Experimental Evidence of Bank Runs as Pure Coordination Failures Jasmina Arifovicy

Janet Hua Jiangzx

Yiping Xu{

March 19, 2013

Abstract This paper investigates how the level of coordination requirement, measured by a coordination parameter, a¤ects the occurrence of bank runs as a result of pure coordination failures in controlled laboratory environments. We …nd that the economy stays close or converges to the run (non-run) equilibrium for high (low) levels of coordination requirement. In addition, there is an indeterminacy region of the coordination parameter such that games with the coordination parameter lying in that region have varying coordination outcomes and exhibit persistent path dependence. We also …nd that the experimental economy may switch between the run and nonrun equilibria even if the economic fundamentals are kept constant. Finally, we show that the behavior of human subjects observed in the laboratory can be well accounted for by a version of the evolutionary algorithm that uses experimentation rates estimated from the experimental data. JEL Categories: D83, G20 Keywords: Bank Runs, Experimental Studies, Evolutionary Algorithm, Coordination Games

For their comments and suggestions, we thank three anonymous referees, the Editor, John Du¤y, Martin Oehmke and seminar participants at the International Monetary Fund, University of Winnipeg, the CEA Meeting, the ESA Asia Paci…c Meeting, the ESA North American Meeting, the Chicago Fed Workshop on Money, Banking, Payments and Finance, the LeeX International Conference on Theoretical and Experimental Macroeconomics at University Pompeu Fabra and the Conference on Experiments in Macroeconomics and Financial Economics at Columbia University. This research was funded by SSHRC, INET, the Humanities and Social Sciences of the Ministry of Education of China (grant number 09YJA790040) and the University of International Business and Economics 211 Grant. The views expressed in this paper are those of the authors. No responsibility for them should be attributed to the Bank of Canada. y Simon Fraser University. E-mail: [email protected]. z Corresponding author. x Bank of Canada, 234 Wellington Street, Ottawa, Ontario, K1A 0G9, Canada. E-mail: [email protected]. { University of International Business and Economics. E-mail: [email protected].

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1

Introduction

A bank run is the situation in which a large number of depositors, fearing that their bank will be unable to repay their deposits, simultaneously try to withdraw their funds even in the absence of liquidity needs. Bank runs were frequently observed in the United States before the establishment of the Federal Deposit Insurance Corporation in 1933. The enactment of the deposit insurance program has greatly reduced the incidence of bank runs. However, a new wave of bank runs has occurred across the world during the recent …nancial turmoil. Examples include the runs on Northern Rock in September 2007, on Bear Stearns in March 2008, on Wamu in September 2008, on IndyMac in July 2008 and on Busan II Savings Bank in February 2011. The theoretical literature on bank runs is largely built on the seminal paper by Diamond and Dybvig (1983) (hereafter DD). The bank is modelled as a liquidity insurance provider that pools depositors’resources to invest in pro…table illiquid long-term assets, and at the same time, issues short-term demand deposits to meet the liquidity need of depositors. The term mismatch between the bank’s assets and liabilities opens the gate to bank runs. There are, broadly speaking, two opposing views about the cause of bank runs. The …rst view (represented by DD) is that bank runs are the result of pure coordination failures. The bank run model in DD has two self-ful…lling symmetric pure-strategy Nash equilibria. In one equilibrium, depositors choose to withdraw only when they need liquidity. In the other equilibrium, in fear that the bank will not be able to repay them, all depositors run to the bank to withdraw money irrespective of their liquidity needs. The run forces the bank to liquidate its long-term investment at …re-sale prices and makes the initial fear a self-ful…lling prophecy. As a result, even banks with healthy assets may be subject to bank runs. The competing view (represented by Allen and Gale, 1998) is that bank runs are caused by adverse information about the quality of the bank’s assets. To empirically test the competing theories of bank runs is challenging. Real-world bank runs tend to involve various factors, which makes it di¢ cult to conclude whether the bank run is due to miscoordination or the deterioration of the quality of the bank’s assets. There are some attempts to empirically test the source of bank runs, nonetheless with mixed results. For example, Gorton (1988), Allen and Gale (1998) and Schumacher (2000) show that bank runs have historically been strongly correlated with deteriorating economic fundamentals, which erode away the value of the bank’s assets. In contrast, Boyd et al. (2001) conclude that bank runs may often be the outcome of coordination failures. An experimental approach has the advantage that it is easier to control the di¤erent factors that may cause bank runs in the laboratory. In this study we focus on coordination failures as a possible source of bank runs. To that goal we …x the rate of return of the bank’s long-term asset throughout the experiment to rule out the deterioration of the quality of the bank’s assets as the source of bank runs. The speci…c design of the experiment is inspired by Temzelides (1997), who applies the evolutionary algorithm to a repeated version of the DD model. He proves a limiting case in which as 2

the probability of experimentation approaches zero, the economy stays at the non-run equilibrium with probability one if and only if it is risk dominant, or less than half of patient consumers are required to coordinate on "wait" so that "wait" gives a higher payo¤ than "withdraw".1 Following Temzelides (1997), we conjecture that the coordination parameter, measured as the fraction of depositors choosing to wait that is required to generate enough complementarity among depositors who wait so that they earn a higher payo¤ than those who withdraw, a¤ects the decision of depositors and in turn the occurrence of bank runs. We are particularly interested in three questions. How does the coordination parameter a¤ect the level of coordination and the frequency of bank runs? Does the path followed by the coordination parameter in‡uence the performance of the economy? Will the economies switch between the equilibria even if the economic fundamentals are constant, and if yes, how does the switching behavior depend on the coordination parameter? To answer these questions, we have run 20 sessions of experiment. In every session a group of 10 subjects acts as depositors deciding whether to withdraw money or wait. A session consists of 7 or 9 phases, and each phase runs for 10 periods. The level of coordination requirement is …xed in each phase but varies across phases ranging from 0.1 to 0.9. The ordering of the coordination parameter is either increasing, decreasing or random. We have three main …ndings. The …rst is that bank runs occur more frequently when the coordination task is more di¢ cult. In particular, we can describe the coordination results in three regions of the coordination parameter values. In the non-run region, where the parameter is

0:5, all experimental economies stay close or converge to the non-run equilibrium. In the

run region, where the parameter is

0:8, all experimental economies stay close or converge to

the run equilibrium. The economies perform very di¤erently and may stay close or converge to either the run or the non-run equilibrium if the coordination parameter falls into the indeterminacy region, where the parameter is equal to 0:6 or 0:7. Second, some order e¤ect is detected, especially in games characterized by a coordination parameter that lies in the indeterminacy region. The experimental economies tend to achieve better coordination outcomes and have less bank runs when the level of coordination requirement increases gradually over time, as compared to the case where the parameter decreases or changes in a random pattern. Third, we observe intra-phase switching between the run and non-run equilibria even if the economic fundamentals are constant within each phase. The switches tend to occur when the coordination parameter is

0:5.

Finally, we evaluate how the evolutionary algorithm can be used to explain the experimental data. Temzelides (1997) proves a limiting case where the probability of experimentation approaches zero and predicts an indeterminacy point of 0:5 for the occurrence of bank runs. The limiting case does not provide a satisfactory explanation of the experimental data, which suggest an indeterminacy region located above 0:5. We show that a modi…ed version of the evolutionary algorithm, which uses experimentation rates estimated from the experimental data, is successful in explaining the behavior of subjects. 1

Ennis (2003) also discusses risk dominance as an equilibrium selection device in the context of the DD model of bank runs.

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The rest of the paper is organized as follows. In Section 2, we discuss related experimental studies on bank runs and coordination games. Section 3 describes the theoretical framework that underlies the experiment, the hypotheses and the experimental design. Section 4 presents the experimental results and major …ndings. Section 5 develops the evolutionary algorithm and evaluates its power in explaining the experimental data. Section 6 concludes and discusses directions for future research.

2

Related Literature

In this section, we discuss related experimental research on bank runs and coordination games. There have been several previous studies of bank runs in controlled laboratory environments, including Madiès (2006), Garrat and Keister (2009), Schotter and Yorulmazer (2009) and Klos and Sträter (2010). Madiès (2006) provides the …rst experimental study of miscoordination-based bank runs within the framework of the DD model. The paper’s emphasis is on the e¤ectiveness of alternative ways to prevent bank runs, including suspension of payments and deposit insurance. In the process, the paper also o¤ers some evidence that the severity of bank runs is a¤ected by the level of coordination requirement. However, the paper studies only two levels of coordination requirement, 30% and 70%. When 30% of coordination is required, the economy stays close to the non-run equilibrium most of the time. When 70% of coordination is required, the economy stays close to the run equilibrium most of the time. Garrat and Keister (2009) study how depositors’ decisions are a¤ected by uncertainty about the aggregate liquidity demand and by the number of opportunities subjects have to withdraw. They show that (i) bank runs are rare when fundamental withdrawal demand is known but occur frequently when it is stochastic, and (ii) subjects are more likely to withdraw when given multiple opportunities to do so than when presented with a single decision. Schotter and Yorulmazer (2009) investigate bank runs in a dynamic context and examine the factors that a¤ect the speed of withdrawals, including the number of opportunities to withdraw and the existence of insiders. They have two main results. The …rst is that the more information laboratory economic agents can expect to learn about the crisis as it develops, the more willing they are to restrain themselves from withdrawing of their funds once a crisis occurs. The second is that the presence of insiders, who know the quality of the bank, signi…cantly mitigates the severity of bank runs. Klos and Sträter (2012) test the prediction of the global game theory of bank runs developed by Morris and Shin (2001) and Goldstein and Pauzner (2005). In the model, depositors receive noisy private signals about the (changing) quality of the bank’s long-term assets. The theory predicts that depositors employ a threshold strategy choosing to withdraw (wait) if the signal is below (above) a cut-o¤ point. Klos and Sträter (2012) …nd that subjects do follow threshold strategies, and the thresholds increase with the short-term rate promised to withdrawers as predicted by the theory.

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However, compared to the theoretical predictions of the global games’approach, the reaction to a change in the repayment rate is less pronounced. They also …nd that the level-k approach, in which level-1 types assume that level-0 types play a random threshold strategy, …ts the data considerably better. Compared with the existing experimental literature on bank runs, our paper has a very di¤erent focus, which is to study how the level of coordination requirement a¤ects the occurrence of bank runs as a result of pure coordination failures. To that goal, throughout the experiment, the rate of return of the bank’s long-term assets is …xed, there is no uncertainty in the demand for liquidity, subjects have only one withdrawing opportunity in each game, all payo¤-relevant variables are public information, and there are no insiders endowed with superior information than others. The only variable that changes during a session is the level of coordination requirement. This is controlled by changing the short-term interest rate promised to withdrawers, which is public information. To investigate the e¤ect of coordination requirement in a systematic way, the coordination parameter takes 7 or 9 values in each session ranging from 0.1 to 0.9. In addition, di¤erent sequences of the coordination parameter are adopted to identify potential order e¤ect. Each value of the coordination parameter is maintained for 10 periods to detect possible switching between the two equilibria with constant economic fundamentals. More generally, our paper is also related to experimental studies of coordination games with strategic complementarity, which feature multiple equilibria that can be Pareto ranked. In these games there is often a tension between e¢ ciency and security. Van Huyck et al. (1990; 1991) show that due to this tension, coordination failures may occur as a result of strategic uncertainty: subjects may conclude that it is too risky to choose the payo¤-dominant strategy. These two studies consider only one game with a tension between e¢ ciency and security, and they do not provide a measure to quantify the riskiness associated with the payo¤-dominant strategy. As a result, the setup cannot be used to determine what levels of risk are "too risky". In our paper, we parameterize and quantify the riskiness of the payo¤-dominant strategy by the coordination parameter. By observing how subjects behave in a series of games characterized by di¤erent levels of coordination requirement, we can study how the coordination behavior is a¤ected by the level of risk and identify the levels of risk that are deemed to be "too risky". In addition, earlier studies (see, for example, Van Huyck et al., 1991) …nd that the initial coordination behavior predicts well the …nal outcomes in repeated coordination games. However, our experimental results show that the result does not hold when the coordination parameter takes certain values. In particular, some experimental economies switch between the two equilibria even if economic fundamentals are kept constant. Weber (2006), Brandts and Cooper (2006) and more recently Romero (2011), have detected path-dependence e¤ect in minimum-e¤ort games. Weber (2006) focuses on the e¤ect of a change in group size, and Brandts and Cooper (2006) study the e¤ect of payo¤ bonuses intended to raise e¤ort levels. Romero (2011) shows that the performance of the experimental economies depends on whether the cost of e¤ort increases or decreases over time. Our study provides additional 5

evidence of path dependence in the context of a bank-run game, with a focus on the e¤ect of the coordination parameter. One of our main …ndings is that there is a strong order e¤ect for games with the coordination parameter lying in the indeterminacy region. Finally, we would like to discuss the experimental studies on global games. The global game equilibrium is developed as a re…nement concept for binary-choice coordination games of complete information. Carlsson and van Damme (1993) show that the multiplicity of these games can be resolved by introducing incomplete information. One way is to assume that players receive private signals about a payo¤-relevant variable. Based on their private signals, players form beliefs about the distribution of the variable. While choosing their actions, players consider the global game, or all possible games each characterized by a di¤erent value in the support of the distribution. The global game has a unique Bayesian perfect equilibrium, where players adopt a threshold strategy choosing the payo¤-dominant action if and only if their signals are favorable enough. Heinemann et al. (2004) provide the …rst experimental study to test the prediction of the global game theory. The main treatment variable is whether there is accurate public information or noisy private information about a payo¤-relevant variable. The principal …nding is that the observed behavior is similar in both situations: individuals follow threshold strategies and choose the payo¤-dominant strategy only if the payo¤-relevant variable is favorable enough (although the threshold values may di¤er). The study suggests that the global game theory can be used as an equilibrium selection device even for coordination games with public information. The result is further substantiated by Cabrales et al. (2007), Heinemann et al. (2009), Du¤y and Ochs (2012) and Klos and Sträter (2012). In our experiment, subjects have common information about all pay-o¤ relevant variables. In view of the experimental studies of global games, we expect the coordination parameter, which is also a payo¤ relevant variable, to function as a coordination device. Our study shows that the experimental economies tend to stay at the non-run equilibrium when the coordination parameter is low. The observation is consistent with the result from the global-game studies that individual players adopt a threshold strategy choosing the payo¤-dominant strategy when the economic fundamental is strong. However, our focus is to study how the level of coordination requirement a¤ects aggregate coordination outcomes. Our experimental design also facilitates the study of the order e¤ect and whether the economy can switch between the two equilibria even with …xed economic fundamentals.

3

Theoretical Framework, Experimental Design and Hypotheses

In this section, we describe the theoretical framework that underlies the experiment, the hypotheses and the main design features of the experiment.

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3.1

Theoretical Framework

The theoretical framework that underlies our study is the DD model of bank runs. There are three dates indexed by 0, 1 and 2. There are D ex ante identical agents in the economy. At date 0, the planning period, each agent is endowed with 1 unit of good and faces a liquidity shock that determines her preferences over goods at date 1 and date 2. The liquidity shock is realized at the beginning of date 1: Among the D agents, N of them become patient agents, who are indi¤erent between consumption at date 1 and date 2, and the rest become impatient agents, who care only about consumption at date 1. Realization of the liquidity shock is private information. Preferences are described by

8 < u(c1 ) U (c1 ; c2 ) = : u(c + c ) 1 2

for impatient consumers, for patient consumers,

where c1 and c2 denote the consumption at date 1 and date 2, respectively. The function u( ) satis…es u00 < 0 < u0 , limc!1 u0 (c) = 0 and limc!0 u0 (c) = 1. The relative risk aversion coe¢ cient

cu00 (c)=u0 (c) > 1 everywhere. There is a productive technology that transforms 1 unit of date 0

output into 1 unit of date 1 output or R > 1 units of date 2 output. At the socially optimal allocation, impatient agents consume only at date 1 and patient agents consume only at date 2. Let ci and cp denote the consumption by impatient consumers and patient consumers, respectively. The optimal allocation, (ci ; cp ), is characterized by 1 < ci (N=D; R; u) < cp (N=D; R; u) < R: A bank, by o¤ering demand deposit contracts, can provide liquidity insurance to agents. The contract requires agents deposit their endowment with the bank at date 0. In return, agents receive a bank security which can be used to demand consumption at either date 1 or 2. The bank promises to pay r > 1 to depositors who choose to withdraw at date 1. If the number of withdrawers, e, exceeds e^ = D=r, the bank will not have enough money to pay every withdrawer the promised rate r. In this case, a sequential service constraint applies: only those who are early in line receive the promised payment. Resources left after paying withdrawers generate a rate of return R > r, and the proceeds are shared by all who choose to roll over their deposits and wait until date 2 to consume. Let z = D to wait and z^ = D

e^ = D(1

prevent bankruptcy. Let

e

e be the number of depositors choosing

1=r) be the minimum number of depositors choosing to wait to

and

z

be the payo¤s to those who choose to withdraw and roll over,

respectively. The deposit contract can be formulated as

e

z

8 < r; if z z^; = : r with probability (D 8 < D r(D z) R; if z z^; z = : 0; if z z^:

z^)=(D

z) and 0 with probability (^ z

The optimal risk-sharing allocation can be achieved by setting r to r 7

c1 .

z)=D

z, if z

z^;

To conduct the experiment, we keep the important features of the DD model. To facilitate the experimental design, we modify the original model along two dimensions. First, in the original model, there are both patient and impatient agents. Impatient agents always withdraw, and only patient agents are "strategic" players. Here we focus on "strategic" players, so we let D = N .2 Second, we abstract from the sequential service constraint for simpli…cation and assume instead that if the bank does not have enough money to pay every withdrawer r, it divides the available resources evenly among all depositors who demand to withdraw. The sequential service constraint is not essential for the existence of multiple equilibria; the fact that r > 1 is su¢ cient to generate a payo¤ externality and panic-based runs. To rule out the possibility that bank runs are caused by weak performance of the bank’s long-term portfolio, we …x the rate of return of the long-term investment, R, throughout the experiment. For the short-term interest rate r, we do not use the optimal rate r . Instead, we set r to be a series of values greater than 1. As will become clear shortly, there is a one-on-one correspondence between r and the coordination parameter. Using r as a control variable allows us to change the coordination parameter in a simple way.3 The resulting payo¤ functions used in the experiment can be represented as e

= min r;

z

= max 0;

N N N

; z r(N z

(1) z)

R :

(2)

The coordination parameter, denoted as , measures the amount of coordination that is required for agents who choose to wait to receive a higher payo¤ than those who choose to withdraw. The parameter can be calculated in two steps. First, solve for the value of z, the number of depositors who choose to wait, that equalizes the payo¤s associated with "withdraw" and "wait", r=

N

(N z

z)r

R,

and denote it by z . Thus, z is given by: z =

R(r r(R

1) N. 1)

Second, divide z by N to get , the fraction of depositors who choose to wait, that equalizes the 2

Madiès (2006) adopts the same arrangement in this regard. For optimal contracting in the DD framework, please refer to Green and Lin (2000, 2003), Andolfatto, Nosal and Wallace (2007), Andolfatto and Nosal (2008) and Ennis and Keister (2009a, 2009b, 2010). The …rst three papers show that the multiple-equilibria result goes away if more complicated contingent contracts –as compared with the simple demand deposit contracts in DD –are used. The three papers by Ennis and Keister show that the multiple-equilibria result is restored if the banking authority cannot commit not to intervene in the event of a crisis, or the consumption needs of agents are correlated. 3

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payo¤s to the two strategy choices: =

z R(r = N r(R

1) : 1)

We can control the coordination parameter, , by changing the short-term interest rate r: The coordination game characterized by the above payo¤ structure has two symmetric purestrategy Nash equilibria.4 In the run equilibrium, every depositor chooses to withdraw and run on the bank expecting others to do the same. As a result, z = 0 and everybody receives a payo¤ of 1. In the non-run equilibrium, every depositor chooses to wait expecting others to make the same choice. In this equilibrium, z = N , and everybody receives a payo¤ of R. Fixing R also has the additional advantage of maintaining the payo¤ di¤erence between the two equilibria …xed at R

1

throughout the experiment.

3.2

Experimental Design and Hypotheses

To examine how the coordination requirement a¤ects the occurrence of bank runs, we design the experiment with the following main features. First, in contrast to earlier experimental literature on coordination games, which adopts context-free phrasing, we phrase the task explicitly as a decision about whether to withdraw money from the bank. The speci…c banking context makes the payo¤ structure more intuitive to understand and helps subjects better comprehend the task they are required to perform. Second, to rule out the possibility that bank runs are caused by weak performance of the bank’s long-term portfolio, we …x the rate of return of the long-term investment, R, at 2 throughout the experiment. The change of coordination requirement is induced by a change in the short-term rate, r. See table 1 for the correspondence between r and . Third, to inspect the e¤ect of coordination requirement, each session of experiment includes 7 or 9 phases, and each phase features a di¤erent value of the coordination parameter ranging from 0:1 to 0:9. Fourth, to detect potential path-dependence e¤ect, we run sessions with di¤erent orderings (increasing, decreasing or random) of the coordination parameter. Fifth, each phase consists of 10 repeated games where the value of the coordination parameter is …xed. This allows us to study whether the economy will switch between the two equilibria even if economic fundamentals are constant, and how the switching behavior is a¤ected by the level of the coordination requirement. In addition, we can also investigate how persistent the order e¤ect is. [Insert Table 1: Correspondence between r and ] These design features allow us to answer the three questions raised in the Introduction: (i) How does the coordination parameter a¤ect the level of coordination and the frequency of bank runs? (ii) Does the path followed by the coordination parameter in‡uence the performance of the 4

There is also a symmetric mixed-strategy equilibrium where each depositor chooses to wait with a probability between 0 and 1 and the expected payo¤ from the two strategies are equalized.

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economy? (iii) Will the economies switch between the equilibria even if the economic fundamentals are constant, and if yes, how does the switching behavior depend on the coordination parameter? For each research question, we form an ex ante hypothesis as a conjecture about the answer. The hypotheses also help us organize the discussion of the experimental results in an e¤ective way. Hypothesis I. There is more coordination at waiting when the coordination task is easier. In view of the study by Temzelides (1997), we expect

= 0:5 as the indeterminacy point for the

occurrence of bank runs: the economy tends to stay close to the non-run (run) equilibrium when < 0:5 (> 0:5) and becomes di¢ cult to predict when Hypothesis II. We expect that as

= 0:5.

takes the increasing order, there is on average more

coordination at waiting and fewer incidences of bank runs. As the level of coordination requirement gradually increases over time, subjects may build up mutual trust while playing games with easy coordination, and the mutual trust may persist into games with more di¢ cult coordination. In contrast, when the coordination parameter starts with a high value and then decreases over time, the pessimism developed in earlier phases may carry forward to later phases with easier coordination. The conjecture is consistent with the results from other studies, including Weber (2006), Brandts and Cooper (2006) and more recently Romero (2011), who show that better coordination can be achieved if subjects play an earlier game with more favorable economic fundamentals. Hypothesis III. The experimental economies will not switch between the run and non-run equilibria in the same phase. This hypothesis follows from earlier studies which suggest that the initial coordination behavior well predicts the …nal outcomes in repeated coordination games (see, for example, Van Huyck et al., 1991). We ran the experiment at three locations: University of International Business and Economics (UIBE), Beijing, China, Simon Fraser University (SFU), Burnaby, Canada, and University of Manitoba (UofM), Winnipeg, Canada. Each session lasted for about an hour. The average earning is about 20 CAD in Canada and 80 RMB in China. All sessions are run in English. To ensure that the subjects in Beijing have su¢ cient English reading and listening skills, we restrict our subject pool to those who have passed the College English Test Grade IV, a standardized national English language test for college students in China. We also ensure that the subjects understand the experimental instructions by giving them 10 trial periods and several opportunities to ask questions before the formal rounds. For each session of the experiment, we recruit 10 subjects to play the role of depositors. Most subjects are recruited from second and third-year undergraduate or graduate economics or business classes. Since the game in the experiment is fairly straightforward, it is important that the subjects have no prior experience with experiments of coordination games. Subjects should also be from mixed sources to guarantee that they do not have close relationships before participating in the experiment. The program used to conduct the experiment is written in z-Tree (Fischbacher, 2007). At the beginning of a session, each subject is assigned a computer terminal, through which they can input

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their decisions to withdraw or to wait.5 Communication among subjects is not allowed during the experiment. At the beginning of each period, every subject starts with 1 experimental dollar in the bank and then makes a decision to withdraw or to wait and roll over their deposits. Payo¤ tables for all phases are provided to list the payo¤ that an individual will receive if he/she chooses to withdraw or wait given that n = 1 s 9 of the other 9 subjects choose to withdraw. The payo¤ tables help to reduce the calculation burden for the subjects so that they can focus on playing the coordination game. Once all subjects make their decisions, the total number of subjects choosing to wait is calculated. Subjects’payo¤s are then determined by equations (1) and (2). At the end of each round, subjects are presented with the history of their own actions, payo¤s and cumulative payo¤s in the current period and all previous periods. A reminder is broadcasted on the subjects’computer screens each time r is changed. We do not show explicitly the information about the total number of withdrawals and the payo¤ to both actions, because we think this is more realistic: depositors do not directly observe the return of other depositors. However, subjects can try to infer such information from the payo¤ tables (as will be shown later in Section 5, subjects can deduce the information most of the time). After the experiment, the total payo¤ that each subject earns is converted from experimental dollars into cash. We ran 20 sessions in total. The …rst 8 sessions feature 7 values of : 0:1, 0:2, 0:3, 0:5, 0:7, 0:8 and 0:9. In sessions 1 to 4, the coordination parameter increases across the 7 phases, which implies coordination becomes increasingly more di¢ cult. Sessions 5 to 8 have decreasing , and coordination becomes easier over time. We …nd that the experimental economies stay close or converge to the non-run (run) equilibrium when when

0:5 (

0:8) but performed very di¤erently

= 0:7 (the details will be shown in the next section where we present the experimental

results). There is also some weak evidence of the order e¤ect. In order to further understand the performance of the economy around the indeterminacy value of 0:7, we run new sessions with a …ner grid of the coordination parameter by adding 0:4 and 0:6 into the spectrum of . In addition, there is the concern that the monotonic change of

is conducive to

the change of coordination behavior in response to the parameter: it may be that after making the same choice for many rounds (40 in sessions with increasing

and 20 in sessions with decreasing

), subjects are bored and waiting for a shift of actions. The alteration of

may have served as a

cue for the change to take place. To address the concern, we include a treatment with a random ordering of .6 We run 12 new sessions (sessions 9 to 20) all with a …ner grid of 0:1. For each of the three orderings –increasing, decreasing and random –we run 4 sessions of experiment. Among the 12 sessions, 2 sessions (sessions 9 and 17) include the full spectrum of 9 values of

ranging from

0:1 to 0:9 with a step of 0:1. In view that almost every subject consistently coordinates at "wait" for

= 0:1 and 0:2, we decided to drop these two values from the other 10 sessions. Excluding the

two smallest values of

also has the additional advantage of having the same number of paying

periods as in the …rst 8 sessions with coarser grids. 5 6

See the Appendix for the experimental instructions. We thank the referees for suggesting these new treatments.

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[Insert Table 2: Experimental Design] Table 2 provides detailed design information for each session of the experiment, including the grid, the ordering of , the speci…c values of

used in the session, and the location and date of the

session. The 12 new sessions of experiment feature an even-spaced grid of 0:1 for the values of the level of coordination requirement. In contrast, in the …rst 8 sessions of experiment, there are two points where the grid is wider at 0:2 (from 0:3 to 0:5 and then from 0:5 to 0:7). In view of this, we say the …rst 8 sessions of experiment have a coarse grid and the 12 sessions have a …ne grid. In total, we have …ve treatments: coarse grid with increasing decreasing

(sessions 5 to 8), …ne grid with increasing

(sessions 13 to 16), and …ne grid with random

4

(sessions 1 to 4), coarse grid with

(sessions 9 to12), …ne grid with decreasing

(sessions 17 to 20).

Experimental Results

In this section, we present and discuss the experimental results. Figure 1 plots the path of z, the number of subjects choosing to wait, for each

value and for each of the 20 sessions of experiment.

The results for the …rst 8 sessions are on the …rst page, and the results for sessions 13 to 20 are on the second page. To facilitate comparison across di¤erent treatments, we plot z according to increasing from 0:1 to 0:9 with a step of 0:1. The sequence of z is thus di¤erent from the actual time path for sessions with decreasing and random treatments. The empty sections in the graphs are due to missing values of : 0.4 and 0.6 for sessions 1 to 8, and 0:1 and 0:2 for sessions 10 to 16 and 18 to 20. [Insert Figure 1: Experimental Results] Table 3 lists the starting, terminal and mean values of the number of subjects choosing to wait (denoted as S, T and M , respectively) for each phase associated with a di¤erent value of . To qualitatively characterize the performance of the experimental economies, we de…ne 10 performance categories. The …rst 8 performance categories are de…ned according to the mean and terminal values of the number of people who choose to wait. If M close to the non-run equilibrium" (denoted as NN); if 8

9, we describe the economy as being "very M < 9, we classify the economy as being

"fairly close to the non-run equilibrium" (denoted as FN); if 5 < M < 8 and T

8, we say that

the economy "converges to the non-run equilibrium" (denoted as CN); if 5 < M < 8 and T < 8, the economy is described to have "moderate high coordination" (denoted as H); if M

1, we mark

the economy as being "very close to the run equilibrium" (denoted as RR); the economy is said to be "fairly close to the run equilibrium" (denoted as FR) if 1 < M

2; if 2 < M < 5 and T

2,

we say that the economy "converges to the run equilibrium" (denoted as CR); and a economy with 2 < M < 5 and T > 2 is classi…ed as showing "moderate low coordination" (denoted as L). In addition, we observe that some of the economies switch from high (low) level of coordination to run (non-run) equilibrium. To capture the switching behavior, we de…ne two more performance 12

categories based on the starting and ending values of z. If S < 5 and T

8, we say the economy

"switches to non-run equilibrium" (denoted as SN); and if S > 5 and T

2, we say the economy

"switches to run equilibrium" (denoted as SR). The de…nitions of the performance categories are summarized in table 4. Table 5 categorizes the performance of all the experimental sessions. [Insert Table 3: Starting, Terminal and Mean Value of the Number of Subjects Choosing to Wait] [Insert Table 4: De…nition of Performance Classi…cation] [Insert Table 5: Classi…cation of Performance of Experimental Economies] In what follows, we …rst discuss the results from the …rst 8 sessions of experiment with the coarse grid. For each of the three hypotheses, we show whether there is evidence to support it. We then discuss the results from the 12 new sessions with the …ner grid and revisit all the main …ndings.

4.1

Findings from Treatments with Coarse Grid

For each of the three questions raised and each of the three hypotheses formulated in Section 3, we discuss the evidence provided by the …rst 8 sessions of experiment. Finding I: There are less withdrawals when a lower level of coordination is required. However, the results suggest an indeterminacy value higher than 0:5, as predicted in Temzelides (1997). As shown in table 3, the average number of subjects choosing to wait in a phase tends to decrease with . A higher coordination requirement means that there is a higher risk associated with waiting, which discourages subjects from choosing the action. As a result, the number of subjects choosing to wait decreases as coordination becomes more di¢ cult. As for the occurrence of bank runs, the experimental data suggest a higher indeterminacy value than 0:5. In table 5, all economies are marked by NN or CN when non-run region. When the run region. When

0:5, which we call the

0:8, all economies are marked by RR, FR or CR. We call

0:8

= 0:7, the experimental economies experience indeterminacy with much

less clear-cut outcomes. The average number of subjects who choose to wait varies from 0:4 to 7:9. The performance categories range from CR and RR to H and CN. Session 1 starts with 6 subjects coordinating on "wait". This number proceeds up to 8 and then back to 6. This session is marked by "H". Session 3 begins with a high level of coordination with 8 subjects choosing to wait, but the number decreases to 2 by the end of the phase. Sessions 2 and 4 start with 7 subjects who coordinate on "wait", but the number eventually decreases to 0. These three sessions are marked by "CR". Among the 4 sessions with decreasing , sessions 5, 6 and 8 begin with very low coordination and stay there; they are marked by "RR". Session 7 starts with 6 subjects who coordinate on "wait" and the number rises to 8 by the end; this session is marked by "CN". The result of an indeterminacy value bigger than 0:5 can be explained as follows. Temzelides 13

(1997) derives 0:5 as the indeterminacy point under the implicit assumption that agents care about strategic uncertainty only. However, the higher payo¤ associated with non-run equilibrium generates an additional force to draw the economy to that equilibrium and raises the indeterminacy value above 0:5. Finding II: There is some weak evidence of the order e¤ect as

takes the indeterminacy value

of 0:7. The results from the …rst 8 sessions of experiment show that the ordering of on the experimental economies when

0:5 (

has little e¤ect

0:8) as all economies manage to stay or converge

to the non-run (run) equilibrium. Given the great variation in performance of di¤erent economies around the indeterminacy value of 0:7, we run the rank-sum test to detect potential order e¤ect for sessions with

= 0:7. The 8 sessions of experiment are separated into two groups of observations:

the …rst group contains the 4 sessions with increasing , and the other group includes the 4 sessions with decreasing . We conduct one-sided Mann-Whitney rank-sum tests on the starting, terminal and mean values of z between the two groups. The null hypothesis is that there is no di¤erence between the two groups. The alternative hypothesis is that the group with increasing

has higher

z or lower number of withdrawals than the other group. The test results are reported in the …rst part of table 6a, including the average values of S; T and M in each group, the value of the test statistic, Z, and the one-sided p-value.7 The initial number of subjects who choose to wait tends to be higher in sessions with increasing

with an average value of 7 versus 2 in the group with

decreasing . The di¤erence is statistically signi…cant with a p-value of 1:38%, which suggests that the order of

a¤ects the performance of the economy at the start of the phase. This is mainly

due to inertia: subjects tend to follow old strategies from the previous phase immediately after a change of . However, the e¤ect of history weakens after the initial periods. The mean and ending values of z are not signi…cantly di¤erent with p-values of 12:41% and 37:05%, respectively. Overall, there is some, albeit weak, evidence that the ordering of

a¤ects the performance of the economy.

[Insert Table 6: Rank-Sum Tests of the Order and Grid E¤ect] Finding III: Some intra-phase switching between the run and non-run equilibria is observed when

is equal to 0:5, 0:7 and 0:8.

As discussed earlier, previous studies …nd that the initial coordination behavior well predicts the …nal outcomes in repeated coordination games. Here we …nd the same result when

= 0:1,

0:2, 0:3 or 0:9: most subjects keep their initial choices, and the experimental economies reach one of the two equilibria instantly. However, we detect intra-phase switching in some sessions when is in the neighborhood of the indeterminacy value of 0:7. In particular, there are 6 incidences 7

In the absence of strong argument (resulted from previous research, theoretical or logical considerations) about the direction that the order e¤ect, two-sided tests should be used with the p-values doubling the values reported in Table 6. One-sided tests are appropriate here, because it is highly unlikely that the order e¤ect would work in the opposite direction. As mentioned in Section 2, previous studies about history dependence (Weber, 2006; Brandts and Cooper, 2006; Romero, 2011) all show that following a previous game with parameters that are conducive to more coordination at the payo¤-dominant strategy, subjects are more likely to choose the strategy in a new game.

14

(see table 5) where the economy starts with high (low) level of coordination but switches to run (non-run) equilibrium at the end of the phase. When

= 0:5; sessions 5 and 8 start with < 5

subjects choosing to wait but the number goes up to 8 and 9 in the last period. These two sessions are marked by "SN". When at "wait" (S

= 0:7, sessions 2, 3 and 4 start with a high level of coordination

7), but the number decreases to

2. These three sessions experience a switch

to the run equilibrium and are marked by "SR". There is also an incidence of switch to the run equilibrium when

= 0:8 in session 7.

The bank-run game involves a tension between safety and e¢ ciency. Withdrawing ensures a payo¤

1 but has a low return ceiling de…ned by r. Waiting is risky with a possible low payo¤

of 0, but has the potential to achieve the e¢ cient allocation of R. If the tension is strong enough, switches are likely to occur. In a switch from a low level of coordination to the non-run equilibrium, the e¢ ciency consideration wins. However, a switch in the opposite direction is also possible if the safety concern becomes dominant. One may also notice a regularity concerning the direction of switches: for sessions with increasing , the switches are always from a high level of coordination at "wait" toward the run equilibrium (marked by "SR"), while for the treatment with decreasing opposite direction (marked by

"SN").8

, the switches are always in the

This regularity is largely due to inertia: the starting level

of coordination in a new phase is often a¤ected by the level of coordination in the preceding phase. In the increasing treatment, this means that subjects tend to start a new phase characterized by a high value of

with a high level of coordination that can not be sustained in the rest of the

phase. For example, in the treatment with increasing , three sessions experience a switch from a high level of coordination to the run equilibrium as subjects play the game with

= 0:7. The

initial high levels of coordination in the phase is due to the high level of coordination exhibited in the preceding phase with

= 0:5. However, the con…dence is not able to last throughout the

whole phase, and the economies experience switches toward the run equilibrium. In contrast, as decreases over time, the history of bad coordination in the past may carry over to a new phase with easier coordination and results in switches in the reverse direction. For example, sessions 5 and 8, which have decreasing , experience switches to the non-run equilibrium in the phase with = 0:5. The low levels of coordination at the start of the phase is directly linked to the low levels of coordination in the preceding phase where

4.2

= 0:7.

Findings Revisited after Treatments with Fine Grid

Given the observed richness of behavior around the indeterminacy value of

= 0:7, we run a new

set of sessions with emphasized focus around the value. In particular, in contrast to the …rst 8 sessions of the experiment, which involve a coarse grid around 0:7, the new sessions feature a …ner grid around the value and include (non-monotonic) ordering of 8

= 0:4 and 0:6. In addition, we run the experiment with random

to study whether the change of coordination behavior is induced by

We thank a referee for pointing out this regularity.

15

the monotonic change of . Altogether, we have three new treatments, all with a …nder grid of of 0:1 and in three di¤erent orderings of : increasing, decreasing and random. The path of z for each session is presented in …gure 1b. The starting, ending and mean values of z are presented in the lower part of table 3. The classi…cation of the performance of the economies is provided in the lower part of table 5. All of our main …ndings are in general con…rmed by the new sessions with the …ner grid. In addition, the new sessions generate stronger evidence for the order e¤ect. In the following we revisit each of the three main …ndings in detail. Finding I revisited. Finding I is robust to the adoption of a …ner grid and non-monotonic ordering of . We still observe more withdrawals in phases with higher values of the coordination parameter, and the indeterminacy value is higher than 0:5. However, the new data with a …ner grid of

captures another indeterminacy value, 0:6, in addition to 0:7, which is identi…ed in the

previous 8 sessions. When

0:5, including

= 0:4, which was not included in the …rst 8 sessions, all economies

stay close or converge to the non-run equilibrium. When

0:8, all economies stay close or

converge to the run equilibrium with one exception. In session 12 when

= 0:8, the economy

seems to converge to the run equilibrium with z = 2 in period 9, but z bounces up to 5 in the last period. As shown in table 5, when

= 0:6 or 0:7, the performance of the experimental economies varies

greatly across di¤erent treatments and di¤erent sessions within the same treatment. When all 4 economies with increasing

= 0:6,

stay close to the non-run equilibrium; 2 out of the 4 economies

with the decreasing and random treatments stay close or converge to the non-run equilibrium, while the other 2 stay close or converge to the run equilibrium. When

= 0:7, all 4 economies with the

increasing treatment again stay close to the non-run equilibrium; all 4 economies with decreasing stay close to the run equilibrium; 3 out of 4 economies with the random ordering stay close to the run equilibrium, and 1 economy stays close to the non-run equilibrium. To summarize, we can characterize the occurrence of bank runs in three regions of the coordination parameter. The new sessions identify the same non-run region,

0:5, and the run region,

0:8. In these two regions, the performance of the experimental economies is more uniform and is robust to the ordering of . When when

lies between 0:5 and 0:8 or the "indeterminacy" region, i.e.,

= 0:6 or 0:7, the economy is hard to predict and the outcome is sensitive to the ordering of

and the magnitude of the grid of . The result that the economy experiences indeterminacy in the region around 0:6 and 0:7 is consistent with a result in Heinemann et al. (2004). In their experimental study of the global game theory, Heinemann et al. (2004) also run sessions with common information, which in essence converts the global game with unique equilibrium into a pure coordination game with multiple equilibria. One of their …ndings is that individual subjects adopt a threshold strategy that constitutes a best response to the belief that other subjects choose the payo¤-dominant strategy with probability 2=3. In the context of the bank-run game, the strategy is to withdraw if

16

> ^, wait if

< ^ and randomize if

= ^ with ^ = 2=3.9 This implies that the indeterminacy region is located

around 2=3. Finding II revisited. The results from the new sessions lend more support to …nding II. With the …ner grid, the order e¤ect becomes much stronger. As in the sessions with the coarse grid, the ordering of 0:5 (or

has little e¤ect on the economy when

0:8) in the sense that all economies manage to stay close or converge to the non-run

(or run) equilibrium. However, the new data suggest a strong order e¤ect when As is obvious from table 5, when

= 0:6 or 0:7.

= 0:6 or 0:7, the experimental economies with the increasing

treatment perform much better than the other two treatments, with all 4 economies staying close to the non-run equilibrium. In contrast, in the decreasing treatment, only 2 sessions manage to stay close or converge to the non-run equilibrium when level of coordination when

= 0:6, and no session achieves a high

= 0:7 (all sessions are close to the run equilibrium). For the random

treatment, only 2 economies are close to the non-run equilibrium when session manages to stay close to the non-run equilibrium when

= 0:6, and only a single

= 0:7 (the other 3 stay close to

the run equilibrium). The order e¤ect is also supported by rank-sum tests on the starting, terminal and mean values of z (denoted as S; T and M , respectively). The test results for

= 0:7 are shown in table 6a. The

increasing treatment achieves signi…cantly better coordination than the decreasing treatment. The average starting, terminal and mean value of the number of subjects choosing to wait are all much higher when

increases. The p-values for S, T and M are 0:76%, 0:41% and 1:01%, respectively.

The p-values of the other two pairwise comparisons, increasing versus random and random versus decreasing, are all below 7%. Now let us turn to the test results for

= 0:6 presented in table 6b.

There is again strong evidence that the increasing treatment exhibits a higher level of coordination than the decreasing treatment. The p-value of M is highly signi…cant at 1:01%. The evidence is less strong for the other two pairwise comparisons. The overall picture is that even with a small sample size of 4 for each treatment, the data provides clear evidence that the performance of the economy is path dependent. Finally, a comparison of the new and old sessions with increasing 9-12) suggests that a more gradual increase in indeterminacy value of 0:7. When

(sessions 1-4 versus sessions

is associated with better coordination at the

increases from 0:5 to 0:7 with a step of 0:2, only 1 out of

4 economies achieves a moderate high coordination when

= 0:7. In contrast, when

increases

from 0:5 to 0:6 and then to 0:7 with a step of 0:1, all 4 economies manage to stay close to the non-run equilibrium when

= 0:7. A possible reason is that with a smaller grid, subjects perceive

the current game to be closer to the previous game with easier coordination. As a result they 9 Under the belief that 9 subjects chooses to wait with probability 2=3, the expected payo¤ P9each of9 the other to "withdraw" is e = (1=3)n (2=3)9 n min[r; 10=(1 + n)] , and the expected payo¤ to "wait" is z = n=0 n P9 9 (1=3)n (2=3)9 n max [0; 2(10 nr)=(10 n)] . The solution to e = z is r = r^ = 1:5, and the implied n=0 n value of the coordination parameter is = ^ = R(^ r 1)=[^ r(R 1)] = 2=3. The best response under the belief that everyone else chooses to wait with probability 2=3 is to withdraw if > ^, wait if < ^ and randomize if = ^.

17

are more likely to maintain the old strategies that were adopted in the previous game. The ranksum tests show that for

= 0:7, the economy tends to have higher starting, terminal and mean

values of z with the …ner grid with p-values

1:3%. The …nding that a gradual change facilitates

coordination reminds us of the work by Steiner and Stewart (2008). The authors study learning in a set of complete information normal form games where players continually face new strategic situations and form beliefs by extrapolation from similar past situations. They show that the use of extrapolations in learning may generate contagion of actions across games as players learn from games with payo¤s very close to the current ones. Our experimental results show that the contagion e¤ect is stronger when the change in the payo¤ structure is smaller and more gradual. Finding III revisited. We continue to observe intra-phase switching between the two equilibria for phases with

0:5. In the increasing treatment, sessions 9-11 start with relatively high

levels of coordination with S

8 and switch to the run equilibrium with T

1 when

= 0:8. When

= 0:9, session 9 starts with S = 6 and switches to the run equilibrium at the end of the phase with T = 0. In the decreasing treatment, sessions 14 and 15 switch to the non-run equilibrium with S = 3 or 4 and T =9 or 10 when run equilibrium when

= 0:6. With the random treatment, session 19 switches to the

= 0:8 and 0:9 with S = 9 and T = 0. Session 20 follows a similar pattern

with S = 6 and T = 1. As explained in the discussion of results from the sessions with the coarse grid, the direction of the switches can be largely accounted for by inertia.

5

The Evolutionary Algorithm

The evolutionary algorithm developed by Young (1993) and Kandori et al. (1993) has been introduced as an equilibrium selection mechanism into many rational expectation models with multiple equilibria and have greatly contributed to the understanding of these models. Temzelides (1997) applies the evolutionary algorithm to a repeated version of the DD model. The algorithm has two main components. The …rst is myopic best response, which means that when agents react to the environment, they respond by choosing the strategy that performed the best in the previous period. In the context of the DD model, the myopic best response is "withdraw" if the number of people choosing to wait in the previous period, zt

1,

is

z , and "wait" otherwise. The second com-

ponent is experimentation, during which agents change their strategies at random with a certain probability. In the context of the DD model, experimentation means the ‡ipping of one’s strategy from "withdraw" to "wait" or vice versa. Temzelides (1997) proves a limiting case where, as the probability of experimentation approaches zero, the economy stays at the non-run equilibrium with probability one if and only if it is risk dominant, or when

< 0:5. In other words,

= 0:5 is

predicted as the watershed for the occurrence of bank runs. The limiting case does not describe the experimental results well. In particular, the experimental results suggest an indeterminacy region located above 0:5. When

= 0:5, all 20 experimental

economies stay close or converge to the non-run equilibrium. Only when

= 0:6 or 0:7 do we

observe large variation across di¤erent treatments and sessions. One way to improve the algorithm 18

is to allow it to feed on the information that agents have about the economy. For example, the coordination parameter,

, captures how di¢ cult the coordination task is, and the number of

subjects choosing to wait in the last period, zt

1,

measures how the group coordinated in the past.

These two variables may in‡uence a subject’s belief about what other subjects will do in the next period and in turn the subject’s strategy choice. In this section, we show that the evolutionary algorithm combined with logit regressions used to estimate the rate of experimentation as functions of

and/or zt

1,

can successfully explain the experimental data. In a related work, Hommes (2011)

shows that evolutionary learning multi-nomial logit models …t data of so-called learning-to-forecast experiments well. In the rest of this Section, we …rst describe the main features of the evolutionary algorithm. We then run logit regressions to estimate the probability of experimentation from the experimental data. Finally, we evaluate the performance of the algorithm.

5.1

Description of the Algorithm

The main feature of our algorithm is that it feeds on the available information about the past period, so …rst we describe the amount of information subjects have in the experiment. We inform subjects about r, and subjects can always infer not directly inform subjects about zt deduce zt

1

1,

from the payo¤ tables. We do

but subjects can potentially refer to the payo¤ tables to

from their own payo¤s in the past period. For the new algorithm, we assume that if

experimental subjects cannot deduce whether zt

1

> z , the arti…cial agents (in our evolutionary

algorithm) skip the …rst part of the algorithm and do not update their strategies; otherwise, they update their strategies to the best response. For experimentation, we assume that the rate of experimentation is a function of the exact value of zt

1,

and/or zt

1.

Speci…cally, if the experimental subject can deduce

then the rate of experimentation depends on both zt

1

and ; otherwise,

the experimentation rate depends only on . The value of the rate of experimentation is to be estimated from the experimental data with logit regression models, where

and/or zt

1

are used

as regressors. Before running the regressions, we provide an overview about how much information is available to the experimental subjects. The 20 sessions of the experiment generate 12; 960 observations for the estimation of the rates of experimentation to be used in intra-phase learning. In each phase of each session, for each of the 10 subjects, there are 9 intra-phase learning periods (actions in the …rst period are taken as given). Sessions 9 and 17 each have 9 phases and these two sessions generate 2x9x10x9 = 1; 620 observations. The remaining 18 sessions each have 7 phases and generate 18x7x10x9 = 11; 340 observations. The …rst 4 columns of table 7 summarize how subjects’ information about the previous period depends on their strategies and payo¤s in the previous period. Suppose a subject chose to wait in the previous period. The subject can deduce zt

1

whenever he/she received a positive payo¤. If the subject receives 0 payo¤, he/she can infer that the bank was bankrupt but cannot deduce the exact value of zt 19

1.

For subjects who chose to

withdraw in the previous period, they can deduce the exact value of zt

1

if their payo¤s were less

than r. If they were paid r, they can infer that the bank did not go bankrupt, but they cannot infer the exact value of zt

1.

As shown in columns 5 and 6 of table 7, the subjects in our experiment can

deduce the best response and the exact value of zt

1

most of the time. To be precise, the subjects

can deduce the best response in 96:34% of the observations, and the exact value of zt of the

1

in 93:19%

observations.10 [Insert Table 7: Information]

Temzelides (1997) assumes agents always update to best response and adopt an exogenously …xed rate of experimentation (which approaches zero in the limiting case). Given that the experimental subjects could deduce the best response most of the time, having arti…cial agents update to best response contingent on the previous period information has negligible e¤ect on the relative performance of the new algorithm. The critical part of the new algorithm is to make the rate of experimentation depend on information about

and/or zt

1.

The success of the new algorithm is

mainly due to the endogenization of the rate of experimentation.

5.2

Estimation of Probability of Experimentation

To estimate the rate of experimentation, we construct each of the 12; 960 observations in …ve steps. First, we collect information about the individual’s last period’s strategy (st (

t 1)

zt

1

1 ),

last period’s payo¤

and current period’s strategy (st ). Second, we determine whether the individual can deduce

> z and the exact value of zt

1

according to table 7. Third, we update individuals’strategies

to best response if they know what the best response was. Those who do not know the best response keep their old strategies. Denote the strategy after the third step as s0t . Fourth, we determine whether the individual ‡ips her strategy by comparing s0t and st and denote it by a binomial variable, f . We count it as a ‡ip (f = 1) if s0t 6= st and a non-‡ip (f = 0) otherwise. Fifth,

we divide observations into three groups according to s0t and whether the individual can deduce the exact value of zt

1.

Group A includes observations with s0t ="withdraw" and involves subjects who

cannot infer the exact value of zt

1;

rate for subjects who do not know zt as

A ).

these observations are used to estimate the experimentation 1

and consider ‡ipping from "withdraw" to "wait" (denoted

Group B includes data used to estimate experimentation rate in the same direction for

subjects who can infer the exact value of zt

1

(denoted as

10

B ).

Group C includes observations with

As two referees point out, one may wonder how the results from the experiment and the evolutionary algorithm depend on whether subjects have full information about the performance of the economy in the past. As shown above, subjects can derive the exact value of zt 1 most of the time. Our experimental economies are thus very close to having full information. Nonetheless, we have run three sessions of experiments with full information, one for each ordering of and all with the …ner grid. To investigate the e¤ect of information availability, we conduct a two-sided rank sum test on the session mean of z between two groups. The …rst group includes the 12 sessions with asymmetric information (sessions 9 to 20), and the second group includes the 3 sessions with full information. While calculating the session mean of z, we omit the two phases with = 0:1 & 0:2 for sessions 9 and 17 to ensure that the same number of periods is included for each session. The p-value is 0:5637 showing that the performance of the economy is not signi…cantly a¤ected by the adoption of full information.

20

s0t ="wait" and are used to estimate the experimentation rate of ‡ipping from "wait" to "withdraw" (denoted as

C ).

Note that subjects who have s0t ="wait" can always infer zt

1

(see table 7).

We run three logit regressions to estimate the experimentation rate for each of the three groups of data. The dependent variable is the binomial variable f (whether a ‡ip of strategy is observed). For those who cannot infer the exact value of zt expected sign of

1

(group A), we use

as the single regressor. The

is negative, meaning that subjects in groups A are less likely to ‡ip strategy

from "withdraw" to "wait" as coordination becomes more di¢ cult. For those who can infer the exact value of zt

1

(groups B and C), we use the di¤erence between zt

1

and z as the single

regressor. According to feedback from subjects (after the sessions we spent a few minutes talking with subjects about their experiences in the experiment), the di¤erence between zt

1

and z plays

an important role in determining the probability that subjects experiment with a strategy, even though they know that the other strategy gave a higher payo¤ in the previous period. Note that z = N so z contains information about the case of as zt

1

B,

as well. The expected sign of (zt

1

z ) is positive in

meaning that agents are more likely to change strategies from "withdraw" to "wait"

moves closer to z from below. The expected sign of (zt

z ) is negative for

1

that the probability of experimentation from "wait" to "withdraw" increases as zt

1

C,

meaning

moves closer

to z from above. The logit regression models are formulated as follows: logit (

A)

= log

logit (

B)

= log

logit (

C)

= log

A

1

A) B

1

=

A

+

1

;

=

B

+

B (zt 1

z );

=

C

+

C (zt 1

z ):

B C

A

C

The results from the logit regression models are listed in table 8. All the coe¢ cients have the expected signs and are highly signi…cant with p-values close to 0. This provides evidence that the subjects do make use of information about

or zt

1

when they decide whether to ‡ip their

strategies. The estimated probability of experimentation is given by: ^A ( ) = ^B (zt

1;

) =

^C (zt

1;

) =

1 ; 1 + exp( 0:91 + 2:45 ) 1 ; 1 + exp[ 0:52 0:50(zt 1 z )] 1 : 1 + exp[2:83 + 0:31(zt 1 z )]

(3) (4) (5)

[Insert Table 8: Logit Regression of Rate of Experimentation –All Data]

5.3

Evaluation of the Algorithm

We evaluate the performance of the evolutionary learning model in two ways. The …rst is to apply the algorithm and use the estimated probability of experimentation in Section 5:2 (equations 3 to 21

5) to simulate the path of the number of subjects choosing to wait and compare the aggregate performance of the simulated economies to that of the experimental economies. In particular, we classify the performance of the simulated economies according to the de…nitions of the 10 performance categories in table 4 and check whether the distribution of the performance categories is consistent with that of the actual experiments. The second is to make out-of-sample predictions of individual strategies. Speci…cally, we use one set of data (the Chinese data) to estimate the rate of experimentation and apply the algorithm to predict the action choice in each of the Canadian observations. The rate of successful predictions can then be used to evaluate the performance of the algorithm. 5.3.1

Simulation

To simulate the path of the number of subjects choosing to wait, we use the same model parameters that were used in the experiment with human subjects: there are 10 simulation subjects, 7 or 9 phases or values of

and 10 rounds in each phase. Each simulation adopts the starting values (S)

in one of the 20 experimental sessions. In the following, we illustrate the simulation process using the starting values of session 3. The starting values of z in the 7 phases are 10, 9, 10, 10, 8, 2 and 1, respectively. In each phase, strategies in the …rst round are set to match the starting value of z. For example, in phase 1, the simulated economy starts with all 10 agents choosing to wait, so that z is equal to 10 in the …rst round. From round 2 on, each arti…cial agent updates its strategy according to the rules of the evolutionary algorithm. In the …rst step, the agent updates its strategy to the best response if it can infer whether or not zt

1

> z ; otherwise, the agent keeps its strategy in the previous period. In the

second step, the agent experiments by ‡ipping a strategy that was decided upon in the …rst step. The probability of experimentation depends on the direction of ‡ipping and whether the agent can deduce zt

1.

For example, if the agent chooses "withdraw" as the strategy in the …rst step and can

infer the value of zt

1,

the probability of changing strategy to "wait" is given by

B (zt 1 ;

) and

can be calculated according to equation (4). After each agent’s strategy is determined, we calculate the number of subjects who choose to wait and the payo¤ to each agent. We conducted 100 runs for each set of starting values. Table 9 lists the frequencies at which the simulated economies fall into each of the 10 performance categories: "NN", "FN", "CN", "H", "RR", "FR", "CR", "L", "SN" and "SR" as de…ned in table 4. An investigation of table 9 shows that the modi…ed evolutionary algorithm captures the main features of the experimental data. For instance, the economies stay close or converge to the non-run equilibrium when

is small. For

0:4, all simulated economies stay close to the non-run equilibrium, spending 100% of the time at "NN" or "FN". When

= 0:5, all economies spend more than 95% of the time at "NN", "FN"

or "CN", and very occasionally at "H", "L" or "CR" when the economy starts with low initial number of agents choosing to wait (S = 4 or 5). When 100% of the time at "RR" or "FR". When

= 0:9, all simulated economies spend

= 0:8, 18 out of the 20 simulated economies spend

22

more than 99% of the time at "RR", "FR" or "CR". With the other 2 economies, due to the high starting value (S = 9), the simulated economies show the possibility of maintaining high level of coordination with a probability

30%. When

= 0:6 or 0:7, as in the actual experiment, the

simulated economies produce very di¤erent results, spending a positive amount of time in each of the …rst 8 performance categories. The evolutionary learning model successfully captures 0:6 and 0:7 as the watershed for coordination. Finally, as in the actual experiment, the simulated economies also exhibit intra-phase switching in games with

0:5.

[Insert Table 9: Performance of Simulated Economies] 5.3.2

Out-of-Sample Prediction

For the out-of-sample prediction, we use the data of the 13 sessions in China to estimate the rates of experimentation, and apply the algorithm to predict the action choice for each of the observations in the 7 sessions of experiment run in Canada. The results for the logit regressions are presented in table 10. Note that the estimated coe¢ cients from regressions with Chinese data are very close to those from regressions with all data (compare tables 8 and 10). This can be taken as evidence that the location does not a¤ect the experimental results in a signi…cant way.11 The process to conduct out-of-sample prediction is as follows. For each Canadian observation, we apply the algorithm using information about the individual’s strategy and payo¤ in the previous period to determine his or her current action s0t . The subject updates strategy to the best response if he or she can deduce the best response and keeps the strategy in the previous period otherwise. Then we calculate the estimated rate of experimentation using the regression results from table 10. The subject ‡ips his or her strategy if and only if the estimated rate of experimentation is > 0:5. If the predicted action, s^t , coincides with the actual action, st , we count it as a successful prediction. The prediction success rates are 63:66% for group A, 90:48% for group B and 97:36% for group C. The overall rate of successful prediction is 91:77%. We take this as another piece of evidence that the evolutionary algorithm performs well in explaining the experimental data. [Insert Table 10: Logit Regression of Rate of Experimentation-Chinese Data]

6

Conclusion

In this paper, we investigate how the level of coordination requirement a¤ects occurrence of bank runs as a result of coordination failures in controlled laboratory environments. We enroll human subjects to play bank-run games de…ned by a demand deposit contract. We …x the rate of return of the bank’s long-term investments to rule out the deterioration of the bank’s assets as the source 11

In addition, we divide the sessions into 2 groups according to location (China versus Canada) and run a two-sided rank-sum test on the mean of the number of subjects choosing to wait across the whole session (excluding the two phases with = 0:1 and 0:2 for sessions 9 and 21). The p-value is 0:663, which suggests the location does not have a signi…cant e¤ect on the experimental results.

23

of bank runs. The only variation in the experiment is that the coordination parameter, which measures the amount of coordination that is required for subjects choosing to wait to receive a higher payo¤ than those who withdraw, changes every 10 periods. We …nd that miscoordination-based bank runs may occur if the required coordination is high enough. Speci…cally, bank runs are frequently (rarely) observed if the coordination parameter is 0:8 (

0:5). When the coordination parameter lies in the indeterminacy region (when it is equal to

0:6 or 0:7), the performance of the experimental economies varies greatly across di¤erent sessions. We also …nd that the sequence followed by the coordination parameter a¤ects the performance of experimental economies, especially for games where the coordination parameter lies in the indeterminacy region. The experimental economy achieves better coordination if the economy starts with low coordination requirement and increases over time, as compared with the case where the parameter is random or decreasing. Finally, the experimental economy exhibits switching between the two equilibria even when economic fundamentals are …xed. The switching tends to occur wen the coordination parameter is

0:5.

In order to capture the experimental results, we combine the evolutionary algorithm in Temzelides (1997) with logit regression models to estimate the rate of experimentation from the experimental data. The evolutionary algorithm is able to replicate the main features of the experimental results and has high out-of-sample predicting power. Finally, DD originally attribute banks runs to the realization of a sunspot variable. In this paper, we do not pursue the sunspot explanation. Instead, we study whether bank runs can occur as a result of pure coordination failures in the absence of a sunspot variable. Sunspot behavior, especially in the context of a model with equilibria that can be Pareto ranked, such as the bank run model, is rarely observed in controlled experiments with human subjects. Du¤y and Fisher (2005) and Fehr et al. (2012) have provided direct evidence of sunspots in the laboratory. However, the payo¤ structure is such that the multiple equilibria are not Pareto rankable or are Pareto equivalent. Most recently, Arifovic et al. (2012) report experimental evidence of sunspot behavior in a general equilibrium type of a model where the payo¤ structure is associated with multiple equilibria that can be Pareto ranked. The experimental results in our paper suggest that the level of coordination requirement may also be an important factor that a¤ects the occurrence of sunspot behavior. The coordination outcomes are clear-cut when the coordination parameter is

0:5 or

0:8 but vary greatly when

the coordination parameter is in the indeterminacy region. This seems to suggest that sunspot behavior is more likely to be observed when the coordination parameter is around 0:6 or 0:7. In our future study, we intend to test directly the sunspot-based theory of bank runs.

24

References Allen, F., Gale, D., 1998. Optimal …nancial crises. Journal of Finance 53(4), 1245-1284. Andolfatto, D. Nosal, E., 2008 Bank incentives, contract design and bank-runs. Journal of Economic Theory 142, 28-47. Andolfatto, D. Nosal, E., Wallace, N., 2007. The role of independence in the Diamond-Dybvig Green-Lin model. Journal of Economic Theory 137, 709-715. Arifovic, J., Evan, G., Kostyshyna, O., 2012. Are sunspots learnable in a general equilibrium model? An experimental investigation. Manuscript. Boyd, J.H., Gomis-Porqueras, P., Kwak, S., Smith, B.D., 2001. A user’s guide to banking crises. Manuscript. Brandts, J., Cooper, D.J., 2006. A change would do you good .... an experimental study on how to overcome coordination failure in organizations. American Economic Review 96, 669-693. Carlsson, H., van Damme, E., 1993. Global games and equilibrium selection. Econometrica 61, 989-1018. Cabrales, A., Nagel, R., Armenter, R., 2007. Equilibrium selection through incomplete information in coordination games: an experimental study. Experimental Economics 10, 221-234. Diamond, D.W., Dybvig, P.H., 1983. Bank runs, deposit insurance, and liquidity. Journal of Political Economy 91, 401-419. Du¤y, J., Fisher, E.O., 2005. Sunspots in the laboratory. American Economic Review 95, 510-529. Du¤y, J., Ochs, J., 2012. Equilibrium selection in static and dynamic entry games. Games and Economic Behavior 76, 97–116. Ennis, H., 2003. Economic fundamentals and bank runs. Federal Reserve Bank of Richmond Economic Quarterly 89, 55-71: Ennis, H., Keister, T., 2009a. Bank runs and institutions: the perils of intervention. American Economic Review 99, 1588-1607. Ennis, H., Keister, T., 2009b. Run equilibria in the Green-Lin model of …nancial intermediation. Journal of Economic Theory 144, 1996-2020. Ennis, H., Keister, T., 2010. Banking panics and policy responses. Journal of Monetary Economics 54, 404-419. Fehr, D., Heinemann, F., Llorente-Saguer, A., 2012. The power of sunspots: an experimental analysis. Working paper, Technische Universität Berlin. Fischbacher, U., 2007. Z-Tree: Zurich toolbox for ready-made economic experiments. Experimental Economics 10(2), 171-178. Garratt, R., Keister, T., 2009. Bank runs as coordination failures: an experimental study. Journal of Economic Behavior and Organization 71, 300-317. Goldstein, I., Pauzner, A., 2005. Demand deposit contracts and the probability of bank runs. Journal of Finance 60(3), 1293-1327. Gorton, G., 1988. Banking panics and business cycles. Oxford Economic Papers 40, 751-781. Green, E.J., Lin, P., 2000. Diamond and Dybvig’s classic theory of …nancial intermediation: what’s missing? Federal Reserve Bank of Minneapolis Quarterly Review 24, 3-13. Green, E.J., Lin, P., 2003. Implementing e¢ cient allocations in a model of …nancial intermediation. Journal of Economic Theory 109, 1-23. 25

Heinemann, F., Nagel, R., Ockenfels, P., 2004. The theory of global games on test: experimental analysis of coordination games with public and private information. Econometrica 72, 1583-99. Heinemann, F., Nagel, R., Ockenfels, P., 2009. Measuring strategic uncertainty in coordination games. Review of Economic Studies 76, 181-221: Hommes, C., 2011. The heterogeneous expectations hypothesis: some evidence from the lab. Journal of Economic Dynamics & Control 35, 1-24. Klos, A., Sträter, N., 2012. How strongly do players react to changes in payo¤ parameters in an experimental bank run game? Working paper, University of Kiel. Kandori, M., Mailath, G.J., Rob, R., 1993. Learning, mutation, and long run equilibria in games. Econometrica 61(1); 29-56: Madiès, P., 2006. An experimental exploration of self-ful…lling bank panics: their occurrence, persistence and prevention. Journal of Business 79, 1836-1866. Morris, S., Shin, H. S., 2001. Rethinking multiple equilibria in macroeconomic modelling. NBER Macroeconomics Annual 15, 139-161. Romero, J., 2011. The e¤ect of hysteresis on equilibrium selection in coordination games. Working Paper, Purdue University. Schotter, A., Yorulmazer, T., 2009. On the dynamics and severity of bank runs: an experimental study. Journal of Financial Intermediation 18, 217-241. Schumacher, L., 2000. Bank runs and currency run in a system without a safety net: Argentina and the ‘Tequila’shock. Journal of Monetary Economics 46(1), 257-277. Steiner, J., Colin, S., 2008. Contagion through learning. Theoretical Economics 3, 431-458. Temzelides, T., 1997. Evolution, coordination and banking panics. Journal of Monetary Economics 40, 163-183. Van Huyck, J., Battalio, R., Beil, R., 1990. Tacit coordination games, strategic uncertainty, and coordination failure. American Economic Review 180(1), 234-48. Van Huyck, J., Battalio, R., Beil, R., 1991. Strategic uncertainty, equilibrium Selection, and coordination failure in average opinion games. Quarterly Journal of Economics 106(3), 885-910. Weber, R. A., 2006. Managing growth to achieve e¢ cient coordination in large groups. American Economic Review 96(1), 114-126. Young, P. H., 1993. The evolution of conventions. Econometrica 61, 57-84.

26

Table 1: Values of r and η Used in the Experiment r η

1.05 1.11 1.18 1.25 1.33 1.43 1.54 1.67 1.82 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

27

Table 2: Experimental Design Session No. 1 2 3 4 5 6 7 8

Grid

17 18 19 20

η

Date

0.1;0.2;0.3;0.5;0.7;0.8;0.9 0.1;0.2;0.3;0.5;0.7;0.8;0.9 0.1;0.2;0.3;0.5;0.7;0.8;0.9 0.1;0.2;0.3;0.5;0.7;0.8;0.9 0.9;0.8;0.7;0.5;0.3;0.2;0.1 0.9;0.8;0.7;0.5;0.3;0.2;0.1 0.9;0.8;0.7;0.5;0.3;0.2;0.1 0.9;0.8;0.7;0.5;0.3;0.2;0.1

2006-11 2008-11 2006-11 2008-11 2007-06 2008-11 2008-11 2008-10

UIBE UIBE UIBE SFU

0.1:0.1:0.9 0.3:0.1:0.9 0.3:0.1:0.9 0.3:0.1:0.9

2012-03 2012-03 2012-05 2012-05

Decrease

UIBE UIBE UIBE SFU

0.9:-0.1:0.3 0.9:-0.1:0.3 0.9:-0.1:0.3 0.9:-0.1:0.3

2012-03 2012-04 2012-06 2012-05

Random

UIBE UIBE UIBE SFU

0.4;0.9;0.1;0.8;0.2;0.7;0.3;0.6;0.5 0.7;0.3;0.9;0.6;0.4;0.8;0.5 0.8;0.4;0.7;0.3;0.6;0.9;0.5 0.8;0.4;0.7;0.3;0.6;0.9;0.5

2012-03 2012-04 2012-05 2012-05

Coarse Decrease

Increase

Fine

Location UIBE UIBE SFU UofM UIBE UIBE SFU UofM

Increase

9 10 11 12 13 14 15 16

Order

28

Table 3: Starting, Terminal and Mean Value of Number of Subjects Choosing to Wait Session 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

S 10 9 10 8 10 10 10 10

0.1 T 10 10 9 10 10 10 10 9

S 10 10 10 9 8 10 10 8

0.3 T 9 10 9 10 10 10 10 9

M S 9.7 10 9.7 9.8 9.7 10 10 9.6

10 9.9 10 10 9 7 10 10 10 10 10 9.6 10 10 10 9 9 9 9

10 10 9 8 10 10 10 10 10 9 10 9

10 10 9 8.5 10 10 10 9.9 9.8 9.5 9.9 9.7

M 10 9.8 9.8 9.7 10 10 10 9.7

S 9 8 9 10 10 10 10 10

10 10 9.9 9

7

0.2 T 10 10 9 10 10 10 10 10

M 9.9 9.5 9.7 9.9 9.9 10 10 10

10 10 10 9 10 10 10 10 10 9 10 10

0.4 T M

10 10 10 9 10 10 10 10 10 9 10 10

10 10 10 8.6 10 10 10 10 10 9.3 10 9.7

S 8 9 10 9 4 6 10 4

0.5 T 10 10 9 10 8 9 10 9

M S 9.3 9.6 9.6 9.9 7.2 8.7 9.8 7.8

10 10 10 10 5 10 9 7 7 9 10 8

10 10 10 10 10 10 10 10 9 9 10 10

10 9.9 10 9 9.3 9.9 9.9 8.2 8.5 9.3 10 9.7

29

9 10 10 9 3 4 3 3 3 2 10 7

0.6 T M

10 10 10 9 0 9 10 0 0 0 10 10

9.8 10 10 8.7 2.7 7.8 8.1 0.4 0.6 0.7 10 9.2

S 6 7 8 7 1 0 6 1

0.7 T 6 0 2 0 0 0 9 0

M 7.5 2.1 3.1 2.2 0.4 0.5 7.9 1

S 2 3 2 1 1 2 6 1

0.8 T 0 0 0 0 0 0 0 1

M 1.2 0.4 1.1 0.7 0.5 0.7 1.9 0.6

S 1 0 1 0 1 1 4 3

0.9 T 0 0 0 0 1 0 2 3

M 0.2 0 0.2 0.1 0.6 0.5 2.2 2

10 10 10 8 3 3 1 3 3 3 10 5

10 10 10 10 0 0 0 0 0 3 10 0

9.7 10 10 8.4 1 0.7 0.2 0.5 0.3 1.7 10 1.4

9 8 9 8 3 1 2 3 5 4 9 6

0 1 0 5 0 1 0 0 0 0 0 1

1.8 2.4 2.3 4.6 0.6 0.7 0.3 0.3 0.8 0.9 3 2.5

6 3 1 4 3 3 3 4 3 3 9 6

0 0 0 2 1 1 0 1 0 0 0 1

0.8 0.9 0.1 2 0.9 1.1 0.8 1.9 1.1 0.7 2.5 1.5

Table 4- Performance Classification Performance Category

Label

Criterion

Very close to the non-run equilibrium

NN

M≥9

Fairly close to the non-run equilibrium

FN

8≤M<9

Converging to the non-run equilibrium

CN

5
Moderate high coordination

H

5
Very close to the run equilibrium

RR

M≤1

Fairly close to the run equilibrium

FR

1
Converging to the run equilibrium

CR

2
Moderate low coordination

L

22

Switch to non-run equilibrium Switch to run equilibrium

SN SR

S<5 and T≥8 S>5 and T≤2

30

Table 5: Classification of Performance of Experimental Economies η Session 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

0.1

0.2

0.3

NN NN NN NN NN NN NN NN

NN NN NN NN NN NN NN NN

NN NN NN NN NN NN NN NN

NN

NN

NN

NN

NN NN NN FN NN NN NN NN NN NN NN NN

0.4

0.5

0.6

NN NN NN NN CN; SN FN NN CN; SN NN NN NN FN NN NN NN NN NN NN NN NN

NN NN NN NN NN NN NN FN FN NN NN NN

NN NN NN FN CR CN; SN FN; SN RR RR RR NN NN

31

0.7

0.8

0.9

H CR; SR CR; SR CR; SR RR RR CN RR

FR RR FR RR RR RR FR; SR RR

RR RR RR RR RR RR CR FR

NN NN NN FN RR RR RR RR RR FR NN FR

FR; SR CR; SR CR; SR L RR RR RR RR RR RR CR; SR CR; SR

RR;SR RR RR FR RR FR RR FR FR RR CR; SR FR; SR

Table 6: Rank-sum Tests of the Order and Grid Effect Table 6a: Rank Sum Tests for η=0.7 Order Effect - Old data Sessions 1-4 Sessions 5-8 Z-value p-value (1-sided)

Sample size 4 4

S 7.000 2.000 2.205 1.38%

T 2.000 2.250 0.331 37.05%

M 3.725 2.450 1.155 12.41%

Sample size 4 4

S 9.500 2.500 2.428 0.76%

T 9.750 0.000 2.646 0.41%

M 9.625 0.600 2.323 1.01%

Order Increase Random

Sample size 4 4

S 9.500 5.500 1.703 4.43%

T 9.750 5.000 2.000 2.28%

M 9.625 5.125 1.479 6.96%

Order Random Decrease

Sample size 4 4

S 5.500 2.500 1.654 4.91%

T 5.000 0.000 1.512 6.53%

M 5.125 0.600 1.443 7.45%

Sample size 4 4

S 9.500 7.000 2.247 1.24%

T 9.750 2.000 2.477 0.66%

M 9.625 3.725 2.323 1.01%

Order Increase Decrease

Order Effect - New data Order Sessions 9-12 Increase Sessions 13-16 Decrease Z-value p-value (1-sided)

Sessions 9-12 Sessions 17-20 Z-value p-value (1-sided)

Sessions 17-20 Sessions 13-16 Z-value p-value (1-sided)

Grid Effect - New and old data Order Sessions 9-12 Increase Sessions 1-4 Increase Z-value p-value (1-sided)

32

Table 6: Rank-sum Tests of the Order and Grid Effect Continued

Table 6b: Rank Sum Tests for η=0.6

Sessions 9-12 Sessions 13-16 Z-value p-value (1-sided)

Sessions 9-12 Sessions 17-20 Z-value p-value (1-sided)

Sessions 17-20 Sessions 13-16 Z-value p-value (1-sided)

Order Increase Decrease

Sample size 4 4

S 9.500 3.250 2.397 0.83%

T 9.750 4.750 1.559 5.95%

M 9.625 4.750 2.323 1.01%

Order Increase Random

Sample size 4 4

S 9.500 5.500 1.488 6.84%

T 9.750 5.000 1 15.87%

M 9.625 5.125 1.183 11.84%

Order Random Decrease

Sample size 4 4

S 5.500 3.250 0.461 32.23%

T 5.000 4.750 0.316 37.91%

M 5.125 4.750 0.577 28.19%

33

Table 7: Information Last period's Last period's Can deduce exact payoff π t-1 value of z t-1 ? choice s t-1 r No Withdraw [1,r) Yes >r Yes Wait (0,r) Yes 0 No

Can deduce Number of Percentage in best response? observations all observations No 474 3.66 Yes 4362 33.66 Yes 7449 57.48 Yes 267 2.06 Yes 408 3.15

34

Choice before experimentation s t '

Group

Withdraw (keep old strategy) Withdraw (update to best response) Wait (update to best response) Withdraw (update to best response) Withdraw (update to best response)

A B C B A

Table 8: Logit Regression of Rate of Experimentation - All Data Table 8a constant η

coefficient 0.91 -2.45

std error 0.26 0.39

t-statistic 3.47 -6.34

p-value 0.00 0.00

coefficient 0.52 0.50

std error 0.13 0.02

t-statistic 3.87 21.76

p-value 0.00 0.00

coefficient -2.83 -0.31

std error 0.26 0.05

t-statistic -11.05 -6.05

p-value 0.00 0.00

Table 8b constant z t‐1 ‐z*

Table 8c constant z t‐1 ‐z*

Notes: Table 8a: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who cannot infer z t-1 Table 8b: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who can infer z t-1 Table 8c: result from regression to esitmate the experimentation rate from "wait" to "withdraw"; subjects can infer z t-1 in this case.

35

Table 9: Performance of Simulated Economies

1

2

3

4

η 0.1 0.2 0.3 0.4 NN 100 100 100 FN 0 0 0 CN 0 0 0 H 0 0 0 RR 0 0 0 FR 0 0 0 CR 0 0 0 L 0 0 0 SN 0 0 0 SR 0 0 0

0.5 81 19 0 0 0 0 0 0 0 0

NN 100 FN 0 CN 0 H 0 RR 0 FR 0 CR 0 L 0 SN 0 SR 0

100 0 0 0 0 0 0 0 0 0

0.7 0 1 0 0 0 32 58 9 0 89

0.8 0 0 0 0 78 22 0 0 0 0

0.9 0 0 0 0 98 2 0 0 0 0

95 5 0 0 0 0 0 0 0 0

1 4 1 3 0 17 65 9 0 84

0 0 0 0 65 33 2 0 0 0

0 0 0 0 100 0 0 0 0 0

NN 100 100 100 FN 0 0 0 CN 0 0 0 H 0 0 0 RR 0 0 0 FR 0 0 0 CR 0 0 0 L 0 0 0 SN 0 0 0 SR 0 0 0

99 1 0 0 0 0 0 0 0 0

37 27 0 12 0 0 22 2 0 27

0 0 0 0 78 22 0 0 0 0

0 0 0 0 98 2 0 0 0 0

NN 100 100 FN 0 0 CN 0 0 H 0 0 RR 0 0 FR 0 0 CR 0 0 L 0 0 SN 0 0 SR 0 0

95 5 0 0 0 0 0 0 0 0

1 4 1 3 0 17 65 9 0 84

0 0 0 0 86 14 0 0 0 0

0 0 0 0 100 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

0.6

5

6

7

8

0.1 0.2 0.3 100 100 99 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.4

0.5 2 44 49 0 0 0 1 4 96 0

0.6

0.7 0 0 0 0 53 42 4 1 0 0

0.8 0 0 0 0 86 14 0 0 0 0

0.9 0 0 0 0 98 2 0 0 0 0

100 100 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

39 53 8 0 0 0 0 0 0 0

0 0 0 0 68 28 3 1 0 0

0 0 0 0 78 22 0 0 0 0

0 0 0 0 98 2 0 0 0 0

100 100 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

0 1 0 0 0 32 58 9 0 89

0 0 0 1 10 76 13 0 0 99

0 0 0 0 86 14 0 0 0 0

100 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2 44 49 0 0 0 1 4 96 0

0 0 0 0 53 42 4 1 0 0

0 0 0 0 86 14 0 0 0 0

0 0 0 0 95 5 0 0 0 0

99 1 0 0 0 0 0 0 0 0

36

Table 9: Performance of Simulated Economies Continued

9

10

11

12

0.1 0.2 0.3 0.4 NN 100 100 100 98 FN 0 0 0 2 CN 0 0 0 0 H 0 0 0 0 RR 0 0 0 0 FR 0 0 0 0 CR 0 0 0 0 L 0 0 0 0 SN 0 0 0 0 SR 0 0 0 0

0.5 99 1 0 0 0 0 0 0 0 0

0.6 84 14 0 1 0 0 1 0 0 1

0.7 76 12 0 8 0 0 4 0 0 10

0.8 0.9 23 0 8 0 0 0 25 0 0 51 0 49 44 0 0 0 0 0 70 100

NN FN CN H RR FR CR L SN SR

100 0 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

96 4 0 0 0 0 0 0 0 0

76 12 0 8 0 0 4 0 0 10

0 0 0 1 0 43 55 1 0 99

0 0 0 0 95 5 0 0 0 0

NN FN CN H RR FR CR L SN SR

100 0 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

96 4 0 0 0 0 0 0 0 0

76 12 0 8 0 0 4 0 0 10

0 0 0 1 0 43 55 1 0 99

0 0 0 0 95 5 0 0 0 0

NN FN CN H RR FR CR L SN SR

90 10 0 0 0 0 0 0 0 0

97 3 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

84 14 0 1 0 0 1 0 0 1

37 27 0 12 0 0 22 2 0 27

0 0 0 1 0 43 55 1 0 99

0 0 0 0 86 14 0 0 0 0

0.1

13

14

15

16

0.2

0.3 0.4 100 98 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.5 23 64 12 1 0 0 0 0 0 0

0.6 0 0 8 0 2 19 28 43 12 0

0.7 0 0 0 0 22 59 15 4 0 0

0.8 0 0 0 0 65 33 2 0 0 0

0.9 0 0 0 0 95 5 0 0 0 0

100 0 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

0 5 11 1 0 16 26 41 20 0

0 0 0 0 22 59 15 4 0 0

0 0 0 0 86 14 0 0 0 0

0 0 0 0 95 5 0 0 0 0

100 0 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

95 5 0 0 0 0 0 0 0 0

0 0 8 0 2 19 28 43 12 0

0 0 0 0 53 42 4 1 0 0

0 0 0 0 78 22 0 0 0 0

0 0 0 0 95 5 0 0 0 0

100 0 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

61 36 3 0 0 0 0 0 0 0

0 0 8 0 2 19 28 43 12 0

0 0 0 0 22 59 15 4 0 0

0 0 0 0 65 33 2 0 0 0

0 0 0 0 86 14 0 0 0 0

37

17

18

19

20

0.1 0.2 0.3 98 100 98 2 0 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.4 98 2 0 0 0 0 0 0 0 0

0.5 61 36 3 0 0 0 0 0 0 0

0.6 0 0 8 0 2 19 28 43 12 0

0.7 0 0 0 0 22 59 15 4 0 0

0.8 0 0 0 0 30 63 7 0 0 0

0.9 0 0 0 0 95 5 0 0 0 0

98 2 0 0 0 0 0 0 0 0

97 3 0 0 0 0 0 0 0 0

95 5 0 0 0 0 0 0 0 0

0 0 8 0 10 21 21 40 12 0

0 0 0 0 22 59 15 4 0 0

0 0 0 0 47 49 4 0 0 0

0 0 0 0 95 5 0 0 0 0

98 2 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

99 1 0 0 0 0 0 0 0 0

96 4 0 0 0 0 0 0 0 0

76 12 0 8 0 0 4 0 0 10

23 8 0 25 0 0 44 0 0 70

0 0 0 0 0 57 43 0 0 100

98 2 0 0 0 0 0 0 0 0

98 2 0 0 0 0 0 0 0 0

81 19 0 0 0 0 0 0 0 0

40 51 2 4 0 0 2 1 0 2

0 0 0 0 2 52 38 8 0 0

0 0 0 1 10 76 13 0 0 99

0 0 0 0 51 49 0 0 0 100

Table 10: Logit Regression of Rate of Experimentation - Chinese Data Table 10a constant η

coefficient 0.62 -2.34

std error 0.37 0.55

t-statistic 1.69 -4.25

p-value 0.09 0.00

coefficient 0.50 0.54

std error 0.18 0.03

t-statistic 2.81 17.51

p-value 0.00 0.00

coefficient -3.77 -0.31

std error 0.53 0.10

t-statistic -7.17 -2.96

p-value 0.00 0.00

Table 10b constant z t‐1 ‐z*

Table 10c constant z t‐1 ‐z*

Notes: Table 10a: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who cannot infer z t-1 Table 10b: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who can infer z t-1 Table 10c: result from regression to esitmate the experimentation rate from "wait" to "withdraw"; subjects can infer z t-1 in this case.

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Figure 1a: Experimental Results (Sessions 1-8)               0.1   0.2  0.3  0.4  0.5   0.6  0.7  0.8  0.9 

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Appendix: Instructions – Increasing Treatment with Fine Grid This experiment has been designed to study decision-making behavior in groups. During today's session, you will earn income in an experimental currency called “experimental dollars” or for short ED. At the end of the session, the currency will be converted into dollars. 1 ED corresponds to 0.2 dollars. You will be paid in cash. The participants may earn different amounts of money in this experiment because each participant's earnings are based partly on his/her decisions and partly on the decisions of the other group members. If you follow the instructions carefully and make good decisions, you may earn a considerable amount of money. Therefore, it is important that you do your best. Please read these instructions very carefully and ask questions whenever you want to. Description of the task You and 9 other people (there are 10 of you altogether) have 1 ED deposited in an experimental bank. You must decide whether to withdraw your 1 ED, or wait and leave it deposited in the bank. The bank promises to pay r>1 EDs to each withdrawer. The money that remains in the bank will earn interest rate R>r. At the end, the bank will divide what it has evenly among people who choose to wait and leave money in the bank. Note that if too many people desire to withdraw, the bank may not be able to fulfill the promise to pay r to each withdrawer (remember that r>1). In that case, the bank will divide the 10 EDs evenly among the withdrawers and those who choose to wait will get nothing. Your payoffs depend on your own decision and the decisions of the other 9 people in the group. Specifically, how much you receive if you make a withdrawal request or how much you earn by leaving your money deposited depends on how many people in the group place withdrawing requests. Let e be the number of people who request withdrawals. The payoff (in ED) to those who withdraw will be: min{r,

10 10 } or the minimum of r and . e e

The payoff (in ED) to those who leave money in the bank will be: max{0,

10  e  r 10  e  r R } or the maximum of 0 and R. 10  e 10  e

In the experiment, R will be fixed at 2, and r will take 8 values: 1.38, 1.18, 1.25, 1.33, 1.43, 1.54, 1.67 and 1.82. To facilitate your decision, the payoff tables 0∼9 list the payoffs if n of the other 9 subjects request to withdraw (n is unknown at the time when you make the withdrawing decision - it can be any integer from 1 to 9 - and you have to guess it) for each value of r. Table 0 will be used for practice, and table 1∼7 will be used for the formal experiment. Let's look at two examples: Example 1. Use table 0 where r=1.38. Suppose that 3 other subjects request withdrawals. Your payoff will be 1.38 if you request to withdraw: the number of withdrawing requests e will 10 10    2.5 =1.38. If you choose to wait and be 3+1=4, so your payoff will be min r  1.38,  4 e   1

leave money in the bank, your payoff will be 1.67: the number of withdrawing requests will be 10  3  1.38  10  e  r  R  2  1.67 =1.67. e=3, so your payoff is max 0, 10 - 3  10  e  Example 2. Continue using table 0 where r=1.38. Suppose that 8 of other subjects request 10 10    1.11 =1.11. Your withdrawals. Your payoff for withdrawing will be min r  1.38,  e 9   payoff for waiting and leaving money in the bank will be e r 10 10 8 1 . 38       max 0, R  2  - 1.04 = 0. 10 - 8  10  e 

Now let's take a closer look at the tables. Notice the following features of tables: 

The payoff to withdrawing is more stable: it is fixed at r for most of the time and is bounded below by 1.



The payoff to waiting and leaving money in the bank is more volatile. When n - the number of other people requesting withdrawals - is small, waiting and leaving money in the bank offers higher payoff than withdrawing. The opposite happens when n is large enough. For your convenience, bold face is used to identify the threshold value of n at which the payoffs to the two choices are equal or almost equal (the difference is ≤0.02 ED).



The threshold values of n vary from table to table. The general pattern is that it is smaller when r is bigger.

Note you are not allowed to ask people what they will do and you will not be informed about the other people's decisions. You must guess what other people will do - how many of the other 9 people will withdraw - and act accordingly. Procedure

You will perform the task described above for 70 times. Each time is called a period. Each period is completely separate. I.e., you start each period with 1ED in the experimental bank. You will keep the money you earn in every period. At the end of each round, the computer screen will show you your decision and payment for that period. Information for earlier periods and your cumulative payment is also provided. Note that the payment scheme changes every 10 periods, so please use the correct payoff table:  Table 1 for period 1-10 (r=1.18),  Table 2 for period 11-20 (r=1.25),  Table 3 for period 21-30 (r=1.33),  Table 4 for period 31-40 (r=1.43),  Table 5 for period 41-50 (r=1.54),  Table 6 for period 51-60 (r=1.67), and  Table 7 for period 61-70 (r=1.82). You will be reminded when you need to change to a new table, pay attention to the message.

2

Beside the 70 paid periods, you will also be given 10 trials (periods -9 to 0) to practice and get familiar with your task. You will not be paid for the trials. Please use table 0 (r=1.38) for the trials. After the practice periods, you will have a chance to ask more questions before the experiment formally start. You will be paid for each of the 70 formal periods. Computer instructions

You will see three types of screens: the decision screen, the payoff screen and the waiting screen. Your withdrawing decisions will be made on the decision screen as shown in figure 1. You can choose to withdraw money or leave money in the bank by pushing one of the two buttons. Note that your decision will be final once you press the buttons, so be careful when you make the move. The header provides information about what period you are in and the time remaining to make a decision. After the time limit is reached, you will be given a flashing reminder “please reach a decision!”. The screen also shows which table you should look at.

Figure 1: The decision screen After all subjects input their decisions, a payoff screen will appear as shown in figure 2. You will see your decision and payoff in the current period. The history of your decisions and payoffs as well as your cumulative payoff is also provided. After finishing reading the information, click on the “continue” button to go to the next period. You will have 30 seconds to review the information before a new period starts.

3

Figure 2: The payoff screen You might see a waiting screen (as shown in figure 3) following the decision or payoff screens - this means that other people are still making decisions or reading the information, and you will need to wait until they finish to go to the next step.

Figure 3: The waiting screen Payment

At the end of the entire experiment, the supervisor will pay you in cash. Your earnings in dollars will be: total payoff (in ED)×0.2.

4

Table 0 (for practice): payoff if n of other 9 subjects withdraw r=1.38 (period -9 to 0)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.38

2.00

1

1.38

1.92

2

1.38

1.81

3

1.38

1.67

4

1.38

1.49

5

1.38

1.24

6

1.38

0.86

7

1.25

0.23

8

1.11

0.00

9

1.00

0.00

Table 1: payoff if n of other 9 subjects withdraw r=1.18 (period 1-10)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.18

2.00

1

1.18

1.96

2

1.18

1.91

3

1.18

1.85

4

1.18

1.76

5

1.18

1.64

6

1.18

1.46

7

1.18

1.16

8

1.11

0.56

9

1.00

0.00

5

Table 2: payoff if n of other 9 subjects withdraw r=1.25 (period 11-20)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.25

2.00

1

1.25

1.94

2

1.25

1.88

3

1.25

1.79

4

1.25

1.67

5

1.25

1.50

6

1.25

1.25

7

1.25

0.83

8

1.11

0.00

9

1.00

0.00

Table 3: payoff if n of other 9 subjects withdraw r=1.33 (period 21-30)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.33

2.00

1

1.33

1.93

2

1.33

1.84

3

1.33

1.72

4

1.33

1.56

5

1.33

1.34

6

1.33

1.01

7

1.25

0.46

8

1.11

0.00

9

1.00

0.00

6

Table 4: payoff if n of other 9 subjects withdraw r=1.43 (period 31-40)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.43

2.00

1

1.43

1.90

2

1.43

1.79

3

1.43

1.63

4

1.43

1.43

5

1.43

1.14

6

1.43

0.71

7

1.25

0.00

8

1.11

0.00

9

1.00

0.00

Table 5: payoff if n of other 9 subjects withdraw r=1.54 (period 41-50)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.54

2.00

1

1.54

1.88

2

1.54

1.73

3

1.54

1.54

4

1.54

1.28

5

1.54

0.92

6

1.43

0.38

7

1.25

0.00

8

1.11

0.00

9

1.00

0.00

7

Table 6: payoff if n of other 9 subjects withdraw r=1.67 (period 51-60)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.67

2.00

1

1.67

1.85

2

1.67

1.67

3

1.67

1.43

4

1.67

1.11

5

1.67

0.66

6

1.43

0.00

7

1.25

0.00

8

1.11

0.00

9

1.00

0.00

Table 7: payoff if n of other 9 subjects withdraw r=1.82 (period 61-70)

n

payoff if withdraw

payoff if wait & leave money in the bank

0

1.82

2.00

1

1.82

1.82

2

1.82

1.59

3

1.82

1.30

4

1.82

0.91

5

1.67

0.36

6

1.43

0.00

7

1.25

0.00

8

1.11

0.00

9

1.00

0.00

8