Simple Markovian equilibria in dynamic spatial legislative bargaining ∗ Jan Z´apal CERGE-EI & IAE-CSIC and Barcelona GSE [email protected] June 6, 2017

Abstract The paper proves, by construction, the existence of Markovian equilibria in a dynamic spatial legislative bargaining model. Players bargain over policies in an infinite horizon. In each period, a sequential protocol of proposal-making and voting, with random proposer recognitions and a simple majority, produces a policy that becomes the next period’s status-quo; the status-quo is endogenous. The construction relies on simple strategies determined by strategic bliss points computed by the algorithm we provide. A strategic bliss point, the dynamic utility ideal, is a moderate policy relative to a bliss point, the static utility ideal. Moderation is strategic and germane to the dynamic environment; players moderate in order to constrain the future proposals of opponents. Moderation is a strategic substitute; when a player’s opponents do moderate, she does not, and when they do not moderate, she does. We provide conditions under which the simple strategies induced by the strategic bliss points computed by the algorithm deliver a Stationary Markov Perfect equilibrium, and we prove its existence in a large class of symmetric games. Because the algorithm constructs all equilibria in simple strategies, we provide their general characterization, and we show their generic uniqueness.

JEL Classification: C73, C78, D74, D78 Keywords: endogenous status-quo; dynamic bargaining; legislative bargaining; spatial collective choice; equilibrium existence

∗ Some of the results presented in this paper originally appeared in my Ph.D. dissertation (Zapal, 2012, chapter 2) and were previously circulated as a working paper entitled ‘Simple equilibria in dynamic bargaining games over policies’. I owe special thanks to my advisors Ronny Razin and Gilat Levy. Further, I would like to thank Avidit Acharya, Vincent Anesi, Enriqueta Aragones, David Baron, Daniel Cardona, John Duggan, Jean Guillaume Forand, Tasos Kalandrakis, Antoine Loeper, Fabio Michelucci, Francesco Nava, Salvatore Nunnari, Clara Ponsati, Ronny Razin, Francesco Squintani and seminar and conference participants at IAE-CSIC, University of Waterloo, University of the Balearic Islands, the 2013 Barcelona GSE Summer Forum Workshop on Dynamic Decisions and the 2015 EEA-ESEM Annual Meeting in Toulouse for helpful comments and discussions. Some of the presented ideas took shape while visiting W. Allen Wallis Institute of Political Economy at the University of Rochester and their hospitality is appreciated. Financial support from the Post-Doc Research Fund of Charles University in Prague is gratefully acknowledged. All remaining errors are my own.

1

Introduction

Many real world policies and spending programs persist and evolve in time, are determined repeatedly, and their changes are enacted under the shadow of the extant legislation that is revised and becomes the new status-quo. Dynamic legislative bargaining models reflect these features. The models build on static non-cooperative models of legislative bargaining in the spirit of Baron and Ferejohn (1989). In these models negotiations follow a sequential protocol of proposal-making and voting, either in distributive bargaining over the allocation of benefits, or in spatial bargaining over choices of policies. The static models assume bargaining terminates upon an agreement being reached. The dynamic models instead embed the static decision-making protocol as a stage game in an infinite horizon repeated interaction. In each stage game the status-quo is the policy last enacted, making the current decision future status-quo and inducing a dynamic, not just repeated, strategic situation. Starting with Baron (1996), the dynamic legislative bargaining literature has been steadily growing (see next section for an overview). Kalandrakis (2004b) was the first to characterize the Markov equilibrium for the dynamic version of the distributive model. In the absence of applicable existence theorems for Markovian equilibria, his characterization constitutes an existence proof as well. In the continuing absence of the existence theorems, and due to the lack of similar characterization for the spatial model, the existence and properties of Markov equilibria in the dynamic spatial model remain unknown.1 In this paper we prove, using constructive arguments, the existence of Markov equilibria in a dynamic spatial legislative bargaining model. A group of legislators repeatedly sets policy in a one-dimensional policy space. The preferences of the legislators are quadratic, characterized by bliss points, the most preferred policies. In each period of infinite horizon a randomly selected legislator puts forward a proposal. Majoritarian voting between the proposal and the status-quo determines the winning alternative that yields utility to the legislators and becomes the status-quo for the subsequent period. The status-quo evolves endogenously and depends on the identity 1

Baron (1996) and Duggan and Kalandrakis (2012), the two papers closest to providing existence and characterization in the spatial model, are discussed in the next section.

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of the proposer and the votes of the entire legislature in every period. We start the equilibrium construction by defining simple stationary Markovian proposal strategies. A Markovian proposal strategy maps the state, the status-quo, into a policy proposal. A simple stationary Markovian proposal strategy depends on a single parameter, the policy a player proposes when the status-quo gives her ample bargaining power. In the static setting this parameter would be the player’s bliss point, the policy maximizing the static utility of a player. In the dynamic setting this parameter is the strategic bliss point, the policy maximizing the dynamic utility of a player. The crux of the construction is an algorithm generating the strategic bliss points. Two conditions guarantee that the construction, the simple proposal strategies in combination with the algorithm, delivers Markov equilibrium. The first one, sufficient, is stronger than necessary but easy to check. The second, necessary and sufficient, is more involved to verify, but still focuses only on a finite set of points in an otherwise infinite policy space. Using these tools, we prove, by construction, the existence of Stationary Markov Perfect equilibrium (SMPE) for any strongly symmetric dynamic spatial legislative bargaining game. Existence is assured under a mild condition on the degree of patience of the players, a condition which ceases to bind as the number of the players increases. For games that are symmetric, a weaker notion, we prove the same result under a stronger condition on the parameters of the game.2 Moreover, the construction works, and hence SMPE exists, for generic games with sufficiently impatient players. Although not generally, the construction also works outside of the cases just discussed. One must, however, specify a meaningful class of games for which it does. We further demonstrate the multiplicity of SMPE in simple strategies. This multiplicity is especially severe in symmetric games with many players; adding two players to a symmetric game increases the number of equilibria twofold, starting from two SMPE with three players. However, any multiplicity of SMPE in simple strategies is non-generic. 2

A game is strongly symmetric if the players’ bliss points are equidistant from each other and the players have equal recognition probabilities. It is symmetric if pairs of players around the median have bliss points equidistant from the median’s bliss point and have equal recognition probabilities. ‘Any’ game discussed below means for any bliss points, recognition probabilities and discounting. See below for formal definitions.

2

We show that all profiles of strategic bliss points that support an SMPE in simple strategies are constructed by our algorithm. By analysing the profiles of strategic bliss points produced by the algorithm, we provide general characterization of all SMPE in simple strategies for any dynamic spatial legislative bargaining game. And the analysis shows that the algorithm produces multiple profiles of strategic bliss points under non-generic conditions. Moderation and its strategic substitute nature are at the core of our equilibrium construction. This is the main insight of the paper. A player moderates when she proposes her strategic bliss point, which is a more moderate policy - closer to the median - than her (static) bliss point. A player moderates in order to constrain opponents; by moving the statusquo closer to median’s bliss point, future proposals are constrained to be moderate as well. When the opponents do moderate, they are effectively constraining themselves, so that the player has no incentive to moderate. If the opponents do not moderate, the player herself has an incentive to do so; that is, moderation is strategic substitute. As a result, all the equilibria that we construct induce asymmetric moderation (in terms of who moderates and to what extent), even if the underlying game is strongly symmetric. Moderation and its intensity are the result of two opposing forces. The first force is standard; proposals are pushed towards the proposers’ stage utility optimum, their bliss points. The second force is strategic; proposals are pushed towards the bliss point of the median player, with proposers aiming to constrain the future policies of all other players. These two forces cancel out at the strategic bliss point. The strategic force gains prominence and the extent of moderation increases with the patience of the players and with the higher probability of recognition of direct opponents - those with bliss points on the other side of the median.

2

Review of the literature

The typical dynamic legislative bargaining model with endogenous statusquo posits a group of players bargaining in an infinite discrete time horizon with discounting. Each period starts with a status-quo, the policy last enacted. A randomly chosen player makes a proposal after which a vote over a binary agenda, consisting of the status-quo and the proposal, follows. The winning alternative determines players’ utility for the period and becomes

3

the status-quo for the next one.3 The original formulation of legislative bargaining as a model with endogenous status-quo are Baron (1996) and Epple and Riordan (1987). Baron (1996) analyses a spatial bargaining model in which the players bargain over a one-dimensional policy space. Epple and Riordan (1987) analyse a distributive bargaining model in which the players bargain to distribute a fixed-sized budget among themselves. In the spatial formulation the utility of players varies in all the dimensions of the policy space. In the distributive setting the players only care about their share of the budget. The model of Baron (1996) is the most closely related to ours. His model is almost identical to our model; he restricts policies to R+ , which we allow for but do not require, and his stage utilities are general, not quadratic.4 He develops partial equilibrium characterization and provides intuition for the strategic forces at play.5 Kalandrakis (2016) constructs a mixed strategy equilibrium in the model we consider, restricted to three symmetric players. Zapal (2016) proves the existence of equilibrium in a three-player version of the same model, without symmetry. In addition to Baron (1996), several other papers analyse spatial models under special constraints. These include restrictions on the policy space (Austen-Smith, Dziuda, Harstad, and Loeper, 2016; Dziuda and Loeper, 2016, 2017; Fong, 2005), restrictions on the number of players (Baron, 2015; Bowen, Chen, Eraslan, and Zapal, 2017; Forand, 2014; Nunnari and Zapal, 2015), time horizon restrictions (Buisseret and Bernhardt, 2016, 2017) or use of numerical computations (Baron and Herron, 2003; Duggan, Kalandrakis, and Manjunath, 2008).6 3 The dynamic legislative bargaining models share many features with the static legislative bargaining models we do not survey here. See, for example, Banks and Duggan (2000, 2006a); Cardona and Ponsati (2007, 2011); Cho and Duggan (2003, 2009); Eraslan (2002); Eraslan and McLennan (2013); Eraslan and Merlo (2002); Herings and Predtetchinski (2010); Kalandrakis (2004a, 2006a,b) for theoretical treatment of the static models. 4 The quadratic utilities cannot be dispensed with because they guarantee that the dynamic median voter theorem applies (see the discussion after Lemma 1). Baron (1996) does not assume quadratic utilities explicitly, but assumes that the dynamic median voter theorem applies. 5 Baron (1996) also includes informal discussion of an example of full equilibrium characterization (his Table 1). The discussion following Proposition 3 explains why the profile of strategies in the example cannot constitute an equilibrium. 6 Papers that embed dynamic spatial models in richer economic or political settings include Bouton, Lizzeri, and Persico (2016) (debt and entitlements), Callander and Krehbiel (2014) (delegation), Callander and Martin (2017); Callander and Raiha (2017) (policy decay), Chen and Eraslan (2017a) (agenda formation), Diermeier, Prato, and Vlaicu (2015,

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Following Epple and Riordan (1987), the analysis of distributive models has focused on equilibrium characterization and properties (Anesi and Seidmann, 2015; Baron, 2016; Baron and Bowen, 2013; Kalandrakis, 2004b, 2010) including investigation of models with risk aversion or alternative decision making protocols (Baron and Bowen, 2013; Battaglini and Palfrey, 2012; Bowen and Zahran, 2012; Diermeier, Egorov, and Sonin, 2016; Jeon, 2015; Nunnari, 2016; Richter, 2014). Models combining distributive and spatial aspects with (Baron, Diermeier, and Fong, 2012; Cho, 2014) or without (Bowen, Chen, and Eraslan, 2014; Cho, 2016; Fong, 2011) electoral competition usually investigate joint public (spatial) and private (distributive) good determination.7,8 General characterization and existence results for Stationary Markov Perfect equilibria, the standard solution concept in the papers surveyed, are scarce and none covers our model. Kalandrakis was the first to provide a characterization of SMPE for the distributive model with three (Kalandrakis, 2004b) or more than five (Kalandrakis, 2010) players. Diermeier and Fong (2011) provide an algorithm leading to SMPE in a model with a persistent agenda setter and a discrete policy space (see Anesi and Duggan, 2017, for further results with discrete policy space). Duggan and Kalandrakis (2012) provide a very general SMPE existence result assuming noise in the preferences and the status-quo between-period transitions. The noise complicates the equilibrium characterization and is absent in our model. Anesi and Duggan (2015), utilizing techniques from Anesi and Seidmann (2015), construct a continuum of SMPE for a class of bargaining games with multi-dimensional policy spaces and patient players. They provide two conditions which ensure their construction works, both of which are violated with the one-dimensional policy space and minimum of three players 2016) (choice of decision-making rules), Fong and Deng (2011) (proposer competition), Levy and Razin (2013) (interest group influence), Ma (2014) (income taxation), Piguillem and Riboni (2016) (capital taxation), Piguillem and Riboni (2015) (present-biased legislators) or Riboni (2010); Riboni and Ruge-Murcia (2008) (monetary policy). 7 Electoral competition in combination with legislative bargaining. However, as Forand (2014) and Nunnari and Zapal (2015) illustrate, the difference between electoral competition and legislative bargaining can be merely a difference in labelling. 8 Two papers, analysing judicial precedents (Anderlini, Felli, and Riboni, 2014) and legislative sunset provisions (Zapal, 2012, Chapter 1), are models with endogenous statusquo, where players bargain jointly over policy and, not necessarily equal, status-quo for the next period. Callander and Clark (2017); Chen and Eraslan (2017b) study judicial precedents more generally.

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considered here. Anesi (2010) shows the existence of SMPE in a model with finite policy space, and proves that the equilibrium absorbing set coincides with the von Neumann-Morgenstern stable set. Duggan (2017) shows the existence of SMPE, allowing for general policy space but restricting the status-quos to a countable set.9,10 The endogenous status-quo literature just discussed is related to but distinct from the models with a single decision to be taken and bargaining proceeding through a series of rounds with evolving default (Anesi and Seidmann, 2014; Bernheim, Rangel, and Rayo, 2006; Diermeier and Fong, 2009; Vartiainen, 2014). Also related but distinct is the literature with dynamic political economy models (for example Azzimonti, 2011; Bai and Lagunoff, 2011; Battaglini and Coate, 2007, 2008; Battaglini, Nunnari, and Palfrey, 2012) where the dynamic link stems not from persistent policies but from the accumulation of a durable public good, (public) debt or capital.

3

Model, notation, solution concept

A game G = hn, x, r, δ, Xi is fully specified by n, x, r, δ, and X all satisfying the assumptions we introduce next, and which are maintained throughout. N = {1, . . . , n} is the set of players with odd n ≥ 3. The stage utility of i ∈ N from policy p is ui (p) = −(p − xi )2 where xi is the bliss point of i. Let x = (x1 , . . . , xn ) be the profile of bliss points of the players. We assume that the bliss points are distinct and ordered such that xi < xi+1 ∀i ∈ N \ {n}. Let m = dn/2e be the median player with bliss point xm = xdn/2e . In each discrete period of infinite horizon, nature first randomly chooses one player to be the proposer, according to the profile of recognition probP abilities r = (r1 , . . . , rn ) with ri > 0 ∀i ∈ N and i∈N ri = 1. Second, the proposer makes a policy proposal p ∈ X where the policy space X is either R or a closed interval centered on xm that includes both x1 and xn . Third, the players vote between the proposal p ∈ X and the status-quo 9

Acharya and Ortner (2017), Hortala-Vallve (2011), Penn (2009) and Roberts (2007) characterize equilibria in models with random, not endogenous and strategically chosen, proposals. 10 Faced with the complex equilibria of the dynamic legislative bargaining models, many authors use, at least partially, numerical computations (Baron and Herron, 2003; Battaglini and Palfrey, 2012; Bowen et al., 2014; Duggan et al., 2008; Piguillem and Riboni, 2016; Riboni and Ruge-Murcia, 2008, among others) or provide numerical computation techniques tailored to these models (Duggan and Kalandrakis, 2011).

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x ∈ X. Fourth, the alternative that receives a simple majority, i.e., at least n+1 2

votes, determines the utility of the players in the current period and

becomes the status-quo for the next period. The utility of player i ∈ N from an infinite policy path {p0 , p1 , . . .} is ∞ X

δ t ui (pt )

(1)

t=0

where δ ∈ [0, 1) is the common discount factor.11 A pure stationary Markov strategy σi = (πi , αi ) of each i ∈ N consists of a proposal strategy πi : X → X and a voting strategy αi : X 2 → {0, 1}. The former specifies the policy i ∈ N proposes given status-quo x, πi (x), the latter specifies how i ∈ N votes given proposed p and status-quo x, αi (p, x), where αi (p, x) = 1 denotes a yes vote and αi (p, x) = 0 a no vote. A profile of strategies σ = (σi )i∈N induces a continuation value function for each i ∈ N , Vi : X → R. Vi (x|σ) denotes the expected utility of i, at the beginning of a period before the proposer is drawn, from an infinite play according to σ starting from status-quo x. For any i ∈ N and x ∈ X, let Ui (x|σ) = ui (x) + δVi (x|σ) be the dynamic (expected) utility of i from (accepted) x. The acceptance set induced by the voting strategies in σ, P given status-quo x ∈ X, is A(x|σ) = {p ∈ X| i∈N αi (p, x) ≥ n+1 2 }. Definition 1 (Stationary Markov Perfect Equilibrium). A stationary Markov perfect equilibrium (SMPE) is a profile of stationary Markov strategies σ ∗ = (πi∗ , αi∗ )i∈N such that, ∀x ∈ X and ∀i ∈ N , πi∗ (x) ∈ arg max ui (p) + δVi (p|σ ∗ ) p∈A(x|σ ∗ )

and, ∀p ∈ X, ∀x ∈ X and ∀i ∈ N , αi∗ (p, x) = 1 if and only if Ui (p|σ ∗ ) ≥ Ui (x|σ ∗ ). The definition requires σ ∗ to constitute a sub-game perfect equilibrium and to be stationary Markov.12 In addition to standard optimality requirements, we focus on equilibria such that any policy proposal is accepted.13 11

Specification of G does not require an initial status-quo, which we leave unspecified. The definition uses, as usual, a one-stage deviation principle to simplify the exposition. 13 Given status-quo x, the proposing player whose utility maximizing proposal is x can

12

7

Moreover, we focus on equilibria in which all players use the stage undominated voting strategies of Baron and Kalai (1993) when voting between the proposed p ∈ X and the status-quo x ∈ X, and vote deferentially, that is, vote for p when indifferent between p and x.14 The following result, independent of the equilibrium construction, implies that in any SMPE, acceptance of a policy is driven by the median. Lemma 1 (Dynamic median voter theorem). For any profile of stationary Markov strategies σ, given p ∈ X and x ∈ X, Um (p|σ) ≥ Um (x|σ) ⇔ |{i ∈ N |Ui (p|σ) ≥ Ui (x|σ)}| ≥

n+1 2 .

Lemma 1 crucially depends on the quadratic stage utilities. The median is the player who is decisive in the vote over deterministic alternatives. However, in our model, voting between alternatives means voting over lotteries the alternatives induce in the future. That decisiveness of the median extends to voting over lotteries, under quadratic preferences, is a well known result (Banks and Duggan, 2006b; Cho and Duggan, 2003; Duggan, 2014). Equally well known is that this result does not extend considerably beyond quadratic utilities (see the example following the proof of Lemma 2.1 in Banks and Duggan, 2006b).15 The following notation is used below. Let d(x) = |x−xm | be the distance between x ∈ R and xm . Let da (x) = xm + d(x) be x mapped above the median’s bliss point and db (x) = xm −d(x) below. Note that x ∈ {db (x), da (x)}. Na = {i ∈ N |xi > xm } and Nb = {i ∈ N |xi < xm } are the sets of players P with bliss points above and below xm , respectively. Let ra = i∈Na ri and P rb = i∈Nb ri be the sums of recognition probabilities for the two groups. Finally, f (a− ) = limx→a− f (x) and f (a+ ) = limx→a+ f (x) denote, respectively, the one-sided limits of a real-valued function from below and above.16 obtain this utility either by proposing x or by making a proposal she knows would be rejected. We assume she does the former. This assumption does not change the set of equilibria that are observationally (outcome) equivalent and is standard in the dynamic bargaining literature. 14 Stage undominated voting is a standard assumption in voting literature and rules out implausible equilibria that can support arbitrary outcomes that are accepted because no voter is pivotal. Assuming that an indifferent voter casts her vote for the proposed policy avoids any open set complications. 15 Alternative voting rules, with a veto player, or decision making protocols, with a representative voter, would not necessitate quadratic stage utilities for the social acceptance set to be driven by preferences of a unique player. Our approach to equilibrium construction would be applicable to these alternative models as well, even with general stage utilities. 16 To avoid any confusion: f : R → R is called increasing if f (x) < f (y) whenever

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4

Equilibrium construction

Our construction relies on simple proposal strategies and on an algorithm that produces the strategic bliss points the simple strategies depend on. Definition 2 (Simple proposal strategies). The simple pure stationary Markov proposal strategy of i ∈ N is  ˆi }   min{da (x), x pi (x|ˆ xi ) = x ˆm   max{db (x), x ˆi }

if i ∈ Na if i = m if i ∈ Nb

ˆ = (ˆ where x ˆi is the strategic bliss point of i. Let x x1 , . . . , x ˆn ) be the profile of strategic bliss points. ˆ induces the profile of proposal Any profile of strategic bliss points x strategies (p(·|ˆ xi ))i∈N . For any i ∈ N and x ∈ X, let the continuation value ˆ be the unique solution to the recursion induced by x Vi (x|ˆ x) =

X

rj [ui (pj (x|ˆ xj )) + δVi (pj (x|ˆ xj )|ˆ x)] .

(2)

j∈N

ˆ , ∀i ∈ N and ∀x ∈ X, let Given the continuation value induced by x Ui (x|ˆ x) = ui (x) + δVi (x|ˆ x) and A(x|ˆ x) = {p ∈ X|Um (p|ˆ x) ≥ Um (x|ˆ x)} be ˆ respectively. Given the dynamic utility and the acceptance set induced by x ˆ by the dynamic utility, construct the voting strategy of i ∈ N induced by x setting, ∀p ∈ X and ∀x ∈ X, αi (p, x) = 1 if and only if Ui (p|ˆ x) ≥ Ui (x|ˆ x). ˆ ). x ˆ satisfies condition C if and only if Definition 3 (Condition C, on x x ˆi ≥ xm ∀i ∈ Na , x ˆm = xm , x ˆi ≤ xm ∀i ∈ Nb .

(C)

ˆ differ from Vi , Ui and A induced In general, Vi , Ui and A induced by x ˆ . However, we claim that these objects are identical when by σ induced by x ˆ satisfies C. To see this, first note that, given the voting strategies induced x ˆ , proposal p ∈ X is accepted under status-quo x ∈ X if and only if it by x is accepted by the median player. This follows by Lemma 1 since, ∀i ∈ N , x < y and non-decreasing if f (x) ≤ f (y) whenever x ≤ y; x ∈ R is positive if x > 0 and non-negative if x ≥ 0. Inverses of these properties are labelled in an obvious way.

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αi (p, x) = 1 if and only if Ui (p|ˆ x) ≥ Ui (x|ˆ x).17 Second, Lemma 2 below, ˆ is single peaked and symmetric around using C, shows that Um induced by x ˆ , given any status-quo x ∈ X, xm . Thus, for the voting strategy induced by x we have αm (p, x) = 1 if and only if Um (p|ˆ x) ≥ Um (x|ˆ x), which is equivalent ˆ such that C holds, we have, ∀i ∈ N and to p ∈ [db (x), da (x)]. Third, given x ∀x ∈ X, pi (x|ˆ xi ) ∈ [db (x), da (x)] directly from Definition 2. Therefore, for ˆ such that C holds, any proposal generated by the proposal strategies any x ˆ is accepted under the voting strategies induced by x ˆ and, thus, induced by x ˆ is identical to Vi induced by x ˆ . Similar ∀i ∈ N , Vi induced by σ induced by x ˆ and holds for Ui and, by the definition of the voting strategies induced by x Lemma 1, for A. ˆ induces a profile of proposal and voting strategies σ. Therefore, any x When C holds, σ induces, ∀i ∈ N , Vi that is identical to Vi in (2). Since ˆ , by construction, similar holds for Ui , the voting strategies induced by x satisfy the SMPE optimality condition in Definition 1. Constructing SMPE ˆ , subject to C, that induces proposal strategies that thus reduces to finding x satisfy the SMPE optimality condition. The following example illustrates the shape of the simple strategies in a strongly symmetric G with three players for a profile of strategic bliss points that constitutes an SMPE, as we prove below.18 Example 1. Consider G with n = 3, xi = i, ri =

1 n

∀i ∈ N and δ =

0.9. Figure 1 illustrates the profile of simple strategies induced by these ˆ = {1.6, 2, 3}. parameters along with a profile of strategic bliss points x Let us first explain the rationale behind calling x ˆi strategic bliss points. x ˆi is the policy i proposes when the status-quo gives her ample bargaining power, that is, when i is not constrained by the acceptance set of the median. In Lemma 2 below we prove that the acceptance set is A(x) = [db (x), da (x)]. Hence i can propose x ˆi when x ∈ / (1.6, 2.4) for i = 1 and x ∈ / (1, 3) for i = 3. Not being constrained means i can propose the policy maximizing her dynamic utility Ui , her strategic bliss point. The reason x ˆi and xi differ is because the former policy maximizes dynamic utility Ui = ui + δVi , whereas the latter policy maximizes (static) 17

ˆ by taking the proposal We can apply Lemma 1 to the dynamic utilities induced by x ˆ and any profile of voting strategies such that any proposal is strategies induced by x accepted, for example, (αi )i∈N such that, ∀i ∈ N , ∀x ∈ X and ∀y ∈ X, αi (y, x) = 1. 18 Unless specified otherwise, the policy space in all the examples is X = R.

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Figure 1: Simple strategies in Example 1 pi (x|ˆ xi ) x3 = x ˆ3 = 3

p3 (x|ˆ x3 )

x2 = x ˆ2 = 2

p2 (x|ˆ x2 )

x ˆ1 = 1.6

p1 (x|ˆ x1 )

x1 = 1 1

2

3

x

utility ui . Take player 1 from Example 1 and suppose the status-quo x = 1. We claim p1 (1) = 1.6 whereas the policy maximizing u1 is x1 = 1. With x = 1, A(1) = [1, 3] hence x1 = 1, if proposed, would be accepted. The reason x1 = 1 6= x ˆ1 = 1.6 is that in the dynamic setting player 1 takes into account the impact of her proposal on the distribution of future policies. Two such distributions, induced by proposing x1 = 1 and p1 (1) = 1.6, are indicated by the (red) circles to left of x = 2 in Figure 1. By proposing p1 (1) = 1.6, as opposed to proposing x1 = 1, player 1 foregoes the chance to maximize her static utility but brings future policy of player 3 from p3 (1) = 3 to p3 (1.6) = 2.4. That is, player 1 moderates her proposal, she foregoes (current) static utility, in an attempt to constrain the future policy of player 3, increasing her future utility when she is not in possession of proposal power. The incentive to moderate is purely strategic; absent the intertemporal link created by persistent policies, player 1 would propose x1 = 1. Furthermore, we claim that player 3 from Example 1 does not moderate and her strategic bliss point coincides with her bliss point. Clearly, the strategic force to moderate is present for player 3 as well. Take status-quo x = 3. We claim player 3 proposes p3 (3) = 3 instead of moderating and proposing, using the same extent of moderation as player 1, p0 = 2.4. Both p3 (3) = 3 and p0 = 2.4 would be accepted with status-quo x = 3 and lead to the distribution over future policies indicated by the (blue) circles to the right of x = 2 in Figure 1. The reason player 3 does not moderate is because 11

proposing p0 = 2.4 or p3 (3) = 3 induces the same future policy by player 1, p1 (2.4) = p1 (3) = 1. In order to constrain the future policy of player 1, player 3 would have to moderate to some policy in [2, 2.4), which is too costly for her in terms of foregone current utility. In other words, moderation is a strategic substitute; when player 1 moderates, the best response for player 3 is not to moderate, and when player 1 does not moderate, player 3 best responds by moderating.19 We now specify the algorithm that derives a profile of strategic bliss ˆ . The simple strategies in combination with x ˆ from the algorithm points x ˆ and the profile of need not constitute an SMPE. At this stage we view x strategies σ it induces as a candidate for SMPE. Algorithm 1 (Strategic bliss points). For the set of players Pt in step t of P P the algorithm, denote rt,a = i∈Pt ∩Na ri and rt,b = i∈Pt ∩Nb ri . step 0 Set x ˆm = xm and P1 = N \ {m} step t For i ∈ Pt compute

x ˆi,t

 x + 2δr (x − x ) i i t,b m = x + 2δr (x − x ) i

t,a

m

i

if i ∈ Na if i ∈ Nb

Define Rt = {i ∈ Pt |(xi − xm )(ˆ xi,t − xm ) ≤ 0} If Rt = ∅, select one j ∈ arg mini∈Pt d(ˆ xi,t ), set x ˆj = x ˆj,t If Rt 6= ∅, select one j ∈ Rt , set x ˆ j = xm Set Pt+1 = Pt \ {j} and if Pt+1 6= ∅, proceed to step t + 1

It is immediate that the algorithm finishes in n − 1 steps and produces ˆ with x a full profile of strategic bliss points x ˆi ∈ [xm , xi ] if i ∈ Na and x ˆi ∈ [xi , xm ] if i ∈ Nb , that is, such that C holds. Further, it is easy to check that x ˆi ≤ x ˆi+1 for i ∈ N \ {n} and that x ˆi = xi for i = 1 or i = n but not both (unless δ = 0). 19

The insight that in any SMPE at least one player does not moderate goes beyond the simple strategies because the following claim can be easily proven. Consider any profile of proposal strategies (πi )i∈N such that, ∀i ∈ N \ {m}, πi (x) = x ˆi ∀x ∈ / (db (ˆ xi ), da (ˆ xi )) with d(ˆ xi ) < d(xi ). That is, ∀i ∈ N \{m}, i moderates to x ˆi whenever the status-quo x satisfies x ≤ db (ˆ xi ) or x ≥ da (ˆ xi ). Then (πi )i∈N cannot be part of an SMPE. The intuition is, using without loss of generality d(x1 ) ≤ d(xn ) and d(ˆ xi ) ≤ d(ˆ xn ) ∀i ∈ N \ {m}, that Vn is constant on X \ (db (ˆ xn ), da (ˆ xn )), Un inherits the shape of un and thus Un (ˆ xn ) < Un (xn ). That is, n has no incentive to moderate to x ˆn .

12

The intuition behind the algorithm is as follows. It starts with a full set of players apart from the median. It conjectures that strategy of all these players is characterized by strategic bliss points equal to +∞ for i ∈ Na and −∞ for i ∈ Nb , that is players in Na proposing da (x) and players in Nb proposing db (x). Calculating Ui for this conjectured strategy, the algorithm computes x ˆi,1 which is a policy at which Ui attains its maximum. At x ˆi,1 it ceases to be optimal for i to propose da (x) or db (x) and the best response, for any status-quo further from xm relative to x ˆi,1 , is to propose x ˆi,1 . The algorithm then drops the player with x ˆi,1 closest to xm as the first player for whom, moving status-quo away from xm , the conjectured strategy ceases to be a best response. Proceeding to step 2, the algorithm conjectures that the strategy of all players not previously dropped is characterized by bliss points equal to +∞ and −∞ and continues similarly. Two possible complications arise. The first one arises when the algorithm arrives at x ˆi,t and x ˆj,t with d(ˆ xi,t ) = d(ˆ xj,t ) and both i and j belong to arg mini∈Pt d(ˆ xi,t ). This implies i ∈ Na and j ∈ Nb or vice versa, the algorithm requires exactly one of the players to be dropped, but we have not specified which one. This reflects the strategic substitute nature of moderation and is the sole reason for SMPE multiplicity. If i is dropped then j does not want to moderate and the algorithm retains j. If j is dropped then i does not want to moderate and is retained. That, say, i is retained means ˆ with i moderating as well. that the algorithm might eventually produce x But this moderation is driven by other players still in the algorithm. In Example 1 dropping j meant i was retained as the sole player, in which case the algorithm produces x ˆi = xi . Example 1 (continued) below illustrates this complication and highlights that the profile of strategic bliss points the algorithm produces need not be unique. The second complication arises when 2δra ≥ 1 or 2δrb ≥ 1 (both cannot hold simultaneously as ra + rb = 1 − rm < 1). Suppose 2δra ≥ 1 holds. Then Rt 6= ∅, Rt ⊆ Nb and Rt ∩ Na = ∅ ∀t ∈ {1, . . . , n−1 2 } which means that the algorithm sequentially drops all the Nb players in steps t ∈ {1, . . . , n−1 2 } and x ˆi = xm ∀i ∈ Nb . That is, the proposal strategies of all the Nb players are identical to the proposal strategy of the median player. Intuitively, when the Na players are very likely to propose, the strategic force pushing the Nb players towards moderation is very strong, dominates any concerns for current utility and the greatest extent of constraint the Nb players can 13

impose on the Na players is by proposing xm . When this happens, the algorithm also produces x ˆi = xi ∀i ∈ Na , that is, the Na players do not moderate. Example 2 below illustrates this complication. The strategic bliss point of player i from Algorithm 1, via the simple strategy pi (x|ˆ xi ), determines the extent of moderation of player i. From the algorithm, unless x ˆ i = xm , x ˆi = xi + 2δr(xm − xi ) where r is the probability of recognition of i’s opponents. Player i thus moderates to a larger extent with increasing δ and r. Both variables reinforce the strategic incentive to moderate and x ˆi increases when i ∈ Nb and decreases when i ∈ Na . Example 1 (continued). In step 0 the algorithm drops the median player and sets x ˆ2 = x2 = 2. In step 1 the algorithm computes x ˆ1,1 = 1.6 and x ˆ3,1 = 2.4 and, by dropping player 1, produces x ˆ3 = x ˆ3,2 = 3 as already anticipated ˆ = {1.6, 2, 3}. Notice that dropping player 3 in in Figure 1, which used x ˆ = {1, 2, 2.4}, which step 1 would produce a profile of strategic bliss points x is distinct but symmetric around xm . Example 2 (Players proposing identically as median). Consider G with n = 5, xi = i ∀i ∈ N , r = {0.4, 0.4, 0.1, 0.05, 0.05} and δ = 0.9. It is easy to confirm that R1 = {4, 5} with the algorithm dropping player 4 and R2 = {5} with the algorithm dropping player 5. After two more steps, the algorithm ˆ = {1, 2, 3, 3, 3}. produces x

5

Equilibrium existence

ˆ and induced σ, let The analysis below uses the following objects.20 Given x N D(σ) = {ˆ xm , db (ˆ x1 ), da (ˆ x1 ), . . . , db (ˆ xn ), da (ˆ xn )} and D(σ) = X \ N D(σ). ˆ are differentiable with respect to the The proposal strategies induced by x status-quo x at any x ∈ D(σ) and might not be at x ∈ N D(σ). For any x ∈ D(σ), define N C(x|σ) = {i ∈ N |p0i (x|ˆ xi ) 6= 0} and C(x|σ) = N \ N C(x|σ). The players in the former set, at x, are on the non-constant part of pi while the players in the latter set are on the constant part. Finally, for any P x ∈ D(σ), define rnc (x|σ) = i∈N C(x|σ) ri as the sum of the recognition probabilities of players in N C(x|σ). Splitting N C(x|σ) into players in Na and Nb , P P we have rnc,a (x|σ) = i∈N C(x|σ)∩Na ri and rnc,b (x|σ) = i∈N C(x|σ)∩Nb ri . 20 The notation is considerable but necessary to state the main results. Readers less interested in technical details can skip to Proposition 1.

14

While we leave N C, C, rnc , rnc,a and rnc,b undefined on N D(σ), we can define one-sided limits at any x ∈ N D(σ). For N C, at any x ∈ N D(σ), let N C(x− |σ) = {i ∈ N |p0i (x− |ˆ xi ) 6= 0} and N C(x+ |σ) = {i ∈ N |p0i (x+ |ˆ xi ) 6= 0} and define one-sided limits of C, rnc , rnc,a and rnc,b using one-sided limits of N C.21,22 ˆ and σ it induces, ∀i ∈ N \ {m}, define the (possibly empty) sets Given x  N D(σ) ∩ (ˆ xi , xi ) Si (σ) = N D(σ) ∩ (x , x ˆ) i

i

if i ∈ Na if i ∈ Nb

Li (σ) = {x ∈ D(σ)|Ui0 (x|ˆ x) = 0}  ((N D(σ) ∪ L (σ)) ∩ (ˆ xi , xi )) ∪ {xi , x ˆi } i Ni (σ) = ((N D(σ) ∪ L (σ)) ∩ (x , x ˆ )) ∪ {x , x ˆ} i

i

i

i

i

(3) if i ∈ Na if i ∈ Nb

with elements of Ni (σ) ordered in increasing (decreasing) order if i ∈ Na (i ∈ Nb ). Si (σ) is the set of points in the interval between x ˆi and xi at which pj is not differentiable for some j ∈ N . Adding x ˆi , xi and Li (σ), ˆ attains a local maximum, to Si (σ) produces points at which Ui induced by x Ni (σ). That the derivative used in the definition of Li (σ) exists follows from the following lemma, which proves several technical properties of Vi and Ui ˆ. induced by x ˆ ). For any x ˆ that satisfies Lemma 2 (Properties of Vi and Ui induced by x condition C, ∀i ∈ N , 1. Vi (db (x)|ˆ x) = Vi (da (x)|ˆ x) ∀x ∈ X; 2. Ui (db (x)|ˆ x) < Ui (da (x)|ˆ x) if i ∈ Na , Ui (db (x)|ˆ x) > Ui (da (x)|ˆ x) if i ∈ Nb and Um (db (x)|ˆ x) = Um (da (x)|ˆ x), ∀x ∈ X \ {xm }; 3. Ui is continuous on X; 21

One-sided limits of N C, C, rnc , rnc,a and rnc,b at any x ∈ D(σ) are defined similarly. Because N C is piecewise ‘constant’ on intervals determined by N D(σ), we have, ∀x ∈ D(σ), N C(x|σ) = N C(x+ |σ) = N C(x− |σ). Similar holds for C, rnc , rnc,a and rnc,b since these are defined using N C. Moreover, since N C is piecewise constant, C, rnc , rnc,a and rnc,b are as well. 22 ˆ and Following easy-to-check properties are used throughout the proofs. Given x σ it induces, pi (da (x)|ˆ xi ) = pi (db (x)|ˆ xi ) ∀x ∈ X and ∀i ∈ N . Hence, ∀x ∈ D(σ), N C(da (x)|σ) = N C(db (x)|σ) and similarly for C, rnc , rnc,a and rnc,b . Thus, ∀x ∈ X, N C(da (x)− |σ) = N C(db (x)+ |σ) and N C(da (x)+ |σ) = N C(db (x)− |σ) and similarly for C, rnc , rnc,a and rnc,b . Furthermore, for any x ∈ D(σ) and y ∈ D(σ) such that d(x) ≤ d(y), N C(y|σ) ⊆ N C(x|σ) and thus rnc,a (y|σ) ≤ rnc,a (x|σ) and rnc,b (y|σ) ≤ rnc,b (x|σ).

15

4. Ui00 (x|ˆ x) < 0 ∀x ∈ D(ˆ x); 5. Um (x|ˆ x) > Um (y|ˆ x) ∀x ∈ X, ∀y ∈ X such that d(x) < d(y); 6. A(x|ˆ x) = [db (x), da (x)] ∀x ∈ X. We are now ready to state two conditions that guarantee that a profile ˆ from Algorithm 1 induces SMPE σ.23 of strategic bliss points x ˆ Definition 4 (Condition S, sufficient). A profile of strategic bliss points x from Algorithm 1 that induces σ satisfies condition S if and only if, ∀i ∈ N \ {m} and ∀x ∈ Si (σ), x − xi − 2δrnc,b (x+ |σ)(xm − xi ) ≥ 0

if i ∈ Na

x − xi − 2δrnc,a (x− |σ)(xm − xi ) ≤ 0

if i ∈ Nb .

(S)

Definition 5 (Condition N, necessary and sufficient). A profile of strategic ˆ from Algorithm 1 that induces σ satisfies condition N if and bliss points x only if, ∀i ∈ N \ {m} and denoting elements of Ni (σ) by {z0 , z1 , . . .}, XJ

h

j=1

XJ j=1

+ izj−1 Ti (x|σ) − ≥ 0 ∀J ∈ {1, . . . , |Ni (σ)| − 1} if i ∈ Na

zj

h

− izj−1 Ti (x|σ) + ≥ 0 ∀J ∈ {1, . . . , |Ni (σ)| − 1} if i ∈ Nb

(N)

zj

where  2  x 2 − ci (x|σ)x Ti (x|σ) = − 1 − δrnc (x|σ) 2  x + 2δr (x|σ)(x − x ) if i ∈ N i m i a nc,b ci (x|σ) = x + 2δr (x|σ)(x − x ) if i ∈ N . i nc,a m i b Proposition 1 (SMPE under S and N conditions). A profile of strategic ˆ from Algorithm 1 induces SMPE σ bliss points x ˆ satisfies condition S; 1. if x ˆ satisfies condition N. 2. if and only if x 23

Both conditions apply to profiles of strategic bliss points from Algorithm 1. By ˆ Proposition 3 below this is without loss of generality, as the algorithm constructs all x ˆ would complicate that support an SMPE for given G. Stating the conditions for general x the exposition.

16

The reason both S and N guarantee that the simple strategies induced ˆ constitute an SMPE is the following. First note that player i ∈ Na by x would never propose policy p < xm due to symmetry, around xm , of the acceptance sets A and of the continuation value functions Vi . Furthermore, in the proof of the proposition we show that Ui is increasing on [xm , x ˆi ] and decreasing on [xi , +∞). However, for the simple strategy with x ˆi to be the best response to the strategies of the other players, Ui must be decreasing on [ˆ xi , xi ] as well. From Lemma 2 we know Ui is piecewise strictly concave, which means ensuring that the right derivative of Ui is non-positive, at any point in N D that falls into (ˆ xi , xi ), suffices for Ui to be decreasing on [ˆ xi , xi ]. This is what condition S does. When it holds, Ui is increasing on [xm , x ˆi ] and decreasing on [ˆ xi , +∞), implying that proposing da (x) when the status-quo x is such that x ˆi ∈ / A(x) and proposing x ˆi otherwise is optimal for i. Condition S is stronger than necessary. It ensures that Ui is decreasing on [ˆ xi , xi ] while for x ˆi to be optimal for i ∈ Na , only Ui (ˆ xi ) ≥ Ui (x) ∀x ≥ x ˆi is required. This is what condition N does. It only checks a finite set of points R ∂ x because Ui is piecewise quadratic and Ui (x) − Ui (y) = U (z)dz . i ∂z y Both conditions guaranteeing the existence of SMPE only need to be checked at a finite set of points. Their disadvantage is that they apply to the strategic bliss points from Algorithm 1. Relating these conditions directly to the parameters defining G is non-trivial due to the complicated ˆ . This is why we study symmetric games mapping from n, x, r and δ to x below. Putting enough structure on the parameters defining G allows us to relate (mainly) condition S to these parameters. We have explained that the incentive of the players to moderate is driven by their concern about the future policy outcomes. It is natural to conjecture ˆ differ that when the players are almost myopic, the strategic bliss points x little from x and hence induce SMPE σ. The following proposition derives conditions such that the conjecture is indeed true. r

Proposition 2 (Condition N holds for small δ). If ri ∈ [ 2j , 2rj ] for every pair of players {i, j} with d(xi ) = d(xj ), then δ¯ ∈ (0, 1) exists, such that, ¯ any x ˆ from Algorithm 1 satisfies condition N. ∀δ ≤ δ, Note that the proposition constitutes proof of the existence of SMPE for generic G with sufficiently impatient players, since in any generic G we have d(xi ) 6= d(xj ) for any i ∈ N and j ∈ N \ {i}. 17

Before we proceed we provide two examples. The first shows that despite the apparent complexity of conditions S and N these can be simple to verify. The second example shows that whether these conditions are satisfied or not can depend non-monotonically on δ. It is also easy to see that both of the conditions hold in Example 2. ˆ = {1.6, 2, 3}, the set of Example 1 (continued). With x = {1, 2, 3} and x points at which differentiability of (at least some of ) the proposal strategies might fail is N D = {1, 1.6, 2, 2.4, 3}. The subset of players on the nonconstant part of their strategy is   {1, 3} N C(x) = {3}   ∅ which induces rnc,a (x) =

1 3

for x ∈ (1.6, 2) ∪ (2, 2.4) for x ∈ (1, 1.6) ∪ (2.4, 3) for x ∈ (−∞, 1) ∪ (3, +∞)

for x ∈ (1, 2) ∪ (2, 3) and rnc,b (x) =

1 3

for x ∈

(1.6, 2)∪(2, 2.4) with both rnc,a and rnc,b equal to 0 for any other x ∈ X \N D. Because S1 = N D ∩ (1, 1.6) = ∅ and S3 = N D ∩ (3, 3) = ∅ and because L1 = L3 = ∅, we have N1 = {1, 1.6} and N3 = {3}. Conditions S and N ˆ = {1.6, 2, 3} constitutes hold, which, by Proposition 1, implies σ induced by x an SMPE. Example 3 (Non-monotonic failure of S and N conditions). Consider G 1 n

∀i ∈ N and δ = 0.5. Then Algorithm 1 produces ˆ (depending on the selection eight different profiles of strategic bliss points x ˆ , condition S, and by implication condition of players to drop). For every x with n = 7, xi = i, ri =

ˆ from Algorithm 1 N, holds. For the same G with δ = 0.9 the number of x reduces to two but both fail both S and N conditions. For the same G with ˆ and for both condition S fails while δ = 0.95 there are again two possible x condition N holds.

6

Equilibrium characterization

This section provides a general characterization of SMPE in the simple proposal strategies. The first lemma implies that in any SMPE in the simple proposal strategies, the strategic bliss point and the bliss point of a player are on the same side of the median.

18

ˆ ). If x ˆ induces SMPE σ, then x ˆ Lemma 3 (Minimal properties of SMPE x satisfies condition C. The following lemma summarizes the key properties of any profile of strategic bliss points produced by Algorithm 1. The real significance of the lemma arises from Proposition 3 that follows. ˆ from Algorithm 1). Let x ˆ be a profile of Lemma 4 (Characterization of x strategic bliss points produced by Algorithm 1. Then ˆ = x; 1. if δ = 0, then x 2. if δ ∈ (0, 1) and 1 ≤ 2δrg for some g ∈ {a, b}, then x ˆi = xm ∀i ∈ N \Ng and x ˆi = xi ∀i ∈ Ng ; 3. if δ ∈ (0, 1), 1 > 2δra and 1 > 2δrb , then x ˆi < x ˆi+1 ∀i ∈ N \ {n} and d(ˆ xi ) 6= d(ˆ xj ) ∀i ∈ N , ∀j ∈ N , i 6= j. ˆ be the set of profiles of strategic bliss points produced Proposition 3. Let X ˆ ˆ constitutes an SMPE, then x ˆ ∈ X. by Algorithm 1. If σ induced by x ˆ that inProposition 3 states that if a profile of strategic bliss points x ˆ is produced by Algorithm 1. Lemma 4 thus duces SMPE σ exists, then x ˆ produced by Algorithm 1, it also constitutes not only characterizes any x a characterization of SMPE in simple proposal strategies.24 In addition, ˆ the number of different profiles of strategic Proposition 3 implies that #X, bliss points produced by the algorithm, puts an upper bound on the number ˆ, of SMPE in simple proposal strategies. If Algorithm 1 produces a unique x then an SMPE in simple strategies is either unique or fails to exist. ˆ ≥ 2 is possible only if ˆ , #X From the way the algorithm constructs x it in step t arrives at x ˆi,t and x ˆj,t with d(ˆ xi,t ) = d(ˆ xj,t ). The equality rewrites as d(xi )(1 − 2δrt,b ) = d(xj )(1 − 2δrt,a ) and is non-generic. That is, a perturbation of x by  > 0, x(), exists, such that Algorithm 1 applied 24

The lemma states that even when G is strongly symmetric and δ ∈ (0, 1), no two strategic bliss points can be the same distance from the median bliss point. The reason is the strategic substitute nature of moderation. If n = 5, player 2 starts moderating when the status-quo is distance d(ˆ x2 ) from x3 = xm . It cannot be SMPE for player 4 to start moderating at d(ˆ x4 ) = d(ˆ x2 ); if player 2 starts at d(ˆ x2 ) it is optimal for player 4 to start at d(ˆ x04 ) > d(ˆ x2 ), if player 4 starts at d(ˆ x4 ) it is optimal for player 2 to start at d(ˆ x02 ) > d(ˆ x4 ). Lemma 4 with Proposition 3 imply that the example of full equilibrium characterization in Baron (1996), based on strategic bliss points in his equation (18) and summarized in his Table 1, cannot constitute an SMPE.

19

ˆ can be ˆ (). In fact, any x ˆ∈X to G() = hn, x(), r, δ, Xi produces unique x ˆ (). The following lemma states this result formally approached by unique x and its proof constructs the claimed perturbation. ˆ from Algorithm 1 applied to G. Then a perˆ ∈ X Lemma 5. Fix any x turbation of x by  > 0, x(), and ¯ > 0 exist, such that lim→0 x() = x and Algorithm 1 applied to G() = hn, x(), r, δ, Xi, ∀ ≤ (0, ¯), produces a ˆ () such that lim→0 x ˆ () = x ˆ. unique profile of strategic bliss points x

7

Equilibrium existence in symmetric games

To define (strongly) symmetric G, denote by dI (i) = |i − m| index ‘distance’ of i ∈ N from m. With this notation, dIa (i) = m+dI (i) and dIb (i) = m−dI (i) is the pair of players index distance dI (i) from median. For j ∈ {1, . . . , n−1 2 }, P j rje = i=1 ri denotes the sum of recognition probabilities of j most extreme players in Nb . By convention rje = 0 when j = 0. Definition 6 (Symmetric G). G is symmetric if and only if, ∀i ∈ N , d(xdI (i) ) = d(xdIa (i) ) and rdI (i) = rdIa (i) . b

b

Definition 7 (Strongly symmetric G). G is strongly symmetric if and only if ri =

1 n

∀i ∈ N and xi − xi−1 = xi+1 − xi ∀i ∈ {2, . . . , n − 1}.

That is, G is symmetric if any pair of players {dIb (i), dIa (i)} has equal recognition probabilities and bliss points at the same distance from xm . This implies ra = rb <

1 2

and that rje , the sum of the recognition probabilities

of the j < m most extreme players {1, . . . , j}, is equal to the sum of the recognition probabilities of players {dIa (j), . . . , n}. G is strongly symmetric if the players’ bliss points are equidistant from each other and the players have equal recognition probabilities. A strongly symmetric G is symmetric. The definition that follows guarantees that Algorithm 1 drops players {m − 1, m + 1} in steps t ∈ {1, 2}. In step t = 1, the algorithm offers an option to drop either one of these two players, and in step t = 2 drops the player not eliminated in step t = 1. In steps t ∈ {3, 4} the algorithm drops players {m − 2, m + 2} in a similar manner and the same happens in any steps {t, t + 1} with t odd. This is what condition G1 ensures. The resulting ˆ along with symmetry of G allows us to write condition G2 structure of x ˆ satisfies condition which, as we prove in Proposition 4, guarantees that x 20

S and hence induces SMPE σ. Notice that both conditions are written in terms of parameters of G. Definition 8 (Pairwise moderation inducing G). G induces pairwise moderation if and only if G is symmetric, ∀i ∈ {1, . . . , n−3 2 } 1 − 2δrie xm − xi ≤ e 1 − 2δri+1 xm − xi+1

(G1 )

and ∀i ∈ {1, . . . , n−3 2 } and ∀j ∈ {1, . . . , i} e 1 − 2δrj−1 xm − xj . ≤ e 1 − 2δrj xm − xi+1

(G2 )

The complexity of the conditions defining pairwise moderation inducing G is driven by our attempt to write them for a general class of symmetric games as much as possible.25 In fact, any symmetric G induces pairwise moderation if the players are sufficiently impatient. ¯ G Lemma 6. For any symmetric G, δ¯ ∈ (0, 1) exists, such that, ∀δ ≤ δ, induces pairwise moderation. Two conditions define pairwise moderation inducing G and we explained their rationale above. However, condition G2 proves to be redundant in certain ‘well behaved’ games satisfying ‘monotonicity’ of the recognition probabilities or of the distances between the bliss points of adjacent players. Lemma 7. If condition G1 co-defining pairwise moderation inducing G holds, then G2 in the same definition holds if at least one of the following conditions are satisfied: 1. ri ≤ ri+1 ∀i ∈ {1, . . . , n−3 2 }; 2. xi − xi−1 ≤ xi+1 − xi ∀i ∈ {2, . . . , n−3 2 } and

1 1−2δr1

≤

xm −x1 xm −x2 .

25

To understand G1 and G2 , after dropping player m−1 in step 1, Algorithm 1 in step 2 e e calculates x ˆm+1,2 = xm+1 +2δrm−2 (xm −xm+1 ) and x ˆm−2,2 = xm−2 +2δrm−1 (xm −xm−2 ). G1 is then the general version of the condition ensuring m + 1 is dropped, d(ˆ xm+1,2 ) ≤ d(ˆ xm−2,2 ). When the algorithm drops player dIa (j) at a further step, db (ˆ xdIa (j) ) ∈ Sm−1 , among other values, needs to satisfy condition S. With db (ˆ xdIa (j) ) = xj + 2δrje (xm − xj ), e the condition requires db (ˆ xdIa (j) ) − xm−1 − 2δrj−1 (xm − xm−1 ) ≤ 0, which rewrites as G2 . Because, say, G1 rewrites as

2δri+1 e 1−2δri+1

≤

xi+1 −xi , xm −xi+1

the incentive to moderate driven by δ and ri+1 .

21

both conditions put an upper bound on

For strongly symmetric games with equidistant bliss points and equal recognition probabilities, the conditions defining pairwise moderation inducing G become trivial to verify. Lemma 8. Symmetric G with n = 3 induces pairwise moderation. Strongly symmetric G with n ≥ 5 and δ ≤

n n+1

induces pairwise moderation.

To state the main result of this section we need the following definition. As we explained above, condition G1 ensures that Algorithm 1 drops pairs of players {dIb (i), dIa (i)} in pairs of steps {t, t + 1}. For knife edge cases when condition G1 holds with equality, the algorithm offers the option, in step t = 1, to drop players {m − 1, m + 1} and dropping m + 1, in step t = 2, ˆ to offers the option to drop players {m − 1, m + 2}. At this point, for x have the structure underlying Proposition 4, we have to ensure that player m − 1 is dropped in step t = 2. That is, we need to ensure that if i ∈ Na is dropped in t = 1 then i ∈ Nb is dropped in t = 2 and vice versa, whenever the algorithm faces multiple players to be dropped. A similar selection is necessary at any step t ≥ 3. Definition 9 (Pairwise path through Algorithm 1). A selection of which players to drop, whenever a non-unique option arises, in Algorithm 1 is called a pairwise path if and only if, in step t ≥ 2, i ∈ Na is dropped when j ∈ Nb has been dropped in step t − 1 and i ∈ Nb is dropped when j ∈ Na has been dropped in step t − 1. Proposition 4 (SMPE with pairwise moderation). Assume G induces pairwise moderation. Then ˆ produced 1. if δ ∈ (0, 1), 2(n−1)/2 distinct profiles of strategic bliss points x ˆ = x; by pairwise paths through Algorithm 1 exist, if δ = 0, x 2. σ induced by any of these profiles of strategic bliss points constitutes an SMPE; 3. σ induced by any of these profiles of strategic bliss points satisfies condition S and, ∀i ∈ N , Ui is single peaked on X. Proposition 4 is the main result of this section. It proves the existence of an SMPE in the large class of games that induce pairwise moderation. To construct an SMPE all that is needed is a profile of strategic bliss points from 22

Algorithm 1 and simple strategies. As already anticipated, the result follows ˆ that satisfies condition because pairwise moderation inducing G delivers x S. Using Lemma 8, Proposition 4 implies the existence of SMPE in any symmetric G with n = 3 and any strongly symmetric G with n ≥ 5 and δ≤

n n+1 ,

a condition which virtually ceases to bind as n increases.

The following examples substantiate our claim that Proposition 4 applies to a large class of games that are not strongly symmetric. The first two examples assume monotonicity in the recognition probabilities (Example 4) or in the distance between bliss points of adjacent players (Example 5). Example 6 takes a strongly symmetric G and increases the distance of bliss points between pairs of players {dIb (j) − 1, dIb (j)} and {dIa (j), dIa (j) + 1}. This produces a G with three ‘clusters’ of players, one around m and two ‘extreme’ clusters. All the examples state conditions guaranteeing that the underlying G induces pairwise moderation. All the conditions put an upper bound on the patience of the players, collapse to δ ≤

n n+1

when G becomes

strongly symmetric, which is allowed by all the examples, and effectively cease to bind when n increases.26 Example 4 (More extreme players less/more likely to propose). Assume G is symmetric with n ≥ 5, xi −xi−1 = xi+1 −xi ∀i ∈ {2, . . . , n− 1} and ri ≤ ri+1 ∀i ∈ {1, . . . , n−3 2 }. Then condition G1 co-defining pairwise moderation inducing G holds if and only if it holds for i =

n−3 27 2 ;

when G1

holds then G2 holds as well; and G induces pairwise moderation if and only if δ ≤

1 2(ra +rm−1 ) ,

which does not bind if rm−1 ≤

1 2

− ra =

rm 2 .

Assume G is symmetric with n ≥ 5, xi −xi−1 = xi+1 −xi ∀i ∈ {2, . . . , n− 1} and ri ≥ ri+1 ∀i ∈ {1, . . . , n−3 2 }. Then condition G1 co-defining pairwise moderation inducing G holds if and only if it holds for i = 1; when G1 holds and δ ≤

1 r1 (n−1)

then G2 holds as well; and G induces pairwise moderation

1 if and only if δ ≤ min{ 2r1 +(n−1)r , 1 }. 2 r1 (n−1)

Example 5 (Increasing/decreasing extremism). Assume G is symmetric with n ≥ 5, xi −xi−1 ≥ xi+1 −xi ∀i ∈ {2, . . . , n−1 2 } and ri = ri+1 ∀i ∈ {1, . . . , n−1 2 }. Then condition G1 co-defining pairwise moderation inducing G holds if and only if it holds for i =

n−3 2 ;

when G1 holds then G2 holds as well; and G induces pairwise moderation 26

Examples 4 and 5 also show that the conditions on δ need not bind at all. This claim, as well as the similar claim for i = 1 below, does not follow immediately. We feel that a formal proof is unnecessary, but are ready to provide it. 27

23

if and only if δ ≤

n(xm−1 −xm−2 ) (n−1)(xm−1 −xm−2 )+2(xm −xm−1 ) ,

which does not bind if

xm−1 − xm−2 ≥ 2(xm − xm−1 ). Assume G is symmetric with n ≥ 5, xi −xi−1 ≤ xi+1 −xi ∀i ∈ {2, . . . , n−1 2 } and ri = ri+1 ∀i ∈ {1, . . . , n−1 2 }. Then condition G1 co-defining pairwise moderation inducing G holds if and only if it holds for i = 1; when G1 holds then G2 holds as well; and G induces pairwise moderation if and only if δ≤

n(x2 −x1 ) 2(xm −x1 +x2 −x1 ) .

Example 6 (Clusters of players). Assume G is symmetric with n ≥ 5, xi − xi−1 = d ∀i ∈ {2, . . . , m} \ {j}, xj − xj−1 = d + e with e ≥ 0 where 2 ≤ j ≤ m and ri = ri+1 ∀i ∈ {1, . . . , n−1 2 }. Then condition G1 co-defining pairwise moderation inducing G holds if and only if it holds for i = 1, when j > 2, and i = 2, when j = 2; when G1 holds then G2 holds as well; and G induces pairwise moderation if and only if δ ≤

n e (n+1)+2 dj

where ej = 0 if j = 2 and ej = e if j ∈ {3, . . . , m}.

Proposition 4 shows that 2(n−1)/2 SMPE exist for any G that induces pairwise moderation, all based on profiles of strategic bliss points delivered ˆ by Algorithm 1. In Lemma 5, we have shown that the multiplicity of x from Algorithm 1 is non-generic and can be perturbed away. The lemma, ˆ () to support SMPE. however, is silent about the ability of the perturbed x The next proposition shows that it is indeed possible to perturb x without upsetting the ability of the profile of strategic bliss points from Algorithm 1 to support an SMPE. ˆ produced Proposition 5. Assume G induces pairwise moderation. Fix any x by pairwise path through Algorithm 1. Then a perturbation of x by  > 0, x(), and ¯ > 0 exist, such that lim→0 x() = x and Algorithm 1 applied to G() = hn, x(), r, δ, Xi, ∀ ≤ (0, ¯), produces a unique profile of strategic ˆ () that satisfies condition S and lim→0 x ˆ () = x ˆ. bliss points x In addition to showing the non-generic nature of the multiplicity of SMPE in pairwise moderation inducing G, Proposition 5 shows that the equilibrium correspondence mapping G into the set of SMPE in simple strategies is upper hemicontinuous in x, SMPE exists as x() → x and continues to exist at the limit of the sequence, at x. However, it fails lower hemicontinuity. Only one of the equilibria that exists at x can be approached by the unique SMPE that exists along x() → x. 24

Throughout this section, we focus on games that induce pairwise moderation. For strongly symmetric G, this implies restricted focus on δ ≤

n n+1 .

This condition becomes rather weak as n increases. However, a natural question arises about the existence and properties of SMPE as δ → 1. The following proposition shows that for high enough δ, at least two SMPE supˆ from Algorithm 1 exist. ported by x Proposition 6 (Patient players in strongly symmetric G). ¯ Assume G is strongly symmetric, n ≥ 5, and δ ≥ δ(n) where ¯ δ(n) = max

n

n n n+1 , n−3

h

2 n−2 n−1 −

q

n3 −n2 −n−7 (n−1)3

io

< 1.

ˆ produced by Algorithm 1 ex1. Two profiles of strategic bliss points x ist with, for g ∈ {a, b}, x ˆm = xm , x ˆi = xi ∀i ∈ Ng and d(ˆ xi ) ∈ (0, d(xm−1 )) ∀i ∈ N \ (Ng ∪ {m}); 2. σ induced by any of these profiles of strategic bliss points constitutes an SMPE. ¯ ˆ that Suppose g = a in Part 1 of the proposition. Then for any δ ≥ δ(n), x ˆ is characterized by x induces SMPE σ exists. This x ˆi = xi ∀i ∈ Na ∪ {m} and x ˆi ∈ (xm−1 , xm ) ∀i ∈ Nb . In words, all the players in Nb moderate while none of the players in Na do. In the proof of the proposition we show that x ˆi = xi + δ n−1 n (xm − xi ) for the moderating players. As a result limn→∞ limδ→1 x ˆi = xm so that proposal behaviour of any moderating player in an SMPE of large G with patient players resembles the proposal behaviour of the median player. From Lemma 8 we know that symmetric G with n = 3 induces pairwise moderation which, by Proposition 4, implies SMPE existence for any δ. n For strongly symmetric G with n ≥ 5, the same lemma requires δ ≤ n+1 . n ¯ Because δ(n) = for n = 5, Propositions 6, 4 and Lemma 8 jointly imply n+1

n SMPE existence in any strongly symmetric G with n = 5. When δ ≤ n+1 , ˆ produced by a pairwise path through Algorithm 1 induces SMPE σ. any x n ˆ from Proposition 6 induces SMPE σ. When δ ≥ n+1 , any x

Corollary 1. An SMPE exists in strongly symmetric G with n = 5. ¯ For strongly symmetric G with more than five players, δ(n) >

n n+1

for any

n ≥ 7, leaving a gap in the range of discount factors covered by Propositions 25

4 and 6.28 On the other hand, from the arguments presented in the proof of Proposition 6, when δ >

n n+1

then Algorithm 1 produces exactly two profiles

of strategic bliss points, the two profiles supporting SMPE in Proposition 6. From Proposition 3 we then know that exactly two SMPE in simple strategies exist. Corollary 2. Exactly two SMPE in simple proposal strategies exist in strongly ¯ symmetric G if n = 5 and δ > n or if n ≥ 7 and δ ≥ δ(n). n+1

Recalling Lemma 5, the duplicity of SMPE the corollary shows is nonˆ that can be approached by the generic. Each SMPE is supported by x sequence of unique profiles of strategic bliss points Algorithm 1 produces for perturbed G. The following proposition shows that it is indeed possible to ˆ in such a way that, along the sequence, all profiles of strategic approach x bliss points support an SMPE. n Proposition 7. Assume G is strongly symmetric with n = 5 and δ > n+1 ¯ ˆ to be one of the two profiles of strategic or with n ≥ 7 and δ ≥ δ(n). Fix x

bliss points from Algorithm 1. Then a perturbation of x by  > 0, x(), and ¯ > 0 exist, such that lim→0 x() = x and Algorithm 1 applied to G() = hn, x(), r, δ, Xi, ∀ ≤ (0, ¯), produces a unique profile of strategic ˆ () that satisfies condition N and lim→0 x ˆ () = x ˆ. bliss points x The proposition shows that for any strongly symmetric G with patient players, an essentially unique SMPE in simple strategies exists. Because ˆ () = x ˆ , we know that in the SMPE, all players with bliss points lim→0 x on one side of the median moderate while their opponents do not. The perturbation required for SMPE uniqueness is constructed in the proof of ˆ the proposition, but its structure is very simple. x() used to approach x with x ˆi = xi ∀i ∈ Na ∪ {m} is xi () = xi ∀i ∈ N \ {m − 1} and xm−1 () = xm−1 − . If the most moderate player in Nb , m − 1, has a stronger incentive to moderate than the most moderate player in Na , m + 1, in the unique SMPE in simple strategies all players in Nb moderate while none of the players in Na do. 28

Example 3, strongly symmetric G with n = 7, illustrates this gap. For n = 7, ¯ = 0.875 and δ(n) ≈ 0.924. When δ = 0.5, eight profiles of strategic bliss points exist, each supporting SMPE by Proposition 4. When δ = 0.95, two profiles of strategic bliss points exist, each supporting SMPE by Proposition 6. n n+1

26

8

Conclusion

This paper provides insights and techniques for studying equilibria in dynamic spatial legislative bargaining. Our results show that the structure of equilibria in these models is simple and intuitive, once we address the formal difficulties for which we provide a resolution. Our approach to equilibrium construction fails when the condition N fails. The existence of equilibria when the condition fails thus remains an open question. In a companion paper (Zapal, 2016), we show that the simple strategies can be adjusted in a way that allows us to prove a general equilibrium existence result for any three-player dynamic spatial legislative bargaining game. We have extensively investigated a similar approach to proving general existence result with more than three players, but so far failed to prove the desired result. Similarly, it remains an open question whether a general existence result can be obtained by extending the mixed strategy equilibrium construction from Kalandrakis (2016) beyond his threeplayer, using our terminology, strongly symmetric game. Several assumptions on the model’s primitives — quadratic stage utilities and simple majority voting rule — raise a natural question regarding the robustness of our results. Jointly, these assumptions imply that the player with the median bliss point is decisive. With general stage utilities, an alternative would be to assume that the player with the median bliss point is decisive by endowing her, for example, with a veto power. In this alternative model, a construction analogous to the one above would use Algorithm 1 with the provisional strategic bliss points calculated in each step so as to reflect the alternative stage utilities. An alternative to the simple majority used throughout would be a more demanding quota voting rule. With a more demanding quota voting rule, a straightforward argument can be used to show that in any SMPE, starting with any status-quo located between the bliss points of the players who are members of all winning coalitions, the equilibrium policy remains constant. That is, any status-quo between the bliss points of the players who are members of all winning coalitions is absorbing in a similar way as the median’s bliss point is absorbing with simple majority. However, a preliminary investigation shows that the way policies approach the absorbing set can be complex, unlike in the model with simple majority.

27

Extending our equilibrium construction beyond one-dimensional policy spaces is possible. In an earlier working paper (Zapal, 2014) we study a dynamic spatial legislative bargaining model with multi-dimensional policy space and use a version of equilibrium construction considered in this paper. The equilibria we construct for multi-dimensional spaces resemble those in one-dimensional policy space; players moderate for strategic reasons and the incentive to moderate is a strategic substitute. For the most part we have failed to stress and comment on the behaviour of policies generated by equilibrium play, focusing instead on the existence of equilibria. Common predictions emerging from our analysis are the convergence to the median’s ideal policy, the convergence path alternation of policies around this policy and asymmetric tendency for moderation towards this policy. With the exception of the first, these predictions concern the dynamics of policies, are distinctive to the dynamic model and cannot be addressed in static legislative bargaining models. Moderation, strategic manipulation of today’s policies driven by concerns over future policies, is a standard observation in dynamic political economy literature. The strategic substitute nature of moderation is, to our knowledge, novel.

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34

A1

Proofs

A1.1

Proof of Lemma 1

Fix a profile of stationary Markov strategies σ. Consider two policies p0 and p00 generating stochastic sequence, via σ, of policies p = {p0 , p1 , . . .} and p0 = {p00 , p01 , . . .} respectively. The utility of player i from voting either for p0 or p00 is "∞ X

Ui (p0 |σ) = E

# −δ t (pt − xi )2

" Ui (p00 |σ) = E

t=0

∞ X

# −δ t (p0t − xi )2 . (A1)

t=0

Differentiating the difference in utility from the two policies with respect to xi gives

" ∞ # X ∂[Ui (p0 |σ) − Ui (p00 |σ)] =E 2 −δ t (p0t − pt ) ∂xi

(A2)

t=0

which is independent of xi and hence Ui (p0 |σ) − Ui (p00 |σ) is linear in xi . When Um (p0 |σ) ≥ Um (p00 |σ), then Ui (p0 |σ) ≥ Ui (p00 |σ) either ∀i ∈ Na or ∀i ∈ Nb . Conversely, when Um (p0 |σ) < Um (p00 |σ), then Ui (p0 |σ) < Ui (p00 |σ) either ∀i ∈ Na or ∀i ∈ Nb .

A1.2



Proof of Lemma 2

To see Part 1, any simple strategy pi with any bliss point x ˆi ∈ R satisfies, ∀x ∈ X, pi (db (x)|ˆ xi ) = pi (da (x)|ˆ xi ). The claim then follows from (2).29 Part 2 follows easily from the symmetry of Vi ∀i ∈ N about xm and asymmetry of the stage utilities ∀i ∈ N \ {m} and symmetry of um . ˆ and To prove Part 3, continuity of the dynamic utilities Ui on X, fix x ˆ satisfies C. As Ui (x|σ) = ui (x) + δVi (x|σ), σ it induces and assume that x we need to prove the continuation value functions Vi are continuous. From pi (x|ˆ xi ) ∈ {db (x), da (x)} for any i ∈ N C(x|σ) and x ∈ D(σ) and from symmetry of Vi about xm , we can write (2), ∀x ∈ D(σ), P Vi (x|ˆ x) =

j∈N

rj ui (pj (x|ˆ xj )) + δ

P

xj )|ˆ x) j∈C(x|σ) rj Vi (pj (x|ˆ

1 − δrnc (x|σ)

(A3)

which is continuous, ∀i ∈ N , by continuity of pj (x|ˆ xj ) ∀j ∈ N , constancy 29 We do not rule out x ˆi = ±∞. The meaning of, say, x ˆi = ∞ in pi is player i ∈ Na proposing da (x) for any status-quo x.

35

of pj (x|ˆ xj ) ∀j ∈ C(x|σ) and by local, that is on any interval induced by N D(σ), constancy of C(x|σ) and rnc (x|σ). What remains is, ∀i ∈ N , Vi (x− |ˆ x) = Vi (x|ˆ x) = Vi (x+ |ˆ x) for any x ∈ N D(σ). For x = xm , the claim follows from Vi (x− x) = Vi (x+ x) m |ˆ m |ˆ ∀i ∈ N (by Part 1), pj (x− xj ) = pj (xm |ˆ xj ) = pj (x+ xj ) = xm ∀j ∈ N , m |ˆ m |ˆ Vi (xm |ˆ x) =

ui (xm ) 1−δ

∀i ∈ N , and Vi (x+ x) = m |ˆ

∀i ∈ N , which uses (A3) and the property that,

ui (xm )+δ(1−rnc (x+ m |σ))Vi (xm |σ) 1−δrnc (x+ m |σ) ∀j ∈ C(x+ ˆj = xm and m |σ), x

hence pj (x|ˆ xj ) = xm ∀x ∈ X. For x ∈ N D(σ)\{xm } let us focus on cases when x > xm . When x < xm the argument is symmetric and omitted. First notice that pj (x− |ˆ xj ) = pj (x|ˆ xj ) = pj (x+ |ˆ xj ) ∀j ∈ N and ∀x ∈ X so that the first sum in the numerator of (A3) is continuous. To prove Vi (x|ˆ x) = Vi (x− |ˆ x), we have i) Vi (pj (x|ˆ xj )|ˆ x) = Vi (pj (x− |ˆ xj )|ˆ x) ∀j ∈ C(x− |σ) and ii) Vi (pj (x|ˆ xj )|ˆ x) = Vi (x|ˆ x) ∀j ∈ N C(x− |σ), so that Vi (x|ˆ x) =

X

rj [ui (pj (x|ˆ xj )) + δVi (pj (x|ˆ xj )|ˆ x)]

j∈N

=

X

rj ui (pj (x− |ˆ xj )) + δ

X

rj Vi (pj (x− |ˆ xj )|ˆ x)

j∈C(x− |σ)

j∈N

(A4)

+ δrnc (x− |σ)Vi (x|ˆ x) = Vi (x− |ˆ x)(1 − δrnc (x− |σ)) + δrnc (x− |σ)Vi (x|ˆ x) and the claim, ∀i ∈ N , follows. To prove Vi (x|ˆ x) = Vi (x+ |ˆ x), we have i) Vi (pj (x|ˆ xj )|ˆ x) = Vi (pj (x+ |ˆ xj )|ˆ x) ∀j ∈ C(x+ |σ) and ii) Vi (pj (x|ˆ xj )|ˆ x) = Vi (x|ˆ x) ∀j ∈ N C(x+ |σ), so that Vi (x|ˆ x) =

X

rj [ui (pj (x|ˆ xj )) + δVi (pj (x|ˆ xj )|ˆ x)]

j∈N

=

X

rj ui (pj (x+ |ˆ xj )) + δ

X

rj Vi (pj (x+ |ˆ xj )|ˆ x)

j∈C(x+ |σ)

j∈N

(A5)

+ δrnc (x+ |σ)Vi (x|ˆ x) = Vi (x+ |ˆ x)(1 − δrnc (x+ |σ)) + δrnc (x+ |σ)Vi (x|ˆ x) and the claim, ∀i ∈ N , follows. Part 4, Ui00 (x|ˆ x) < 0 ∀x ∈ D(σ) and ∀i ∈ N , follows by i) u00i (x) = −2, ii) P u00i (pj (x|ˆ xj )) = −2[p0j (x|ˆ xj )]2 , iii) j∈N C(x|σ) rj ui (pj (x|ˆ xj )) being the only (locally) non-constant term in (A3) and iv) p0j (x|ˆ xj ) = ±1 ∀j ∈ N C(x|σ). 36

−2rnc (x|σ) Thus, ∀x ∈ D(σ) and ∀i ∈ N , Ui00 (x|σ) = −2+δ 1−δr = nc (x|σ)

−2 1−δrnc (x|σ)

< 0.

To prove Part 5, we only need to show that Um (x|ˆ x) is increasing for x < xm and decreasing for x > xm . For any i ∈ N and x ∈ D(σ) we have, using (A3) and p0j (x|ˆ xj ) = ±1 ∀j ∈ N C(x|σ) depending on x ≷ xm and j ∈ Na or j ∈ Nb in obvious manner,  −2[x − xi − 2δrnc,a (x|σ)(xm − xi )]   if x < xm  1 − δrnc (x|σ) 0 Ui (x|ˆ x) =  −2[x − xi − 2δrnc,b (x|σ)(xm − xi )]   if x > xm . 1 − δrnc (x|σ)

(A6)

Evaluating the derivative for m shows that Um is, ∀x ∈ D(σ), increasing for x < xm and decreasing for x > xm . By continuity of Um the claim follows. Part 6, A(x|ˆ x) = [db (x), da (x)] ∀x ∈ X, is a consequence of Part 5.

A1.3



Proof of Proposition 1

ˆ from Algorithm 1 and the induced Fix a profile of strategic bliss points x ˆ . Thus, ∀i ∈ N , profile of strategies σ. We know that condition C holds for x ˆ and Vi , Ui and A induced by σ induced by x ˆ Vi , Ui and A induced by x coincide and have the properties stated in Lemma 2. Moreover, the voting strategies in σ satisfy the SMPE optimality condition. We now argue that the proposal strategies in σ satisfy the SMPE optimality condition. For m, we have x ˆm = xm , hence the proposal strategy of the median player is optimal by Lemma 2 part 5. Fix i ∈ Na . The argument for i ∈ Nb is symmetric and omitted. By Lemma 2 part 2, it is never optimal for i to propose policy p < xm . Using the shape of A from Lemma 2 part 6, we need to make sure that proposing da (x) for any x ∈ [db (ˆ xi ), da (ˆ xi )] and x ˆi otherwise is optimal for i. Ui making this proposal strategy optimal has to satisfy Ui (x|σ) ≤ Ui (y|σ) for any x ∈ [xm , x ˆi ] and y ∈ [xm , x ˆi ] such that x < y and Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y > x ˆi . The first inequality follows from the way Algorithm 1 constructs the strategic bliss points; it generates 0 − ˆ such that Ui0 (ˆ x x− ˆi ) i |σ) = 0 and Ui (x |σ) ≥ 0 for any x ∈ N D(σ) ∩ (xm , x which, combined with the piecewise strict concavity of Ui , shows the claim. To ensure the second inequality, notice that from (A6) we have Ui0 (x|σ) ≤ 0 for x ∈ D(σ) and x ≥ xi so that Ui (xi |σ) ≥ Ui (y|σ) for any y > xi . Hence we need to make sure that Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ [ˆ xi , xi ] in order for σ to constitute an SMPE. 37

To prove that condition S is sufficient, Part 1, note that Ui0 (ˆ x+ i |σ) ≤ 0. To see this, when x ˆi = xm , Algorithm 1 drops i because Ui0 (ˆ x+ i |σ) ≤ 0. When x ˆi > xm , Algorithm 1 drops i because Ui0 (ˆ x− i |σ) = 0 and we have 0 x+ |σ) from (A6), because exactly one player is dropped in Ui0 (ˆ x− i |σ) = Ui (ˆ i

any step of the algorithm, from d(ˆ xj ) > d(ˆ xi ) for any player j dropped subsequently, and from rnc,b (ˆ x− x+ i |σ) = rnc,b (ˆ i |σ) when i ∈ Na is dropped. Because we have Ui0 (ˆ x+ i |σ) ≤ 0, by the piecewise strict concavity of Ui , we need to ensure that Ui0 (x+ |σ) ≤ 0 ∀x ∈ N D(σ) ∩ (ˆ xi , xi ) = Si (σ). Using (A6) this condition becomes x − xi − 2δrnc,b (x+ |σ)(xm − xi ) ≥ 0, which is what the condition S requires. Hence if S holds, we have Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ [ˆ xi , xi ] and σ constitutes an SMPE. To prove that condition N is necessary and sufficient, Part 2, we note that Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ [ˆ xi , xi ] is equivalent to Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ ((N D(σ) ∪ Li (σ)) ∩ (ˆ xi , xi )) ∪ {xi , x ˆi } = Ni (σ). To see this, take two adjacent elements of N D(σ) from [ˆ xi , xi ], x0 and x00 , with x0 < x00 . If Ui has no local maximum on [x0 , x00 ], that is when [x0 , x00 ] ∩ Li (σ) = ∅, then Ui (x0 |σ) > Ui (x00 |σ) ⇔ Ui (x0 |σ) > Ui (y|σ) and Ui (x0 |σ) < Ui (x00 |σ) ⇔ Ui (x0 |σ) < Ui (y|σ) for any y ∈ [x0 , x00 ] (equality cannot occur by the strict concavity of Ui ). If Ui has local maximum on [x0 , x00 ] then exactly one and we can set x000 = [x0 , x00 ] ∩ Li (σ) and proceed with a similar argument using x000 instead of x00 . To show that Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ Ni (σ) is equivalent to N, for R any differentiable continuous function f , f (x) − f (z) = [ f 0 (a)da]xz . When f is not differentiable at x, y, z with x < y < z but possesses one-sided R R + + derivatives at x, y, z, we have f (x) − f (z) = [ f 0 (a)da]xy− + [ f 0 (a)da]yz − . [x − ci (x|σ)] Now, (A6) for x > xm can be rewritten as Ui0 (x|σ) = 1−δr−2 R 0 nc (x|σ) where ci (x|σ) h 2 = xi + 2δrnc,b i (x|σ)(xm − xi ). Hence Ui (x|σ)dx = Ti (x|σ) = −2 x 1−δrnc (x|σ) 2 − ci (x|σ)x as rnc,b (x|σ) and rnc (x|σ) are both constant on any interval induced by N D(σ). Condition N then takes into account that Ni (σ) can have an arbitrary number of elements. When N holds, we have Ui (ˆ xi |σ) ≥ Ui (y|σ) for any y ∈ [ˆ xi , xi ] and σ constitutes an SMPE. When N fails, we have Ui (ˆ xi |σ) < Ui (y|σ) for some y ∈ [ˆ xi , xi ] and σ cannot constitute an SMPE, as i would prefer to deviate to proposing y when the status-quo is y, as opposed to proposing x ˆi that σ requires.

38



A1.4

Proof of Proposition 2

Algorithm 1 in step t calculates x ˆi,t = xi + 2δrt,a (xm − xi )

if i ∈ Nb

x ˆi,t = xi + 2δrt,b (xm − xi )

if i ∈ Na

(A7)

and drops i ∈ arg minj∈Pt d(ˆ xj,t ) if Rt = ∅. Throughout the proof let us assume δ ≤ 12 , so that 1 > 2δra and 1 > 2δrb , which implies Rt = ∅. Suppose first that d(xi ) 6= d(xj ) ∀i ∈ N and ∀j ∈ N \ {i}. Writing d(ˆ xi,t ) = d(xi )(1 − 2δrt,a ) for i ∈ Nb and d(ˆ xi,t ) = d(xi )(1 − 2δrt,b ) for i ∈ Na shows that d(ˆ xi,t ) ∈ (d(xi )(1 − 2δ), d(xi )] ∀i ∈ N \ {m} and ∀t ∈ ¯ d(xj ) < d(xi ) {1, . . . , n − 1}. Hence, δ¯ ∈ (0, 1) exists, such that, ∀δ ≤ δ, implies d(xj ) < d(xi )(1 − 2δ) and hence d(ˆ xj,t ) < d(ˆ xi,t ) ∀t ∈ {1, . . . , n − 1}. ¯ drops Since d(xi ) 6= d(xj ) for any pair of players, Algorithm 1, ∀δ ≤ δ, the player with the smallest d(xi ) in step 1 and the player with the second smallest d(xi ) in step 2. The algorithm continues in a similar manner, dropping the player with the tth smallest d(xi ) in step t − 1, until step n − 1 ˆ be the when it drops the player with the largest d(xi ). Fix δ ≤ δ¯ and let x unique profile of strategic bliss points produced by the algorithm and σ the profile it induces. ˆ satisfies condition S. Let it denote the player We now argue that x dropped in step t ∈ {0, . . . , n − 1}. For i0 = m we do not need to verify S since it does not apply to m. For in−1 , x ˆin−1 = xin−1 and hence Sin−1 (σ) = ∅ and condition S holds for in−1 . For it with t ∈ {1, . . . , n − 2}, we know that d(ˆ xit−1 ) ≤ d(xit−1 ) < d(ˆ xit ) ≤ d(xit ) < d(ˆ xit+1 ) ≤ d(xit+1 ), so that Sit (σ) = ∅ and condition S holds for it for any t ∈ {1, . . . , n − 2}. Suppose now that a pair of players {i0 , j 0 } with d(xi0 ) = d(xj 0 ) exists. Without loss of generality let i0 ∈ Nb and j 0 ∈ Na . If multiple such pairs exist, let {i0 , j 0 } be the one with the largest i0 and hence the smallest j 0 . ¯ Algorithm 1 drops players {i0 + 1, . . . , j 0 − 1} in steps t ∈ Because δ ≤ δ, {0, . . . , j 0 −i0 −2}, drops players i0 and j 0 in steps t0 = j 0 −i0 −1 and t0 +1, and drops players {1, . . . , i0 − 1} ∪ {j 0 + 1, . . . , n} in steps t ∈ {t0 + 2, . . . , n − 1}. ¯ we ˆ the algorithm produces and σ induced by x ˆ . Because δ ≤ δ, Fix x have d(xi ) < d(ˆ xi0 ) and d(xi ) < d(ˆ xj 0 ) ∀i ∈ {i0 + 1, . . . , j 0 − 1} and d(xi0 ) = d(xj 0 ) < d(ˆ xi ) ∀i ∈ {1, . . . , i0 −1}∪{j 0 +1, . . . , n}. This implies that condition S holds ∀i ∈ {i0 + 1, . . . , j 0 − 1} and that Si0 (σ) and Sj 0 (σ) include at most 39

unique element db (ˆ xj 0 ) and da (ˆ xi0 ) respectively. We now need to verify condition N for i0 and j 0 . Suppose i0 has been dropped in step t0 and j 0 in step t0 + 1. In step t0 of the algorithm, Pt0 = P0 P {1, . . . , i0 } ∪ {j 0 , . . . , n}, rt0 ,b = ik=1 rk and rt0 ,a = nk=j 0 rk and i0 can be dropped only if rt0 ,b ≤ rt0 ,a . This implies x ˆi0 = xi0 + 2δrt0 ,a (xm − xi0 ) x ˆj 0 = xj 0 + 2δ(rt0 ,b − ri0 )(xm − xj 0 )

(A8)

which gives db (ˆ xj 0 ) = xi0 + 2δ(rt0 ,b − ri0 )(xm − xi0 ) from d(xi0 ) = d(xj 0 ) ⇔ (xm − xi0 ) = −(xm − xj 0 ). Thus xi0 ≤ db (ˆ xj 0 ) < x ˆi0 , and hence da (ˆ x i0 ) < x ˆj 0 ≤ xj 0 . If x ˆj 0 = xj 0 , Si0 (σ) = Sj 0 (σ) = ∅ so that condition S and hence N holds for i0 and j 0 . Suppose x ˆj 0 < xj 0 . Then Si0 (σ) = {db (ˆ xj 0 )} and Sj 0 (σ) = ∅ and we need to verify condition N for i0 . Denote z0 = xi0 + 2δrt0 ,a (xm − xi0 )

z2 = xi0 + 2δ(rt0 ,a − rj 0 )(xm − xi0 )

z1 = xi0 + 2δ(rt0 ,b − ri0 )(xm − xi0 )

z3 = x i0

and note that z0 = x ˆi0 and z1 = db (ˆ xj 0 ). From definitions of rnc,a and rnc,b , rnc,a (x|σ) = rt0 ,a ∀x ∈ (z1 , z0 ), rnc,a (x|σ) = rt0 ,a − rj 0 ∀x ∈ (z3 , z1 ) and rnc,b (x|σ) = rt0 ,b − ri0 ∀x ∈ (z3 , z1 ) ∪ (z1 , z0 ). To verify condition N for i0 , we first verify condition S, which suffices for N, and only when it fails we directly verify N. From Si0 (σ) = {db (ˆ xj 0 )}, condition S for i0 writes db (ˆ xj 0 ) − xi0 − 2δ(rt0 ,a − rj 0 )(xm − xi0 ) ≤ 0

(A9)

which is equivalent to 2δ(xm − xi0 )(rt0 ,b − rt0 ,a + rj 0 − ri0 ) ≤ 0. The condition holds if rj 0 ≤ ri0 because rt0 ,b ≤ rt0 ,a and xm − xi0 > 0. Assume rj 0 > ri0 and that condition S fails for i0 , that is rt0 ,b − rt0 ,a + rj 0 − ri0 > 0. Because rt0 ,a > rt0 ,b − ri0 , we have rt0 ,a > rt0 ,b − ri0 > rt0 ,a − rj 0 ≥ 0 so that z0 > z1 > z2 ≥ z3 . To verify condition N, Ni0 (σ) = {z0 , z1 , z2 , z3 } when z2 > z3 and Ni0 (σ) = {z0 , z1 , z2 } when z2 = z3 is easy to see from the definition of Ni .

40

Direct substitution of expressions for rnc,a and rnc,b into Ti0 (x|σ) gives  x2 − x · z0 t 2   2 x 2 Ti0 (x|σ) = − 1−δ(r 0 0 +r 0 −r 0 −r 0 ) − x · z2 t ,a t ,b i j 2 Ti0 (x|σ) = − 1−δ(r 0

Condition N writes

2 ,a0 +rt0 ,b −ri0 )



− j=1 Ti0 (zj−1 |σ)

PJ

if x ∈ (z1 , z0 ) (A10) if x ∈ (z3 , z1 ).

− Ti0 (zj+ |σ) ≥ 0 for J ∈ {1, 2, 3} when

z2 > z3 and J ∈ {1, 2} when z2 = z3 . Each of the (at most) three terms in the condition rewrites Ti0 (z0− |σ) − Ti0 (z1+ |σ) =

(z0 − z1 )2 1 − δ(rt0 ,a + rt0 ,b − ri0 )

Ti0 (z1− |σ) − Ti0 (z2+ |σ) =

−(z1 − z2 )2 1 − δ(rt0 ,a + rt0 ,b − ri0 − rj 0 )

Ti0 (z2− |σ) − Ti0 (z3+ |σ) =

(z2 − z3 )2 1 − δ(rt0 ,a + rt0 ,b − ri0 − rj 0 )

(A11)

When z2 > z3 , the first and the third term are clearly positive. Condition N thus holds if Ti0 (z0− |σ) − Ti0 (z1+ |σ) + Ti0 (z1− |σ) − Ti0 (z2+ |σ) ≥ 0. The same condition applies when z2 = z3 as the first term is positive and the third term is not part of condition N. Dropping positive multiplicative constants, the condition writes (rt0 ,b − rt0 ,a − ri0 + rj 0 )2 (rt0 ,a − rt0 ,b + ri0 )2 − ≥ 0. 1 − δ(rt0 ,a + rt0 ,b − ri0 ) 1 − δ(rt0 ,a + rt0 ,b − ri0 − rj 0 )

(A12)

The denominator of the first term is smaller than the denominator of the second one, so the condition holds if (rt0 ,a − rt0 ,b + ri0 )2 − (rt0 ,b − rt0 ,a − ri0 + rj 0 )2 ≥ 0 or ri0 + rt0 ,a − rt0 ,b ≥ for player

rj 0 2 .

Because rt0 ,a ≥ rt0 ,b , ri0 ≥

and

rj 0 ≥

j0 ri0 2 ,

suffices for N to hold

i0 .

To finish the proof, we know that if ri0 ≥ i0

rj 0 2

(A13)

rj 0 2 ,

then condition N holds for

if rt0 ,a ≥ rt0 ,b . For rt0 ,a ≤ rt0 ,b , symmetric argument would lead to rj 0 0 2 , 2rj ]. i0 among

or ri0 ≤ 2rj 0 . These two conditions jointly require ri0 ∈ [

Finally, we assumed that {i0 , j 0 } is pair of players with the largest

the pairs of player with d(xi ) = d(xj ). The proof can now proceed to a pair of players {i00 , j 00 } such that d(xi00 ) = d(xj 00 ) and i00 < i0 . Identical 41

argument gives ri00 ∈ [

rj 00 00 2 , 2rj ]

and considering any further pair of players

with d(xi ) = d(xj ) leads to the very same condition.

A1.5



Proof of Lemma 3

ˆ and assume σ it induces constitutes an SMPE. By Lemma 1, given σ, Fix x proposal p ∈ X is accepted under status-quo x ∈ X if and only if m votes for p. Because m can enforce xm as an outcome in any future period by rejecting any proposal p 6= xm when status-quo is xm , we have Vm (xm |σ) = 0. This implies Um (xm |σ) > Um (x|σ) ∀x ∈ X \ {xm } and, by Lemma 1, A(xm |σ) = {xm }. The proposal strategies in σ thus have to satisfy πi (xm ) = xm ∀i ∈ N , or, in terms of the simple strategies, pi (xm |ˆ xi ) = xm ∀i ∈ N , which rewrites as x ˆi ≥ xm ∀i ∈ Na , x ˆi ≤ xm ∀i ∈ Nb and x ˆ m = xm .

A1.6



Proof of Lemma 4

ˆ be a profile of strategic bliss points from Algorithm 1. To see Part Let x 1, if δ = 0, the algorithm in step t computes x ˆi,t = xi ∀t ∈ {1, . . . , n − 1} and ∀i ∈ N . Hence Rt = ∅ ∀t ∈ {1, . . . , n − 1} since the condition defining Rt , (xi − xm )(ˆ xi,t − xm ) ≤ 0, rewrites as (xi − xm )2 ≤ 0 and is violated. The algorithm thus sets x ˆi = xi in every step t ∈ {1, . . . , n − 1} and because ˆ = x follows. x ˆ m = xm , x To prove Part 2, assume 1 ≤ 2δra . When 1 ≤ 2δrb the argument is symmetric and omitted. 1 ≤ 2δra implies 1 > 2δrb ; 1 ≤ 2δrb and 1 ≤ 2δra sum to 1 ≤ δ(ra + rb ), which contradicts δ < 1 and ra + rb = 1 − rm < 1. In step 0, the algorithm produces x ˆm = xm . In step 1, the algorithm computes x ˆi,1 ∀i ∈ N \ {m} using r1,a = ra and r1,b = rb . Now notice that, in general step t of the algorithm, (xi − xm )(ˆ xi,t − xm ) used to construct Rt rewrites as (xi − xm )2 (1 − 2δrt,a ) if i ∈ Nb and as (xi − xm )2 (1 − 2δrt,b ) if i ∈ Na . In step 1 this means R1 = Nb when 1 ≤ 2δra and 1 > 2δrb . At this point the algorithm drops one of the players in R1 = Nb , say j 0 , and sets x ˆ j 0 = xm , which implies that P2 = Na ∪Nb \{j 0 } and hence r2,a = ra and r2,b = rb −rj 0 . Clearly R2 = Nb \ {j 0 }, the algorithm in step 2 drops j 00 ∈ R2 ( Nb and sets x ˆj 00 = xm , which implies P3 = Na ∪ Nb \ {j 0 , j 00 } and hence r3,a = ra and r3,b = rb − rj 0 − rj 00 . The algorithm continues in similar manner, dropping j ∈ Nb and setting x ˆj = xm , until step

n−1 2 ,

in which it drops last player from

Nb . This implies P n−1 +1 = Na and hence r n−1 +1,a = ra and r n−1 +1,b = 0. 2

2

42

2

For the remaining steps the algorithm thus sets x ˆi = xi for all i ∈ Na . To prove Part 3, because the algorithm is dropping players and rt,a and rt,b are sums of recognition probabilities of the players that remain in the algorithm, rt,a ≥ rt+1,a and rt,b ≥ rt+1,b ∀t ∈ {1, . . . , n − 2}. 1 > 2δra and 1 > 2δrb with r1,a = ra and r1,b = rb thus imply 1 > 2δrt,a and 1 > 2δrt,b ∀t ∈ {1, . . . , n − 1}. For any step t ∈ {1, . . . , n − 1} of the algorithm, this implies Rt = ∅, x ˆi,t > xm if i ∈ Na and x ˆi,t < xm if i ∈ Nb and hence x ˆm−1 < x ˆ m = xm < x ˆm+1 . To prove x ˆi < x ˆi+1 ∀i ∈ N \ {n}, we thus need to show x ˆi < x ˆi+1 ∀i ∈ Na \ {n} and ∀i ∈ Nb \ {m − 1}. We do so for i ∈ Na \ {n}. For i ∈ Nb \ {m − 1} the argument is similar and omitted. Note that, if i ∈ Na and t ∈ {1, . . . , n − 1}, and

∂x ˆi,t ∂rt,b

∂x ˆi,t ∂xi

= 1 − 2δrt,b > 0

= 2δ(xm − xi ) < 0. The first inequality implies x ˆi,t < x ˆi+1,t

if i ∈ Na \ {n} and t ∈ {1, . . . , n − 1}. The second inequality implies x ˆi,t ≤ x ˆi,t+1 if i ∈ Na and t ∈ {1, . . . , n − 2}. Hence, if the algorithm drops player i ∈ Na \ {n} in step t and player i + 1 in step t0 , then t < t0 , which allows us to write x ˆi = x ˆi,t < x ˆi+1,t ≤ x ˆi+1,t0 = x ˆi+1 . To prove d(ˆ xi ) 6= d(ˆ xj ) for any pair of players {i, j} with i 6= j, ∀t ∈ {1, . . . , n − 1}, d(ˆ xi,t ) = (xi − xm )(1 − 2δrt,b )

if i ∈ Na

d(ˆ xi,t ) = (xm − xi )(1 − 2δrt,a )

if i ∈ Nb

(A14)

and hence d(ˆ xi,t ) < d(ˆ xi+1,t ) if i ∈ Na \ {n} and d(ˆ xi,t ) < d(ˆ xi−1,t ) if i ∈ xi,t ) thus Nb \ {1}. In step t ∈ {1, . . . , n − 1} of the algorithm, arg mini∈Pt d(ˆ either includes unique player i0 or pair of players {i0 , j 0 } such that i0 ∈ Na and j 0 ∈ Nb . In the former case, x ˆ i0 = x ˆi0 ,t and d(ˆ xi0 ) < d(ˆ xi,t ) ≤ d(ˆ xi,t+1 ) ∀i ∈ Pt \{i0 }, where the weak inequality follows because rt,a and rt,b are nonincreasing in t and thus d(ˆ xi,t ) ≤ d(ˆ xi,t+1 ) ∀t ∈ {1, . . . , n − 2} for any i ∈ N . When the algorithm drops i00 ∈ Pt+1 = Pt \{i0 } in step t+1, x ˆi00 = x ˆi00 ,t+1 and hence d(ˆ xi0 ) < d(ˆ xi00 ). In the latter case, suppose, without loss of generality, that i0 is dropped. Then x ˆ i0 = x ˆi0 ,t and d(ˆ xi0 ) < d(ˆ xi,t ) ≤ d(ˆ xi,t+1 ) ∀i ∈ Pt \ {i0 , j 0 }. It thus suffices to show that d(ˆ xi0 ,t ) < d(ˆ xj 0 ,t+1 ). This follows from d(ˆ xi0 ,t ) = d(ˆ xj 0 ,t ), because, when i0 ∈ Na is dropped, rt,a > rt+1,a and, thus, d(ˆ xi,t ) < d(ˆ xi,t+1 ) for any i ∈ Nb , including j 0 ∈ Nb .

43



A1.7

Proof of Proposition 3

ˆ induces SMPE σ, then x ˆ satisfies condition C. From From Lemma 3, if x ˆ produced by Algorithm 1 satisfies C as well. Lemma 2 thus Lemma 4, any x ˆ that constitutes an SMPE or is applies irrespective of whether we refer to x produced by Algorithm 1. ˆ that induces SMPE σ exists, Case 1: When δ = 0, clearly unique x ˆ = x, and we know, by Lemma 4 part 1, that Algorithm 1 produces x ˆ = x. x Case 2: When δ ∈ (0, 1) and 1 ≤ 2δra , by Lemma 4 part 2, we need to ˆ induces SMPE σ, then it satisfies x show that if x ˆi = xm ∀i ∈ N \ Na and x ˆi = xi ∀i ∈ Na . Note that 1 ≤ 2δra implies 1 > 2δrb as shown in the proof ˆ and suppose it induces SMPE σ. We proceed with a of Lemma 4. Fix x series of claims. First, we claim x ˆi > xm ∀i ∈ Na . Suppose, towards a contradiction, that x ˆi = xm for some i ∈ Na . Using (A6) and rnc,b (x+ m |σ) ≤ rb , we have Ui0 (x+ m |σ) =

−2(xm −xi )(1−2δrnc,b (x+ m |σ)) 1−δrnc (x+ m |σ)

> 0. Hence, 0 > 0 exists, such that,

∀ ∈ (0, 0 ), Ui (xm |σ) < Ui (xm + |σ), pi (xm + |ˆ xi ) = xm and xm +  ∈ ˆ that induces SMPE A(xm + |σ), which contradicts x ˆi = xm being part of x σ. Second, we claim x ˆi = xm ∀i ∈ Nb . Suppose, towards a contradiction, that x ˆi < xm for some i ∈ Nb . Using (A6) and rnc,a (x− m |σ) = ra ≥

1 2δ ,

where the equality follows from x ˆj > xm ∀j ∈ Na proven in the previous claim, Ui0 (x− m |σ) = by Lemma 2 part

−2(xm −xi )(1−2δra ) ≥ 0. Because Ui00 (x|σ) < 0 ∀x ∈ D(σ) 1−δrnc (x− m |σ) 4, 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Ui (xm − |σ) <

Ui (xm |σ), pi (xm − |ˆ xi ) = xm −  and xm ∈ A(xm − |σ), which contradicts ˆ that induces SMPE σ. x ˆi < xm being part of x Third, we claim x ˆi = xi ∀i ∈ Na . Suppose, towards a contradiction, that x ˆi 6= xi for some i ∈ Na . By the first claim, this implies x ˆi ∈ (xm , xi ) ∪ (xi , ∞). Using (A6) and rnc,b (x|σ) = 0 ∀x ∈ D(σ), where the equality follows from x ˆj = xm ∀j ∈ Nb proven in the previous claim, sgn [Ui0 (ˆ x− i |σ)] = sgn [Ui0 (ˆ x+ ˆi ]. If x ˆi ∈ (xm , xi ), 0 > 0 exists, such that, i |σ)] = sgn [xi − x ∀ ∈ (0, 0 ), Ui (ˆ xi |σ) < Ui (ˆ xi +|σ), pi (ˆ xi +|ˆ xi ) = x ˆi and x ˆi + ∈ A(ˆ xi +|σ). If x ˆi ∈ (xi , ∞), 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Ui (ˆ xi |σ) < Ui (ˆ xi − |σ), pi (ˆ xi |ˆ xi ) = x ˆi and x ˆi −  ∈ A(ˆ xi |σ). Each case contradicts x ˆi being part of ˆ that induces SMPE σ. x Case 3: When δ ∈ (0, 1) and 1 ≤ 2δrb , by Lemma 4 part 2, we need

44

ˆ induces SMPE σ, then it satisfies x to show that if x ˆi = xm ∀i ∈ N \ Nb and x ˆi = xi ∀i ∈ Nb . The proof is analogous to the proof of Case 2 and is omitted. Case 4: When δ ∈ (0, 1), 1 > 2δra and 1 > 2δrb , we need to show ˆ where X ˆ is the set of profiles of ˆ induces SMPE σ, then x ˆ ∈ X, that if x strategic bliss points produced by Algorithm 1. We start by proving several ˆ that induces SMPE σ. properties of x ˆ induces SMPE Lemma A1. Assume δ ∈ (0, 1), 1 > 2δra and 1 > 2δrb . If x σ, then 1. x ˆi > xm ∀i ∈ Na and x ˆi < xm ∀i ∈ Nb ; 0 x+ |σ) = 0 ∀i ∈ N ; 2. Ui0 (ˆ x− b i |σ) = 0 ∀i ∈ Na and Ui (ˆ i 0 (x− |σ) and U 0 (x+ |σ) < U 0 (x+ |σ) ∀x ∈ X and 3. Ui0 (x− |σ) < Ui+1 i i+1

∀i ∈ N \ {n}; 4. x ˆi < x ˆi+1 ∀i ∈ N \ {n} and d(ˆ xi ) 6= d(ˆ xj ) ∀i ∈ N , ∀j ∈ N , i 6= j. Proof. To show Part 1 of the lemma, x ˆi > xm ∀i ∈ Na follows, since 1 > 2δrb , by an argument identical to the one used to prove the first claim in Case 2. An analogous argument can be used to prove x ˆi < xm ∀i ∈ Nb if 1 > 2δra . To show Part 2, we show Ui0 (ˆ x− i |σ) = 0 ∀i ∈ Na . The argument proving Ui0 (ˆ x+ i |σ) = 0 ∀i ∈ Nb is analogous and omitted. Suppose, towards a first contradiction, that Ui0 (ˆ x− i |σ) < 0 for some i ∈ Na . By Part 1, x ˆi > xm . Hence, 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Ui (ˆ xi |σ) < Ui (ˆ xi −|σ), ˆ pi (ˆ xi |ˆ xi ) = x ˆi and x ˆi −  ∈ A(ˆ xi |σ), which contradicts x ˆi being part of x that induces SMPE σ. Suppose now, towards a second contradiction, that Ui0 (ˆ x− ˆi > xm , i |σ) > 0 for some i ∈ Na . Using (A6) and x   x ˆi − xi − 2δrnc,b (ˆ x− i |σ)(xm − xi )   −2 Ui0 (ˆ x+ x ˆi − xi − 2δrnc,b (ˆ x+ i |σ) = 1−δr (ˆ i |σ)(xm − xi ) . x+ |σ)

Ui0 (ˆ x− i |σ) =

−2 1−δrnc (ˆ x− i |σ) nc

(A15)

i

Because rnc,b (x− |σ) ≥ rnc,b (x+ |σ) for any x > xm , Ui0 (ˆ x− i |σ) > 0 implies 0 0 Ui0 (ˆ x+ xi |σ) < i |σ) > 0. Hence,  > 0 exists, such that, ∀ ∈ (0,  ), Ui (ˆ

Ui (ˆ xi + |σ), pi (ˆ xi + |ˆ xi ) = x ˆi and x ˆi +  ∈ A(ˆ xi + |σ), which contradicts ˆ that induces SMPE σ. x ˆi being part of x

45

For Part 3, taking limits from below and from above in (A6) and differentiating with respect to xi gives, ∀x ∈ X,  

  1 − 2δrnc,a (x− |σ) 0 − ∂   ∂xi Ui (x |σ) =  2 1 − 2δrnc,b (x− |σ) 1−δrnc (x− |σ)    2  1 − 2δrnc,a (x+ |σ) 1−δrnc (x+ |σ) 0 + ∂   ∂xi Ui (x |σ) =  2 1 − 2δrnc,b (x+ |σ) 1−δrnc (x+ |σ) 2 1−δrnc (x− |σ)

which, by rnc,a (x|σ) ≤ ra < hence rnc,g

(x− |σ)

implies, ∀i ∈ N ,

≤ rg and ∂ 0 − ∂xi Ui (x |σ)

1 2δ

if x ≤ xm if x > xm (A16) if x < xm if x ≥ xm

and rnc,b (x|σ) ≤ rb <

rnc,g

(x+ |σ)

> 0 and

1 2δ

≤ rg ∀x ∈ X ∂ 0 + ∂xi Ui (x |σ) > 0.

∀x ∈ D(σ) and

and ∀g ∈ {a, b},

To show Part 4, we first prove x ˆi < x ˆi+1 ∀i ∈ N \ {n}. By Part 1, x ˆi < xm ∀i ∈ Nb and x ˆi > xm ∀i ∈ Na . It thus suffices to prove x ˆi < x ˆi+1 ∀i ∈ Na \ {n} and ∀i ∈ Nb \ {m − 1}. We do so for i ∈ Na \ {n}. For i ∈ Nb \ {m − 1} the argument is similar and omitted. Suppose, towards a first contradiction, that x ˆi = x ˆi+1 for some i ∈ Na \{n}. By Part 1, x ˆi > xm , 0 which by Part 2 implies Ui0 (ˆ x− x− i |σ) = 0 and hence, by Part 3, Ui+1 (ˆ i |σ) > 0. 0 (ˆ The last inequality contradicts Ui+1 x− i |σ) = 0, which follows by Part 2 and

x ˆi = x ˆi+1 . Suppose, towards a second contradiction, that x ˆi+1 < x ˆi . By 0 (ˆ 0 x− |σ) < 0. Because Part 2, Ui+1 x− i+1 |σ) = 0, which by Part 3 implies Ui (ˆ i+1

x ˆi+1 > xm , 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Ui (ˆ xi+1 |σ) < Ui (ˆ xi+1 − |σ), pi (ˆ xi+1 |ˆ xi ) = x ˆi+1 and x ˆi+1 −  ∈ A(ˆ xi+1 |σ), which contradicts x ˆi being part ˆ that induces SMPE σ. of x To prove d(ˆ xi ) 6= d(ˆ xj ) for any pair of players {i, j} such that i 6= j, because x ˆi < x ˆi+1 ∀i ∈ N \ {n}, it suffices to rule out d(ˆ xi ) = d(ˆ xj ) ∀i ∈ Nb and ∀j ∈ Na . Suppose, towards a contradiction, that i ∈ Nb and j ∈ Na such that d(ˆ xi ) = d(ˆ xj ) exist. By Part 2, Uj0 (ˆ x− xi ) = d(ˆ xj ) j |σ) = 0. Because d(ˆ 0 x+ |σ) > and i ∈ Nb , rnc,b (ˆ x− x+ j |σ) > rnc,b (ˆ j |σ), which from (A15) implies Uj (ˆ j

0. Hence, 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Uj (ˆ xj |σ) < Uj (ˆ xj + |σ), pj (ˆ xj + |ˆ xj ) = x ˆj and x ˆj +  ∈ A(ˆ xj + |σ), which contradicts x ˆj being part ˆ that induces SMPE σ. of x  ˆ that constitutes an SMPE or is produced Returning to Case 4, for any x by Algorithm 1, define iteratively, for t ∈ {0, . . . , n−1} starting with ixˆ (0) = m, ixˆ (t) =

arg min i∈N \{ix ˆ (0),...,ix ˆ (t−1)}

46

d(ˆ xi )

(A17)

with the equal sign justified by d(ˆ xi ) 6= d(ˆ xj ) for any pair of players {i, j} ˆ for which we define ixˆ . ixˆ (t) is index of player with (t + 1)th smallest in x ˆ , starting from t = 0. Using ixˆ define for t ∈ {0, . . . , n − 1} d(ˆ xi ) in x o(ˆ x, t) = (ixˆ (0), ixˆ (1), . . . , ixˆ (t))

(A18)

and write o(ˆ x, t) = o(ˆ x0 , t) if and only if ixˆ (k) = ixˆ 0 (k) ∀k ∈ {0, . . . , t}. ˆ , so that d(ˆ o(ˆ x, n−1) is the set of players in N ordered by d(ˆ xi ) in x xixˆ (k) ) < d(ˆ xixˆ (k+1) ) ∀k ∈ {0, . . . , n − 2}. ˆ o induces Lemma A2. Assume δ ∈ (0, 1), 1 > 2δra and 1 > 2δrb . If x ˆ then o(ˆ ˆ ∈ X, SMPE σ and x x, t0 ) = o(ˆ xo , t0 ) for some t0 ∈ {0, . . . , n − 1} ˆoixˆ o (t) ∀t ∈ {0, . . . , t0 }. implies x ˆixˆ (t) = x ˆ produced by Algorithm 1. ˆ o that induces SMPE σ o and x ˆ∈X Proof. Fix x Suppose o(ˆ xo , t0 ) = o(ˆ x, t0 ) for some t0 ∈ {0, . . . , n − 1}. The proof proceeds by induction on t. For t = 0, we have ixˆ (0) = ixˆ o (0) = m and we know x ˆom = x ˆm = xm . Suppose that x ˆixˆ (t00 ) = x ˆoixˆ o (t00 ) ∀t00 ∈ {0, . . . , t} for some t < t0 . We need to show x ˆixˆ (t+1) = x ˆoixˆ o (t+1) . Because o(ˆ xo , t0 ) = o(ˆ x, t0 ) and t + 1 ≤ t0 , let us use only the ixˆ indexing. Denote j 0 = ixˆ (t) and j 00 = ixˆ (t + 1). We need to show x ˆj 00 = x ˆoj00 . Assume that j 00 ∈ Na . When j 00 ∈ Nb , the proof is similar and omitted. Denote Nj 0 = ∪ti=0 ixˆ (i) and Nj 00 = N \ Nj 0 . xoj00 ). Because By definition of j 0 and j 00 , d(ˆ xj 0 ) < d(ˆ xj 00 ) and d(ˆ xoj0 ) < d(ˆ x ˆixˆ (t00 ) = x ˆoixˆ (t00 ) ∀t00 ∈ {0, . . . , t}, we know x ˆi = x ˆoi ∀i ∈ Nj 0 , so that d(ˆ xi ) < d(ˆ xj 0 ) and d(ˆ xoi ) < d(ˆ xoj0 ) ∀i ∈ Nj 0 \ {j 0 }. From o(ˆ xo , t + 1) = o(ˆ x, t + 1), we xj 00 ) < d(ˆ know j 00 = ixˆ (t + 1) = ixˆ o (t + 1), so that d(ˆ xi ) and d(ˆ xoj00 ) < d(ˆ xoi ) ∀i ∈ Nj 00 \ {j 00 }. From these rnc,a (x|σ o ) =

P

and rnc,b (x|σ o ) =

P

i∈Nj 00 ∩Nb ri o− o o o o 0 ∀x ∈ (da (ˆ xj 0 ), x ˆj 00 ) ⊂ D(σ ). Using (A6) and Uj 00 (ˆ xj 00 |σ ) = 0 from Lemma P o A1 part 2, x ˆj 00 = xj 00 + 2δ i∈N 00 ∩Nb ri (xm − xj 00 ). j To calculate x ˆj 00 , Algorithm 1 drops player j 0 in step t, which means the algorithm uses, in step t + 1 when j 00 is dropped and x ˆj 00 set, Pt+1 = Nj 00 .

This gives rt+1,b = Clearly, x ˆj 00 =

P

i∈Nj 00 ∩Nb ri

i∈Nj 00 ∩Na ri

and x ˆj 00 = xj 00 +2δ

x ˆoj00 .

P

i∈Nj 00 ∩Nb ri (xm −xj 00 ).



ˆ o that induces SMPE σ o . We need to show Returning to Case 4, fix x

47

ˆ Suppose x ˆ For t ∈ {0, . . . , n − 1} define ˆ o ∈ X. ˆo ∈ x / X. ˆ t = {ˆ ˆ x, t) = o(ˆ X x ∈ X|o(ˆ xo , t)}.

(A19)

ˆ t is the set of profiles of strategic bliss points from Algorithm 1 that satisfy X ˆ t0 , then ˆ ∈ X ixˆ (k) = ixˆ o (k) for all k ∈ {0, . . . , t}. By Lemma A2, if x ˆ t+1 ⊆ X ˆ t ∀t ∈ {0, . . . , n − 2}. ˆo ∀t ∈ {0, . . . , t0 }. Clearly, X x ˆi (t) = x ix ˆ o (t)

ˆ x

ˆ X ˆ 0 = X. ˆ From x ˆ we also ˆo ∈ Because x ˆom = xm and x ˆm = xm ∀ˆ x ∈ X, / X, ˆ n−1 = ∅; if X ˆ n−1 6= ∅ we would have o(ˆ have X x, n − 1) = o(ˆ xo , n − 1) for ˆ n−1 and hence, by Lemma A2, x ˆ=x ˆo. ˆ∈X x ˆ t 6= ∅ and X ˆ t+1 = ∅ and fix x ˆ t . Clearly, ˆ ∈ X Now pick t such that X ˆ t . Denote t ∈ {0, . . . , n−2} and o(ˆ x, t) = o(ˆ xo , t) follows from definition of X j 0 = ixˆ (t) = ixˆ o (t), ja00 = ixˆ (t + 1), jo00 = ixˆ o (t + 1), Nj 0 = ∪ti=0 ixˆ (i) and Nj 00 = N \ Nj 0 . xi ) < d(ˆ xj 0 ) < d(ˆ xja00 ) < d(ˆ By definition of j 0 and ja00 , d(ˆ xj ) ∀i ∈ Nj 0 \ {j 0 } xoj ) ∀i ∈ xojo00 ) < d(ˆ xoi ) < d(ˆ xoj0 ) < d(ˆ and ∀j ∈ Nj 00 \ {ja00 }. Similarly, d(ˆ

Nj 0 \ {j 0 } and ∀j ∈ Nj 00 \ {jo00 }. From Lemma A2 and o(ˆ x, t) = o(ˆ xo , t), we

also know x ˆi = x ˆoi ∀i ∈ Nj 0 . o i∈Nj 00 ∩Nb ri i∈Nj 00 ∩Na ri and rnc,b (x|σ ) = o o o o o ∀x ∈ (db (ˆ xjo00 ), db (ˆ xj 0 )) ∪ (da (ˆ xj 0 ), da (ˆ xjo00 )) ⊂ D(σ ). Also, Algorithm 1 drops player j 0 in step t, which means it uses, in step t + 1 when ja00 is dropped

From these rnc,a (x|σ o ) =

P

P

and x ˆja00 set, Pt+1 = Nj 00 . This gives rt+1,a = P i∈N 00 ∩Nb ri .

P

i∈Nj 00 ∩Na ri

and rt+1,b =

j

We now show d(ˆ xja00 ) = d(ˆ xojo00 ). Suppose, towards a first contradiction,

o 00 that d(ˆ xojo00 ) < d(ˆ xja00 ). From Lemma A1 part 2, Uj0o00 (ˆ xo− j 00 |σ ) = 0 if jo ∈ Na

o 00 and Uj0o00 (ˆ xo+ j 00 |σ ) = 0 if jo ∈ Nb . Using (A6), we get

o

o

x ˆojo00

 P  xjo00 + 2δ i∈N 00 ∩N ri (xm − xjo00 ) if jo00 ∈ Na b j = P 00  x 00 + 2δ jo i∈N 00 ∩Na ri (xm − xjo00 ) if jo ∈ Nb

(A20)

j

Algorithm 1 in step t + 1 calculates x ˆjo00 ,t+1 and x ˆja00 ,t+1 and, since ja00 is dropped and x ˆja00 set, we know d(ˆ xja00 ) ≤ d(ˆ xjo00 ,t+1 ). Because the algorithm in step t + 1 uses Pt+1 = Nj 00 , clearly x ˆjo00 ,t+1 = x ˆojo00 and hence d(ˆ xja00 ) ≤ d(ˆ xojo00 ), which yields the desired contradiction. Suppose now, towards a second

48

xojo00 ). From Algorithm 1, contradiction, that d(ˆ xja00 ) < d(ˆ

x ˆja00

 P  xja00 + 2δ i∈N 00 ∩N ri (xm − xja00 ) if ja00 ∈ Na b j = 00  x 00 + 2δ P ja i∈N 00 ∩Na ri (xm − xja00 ) if ja ∈ Nb

(A21)

j

Because d(ˆ xoj0 ) = d(ˆ ˆja00 in (A6) to show that xj 0 ) < d(ˆ xja00 ), we can use x Uj0a00 (ˆ xja00 |σ o ) = 0. Assume ja00 ∈ Na . When ja00 ∈ Nb the argument is similar

ˆoja00 . and omitted. From ja00 ∈ Nj 00 , we have d(ˆ xojo00 ) < d(ˆ xoja00 ) and hence x ˆja00 < x

Uj0a00 (ˆ xja00 |σ o ) = 0 and Uj00a00 (x|σ o ) < 0 ∀x ∈ D(σ o ) from Lemma 2 part 4 then xja00 +|σ o ), xja00 |σ o ) > Uja00 (ˆ imply that 0 > 0 exists, such that, ∀ ∈ (0, 0 ), Uja00 (ˆ ˆoja00 xja00 + |σ o ), which contradicts x ˆja00 ∈ A(ˆ xoja00 ) = x ˆja00 +  and x xja00 + |ˆ pja00 (ˆ ˆ o that induces SMPE σ o . being part of x Having shown d(ˆ xja00 ) = d(ˆ xojo00 ), Algorithm 1 in step t + 1 calculates

x ˆja00 ,t+1 and x ˆjo00 ,t+1 and, since ja00 is dropped and x ˆja00 set, d(ˆ xja00 ) = d(ˆ xja00 ,t+1 ).

Because the algorithm in step t + 1 uses Pt+1 = Nj 00 , x ˆjo00 ,t+1 = x ˆojo00 so that ˆ exists, such that, ∀k ∈ {0, . . . , t}, ˆ0 ∈ X d(ˆ xja00 ,t+1 ) = d(ˆ xjo00 ,t+1 ). Thus x ixˆ (k) = ixˆ 0 (k) and ixˆ 0 (t + 1) = jo00 , created by dropping jo00 instead of ja00 in step t + 1. Because o(ˆ x, t) = o(ˆ xo , t) = o(ˆ x0 , t) and ixˆ 0 (t + 1) = jo00 , ˆ t+1 , a contradiction to ˆ0 ∈ X o(ˆ x0 , t + 1) = o(ˆ xo , t + 1), which implies x ˆ t+1 = ∅. X 

A1.8

Proof of Lemma 5

By Lemma 4, it suffices to show the lemma only for δ ∈ (0, 1), 1 > 2δra and 1 > 2δrb ; if δ = 0 or δ ∈ (0, 1) and 1 ≤ 2δrg for some g ∈ {a, b}, then ˆ . Fix x ˆ from Algorithm 1 applied to G with Algorithm 1 produces unique x 0 ˆ produced by the algorithm exists. x and assume another x ˆ . In step 0, the We follow the steps of Algorithm 1 when producing x algorithm sets x ˆm = xm . From 1 > 2δra and 1 > 2δrb , Rt = ∅ for any ˆ 0 6= x ˆ , t0 exists, such that remaining step t ∈ {1, . . . , n − 1}. Because x the algorithm in step t0 calculates x ˆi0 ,t0 and x ˆj 0 ,t0 with d(ˆ xi0 ,t0 ) = d(ˆ xj 0 ,t0 ), drops i0 and retains j 0 . Suppose t0 is the first such step, that is, in all steps t ∈ {0, . . . , t0 − 1} the algorithm uniquely selects a player to drop. Assume i0 ∈ Na . When i0 ∈ Nb the argument is similar and omitted. We start construction of the claimed perturbation x() by setting xi () =

49

xi ∀i ∈ N \ {i0 } and xi0 () = xi0 − .30 Because xi0 −1 < xi0 , ¯ > 0 exists, such that, ∀ ∈ (0, ¯), xi0 −1 () < xi0 (). Clearly, lim→0 x() = x. We claim that ¯ > 0 exists, such that, ∀ ∈ (0, ¯), Algorithm 1 applied to x() drops players in the same order as Algorithm 1 applied to x, uniquely selects player i0 to ˆ () = x ˆ. ˆ () such that lim→0 x drop in step t0 , and produces x To see that players are dropped in the same order for x and x(), we know that in any step t ∈ {0, . . . , t0 − 1} Algorithm 1 applied to x uniquely selects a player to drop and does not drop player i0 . This implies that, ∀t ∈ {0, . . . , t0 − 1}, i ∈ Pt such that d(ˆ xi,t ) < d(ˆ xi0 ,t ) = d(xi0 )(1 − 2δrt,b ) exists. Because the perturbation affects only the bliss point of player i0 , we have, ∀t ∈ {0, . . . , t0 − 1}, x ˆi,t () = x ˆi,t ∀i ∈ Pt \ {i0 } and d(ˆ xi0 ,t ()) = (d(xi0 ) − )(1 − 2δrt,b ). Clearly, ¯ > 0 exists, such that, ∀ ∈ (0, ¯) and ∀t ∈ {0, . . . , t0 − 1}, i ∈ Pt such that d(ˆ xi,t ()) < d(ˆ xi0 ,t ()) exists. That is, players are dropped in the same order for x and x() in steps t ∈ {0, . . . , t0 −1}. The same holds for steps t ∈ {t0 +1, . . . , n−1}, because the perturbation does not affect the bliss points of any of the players still in the algorithm in these steps. What remains is to show that Algorithm 1 applied to x() drops player i0 in step t0 . To see this, we know that d(ˆ xi0 ,t0 ()) < d(ˆ xi0 ,t0 ), d(ˆ xi0 ,t0 ) = d(ˆ xj 0 ,t0 ) and d(ˆ xj 0 ,t0 ) = d(ˆ xj 0 ,t0 ()). This implies d(ˆ xi0 ,t0 ()) < d(ˆ xj 0 ,t0 ()) so that i0 is dropped in step t0 . Because d(ˆ xi0 ,t0 ()) < d(ˆ xj 0 ,t0 ()), the algorithm uniquely selects player i0 to drop in step t0 and since the perturbation affects only the ˆ () = x ˆ. bliss point of player i0 , clearly lim→0 x ˆ until step t0 , the first step at We followed Algorithm 1 when producing x which the algorithm offers an option regarding the player to drop. At that point we constructed x() such that the algorithm applied to x() drops a unique player in step t0 and the order of players dropped is the same for x and x(). We can now proceed iteratively, find step t00 > t0 , the second step of the algorithm applied to x at which it gives an option regarding the player to drop, and set xi00 () = xi00 −  in x() for player i00 dropped in step t00 . The order of players dropped again remains the same and the algorithm ˆ (). drops a unique player i00 in step t00 when constructing x  30

If i0 ∈ Nb the perturbation required is xi0 () = xi0 + .

50

A1.9

Proof of Lemma 6

Conditions G1 and G2 clearly hold for δ = 0. In both conditions, the right hand side is strictly greater than unity, the left hand side is equal to unity for δ = 0 and is increasing in δ.

A1.10



Proof of Lemma 7

To prove Part 1, that G1 implies G2 when ri ≤ ri+1 ∀i ∈ {1, . . . , n−3 2 }, we have ∀i ∈ {1, . . . , n−3 2 } and ∀j ∈ {1, . . . , i} e 1 1 − 2δr e 2 xm − xi 3 xm − xj 1 − 2δrj−1 i ≤ ≤ ≤ . e 1 − 2δrje 1 − 2δri+1 xm − xi+1 xm − xi+1

(A22)

3

2

1 xm −xj xm −xi+1 decreasing in j. To see ≤, note 1−2δre 1−2δrie that 1−2δri−1 holds ∀i ∈ {1, . . . , n−3 e ≤ 1−2δr e 2 }. It rewrites as (ri+1 −ri )(1− i i+1 n−3 e 2δri ) + 2δri ri+1 ≥ 0 for i ∈ {1, . . . , 2 } and clearly holds when ri ≤ ri+1 1 ∀i ∈ {1, . . . , n−3 2 }. Subsequently ≤ must hold for any j ∈ {1, . . . , i}. The

≤ is condition G1 . ≤ follows from

outer inequality in (A22) is condition G2 . To prove Part 2, that G1 implies G2 when xi − xi−1 ≤ xi+1 − xi ∀i ∈ {2, . . . , n−3 2 } and

1 1−2δr1

≤

xm −x1 xm −x2 ,

we have ∀j ∈ {2, . . . , n−3 2 } and ∀i ∈

{j, . . . , n−3 2 } e 1 xm − xj−1 2 xm − xj 3 xm − xj 1 − 2δrj−1 ≤ . ≤ ≤ e 1 − 2δrj xm − xj xm − xj+1 xm − xi+1 1

3

≤ is condition G1 . ≤ follows from that

xm −xi−1 xm −xi

≤

xm −xi xm −xi+1

xm −xj xm −xi+1

(A23) 2

increasing in i. To see ≤, note

holds ∀i ∈ {2, . . . , n−3 2 }. It rewrites as (xm −

xi )(di+1 − di ) + di+1 di ≥ 0 for i ∈ {2, . . . , n−3 2 } where di = xi − xi−1 and clearly holds when xi+1 − xi = di+1 ≥ di = xi − xi−1 . The outer equality in (A23) is condition G2 except when j = 1 and i ∈ {1, . . . , n−3 2 }. For these values of j and i, G2 reads

1 1−2δr1

≤

xm −x1 xm −xi+1

and holds, because the right

hand side of the inequality is increasing in i and we have

A1.11

1 1−2δr1

≤

xm −x1 xm −x2 .



Proof of Lemma 8

Symmetric G with n = 3 induces pairwise moderation because it is symmetric and the parametric conditions in Definition 8 apply only for n ≥ 5.

51

For strongly symmetric G, rie =

i n

and xm − xi = ( n+1 2 − i)(xm − xm−1 )

for any i ∈ {1, . . . , n−1 2 }. Substituting into G1 in Definition 8, which by Lemma 7 suffices, gives δ ≤

A1.12

n n+1 .



Proof of Proposition 4

ˆ = x. Hence, assume δ ∈ (0, 1). To show that When δ = 0 in Part 1, clearly x ˆ and that Algorithm 1 produces, by pairwise paths, 2(n−1)/2 distinct sets of x ˆ produced has any of these constitutes an SMPE, we first show that any x a special structure. Recall that the algorithm starts with step 0 in which it drops player m and that it finishes in n − 1 steps. We want to show that, for any pairwise moderation inducing G, the algorithm in every odd step t ∈ {1, 3, . . . , n − 2} gives an option to drop players {m − t0 , m + t0 } where t0 =

t+1 2 .

Dropping one of the players we want the other player to be

dropped in the subsequent step t + 1. This implies that in any odd step t, the number of players still in the algorithm is even and half come from Na while the other half come from Nb . Suppose the algorithm exhibited such behaviour in all steps until step t ∈ {1, 3, . . . , n−4} and hence already dropped players {m−t0 +1, . . . , m+t0 −1}. e In t, the algorithm computes x ˆi,t = xi + 2δrm−t 0 (xm − xi ) for all players still

in the algorithm and gives an option to drop players {m−t0 , m+t0 }. Assume, without loss of generality, that m + t0 ∈ Na is dropped. Then in t + 1, the algorithm computes, for the retained players, e x ˆi,t+1 = xi + 2δrm−t 0 (xm − xi )

if i ∈ Na

e x ˆi,t+1 = xi + 2δrm−t 0 −1 (xm − xi )

if i ∈ Nb .

(A24)

The algorithm at this point drops the player with x ˆi,t+1 closest to xm . Two possible candidates are m − t0 ∈ Nb not dropped in t and m + t0 + 1 ∈ Na . We want the algorithm to drop m − t0 .31 This is the case whenever xm − x ˆm−t0 ,t+1 ≤ x ˆm+t0 +1,t+1 − xm . This inequality rewrites as e 1 − 2δrm−t xm − xm−t0 −1 0 −1 ≤ e 1 − 2δrm−t0 xm − xm−t0

(A25)

where we have already used xm+t0 +1 − xm = xm − xm−t0 −1 , which follows 31 No condition is necessary for the final odd step, n − 2, which is followed by the final step of the algorithm with only one player remaining.

52

from the symmetry of G. Setting i = m−t0 −1 and using t ∈ {1, 3, . . . , n−4}, we have i ∈ {1, . . . , n−3 2 }. (A25) is thus equivalent to condition G1 . The pairwise path through the algorithm from Definition 9 then ensures that the ˆ arises even when (A25) holds with equality. Because desired structure of x the algorithm goes through

n−1 2

odd steps, and each gives an option to drop ˆ evaluates at 2(n−1)/2 . one of two players, the multiplicity of x ˆ produced constitutes an SMPE, we show that it satTo see that any x ˆ produced isfies condition S when G induces pairwise moderation. Fix x by pairwise path through Algorithm 1 for pairwise moderation inducing G and induced σ. Take player i ∈ {1, . . . , n−1 2 } = Nb . For players in Na the argument is symmetric and omitted. Suppose the algorithm dropped player i producing x ˆi . The set of players dropped subsequently is a subset of {1, . . . , i − 1} ∪ {dIa (i), . . . , n}. Only these players can produce points in N D(σ) in the interval [xi , x ˆi ], that is points defining Si (σ) = N D(σ)∩(xi , x ˆi ) used in condition S. Furthermore, from (A6) we know that for any j 0 ∈ Nb 0 x+ |σ)]. Thus, we only need to check and i ∈ Nb , sgn [Ui0 (ˆ x− j |σ)] = sgn [Ui (ˆ j

condition S for those points in Si (σ) induced by players j ∈ {dIa (i), . . . , n} being dropped by Algorithm 1. If condition S holds for these points, it must hold for all points in Si (σ). For any j ∈ {dIa (i), . . . , n} Algorithm 1, by pairwise moderation, produces either x ˆj = xj + 2δrdeI (j) (xm − xj ) or x ˆj = xj + 2δrdeI (j)−1 (xm − xj ). By b

b

the symmetry of G we can map x ˆj below xm into db (ˆ xj ) = xj 0 +2δrje0 (xm −xj 0 ) or db (ˆ xj ) = xj 0 + 2δrje0 −1 (xm − xj 0 ) for j 0 = dIb (j) ∈ {1, . . . , i}. Condition S evaluated for i ∈ Nb and db (ˆ xj ) becomes xj 0 + 2δrje0 (xm − xj 0 ) − xi − 2δrje0 −1 (xm − xi ) ≤ 0 xj 0 + 2δrje0 −1 (xm − xj 0 ) − xi − 2δrje0 −1 (xm − xi ) ≤ 0

(A26)

where we used rnc,a (db (ˆ xj )− |σ) = rje0 −1 ; when j is dropped by the algorithm, j 0 − 1 players in Na remain on the non-constant part of their strategy as we approach db (ˆ xj ) from below. When j 0 = i, i must have been dropped by Algorithm 1 first out of pair {i, dIa (i)} of players. This implies db (ˆ xj ) = xj 0 + 2δrje0 −1 (xm − xj 0 ) so that only the second line of (A26) applies and the left hand side equals 0. When j 0 < i both lines of (A26) apply but from rje0 (xm − xj 0 ) > rje0 −1 (xm − xj 0 ), if the first line holds the second one must hold as well. The first line rewrites

53

as

1 − 2δrje0 −1 1−

2δrje0

≤

xm − xj 0 xm − xi

(A27)

0 and needs to hold for i ∈ {2, . . . , n−1 2 } and j ∈ {1, . . . , i − 1}, where we have

already adjusted for the fact that we only need to take care of cases when i > j 0 . Rewriting the condition as e 1 − 2δrj−1 xm − xj ≤ 1 − 2δrje xm − xi+1

(A28)

∀i ∈ {1, . . . , n−3 2 } and ∀j ∈ {1, . . . , i}, we get condition G2 . To summarize, when G induces pairwise moderation, conditions G1 and ˆ produced by a G2 hold by Definition 8. Condition G1 implies that any x pairwise path through Algorithm 1 has a special structure that allowed us to use condition G2 to show that condition S holds, which by Proposition 1 ˆ constitutes an SMPE. implies that σ induced by x What remains is to show that Ui is single peaked on X ∀i ∈ N . For m we already know the claim is true by Lemma 2 part 5. Consider i ∈ Na omitting again the symmetric argument for players in Nb . By condition S, Ui is single peaked for x ≥ xm . For x ≤ xm and any x ∈ D(σ), from (A6) we need x − xi − 2δrnc,a (x|σ)(xm − xi ) ≤ 0. This follows from x ≤ xm and 1 − 2δrnc,a (x|σ) > 0 as rnc,a (x|σ) ≤

A1.13

1 2

for any symmetric G.



Proof of Proposition 5

ˆ produced by a pairwise path through Algorithm 1. Denote by ti Fix x ∀i ∈ N step of the algorithm at which i has been dropped. Note that ti is decreasing in i for i ∈ Nb ∪ {m} and increasing in i for i ∈ Na ∪ {m}. We construct the perturbation of x by  > 0, x(), the proposition postulates as x() = {x1 +

 t1 , . . . , xm−1

+

 tm−1 , xm , xm+1

−

 tm+1 , . . . , xn

−

 tn }

(A29)

where lim→0 x() = x is immediate. Note also that ¯ > 0 exists, such that, ∀ ≤ (0, ¯), xm−1 () < xm < xm+1 () and hence xi () < xi+1 () ∀i ∈ N \{n}. We now show that Algorithm 1 for G() = hn, x(), r, δ, Xi produces ˆ () and that the order in which players are dropped during construcunique x ˆ () and x ˆ is the same. Recall that, when producing x ˆ , Algorithm tion of x 1 in step t ∈ {1, 3, . . . , n − 2} dropped one of players from {m − t0 , m + t0 }, 54

where t0 =

t+1 2 ,

and the other player in step t + 1. We need to show the ˆ (). algorithm (uniquely) mimics this behaviour when constructing x Assume the algorithm has done so until step t ∈ {1, 3, . . . , n − 2} and

hence has already dropped players {m − t0 + 1, . . . , m + t0 − 1}. In t, the  e e algorithm computes x ˆi,t () = xi + 2δrm−t 0 (xm − xi ) + t (1 − 2δrm−t0 ) ∀i ∈ Nb i  e e and x ˆi,t () = xi + 2δrm−t 0 (xm − xi ) − t (1 − 2δrm−t0 ) ∀i ∈ Na . Only players i

m − t0 ∈ Nb or m + t0 ∈ Na can be dropped in t and we need to show the former is dropped if tm−t0 < tm+t0 and the latter is dropped if tm−t0 > tm+t0 . Calculating d(ˆ xm−t0 ,t ()) and d(ˆ xm+t0 ,t ()), e d(ˆ xm−t0 ,t ()) = d(xm−t0 )(1 − 2δrm−t 0) −

 tm−t0 (1

e d(ˆ xm+t0 ,t ()) = d(xm+t0 )(1 − 2δrm−t 0) −

 tm+t

e − 2δrm−t 0)

e (1 − 2δrm−t 0 ). 0

(A30)

e Because d(xm−t0 ) = d(xm+t0 ) and 1 − 2δrm−t 0 > 0, tm+t0 < tm−t0 implies the

required d(ˆ xm+t0 ,t ()) < d(ˆ xm−t0 ,t ()) and tm+t0 > tm−t0 implies the required d(ˆ xm+t0 ,t ()) > d(ˆ xm−t0 ,t ()). We now show that from the pair of players {m − t0 , m + t0 }, the one not dropped in step t is uniquely dropped in step t + 1. Assume, without loss of generality, that m + t0 ∈ Na is dropped in step t. In step t + 1 the algorithm computes, for the retained players, e x ˆi,t+1 () = xi + 2δrm−t 0 −1 (xm − xi ) + e x ˆi,t+1 () = xi + 2δrm−t 0 (xm − xi ) −

 ti (1

 ti (1

e − 2δrm−t 0 −1 )

e − 2δrm−t 0)

if i ∈ Nb if i ∈ Na .

which, for the pair of players {m − t0 , m + t0 + 1} that can be dropped, gives e d(ˆ xm−t0 ,t+1 ()) = d(xm−t0 )(1 − 2δrm−t 0 −1 ) − e d(ˆ xm+t0 +1,t+1 ()) = d(xm+t0 +1 )(1 − 2δrm−t 0) −

 tm−t0 (1

e − 2δrm−t 0 −1 )

 tm+t0 +1 (1

e − 2δrm−t 0 ).

e e We know d(xm−t0 )(1 − 2δrm−t 0 −1 ) ≤ d(xm+t0 +1 )(1 − 2δrm−t0 ) because G

induces pairwise moderation. To show d(ˆ xm−t0 ,t+1 ()) < d(ˆ xm+t0 +1,t+1 ()), e 1−2δrm−t 0 −1 tm−t0 e e rm−t 0 −1 < rm−t0 .

it thus suffices to show that tm−t0 < tm+t0 +1 and

>

e 1−2δrm−t 0 tm+t0 +1 ,

which follows from

ˆ and x ˆ (), dropped players Because Algorithm 1, when constructing x in an identical order, we have, for any i ∈ Na , x ˆi = xi + 2δr0 (xm − xi ) and x ˆi () = xi + 2δr0 (xm − xi ) −

 ti (1

55

− 2δr0 ), where r0 is the probability

the algorithm used in step ti . Clearly lim→0 x ˆi () = x ˆi ∀i ∈ Na . Using a ˆ () = x ˆ. similar argument for i ∈ Nb and noting x ˆm = x ˆm () shows lim→0 x ˆ () satisfies condition S, take player i ∈ {1, . . . , n−1 To show that x 2 } = Nb . For players in Na the argument is symmetric and omitted. The set of players subsequently dropped is {1, . . . , i − 1} ∪ {dIa (i), . . . , n}. Using a ˆ () induces σ(), we similar argument as in the proof of Proposition 4, when x only need to check condition S for those points in Si (σ()) induced by players j ∈ {dIa (i), . . . , n} being dropped by Algorithm 1. For any j ∈ {dIa (i), . . . , n} Algorithm 1 produces either x ˆj () = xj + 2δrdeI (j) (xm − xj ) − tj (1 − 2δrdeI (j) ) b

b

or x ˆj () = xj +2δrdeI (j)−1 (xm −xj )− tj (1−2δrdeI (j)−1 ). Mapping x ˆj () below b

b

xm and using j 0 = dIb (j) gives db (ˆ xj ()) = xj 0 + 2δrje0 (xm − xj 0 ) + tj (1 − 2δrje0 ) or db (ˆ xj ()) = xj 0 +2δrje0 −1 (xm −xj 0 )+ tj (1−2δrje0 −1 ). Condition S evaluated for i ∈ Nb and db (ˆ xj ()) becomes xj 0 + 2δrje0 (xm − xj 0 ) − xi − 2δrje0 −1 (xm − xi ) +   1−2δrje0 −1 1−2δrje0 − ≤0  tj ti xj 0 + 2δrje0 −1 (xm − xj 0 ) − xi − 2δrje0 −1 (xm − xi ) +   1−2δrje0 −1 1−2δrje0 −1  − ≤0 tj ti

(A31)

and we know, since G induces pairwise moderation, that it holds ∀i ∈ 0 {1, . . . , n−1 2 } and ∀j ∈ {1, . . . , i} when  = 0. Noting that ti < tj and

rje0 −1 < rje0 , each of the terms in the square brackets in the condition is ˆ () as well. non-positive, showing that condition S holds for x 

A1.14

Proof of Proposition 6

Throughout the proof assume G is strongly symmetric with n ≥ 5 and δ≥

n n+1 .

Algorithm 1 in step 0 sets x ˆm = xm and in step 1 gives an option

to drop one of the players in {m − 1, m + 1}. For these two players 1 x ˆm−1,1 = xm−1 + 2δ n−1 2 n (xm − xm−1 ) 1 x ˆm+1,1 = xm+1 + 2δ n−1 2 n (xm − xm+1 )

(A32)

and d(ˆ xm−1,1 ) = d(ˆ xm+1,1 ) follows from the strong symmetry of G. Assume the algorithm drops m − 1. The argument for m + 1 is similar and omitted. We claim the algorithm in steps t ∈ {2, . . . , n−1 2 } drops all the remaining 56

players from Nb . Suppose the algorithm has been dropping players from Nb until step t−1 ∈ {1, . . . , n−3 2 }. We need to show that it drops the player from 0 0 Nb in step t ∈ {2, . . . , n−1 2 }. From Pt = Na ∪ {1, . . . , i } where i =

rt,a =

n−1 1 2 n

and rt,b =

i0 n1 .

Since only players in

{i0 , m + 1}

n+1 2

− t,

can be dropped,

we need to show d(ˆ xm+1,t ) ≥ d(ˆ xi0 ,t ) for 0

x ˆm+1,t = xm+1 + 2δ in (xm − xm+1 ) 0 x ˆi0 ,t = xi0 + 2δ n−1 2n (xm − xi )

(A33)

∀i0 ∈ {1, . . . , n−3 xm+1,t ) = 2 }. Denoting xi − xi+1 = l > 0 ∀i ∈ N \ {n}, d(ˆ 0

n−1 0 l(1 − 2δ in ) and d(ˆ xi0 ,t ) = ( n+1 xm+1,t ) ≥ d(ˆ xi0 ,t ) 2 − i )l(1 − δ n ) so that d(ˆ

is equivalent to δ ≥ drop

i0

n n+1 .

When δ =

n n+1 ,

the algorithm gives an option to

or m+1 and we assume the former player is dropped. When δ >

n n+1

the algorithm uniquely selects player i0 to drop. Because the algorithm drops all players from Nb in steps t ∈ {1, . . . , n−1 2 }, in steps t ∈ { n−1 2 + 1, . . . , n − 1} it drops all players from Na . The resulting ˆ thus satisfies x x ˆ m = xm , x ˆi = xi ∀i ∈ Na and x ˆi = xi + δ n−1 n (xm − xi ) ∀i ∈ Nb . To finish the proof of Part 1, what remains is to show that d(ˆ xi ) ∈ (0, d(xm−1 )) ∀i ∈ Nb . Because d(ˆ xi ) = d(xi )(1 − δ n−1 xi ) > 0 n ), d(ˆ ∀i ∈ Nb is immediate. To show d(ˆ xi ) < d(xm−1 ) = d(xm+1 ), it suffices to show d(ˆ x1 ) < d(xm+1 ) since d(ˆ xi ) ≤ d(ˆ x1 ) ∀i ∈ Nb . And d(ˆ x1 ) < d(xm+1 ) follows because in step

n−1 2

the algorithm dropped player 1 due to

d(ˆ x1, n−1 ) ≤ d(ˆ xm+1, n−1 ) < d(xm+1 ). 2 2 ˆ with x Fix x ˆm = xm , x ˆi = xi ∀i ∈ Na and x ˆi = xi + δ n−1 n (xm − xi ) ∀i ∈ Nb and σ it induces. By the strong symmetry of G we have N D(σ) = {x1 , . . . , xm−1 , x ˆ1 , . . . , x ˆm−1 , xm , . . .} and from definitions rnc,a (x|σ) =

j n

and rnc,b (x|σ) = 0 ∀x ∈ (xj , xj+1 ) where j ∈ {1, . . . , n−3 2 }. Furthermore, rnc,a (x|σ) =

n−1 2n

∀x ∈ (xm−1 , xm ) \ N D(σ). To prove Part 2, we need to

show σ constitutes an SMPE. For players in Na ∪ {m}, the optimality of their strategies is easy to see; xm is clearly optimal for m and for any i ∈ Na , Si (σ) = ∅ follows from xi = x ˆi and hence condition S holds for any i ∈ Na . Because condition S in general fails for players in Nb , we need to check condition N for these players. We now argue that it suffices to check condition N for player 1. To see this we first claim that Ui (x|σ) ≤ Ui (ˆ xi |σ) ∀x ∈ [ˆ x1 , x ˆi ] and ∀i ∈ Nb . The claim

57

follows from the piecewise strict concavity of Ui proven in Lemma 2 part 4, 0 x− |σ) = 0 ∀i ∈ N , which follows from the x ˆi < x ˆi+1 ∀i ∈ Nb , Ui0 (ˆ x+ b i |σ) = Ui (ˆ i 0 x− |σ)] ∀i ∈ N way Algorithm 1 constructs x ˆi , and sgn [Ui0 (ˆ x+ b j |σ)] = sgn [Ui (ˆ j

and ∀j ∈ Nb , which follows from rnc,a (x|σ) =

n−1 2n

∀x ∈ (xm−1 , xm ) \ N D(σ)

and inspection of (A6). Suppose now that condition N holds for player 1. This means U1 (x|σ) ≤ U1 (ˆ x1 |σ) ∀x ≤ x ˆ1 . From Lemma 2 part 5 and x ˆ1 < xm we know Um (x|σ) ≤ Um (ˆ x1 |σ) ∀x ≤ x ˆ1 . Using an argument similar to the one used to prove Lemma 1, we thus have Ui (x|σ) ≤ Ui (ˆ x1 |σ) ∀x ≤ x ˆ1 and ∀i ∈ Nb , or, using the claim above, Ui (x|σ) ≤ Ui (ˆ x1 |σ) ≤ Ui (ˆ xi |σ). Thus, if condition N holds for player 1 it must hold for all players in Nb . What remains is to show that condition N holds for player 1. The set of points in [x1 , x ˆ1 ] at which U1 is not differentiable is {x1 , x2 , . . . , xm−1 , x ˆ1 }. We first show that, for j ∈ {1, . . . , n−3 2 }, U1 has a unique local maximizer on (xj , xj+1 ), which we denote by x0j . Using rnc,a (x|σ) = ∀x ∈ (xj , xj+1 ) and ∀j ∈ {1, . . . , U10 (x|σ) = −

n−3 2 }

2

h

1 − δ nj

j n

and rnc,b (x|σ) = 0

in (A6) gives

i x − x1 − 2δ nj (xm − x1 ) .

(A34)

x0j = x1 + 2δ nj (xm − x1 ) is the local maximizer of U1 if x0j ∈ (xj , xj+1 ). We show that this is the case ∀j ∈ {1, . . . , n−3 2 }. Noting x1 = xm − xj = xm − ( n+1 2 and xj < x0j ⇔

− j)l, straightforward algebra shows δ >

n j−1 n−1 j .

x0j

n−1 2 l

and

< xj+1 ⇔ δ <

n n−1

The first inequality clearly holds. The right

hand side of the second inequality is increasing in j so it must hold for any n−3 n n−5 2 . Evaluation gives δ > n−1 n−3 and because, as is n n n−5 easily checked, n+1 > n−1 n−3 , shows that the inequality holds. We thus 0 0 have N1 (σ) = {x1 , x1 , x2 , x2 , . . . , xm−2 , x0m−2 , xm−1 , x ˆ1 }. We now make two

j if it holds for j =

claims that jointly imply that in order to check condition N for player 1, it suffices to ensure U1 (x0m−2 |σ) ≤ U1 (ˆ x1 |σ), that is, to check condition N only for J = 2. First, we claim that rewrites as δ ≥ n−3 2

n n−1

j− 21 j

reads δ ≥ ¯ holds when δ ≥ δ(n).

at j =

xj +xj+1 2

≤ x0j ∀j ∈ {1, . . . , n−3 2 }. The condition

, its right hand side is increasing in j and evaluated n n−4 n−1 n−3 .

Below we show that δ ≥

n n−4 n−1 n−3

indeed

Second, we claim that U10 (x|σ) > U10 (x + l|σ) ∀x ∈ (xj , xj+1 ) and ∀j ∈

58

{1, . . . , n−5 2 }. Using (A34) the condition is

−

i h 2 x − x1 − 2δ nj (xm − x1 ) 1 − δ nj

>−

i h (x − x ) 2 x + l − x1 − 2δ j+1 m 1 n 1 − δ j+1 n

(A35)

and rewrites as d(x) + d(x1 ) < l( nδ − j). Because d(x) ≤ ( n+1 2 − j)l and d(x1 ) =

n−1 2 l,

d(x) + d(x1 ) ≤ (n − j)l, so the inequality holds.

From the second claim, ∀j ∈ {1, . . . , n−5 2 } and any y ∈ [xj , xj+1 ], Z

xj+1

U1 (xj+1 |σ) − U1 (y|σ) =

U10 (z|σ)dz

y

Z

xj+1

≥

U10 (z + l|σ)dz

y

Z

xj+2

=

(A36)

U10 (w|σ)dw

y+l

= U1 (xj+2 |σ) − U1 (y + l|σ). From the first claim, ∀j ∈ {1, . . . , n−3 2 }, because U1 is quadratic on (xj , xj+1 ) and hence symmetric about x0j ≥

xj +xj+1 , 2

U1 (xj |σ) ≤ U1 (xj+1 |σ). Combin-

ing the inequalities we get 0 ≥ U1 (xj+1 |σ) − U1 (xj+2 |σ) ≥ U1 (y|σ) − U1 (y + l|σ)

(A37)

so that U1 (y + l|σ) ≥ U1 (y|σ) ∀y ∈ [xj , xj+1 ] and ∀j ∈ {1, . . . , n−5 2 }. Since U1 (x0m−2 |σ) ≥ U1 (y|σ) ∀y ∈ [xm−2 , xm−1 ] and y + l ∈ [xm−2 , xm−1 ] when y ∈ [xm−3 , xm−2 ], it must be the case that U1 (x0m−2 |σ) ≥ U1 (y|σ) ∀y ∈ [x1 , xm−1 ]. Hence, if we prove U1 (x0m−2 |σ) ≤ U1 (ˆ x1 |σ), we can conclude that U1 (x|σ) ≤ U1 (ˆ x1 |σ) ∀x ∈ N1 (σ), that is, that condition N holds for player 1. To prove U1 (x0m−2 |σ) ≤ U1 (ˆ x1 |σ), we evaluate condition h 2 N for J = i2. For 2 x ∈ (xm−1 , x ˆ1 ), c1 (x|σ) = x ˆ1 and T1 (x|σ) = − 1−δ n−1 x2 − c1 (x|σ)x . For 2n h 2 i 2 x 0 x ∈ (xm−2 , xm−1 ), c1 (x|σ) = xm−2 and T1 (x|σ) = − 1−δ n−3 2 − c1 (x|σ)x . Substitution into the condition gives h

ixˆ− 1 T1 (x|σ) +

h

i x− m−1 T1 (x|σ) 0+ = −

xm−1

xm−2

=

2n

1 (ˆ x1 − xm−1 )2 1 − δ n−1 2n 1 (xm−1 − x0m−2 )2 1 − δ n−3 2n 59

(A38)

n−1 n−1 n−3 n−3 n−1 0 and x ˆ1 −xm−1 = −l( n−3 2 −2δ 2n 2 ), xm−2 −xm−1 = −l( 2 −2δ 2n 2 ). − h i h ixˆ− xm−1 1 The condition reads T1 (x|σ) + + T1 (x|σ) 0+ ≥ 0, which after some xm−1 i h q xm−2 3 2 −n−7 n n−2 algebra rewrites as δ ≥ δ 0 (n) = n−3 2 n−1 − n −n . Checking that 3 (n−1)

1 > δ 0 (n) >

n n−4 n−1 n−3

∀n ≥ 5 is routine algebra. The proposition claims the

n n−4 n 0 first inequality and we required δ ≥ n−1 n−3 above. Because δ (n) < n+1 ¯ holds if and only if n = 5, δ ≥ δ(n) = max { n , δ 0 (n)} guarantees that n+1

condition N holds for all players in Nb and hence σ constitutes an SMPE. 

A1.15

Proof of Proposition 7

Throughout the proof assume G is strongly symmetric with n = 5 and 32 From Proposition 6, Algorithm 1 ¯ δ > n or with n ≥ 7 and δ ≥ δ(n). n+1

ˆ with x produces two x ˆi = xi ∀i ∈ Ng ∪ {m} and x ˆi = xi + δ n−1 n (xm − xi ) ∀i ∈ N \ (Ng ∪ {m}), where g ∈ {a, b}. From the proof of that proposition, ˆ produced by Algorithm 1 exists. no other x ˆ with g = a, that is x ˆ with x Fix x ˆi = xi ∀i ∈ Na ∪ {m} and x ˆi = xi +δ n−1 n (xm −xi ) ∀i ∈ Nb . For the other profile the argument is similar and omitted. We construct the perturbation of x by  > 0, x(), the proposition postulates as x() = {x1 , . . . , xm−2 , xm−1 + , xm , xm+1 , xm+2 , . . . , xn }

(A39)

where lim→0 x() = x is immediate.33 Clearly, ¯ > 0 exists, such that, ∀ ≤ (0, ¯), xm−1 () < xm . We now show that Algorithm 1 for G = hn, x(), r, δ, Xi produces a n−1 ˆ () with x unique x ˆm−1 () = xm−1 + δ n−1 n (xm − xm−1 ) + (1 − δ n ) and x ˆi () = x ˆi ∀i ∈ N \ {m − 1}. In step 1, the algorithm calculates, for the players in {m − 1, m + 1} that can in principle be dropped, n−1 x ˆm−1,1 () = xm−1 + δ n−1 n (xm − xm−1 ) + (1 − δ n )

x ˆm+1,1 () = xm+1 + δ n−1 n (xm − xm+1 ). 32

(A40)

n The perturbation of x we are about to construct is extremely simple when δ > n+1 ˆ for the unperturbed x, giving an option regarding as Algorithm 1 produces exactly two x n which player to drop only in the first step. When δ = n+1 , we would have to construct ¯ more complex perturbation of x. Since δ ≥ δ(n), which we need to show that condition N n n holds, implies δ > n+1 for any n ≥ 7, for n = 5 we assume δ > n+1 . 33 ˆ from Proposition 6 with g = b would be identical The perturbation required for x except for xm−1 () = xm−1 and xm+1 () = xm − .

60

Because d(xm−1 ) = d(xm+1 ), m − 1 is dropped with x ˆm−1 () = x ˆm−1,1 (). From the arguments presented in the proof of Proposition 6, it follows that the algorithm uniquely drops all the remaining players from Nb in steps n−1 t ∈ {2, . . . , n−1 2 } and all the players from Na in steps t ∈ { 2 +1, . . . , n−1}.

This implies x ˆi () = xi + δ n−1 ˆi () = xi n (xm − xi ) ∀i ∈ Nb \ {m − 1} and x ∀i ∈ Na . Clearly, x ˆi () = x ˆi ∀i ∈ N \ {m − 1} and since lim→0 x ˆm−1 () = ˆ () = x ˆ. x ˆm−1 , lim→0 x ˆ () induces σ() that supports SMPE. What remains is to show that x The argument is essentially identical to the one used to prove Proposition 6 and for space consideration we include here only the key steps. For any i ∈ Na condition S holds as Si (σ()) = ∅ and xm is optimal for m. That Ui (ˆ x1 ()|σ()) ≤ Ui (ˆ xi ()|σ()) ∀i ∈ Nb follows by the same argument as in the proof of Proposition 6 and hence condition N only needs to hold for player 1 for σ() to support SMPE. That the condition indeed holds when ¯ δ ≥ δ(n) follows again by the argument used in the proof of Proposition 6. The entire argument there relied only on the derivative of U1 and it ˆ , ∀x ∈ is easy to see that U10 (x|σ()) = U10 (x|σ), where σ is induced by x [x1 , x ˆ1 ] \ N D(σ) = [x1 (), x ˆ1 ()] \ N D(σ()).

61