allows translating the result for Martingale diffusions to the case of an arbitrary diffusion satisfying the conditions imposed on (1). As a result, the continuation/stopping region for a general diffusion follows by ‘translating’ the preferences of the agent with the same procedure as the rescaling of the prize space.

3.2

The Sophisticated Case.

For the case of a sophisticated agent we restrict the set of policies to the set of Markov policies and we model the behavior of the sophisticated agent as a Markov Nash equilibrium, just as in Ebert, Strack (2016) who focus on Cumulative Prospect Theory preferences. This assumption is made for tractability. In the next subsection, we clarify in part how strong the Markovian assumption on policies is. The Nash assumption has already been used in related work, see for ex. Ebert, Strack (2016). Karni, Safra (1990) on the other hand, consider a finite horizon discrete time model and thus can solve through backward induction for the Markov perfect equilibrium (MPE) in randomized Markov strategies. Given concerns of tractability, the Nash requirement is a good compromise. Definition 4. 1) A pure Markov policy is a function s : [w, b]→{continue, stop}. It constitutes an equilibrium if at every point in time t it is optimal to take the decision s(Xt ) given future selves use the strategy s. 2) A mixed Markov stopping policy for the sophisticated agent is a probability distribution σ over pure Markov policies. It constitutes an equilibrium if • it is optimal for the self at time zero to randomize according to σ and • any realization s of σ constitutes an equilibrium according to 1). A Markov Nash equilibrium in pure Markov policies always exists. Namely, it is given by choosing with probability one at the start the pure Markov policy s0 ≡ stop for all. This instructs all selves to never start. The interesting question is then to find conditions under which there is another equilibrium of the game played by the selves of the sophisticated agent, which is preferred by the self at time zero to the never starting one. Given the set of all pure Markov Nash equilibria we assume the sophisticated agent at time zero chooses the best mixture of them in terms of the final prize lottery they induce. The concept of randomization used is a particular one and is not equivalent to other possible randomization concepts. For example, one could ask for time-homogeneous randomization which would be equivalent to studying Markov stopping policies of the type s : [w, b]→[0, 1], where s(x) denotes the probability that the agent continues whenever 13

she reaches x. This is a different randomization concept from the one used in this paper, since it precludes time-inhomogeneous behavior. Given tractability issues, we focus on the simpler concept where in an equilibrium only the agent at time zero potentially randomizes among pure Markov policies. As we will see when we apply Definition 4 and its consequences, our concept of an equilibrium in mixed Markov stopping policies has considerable behavioral content among different models of static risk behavior. We can use the characterization of Markov stopping policies from Proposition 1 through simple threshold stopping times to understand when the game has a different Markov Nash equilibrium from the not-starting one, which is preferred by the self at time 0. Take a fixed y0 ∈ (w, b). The proof of Proposition 1 implies that when a Markov policy at y0 for the sophisticated agent exists that is better than s0 (always stopping), there is a simple threshold stopping time, which has the form τa,c (see 3) in Definition 1) with some w ≤ a < c ≤ b and is so that the agent continues at each y ∈ (a, c) and stops otherwise. Any mixed Markov stopping policy corresponds then to a randomized stopping time κ according to Definition 1, where for each r ∈ [0, 1], κ(r, ·) is a simple threshold stopping time. Note that randomized stopping times may lead to time-inhomogeneous behavior because the stopping strategy started at zero prescribes a different action in the case of a return to the starting point, whenever the randomized stopping time is non-constant in the first argument. Nevertheless, from the perspective of period zero, the set of such randomized stopping times exhausts the set F(y0 ) of prize distributions achievable by stopping strategies. This follows from Lemma 3 from the Appendix. The equilibrium concept for mixed Markov policies requires each pure Markov policy in its support to be an equilibrium, which then naturally restricts the set of implementable distributions from F(y0 ). We first focus on characterizing the simple threshold stopping times which are equilibria according to 1) of Definition 4. Given a diffusion X as in (1), started at some point y ∈ (a, c) and τa,c a simple threshold stopping time, the self at y faces lottery of the form L(p(y), a, c) = (1−p(y))δa +p(y)δc , where p(y) is given by (see for example Revuz, Yor (2013), pp. 303) 1 − pX a,c (y) =

S(c, y0 ) − S(y, y0 ) , S(c, y0 ) − S(a, y0 )

(4)

where S(y, y0 ) is the scale function for the stopping problem (X, y0 ). pX a,c is strictly X X increasing, smooth with the properties pa,c (a) = 1, pa,c (c) = 0. Definition 5. For a fixed diffusion X whose dynamic is given by (1) and interval (a, c) ⊂ S(y,y0 )−S(a,y0 ) X [w, b] we denote the function pX a,c : (a, c)→[0, 1] given by pa,c (y) = S(c,y0 )−S(a,y0 ) the win probability of X for the interval (a, c). The following definition is central as it helps characterize when a strategy is feasible for a sophisticated agent. Definition 6. The curve qa,c : [a, c]→[0, 1] given by L(1 − qa,c (y), a, c) ∼ δy ,

y ∈ [a, c].

is called the calibration function of the agent for (a, c). The calibration function of an agent for (a, c) gives for each element y ∈ (a, c) in the interval the probability of c which ‘calibrates’ y, i.e. the probability q(y) in the lottery 14

L(1 − q(y), a, c) which makes the agent indifferent between L(1 − q(y), a, c) and δy . Obviously, qa,c is continuous and strictly increasing. The two definitions above allow us to formulate the necessary and sufficient condition for never starting any diffusion. Definition 7. Say that the agent exhibits extreme sensitivity to risk at y0 if and only if the following holds: (C)

for every a < y0 < c the slope of qa,c : [a, c]→[0, 1] is + ∞ at a or 0 at c.

To characterize formally the optimal solution for the sophisticate agent, we additionally introduce the following definition for a diffusion with normalized drift z→ σµ(z) 2 (z) . Definition 8. Let16 Fsoph (y0 ) = {L(1 − p(y0 ), a, c) : p(z) = pX a,c (z) ≥ qa,c (z), for all z ∈ [a, c]}. This is a compact subset of FX (y0 ). Denote also by conv(Fsoph (y0 )) its convex hull, which is again a compact set. Having defined the necessary machinery, the following Theorem gives a full characterization of the never stopping result in the case of the sophisticated agent. Theorem 2. For a sophisticated agent at current wealth y0 the following hold. 1. y0 is contained in the stopping region for all diffusions if and only if the agent exhibits extreme sensitivity to risk at y0 . 2. y0 is contained in the continuation region of X if and only if there exists a, c with w ≤ a < y0 < c ≤ b such that pX a,c dominates pointwise qa,c . 3. Under the tie-breaking rule, that a sophisticated agent will start whenever she is indifferent between starting and stopping, the solution to the stopping problem (X, y0 ) of the sophisticated agent is given by max

F ∈conv(Fsoph (y0 ))

V (F ), so that V (F ) ≥ V (δy0 ).

Intuitively, the calibration function for an interval is a preference constraint the win probability of a diffusion has to overcome so that the sophisticated agent can set up a consistent plan, which will be followed by all possible future selves. The higher the calibration function for an interval, the higher the constraint the diffusion has to overcome for the sophisticated agent to find it optimal to start. If no diffusion can overcome the calibration function for any interval, then for each potential optimal strategy τa,c , there exists a self y ∈ (a, c) such that if diffusion lands at her wealth, she’ll stop, because she prefers the certainty of current wealth rather than the continuation lottery offered by the diffusion. When the diffusion can overcome the calibration function for some interval, there are consistent plans the sophisticated agent can set up and which will be followed by future selves. Allowing for a randomization device at time zero and before the starting/stopping decision of self zero, the agent at time zero chooses the best mixture out of all pure Markov policies which constitute a pure Markov Nash equilibrium of the game faced by all selves. We use Theorem 2 extensively in Section 4 to characterize sophisticated behavior across a variety of risk preferences. 16

Here we again exclude X from the notation Fsoph (y0 ). In any application, it will be clear from the context which diffusion X underlies Fsoph (y0 ).

15

3.3

On the optimality of simple threshold stopping times.

In general, whether a strategy prescribes stopping at a particular value of X at a particular time t will depend on the whole path of X up to time t. This is for example the case if the optimal continuation lottery a naive agent chooses is not binary. The latter are implemented through simple threshold stopping times. Simple threshold stopping times have the property of corresponding to simple stopping strategies: cut high losses and stop after high gains. Besides being also simple to encode17 it is also a well-known fact, that for Expected Utility preferences the solution to the optimal stopping problem is of this form. In this section we generalize this result by uncovering the preference class which ensures that simple threshold stopping times are optimal, no matter whether the agent has commitment, is naive or sophisticated about possible deviations of dynamic consistency. We note here a condition on the preference , which implies the existence of optimal policies which satisfy the Markovian property for all stopping problems. It follows, that in those cases we can restrict the set of feasible strategies to simple threshold stopping times without loss of generality. In section 4 and in the online Appendix 2, we apply our general results to several preferences satisfying this condition. Definition 9.  is called quasi-convex18 if for F, F 0 , F 00 ∈ ∆([w, b]) F, F 0 F 00

implies

αF + (1 − α)F 0 F 00 , for all α ∈ (0, 1)

and F F 00 , F 0 ≺F 00

implies

αF + (1 − α)F 0 ≺F 00 , for all α ∈ (0, 1).

The class of quasi-convex preferences contains Betweenness preferences and thus in particular Gul preferences but also a subclass of quadratic preferences, including the CPE preferences of K¨oszegi, Rabin (2007).19 When preferences are quasi-convex it is sufficient to check violations of wRA only for binary bets, as the following Proposition shows. Proposition 2. Assume that  is quasi-convex. Then the agent exhibits wRA at y0 ∈ [w, b] if and only if for all binary lotteries F with mean y0 it holds F δE[F ] . The agent violates wRA at y0 if and only if there exists a binary lottery F with mean x such that F δE[F ] . Moreover, up to tie-breaking considerations, a quasi-convex agent of all three types: naive, sophisticated with commitment and sophisticated without commitment will use simple threshold stopping times whenever she starts. 17

Namely, through the two boundary points of the interval where the agent continues. This implies quasi-convexity of the utility function V . In the literature, usually quasi-convexity is defined as L ∼ L0 =⇒αL + (1 − α)L0 L. Under our Continuity assumption this definition is equivalent with the first part of Definition 9. 19 See section 4 and online Appendix 2 for details on these risk preferences. 18

16

Intuitively, an agent with quasi-convex preferences is averse to mixtures of lotteries. Therefore, mixing binary lotteries never raises her utility. But binary lotteries are ‘extremal’ in the sense that any lottery with finite support is a mixture of binary lotteries. It follows that: (1) whether the agent is weakly risk averse or not depends on comparisons involving binary lotteries only, and (2) since the set of implementable distributions through stopping strategies is convex and binary lotteries are ‘extremal’ in this set as well, an optimizing agent with quasi-convex preferences will pick simple threshold stopping times, up to tie-breaking considerations. The latter holds regardless of her perceptions about possible Dynamic Inconsistency of her behavior over time. In particular, a sophisticated agent will not randomize at time zero. Note, that for a preference which is represented by a quasi-concave utility function (these are known as convex preferences20 in the literature; here we also call them quasiconcave preferences) it is in general not true, that optimal stopping policies for both naive agent and the agent with commitment are simple threshold stopping times. An instance of this is again Example 4.1 in Section 4.

4

Applications

The well-known Allais paradox and other related paradoxes challenge the main behavioral implication of Expected Utility: the independence axiom.21 We restate it here for the reader’s convenience. Axiom: Independence For G1 , G2 , F ∈ ∆([w, b]) with G1 G2 and any α ∈ [0, 1] we have that αH + (1 − α)G1 αH + (1 − α)G2 . As a response to violations of Independence, a wide variety of risk preference models have been offered as alternatives in the decision theory literature. Barberis (2012), Ebert, Strack (2015), Ebert, Strack (2016) and He et al. (2016) pick CPT as a non-EU preference, when studying the optimal stopping problem. The behavior CPT induces in the optimal stopping problem may be used to reflect on the value of using CPT as a model of choice among risky prospects in general. The general results proven in previous sections allow a comparison of CPT within the optimal stopping framework with many other well-known risk preference models. This enables a better understanding as to how much of the optimal stopping behavior under a CPT preference depends actually on the CPT functional form assumption. We consider different risk preference classes one after the other. We first focus on models of probability weighting like CPT, which has known the most applications in the literature, and Rank-Dependent Utility (RDU), an important subclass of CPT. RDU switches off the reference dependence in the CPT model and focuses on probability weighting. As a second class, we consider preferences which satisfy Betweenness and among them, we focus on the special case of Disappointment Aversion preferences (otherwise known as Gul preferences). Finally, we consider a new class of convex risk preferences called Cautious Expected Utility (CEU), which has been introduced and axiomatized in Cerreia-Vioglio 20

A preference is convex if for every F, G ∈ ∆([w, b]) and λ ∈ (0, 1) F ∼ G implies λF + (1 − λ)GF . See Machina (1982) and Machina (1989) for a discussion of Allais paradox and its implications for economic theory. 21

17

et al. (2015) and which can explain both the Allais Paradox and exhibit the so-called Certainty Effect. In the online Appendix 2 we give sufficient conditions for a CEU sophisticated agent to start some diffusions, i.e. for the agent to not exhibit extreme sensitivity to risk. Finally, we also provide there additional applications of the general characterization theorems from section 3 to the main well-known subclass of Smooth Preferences as introduced in Machina (1982) and axiomatized in Chew et al. (1991): quadratic utility. Within the quadratic class, we also look at CPE, a behavioral model first introduced in K¨oszegi, Rabin (2007), and which enjoys considerable popularity in the applied behavioral theory literature. Some of the preference classes we consider overlap: CPE is both an RDU and a quadratic preference,22 Disappointment Aversion is sometimes contained in the CEU class, etc.23 Besides showing the reach in terms of application of the general results proven in the previous sections, there are two main take-aways of interest to investment/gambling theory from the characterizations and examples we display in the rest of this section. First, unless one is ready to violate our two basic assumptions of continuity and monotonicity on the static preference from section 2, it seems only some specific classes of probability weighting are the only preferences which exhibit extreme behavior in the following sense: a naive agent always continues any diffusion with positive probability and a sophisticated agent always stops any diffusion. Second, and related to the first, all other static models of risk preference we consider are closer to Expected Utility in the sense that, when the diffusion process is ‘favorable’ enough, even a sophisticated risk averse agent will find it profitable to continue with positive probability and when the diffusion process is ‘unfavorable’ enough, even a naive risk loving agent will find it optimal to stop. These results can be considered a robustness analysis on the preference assumptions of the models contained in Barberis (2012), Ebert, Strack (2015), Ebert, Strack (2016) and He et al. (2016), among others.

4.1

Models of Probability Weighting

Barberis (2012), Ebert, Strack (2015), Ebert, Strack (2016) and He et al. (2016) use preference of the Prospect Theory type in their optimal stopping models. These preferences were introduced in Tversky, Kahneman (1992) to explain several well-known empirical facts of risk preferences. If the reference point of the agent is r, they are given for a binary lottery L(p, c, a) = pδc + (1 − p)δa with a < c by  + +  if r ≤ a (1 − ν (p))U (a) + ν (p)U (c) − + CP T (L(p, c, a)) = ν (1 − p)U (a) + ν (p)U (c) (5) if a < r ≤ c   − − ν (1 − p)U (a) + (1 − ν (1 − p))U (c) if c < r. Here, U is the value function and ν the probability weighting function: a strictly increasing, continuous function ν : [0, 1]→[0, 1]. The continuity requirement on the preference is equivalent to the restriction ν − (q) = 1 − ν + (1 − q) for all q ∈ [0, 1]. This implies 22

In fact, as Masatlioglu, Raymond (2016) show, when considered as a static risk preference, CPE is precisely the intersection of quadratic and RDU preferences. 23 See Section 5 of Cerreia-Vioglio et al. (2015).

18

that in (5) the reference point doesn’t play a role in the evaluation of a lottery’s probabilities. It does play a role in the evaluation of the outcomes w.r.t. the reference level. Ebert, Strack (2015) assume the following for the value function U . Assumption from Ebert, Strack (2015): The value function U is continuous, strictly monotonic, it has finite left and right derivatives, ∂− U (x), ∂+ U (x), at every wealth level (x) x. Further, λ = supx∈R ∂∂−+ UU (x) < ∞. +,− ν : [0, 1]→[0, 1] are continuously differentiable and have ν +,− (0) = 0, ν +,− (1) = 1. If λ > 1, then we say the agent exhibits loss aversion. For continuous random variables, reference level of r and U with U (r) = 0, the utility function for the lottery induced by some random variable X is given by Z Z + CP T (X) = ν (P(U (X) > y))dy − ν − (P(U (X) < y))dy. R+

R−

Ebert, Strack (2015) prove that for a wide range of specifications of CPT which are used in applications, a naive agent continues with positive probability any diffusion.24 Their CPT specifications are so that the agent exhibits no wRA everywhere. The gist of the proof of Ebert, Strack (2015) is to construct for each y0 ∈ R a binary lottery L(p, a, c) such that E[L] = y0 and CP T (L) > U (y0 ). Thus, at each current wealth level, the agent has a simple threshold continuation strategy which she prefers to stopping. Intuitively, the functional form assumptions in their paper can be summarized as ’probability weighting part of CPT is stronger than the loss aversion part’. Probability weighting implies a preference for skewed bets in the small, which is what drives the result on an intuitive level. Ebert, Strack (2016) consider the extreme result of never-starting for the sophisticate. Their sufficient conditions for never starting, for the case of continuous CPT preferences they discuss, are that the weighting function for gains has infinite slope at one and the weighting function for losses has infinite slope at zero. In the previous section, we have used their modeling choice of studying sophisticated behavior through the Markov Nash equilibrium concept to our more general set up of arbitrary monotonic and continuous risk preferences. Our analysis allows for a complete characterization of all CPT preferences exhibiting the never starting result, under a weak differentiability assumption on the value function U. Proposition 3 (CPT-never starting). Assume that U in (5) is differentiable, strictly increasing and has a bounded derivative. Then the CPT sophisticated agent never starts if and only if one of the following two cases is true: 1. (ν + )0 (0+) = 0 and (ν − )0 (1−) = 0 2. (ν + )0 (1−) = +∞ and (ν − )0 (0+) = +∞. 24

Ebert, Strack (2015) look at pure strategies according to our Definition 1 and thus formulate their result as a never stopping result. As Henderson et al. (2017) shows, once mixed strategies are allowed, the right interpretation of the main result in Ebert, Strack (2015) is that of continuing with positive probability any diffusion.

19

The sufficient conditions identified by Ebert, Strack (2016) correspond to the second case in the proposition.25 The first case shows that their conditions are not necessary and that a ’dual’ counterpart of their condition also leads to never-starting. Several important functional forms used in the applied behavioral literature fit the conditions of Proposition 3. See for example Wu et al. (2004) for an extensive list of popular functional forms.26 Rank-Dependent Utility RDU is a special case of CPT, where the restriction ν − (q) = 1 − ν + (1 − q) in (5) is valid. It focuses on the probability weighting part of CPT and disregards reference dependence. One can represent the RDU model by Z V (F ) = u(x)dν(1 − F (x)), (6) where ν : [0, 1]→[0, 1] is a strictly increasing, onto function and u is any continuous, increasing function. Assume in the setting of (6), that u : R+ →R+ is differentiable, strictly increasing and that ν is differentiable as well. For the case of the sophisticated RDU agent, Proposition 3 immediately gives the full characterization for never-starting any diffusion. Proposition 4 (RDU-Never Starting). For a sophisticated RDU agent the stopping region of any diffusion is the whole prize space [w, b] if and only if one of the following conditions is satisfied 1. ν 0 (0+) = 0 2. ν 0 (1−) = +∞. Thus, the never-starting result in the RDU case again depends only on probability weighting. We see that then ν cannot be concave, so that the agent cannot be globally risk loving and her static preferences cannot be globally convex. Many specifications of probability weighting in the literature satisfy the requirements Proposition 4. We mention ν(p) =

pα (pα + (1 − p)α )

1 α

,

apδ ν(p) = δ , ap + (1 − p)δ

ν(p) = exp(−(− ln(p))α ),

p ∈ [0, 1]

(7) where α ∈ (0, 1), a, δ > 0, δ 6= 1. These specifications are discussed at length in Prelec (1998). The first specification in (7) is used in Barberis (2012), but also in Benartzi, Thaler (1995). The latter show in a model of a CPT agent that a combination of (7) as probability weighting function, loss aversion and the fact that agents evaluate their portfolios frequently, leads to an explanation of the equity premium puzzle. Proposition 4 gives another explanation for this puzzle and the related empirical fact of stock-market nonparticipation. Namely, sophisticated RDU agents with probability weighting functions satisfying the requirements in Proposition 4 may avoid participation in the stock market 25 They also consider a special case of discontinuous CPT preferences. We have omitted the study of discontinuous preferences already through our basic assumptions. δpγ 26 Well-known examples are ν + (p) = ν − (p) = pα for α ∈ (0, 1) or w+ (p) = ν − (p) = δpγ +(1−p) for γ ∈ (0, 1) and δ > 0.

20

due to their inability to construct an investment plan which will be followed by future selves. We now turn to the case of the naive agent. Proposition 4 from Segal, Spivak (1990) shows, that for all RDU models, if w is strictly concave, the preferences violate wRA.27 We can use this fact to find cases where, similar to the result of Ebert, Strack (2015), the RDU naive agent never stops with probability one, i.e. for all y0 and diffusions X, y0 is never in the stopping region of X. Proposition 5. Suppose a probability weighting function ν : [0, 1]→[0, 1] is so that the static RDU preference (u, w) violates wRA everywhere, irrespective of u : [w, b]→R. Then for any y0 ∈ [w, b] and any diffusion X, y0 is never in the stopping region of a naive RDU agent with probability weighting w. The proof of this proposition follows directly from the proof of Theorem 2 in Ebert, Strack (2015). They consider the more general set up of CPT. To get a feeling for the stopping times a naive RDU agent chooses, consider the following example. Example 4.1: Naive RDU agent who always picks non-binary distributions. Consider an RDU agent with Z b V (F ) = u(x)d(ν(1 − F (x))), (8) 0

for some finite b > 0 large enough and with u(x) = xr , r ∈ (0, 1] and ν(x) = xα , α ∈ (0, 1). In particular, this agent has quasi-concave preferences.28 Thus, we shouldn’t expect the naive agent to choose simple threshold stopping times in the optimal stopping of a diffusion. Assume the agent faces a geometric Brownian motion dXt = µXt dt + σXt dWt , 2

X0 = y0 > 0.

2

and assume that the stopping problem and the preference Define γ = σ2rσ−2µ and β = σ σ−2µ 2 parameters are such that γ ∈ (0, α). One finds, that the agent’s optimal distribution out of F(y0 ) is   if x = b  1, β−r
1−α − Fy0 (x) = 1 − (1 − xb ) xb 1−α , if b > x ≥ b(1 − xb ) β−r (9)  1−α  0, if 0 ≤ x < b(1 − xb ) β−r , where xb is dependend on y0 . This distribution is a truncated Pareto distribution with an atom at b. One possible optimal stopping time inducing Fy0 is the so-called Azema-Yor stopping time, which is given as follows. 1  α−r β τAY,y0 = inf{t ≥ 0| Xt ≤ min{b, max Xs }}. (10) 0≤s≤t 1−r 27

This is because their Proposition 4 implies that the preferences exhibit first-order risk loving behavior, which trivially implies risk loving behavior and thus also a violation of wRA. 28 Wakker (1994) proves that a RDU preference with convex weighting function is quasi-convex and a RDU preference with concave weighting function is quasi-concave.

21

When using this strategy, the commitment agent sells as soon as the price has fallen some percentage below the historical maximum. The fact that under µ > 0 there is positive
β1 probability that Xt reaches b before it falls below α−r max0≤s≤t Xs explains the atom 1−r at b, i.e. that the optimal distribution has a jump at b: the probability of b is 1 − xb . The formulas in the Appendix show that 1 − xb is increasing in y0 as can be expected intuitively. This example also illustrates the fact that in some cases it might be possible in principle for an outside analyst to derive the correct level of the agent’s sophistication only from the observed stopping behavior. Namely, if one considers the sophisticated agent with preferences as above but r = 1 and facing a stopping problem with µ < 0 and so that γ < α, it follows that the sophisticated agent will never start at any y0 . This, and the fact that the naive agent starts with positive probability, allows an outside analyst in our model to infer the level of sophistication of the agent under the parameter restrictions assumed. Finally, additional results for the case of the naive RDU agent based on the results from Xu, Zhou (2013) are recorded in the online Appendix 2. The next example illustrates the fact that many specifications of RDU exhibit similar qualitative behavior as Expected Utility, in the sense that, a risk averse sophisticated agent never stops diffusions which have a ‘favorable enough’ dynamic. Example 4.2: Sophisticated RDU agent who never stops some diffusions. Assume RDU with u(x) = x, w ≥ 0 and some strictly convex and twice continuously differentiable ν with ν 0 (0+) > 0, ν 0 (1−) < +∞. Note, that this agent is risk averse and her utility function V over lotteries in ∆([w, b]) is quasi-convex.29 We focus on the case of simple geometric Brownian motion with µ ∈ R, σ > 0. We show, that there are specifications such that the sophisticated agent chooses τw,b as an optimal stopping rule (bang-bang). In a casino gambling model this means that she never leaves the casino. For fixed y0 the diffusion chosen gives for the win probability for y0 ∈ (a, c)30 p(y) =

2µ

2µ

2µ

2µ

e−a σ2 − e−y σ2 e−a σ2 − e−c σ2

.

The sophisticated agent solves then the following problem, if she starts max

a,c:ν≤a≤y0
ν(1−p(y0 ))a+(1−ν(1−p(y0 )))c, such that

c−y ≥ ν (1 − p(y)) , c−a

y ∈ [a, c].

Focus on the case µ > 0. The agent will always start such a geometric Brownian motion and one can show, that she will choose τw,b as an optimal policy whenever the normalized drift σµ2 is large enough. In particular, there are stopping problems, for which this class of RDU agents continues to gamble even if in the neighborhood of the worst wealth level w, despite being sophisticated and risk averse. Finally, we note that the respective Expected utility agent, i.e. with Bernoulli utility function u(x) = x is risk neutral and thus never stops any diffusion with non-negative normalized drift. 29

See footnote 28. In the following we suppress the sub-and-superscripts a, c, X in the notation of pX a,x . The simple threshold time and the diffusion will be clear from the context. 30

22

4.2

Betweenness preferences

Dekel (1986) considers relaxing the Independence axiom of Expected Utility to Betweenness: pq (p ∼ q) implies pλp + (1 − λ)qq (p ∼ λp + (1 − λ)q ∼ q). Note that Betweenness preferences are quasi-convex, because their indifference curves are linear (though in general not parallel). In particular, due to Proposition 2, the agent can be assumed to pick simple threshold stopping times. Theorem 1 in the online Appendix establishes, that unless Betweenness preferences are strictly Expected Utility, they don’t satisfy Dynamic Consistency.31 A special case of Betweenness preferences are Gul preferences as defined in Gul (1991). They satisfy Betweenness and are disjoint from RDU preferences, except for Expected Utility.32 They are encoded uniquely by a pair (u, β) consisting of a strictly increasing Bernoulli utility function u : [w, b]→R and a parameter β ∈ (−1, ∞), which measures the degree with which the probability of those prizes in the support of a lottery is overvalued, which are less preferred by the agent than the lottery itself (‘disappointing’ prizes). An agent with β > 0 is said to exhibit Disappointment Aversion, whereas an agent with β < 0 is said to be Elation Loving. The case β = 0 corresponds to Expected Utility with Bernoulli utility u. Here we exhibit the representation of Gul preferences for binary lotteries only. According to Proposition 2 this is sufficient for optimal stopping purposes.33 For binary lotteries of the form L(1 − p, x, y) with x ≤ y the representation of Gul is given by V ((1 − p)δx + pδy ) =

(1 − p)(1 + β) p u(y) + u(x).34 1 + (1 − p)β 1 + (1 − p)β

(11)

The intuition is that the agent feels elation for any realized prize from the lottery which is better than the lottery itself and feels disappointment for any realized prize from the lottery which is worse than the lottery itself. For the case of the naive Gul agent we use the general result in Theorem 1 to give a full characterization of the behavior of the naive Gul agent when X is a geometric Brownian motion (µ(x) = µx, σ(x) = σx) or an arithmetic Brownian motion (µ(x) = µ, σ(x) = σ). We have the following characterization of naive stopping and continuation regions. Proposition 6. For either of the two following cases 1. X is a geometric Brownian motion with parameters µ ∈ R, σ ∈ R+ and the worst prize w is positive, 2. X is an arithmetic Brownian motion with parameters µ ∈ R, σ ∈ R+ , the continuation region of X for a naive agent consists of 2µ

e σ2 (y0 −x) − 1 u(y) − u(y0 ) > 1 + β}. CX = {y0 ∈ (w, b) : there exists x < y0 < y with µ 1 − e−2 σ2 (y−y0 ) u(y0 ) − u(x) 31

Gul, Lantto (1990) show that in dynamic choice problems involving finite lottery trees, if  satisfies strict FOSD-monotonicity and Continuity, then it is a Betweenness preference if and only if it satisfies another dynamic axiom, called weak Consequentialism. This axiom is related to backwards induction solution procedure for finite horizon dynamic problems. 32 See Masatlioglu, Raymond (2016). 33 See Theorem 1 in Gul (1991) and the surrounding discussion there for the general representation of Gul preferences. 34 This equation is on page 677 of Gul (1991).

23

The continuation region CX is non-empty and strictly smaller than the whole prize space [w, b] for a wide variety of parameter values. One can deduce easily from the Proposition that ceteris paribus a higher β, which corresponds to higher Disappointment Aversion, shrinks the continuation region. Moreover, ceteris paribus a more concave u, shrinks the continuation region as well. Finally, a higher normalized drift σµ2 ceteris paribus expands the continuation region. We now turn to the sophisticated case. Assume for simplicity, that the Bernoulli utility u in the Gul representation is strictly increasing. The following Proposition is then immediate. Proposition 7. If the Gul agent is sophisticated and in the representation (u, β) with β > −1 u is differentiable in (w, b), there is some diffusion with non-empty continuation region. The following example illustrates further the behavior of a sophisticated Gul agent. Just as Example 4.2, it illustrates the fact that, similarly to an EU agent, a risk averse sophisticated Gul agent never stops diffusions which have a ‘favorable’ enough dynamic. Example 4.3: Sophisticated, risk averse Gul agent. Assume that u(x) = x, β ∈ (0, +∞) and that X follows a Brownian motion (or a geometric Brownian motion); with variance σ > 0 (σXt , σ > 0) and drift µ ∈ R (µXt ). Assume also, that the worst prize w is greater than zero. According to Gul (1991) this agent is risk averse. The continuation condition for the stopping time τa,c is p(y)c + (1 + β)(1 − p(y))a ≥ y(1 + (1 − p(y))β),

y ∈ (a, c).

(12)

In the Appendix we first find that the feasibility of a simple threshold stopping times τa,c depends on d0 = σµ2 (c − a) and β and not on the starting point y0 . τa,c is feasible for the sophisticated agent if and only if d0 is high enough in comparison to β. In particular, for non-positive drift µ this sophisticated agent will not start. The sophisticated agent with Gul preferences and starting wealth y0 ∈ (w, b) solves the following problem. p(y0 )c (1 + β)(1 − p(y0 ))a + , w≤a
such that (12) holds.

We show in Appendix D, that for a region of parameter values the agent’s optimal stopping time is τw,b . This means the agent gambles till either ruin or the highest price. In particular, the same remark is true as for the sophisticated RDU agent in Example 5.1: there are diffusions, for which this Gul agent continues to gamble even if in the neighborhood of the best prize b, despite being sophisticated and the risk faced by continuing.

4.3

Cautious Expected Utility

We say that the preference has a Cautious Expected Utility (CEU) representation if there exists a compact, convex set U of strictly increasing and continuous functions u : [w, b]→R such that the preferences are represented by V (F ) = inf u−1 (Ex∼F [u(x)]) . u∈U

24

(13)

Without loss of generality one can assume that all u ∈ U are normalized: u(w) = 0, u(b) = 1. These preferences are convex, they can explain the Allais Paradox and exhibit the Certainty Effect, which is related to the common ratio version of the Allais paradox. They were introduced and axiomatized in Cerreia-Vioglio et al. (2015). Cautious Expected Utility representations where the set U has only finitely many elements don’t satisfy Betweenness.35 The main behavioral axiom satisfied by a cautious EU preference is Negative Certainty Independence (NCI): for all x ∈ [w, b], F, G ∈ ∆([w, b]) and λ ∈ [0, 1] F δx

=⇒ λF + (1 − λ)Gλδx + (1 − λ)G.

For an agent satisfying NCI, if the certain outcome x is not able to compensate for F despite the riskiness in F , then mixing both F and δx with another lottery G will further (weakly) lower the appeal of x vis-a-vis F . This relaxation of Independence allows for the Certainty Effect.36 The following Proposition is a partial characterization of the optimal stopping behavior of a naive CEU agent. Proposition 8. Let the preference have a Cautious Expected Utility representation parametrized by the set of Bernoulli functions U and assume 0 ∈ (w, b). Assume the naive agent faces a stopping problem (X, y0 ) with its respective scaling function S. 1. If for all the functions u ∈ U the function u ◦ S −1 is concave, then y0 is in the stopping region of X. 2. If for all the functions u ∈ U the function u◦S −1 is convex, then y0 is in the stopping region of X. 3. (S−shaped case) If all functions u in U are convex for x < 0, concave for x > 0, , then the agent u(0) = 0 and have a one-sided derivative at b with u0 (b) ≥ u(b)−u(w) b−w violates wRA everywhere and so, if X is a Martingale diffusion, the continuation region of X is (w, b). 4. (S−shaped case) If all functions u ∈ U are so that u ◦ S −1 is convex for x < 0, concave for x > 0, u(0) = 0 and have a one-sided derivative at b with u0 (b) ≥ u(b)−u(w) S 0 (b, y0 ) S(b,y , then y0 is in the continuation region of X. 0 )−S(w,y0 ) This result shows that cautious EU preferences can accommodate a wide range of behavior, just as EU agents and Gul agents. For an illustration, take part 2. of the Proposition: even for a risk averse CEU naive agent, if a geometric Brownian motion X has high enough normalized drift, the agent will start with positive probability.

5

Conclusion

This paper considers the optimal stopping problem in a general setting without assuming any particular functional restrictions about how the agent evaluates risky prospects she faces when continuing. We have modeled the sequence of gambles facing an agent through 35 36

Personal communication with Pietro Ortoleva. See Cerreia-Vioglio et al. (2015) for examples.

25

a diffusion. We have characterized in terms of conditions on her risk preference her continuation and stopping region, both when she is unaware of the possible dynamic inconsistency in her behavior (naive agent) and when she is aware of that (sophisticated agent). We study the sophisticated case under some relatively strong assumptions on the policies the agent can use. In general, studying sophisticated behavior when dynamic inconsistency of preferences is a possibility is a hard task. In our setting, asking for a stronger solution concept like Markov Perfect Equilibrium or considering a larger set of possible strategies, turned out to be technically challenging and is left for future research. Finally, the model presented in this paper assumes that the agent’s preferences are defined only on lotteries about the final outcome of the gambling/investment decision she faces. Here, the agent cares about the path of the resolution of uncertainty only because of instrumental reasons. One can imagine that an agent may have preferences defined directly on the paths of the uncertainty tree she faces. For example, given two time paths of the realized gambling/investment outcomes, which lead to the same final lottery on wealth from the perspective of an agent at some fixed time t, the agent might prefer the path which has more, or less ‘variation’ on lottery outcomes before time t. Another natural case of stopping problems with many applications is when the agent’s preferences are defined on lotteries on the best value achieved by the process till it is stopped (running maximum). Think of a seller trying to figure out the best selling price for an item she owns and stopping the flow of price demands at some period t to sell the item to the highest bidder up to time t. While the running maximum is not a Markov process, the pair consisting of the process and the running maximum is a Markov process, albeit two-dimensional. Extending the current results to the case where the prize space is multidimensional is thus interesting not only from a purely technical perspective. More generally, studying optimal stopping of Markov processes with preferences which depend on the path toward final wealth and thus violate Consistency with static preference as defined in online Appendix 1, remains an exciting topic for future research.

6

Appendix

A On the stochastic process Here we discuss some details about the stochastic process used in the paper. Throughout the paper, we consider the diffusion on the Wiener space of continuous functions of the non-negative real line equipped with its natural sigma-algebra. We also consider the natural filtration, which is determined by the paths of the Brownian motion. Furthermore, to avoid cumbersome arguments in measure theory we assume that the underlying sigma-algebra of the Wiener space is complete. See Chapter 2.7 of Karatzas, Shreve (2012) for details. Finally, for the definition of progressive measurability used in Definition 1, which is needed to ensure that the stopped process is measurable, see Chapter 1 of Karatzas, Shreve (2012). Proof of Proposition 1. 1)-2) The first statement of the Lemma is a corollary of Ankirchner et al. (2015). We elaborate this in the following. 26

For the case of Martingale diffusions µ ≡ 0, we combine the statement of their Theorem 2 in Section 1 and the first statement in their Proposition 2 in the second Section. These state that for Martingale diffusions, the set of implementable distributions through stopping times (pure strategies) is precisely FX (y0 ) = {F ∈ ∆([w, b]) : Ex∼F [x] = y0 }. Section 6 of their paper from the beginning till the statement of Theorem 6, but excluding it, discusses how the result for the martingale case can be used to arrive at the result for general diffusions. The scaling function S just rescales the diffusion to a Martingale diffusion and the interval of prizes [w, b] to [S(w), S(b)]. Thus the set of implementable distributions through stopping times (strategies) is FX (y0 ) = {F ∈ ∆([w, b]) : Ex∼F [S(x, y0 )] = y0 }. It is easily seen that this set is convex. This and equation (2) imply that no randomized stopping time (mixed strategy) can achieve a distribution which can’t be achieved through a stopping time (pure strategy). Finally, since S is a bounded, continuous function, FX (y0 ) is closed in ∆([w, b]), which is compact in the topology of convergence in distribution, since [w, b] is a compact interval. 3) It is clear that binary distributions can be induced by simple threshold stopping times, since if the support of F is {a, c}, then Xτa,c ∈ {a, c} with probability one. We show that a pure Markovian stopping policy corresponds to a unique threshold stopping time. For the second part, let s : [w, b]→{stop, continue} be a pure Markovian policy and assume that current wealth is y0 . If s(y0 ) = continue, pick a = sup{y ∈ [w, y0 ] : s(y) = stop} and c = inf{y ∈ [y0 , b] : s(y) = stop}. Then the strategy s can be implemented by the simple threshold stopping time τa,c . Moreover, any stopping time which implements s has to be a simple threshold stopping time and it has to be equal to the τa,c defined above. This follows from the continuity of the paths of the process. For the other direction, let τa,c be a simple threshold stopping time. Define s(x) to be continue if x ∈ (a, c) and stop otherwise. This is obviously a pure Markovian stopping policy and it is implemented by τa,c . Ankirchner et al. (2015) talk about weak solutions in their paper, but this is w.l.o.g. in our setting because strong solutions, (which are ensured by our assumptions on the diffusion), are also weak solution and the embeddings found in their paper (i.e. the respective stopping times) are always measurable w.r.t. the filtration of the weak solution assumed in their set up. Proof of Lemma 1. Let f ∈ C ([0, ∞), [w, b]) with f (0) = y0 and consider it as a function in the larger space C ([0, ∞), R). Then according to the usual version of the Stroock-Varadhan Support Theorem (Theorem 6.1. in Pinsky (1995)) there exists a sequence of diffusion paths fn with fn converging to f in the maximum norm for all compact subintervals of [0, ∞). For an arbitrary function in C([0, ∞), R) define the hitting times τx = inf{t ≥ 0 : f (t) = x}. Consider the Lipschitz continuous mapping

27

π : C ([0, ∞), R) →C ([0, ∞), [w, b]) given by   f (t) if t ≤ min{τw , τb } π(f )(t) = w if t > τw   b if t > τb . This is an onto map of C ([0, ∞), R) into C ([0, ∞), [w, b]). The paths fn are mapped to paths π(fn ) of the diffusion stopped when leaving [w, b]. Moreover, Lipschitz continuity of π implies that π(fn ) converges to π(f ) in the topology of uniform convergence in compact subsets of [0, ∞). Remark 1. The assumptions made for the SDE (1) ensure that it has a strong unique solution, i.e. there is also path-wise uniqueness of the process in the space of continuous functions from [0, ∞) with starting point at y0 . A strong solution is adapted to the natural filtration of the Brownian motion, which drives the exogenous uncertainty in this model. Strulovici, Szydlowski (2015) make the point, that in economics only strong solutions of SDEs should be considered, as weak solutions are defined on larger spaces and thus account for more (uncontrolled) uncertainty than is intended to be modeled by the analyst (the analyst usually assumes all uncertainty is modeled through Wt ). In line with their logic we require throughout that the stopping times of the agents are adapted to the Brownian filtration of the Brownian motion driving the diffusion. Our assumption of Lipschitz continuity of the coefficients of the SDE in (1) can be relaxed along the lines of Le Gall (1983) (see Lemme 1.0, Corollaire 1.1, Corollaire 1.2 there and the subsequent discussion) and the general result for the naive agent as well as Theorem 1 in the online appendix can be proven for this more general class of diffusions. Details are available upon request. On the other hand, besides being standard in the literature in economics and beyond37 , Lipschitz continuity of the coefficients of (1) is a minimal assumption in the sense that there are examples of SDE’s with non-Lipschitz σ s.t. no strong solution exists (see for example Tanaka’s equation in Kallenberg (2006)). Our assumptions on (1) imply that there exists a constant  > 0 s.t. the variance process σ is always above . This fact is the only technical assumption needed in the proof of Theorem 2 to arrive at the characterization of the behavior of the sophisticated agent, besides conditions ensuring unique strong solutions for (1). This assumption corresponds to cases 2 and 3 in Theoreme 1.3 of Le Gall (1983) and thus is consistent with the existence of unique strong solutions for (1) as well. On the level of economic intuition, it means that we are only looking at prize processes whose uncertainty remains non-negligible over time.

B Proofs for section 3 6.0.1

Proof of Theorem 1

We will use in the proofs an equivalent way to rewrite the condition F δE[F ] for every lottery F with E[F ] = x: Fx+ δx for every zero-mean random variable  such that the random variable x +  has support within [w, b]. 37

See e.g. Strulovici, Szydlowski (2015).

28

Proof of Theorem 1. 1) Since F(y0 ) is the feasible set of lotteries induced by stopping times chosen by the agent this is obvious. 2-a) We will use in the proof the equivalent formulation of wRA through zero-mean noises. Assume first that X is a martingale38 and let for a fixed y0 ∈ (w, b) be  the zero-mean lottery, such that V (Fy0 + ) > V (δy0 ). Recall that we assumed that the preferences are always continuous. Assume that this y0 is current wealth. If the agent has a stopping time, such that Xτ ∼ y0 + , then the agent won’t stop. But  has bounded support and σ is bounded away uniformly from zero, so the conditions of Lemma 1 are fulfilled and the existence of the stopping time is assured. The construction in Ankirchner et al. (2015) is done for every filtration for which a weak solution exists, while here we have assured that the diffusion has a strong solution. This is w.l.o.g. We comment more on this technical detail in the Appendix A. The support of x +  being bounded and it having mean x, it follows that {Xt∧τ }t is a bounded Martingale39 converging in L1 (and a.s. of course, too). Therefore, the embedding from Ankirchner et al. (2015) is also uniformly integrable. 2-b) Note that we have V (δy0 ) ≥ V (Fy0 + ), (14) for all integrable zero-mean , provided x +  has support in I. Assume she doesn’t stop when some martingale diffusion X starts at y0 . This means there exists a stopping time τ such that Xτ ∼ F for some F ∈ F(y0 ) with V (F ) > V (δy0 ). But then Xτ − x fulfills the definitions of  in (14) and we have a contradiction. The second claim follows from the Stroock-Varadhan support theorem.40 To formulate this result in our setting, consider the space of continuous functions with domain [0, ∞) and values in [w, b]: C ([0, ∞), [w, b]). Note that every path of the diffusion X is an element from this set. Equip this space with the Maximum norm. Lemma 1 (Stroock-Varadhan). Assume the diffusion starts at y0 ∈ (w, b) and consider Cy0 := {f ∈ C ([0, ∞), [w, b]) : f (0) = y0 }. The set of possible paths of the diffusion X started at y0 is dense in Cy0 w.r.t. Maximum norm. This Lemma is proven in Appendix A. It implies, that for every f ∈ C ([0, ∞), [w, b]) with f (0) = y, t > 0,  > 0, y ∈ [w, b], under the assumptions we made on the diffusion the following holds Py ( sup |X(s) − f (s)| < ) > 0. (15) 0≤s≤t

This can be seen as follows: The set Bft () = {g ∈ Cy : sup0≤s≤t |g(s) − f (s)| < } is an open neighborhood of f in the metric space Cy (equipped with the maximum-norm). Lemma 1 shows that the paths of the diffusion X are dense in Cy (w.r.t. the topology induced by the maximum-norm). In particular, the measure Py has full support on Cy . 38

I.e. µ = 0 everywhere in the diffusion formulation (1). Recall, that we have assumed that X only lives in a bounded interval. 40 See for example the version in Pinsky (1995), p. 65. There it is stated for the general version of diffusions with values in R, but it can be extended easily to the case of diffusions stopped when leaving a bounded interval. 39

29

This, the definition of the support of a measure and the openness of the set Bft () imply (15). To complete the proof, take for y0 ∈ (w, b) a continuous function f with f ( 3t ) = y0 − 4 |y0 | and f ( 2t3 ) = y0 + 4 |y0 | with  > 0 small enough, we have that the probability that X reaches y0 when starting from y is positive. 3) For the case that X is a Martingale the result is trivial as then S(x) = x. For the case that X is not a martingale, we kill the drift using the scale function (an increasing homeomorphism)   Z t Z y µ(z) dz dt. S(y) = exp −2 2 y0 x σ (z) Then M = S(X) is a martingale and fulfills dMt = σ ˆ (Mt )dWt ,

M0 = 0.

with σ ˆ = (S 0 · σ) ◦ S −1 . This can be seen by applying Ito-s formula.41 Note that σ ˆ is 0 −1 bounded away from zero locally, because S , σ are. Define ρ = (y0 + ) ◦ S . Then ρ has d d again a bounded support. It holds that Xτ = (y0 + ) is equivalent to Mτ = ρ. It follows from this, that E[ρ] = 0, because X being a bounded process, so is M and we can use again the bounded convergence Theorem for Martingales. Lemma 1 is again applicable. Note for the following argument that ∆([S(w), S(b)]) = {L = α ◦ S −1 : α ∈ ∆([w, b])}. It follows, that the agent will continue at y0 , if the agent whose utility is given for each L of the form L = α ◦ S −1 for some α ∈ ∆([w, b]), by VS (L) = V ◦ S −1 (L) = V (α), continues at 0. This is given, since the latter agent prefers L to 0 if and only if the agent with utility function V prefers α to y0 .

6.1

Proof of Theorem 2

2,L Proof. Let Cinc ([w, b]) be the subset of twice continuously differentiable, strictly increasing functions f : [w, b]→R such that the second derivative f 00 is also Lipschitz continuous. For a fixed interval (a, c), which contains current wealth y0 , there is a correspondence between diffusions started in (a, c) and the set 2,L Pa,c = {p : [a, c]→[0, 1]|p(a) = 0, p(c) = 1, p ∈ Cinc ([a, c]), p0 (0+) < +∞, p0 (1−) > 0}.

This is the set of all possible win probabilities pX a,c for (a, c) as one varies through regular diffusions X. In this setting, the set of technologically feasible lotteries, given a fixed diffusion X with normalized drift σµ(z) 2 (z) , which corresponds uniquely to some scaling function S is given by X FX (y0 ) = {L(1 − pX a,c (y0 ), a, c) : a < y0 < c, 1 − pa,c (y0 ) = 41

For details see Chapter 7.3 in Revuz, Yor (2013).

30

S(c) − S(y0 ) }. S(c) − S(a)

Note, that for all diffusions FX (y0 ) ⊂ FX (y0 ) and even more is true: FX (y0 ) ∪ {δy0 } = ext (FX (y0 )), the set of extreme points of FX (y0 ). For fixed y0 , S, due to continuity of S it is easy to see that FX (y0 ) is closed and thus also compact. We establish the following 2,L useful correspondence between Cinc and diffusions. Lemma 2. Each scaling function S of a diffusion with normalized drift σµ2 is a member 2,L 2,L of Cinc ([w, b]). Conversely, for each function S in Cinc ([w, b]) such that S(y0 ) = 0 for some y0 ∈ [w, b] there exists a diffusion such that S is its scaling function. Proof. Checking the first statement is routine, except for perhaps the Lipschitz continuity of S 00 (·, y0 ). We give arguments for this in the following. Note that it holds  Z x  µ(t) µ(x) 00 S (x, y0 ) = (−2) exp −2 dt . 2 σ 2 (x) y0 σ (t) We note first that x 7→ σ21(x) is Lipschitz continuous. This uses the fact that σ as a function is bounded away from zero and from above. To close the argument, we use the fact that the product of two bounded, Lipschitz continuous functions is again Lipschitz continuous two times: first this delivers that x 7→ σµ(x) 2 (x) is Lipschitz continuous and second, that   R x µ(t) since x 7→ exp −2 y0 σ2 (t) dt is Lipschitz continuous and bounded, the second derivative x 7→ S 00 (x, y0 ) is Lipschitz continuous as well. 2,L For the second statement of the Lemma, take S ∈ Cinc ([w, b]) and y0 ∈ [w, b] such that S(y0 ) = 0. Then, obviously Z y S(y) = S 0 (z)dz, y ∈ [w, b] y0

and S 0 (z) = exp(− log( S 01(z) )). Consider then µ : [w, b]→R given by µ(z) = −

S 00 (z) . 2S 0 (z)

2,L This function is Lipschitz continuous, since S is in Cinc ([w, b]) and it is trivial to check that for any diffusion with normalized drift equal to µ the stopping problem (X, y0 ) has scaling function equal to S.

The requirement of never starting any diffusion X, identified with its respective p, boils down to: for every a < x < c and every p ∈ Pa,c the following holds: {y ∈ (a, c)|L(1 − p(y), a, c)δy } = 6 ∅. S(c)−S(y) Recall now that for fixed a < x < c we have p(y) = S(c)−S(a) and thus p0 (y) is proportional to −S 0 (y).42 But note also that  Z y  2µ(z) 0 S (y) = exp − dz . 2 x σ (z) 42

In particular, it is differentiable.

31

Due to our regularity assumption on the diffusion processes, we know that S 0 (y) is always bounded away from zero and infinity on each interval (a, c) with x ∈ (a, c). Now the statements of the theorem follow straightforwardly with the help of the following simple mathematical fact: for any strictly increasing f : [a, c]→[0, 1] with f (a) = 0, f (c) = 0 (a+) = 1 there exists some p ∈ Pa,c with p(x) > f (x), x ∈ (a, c) if and only if qa,c 0 +∞, or qa,c (c−) = 0.

6.2

Proof of Proposition 2

Before proving Proposition 2 we prove a technical Lemma which is needed in the proof and is of technical interest on its own. Lemma 3. 1) The set F(y0 ) is a compact, convex set.43 2) Every finite support element of F(y0 ) can be written as a convex combination of δy0 and binary lotteries from F(y0 ). 3) The subset of finite support measures in F(y0 ) is dense. Proof. Step 1. Let us show the result first for Brownian motion. In this case F bm (y0 ) = {F ∈ ∆([w, b]) : Ex∼F [x] = y0 }. This is clearly a convex set. Compactness follows from the fact that the set F(y0 ) is closed. The latter fact follows from the continuity of the function F 7→ E[F ], which is clear by the properties of weak convergence of probability measures and the fact that the function id : [w, b]→[w, b] with id(x) = x is continuous and bounded. 2) Let F be a finite support distribution in F(y0 ) and denote by p the finite support lottery it induces over [w, b]. Assume first that p(y0 ) = 0. The result then follows from Lemma C.1. and Corollary C.2. in the appendix of Xu, Zhou (2013). Their set of feasible distributions D is defined through an inequality, because the space of possible prizes there is [0, +∞), but a look at their proof shows that the proof is valid word-for-word in the case of our model as well, where the prize space is [w, b]. 3) Take a sequence of finite support distributions Fn on [w, b] s.t. Fn converges weakly to F . Because the function id : [w, b]→[w, b] with id(x) = x is continuous and bounded it follows for xn := E[Fn ] that xn →x. Take 0 <  < 21 min{|x − w|, |x − b|}. From now on, consider only sequences Fn s.t. xn ∈ (x − , x + ) for all n. If xn ≥ x define zn = − + x and xn < x define zn = + + x. Define then λn ∈ [0, 1] s.t. x = λn xn + (1 − λn )zn for all n. Looking at Gn = λn Fn + (1 − λn )δzn it follows λn →1, n→∞ from xn →x and so that Gn →F weakly. But E[Gn ] = x by construction and Gn are again probability distributions with finite support. Step 2. We now take an arbitrary diffusion which satisfies our regularity assumption. Due to Ankirchner et al. (2015) we know F(y0 ) = {F ∈ ∆([w, b]) : Ex∼F [S(x, y0 )] = 0}. Note that the argument in Step 1 didn’t use the particular form of the interval [w, b]. So in the following for the set F bm (y0 ) defined in Step 1 use the interval [S −1 (w), S −1 (b)]. This interval contains obviously zero (S −1 (y0 )). 43

Here the mixture operation is the mixture operation on distributions over [w,b].

32

Define the map ψ : F(y0 )→F bm (0), given by ψ(F )(x) = F (S −1 (x)). It is easy to see that this map is continuous, linear and a homeomorphism between the two compact sets F(y0 ) and F bm (0). Using this fact and Step 1 we are done. Proof of Proposition 2. We prove: if the agent prefers the certain expected value of a binary lottery to the binary lottery itself and is quasi-convex then she exhibits wRA everywhere. The case of not wRA is similar. Fix x ∈ (w, b) and a finite sequence of distributions Fi , i = 1, . . . , n with E[Fi ] = x such that each has a support of two elements only and so that Fi δx , (16) P Proper quasi-convexity then implies, that for all αi ≥ 0, i = 1, . . . , n with i=1 αi = 1 we have X αi Fi δx . i

By varying the Fi -s and the αi -s, it is easy to see that we can extend (16) to all finite support distributions with mean x. This uses Lemma 3. Due to Assumption 1 and Continuity (16) can then be extended to all distributions F with mean x. The above argument can be modified to show, that the agent whose current wealth is y0 and whose preferences satisfy quasi-convexity will always find it optimal to choose either δy0 or a binary lottery out of F(y0 ). To see this, let w.l.o.g. F ∗ 6= δy0 be an element which maximizes preference over F(y0 ). This implies in particular, that the agent starts. Assume that L≺F ∗ for all binary L ∈ F(y0 ). From quasi-convexity of preference, completeness and Lemma 3 it follows for all L in F(y0 ) with finite support that L≺F ∗ . This can be seen as follows: P if it weren’t true then for some finite support L with F ∗ L and decomposition L = ni=1 αi Li where αi ≥ 0, Li ∈ F(y0 ) have support of at most two elements, we can pick Li with F ∗ Li , contradicting either L≺F ∗ for all binary L ∈ F(y0 ) or that F ∗ 6= δy0 . Take a sequence of finite support Ln ∈ F(y0 ) with Ln converging to F ∗ . The existence of this sequence is assured due to Lemma 3. Again due to quasi-convexity and completeness of preference, there exists some Bn ∈ F(y0 ), binary lottery, or Bn = δy0 such that Ln Bn for all n. It follows in all Ln Bn ≺F ∗ . F(y0 ) being compact, there exists a converging subsequence of Bn . In particular, its limit B has to be either a binary lottery or δy0 . Continuity of preference then implies, that B ∼ F ∗ . But this contradicts the assumption, that either L≺F ∗ for all binary L ∈ F(y0 ) or that F ∗ 6= δy0 . It follows that B ∼ F ∗ has to be true. In all, the maximand in F(y0 ) can always be chosen to be a binary lottery in the case of quasi-convex preferences.

C Proofs for section 4 CP T (L) > U (x0 )

and E[L] = x0 .

Proof of Proposition 3. Pure-gain bet: r ≤ a < c. In this case the definition of qa,c (y) leads by the usual calculations to 0 (Pure Gain) (ν + )0 (qa,c (y))qa,c (y) =

33

U 0 (y) , U (c) − U (a)

y ∈ (a, c).

Mixed-gain bet: a < r ≤ c. In this case the definition of q(y) leads by the usual calculations to 0 (Mixed) qa,c (y)[U (c)(ν + (qa,c (y)))0 − U (a)(ν − (1 − qa,c (y)))0 ] = U 0 (y),

y ∈ (a, c).

Pure-loss bet: r ≤ c < r. In this case the definition of q(y) leads by the usual calculations to U 0 (y) 0 (Pure Loss) (ν − )0 (1 − qa,c (y))qa,c (y) = , y ∈ (a, c). U (c) − U (a) 0 Now we have to find sufficient and necessary conditions which ensure that either qa,c (a+) = 0 +∞ for all three kinds of bets or qa,c (c−) = 0 for all three kinds of bets. Taking the limits in the equations above and using that U is strictly increasing and U 0 (a), U 0 (c) ∈ (0, ∞) always, we see the following. 0 • Necessary for qa,c (a+) = +∞ it is that (ν + )0 (0+) = 0 and (ν + )0 (1−) = 0. 0 • Necessary for qa,c (c−) = 0 for for pure gains and pure losses bets it is that (ν + )0 (1−) = + 0 +∞ and (ν ) (0+) = +∞.

This establishes necessity. For sufficiency, consider first case 1. Fix a < y0 < c for some y0 ∈ (w, b). It follows 0 from the fact that U 0 (y) is bounded for y ∈ (a, c) and 1. that qa,c (a+) = +∞ in the case of all three kinds of bets. Now consider case 2. It follows easily, again from the fact that U 0 (y) is bounded for 0 (c−) = +∞ for the case of a pure gain and pure loss. For the case of y ∈ (a, c) that qa,c a mixed bet we have from the relation ν + (1 − p) + ν − (p) = 1 44 , replaced in the relation for mixed bets above, that U 0 (y) = +∞. 0 (y))(U (c) − U (a)) y→c− (ν + )0 (qa,c

0 lim qa,c (y) = lim

y→c−

Details for Example 4.1. The commitment case for geometric Brownian motion with w = 0, b = +∞, i.e. arbitrarily high prizes possible, has been studied in Xu, Zhou (2013). One can see from the paper that their results can be adapted to the case of b < +∞. In particular, Theorem 5.1 there, holds true again with b < +∞. To expand on details: the main change in the case b < +∞ in the Xu, Zhou (2013) set-up is that the set of possible distributions which can be implemented by uniformly integrable stopping times is given now by D = {F : [w, b]→[0, 1] : F is a CDF and EF [S(·, y0 )] = 0}. This is the same set as in Lemma 3.2 in Xu, Zhou (2013), except that the inequality EF [S(·, y0 )] ≤ 0 is strengthened to an equality.45 The proof of Theorem 5.1 in Xu, Zhou 44

This relation holds due to continuity of the CPT preference. There, the authors work with the geometric Brownian motion transformed into a Martingale, but that restriction is w.l.o.g. by the same steps as we have used here when proving the general results for naive agents. Moreover, the weak inequalities in Lemma 3.2 of Xu, Zhou (2013) are due to their choice of b = +∞. See Ankirchner et al. (2015) for more on this too. 45

34

(2013) can be repeated word for word for the case b < +∞ as well and the defining equation (5.3) for the optimal quantile function becomes G∗ (x) = min{b, G∗XZ (x)}, where G∗XZ (·) is given by equation (5.3) in Xu, Zhou (2013). We state the version of the Proposition we need in the following. β is as defined in the main body of the paper 1

monotone Proposition 9. Assume that v(x) = u(x β ) is concave and and with a strictly  λ 0 −1 derivative, as well as ν. If there exists a λ ≥ 0 such that (v ) > 0, ∀x ∈ (0, 1) ν 0 (1−x) and   Z 1 λ 0 −1 min{(v ) , b}dx = y0β , (17) 0 ν (1 − x) 0   λ 0 −1 then G(x) = min{(v ) , b} is the quantile function of an optimal distribution ν 0 (1−x) for the agent with commitment in the optimal stopping problem (X, y0 ). For our specification, denoting xb ∈ (0, 1] the smallest number with G(xb ) = b, the relation between xb and λ, if λ as in Proposition 9 exists is xb = 1 −

1  α  1−α

β−r

b− 1−α .

λ There is one-to-one relation between xb and λ > 0. The feasibility condition can then be written in terms of xb as 1−α βα−r yβ β−r r (1 − xb ) 1− β (1 − (1 − xb ) β−r ) + (1 − xb ) = β0 . βα − r b

(18)

It can be easily seen that for all y0 ∈ (0, b) an xb ∈ (0, 1) to fulfill the above equation exists and thus also a respective λ > 0 exists. The quantile function of the optimal distribution as given by Proposition 9 is  G(x) = b min{1,

1 − xb 1−x

 1−α β−r }.

Using the relation between λ and xb one gets by inverting the quantile function the optimal distribution   if x = b  1, β−r
1−α Fy0 (x) = 1 − (1 − xb ) xb − 1−α , if b > x ≥ b(1 − xb ) β−r  1−α  0, if 0 ≤ x < b(1 − x ) β−r , b

In particular, due to the one-to-one correspondence between quantile functions and distributions, the optimal distribution chosen by the naive agent is a Pareto distribution truncated at b < +∞ and with an atom at b. Proof of the last statement for the sophisticated agent. We prove the following slightly more general claim.

35

Claim: A sophisticated RDU agent never starts if u : [w, b]→R+ is (weakly) concave, µ is negative and ν is strictly concave. Proof of the Claim. The respective preference constraint that has to be satisfied for a sophisticated agent to implement a simple threshold stopping time is u(c) − u(y) ≥ ν(1 − p(y)). u(c) − u(a) Here p(y) is the win probability for an interval (a, c) (we omit for notational simplicity the superscripts and subscripts denoting resp. the diffusion and the interval). One can easily show that 1 − p(y) is a strictly concave and decreasing function whenever µ < 0. The right-hand side of the above inequality is thus decreasing and strictly concave, since d2 S 0 (y, y0 )2 S 00 (y, y0 ) 00 0 <0 (ν(1 − p(y))) = ν (1−p(y)) −ν (1−p(y)) dy 2 (S(c, y0 ) − S(a, y0 ))2 S(c, y0 ) − S(a, y0 ) (19) Here we have used that S is a strictly convex function for µ < 0. Since the left-hand side is (weakly) convex and decreasing and both sides coincide at y = a, c and are continuous, it follows that the required inequality can never be satisfied under the conditions mentioned. End of Proof of Claim. Returning to the parametric functions of the example: The claim covers the case r = 1. The result remains valid for all r < 1 but near enough 1, because the slope of the left-hand side of (19) doesn’t depend on r (the necessary parameter restriction consisting of γ < α remains valid whenever µ is negative enough). Details for Example 4.2. For µ > 0, 1 − p(y) is a convex, decreasing function 46 , and one can check by taking derivatives (see (19)) that ν(1 − p(y)) is convex and decreasing in y. Here we have used that S is a concave function for µ > 0. In general it is given by  σ2  µ 1 − exp(−2 2 (x − y0 )) . S(x, y0 ) = 2µ σ Moreover, clearly ν(1 − p(a)) = 1, ν(1 − p(c)) = 0. It follows, that the agent will start any geometric Brownian motion with µ > 0. More so, the set of possible stopping times for the agent are all τa,c with w ≤ a < y0 < c ≤ b. Given that the continuation requirement is void, the problem of the agent reduces to max

a:ν≤a≤y0 ≤c≤b

ν(1 − p(y0 ))a + (1 − ν(1 − p(y0 )))c.

Denote the objective function by Obj(a, c, y0 ) = ν(1 − p(y0 ))a + (1 − ν(1 − p(y0 )))c.

(20)

Its derivative w.r.t. a is dObj d = ν 0 (1 − p(y0 )) (1 − p(y0 )) (a − c) + ν(1 − p(y0 )), da da

(21)

while its derivative w.r.t. c is dObj d = ν 0 (1 − p(y0 )) (1 − p(y0 )) (a − c) + (1 − ν(1 − p(y0 ))). dc dc 46

In fact, 1 − p(y) becomes more and more concave as

36

µ σ2

is increased.

(22)

As long as respectively that

dObj da

< 0 or

dObj dc

> 0, it is profitable to lower a and raise c. Note,

2µ

2µ

2µ

d 2µ e−a σ2 2µ e−(y+a) σ2 − e−(c+a) σ2 (1 − p(y)) = 2 (1 − p(y)) =  2 > 0, da σ e−a σ2µ2 − e−b σ2µ2 σ2 −a 2µ2 −c 2µ2 σ σ e −e

(23)

and that 2µ

2µ

d 2µ 1 2µ e(c−a) σ2 − e(c−y) σ2 (1 − p(y)) = 2 p(y) = 2  2 > 0. 2µ dc σ e(c−a) σ2µ2 − 1 σ e(c−a) σ2 − 1

(24)

Plugging (23) and (24) in the derivative expressions (21) and (22), it follows after routine cancellations that a sufficient condition for dObj > 0 and dObj < 0 for all a < y0 < c is for dc da µ 47 c to be large enough (with a fixed). The argument establishing this is independent σ2 of the shape of w as long as σµ2 is large enough. It follows, that whenever τw,b is feasible, it will be chosen as long as σµ2 is large enough. Proof of Proposition 6. Fix a stopping problem (X, y0 ) of the type prescribed in the statement. Note that in both of diffusion classes considered, the scaling function for the stopping problem (X, y0 ) is as follows. S(x, y0 ) =

 σ2  µ 1 − exp(−2 2 (x − y0 )) . 2µ σ

Because the preference is quasi-convex (it satisfies Betweenness), we can restrict our analysis of feasibility of stopping times to simple threshold ones and only look at elements from F(y0 ) which are binary lotteries. The feasibility condition for a lottery L(p, y, x) with x < y0 < y can then be written as µ

µ

1 = (1 − p)e−2 σ2 (x−y0 ) + pe−2 σ2 (y−y0 ) . We solve here for p and plug the resulting equation in the formula of the utility of Gul preferences for binary lotteries, given by (11). It holds V (L(p, y, x)) > u(y0 ) if and only if y0 is not in the stopping region of X. Rearranging gives the condition 2µ

e σ2 (y0 −x) − 1 u(y) − u(y0 ) > 1 + β. µ 1 − e−2 σ2 (y−y0 ) u(y0 ) − u(x)

47

To check for

dObj dc

dObj da

µ σ 2 (c

< 0 it helps to look at the expression

> 0 it helps to look at the expression

µ σ 2 (c

− a) e

(c−a)



2µ σ2

−e

− a) e

2µ (c−y0 ) 2 σ 2

2µ (c−a) 2 σ e

2µ −(y0 +a) 2 σ

 e

−a

2µ σ2

−e

−e

−(c+a)

−c

2µ σ2

2 2µ

, while for

σ2

and to recognize, that

−1

1 − ν(1 − p(y0 )) = ν 0 (1 − p(y0 ))ξ(y0 ), with some ξ(y0 ) ∈ (1 − p(y0 ), 1). Routine cancellations show then, dObj that σµ2 (c − a) large enough is sufficient for dObj da < 0 and dc > 0.

37

Proof of Proposition 7. For fixed a < y0 < c, that the slope of the q := qa,c function fulfills u0 (y)(1 + (1 − q(y)))β − u(y)q 0 (y)β = u(c)q 0 (y) − (1 + β)u(a)q 0 (y). In particular, it follows q 0 (c−) =

u0 (c−) , (1 + β)(u(c) − u(a))

q 0 (a+) =

u0 (a+)(1 + β) . u(c) − u(a)

(25)

Result now follows directly from (25) and from Theorem 2. Details for Example 4.3. We first show that strategy τw,b is feasible for the sophisticate, i.e. it satisfies (12) with c = b, a = w for all y ∈ (w, b). For this, rewrite (12) as (1 + β)(y − a) p(y) ≥ . (26) c − a + β(y − a) Define the function g(y, a, c) as the LHS of (26). Given that p(y) and g(y, a, c) are both strictly concave and strictly increasing for y ∈ (a, c), τa,c is feasible if and only if its respective win probability function p(·) satisfies d d p(y)|y=a > g(y, a, c)|y=a , dy dy

d d p(y)|y=c < g(y, a, c)|y=c . dy dy

These are ensured if and only if 2µ 1 (c − a) 2µ > 1 + β, 2 σ 1 − e−(c−a) σ2

2µ 1 (c − a) < 1. 2µ 2 σ e(c−a) σ2 − 1

Since the function f1 : [0, ∞)→Rt 7→ 1−et −t is increasing in t and the function f2 : t [0, ∞)→R, t 7→ et −1 is decreasing in t we see that d0 is determined as the maximum of d1 and d2 which satisfy respectively f1 (d1 ) = 1 + β and f2 (d2 ) = 1. To get a sufficient condition which implies that τw,b is optimal for the agent we require first the bang-bang lottery to be preferred to current wealth. 1. For q =

2µ −(y0 −w) 2 σ 2µ −(b−w) 2 σ 1−e

1−e

it holds

qb+(1+β)(1−q)w 1+(1−q)β

> y0 .

Now we find conditions that ensure that given any fixed policy τa,c with w < a < y0 < c < b, the agent who has started would always want to increase c and lower a. We rewrite the objective function Obj(a, c) =

p(y0 )c (1 + β)(1 − p(y0 ))a + 1 + (1 − p(y0 ))β 1 + (1 − p(y0 ))β

by making the change of variables d = c − a into ˆ Obj(a, d) = a +

p(y0 )d . 1 + (1 − p(y0 ))β

Note that now the win probability of the diffusion can be written as 2µ

p(y0 ) =

1 − e−(y0 −a) σ2 2µ

1 − e−d σ2 38

.

We calculate

2µ

ˆ dObj 2µ (1 + β)d e−(y0 −a) σ2 =1− 2 . da σ (1 + (1 − p)β)2 1 − e−d σ2µ2 and

" # 2µ ˆ p(y0 ) 2µ (1 + β)d e−d σ2 dObj = 1− 2 . dd 1 + (1 − p(y0 ))β σ 1 + (1 − p)β 1 − e−d σ2µ2

One can then see the following facts 2. Whenever 2µ is large enough: if the agent starts a simple threshold stopping time σ2 τa,c , i.e. in particular a < y0 then she wants optimally to set d = c − a as large as possible.48 is large enough: if the agent starts a simple threshold stopping time 3. Whenever 2µ σ2 τa,c , i.e. in particular a < y0 and d is large enough she wants to optimally set a as small as possible. Combining now 1., 2. and 3. it is easy to see that for large enough sophisticated agent chooses optimally τw,b , i.e. she never stops.

µ (b σ2

− w) the Gul

Proof of Proposition 8. We first prove the following claim. For a stopping problem (X, y0 ) define Claim The preference defined by the functional VS : ∆([S(w, y0 ), S(b, y0 )])→R through VS (G) = V (G ◦ S) is a CEU preference with space of Bernoulli utilities Uˆ = {u ◦ S −1 : u ∈ U }. Proof of Claim. It is easy to see that EF [u] = EF ◦S −1 [u ◦ S −1 ],

F ∈ ∆([w, b]), u ∈ U.

From this it follows
−1 −1 −1 −1 −1 [u ◦ S V (F ) = inf u−1 (EF [u]) = inf S ◦ (u ◦ S ) ◦ E ] F ◦S u∈U u◦S −1 :u∈U  

−1 −1 −1 −1 −1 −1 −1 [u ◦ S =S inf (u ◦ S ) ◦ E ] = S V ) . S (F ◦ S F ◦S −1 u◦S

:u∈U

Note that the set Uˆ as defined in the statement of the claim, is again convex and compact. The latter follows because the map ψ : C([w, b])→C([S(w, y0 ), S(b, y0 )]),

ψ(u) = u ◦ S −1

is a homeomorphism between the two spaces of continuous functions, considered as normed spaces with the maximum norm. It follows that VS has a CEU representation with space of Bernoulli utilities given by Uˆ . 48

This uses the fact that p(y0 ) for the interval (a, c) remains bounded away from zero as d is varied, whenever a < y0 .

39

From part (3) of Theorem 1 it follows that y0 is in the stopping region of X if and only if VS exhibits wRA at zero. With the characterization of risk averse and risk loving behavior from Theorem 3 of Cerreia-Vioglio et al. (2015) (1)-(2) follow directly. (3) Let y0 ∈ (w, b) arbitrary. Note that for every u ∈ U the condition u0 (b) ≥ u(b)−u(w) implies that there exists a(y0 ) < b, c(y0 ) > y0 such that the line connecting b−w (a(y0 ), u(a(y0 ))) and (c(y0 ), u(a(y0 ))) lies above the graph of u restricted to the interval (a(y0 ), c(y0 )). Then choosing probability p such that L = L(p, a(y0 ), c(y0 )) has mean y0 we have that EL [u] > u(y0 ). This shows that V (L(p, a(y0 ), c(y0 )) > V (δy0 ) = y0 . It follows that the agent violates wRA at y0 . The result now follows from part 2-a) of Theorem 1. (4) Due to the claim, we can check whether 3) is satisfied for the preference represented by VS at the point 0 ∈ [S(w, y0 ), S(b, y0 )]. Obviously, that would be sufficient. Writing out the condition required in (3) yields the condition stated in (4).

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42

Online Appendix to Optimal Stopping with General Risk Preferences Jetlir Duraj∗ The online appendix is organized as follows. In part 1 we present behavioral foundations of the model considered in the main body of the paper. The main result of part 1 characterizes Expected Utility as the only risk preference model which is consistent with Dynamic Consistency of Preferences in our setting. In part 2 we relate behaviorally weak Risk Aversion to the classical Risk Aversion concept, which is defined as aversion to mean-preserving spreads. Finally, part 3 presents additional applications and examples of Theorems 1 and 2 from the main body of the paper.

1

Behavioral foundations of optimal stopping in continuous time.

In this part we give behavioral foundations in terms of axioms on dynamic choice behavior of the agent for the general optimal stopping problem we have considered in the previous parts. The motivation for this analysis is the following. First, it helps us establish that the assumptions about Naivet´e or Sophistication are indeed needed as soon as  does not have an Expected Utility representation. For dynamic choice over finite lottery trees (a discrete time setting) the following result is well-known: under some weak technical requirements, two axioms of dynamic behavior: ’Consequentialism’ and ’Dynamic Consistency of Preferences’1 , are equivalent to the agent being an Expected Utility-maximizer (see Hammond (1988) for a formal proof and Hammond (1989), Gul, Lantto (1990) but also Machina (1989) for more on the interpretation of this result).2 It follows, that to be able to pin down uniquely the dynamic behavior of the agent in finite lottery trees, when the agent is not an Expected Utility-maximizer, additional assumptions on dynamic behavior are needed, besides the existence of the preference relation  on lotteries. In this part, we formulate Dynamic Consistency of Preferences (DCP) for our setting and extend the classical result to our continuous time diffusion setting where the choice objects are (uniformly integrable) stopping times.3 This ∗

[email protected] In our setting we don’t need a Consequentialism axiom as it follows from our Consistency axiom, since the prize process, it being a diffusion, is Markovian; see details in the following. 2 The result we prove is not a special case of the classical result about finite lottery trees. Besides the different set up of continuous time, in the classical Theorem as stated in Gul, Lantto (1990) the agent faces finite lottery trees of arbitrary length, but the horizon of the decision problem is fixed once the tree she faces is fixed. Here the horizon of the decision problem is endogenous due to stopping. 3 We haven’t found any similar result for continuous time processes in the literature. Moreover, adapting the proof techniques from the discrete time setting in Hammond (1988) to ours is impossible. 1

1

means that DCP has to be relaxed in our setting as well, and that additional rules are needed to fully specify dynamic behavior. Naivet´e and Sophistication are precisely these additional rules.

A behavioral model of optimal stopping. We start explaining the behavioral foundations of the model of this paper by introducing some notation needed to state the axioms. Unless otherwise stated, in the following the index X runs over regular diffusions as defined in Set Up part of the main body of the paper. Fix such a diffusion X and a starting point y0 ∈ (w, b). Denote the elements of the filtration of the diffusion X by FXT (y0 ), T ∈ [0, ∞). Recall, that FXT (y0 ) encodes the prize uncertainty resolved till time T . An arbitrary element n from FXT (y0 ) can be interpreted as a ‘node’ of depth T of the ‘uncertainty tree’ defined by the diffusion X started at y0 . We identify the singleton set 0 FX0 (y0 ) with y0 . We say that n0 ∈ FXT (y0 ) is a continuation of n ∈ FXT (y0 ) if T 0 > T and the occurrence of n0 implies that of n. Denote by FX (y0 ) the union of FXT (y0 ) as T ranges across all positive time periods and by F(y0 ) the union of all FX (y0 ) as X ranges across all regular diffusions. The former encodes all possible histories of prize path realizations when fixing a particular regular diffusion and the latter when considering all regular diffusions started at y0 . FX (y0 ) is a well-defined object because all diffusions X are adapted to the filtration of the underlying Brownian motion W of the diffusion. The choice objects of the agent at each moment in time are given by the set S of (uniformly integrable) stopping times. We recall here the convention from Set Up section of the main body of the paper: all stopping times are assumed to be adapted to the filtration of the underlying Brownian motion which drives the diffusions.4 For each n ∈ F(y0 ) we assume that the agent has a complete and transitive preference relation n over S. An agent can thus be identified with the collection of preference relations A = (n )n∈F (y0 ) : for each diffusion X a collection of preference relations for each node that can be reached by the diffusion. A stopping time can then also be identified with a stopping policy, which is a function from FX (y0 ) to {stop, continue} telling the agent to stop or continue after any event from FX (y0 ). In our model, the preference relations n are related to each other by the existence of a fixed, static risk preference, which evaluates prize lotteries in a history-independent way. To simplify notation in the following we denote for a real-valued random variable Y and n an event from some sigma-algebra σ by FY |n the distribution of the random variable conditional on the event n occurring. The following definition is an adaptation of the similar to the Definition of Consistency in Gul, Lantto (1990), who consider choice in discrete time problems modeled as finite lottery trees. Definition 1. The agent exhibits Consistency with static preference if there exists a risk preference functional V : ∆([w, b])→R such that for all diffusions X and all events n 4

In particular, this also ensures that all diffusions live in the same filtered probability space.

2

possible under FX (y0 ), it holds n is represented by V (FXτ |n ). Thus, the value of stopping time τ if event n has occurred is given by the static utility of the distribution of Xτ , evaluated conditional on n having occurred. We maintain Consistency with static preference in the following. The next definition gives a formal statement of the well-known Dynamic Consistency axiom in our setting. Definition 2. An agent satisfies Dynamic Consistency of Preferences (DCP) if for all diffusions X, all y0 ∈ (w, b) and three different n, n1 , n2 ∈ FX (y0 ) such that i) n1 , n2 are continuations of n and ii) the probability that either n1 or n2 happens conditional on n happening, is one, the following implication is true for any four stopping times τ1 , τ10 , τ2 , τ20 from S: ( ( τ , if n τ10 , if n1 1 1 . n τ 0 = if τ1 n1 τ10 and τ2 n2 τ20 then τ = τ20 , if n2 τ2 , if n2 Dynamic Consistency of Preferences says, that if for two mutually exclusive and exhaustive continuation events n1 , n2 of n, stopping time τ1 is preferred to τ10 at event n1 and τ2 is preferred to τ20 at event n2 , it should also hold that the combined stopping time τ is preferred to the stopping time τ 0 after event n. DCP is violated when there is an event n and two stopping times τ, τ 0 from S such that at n it holds τ 0 n τ even though the stopping time τ leads to (weakly) better prospects than τ 0 in all future continuations of event n. DCP puts very strong restrictions on the preference  which the agent uses to evaluate lotteries from ∆([w, b]), as the following Theorem shows. Theorem 1. The static risk preference functional V of an agent has an Expected Utility representation with a strictly increasing, bounded and continuous Bernoulli utility function u : [w, b]→R if and only if the following requirements are met: 1. V satisfies FOSD-monotonicity and is continuous in the topology of convergence in distribution. 2. The agent satisfies Dynamic Consistency of Preferences. Among other things, this result implies that if the agent’s risk preferences are not Expected Utility, the agent is time inconsistent in some optimal stopping problem. This occurs in the case of a naive agent at an event n, because she projects the ‘current’ preference n into all preferences of continuation events: she decides on the stopping time at event n by assuming that n0 = n for all continuation events n0 of n. The sophisticated agent on the other hand, restricts her choice set of stopping times S at event n to the set S(n) of stopping times she knows will not lead to preference reversals, no matter the continuation of the process. 3

Before giving the full proof of Theorem 1, which is technically involved, we sketch the main idea behind the hardest part of the proof: sufficiency of DCP for the risk preference to have an Expected Utility representation, i.e. to satisfy Independence. Suppose the agent prefers distribution F to G and assume that she faces the choice between λF + (1 − λ)H and λG + (1 − λ)H for some other arbitrary lottery H and λ ∈ (0, 1). Suppose agent satisfies DCP and there is a stopping problem where the agent has two stopping strategies τF,H and τG,H which, if implemented, lead to a history h1 where H is realized with probability 1 − λ and otherwise to a history h2 where respectively F or G is realized with probability λ. Under this situation DCP will imply that the agent prefers τF,H to τG,H at the current moment of time, because conditional on either history h1 or h2 the agent prefers τF,H to τG,H . But the distribution induced by, respectively τF,H or τG,H , is λF + (1 − λ)H or λG + (1 − λ)H! Now consistency will imply that the agent prefers λF + (1 − λ)H to λG + (1 − λ)H in a static problem as well! This implies that the risk preference of the agent satisfies Independence. The latter fact and the other assumed technical conditions imply due to classical results, that the risk preference has an Expected Utility representation.5 Proof of Theorem 1. Necessity: Let V be given by V (F ) = EF [u] for some u : [w, b]→R strictly increasing, bounded and continuous. Requirement 1. is then standard. Regarding Requirement 2: take τ1 , τ10 , τ2 , τ20 , n, n1 , n2 as in Definition 2. It follows for τ¯ either τ or τ 0 that E[u(Xτ¯ )|n] = E[u(Xτ¯ ), n1 |n] + E[u(Xτ¯ ), n2 |n] = E[E[u(Xτ¯ )|n1 ]|n] + E[E[u(Xτ¯ )|n2 ]|n], where the last equality follows from the Markov property of diffusion processes. Both summands at the end of the calculation above are weakly higher for τ¯ = τ than τ¯ = τ 0 by hypothesis. This shows necessity of the requirements. Sufficiency: The proof consists of two steps. First, we establish the following important Lemma which states that under the three Requirements, V satisfies the Independence axiom of Expected Utility (cf. the discussion preceding Proposition 1 in the Set Up section of the main body of the paper). Lemma 1. Under the Requirements 1−2, V satisfies Independence, i.e. for all H, G1 , G2 ∈ ∆([w, b]) and α ∈ [0, 1] we have V (G1 ) ≥ V (G2 ) implies V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Proof. Take H as in the statement and assume w.l.o.g. that α ∈ (0, 1). We assume for now additionally that Gi , i = 1, 2 are step functions, i.e. correspond to lotteries of finite support. We relax this assumption at the very end. We divide the proof for this case in several steps. Step 1. Assume first, that {G1 , G2 } is ordered by FOSD-monotonicity. In particular, it holds G1 >F OSD G2 , the other case being excluded by V (G1 ) ≥ V (G2 ). It then follows that αH + (1 − α)G1 >F OSD αH + (1 − α)G2 , 5

See Theorem 3 in Grandmont (1972).

4

and by Requirement 1 that V (αH + (1 − α)G1 ) > V (αH + (1 − α)G2 ). Step 2. Assume now that {G1 , G2 } is not ordered by FOSD-monotonicity and that H has support contained in (w, b). 2,L Recall from the statement and proof of Lemma 2 the set Cinc . Equip it with the metric 2,L given by the maximum norm. Consider the map ψ : Cinc →R given by ψ(S) = EG1 [S] − EG2 [S]. 2,L This map is continuous and since Cinc ([w, b]) is a convex metric space (in particular it is connected), it follows that the image of ψ in R is connected. In particular, it is an interval. We need the following auxiliary Claim. 2,L Claim 1: There exists a S ∈ Cinc ([w, b]) such that ψ(S) = 0. Proof of Claim 1. Assume this is not the case. It then follows that the whole image of ψ, it being connected, consists of either only negative or only positive numbers. It then follows that 2,L either Case 1: EG1 [S] > EG2 [S] for all S ∈ Cinc ([w, b])

or 2,L Case 2. EG1 [S] > EG2 [S] for all S ∈ Cinc ([w, b])

We close the proof of the Claim by showing that this implies that {G1 , G2 } is ordered by FOSD. Focus on Case 1, the other one being analogous. Pick a finite sequence x1 < x2 < · · · < xn in [w, b] which consists of the union of the support of the lotteries corresponding 2,L to G1 and G2 . For a k ∈ {2, . . . , n} pick Sk ∈ Cinc ([w, b]) with Sk (xj ) = 1 − (n − j) for k ≤ j ≤ n and Sk (xj ) = j for all 1 ≤ j < k. This is possible for  > 0 small enough. It follows from Case 1 that n X

(1 − (n − j))(G1 (xj ) − G1 (xj−1 )) + 

>

G1 (xj ) − G1 (xj−1 )

j=1

j=k n X

k−1 X

(1 − (n − j))(G2 (xj ) − G2 (xj−1 )) + 

k−1 X

G2 (xj ) − G2 (xj−1 )

j=1

j=k

for all  > 0 small enough. Letting now  go to zero we recover for all k ∈ {2, . . . , n} that G1 (xj ) ≤ G2 (xj ), j = 1, . . . , n. But this implies that G1 FOSD-dominates G2 .6 End of Proof of Claim 1. 2,L It follows, that there exists some S ∈ Cinc ([w, b]) with EG1 [S] = EG2 [S]. As in the proof of Lemma 2 of the Appendix of the paper it follows that there exists some y2 and some diffusion X, started at y2 ∈ [w, b] with scaling function S : [w, b] × [w, b]→R, so that S(y2 , y2 ) = 0 and Ex∼G1 [S(x, y2 )] = Ex∼G2 [S(x, y2 )] = 0. 6

cf. Proposition 6.D.1. in pg. 195 of Mas-Colell, Whinston, Green (1995).

5

Consider now the function ρ : [w, b]→R given by ρ(z) = Ex∼H [S(x, z)]. This is a continuous function with ρ(b) ≥ 0 and ρ(w) ≤ 0. In particular, it follows, there exists y1 ∈ (w, b) with ρ(y2 ) = 0 (that y1 can be chosen different from w, b follows from the intermediate assumption that supp(H) ⊂ (w, b) ). Due to Proposition 1 in the main body of the paper there exists τH , τG1 and τG2 such that if the diffusion X is started at y1 then FXτH = H and if X is started at y2 then FXτG = Gi , i = 1, 2. i Step 2a. Assume first, that y1 > y2 and consider the stopping time τy2 ,y1 . We show that Independence holds for these kinds of H. The case of y2 > y1 is similar. We know from the proof of Theorem 2 in the main body of the paper and its preceding discussion, that for y ∈ (y1 , y2 ) the probability p(y) that X started at y reaches y1 before it reaches y2 is a strictly increasing continuous function with p(y1 ) = 1, p(y2 ) = 0. There exists thus a y0 ∈ (y1 , y2 ) with p(y0 ) = α. Consider now the stopping times τi for i = 1, 2 given by7 ( τH , if Xτy1 ,y2 = y1 τi = τGi , if Xτy1 ,y2 = y2 . This is again a uniformly integrable stopping time, i.e. an element of S.8 The Markov property of X shows that Xτi has the distribution αH + (1 − α)Gi . Take the ‘root’ event y0 ∈ FX0 (y0 ). It follows that on the event n = {Xτy1 ,y2 = y1 } V (FXτi |n ) = V (FXτH |n ) = V (H), while on the event m = {Xτy1 ,y2 = y2 } V (FXτi |m ) = V (FXτG

i

|m )

= V (Gi ).

Here we have used the fact that diffusions satisfy the strong Markov property which implies that the distribution of Xτ conditional on an event A in the sigma-algebra of another stopping time τˆ, which is finite with probability one, depends on its sigma-algebra σ(ˆ τ ) only through Xτˆ . Note that the union of m and n is the whole sample space and thus has probability one of occurring. This uses continuity of the diffusion process X. Because V (G1 ) ≥ V (G2 ) DCP yields that τ1 y0 τ2 . In all, due to the Definition 1 and DCP, it follows V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Step 2b. We now assume that H with supp(H) ⊂ (w, b) has y1 = y2 instead. In this case, one can find a sequence of distributions Hn , n ∈ N with (1) supp(Hn ) ⊂ (w, b), (2) Hn →H, n→∞ in distribution and (3) so that the y1n in (w, b) defined by Ex∼Hn [S(x, y1n )] = 0 7

Intuitively, we paste together the two distributions Gi and H in such a way so that from the perspective of the agent at time 0, when diffusion is started at y0 the probability that τH is realized is precisely α. 8 One checks easily that the events {τi > t} for t ≥ 0 depend only on the evolution of the process till time t.

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have y1n > y2 (since S(·, ·) is increasing in the first argument and decreasing in the second, it suffices to perturb H by shifting some probability on higher values within the support of H). The case above gives V (αHn + (1 − α)G1 ) ≥ V (αHn + (1 − α)G2 ) and Continuity of V establishes that V (αH + (1 − α)G1 ) ≥ V (αH + (1 − α)G2 ). Step 3. Finally, it remains to consider the case of general H ∈ ∆([w, b]) with support possibly including w or b. In this case, we can again find a sequence Hn , n ∈ N with (1) supp(Hn ) ⊂ (w, b), (2) Hn →H, n→∞ in distribution. The arguments above then give V (αHn + (1 − α)G1 ) ≥ V (αHn + (1 − α)G2 ) and Continuity of V establishes again that V (αH +(1−α)G1 ) ≥ V (αH +(1−α)G2 ). This establishes the proof for the case of step distributions G1 , G2 . We now use the following Claim and continuity to close the proof for the case when G1 , G2 are not necessarily step distributions (i.e. their respective lotteries don’t have finite support). Claim 2 For G a cdf, there exists sequences of step cdf-s Gi,n , n ∈ N, i = 1, 2 so that Gi,n →Gi in distribution with Gi,n V (G1 ) ≥ V (G2 ) > V (G2,n ) and Gi,n →Gi , weakly for n→∞. This, Continuity of V and the fact that Independence holds for the triplets G1,n , G2,n , H finishes the proof of the Lemma. Given the result of Lemma 1 and the Continuity assumption on , Theorem 3 of Grandmont (1972) yields a representation V (F ) = EF [u] with a bounded and continuous u : [w, b]→R. Requirement 1 then establishes that u is also strictly increasing. This finishes the proof of sufficiency.

2

Relation between weak Risk Aversion and Risk Aversion

Weak Risk Aversion (wRA) is crucial concept for the characterization of the optimal stopping behavior of a naive agent. It is an implication of risk aversion, defined as aversion to mean-preserving spreads.9 Here we clarify precisely the behavioral relation between these two properties of preference. The axiom needed to establish the relation is a relaxation of the Independence Axiom from Expected Utility Theory, which we restate here for comparison and reader’s reference. 9

See Section 6.D. in Mas-Colell, Whinston, Green (1995) for a definition and discussion of this concept.

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Axiom: Independence For G1 , G2 , F ∈ ∆([w, b]) with G1 G2 and any α ∈ [0, 1] we have that αH + (1 − α)G1 αH + (1 − α)G2 . The following relaxation is needed for the proof of Proposition 1. Axiom: Mixture Monotonicity w.r.t. Certainty (MMC) Let for i = 1, . . . , n be P xi ∈ [w, b], Fi be lotteries with E[Fi ] = xi and αi ≥ 0 with i αi = 1. If Fi δxi for all i = 1, . . . , n, then X X αi F i  α i δx i . i

i

If Fi δxi for all i = 1, . . . , n, then X

αi F i 

X

i

α i δx i ,

i

Here, mixture operator is in the sense of distributions. This axiom says that one can aggregate preference comparisons as long as one (and the same) side of the comparisons concern the certain expected value of the other respective side of the comparisons. Obviously, MMC is implied by Independence. It is well known in the decision theory literature that for Expected Utility preferences risk aversion is equivalent to wRA and that for non-Expected Utility preferences risk aversion is stronger than wRA. The following Proposition states, that in the case of non-Expected Utility risk preferences, MMC is precisely the weakening of Independence needed, under which wRA implies risk aversion. Proposition 1. 1) The following are equivalent. (a)  satisfies Risk Aversion in the sense of aversion to mean-preserving spreads (b)  satisfies Mixture Monotonicity w.r.t. Certainty and weak Risk Aversion everywhere. 2) The following are equivalent. (a)  satisfies Mixture Monotonicity w.r.t. Certainty and strong not wRA everywhere: for all x ∈ (w, b) and F ∈ ∆([w, b]) with E[F ] = x we have F δE[F ] . (b)  satisfies Risk Loving in the sense of preference for mean-preserving spreads. Proof of Proposition 1. 1) We show that (b) implies (a) first. Assume  satisfies wRA and MMC. Let F be a mean preserving spread of G. Then there exists a probability kernel K : [w, b] × [w, b]→[0, 1] 10 such that  Z Z F (A) = K(dz, y) dG(y), A 10

I.e. K(z, ·) is measurable for each z ∈ [w, b] and z 7→ K(z, y) is a probability distribution function for all y ∈ [w, b].

8

and

Z zK(dz, y) = y, for all y ∈ [w, b].

This characterization follows from the arguments in Example 6.D.2 in Chapter 6 of MasColell, Whinston, Green (1995). In particular, we have that the distribution K(·, y) is different from δy for y ∈ [w, b] only by a zero-mean bet. It follows from wRA, that K(·, y)δy , for all y ∈ [w, b].

(1)

Now MMC implies that F G, if G is a step function. For a general distribution G there is a sequence of step functions Gn , which are also probability distributions, such that Gn converges weakly to G. It follows for Z Fn (z) = K(z, y)dGn (y), z ∈ [w, b]. that Fn is a mean preserving spread of Gn and that therefore Fn Gn due to the previous argument. We now use the following Fact 1 If a sequence of distributions Gn , n ≥ 1 converges weakly to G, then for all upper-semicontinuous functions f : [w, b]→R we have Z Z lim sup f (z)dGn (z) ≤ f (z)dG(z). (2) n→∞

Proof of Fact 1. For f an indicator of a closed set, the result follows from the socalled Portmanteau Theorem.11 Otherwise, the result is standard, once it is recalled that an upper-semicontinuous function over a compact set has a maximum and thus is bounded from above. A reference is for example Theorem 1.3.4 in Van Der Vaart, Wellner (1996). It follows that due to weak convergence of Gn to G and the fact that K(·, y) is uppersemicontinuous for all y ∈ [w, b], it being a probability distribution, one has due to the Fact just proved, that lim sup Fn (y) = lim sup Ex∼Gn [K(x, y)] ≤ Ex∼G [K(x, y)] = F (y), n→∞

y ∈ [w, b]

(3)

n→∞

Assume by contradiction that F G. Due to continuity, there is a natural number N s.t. F Gn for all n ≥ N . Furthermore, the space ∆([w, b]) being compact w.r.t. convergence in distribution, we have a subsequence Fnk , k ≥ 1 converging weakly to some F¯ ∈ ∆([w, b]). Now we note the following fact. 11

See Theorem 4.25 in Kallenberg (2006).

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Fact 2 For a distribution F ∈ ∆([w, b]) the set of discontinuities is countable and each open interval of [w, b] has a point where F is continuous. Proof of Fact 2. It suffices to show that the number of discontinuities is countable as the second part follows from the fact that any open, non-empty interval has uncountably many points. Any discontinuity of f is in ∪n≥1 An := ∪n≥1 {x ∈ [w, b] : f (x+) − f (x−) >

1 }. n

Since f (b) − f (w) = 1, any of the An sets cannot contain infinitely many elements. We use this claim on the implication of (3) which implies that F¯ (y) ≤ F (y), for all continuity points y of F¯ .

(4)

Given this, we note that for arbitrary y ∈ [w, b] we have F¯ (y) = lim F¯ (y + n ) ≤ lim F (y + n ) = F (y). n →0+

n →0+

Here, in the first equality we have used Claim 2 to construct a sequence of positive numbers n going to zero, s.t. F¯ is continuous in the points y + n as well as right-continuity of F¯ , whereas the inequality follows from (4) and the last equality again from right-continuity of F . This gives that F¯ FOSD-dominates F . In particular, for all nk ≥ N we have F¯ Gnk . But Fnk converges weakly to F¯ , so that there exists nk , k large enough with Fnk Gnk . This is a contradiction to the assumption. It follows that F G and thus the conclusion for arbitrary F as well. Now we show that (a) implies (b). Risk aversion implies directly wRA. Given this, the case Fxi +i Fxi never occurs in the strict form, so that for MMC we can focus on the case Fxi +i Fxi . But it is easy to see that X αi Fxi +i i

is a mean preserving spread of the distribution of the lottery 2) The proof is analogous to the proof of 1).

3

P

i

α i δx i .

Additional Applications of Theorems 1 and 2

S2.1: Applications to Quadratic Preferences Quadratic preferences are a special case of smooth preferences, which were introduced in Machina (1982) as a response to Allais paradox and several other related paradoxes of Expected Utility. Smooth preferences are precisely those risk preferences which can be represented by a functional which is Fr´echet-differentiable.12 This implies that locally, preferences look like Expected Utility, even though globally Independence Axiom may be violated. This pattern can be used to explain Allais paradox. 12

We refer to Machina (1982) for the formal definition and the properties.

10

Here we concentrate on the main example of smooth preferences from Machina (1982): quadratic preferences. Given functions R : [w, b]→R and T : [w, b]→R these are given by the preference functional Z Z 1 1 V (F ) = RdF + [ T dF ]2 = EF [R] + EF [T ]2 . (5) 2 2 One says the preference is properly quadratic if T is a non-constant function.13 They were axiomatized in Chew et al. (1991). Another way to represent quadratic preferences used in Chew et al. (1991) is given by Z Z 1 V (F ) = φ(x, y)dF (x)dF (y).14 (6) 2 A sufficient condition in the case of the naive agent for y0 to be in the continuation region of X, is the existence of a stopping time τ such that 1 1 E[R(Xτ )] + [E[T (Xτ )]2 > R(x0 ) + T (x0 )2 . 2 2 Similarly, the naive agent will stop immediately if R, T are such that for every stopping time τ which is strictly positive with positive probability, we have 1 1 E[R(Xτ )] + [E[S(Xτ )]2 < R(x0 ) + S(x0 )2 . 2 2 From this we can derive immediately classical results for the case the agent’s preferences are Expected Utility.15 A special case: Expected Utility Set T ≡ 0 in (5). From the previous paragraph, the following Proposition is immediate. Proposition 2. An EU-agent with Bernoulli utility u always stops a diffusion Xt with probability one if u(Xt ) is a Supermartingale. A EU agent never stops a diffusion with probability one if u(Xt ) is a Submartingale. This generalizes Proposition 1 from Ebert, Strack (2015), because their assumptions on the Bernoulli utility function and the process Xt are precisely so that R(Xt ) in their setting is a strict Supermartingale. A formal proof of this result, which we omit since it is easy, uses the fact that the Bernoulli utility of an EU agent is bounded, if her risk preference is defined on the set of all possible distributions over [w, b].16 Sufficient conditions for the Supermartingale or Submartingale property are easy to find, for example by employing Ito-s Lemma.17 13 14

This definition corresponds to a function being properly quadratic in Chew et al. (1991). To see this rewrite from the definition in equation (6) of Machina (1982) as Z Z 1 (R(x) + R(y) + T (x)T (y)) dF (x)dF (y). V (F ) = 2

15

This corresponds to the case that the preferences as represented in (5) are called not properly quadratic. 16 See Theorem 3 in Grandmont (1972). 17 See Chapter 3 of Karatzas, Shreve (2012).

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Corollary 1. Assume that u : [w, b]→R is twice continuously differentiable. Then, u(Xt ) 2 is a Supermartingale if µ(x)u0 (x) + σ 2(x) u00 (x) ≤ 0 and the EU-agent stops with probability 2 one. u(Xt ) is a Submartingale if µ(x)u0 (x) + σ 2(x) u00 (x) ≥ 0 and the EU-agent never stops with probability one. This Corollary has the following nice interpretation in terms of the00 traditional concept (x) of the coefficient of absolute risk aversion, defined as A(x) = − uu0 (x) . Whenever the 1 normalized drift is below 2 A(x) for all x ∈ [w, b] the agent never starts and whenever it is above the agent never stops with probability one. So the preference constraint in the case of an agent with Expected Utility can be described uniquely by the coefficient of absolute risk aversion A. Another special case of quadratic preferences: Choice-Acclimating Personal Equilibrium (CPE) This equilibrium concept was introduced in K¨oszegi, Rabin (2007) to model preferences under risk for an agent who experiences expectation-based loss aversion. They have become very popular in the applied behavioral economic theory due to their tractability compared to other models of stochastic reference dependence. Due to the work in Masatlioglu, Raymond (2016) we know that CPE, taken as a static risk preference, is the intersection of quadratic and rank-dependent preferences. As quadratic preferences they are encoded by the function φ : [w, b] × [w, b]→R given by φ(x, y) =

1 (u(x) + u(y) + (1 − λ)|u(x) − u(y)|) , 2

where u : [w, b]→R is an increasing function and 2 ≥ λ > 1 is the loss aversion parameter.18 This parameter determines the magnitude of the disutility the agent experiences when comparing each possible realized outcome of the lottery with a better outcome which could have been realized. Thus CPE preferences are defined uniquely by the pair (u, λ). Masatlioglu, Raymond (2016) establish that CPE preferences are quasiconvex. Theorem 1 in the main body of the paper and the results in Masatlioglu, Raymond (2016) give the following characterization for the case of a naive CPE agent. Proposition 3. Consider a CPE agent with loss aversion parameter λ ∈ (1, 2] and utility u : [w, b]→R, who is naive about the dynamic inconsistency she faces. Then the following holds for any stopping problem (X, y0 ) with scaling function S 1. If u ◦ S −1 is concave, then y0 is in the stopping region of X. 2. If u ◦ S −1 is strictly convex at 0, then y0 is in the continuation region of X. We now turn to the sophisticated case general for quadratic preferences. Recall the specification (5) of quadratic preferences. Assume that R, T : [w, b]→R are differentiable and so, that for every F ∈ ∆([w, b]) we have R0 (x) + T 0 (x)EF [T ] > 0. 18

The restriction 2 ≥ λ > 1 is needed to ensure that the preferences are FOSD-monotonic.

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(7)

This condition ensures, via the characterization of smooth preferences in Machina (1982), that the quadratic preferences we consider fulfill FOSD-monotonicity.19 In line with Axiom 2 from the Set Up section in the main body of the paper, we assume (7) in the following, as it is part of the basic assumptions we have imposed on the preference. We prove the following in the appendix. Proposition 4. 1. If R, T are continuously differentiable it follows that the sophisticated agent always starts some diffusion. 2. A sophisticated agent with monotonic CPE preferences (λ ∈ (1, 2]) and differentiable u always starts some diffusion. Besides the case when R, T are differentiable, one could hope to get the never stopping result if we restrict to quadratic preferences whose representation has non-differentiable R or T . As Proposition 4 shows, this is not true for the subclass of CPE preferences.

S2.2: Proofs for the results on Quadratic Preferences Proof of Corollary 1. Ito-s lemma applied to u(Xt ) gives Z t Z t σ 2 (Xs ) 00 0 u (Xs )ds + σ(Xs )u0 (Xs )dWs . µ(Xs )u (Xs ) + u(Xt ) = u(y0 ) + 2 0 0 The last term in the RHS is a zero-mean Martingale. The result now follows immediately by taking expectations. Proof of Proposition 3. We first prove the following Claim, which holds more generally for quadratic preferences as defined in (6). Claim Given a quadratic preference over ∆([w, b]) identified with the properly quadratic function φ, the preference over ∆([S(w, y0 ), S(b, y0 )]) given by VS (G) = V (G ◦ S) is again a proper quadratic preference encoded by φˆ : [S(w, y0 ), S(b, y0 )] × [S(w, y0 ), S(b, y0 )]→R ˆ r) = φ(S −1 (z), S −1 (r)). given by φ(z, Proof of Claim. One writes Z Z Z Z 1 b b 1 b bˆ V (F ) = φ(x, y)dF (x)dF (y) = φ(S(x), S(y))dF ◦ S −1 (S(x))dF ◦ S −1 (S(y)) 2 w w 2 w w Z Z 1 S(b) S(b) ˆ = φ(z, r)dF ◦ S −1 (z)dF ◦ S −1 (r). 2 S(w) S(w)

This claim implies directly, that for CPE preference the preference VS is given by the pair (u ◦ S −1 , λ). 1. According to Proposition 6 in Masatlioglu, Raymond (2016) the CPE agent (u ◦ S −1 , λ) is then risk averse everywhere. 19

Details are in the Appendix in the proof of Proposition 4.

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2. If uˆ := u ◦ S −1 is strictly convex at 0 we have for the lottery 1 1 L = δ− + δ+ , 2 2 that 1 1 1 V (L) = (ˆ u(−) + uˆ(+) − (λ − 1)|ˆ u(+) − uˆ(−)|) + uˆ(−) + uˆ(+) > uˆ(0), 4 4 4 for all  > 0 small enough, due to strict convexity and continuity of uˆ (the latter is ensured by convexity). I.e. we have just showed that the agent with (u ◦ S −1 , λ) violates wRA at 0. In particular, part (3) of Theorem 1 in the main body of the paper is applicable.

Proof of Proposition 4. 1. We first establish the following. Claim The quadratic preference given by (5) satisfies FOSD-monotonicity if and only if (7) holds true. Proof of Claim. Given a functional V representing preferences over lotteries, V is called Fr´echet differentiable or smooth if for all F there exists a local Bernoulli utility function u(·, F ), such that for all lotteries G one has V (G) = V (F ) + EG [u(x, F )] − EF [u(x, F )] + o(||F − G||),

F →G

where the norm used in the o-term is Z ||F − G|| =

|F (z) − G(z)|dz.

One calculates that the local utility function for quadratic preferences is x 7→ u(x, F ) = R(x) + T (x)EF [T ]. Theorem 1 in Machina (1982) gives now the claim. Let a < x < c be fixed and let’s calculate the slope of the q-curve. It fulfills the equation 1 (1 − q(y))R(a) + q(y)R(c) + (1 − q(y))2 T (a)2 + (1 − q(y))2 T (c)2 2 1 + 2q(y)(1 − q(y))T (a)T (c)) = R(y) + T (y)2 . 2 Using the Implicit Function Theorem from Real Analysis and letting y→a one gets the equation characterizing q 0 (a): −q 0 (a+)(R(a) − R(c)) − q 0 (a+)T (a)2 − 2q 0 (a+)T (c)2 − 2q 0 (a)T (a)T (c) = R0 (a) + T (a)T 0 (a). Here we see that q 0 (a+) can’t be +∞. We calculate the slope at c in a similar way to find, that it is characterized by −q 0 (c−)(R(a) − R(c)) − 2q 0 (c−)T (a)T (c) = R0 (c) + T (c)T 0 (c). 14

We see that q 0 (c−) = 0 would imply R0 (c) + T (c)T 0 (c) = 0 and we need this to hold for all b > c > x. Plugging this identity into (7) implies though, that we need for all possible F T 0 (c)[EF [T ] − T (c)] > 0. We have a contradiction as soon as we pick F = δc . The result now follows from the general results in Theorem 2 of the main body of the paper. 2. We know from Masatlioglu, Raymond (2016), that a monotonic CPE agent is RDU with the probability weighting function wλ (z) = (2 − λ)z + (λ − 1)z 2 . Calculate from this, that wλ0 (0+) = 2 − λ, wλ0 (1−) = λ and recall that it is assumed λ ∈ (1, 2]. Now Theorem 2 of the main body of the paper completes the proof.

S2.3: Other examples for the naive RDU agent Other examples for RDU naive behavior can be constructed with the help of results from Xu, Zhou (2013), who for example show that, an agent with commitment stops always, 1 if w is convex and U (x) := u(x β ) is concave. The latter is the case if the dynamics of the diffusion Xt are unfavorable (low normalized drift σµ2 ). In particular, there is a whole class of naive RDU agents who are first-order risk averse everywhere in the sense of Segal, Spivak (1990)20 and who stop a whole class of geometric Brownian motions. For a more detailed class of examples in this spirit: it is easy to see, that u concave, µ ∈ (0, 12 σ 2 ) and w convex are sufficient conditions for the naive to stop a whole class of geometric Brownian motions.21

S2.4: Applications to Cautious Expected Utility: a result for the sophisticated agent Here we give a sufficient condition for the sophisticated CEU agent to actually start some diffusion. These conditions are rather weak, which implies that extreme sensitivity is generically not satisfied for CEU models. Proposition 5. Assume that the set U is a compact set of strictly increasing functions and that the one-sided derivatives of all u ∈ U are bounded uniformly away from zero and infinity: i.e. the following is fulfilled sup sup u0 (x) < +∞, u∈U x∈[w,b]

inf inf u0 (x) > 0.

u∈U x∈[w,b]

Then for all y0 ∈ [w, b] there are diffusions X such that y0 is in the continuation region of X. 20 21

See Proposition 4 in their paper Calculate 4µ 1 1 1 1 2µ − 2µ2 −1 U 00 (x) = u00 (x β ) 2 x− σ2 − u0 (x β ) x σ . β β σ2

If u is concave and µ ∈ (0, 12 σ 2 ) is concave then U is concave as well.

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Proof. Note first, that to ensure FOSD-monotonicity of V , the minimal set U , in the sense of inclusion, in the representation22 has to contain only strictly increasing functions. Given that U is compact, for each y0 ∈ (w, b) there exists uy some element in the U such that V ((1 − q(y))δw + q(y)δb ) = u−1 y (q(y)) = y. It follows q(y) = uy (y). We know that q(y) is non-decreasing so that it has one-sided derivatives at w, b. It follows Z b q(b) − q(y) 1 − uy (y) 1 = = u0 (s)ds, b−y b−y b−y y y where we have used that the functions u ∈ U are absolutely continuous due to being non-decreasing. The assumption in the statement implies then that q 0 (b−) > 0. One can show similarly, that q 0 (w−) < +∞. The result follows then from the general results for the sophisticated case.

References Cerreia-Vioglio, S., Dillenberger, D. and Ortoleva, P., 2015. Cautious Expected Utility and the Certainty Effect, Econometrica, 83(2), pp.693-728. Chew, S.H., Epstein, L. G. and Segal, U., 1990. Mixture Symmetry and Quadratic Utility, Econometrica, pp.139-163. Ebert, S. and Strack, P., 2015. Until the Bitter End: On Prospect Theory in a Dynamic Contest, The American Economic Review, 105(4), pp.1618-1633. Grandmont, J-M., 1972. Continuity Properties of Paretian Utility, Journal of Economic Theory, 4(1), pp.45-57. Gul, F. and Lantto, O., 1990. Betweenness Satisfying Preferences and Dynamic Choice, Journal of Economic Theory, 52(1), pp.162-177. Hammond, P.J., 1988. Consequentialist foundations for expected utility, Theory and decision, 25(1), pp.25-78. Hammond, P.J., 1989. Consistent Plans, Consequentialism and Expected Utility, Econometrica: Journal of the Econometric Society, pp.1445-1449. Kallenberg, O., 2006. Foundations of Modern Probability, Springer Science and Business Media. Karatzas, I and Shreve, S., 2012. Brownian motion and stochastic calculus (Vol. 113). Springer Science and Business Media. K¨oszegi, B. and Rabin, M., 2007. Reference Dependent Risk Attitudes, The American Economic Review, 97(4), pp.1047-1073. Machina, M., 1982. Expected Utility Analysis without the Independence Axiom, Econometrica: Journal of the Econometric Society, pp.277-323 22

See section 2.5 of and in particular Theorem 2 in Cerreia-Vioglio et al. (2015) for the meaning of ‘minimal set U ’.

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Machina, M., 1989. Dynamic Consistency and Non-Expected Utility models of Choice under Uncertainty, Journal of Economic Literature, 27(4), pp.1622-1668. Mas-Colell, A., Whinston, M.D. and Green, J.R., 1995. Microeconomic theory (Vol. 1). New York: Oxford University Press. Masatlioglu, Y. and Raymond, C., 2016. A Behavioral Analysis of Stochastic Reference Dependence, The American Economic Review, 106(9), pp.2760-2782. Segal, U. and Spivak, A., 1990. First Order vs. Second Order Risk Aversion, Journal of Economic Theory, 51(1), pp.111-125. Van der Vaart, A.W. and Wellner, Wellner, J.A., 1996. Weak Convergence and Empirical Processes, Springer Verlag Xu, Z.Q. and Zhou, X.Y., 2013. Optimal Stopping under Probability Distortion, The Annals of Applied Probability, 23(1), pp.251-282.

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