Experimental Evidence of Bank Runs as Pure Coordination

Failures∗

Jasmina Arifovic† Janet Hua Jiang‡§ Yiping Xu¶

August 5, 2012

Abstract

This paper investigates how the level of coordination requirement affects the occurrence

of bank runs as a result of pure coordination failures in controlled laboratory environments.

The finding is that miscoordination-based bank runs can be observed when the coordination

parameter, defined as the amount of coordination among depositors choosing to wait that is

required for them to receive a higher payoff than depositors who choose to withdraw, is high

enough. In particular, we find a threshold result for the occurrence of bank runs: when the

coordination parameter is below (above) certain values, the experimental economies stay close

to or converge to the non-run (run) equilibrium. When the coordination parameter takes the

threshold values, the outcomes of the experimental economies vary widely and are hard to

predict. Some learning and hysteresis effects are also detected when the coordination parameter

lies around the threshold values. Finally, we show that behavior of human subjects observed in

the laboratory can be well accounted for by the logit evolutionary algorithm.

JEL Categories: D83, G20

Keywords: Bank Runs, Experimental Studies, Evolutionary Algorithm, Coordination Games

∗We appreciate the comments and suggestions from John Duffy, three anonymous referees and the Editor. Wewould also like to thank participants at the CEA Meeting, the ESA Asia Pacific Meeting, the ESA North AmericanMeeting, the Chicago Fed Workshop on Money, Banking, Payments and Finance, and the seminar participants atthe International Monetary Fund and the University of Winnipeg. This research was funded by SSHRC and theUniversity of International Business and Economics 211 grant. The views expressed in this paper are those of theauthors. No responsibility for them should be attributed to the Bank of Canada.†Department of Economics, Simon Fraser University. E-mail: arifovic@sfu.ca.‡Corresponding author.§Funds Management and Banking Department, Bank of Canada. Address: Bank of Canada, 234 Wellington Street,

Ottawa, Ontario, K1A 0G9, Canada. Tel: 1(613)782-7437. Fax: 1(613)782-8751. E-mail: jjiang@bankofcanada.ca.¶School of International Trade and Economics, University of International Business and Economics. E-mail:

xuyiping@uibe.edu.cn.

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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 financial 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 profitable 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 first view

(represented by DD) is that bank runs are the result of pure coordination failures. The bank run

model in DD has two symmetric self-fulfilling Nash equilibria. In one equilibrium, depositors choose

to withdraw only when they need the 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 fire-sale prices

and makes the initial fear a self-fulfilling 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 and this makes it diffi 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 different factors that

may cause bank runs in the laboratory. In this study, we would like to focus on coordination failures

as a possible source for bank runs. To that goal, we fix 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 specific design of the experiment is inspired by the evolutionary learning literature. The

evolutionary algorithm developed by Young (1993) and Kandori et al. (1993) has been introduced as

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an equilibrium selection mechanism in many rational expectation models with multiple equilibria.

Temzelides (1997) applies the algorithm to a repeated version of the DD model. He proves a

limiting case in which 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 less than half

of patient consumers are required to coordinate on "wait" so that "wait" gives a higher payoff

than "withdraw".1 Following Temzelides (1997), we conjecture that the coordination parameter

—which measures the amount of coordination (the fraction of depositors choosing to wait) that

is required to generate enough complementarity among depositors who wait so that they earn a

higher payoff than those who choose to withdraw —affects the decision of depositors and in turn

the occurrence of bank runs. In each of the 20 sessions of experiment we run, a group of 10 subjects

acts as depositors deciding whether to withdraw money early or wait. Each session has 7 or 9

phases that feature different levels of coordination requirement ranging from 0.1 to 0.9. Within

each phase, subjects repeat the same game for 10 periods. This allows us to check how the value

of the coordination requirement affects the occurrence of bank runs. The combination of including

a wide range of values of the coordination parameter and repeating the same game with the same

level of coordination requirement also helps us catch its effect on the learning process and the

evolution of the coordination behavior. In addition, we conduct sessions with different orderings of

the coordination parameter to detect potential hysteresis effect.

There are three main findings from the experiment. The first is that there is a threshold

phenomenon concerning the occurrence of bank runs. When the required amount of coordination

is ≤ 0.5, all experimental economies stay close or converge to the non-run equilibrium. When

the required amount of coordination is ≥ 0.8, all experimental economies stay close or convergeto the run equilibrium. The economies perform very differently when η = 0.6 or 0.7. Second,

some hysteresis effect is detected. The experimental economies tend to coordinate better and

have less bank runs, especially at threshold values of 0.6 and 0.7, when the economy starts with

easier coordination and increases gradually over time in contrast to the cases where the level

of coordination requirement decreases or changes in a random pattern. Third, we detect some

learning effect when the coordination parameter is close to the threshold values: some experimental

economies experience switches starting with low (high) numbers of people choosing to wait but

evolving to the non-run (run) equilibrium.

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 goes to

zero and predicts a threshold of 0.5 for the occurrence of bank runs. The limiting case therefore

does not provide a satisfactory explanation of the experimental data. In particular, the threshold

observed in the experiment seems to be higher than 0.5 (at around 0.6 and 0.7). We show that the

logit evolutionary algorithm, which estimates the rate of experimentation from experimental data

through logit regressions, is successful in explaining the behavior of subjects.

1Ennis (2003) also includes a discussion of risk dominance as an equilibrium selection device in the context of theDD model of bank runs.

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The rest of the paper proceeds as follows. In Section 2, we discuss related experimental works

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 findings. Section 5 develops the logit 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 works 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 first experimental study of miscoordination-induced bank runs based

on the DD model. The paper’s emphasis is on the effectiveness of alternative ways to prevent bank

runs, including suspension of payments and deposit insurance. In the process, the paper also offers

some evidence that the severity of bank runs is affected by the level of coordination requirement.2

However, the paper studies only two levels of coordination requirement, 30% and 70%. When 30%

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 affected 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 examines the

factors that affect the speed of withdrawals, including the number of opportunities to withdraw

and the existence of insiders. The results are (i) 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, and (ii) the presence of insiders, who know

the quality of the bank, significantly affects the dynamics of bank runs and mitigates the severity

of bank runs.

Klos and Sträter (2010) 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 (varying) quality of the bank’s long-term assets. The theory predicts that

2The paper also shows the payoff differential between the run and non-run equilibria influences the severity ofbank runs.

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depositors employ a threshold strategy choosing to withdraw (wait) if the signal is below (above) a

cut-off point. Klos and Sträter find that subjects do follow threshold strategies and the thresholds

increase with the short-term rate promised to withdrawers as predicted by the theory. However,

compared to the theoretical predictions of the global games approach, the reaction to an increase

in the repayment rate is less pronounced. They also find that the level-k approach, in which level-1

types assume that level-0 types play a random threshold strategy, fits the data considerably better.

Compared with the existing experimental literature on bank runs, our paper has a very different

focus, which is to study how the level of coordination requirement affects 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 fixed, there is no uncertainty in the demand for liq-

uidity, subjects only have one withdrawing opportunity in each game, all payoff-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 (achieved

through a change in the short-term repayment rate promised to withdrawers) which is also public

information. To investigate the effect of coordination requirement in a systematic way, we have the

coordination parameter taking 7 or 9 values in each session ranging from 0.1 to 0.9. Each value

of the coordination parameter is maintained for 10 periods to facilitate the study of the learning

effect. In addition, different orderings of the coordination parameter is adopted to detect potential

hysteresis effect.

More generally, our paper is also related to experimental studies of coordination games with

strategic complementarity, which generates multiple equilibria that can be Pareto ranked. In these

games there is often a tension between effi ciency and security. The experimental studies of 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 payoff-dominant

strategy.3 These two studies consider only one game with a tension between effi ciency and security,

and they do not provide a measure to quantify the riskiness associated with the payoff-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 payoff-dominant strategy by the

coordination parameter. By observing how subjects behave in a series of games characterized by

different levels of coordination requirement, we can study how the coordination behavior is affected

by the level of risk and identify the levels of risk that are deemed to be "too risky". In addition,

the evolution of the coordination behavior may well be affected by the level of the coordination

requirement. Van Huyck et al. (1991) find that the initial coordination behavior well predicts the

final outcomes in repeated coordination games. However, our experimental results show that the

result does not hold when the coordination parameter takes certain values. 4

3Van Huyck et al. (1990) study the average opinion game, and Van Huyck et al. (1991) study the minimum-effortgame.

4The earlier experimental works on coordination games have offered systematic studies of the effects of the numberof players and the optimization premium by including them as treatment variables. Van Huyck et al. (1990, 1991)show that subjects tend to choose the secure action more often with a larger group size. Battalio et al. (2001)

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Finally, we would like to discuss several experimental papers on global games. The global

game equilibrium is proposed as a refinement concept for the so called "entry" games: binary-

choice games ("enter" or "not enter") of complete information with strategic complementarity and

multiple equilibria. The payoff to "enter" exceeds that to "not enter" if the number of entrants

exceeds a threshold. Carlsson and van Damme (1993) show that the multiplicity can be resolved by

the introduction of incomplete information.5 One way is to assume players receive private signals

about a payoff-relevant variable (all players start with a common prior belief about the distribution

of the variable). After a player receives her private signal, she forms a posterior belief about the

distribution of the variable. When she chooses her action, she considers the global game, or all

possible games each characterized by a different value in the support of the posterior distribution.

Unlike the original game of complete information with multiple equilibria, the global game of

incomplete information has a unique Bayesian perfect equilibrium, where players adopt a threshold

strategy choosing to enter if and only if their signals are favorable enough. Heinemann et al. (2004)

provide the first experimental study to test the prediction of the global game theory. They conduct

a context-free experiment to test the currency attack model of Morris and Shin (1998). The main

treatment variable is whether there is accurate public information or noisy private information

about a payoff-relevant variable. The principal finding is that the observed behavior is similar

in both situations: individuals follow threshold strategies and choose to enter only if the payoff-

relevant variable is favorable enough (although the threshold values may differ). The study by

Heinemann et al. 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) and Duffy and Ochs (2010).6

To study miscoordination-induced bank runs, we provide subjects with common information

about all payoff-relevant variables. In view of earlier studies of global games, we expect to observe

threshold behaviors as well in our experiment as the coordination requirement parameter (which

is also a payoff relevant variable) changes in a session. However, we have a different focus: we

would like to investigate systematically the effect of the coordination requirement as measured by

the coordination parameter. Our experimental design also has features that facilitate the study of

learning and hysteresis effects. As discussed earlier, the combination of including a wide range of

values of the coordination parameter and repeating the same game with the same level of coordi-

nation requirement allows us to catch its effect on the learning process. The adoption of different

investigate the effect of optimization premium, which measures the penalty attached to not playing a best-response.The main finding is that a higher optimization premium is associated with more sensitive responses to the historyof opponents’ play, faster convergence and more frequent occurence of the risk-dominant equilibrium. Our workcomplements ealier studies by focusing on the effect of the coordination requirement.

5Carlsson and van Damme (1993) study the case of 2x2 games. Morris and Shin (2003) generalize the analysisto a class of 2xN symmetric games. Notable applications of the theory include Morris and Shin (1988) to currencyattacks and Goldstein and Pauzner (2005) to bank runs.

6Cabrales et al. (2007) investigate the effect of payoff uncertainty in a 2-person coordination game. The mainpurpose of Heinemann et al. (2009) is to measure strategic uncertainty when agents have common information aboutpayoff relevant variables. Duffy and Ochs (2010) improve upon the experimental methodology in Heinemann et al.(2004) by introducing a within-subject design, and investigate dynamic entry games.

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orderings of the coordination parameter allows us to detect potential hysteresis effect. In contrast,

in the experiment by Klos and Sträter (2010), Heinemann et al. (2004, 2009) and Duffy and Ochs

(2010), subjects face a constantly changing environment. In Heinemann et al. (2004, 2009), in each

period, subjects are presented with a table of 10 situations each characterized by a different value

of a payoff-relevant variable and choose whether to enter or not under each situation. In Duffy

and Ochs (2010), a payoff-relevant variable changes in every period. Klos and Sträter (2010) try

both. Their experimental designs are more suitable for studying how individuals respond when a

payoff-relevant variable changes constantly, and less so for the study of the learning process or the

hysteresis effect. In fact, Heinemann et al. (2004) and Klos and Sträter (2010) find no evidence of

learning effect as the thresholds adopted by individual subjects change little across time. Citing

the result of lack of learning effect in previous studies, Heinemann et al. (2009) study non-repeated

games and focus solely on the initial coordination behavior.

3 Theoretical Framework, Hypotheses and Experimental Design

In this section, we describe the theoretical framework that underlies the experiment, the hypotheses

and the main design features of the experiment.

3.1 The Theoretical Framework

The theoretical framework that underlies our experimental studies 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

indifferent 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

U(c1, c2) =

u(c1) for impatient consumers,

u(c1 + c2) for patient consumers,

where c1 and c2 denote the consumption at date 1 and date 2, respectively. The function u(·)satisfies u′′ < 0 < u′, limc→∞ u′(c) = 0 and limc→0 u′(c) =∞. The relative risk aversion coeffi cient−cu′′(c)/u′(c) > 1 everywhere. There is a productive technology that transforms 1 unit of date 0output 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, (c∗i , c∗p), is characterized by 1 < c∗i (N/D,R, u) <

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c∗p(N/D,R, u) < R. A bank, by offering 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 = Dr , the bank will not have enough money to pay every

withdrawer the promised rate r. In this case, a sequential service constraint (SSC) 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 − e be the number of depositors choosingto wait and z = D − e = D(1 − 1/r) be the minimum number of depositors choosing to wait to

prevent bankruptcy. Let πe and πz be the payoffs to those who choose to withdraw and roll over,

respectively. The deposit contract can be formulated as

πe =

r, if z ≥ z,

r with probability D−zD−z , and 0 with probability

z−zD−z , if z ≤ z;

πz =

D−r(D−z)

z R, if z ≥ z,

0, if z ≤ z

The optimal risk-sharing allocation can be achieved by setting r to r∗ ≡ c∗1.

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 early and

only patient agents are "strategic" players. Here we focus on "strategic" players so we let D = N .7

Second, we abstract from SSC for simplification and assume instead that when the bank does not

have enough money to pay every withdrawer r, it divides the available resource evenly among

all depositors who demand to withdraw. The SSC is not essential for the existence of multiple

equilibria; the fact that r > 1 is suffi cient to generate a payoff 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 fix 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.8 The resulting payoff function used in the experiment can

7Madiès (2006) adopts the same arrangement in this regard.8For 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, 2009c). The first few papers showthat the multiple-equilibria result goes away if more complicated contingent contracts —as compared with the simpledemand deposit contracts in DD —are used. The three papers by Ennis and Keister show that the multiple-equilibriaresult is restored if the banking authority cannot commit not to intervene in the event of a crisis, or the consumptionneeds of agents are correlated.

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be represented as

πe = min

{r,

N

N − z

}; (1)

πz = max

{0,N − r(N − z)

zR

}. (2)

The coordination parameter (denoted as η) measures the amount of coordination that is required

for agents who choose to wait to receive a higher payoff 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 payoffs associated with "withdraw" and "wait",

r =N − (N − z)r

zR;

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

z∗ =R(r − 1)r(R− 1)N .

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

payoffs to both strategy choices:

η =z∗

N=R(r − 1)r(R− 1) .

We can control the coordination parameter, η, by changing the short-term rate r.

The coordination game characterized by the above payoff structure has two symmetric Nash

equilibria. 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 payoff of 1. In

the non-run equilibrium, every depositor chooses to wait expecting others to do the same. In

this equilibrium, z = N and everybody receives a payoff of R. Fixing R also has the additional

advantage of maintaining the payoff difference between the two equilibria fixed at R−1 throughoutthe experiment.

3.2 Hypotheses

We test the following three hypotheses through the experiment.

Hypothesis I. We expect to observe a threshold result: the economy tends to stay close tothe non-run (run) equilibrium when η is very small (large) and there exist threshold values of η

around which the economy becomes more diffi cult to predict. A higher coordination requirement

means the there is a higher risk associated with waiting and therefore, discourages subjects from

choosing the action. As a result, one would expect that the number of subjects choosing to wait

decreases as coordination becomes more diffi cult. This has been verified by the experimental studies

of Heinemann et al. (2004, 2009), Duffy and Ochs (2010) and Klos and Sträter (2010) showing

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that individuals adopt threshold strategies in response to changes in payoff-relevant variables. Here

we conjecture that a threshold phenomenon may also show up at the aggregate level. About

the value of the threshold, Temzelides (1997) suggests η = 0.5 assuming that agents adopt the

evolutionary algorithm and the rate of experimentation approaches zero. We conjecture that the

threshold value should be higher than 0.5. The value of 0.5 would be the threshold if subjects only

care about strategic uncertainty. However, the higher payoff associated with non-run equilibrium

generates an additional force to draw people to that equilibrium. Heinemann et al. (2004) seem

to have a similar suggestion. In their experimental study of 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. They find the thresholds

adopted by subjects constitute a best response to the belief that other subjects choose the payoff-

dominant strategy ("wait" in the context of bank runs) with probability 2/3. The adoption of

the threshold by individuals implies that at the aggregate level, the run equilibrium tends to occur

when η > 2/3. In view of the discussion above, we hypothesize that the threshold for the occurrence

of bank runs is around 2/3.

Hypothesis II. There is a hysteresis effect in the sense that the ordering of η affects the perfor-mance of the economy (especially at the threshold values). If the level of coordination requirement

gradually increases over time, subjects may build mutual trust while playing games with easy coor-

dination and the mutual trust may persist in games with more diffi cult coordination. As a result,

one may expect that the increasing ordering may induce on average more coordination at "wait"

and fewer incidences of bank runs. Furthermore, we expect that the hysteresis effect to be small

for extreme values of coordination requirement because subjects know clearly what they should do

and the effect of preceding games may be minimal. In contrast, when η is around the threshold

values, there is more uncertainty and the performance of the economy may become sensitive to its

performance in previous phases with different levels of coordination requirement.

Hypothesis III. We expect to detect stronger learning effect when the level of coordinationrequirement is close to the threshold values. Earlier studies (for example, Van Huyck et al., 1991)

find that the initial coordination behavior well predicts the final outcomes in repeated coordination

games. Heinemann et al. (2004) and Klos and Sträter (2010) find the value of the thresholds

adopted by individual subjects changes little across time. Citing the result of weak learning effect

in previous studies, Heinemann et al. (2009) study non-repeated games and focus solely on the

initial coordination behavior. As discussed in Section 2, Van Huyck et al. (1991) only consider one

game with a tension between effi ciency and security. The conclusion that the initial behavior well

predicts final outcomes in repeated games may not hold for all levels of coordination requirement.

The papers on global games involve subjects making decisions in environments with ever-changing

payoff-relevant variables and may not be inducive to learning. Our experiment includes a wide

range of values of the coordination parameter, and at the same time, subjects have the opportunity

to repeat the same game with the same level of coordination requirement for 10 periods. As

compared with the studies of global games, the more stable environment in our experiment may

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induce stronger learning effect. As compared with Van Huyck et al. (1991), we can inspect how

the learning process depends on the level of coordination requirement. We expect that for extreme

values of the coordination requirement, subjects have little doubt about what they should do and

the learning effect would be weak. However, if the level of coordination requirement is close to the

threshold value, subjects are less certain about what they should do and their actions may switch

as they learn about the coordination history. As a result, the initial coordination behavior may not

predict the final outcomes as well.

3.3 Experimental Design

To examine how the coordination requirement affects the occurrence of bank runs, we design the

experiment with the following 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 specific banking context makes it easier for subjects to comprehend the task they

are required to perform and makes the payoff structure more intuitive to understand. Second, to

rule out the possibility that bank runs are caused by weak performance of the bank’s long-term

portfolio, we fix 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 η. Third, to inspect the effect of coordination requirement,

each session of experiment includes 7 or 9 phases, with each phase corresponding to a different value

of the coordination parameter. Fourth, each phase consists of 10 repeated games with the same

value of the coordination parameter. The combination of features three and four allow us to study

how the value of coordination requirement affects the evolution of the experimental economy and

the learning process. Fifth, to detect the hysteresis effect, we run sessions with different orderings

(increasing, decreasing and random) of the coordination parameter.

[Insert Table 1: Correspondence between r and η]

For each session of experiment, we enroll 10 subjects to act as depositors. Most subjects are

enrolled from 2nd or 3rd-year undergraduate or graduate economics or business classes. Since the

game in the experiment is fairly straightforward, it is important to have subjects with no prior

experience with experiment of coordination games. It is also important that the subjects are from

mixed sources to guarantee that they do not have close relationship before participating in the

experiment.

The experiment is run at three locations: University of International Business and Economics

(UIBE), Beijing, China; Simon Fraser University (SFU), Burnaby, Canada; University of Manitoba

(UofM), Winnipeg, Canada. Each experiment lasts for about an hour. Subjects earn on average

20 CAD in Canada, and 80 RMB in China. All sessions are run in English. To ensure that the

subjects in Beijing have suffi cient English reading and listening skills, we restrict our subject pool to

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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.

The program used to conduct the experiment is written in z-Tree (Fischbacher, 2007). During

the experiment, subjects are each assigned a computer terminal through which they can input their

decisions to withdraw or to wait (see the Appendix for the experimental instructions). Communi-

cation among subjects is not allowed during the experiment.

At the beginning of each period, every subject starts with 1 experimental dollar (ED) in the

bank and then makes a decision to withdraw or to wait and roll over their deposits. Payoff tables

for all phases are provided to list the payoff that an individual will receive if he/she chooses to

withdraw early or to leave money in the bank given that n = 1 ∼ 9 of the other 9 subjects chooseto withdraw early. The payoff tables help to reduce the calculation burden of 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’payoffs are then determined by

equations (1) and (2). At the end of each round, subjects are presented with the history of their

own actions, payoffs, and cumulative payoffs for 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 payoff 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 deduce such information from the payoff tables (as

will be shown later in Section 5, subjects can deduce the information most of the time). After the

experiment, the total payoff that each subject earns is converted from EDs into cash.

The first 8 sessions (sessions 1 to 8) we run include 7 values of η (0.1, 0.2, 0.3, 0.5, 0.7, 0.8, 0.9). In

sessions 1 to 4, the coordination parameter increases across the 7 phases which implies coordination

becomes increasingly more diffi cult over time. Sessions 5 to 8 have decreasing η and coordination

becomes increasingly easier over time. We find the experimental economies performed very differ-

ently when η = 0.7 (details will be shown in the next section where we present the experimental

results). There is also some weak evidence of hysteresis effect. To gain more understanding about

the performance of the economy around the threshold value of 0.7, we run new sessions with a finer

grid of η by adding 0.4 and 0.6 into the spectrum of η. In addition, there is the concern that the

threshold result may be induced by the monotonic change of η: it may be that after making the

same choice for many rounds (40 in sessions with increasing η and 20 in sessions with decreasing

case), subjects are waiting for a change of actions. The alteration of η may have served as a cue for

the change to take place. To address the concern, we include a treatment with random ordering of

η.9 In total, we run 12 new sessions (sessions 9 to 20) all with a finer grid of 0.1. For each of the

three orderings, increase, decrease and random, we run 4 sessions of experiment. Among the 12 new

sessions, 2 sessions (sessions 9 and 17) include the full spectrum of 9 values of η ranging from 0.1

9We thank the referees for suggesting these new treatments.

12

to 0.9 with a step of 0.1. In view that almost every subject consistently coordinates at "wait" for

η = 0.1 and 0.2, we decided to drop these two values from the other 10 sessions. Excluding the two

smallest values of η also has the additional advantage of having the same number of paying periods

as in the first 8 sessions with more coarse grids. Table 2 provides detailed design information for

each session of experiment, including the grid of η, the ordering of η, the specific values of η, and

the location and date for the session.

[Insert Table 2: Experimental Design]

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 first 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 first 8 sessions of experiment have a coarse grid and the 12 sessions have a fine grid. In total,

we have five treatments: coarse grid and increasing η (sessions 1 to 4); coarse grid and decreasing

η (sessions 5 to 8); fine grid and increasing η (sessions 9 to12); fine grid and decreasing η (sessions

13 to 16); and fine grid and random η (sessions 17 to 20).

4 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 η value and for each of the 20 sessions of experiment.

The results for the first 8 sessions are on the first page, and the results for sessions 13 to 20 are

on the second page. To facilitate comparison across different treatments, we plot z according to η

increasing from 0.1 to 0.9 with a step of 0.1. The sequence of z is thus different from the actual

time path for sessions with decreasing and random ordering of η. The empty sections in the graphs

are due to missing values of η (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 marked by a different value of η. To

qualitatively characterize the performance of the experimental economies, we define 10 performance

categories. The first 8 performance categories are defined according to the mean and terminal value

of the number of people who choose to wait. If M ≥ 9, we say the economy is "very close to thenon-run equilibrium" (denoted as NN); if 8 ≤ M < 9, we say the economy is "fairly close to the

non-run equilibrium" (denoted as FN); if 5 < M < 8 and T ≥ 8, we say the economy "convergesto the non-run equilibrium" (denoted as CN); if 5 < M < 8 and T < 8, we say the economy is

characterized by "moderate high coordination" (denoted as H); if M ≤ 1, we say the economy is"very close to the run equilibrium" (denoted as RR); if 1 < M ≤ 2, we say the economy is "fairly

13

close to the run equilibrium" (denoted as FR); if 2 < M < 5 and T ≤ 2, we say the economy

"converges to the run equilibrium" (denoted as CR); if 2 < M < 5 and T > 2, we say the economy

shows "moderate low coordination" (denoted as L). In addition, we observe that some economies

switch from high (low) level of coordination to run (non-run) equilibrium. To capture the switch

of behavior, we define two more performance 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); if S > 5 and T ≤ 2, we say the economy "switches to run equilibrium" (denoted as SR).

The definitions of the performance categories are summarized in table 4. Table 5 categorizes the

performance of the 20 sessions of experiment.

[Insert Table 3: Starting, Terminal and Mean Value of the Number of SubjectsChoosing to Wait]

[Insert Table 4: Definition of Performance Classification]

[Insert Table 5: Classification of Performance of Experimental Economies]

In the following, we first discuss the results from the first 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 finer grid and revisit all the main

findings.

4.1 Findings from Treatments with Coarse Grid

Let us first investigate what kind of evidence the first 8 sessions of experiment provide for the three

hypotheses raised in Section 3.2.

Finding I (Threshold result). The experimental economies exhibit a threshold phenomenon.

When the required amount of coordination is ≤ 0.5, all experimental economies stay close orconverge to the non-run equilibrium. When the required amount of coordination is ≥ 0.8, all

experimental economies stay close or converge to the run equilibrium. The economies perform very

differently when η = 0.7.

When η ≤ 0.5, all economies are marked by NN or CN. When η ≥ 0.8, all economies are markedby RR, FR or CR. When η = 0.7, the experimental results are much less clear-cut. The average

number of people 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 who coordinate 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 at first, but the number decreases to

2. 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

η, 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".

14

Finding II (Hysteresis effect). There is some weak evidence of hysteresis effect as η takesthe threshold value of 0.7.

As conjectured in Section 3.2, the results from the first 8 sessions of experiment show that effect

of the ordering of η has little effect on the experimental economies when η ≤ 0.5 (or η ≥ 0.8) asall economies manage to stay or converge to the non-run (or run) equilibrium. Given the great

variation in performance of different economies around the threshold value of 0.7, we run the rank

sum test to detect potential order effect for the phase with η = 0.7. To do the test, we separate

the 8 sessions of experiment into two groups of observations: the first group contains the 4 sessions

with increasing η and the other group includes the 4 sessions with decreasing η. We conduct one-

sided Mann-Whitney rank sum tests on the starting, ending and mean value of z between the two

groups, with the hypothesis that more subjects choose to wait in sessions with increasing η. The

test results are in the first 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 p-value. The initial number of people who choose

to wait tends to be higher in sessions with increasing η with an average of 7 versus an average of

2 in the group with decreasing η. The difference is statistically significant with a p-value of 1%.

However, the ending value of z is not significantly different with a high p-value of 37.05%. The

mean value of z is significant at the 12% level. Overall, there is some, albeit weak, evidence that

the ordering of η affects the performance of the economy.

[Insert Table 6: Rank Sum Tests of the Order and Grid Effect]

Finding III (Intra-phase learning). When the coordination parameter is close to the thresh-old value, the initial coordination behavior may not predict the final outcomes well.

As discussed earlier, previous studies find that the initial coordination behavior well predicts

the final outcomes in repeated coordination games. Here we find the same result when η = 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 learning for some sessions when η

is close to the threshold value of of 0.7. In particular, there are 6 incidences (see table 5) that the

economy starts with high (or low) level of coordination but switches to run (or non-run) equilibrium

at the end of the phase. When η = 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 η = 0.7, three sessions (sessions 2, 3 and 4) start with a high level of coordination at "wait"

(S ≥ 7), but the number decreases to ≤ 2. These three sessions experience a switch to the run

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

when η = 0.8 in session 7. In these 6 sessions, there is a significant difference between the starting

and ending values of the number of subjects choosing to wait as subjects learn from previous rounds

and change their actions.

15

4.2 Findings Revisited after Treatments with Fine Grid

Given the observed richness of activities around the threshold value of η = 0.7, we run a new

set of sessions with emphasized focus around the value. In particular, in contrast to the first 8

sessions of experiment which involve a coarse grid around 0.7, the new sessions feature a finer

grid around the value and include η = 0.4 and 0.6. In addition, we run experiment with random

(non-monotonic) ordering of η to study whether the threshold result is induced by the monotonic

change of η. Altogether, we have three new treatments, all with a finder grid of η of 0.1 but in

three different orderings of η: increasing, decreasing and random.

The path of z for each session is presented in figure 1b. The starting, ending and mean value

of z are presented in the lower part of table 3. The characterization of the performance of the

economies is provided in the lower part of table 5.

All main findings are in general confirmed by the new sessions with the finer grid. In addition,

the new sessions generate stronger evidence for the hysteresis effect. In the following, we revisit

each of the three main findings in details.

Finding I (Threshold result) revisited. The threshold result in finding I is robust to theadoption of a finer grid and non-monotonic ordering of η.

When η ≤ 0.5 (including η = 0.4 which was not included in the 8 old sessions) all economies stayclose or converge to the non-run equilibrium; when η ≥ 0.8, all economies stay close or convergeto the run equilibrium with a small exception. In session 12 when η = 0.8, the economy seems to

converge to the run equilibrium with z = 2 in period 9 but bounces up to 5 in the last period.

As shown in table 5, when η = 0.6 or 0.7, the performance of the experimental economies varies

greatly across different treatments and different sessions within the same treatment. When η = 0.6,

all 4 economies with increasing η 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 η = 0.7, all 4 economies with the

increasing treatment again stay close to the non-run equilibrium; all 4 economies with decreasing

η 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.

In summary, when η ≤ 0.5 or η ≥ 0.8, the performance of the experimental economies is moreuniform and is robust to the ordering of η. When η = 0.6 or 0.7, the economy is hard to predict

and is sensitive to the ordering of η, the magnitude of the grid of η and the coordination result in

the past.

Finding II (Hysteresis effect) revisited. The results from the new sessions lend more

support to finding II. With the finer grid, the hysteresis effect becomes much stronger.

As in the sessions with the coarse grid, the ordering of η has little effect on the economy when

η ≤ 0.5 (or η ≥ 0.8) in the sense that all economies manage to stay close to or converge to thenon-run (or run) equilibrium. However, the new data shows a strong hysteresis effect when η = 0.6

16

or 0.7.

As is obvious from table 5, when η = 0.6 or 0.7, the experimental economies with the increasing

treatment perform much better than the other two treatments, with all 4 economies close to the

non-run equilibrium. In contrast, for the decreasing treatment, only 2 sessions manage to stay

close or converge to the non-run equilibrium when η = 0.6 and no session achieves a high level of

coordination when η = 0.7 (they are all close to the run equilibrium). For the random treatment,

only 2 economies are close to the non-run equilibrium when η = 0.6 and only a single session

manages to stay close to the run equilibrium when η = 0.7 (the other 3 stay close to the run

equilibrium).

The strong order effect is also supported by rank sum tests on the starting, ending and mean

value of z (denoted as S, T andM, respectively). The test results for η = 0.7 are shown in table 6a.

The increasing treatment achieves significantly 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 η increases. All three p-values are below or equal to 1%. The p-values of the

other two pairwise comparisons, increasing versus random and random versus decreasing, are all

below or equal to 7%. Now let us turn to the test results for η = 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 with all three p-values below 6%. The evidence is less strong for the other

two pairwise comparisons.

Finally, a comparison of the new and old sessions with increasing η (sessions 1-4 versus sessions

9-12) suggests that a more gradual increase in η is associated with better coordination at the

threshold value of 0.7. When η 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 η = 0.7. In contrast, when η increases from

0.5 to 0.6 and then to 0.7 with a step of 0.1, all 4 out of 4 economies manage to stay close to the

non-run equilibrium when η = 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

are more likely to maintain the old strategies that were adopted in the previous game. The rank

sum tests also show that for η = 0.7, the economy tends to have higher starting, ending and mean

value of z with the finer grid with p-values ≤ 1%. The finding that a gradual change facilitatescoordination 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 when players learn

from games with payoffs very close to the current ones. Our experiments show that the contagion

effect is stronger when the change in the payoff structure is smaller and more gradual.

Finding III (Intra-phase learning) revisited. For finding III, we still observe some learningeffect when η is close to the threshold values. With the increasing treatment, 3 sessions (sessions

9-11) start with relatively high level of coordination (S ≥ 8) to the run equilibrium (T ≤ 1) whenη = 0.8. When η = 0.9, session 9 starts with S = 6 and switches to the run equilibrium at the

17

end of the phase (T = 0). For the decreasing treatment, two economies (sessions 14 and 15) switch

to the non-run equilibrium (with S = 3 or 4 and T = 9 or 10) when η = 0.6. With the random

treatment, session 19 switches to the run equilibrium when η = 0.8 and 0.9 (with S = 9 and T = 0).

Session 20 follows a similar pattern with S = 6 and T = 1.

5 The Logit Evolutionary Algorithm

The evolutionary algorithm developed by Young (1993) and Kandori et al. (1993) has been intro-

duced 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 first 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 to flip one’s strategy from

"withdraw" to "wait" or vice versa. Temzelides (1997) proves a limiting case that 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 η < 0.5. In other words, η = 0.5 is predicted to be

the watershed for the occurrence of bank runs in the limiting case studied by Temzelides (1997).

Obviously, the limiting case does not describe the experimental results well. In particular,

the experimental results suggest a higher threshold than 0.5. When η = 0.5, all 20 experimental

economies stay close or converge to the non-run equilibrium. Only when η = 0.6 or 0.7 do we

observe large variation across different treatments and sessions. One way to improve the algorithm

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

coordination parameter η captures how diffi 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 influence 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 models to estimate the rate of experimentation as functions of η and/or zt−1,

can successfully explain the experimental data. Recently in a similar spirit, Hommes (2011) shows

that the evolutionary learning logit model fits data of so-called learning-to-forecast experiments

well.

In the rest of this Section, we first describe the main features of the logit evolutionary algorithm.

We then logit regressions to estimate the probability of experimentation from the experimental data.

Finally, we evaluate the performance of the algorithm.

18

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 we first describe the amount of information subjects have in the experiment.

In the experiment, we inform subjects about r and subjects can always deduce η from the payoff

tables. We do not directly inform subjects about zt−1, but subjects can potentially refer to the

payoff tables to deduce zt−1 from their own payoffs in the past period. For the new algorithm, we

assume that if a subject cannot deduce whether zt−1 > z∗, she skips the first part of the algorithm

and does not update her strategy; otherwise, she updates her strategy to the best response. For

experimentation, we assume that the rate of experimentation is a function of η and/or zt−1. Specif-

ically, if the subject can deduce the exact value of zt−1, then the rate of experimentation depends

on both zt−1 and η; otherwise, her experimentation rate depends only on η. The value of the rate

of experimentation is to be estimated from the experimental data with logit models using η and/or

zt−1 as regressors.

Before running the regressions, we provide an overview about how much information is available

to subjects in the experiment. The 20 sessions of experiment generate 12, 960 observation for the

estimation of the rate of experimentation and the study of 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 first

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 first 4 columns of table 7 summarize how a subject’s

information about the past period depends on her strategy and payoff in the past period. Suppose

a subject chose to wait in the previous period. The subject can deduce zt−1 whenever she received

a positive payoff. If she got 0 payoff, she can infer the bank was bankrupt, but she cannot deduce

the exact value of zt−1. If the subject chose to withdraw in the previous period, she can deduce the

exact value of zt−1 if her payoff was less than r. If she was paid r, she can infer the bank did not

go bankrupt, but she cannot infer the exact value of zt−1. As shown in columns 5 and 6 of table 7,

subjects can deduce the best response and the exact value of zt−1 most of the time. To be precise,

subjects can deduce the best response in 96.34% of the observations, and the exact value of zt−1 in

93.19% of the observations.10

[Insert Table 7: Information]

10As two referees point out, one may wonder how the results from the experiment and the evolutionary algorithmdepend on whether subjects have full information about the performance of the economy in the past. As shownabove, most of the time, subjects can derive the exact value of zt−1. Our experimental economies are thus very closeto having full information. Nonetheless, we have also run three sessions of experiments with full information, onefor each ordering of η and all with the finer grid. To investigate the effect of information availability, we conduct atwo-sided rank sum test on the session mean of z between two groups. The first group includes the 12 sessions withasymmetric information (sessions 9 to 20), and the second group includes the 3 sessions with full information. Whilecalculating the session mean of z, we omit the two phases with η = 0.1 & 0.2 for sessions 9 and 17 to ensure the samenumber of periods is included for each session. The p-value is 0.5637 showing that the performance of the economyis not significantly affected by the adoption of full information.

19

Temzelides (1997) assumes agents always update to best response and adopt an exogenously

fixed rate of experimentation (which approaches zero in the limiting case). Given that subjects can

deduce the best response most of the time, making whether to update to best response information

contingent has negligible effect 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 η

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 five

steps. First, collect information about the individual’s last period’s strategy (st−1), last period’s

payoff (πt−1) and current period’s strategy (st). Second, decide whether the individual can deduce

zt−1 > z∗ and the exact value of zt−1 according to table 7. Third, 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 s′t. Fourth, decide whether

the individual flips her strategy by comparing s′t and st and denote it by a binomial variable, f .

Count it as a flip (f = 1) if s′t 6= st and a non-flip (f = 0) otherwise. Fifth, divide observations

into three groups according to s′t and whether the individual can deduce the exact value of zt−1.

Group A includes observations with s′t ="withdraw" and involves subjects who cannot infer the

exact value of zt−1; these observations are used to estimate the experimentation rate (denoted as

δA) for subjects who have information about zt−1 and consider flipping from "withdraw" to "wait".

Group B includes data to estimate experimentation rate (denoted as δB) in the same direction

for subjects who cannot infer the exact value of zt−1. Group C includes observations with s′t =

"wait" and are used to estimate the experimentation rate (denoted as δC) of flipping from "wait"

to "withdraw". Note that subjects who have s′t ="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 flip of strategy is observed).

For those who cannot infer the exact value of zt−1 (group A), we use η as the single regressor. The

expected sign of η is negative, meaning that subjects in groups A are less likely to flip strategy

from "withdraw" to "wait" as coordination becomes more diffi cult. For those who can infer the

exact value of zt−1 (groups B and C), we use the difference 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 difference 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 payoff in the previous period. Note that

z∗ = ηN so z∗ contains information about η as well. The expected sign of (zt−1− z∗) is positive inthe case of δB, meaning that agents are more likely to change strategies from "withdraw" to "wait"

as zt−1 moves closer to z∗ from below. The expected sign of (zt−1− z∗) is negative for δC , meaning

20

that the probability of experimentation from "wait" to "withdraw" increases as zt−1 moves closer

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

logit (δA) = log

(δA

1− δA)

)= αA + βAη;

logit (δB) = log

(δB

1− δB

)= αB + βB(zt−1 − z∗);

logit (δC) = log

(δC

1− δC

)= αC + βC(zt−1 − z∗).

The results from the logit regression models are listed in table 8. All the coeffi cients are highly

significant with a p-values close to 0 and all have the expected signs. This provides evidence that

the subjects do make use of information about η or zt−1 when they decide whether to flip their

strategies. The estimated probability of experimentation is given by:

δA(η) =1

1 + e−0.91+2.45 η; (3)

δB(zt−1, η) =1

1 + e−0.52−0.50(zt−1−z∗); (4)

δC(zt−1, η) =1

1 + e2.83+0.31(zt−1−z∗). (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 logit model in two ways. The first is to

apply the algorithm and use the estimated probability of experimentation in Section 5.2 (equations

3 to 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 definitions 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. Specifically, 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 subjects: there are 10 simulation subjects, 7 or 9 phases or

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

21

the 20 experimental sessions.

Here we illustrate the simulation process using the starting values of session 3. The starting

values of z in the 7 phases are respectively 10, 9, 10, 10, 8, 2 and 1. In each phase, strategies in the

first round are set to match the starting value of z. For example, in phase 1, the simulation economy

starts with all 10 agents choosing to wait so that z is equal to 10 in the first round. From round

2 on, each simulation agent updates its strategy according to the rules of the logit evolutionary

algorithm. In the first step, the agent updates its strategy to 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 flipping strategies developed in the first step. The calculated probability

of experimentation depends on the direction of flipping and whether the agent can deduce zt−1. For

example, if the agent chooses "withdraw" as the strategy in the first step and can infer the value of

zt−1, the probability of changing strategy to "wait" is given by δB(zt−1, η) 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. Each simulation is run for 70 periods, generating 70 values of z.

We run 100 simulations 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 defined in table 4. An investigation of table 9 shows

that the modified evolutionary algorithm captures the main features of the experimental data. For

instance, the economies stay close or converge to the non-run equilibrium when η is small. For

η ≤ 0.4, all simulated economies stay close to the non-run equilibrium, spending 100% of the time

at "NN" or "FN". When η = 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 η = 0.9, all simulated economies spend 100% of the

time at "RR" or "FR". When η = 0.8, 18 out of the 20 simulated economies spend 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 about 30% probability. When η = 0.6 or 0.7, as in the actual experiment, the simulated

economies produce very different results, spending a positive amount of time in each of the first

8 performance categories. The evolutionary learning logit model successfully captures 0.6 and 0.7

as the watershed for coordination. Finally, as in actual experiment, the simulated economies may

start with low level of coordination (S < 5) and switch to non-run equilibrium by the end, of start

with high level of coordination (S > 5) but switch to the run equilibrium at the end of the phase.

[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 rate of

experimentation, and apply the algorithm to predict the action choice for each of the observation

in the 7 sessions of experiment run in Canada. The results for the logit regressions are presented

22

in table 10. Note that the estimated coeffi 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 of experiment does not affect the experimental results in a significant 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 payoff in the previous period

to determine his current action s′t. The subject updates strategy to best response if she can deduce

the best response and keeps the strategy in the previous period if otherwise. Then we calculate

the estimated rate of experimentation using the regression results in table 10. The subject flips her

strategy if and only if the estimated rate of experimentation is > 0.5. If the predicted action, st,

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 logit evolutionary

algorithm does a good job in explaining the experimental data.

[Insert Table 10: Logit Regression of Rate of Experimentation-Chinese Data]

6 Conclusion

In this paper, we take an experimental approach to study how the level of coordination requirement

affects occurrence of bank runs as a result of coordination failures. To do that, we enroll human

subjects to act as depositors and play a repeated version of the game defined by a demand deposit

contract. We fix the rate of return of the bank’s long-term investments to rule out the deterioration

of the bank’s assets as the source 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 payoff than those who withdraw, changes every 10 periods.

We find that bank runs can happen as the result of pure coordination failures if the required

coordination is high enough. In particular, bank runs are frequently (rarely) observed if the coordi-

nation parameter is ≥ 0.8 (≤ 0.5). When the coordination parameter is in the middle (for example,0.6 or 0.7), the performance of the experimental economies becomes less predictable. We also find

evidence of hysteresis effect: the ordering of the coordination parameter affects the performance

of experimental economies, especially when the coordination parameter is around to the threshold

values. In particular, when the economy starts with low coordination requirement and increases

over time, the experimental economies achieve better coordination (as compared with the case in

which the coordination parameter decreases or is random). We also observe some learning effect

in the sense that the initial coordination behavior may not predict the final outcomes in repeated

coordination games. In particular, we observe some experimental economies start a phase (which

11We also divide the sessions into 2 groups according to location (China versus Canada), and run a two-sidedrank sum test of the mean of the number of subjects choosing to wait across the whole session (excluding the twophases with η = 0.1 and 0.2 for sessions 9 and 21). The p-value is 0.663, which suggests the location does not have asignificant effect on the experimental results.

23

consists of a 10-period repeated game) with high (low) level of coordination but switch to the

run (non-run) equilibrium at the end of the phase. The switch is more likely to occur when the

coordination parameter is close to the threshold values.

In order to capture the learning effect in the experimental data, we combine the evolutionary

algorithm in Temzelides (1997) with logit regression models to estimate the rate of experimentation

from the experimental data. The logit 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. Duffy and Fisher (2005)

and Fehr et al. (2012) have provided direct evidence of sunspots in the laboratory. However, the

payoff structure is such that the multiple equilibria are not Pareto rankable or are Pareto equivalent.

Most recently, Arifovic et al. (2012) generate some initial experimental evidence of sunspot behavior

when the payoff structure is associated with multiple equilibria that can be Pareto ranked. One

important finding is that training with respect to the sunspot variable during the practice period

is crucial for the observation of sunspot behavior in later periods.12 The experimental results in

our paper suggest that the level of coordination requirement may also be an important factor that

affects the occurrence of sunspot behavior. In the experiment, there is little ambiguity when the

coordination parameter is≤ 0.5 or≥ 0.8, but a huge variation in performance when the coordinationparameter is equal to 0.6 or 0.7. This seems to suggest that sunspot behavior is more likely to be

observed when the coordination parameter is around 0.6 or 0.7. We intend to pursue the studies

in the future to test directly the sunspot-based theory of bank runs, and at the same time, provide

more experimental evidence about sunspot behavior in games with multiple equilibria that can be

Pareto-ranked.

12The training is achieved by building a correlation between the realization of the sunspot variable and the aggregateoutcome. The process is determined by the experimenter and is not known to the subjects.

24

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27

Table 1: Values of r and η Used in Experiments

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

28

Table 2: Experimental Design

Session No. Grid Order Location η Date

1 UIBE 0.1;0.2;0.3;0.5;0.7;0.8;0.9 2006-112 UIBE 0.1;0.2;0.3;0.5;0.7;0.8;0.9 2008-113 SFU 0.1;0.2;0.3;0.5;0.7;0.8;0.9 2006-114 UofM 0.1;0.2;0.3;0.5;0.7;0.8;0.9 2008-115 UIBE 0.9;0.8;0.7;0.5;0.3;0.2;0.1 2007-066 UIBE 0.9;0.8;0.7;0.5;0.3;0.2;0.1 2008-117 SFU 0.9;0.8;0.7;0.5;0.3;0.2;0.1 2008-118 UofM 0.9;0.8;0.7;0.5;0.3;0.2;0.1 2008-10

9 UIBE 0.1:0.1:0.9 2012-0310 UIBE 0.3:0.1:0.9 2012-0311 UIBE 0.3:0.1:0.9 2012-0512 SFU 0.3:0.1:0.9 2012-05

13 UIBE 0.9:-0.1:0.3 2012-0314 UIBE 0.9:-0.1:0.3 2012-0415 UIBE 0.9:-0.1:0.3 2012-0616 SFU 0.9:-0.1:0.3 2012-05

17 UIBE 0.4;0.9;0.1;0.8;0.2;0.7;0.3;0.6;0.5 2012-0318 UIBE 0.7;0.3;0.9;0.6;0.4;0.8;0.5 2012-0419 UIBE 0.8;0.4;0.7;0.3;0.6;0.9;0.5 2012-0520 SFU 0.8;0.4;0.7;0.3;0.6;0.9;0.5 2012-05

Coarse

Fine

Increase

Decrease

Increase

Decrease

Random

29

Table 3: Starting, Terminal and Mean Value of Number of Subjects Choosing to Wait

Session 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9S T M S T M S T M S T M S T M S T M S T M S T M S T M

1 10 10 10 9 10 9.9 10 9 9.7 8 10 9.3 6 6 7.5 2 0 1.2 1 0 0.22 9 10 9.8 8 10 9.5 10 10 10 9 10 9.6 7 0 2.1 3 0 0.4 0 0 03 10 9 9.8 9 9 9.7 10 9 9.7 10 9 9.6 8 2 3.1 2 0 1.1 1 0 0.24 8 10 9.7 10 10 9.9 9 10 9.8 9 10 9.9 7 0 2.2 1 0 0.7 0 0 0.15 10 10 10 10 10 9.9 8 10 9.7 4 8 7.2 1 0 0.4 1 0 0.5 1 1 0.66 10 10 10 10 10 10 10 10 10 6 9 8.7 0 0 0.5 2 0 0.7 1 0 0.57 10 10 10 10 10 10 10 10 10 10 10 9.8 6 9 7.9 6 0 1.9 4 2 2.28 10 9 9.7 10 10 10 8 9 9.6 4 9 7.8 1 0 1 1 1 0.6 3 3 2

9 10 10 9.9 9 10 9.9 10 10 10 10 10 10 10 10 10 9 10 9.8 10 10 9.7 9 0 1.8 6 0 0.810 10 10 10 10 10 10 10 10 9.9 10 10 10 10 10 10 8 1 2.4 3 0 0.911 9 9 9 10 10 10 10 10 10 10 10 10 10 10 10 9 0 2.3 1 0 0.112 7 8 8.5 9 9 8.6 10 10 9 9 9 8.7 8 10 8.4 8 5 4.6 4 2 213 10 10 10 10 10 10 5 10 9.3 3 0 2.7 3 0 1 3 0 0.6 3 1 0.914 10 10 10 10 10 10 10 10 9.9 4 9 7.8 3 0 0.7 1 1 0.7 3 1 1.115 10 10 10 10 10 10 9 10 9.9 3 10 8.1 1 0 0.2 2 0 0.3 3 0 0.816 10 10 9.9 10 10 10 7 10 8.2 3 0 0.4 3 0 0.5 3 0 0.3 4 1 1.917 7 10 9.6 10 10 10 9 10 9.8 10 10 10 7 9 8.5 3 0 0.6 3 0 0.3 5 0 0.8 3 0 1.118 9 9 9.5 9 9 9.3 9 9 9.3 2 0 0.7 3 3 1.7 4 0 0.9 3 0 0.719 9 10 9.9 10 10 10 10 10 10 10 10 10 10 10 10 9 0 3 9 0 2.520 9 9 9.7 10 10 9.7 8 10 9.7 7 10 9.2 5 0 1.4 6 1 2.5 6 1 1.5

30

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<M<8 and T≥8

Moderate high coordination H 5<M<8 and T<8

Very close to the run equilibrium RR M≤1

Fairly close to the run equilibrium FR 1<M≤2

Converging to the run equilibrium CR 2<M<5 and T≤2

Moderate low coordination L 2<M<5 and T>2

Switch to non-run equilibrium SN S<5 and T≥8

Switch to run equilibrium SR S>5 and T≤2

31

Table 5: Classification of Performance of Experimental Economies

ηSession

1 NN NN NN NN H FR RR2 NN NN NN NN CR; SR RR RR3 NN NN NN NN CR; SR FR RR4 NN NN NN NN CR; SR RR RR5 NN NN NN CN; SN RR RR RR6 NN NN NN FN RR RR RR7 NN NN NN NN CN FR; SR CR8 NN NN NN CN; SN RR RR FR

9 NN NN NN NN NN NN NN FR; SR RR;SR10 NN NN NN NN NN CR; SR RR11 NN NN NN NN NN CR; SR RR12 FN FN NN FN FN L FR13 NN NN NN CR RR RR RR14 NN NN NN CN; SN RR RR FR15 NN NN NN FN; SN RR RR RR16 NN NN FN RR RR RR FR17 NN NN NN NN FN RR RR RR FR18 NN NN NN RR FR RR RR19 NN NN NN NN NN CR; SR CR; SR20 NN NN NN NN FR CR; SR FR; SR

0.1 0.2 0.3 0.90.80.4 0.5 0.6 0.7

32

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

Table 6a: Rank Sum Tests for η=0.7

Order Effect - Old data

Order Sample size S T MSessions 1-4 Increase 4 7.000 2.000 3.725Sessions 5-8 Decrease 4 2.000 2.250 2.450Z-value 2.205 0.331 1.155p-value 1.38% 37.05% 12%

Order Effect - New data

Order Sample size S T MSessions 9-12 Increase 4 9.500 9.750 9.625Sessions 13-16 Decrease 4 2.500 0.000 0.600Z-value 2.428 2.646 2.323p-value 0.76% 0.41% 1%

Order Sample size S T MSessions 9-12 Increase 4 9.500 9.750 9.625

Sessions 17-20 Random 4 5.500 5.000 5.125Z-value 1.703 2.000 1.479p-value 4.43% 2.28% 7%

Order Sample size S T M

Sessions 17-20 Random 4 5.500 5.000 5.125Sessions 13-16 Decrease 4 2.500 0.000 0.600Z-value 1.654 1.512 1.443p-value 4.91% 6.53% 7%

Grid Effect - New and old data

Order Sample size S T MSessions 9-12 Increase 4 9.500 9.750 9.625Sessions 1-4 Increase 4 7.000 2.000 3.725Z-value 2.247 2.477 2.323p-value 1.24% 0.66% 1%

33

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

Table 6b: Rank Sum Tests for η=0.6

Order Sample size S T MSessions 9-12 Increase 4 9.500 9.750 9.625Sessions 13-16 Decrease 4 3.250 4.750 4.750Z-value 2.397 1.559 2.323p-value 0.83% 5.95% 1%

Order Sample size S T MSessions 9-12 Increase 4 9.500 9.750 9.625Sessions 17-20 Random 4 5.500 5.000 5.125Z-value 1.488 1 1.183p-value 6.84% 15.87% 12%

Order Sample size S T MSessions 17-20 Random 4 5.500 5.000 5.125Sessions 13-16 Decrease 4 3.250 4.750 4.750

Z-value 0.461 0.316 0.577

p-value 32.23% 37.91% 28%

34

Table 7: Information

Last period's choice s t-1

Last period's payoff π t-1

Can deduce exact value of z t-1 ?

Can deduce best response?

Number of observations

Percentage in all observations

Choice before experimentation s t ' Group

r No No 474 3.66 Withdraw (keep old strategy) A[1,r) Yes Yes 4362 33.66 Withdraw (update to best response) B>r Yes Yes 7449 57.48 Wait (update to best response) C

Wait (0,r) Yes Yes 267 2.06 Withdraw (update to best response) B0 No Yes 408 3.15 Withdraw (update to best response) A

Withdraw

35

Table 8: Logit Regression of Rate of Experimentation - All Data

Table 8a

coefficient std error t-statistic p-valueconstant 0.91 0.26 3.47 0.00η -2.45 0.39 -6.34 0.00

Table 8b

coefficient std error t-statistic p-valueconstant 0.52 0.13 3.87 0.00z t‐1 ‐z* 0.50 0.02 21.76 0.00

Table 8c

coefficient std error t-statistic p-valueconstant -2.83 0.26 -11.05 0.00z t‐1 ‐z* -0.31 0.05 -6.05 0.00

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.

36

Table 9: Performance of Simulated Economies

η 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9NN 100 100 100 81 0 0 0 100 100 99 2 0 0 0FN 0 0 0 19 1 0 0 0 0 1 44 0 0 0CN 0 0 0 0 0 0 0 0 0 0 49 0 0 0H 0 0 0 0 0 0 0 0 0 0 0 0 0 0

1 RR 0 0 0 0 0 78 98 5 0 0 0 0 53 86 98FR 0 0 0 0 32 22 2 0 0 0 0 42 14 2CR 0 0 0 0 58 0 0 0 0 0 1 4 0 0L 0 0 0 0 9 0 0 0 0 0 4 1 0 0

SN 0 0 0 0 0 0 0 0 0 0 96 0 0 0SR 0 0 0 0 89 0 0 0 0 0 0 0 0 0

NN 100 99 100 95 1 0 0 100 100 100 39 0 0 0FN 0 1 0 5 4 0 0 0 0 0 53 0 0 0CN 0 0 0 0 1 0 0 0 0 0 8 0 0 0H 0 0 0 0 3 0 0 0 0 0 0 0 0 0

2 RR 0 0 0 0 0 65 100 6 0 0 0 0 68 78 98FR 0 0 0 0 17 33 0 0 0 0 0 28 22 2CR 0 0 0 0 65 2 0 0 0 0 0 3 0 0L 0 0 0 0 9 0 0 0 0 0 0 1 0 0

SN 0 0 0 0 0 0 0 0 0 0 0 0 0 0SR 0 0 0 0 84 0 0 0 0 0 0 0 0 0

NN 100 100 100 99 37 0 0 100 100 100 99 0 0 0FN 0 0 0 1 27 0 0 0 0 0 1 1 0 0CN 0 0 0 0 0 0 0 0 0 0 0 0 0 0H 0 0 0 0 12 0 0 0 0 0 0 0 1 0

3 RR 0 0 0 0 0 78 98 7 0 0 0 0 0 10 86FR 0 0 0 0 0 22 2 0 0 0 0 32 76 14CR 0 0 0 0 22 0 0 0 0 0 0 58 13 0L 0 0 0 0 2 0 0 0 0 0 0 9 0 0

SN 0 0 0 0 0 0 0 0 0 0 0 0 0 0SR 0 0 0 0 27 0 0 0 0 0 0 89 99 0

NN 100 100 98 95 1 0 0 100 100 99 2 0 0 0FN 0 0 2 5 4 0 0 0 0 1 44 0 0 0CN 0 0 0 0 1 0 0 0 0 0 49 0 0 0H 0 0 0 0 3 0 0 8 0 0 0 0 0 0 0

4 RR 0 0 0 0 0 86 100 0 0 0 0 53 86 95FR 0 0 0 0 17 14 0 0 0 0 0 42 14 5CR 0 0 0 0 65 0 0 0 0 0 1 4 0 0L 0 0 0 0 9 0 0 0 0 0 4 1 0 0

SN 0 0 0 0 0 0 0 0 0 0 96 0 0 0SR 0 0 0 0 84 0 0 0 0 0 0 0 0 0

37

Table 9: Performance of Simulated Economies Continued

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9NN 100 100 100 98 99 84 76 23 0 100 98 23 0 0 0 0 98 100 98 98 61 0 0 0 0FN 0 0 0 2 1 14 12 8 0 0 2 64 0 0 0 0 2 0 2 2 36 0 0 0 0CN 0 0 0 0 0 0 0 0 0 0 0 12 8 0 0 0 0 0 0 0 3 8 0 0 0H 0 0 0 0 0 1 8 25 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0

9 RR 0 0 0 0 0 0 0 0 51 13 0 0 0 2 22 65 95 17 0 0 0 0 0 2 22 30 95FR 0 0 0 0 0 0 0 0 49 0 0 0 19 59 33 5 0 0 0 0 0 19 59 63 5CR 0 0 0 0 0 1 4 44 0 0 0 0 28 15 2 0 0 0 0 0 0 28 15 7 0L 0 0 0 0 0 0 0 0 0 0 0 0 43 4 0 0 0 0 0 0 0 43 4 0 0

SN 0 0 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 12 0 0 0SR 0 0 0 0 0 1 10 70 100 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

NN 100 98 99 96 76 0 0 100 98 99 0 0 0 0 98 97 95 0 0 0 0FN 0 2 1 4 12 0 0 0 2 1 5 0 0 0 2 3 5 0 0 0 0CN 0 0 0 0 0 0 0 0 0 0 11 0 0 0 0 0 0 8 0 0 0H 0 0 0 0 8 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0

10 RR 0 0 0 0 0 0 95 14 0 0 0 0 22 86 95 18 0 0 0 10 22 47 95FR 0 0 0 0 0 43 5 0 0 0 16 59 14 5 0 0 0 21 59 49 5CR 0 0 0 0 4 55 0 0 0 0 26 15 0 0 0 0 0 21 15 4 0L 0 0 0 0 0 1 0 0 0 0 41 4 0 0 0 0 0 40 4 0 0

SN 0 0 0 0 0 0 0 0 0 0 20 0 0 0 0 0 0 12 0 0 0SR 0 0 0 0 10 99 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

NN 100 98 99 96 76 0 0 100 98 95 0 0 0 0 98 98 99 96 76 23 0FN 0 2 1 4 12 0 0 0 2 5 0 0 0 0 2 2 1 4 12 8 0CN 0 0 0 0 0 0 0 0 0 0 8 0 0 0 0 0 0 0 0 0 0H 0 0 0 0 8 1 0 0 0 0 0 0 0 0 0 0 0 0 8 25 0

11 RR 0 0 0 0 0 0 95 15 0 0 0 2 53 78 95 19 0 0 0 0 0 0 0FR 0 0 0 0 0 43 5 0 0 0 19 42 22 5 0 0 0 0 0 0 57CR 0 0 0 0 4 55 0 0 0 0 28 4 0 0 0 0 0 0 4 44 43L 0 0 0 0 0 1 0 0 0 0 43 1 0 0 0 0 0 0 0 0 0

SN 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0SR 0 0 0 0 10 99 0 0 0 0 0 0 0 0 0 0 0 0 10 70 100

NN 90 97 99 84 37 0 0 100 98 61 0 0 0 0 98 98 81 40 0 0 0FN 10 3 1 14 27 0 0 0 2 36 0 0 0 0 2 2 19 51 0 0 0CN 0 0 0 0 0 0 0 0 0 3 8 0 0 0 0 0 0 2 0 0 0H 0 0 0 1 12 1 0 0 0 0 0 0 0 0 0 0 0 4 0 1 0

12 RR 0 0 0 0 0 0 86 16 0 0 0 2 22 65 86 20 0 0 0 0 2 10 51FR 0 0 0 0 0 43 14 0 0 0 19 59 33 14 0 0 0 0 52 76 49CR 0 0 0 1 22 55 0 0 0 0 28 15 2 0 0 0 0 2 38 13 0L 0 0 0 0 2 1 0 0 0 0 43 4 0 0 0 0 0 1 8 0 0

SN 0 0 0 0 0 0 0 0 0 0 12 0 0 0 0 0 0 0 0 0 0SR 0 0 0 1 27 99 0 0 0 0 0 0 0 0 0 0 0 2 0 99 100

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Table 10: Logit Regression of Rate of Experimentation - Chinese Data

Table 10a

coefficient std error t-statistic p-valueconstant 0.62 0.37 1.69 0.09η -2.34 0.55 -4.25 0.00

Table 10b

coefficient std error t-statistic p-valueconstant 0.50 0.18 2.81 0.00z t‐1 ‐z* 0.54 0.03 17.51 0.00

Table 10c

coefficient std error t-statistic p-valueconstant -3.77 0.53 -7.17 0.00z t‐1 ‐z* -0.31 0.10 -2.96 0.00

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)  

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

 

   

 

 

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Figure 1b: Experimental Results (Sessions 9-20)   

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

 

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Session 9  Session 13

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Session 11  Session 15

Session 12  Session 16

1

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, e

10} or the minimum of r and

e

10.

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

max{0, Re

re

10

10} or the maximum of 0 and R

e

re

10

10.

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

be 3+1=4, so your payoff will be

5.2

4

1010,38.1min

er =1.38. If you choose to wait and

2

leave money in the bank, your payoff will be 1.67: the number of withdrawing requests will be

e=3, so your payoff is

1.67 23-10

38.1310

10

10,0max R

e

re=1.67.

Example 2. Continue using table 0 where r=1.38. Suppose that 8 of other subjects request

withdrawals. Your payoff for withdrawing will be

11.1

9

1010,38.1min

er =1.11. Your

payoff for waiting and leaving money in the bank will be

1.04- 28-10

38.1810

10

10,0max R

e

re = 0.

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.

3

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.

4

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.

5

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

6

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

7

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

8

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