Experimental Evidence of Bank Runs as Pure Coordination
Failures
Jasmina Arifovicy
Janet Hua Jiang
z
Yiping Xux
February 15, 2013
Abstract
This paper investigates how the level of coordination requirement a¤ects the occurrence of
bank runs as a result of pure coordination failures in controlled laboratory environments. We
…nd that miscoordination-based bank runs can be observed when the coordination parameter,
de…ned as the amount of coordination among depositors choosing to wait that is required for
them to receive a higher payo¤ than depositors who choose to withdraw, is high enough. In
particular, we can divide the values of the coordination parameter into three regions: the run
region characterized by high values of the parameter, the non-run region characterized by low
values of the parameter and the indeterminacy region characterized by intermediate values of the
parameter. In the run (non-run) region, all experimental economies stay close or converge to the
run (non-run) equilibrium. When the coordination parameter lies in the indeterminacy region,
the outcomes of the experimental economies vary widely and are hard to predict. Some learning
and path-dependence e¤ects are also detected when the coordination parameter is located in or
around the indeterminacy region. 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 Du¤y, three anonymous referees and the Editor. We
would also like to thank participants at the CEA Meeting, the ESA Asia Paci…c Meeting, the ESA North American
Meeting, the Chicago Fed Workshop on Money, Banking, Payments and Finance, and the seminar participants at
the International Monetary Fund and the University of Winnipeg. This research was funded by SSHRC, INET, the
Humanities and Social Sciences of the Ministry of Education of China Grant and the University of International
Business and Economics 211 Grant. The views expressed in this paper are those of the authors. No responsibility
for them should be attributed to the Bank of Canada.
y
Simon Fraser University. E-mail: arifovic@sfu.ca.
z
Bank of Canada. Email: jjiang@bankofcanada.ca.
x
University of International Business and Economics. Email: xuyiping@uibe.edu.cn.
1
1
Introduction
A bank run is the situation in which a large number of depositors, fearing that their bank will be
unable to repay their deposits, simultaneously try to withdraw their funds even in the absence of
liquidity needs. Bank runs were frequently observed in the United States before the establishment
of the Federal Deposit Insurance Corporation in 1933. The enactment of the deposit insurance
program has greatly reduced the incidence of bank runs. However, a new wave of bank runs
has occurred across the world during the recent …nancial turmoil. Examples include the runs on
Northern Rock in September 2007, on Bear Stearns in March 2008, on Wamu in September 2008,
on IndyMac in July 2008, and on Busan II Savings Bank in February 2011.
The theoretical literature on bank runs is largely built on the seminal paper by Diamond and
Dybvig (1983) (hereafter DD). The bank is modelled as a liquidity insurance provider that pools
depositors’resources to invest in pro…table illiquid long-term assets, and at the same time, issues
short-term demand deposits to meet the liquidity need of depositors. The term mismatch between
the bank’s assets and liabilities opens the gate to bank runs.
There are, broadly speaking, two opposing views about the cause of bank runs. The …rst view
(represented by DD) is that bank runs are the result of pure coordination failures. The bank run
model in DD has two symmetric self-ful…lling 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 …re-sale prices
and makes the initial fear a self-ful…lling prophecy. As a result, even banks with healthy assets may
be subject to bank runs. The competing view (represented by Allen and Gale, 1998) is that bank
runs are caused by adverse information about the quality of the bank’s assets.
To empirically test the competing theories of bank runs is challenging. Real-world bank runs
tend to involve various factors and this makes it di¢ cult to conclude whether the bank run is due to
miscoordination or the deterioration of the quality of the bank’s assets. There are some attempts
to empirically test the source of bank runs, nonetheless with mixed results. For example, Gorton
(1988), Allen and Gale (1998) and Schumacher (2000) show that bank runs have historically been
strongly correlated with deteriorating economic fundamentals which erode away the value of the
bank’s assets. In contrast, Boyd et al. (2001) conclude that bank runs may often be the outcome
of coordination failures.
An experimental approach has the advantage that it is easier to control di¤erent 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 …x the rate of return of the bank’s long-term
asset throughout the experiment to rule out the deterioration of the quality of the bank’s assets as
the source of bank runs.
The speci…c design of the experiment is inspired by the evolutionary learning literature. The
evolutionary algorithm developed by Young (1993) and Kandori et al. (1993) has been introduced as
2
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 payo¤
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 payo¤ than those who choose to withdraw –a¤ects the decision of depositors and in turn the
occurrence of bank runs. We focus on three questions: (1) How does the coordination parameter
a¤ect the level of coordination and the frequency of bank runs? (2) Does the path followed by
the coordination parameter a¤ect the performance of the economy? (3) How does the level of
coordination a¤ect the learning process and speed of convergence of the economy? To answer
these questions, we run 20 sessions of experiment. In each session, 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 di¤erent 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 a¤ects 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 e¤ect on the learning process and the
evolution of the coordination behavior. In addition, we conduct sessions with di¤erent sequences
of the coordination parameter to detect potential order e¤ect.
There are three main …ndings from the experiment. The …rst is that bank runs occur more
frequently when the coordination task is more di¢ cult. In particular, we can describe the coordination results in three regions of the coordination parameter. In the non-run region where the
parameter is
0:5, all experimental economies stay close or converge to the non-run equilibrium.
In the run region where the parameter is
0:8, all experimental economies stay close or converge
to the run equilibrium. The economies perform very di¤erently and may converge to either the
run or the non-run equilibrium if the coordination parameter falls into the indeterminacy region
where the parameter is equal to 0:6 or 0:7. Second, some order e¤ect is detected, especially when
the coordination parameter is in the indeterminacy region. The experimental economies tend to
coordinate better and have less bank runs, when the level of coordination requirement increases
gradually over time in contrast to the case where the parameter decreases or changes in a random
pattern. Third, we detect some learning e¤ect when the coordination parameter is in or around
the indeterminacy region: it takes time for some experimental economies to converge to one of the
two equilibria, and some sessions experience switches starting with low (high) numbers of people
choosing to wait but evolving to the non-run (run) equilibrium.
1
Ennis (2003) also includes a discussion of risk dominance as an equilibrium selection device in the context of the
DD model of bank runs.
3
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 an indeterminacy point of 0:5 for the occurrence of bank runs. The limiting case
therefore does not provide a satisfactory explanation of the experimental data, which suggests an
indeterminacy region located above 0:5. We show that the logit evolutionary algorithm (refer to
Hommes, 2011, for a survey of the method), where the rate of experimentation is estimated from
experimental data, is successful in explaining the behavior of subjects.
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 …ndings. 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 …rst experimental study of miscoordination-based bank runs based
on the DD model. The paper’s emphasis is on the e¤ectiveness of alternative ways to prevent bank
runs, including suspension of payments and deposit insurance. In the process, the paper also o¤ers
some evidence that the severity of bank runs is a¤ected by the level of coordination requirement.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 a¤ected by uncertainty about
the aggregate liquidity demand and by the number of opportunities subjects have to withdraw.
They show that (i) bank runs are rare when fundamental withdrawal demand is known but occur
frequently when it is stochastic, and (ii) subjects are more likely to withdraw when given multiple
opportunities to do so than when presented with a single decision.
Schotter and Yorulmazer (2009) investigate bank runs in a dynamic context and examine the
factors that a¤ect the speed of withdrawals, including the number of opportunities to withdraw
and the existence of insiders. The results are (i) the more information laboratory economic agents
2
The paper also shows the payo¤ di¤erential between the run and non-run equilibria in‡uences the severity of
bank runs.
4
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, signi…cantly a¤ects the dynamics of bank runs and mitigates the severity
of bank runs.
Klos and Sträter (2012) test the prediction of the global game theory of bank runs developed by
Morris and Shin (2001) and Goldstein and Pauzner (2005). In the model, depositors receive noisy
private signals about the (varying) quality of the bank’s long-term assets. The theory predicts that
depositors employ a threshold strategy choosing to withdraw (wait) if the signal is below (above) a
cut-o¤ point. Klos and Sträter …nd 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 …nd that the level-k approach, in which level-1
types assume that level-0 types play a random threshold strategy, …ts the data considerably better.
Compared with the existing experimental literature on bank runs, our paper has a very di¤erent
focus, which is to study how the level of coordination requirement a¤ects the occurrence of bank
runs as a result of pure coordination failures. To that goal, throughout the experiment, the rate
of return of the bank’s long-term assets is …xed, there is no uncertainty in the demand for liquidity, subjects have only one withdrawing opportunity in each game, all payo¤-relevant variables
are public information, and there are no insiders endowed with superior information than others.
The only variable that changes during a session is the level of coordination requirement (achieved
through a change in the short-term repayment rate promised to withdrawers), which is also public
information. To investigate the e¤ect 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 e¤ect.
In addition, di¤erent sequences of the coordination parameter are adopted to detect potential order
e¤ect.
More generally, our paper is also related to experimental studies of coordination games with
strategic complementarity, which feature multiple equilibria that can be Pareto ranked. In these
games there is often a tension between e¢ ciency and security. 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 payo¤-dominant
strategy.3 These two studies consider only one game with a tension between e¢ ciency and security,
and they do not provide a measure to quantify the riskiness associated with the payo¤-dominant
strategy. As a result, the setup cannot be used to determine what levels of risk are "too risky".
In our paper, we parameterize and quantify the riskiness of the payo¤-dominant strategy by the
coordination parameter. By observing how subjects behave in a series of games characterized by
di¤erent levels of coordination requirement, we can study how the coordination behavior is a¤ected
3
Van Huyck et al. (1990) study the average-opinion game, and Van Huyck et al. (1991) study the minimum-e¤ort
game.
5
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 a¤ected by the level of the coordination
requirement. Van Huyck et al. (1991) …nd that the initial coordination behavior well predicts
the …nal outcomes in repeated coordination games. However, our experimental results show that
the result does not hold when the coordination parameter takes certain values.4 Weber (2006),
Brandts and Cooper (2006), and most recently, Romero (2011) have detected path dependence
e¤ect in minimum-e¤ort games. Weber (2006) focuses on the e¤ect of a change in group size, and
Brandts and Cooper (2006) study the e¤ect of payo¤ bonuses intended to raise e¤ort levels. Romero
(2011) shows that the performance of the experimental economies depends on whether the cost of
e¤ort increases or decreases over time. Our study provides additional evidence of path dependence
in the context of a bank run game, with a focus on the e¤ect of the coordination parameter. One
of our …ndings is that path dependence is more likely to occur when the coordination parameter is
in the indeterminacy region.
Finally, we would like to discuss several experimental papers on global games. The global
game equilibrium has been proposed in the literature as a re…nement concept for the so called
"entry" games: binary-choice games ("enter" or "not enter") of complete information with strategic
complementarity and multiple equilibria. The payo¤ to "enter" exceeds that to "not enter" if the
number of entrants exceeds a threshold value. 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 payo¤-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 di¤erent 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 …rst 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 payo¤-relevant variable. The principal …nding is that the
observed behavior is similar in both situations: individuals follow threshold strategies and choose
to enter only if the payo¤-relevant variable is favorable enough (although the indeterminacy values
4
The earlier experimental works on coordination games have o¤ered systematic studies of the e¤ects of the number
of 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)
investigate the e¤ect of optimization premium, which measures the penalty attached to not playing a best-response.
The main …nding is that a higher optimization premium is associated with more sensitive responses to the history
of opponents’ play, faster convergence and more frequent occurrence of the risk-dominant equilibrium. Our work
complements earlier studies by focusing on the e¤ect of the coordination requirement.
5
Carlsson and van Damme (1993) study the case of 2x2 games. Morris and Shin (2003) generalize the analysis
to a class of 2xN symmetric games. Notable applications of the theory include Morris and Shin (1988) to currency
attacks and Goldstein and Pauzner (2005) to bank runs.
6
may di¤er). 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 Du¤y and Ochs
(2012).6
To study miscoordination-induced bank runs, we provide subjects with common information
about all payo¤-relevant variables. In view of earlier experimental studies of global games, we
expect the coordination parameter, which is also a payo¤ relevant variable, to function as a coordination device as it changes within a session. However, we have a di¤erent focus here: we
would like to investigate systematically the e¤ect of the coordination requirement as measured by
the coordination parameter. Our experimental design also has features that facilitate the study
of learning and order e¤ects. 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 coordination requirement allows us to catch its e¤ect on the learning process. The adoption of di¤erent
sequences of the coordination parameter allows us to detect potential order e¤ect. In contrast, in
the experiment by Klos and Sträter (2012), Heinemann et al. (2004, 2009) and Du¤y and Ochs
(2012), 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 di¤erent value
of a payo¤-relevant variable and choose whether to enter or not under each situation. In Du¤y and
Ochs (2012), a payo¤-relevant variable changes in every period. Klos and Sträter (2012) try both.
Their experimental designs are more suitable for studying how individuals respond when a payo¤relevant variable changes constantly, and less so for the study of the learning process or the order
e¤ect. In fact, Heinemann et al. (2004) and Klos and Sträter (2012) …nd no evidence of learning
e¤ect as the threshold levels adopted by individual subjects change little across time. Citing the
result of lack of learning e¤ect in previous studies, Heinemann et al. (2009) study non-repeated
games and focus solely on the initial coordination behavior. Finally, we focus on the e¤ect of the
coordination parameter on aggregate coordination outcomes, while the papers on global games are
mainly concerned with individual strategies.
3
Theoretical Framework, Experimental Design and Hypotheses
In this section, we describe the theoretical framework that underlies the experiment, the hypotheses
and the main design features of the experiment.
6
Cabrales et al. (2007) investigate the e¤ect of payo¤ uncertainty in a 2-person coordination game. The main
purpose of Heinemann et al. (2009) is to measure strategic uncertainty when agents have common information about
payo¤ relevant variables. Du¤y and Ochs (2012) improve upon the experimental methodology in Heinemann et al.
(2004) by introducing a within-subject design, and investigate dynamic entry games.
7
3.1
Theoretical Framework
The theoretical framework that underlies our study is the DD model of bank runs. There are three
dates indexed by 0, 1, and 2. There are D ex ante identical agents in the economy. At date 0
(the planning period), each agent is endowed with 1 unit of good and faces a liquidity shock that
determines her preferences over goods at date 1 and date 2. The liquidity shock is realized at the
beginning of date 1: Among the D agents, N of them become patient agents who are indi¤erent
between consumption at date 1 and date 2, and the rest become impatient agents who care only
about consumption at date 1. Realization of the liquidity shock is private information. Preferences
are described by
8
< u(c1 )
U (c1 ; c2 ) =
: u(c + c )
1
2
for impatient consumers,
for patient consumers,
where c1 and c2 denote the consumption at date 1 and date 2, respectively. The function u( )
satis…es u00 < 0 < u0 , limc!1 u0 (c) = 0 and limc!0 u0 (c) = 1. The relative risk aversion coe¢ cient
cu00 (c)=u0 (c) > 1 everywhere. There is a productive technology that transforms 1 unit of date 0
output into 1 unit of date 1 output or R > 1 units of date 2 output.
At the socially optimal allocation, impatient agents consume only at date 1 and patient agents
consume only at date 2. Let ci and cp denote the consumption by impatient consumers and patient
consumers, respectively. The optimal allocation, (ci ; cp ), is characterized by 1 < ci (N=D; R; u) <
cp (N=D; R; u) < R: A bank, by o¤ering demand deposit contracts, can provide liquidity insurance
to agents. The contract requires agents deposit their endowment with the bank at date 0. In
return, agents receive a bank security which can be used to demand consumption at either date
1 or 2. The bank promises to pay r > 1 to depositors who choose to withdraw at date 1. If the
number of withdrawers, e, exceeds e^ =
D
r,
the bank will not have enough money to pay every
withdrawer the promised rate r: In this case, a sequential service constraint (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
to wait and z^ = D
e^ = D(1
prevent bankruptcy. Let
e
and
e be the number of depositors choosing
1=r) be the minimum number of depositors choosing to wait to
z
be the payo¤s to those who choose to withdraw and roll over,
respectively. The deposit contract can be formulated as
e
z
8
< r; if z z^;
=
: r with probability D z^ , and 0 with probability
D z
8
< D r(D z) R; if z z^;
z
=
: 0; if z z^:
The optimal risk-sharing allocation can be achieved by setting r to r
8
z^ z
D z,
c1 .
if z
z^;
To conduct the experiment, we keep the important features of the DD model. To facilitate the
experimental design, we modify the original model along two dimensions. First, in the original
model, there are both patient and impatient agents. Impatient agents always withdraw 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 simpli…cation and assume instead that if 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 su¢ cient to generate a payo¤ externality and panic-based runs.
To rule out the possibility that bank runs are caused by weak performance of the bank’s long-term
portfolio, we …x the rate of return of the long-term investment, R, throughout the experiment.
For the short-term interest rate r, we do not use the optimal rate r . Instead, we set r to be a
series of values greater than 1. As will become clear shortly, there is a one-on-one correspondence
between r and the coordination parameter. Using r as a control variable allows us to change the
coordination parameter in a simple way.8 The resulting payo¤ function used in the experiment can
be represented as
e
= min r;
z
= max 0;
N
N
N
;
z
r(N
z
(1)
z)
R :
(2)
The coordination parameter (denoted as ) measures the amount of coordination that is required
for agents who choose to wait to receive a higher payo¤ than those who choose to withdraw. The
parameter can be calculated in two steps. First, solve for the value of z, the number of depositors
who choose to wait, that equalizes the payo¤s associated with "withdraw" and "wait",
r=
N
(N
z
z)r
R;
and denote it by z . Thus, z is given by:
z =
R(r
r(R
1)
N.
1)
Second, divide z by N to get , the fraction of depositors who choose to wait, that equalizes the
payo¤s to both strategy choices:
=
z
R(r
=
N
r(R
7
1)
:
1)
Madiès (2006) adopts the same arrangement in this regard.
For optimal contracting in the DD framework, please refer to Green and Lin (2000, 2003), Andolfatto, Nosal and
Wallace (2007), Andolfatto and Nosal (2008), and Ennis and Keister (2009a, 2009b, 2009c). The …rst few papers show
that the multiple-equilibria result goes away if more complicated contingent contracts –as compared with the simple
demand deposit contracts in DD –are used. The three papers by Ennis and Keister show that the multiple-equilibria
result is restored if the banking authority cannot commit not to intervene in the event of a crisis, or the consumption
needs of agents are correlated.
8
9
We can control the coordination parameter, , by changing the short-term rate r:
The coordination game characterized by the above payo¤ structure has two symmetric purestrategy Nash equilibria.9 In the run equilibrium, every depositor chooses to withdraw and run on
the bank expecting others to do the same. As a result, z = 0 and everybody receives a payo¤ of
1. In the non-run equilibrium, every depositor chooses to wait expecting others to do the same.
In this equilibrium, z = N and everybody receives a payo¤ of R. Fixing R also has the additional
advantage of maintaining the payo¤ di¤erence between the two equilibria …xed at R
1 throughout
the experiment.
3.2
Experimental Design and Hypotheses
To examine how the coordination requirement a¤ects the occurrence of bank runs, we design the
experiment with the following main features.
First, in contrast to earlier experimental literature on coordination games, which adopts contextfree phrasing, we phrase the task explicitly as a decision about whether to withdraw money from
the bank. The speci…c banking context makes it easier for subjects to comprehend the task they
are required to perform and makes the payo¤ 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 …x the rate of return of the long-term investment, R, at 2; throughout the experiment.
The change of coordination requirement is induced by a change in the short-term rate, r. See table
1 for the correspondence between r and . Third, to inspect the e¤ect of coordination requirement,
each session of experiment includes 7 or 9 phases, with each phase corresponding to a di¤erent value
of the coordination parameter ranging from 0:1 to 0:9. 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 a¤ects the evolution of the
experimental economy and the learning process. Fifth, to detect potential path-dependence e¤ect,
we run sessions with di¤erent orderings (increasing, decreasing and random) of the coordination
parameter.
[Insert Table 1: Correspondence between r and ]
These above design features allow us to answer the three questions raised in Introduction: (1)
How does the coordination parameter a¤ect the level of coordination and the frequency of bank
runs? (2) Does the path followed by the coordination parameter a¤ect the performance of the
economy? (3) How does the level of coordination a¤ect the learning process and the speed of
convergence of the economy? For each research question, we form an ex ante hypothesis as a
conjecture about the experimental results.
9
There is also a symmetric mixed-strategy equilibrium where each depositor chooses to wait with a probability
between 0 and 1 and the expected payo¤ from the two strategies are equalized.
10
Hypothesis I. There is more coordination at "wait" when the coordination task is easier. In
view of the study by Temzelides (1997), we expect to see
= 0:5 as the indeterminacy point for the
occurrence of bank runs: the economy tends to stay close to the non-run (run) equilibrium when
< 0:5 (> 0:5) and becomes di¢ cult to predict when
= 0:5.
Hypothesis II. There is an order e¤ect: the path followed by
a¤ects the performance of
the economy, especially when the level of the coordination requirement is at the indeterminacy
value. We expect that as
takes the increasing order, there is on average more coordination
at "wait" and fewer incidences of bank runs. As the level of coordination requirement gradually
increases over time, subjects may build up mutual trust while playing games with easy coordination
and the mutual trust may persist into games with more di¢ cult coordination. In contrast, when
the coordination parameter starts with a high value and then decreases over time, the pessimism
developed in earlier phases may carry forward to later phases with easier coordination.
Romero (2011) …nds that in the minimum-e¤ort game, the e¤ort level tends to be higher (lower)
if the cost of e¤ort increases (decreases) over time. Our hypothesis is consistent with Romero’s
…ndings. However, we expect to see a strong order e¤ect only for games with the coordination
parameter around the indeterminacy value when agents are less certain about their strategies.
Hypothesis III. The learning process of repeated games is a¤ected by the level of coordination
requirement. For a game with the coordination parameter around the indeterminacy value, there is
a stronger learning e¤ect: the economy is likely to evolve over time and may even switch between
the two equilibria, and the initial and …nal coordination behaviors may be very di¤erent.
Earlier studies (for example, Van Huyck et al., 1991) suggest the initial coordination behavior
well predicts the …nal outcomes in repeated coordination games. Heinemann et al. (2004) and
Klos and Sträter (2012) …nd the value of the thresholds adopted by individual subjects changes
little across time. Citing the result of weak learning e¤ect 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) consider only one game with a tension between e¢ ciency
and security. The conclusion that the initial behavior well predicts …nal 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 payo¤-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 induce stronger learning e¤ect. 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 if the level of coordination requirement is around the indeterminacy value,
subjects may become uncertain about what they should do and their actions may switch as they
learn about the coordination history. As a result, the …nal outcomes may deviate greatly from the
11
initial coordination behavior.
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 su¢ cient English reading and listening skills, we restrict our subject pool to
those who have passed the College English Test Grade IV, a standardized national English language
test for college students in China. We also ensure that the subjects understand the experimental
instructions by giving them 10 trial periods and several opportunities to ask questions before the
formal rounds.
For each session of experiment, we recruit 10 subjects play the role of depositors. Most subjects
are recruited from 2nd or 3rd-year undergraduate or graduate economics or business classes. Since
the game in the experiment is fairly straightforward, it is important that the subjects have no prior
experience with experiments of coordination games. It is also important that the subjects be from
mixed sources to guarantee that they do not have close relationship before participating in the
experiment.
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). Communication 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. Payo¤ tables for all phases are provided to list
the payo¤ that an individual will receive if he/she chooses to withdraw early or to leave money
in the bank given that n = 1 s 9 of the other 9 subjects choose to withdraw early. The payo¤
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’payo¤s are then determined by equations (1) and (2). At the end
of each round, subjects are presented with the history of their own actions, payo¤s, and cumulative
payo¤s 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 payo¤ to both actions because we think this is more realistic:
depositors do not directly observe the return of other depositors. However, subjects can try to
deduce such information from the payo¤ tables (as will be shown later in Section 5, subjects can
deduce the information most of the time). After the experiment, the total payo¤ that each subject
earns is converted from EDs into cash.
In total, we have run 20 sessions. The …rst 8 sessions (sessions 1 to 8) include 7 values of
(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 di¢ cult over time. Sessions
5 to 8 have decreasing
and coordination becomes easier over time. We …nd the experimental
12
economies stayed close or converge to the non-run (run) equilibrium when
performed very di¤erently when
0:5 (
0:8), but
= 0:7 (details will be shown in the next section where we present
the experimental results). There is also some weak evidence of the order e¤ect. To gain more
understanding about the performance of the economy around the indeterminacy value of 0:7, we
run new sessions with a …ner grid of
by adding 0:4 and 0:6 into the spectrum of . In addition,
there is the concern that the monotonic change of
is conducive to the change of coordination
behavior in response to the parameter: it may be that after making the same choice for many
rounds (40 in sessions with increasing
and 20 in sessions with decreasing case), subjects are bored
and waiting for a change of actions. The alteration of
may have served as a cue for the change
to take place. To address the concern, we include a treatment with random ordering of .10 In
total, we run 12 new sessions (sessions 9 to 20) all with a …ner grid of 0:1. For each of the three
orderings, increasing, decreasing and random, we run 4 sessions of experiment. Among the 12 new
sessions, 2 sessions (sessions 9 and 17) include the full spectrum of 9 values of
ranging from 0:1
to 0:9 with a step of 0:1. In view that almost every subject consistently coordinates at "wait" for
= 0:1 and 0:2, we decided to drop these two values from the other 10 sessions. Excluding the two
smallest values of
also has the additional advantage of having the same number of paying periods
as in the …rst 8 sessions with more coarse grids. Table 2 provides detailed design information for
each session of experiment, including the grid of , the ordering of , the speci…c values of , and
the location and date of 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 …rst 8 sessions of experiment, there are two points
where the grid is wider at 0:2 (from 0:3 to 0:5 and then from 0:5 to 0:7). In view of this, we say
the …rst 8 sessions of experiment have a coarse grid and the 12 sessions have a …ne grid. In total,
we have …ve treatments: coarse grid and increasing
(sessions 5 to 8); …ne grid and increasing
13 to 16); and …ne grid and random
4
(sessions 1 to 4); coarse grid and decreasing
(sessions 9 to12); …ne grid and decreasing
(sessions
(sessions 17 to 20).
Experimental Results
In this section, we present and discuss the experimental results. Figure 1 plots the path of z, the
number of subjects choosing to wait, for each
value and for each of the 20 sessions of experiment.
The results for the …rst 8 sessions are on the …rst page, and the results for sessions 13 to 20 are
on the second page. To facilitate comparison across di¤erent treatments, we plot z according to
increasing from 0:1 to 0:9 with a step of 0:1. The sequence of z is thus di¤erent from the actual
time path for sessions with decreasing and random ordering of . The empty sections in the graphs
10
We thank the referees for suggesting these new treatments.
13
are due to missing values of
(0.4 and 0.6 for sessions 1 to 8, and 0:1 and 0:2 for sessions 10 to 16
and 18 to 20).
[Insert Figure 1: Experimental Results]
Table 3 lists the starting, terminal and mean values of the number of subjects choosing to
wait (denoted as S, T , and M , respectively) for each phase marked by a di¤erent value of . To
qualitatively characterize the performance of the experimental economies, we de…ne 10 performance
categories. The …rst 8 performance categories are de…ned according to the mean and terminal value
of the number of people who choose to wait. If M
non-run equilibrium" (denoted as NN); if 8
9, we say the economy is "very close to the
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 "converges
to 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
"very close to the run equilibrium" (denoted as RR); if 1 < M
1, we say the economy is
2, we say the economy is "fairly
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 switching
behavior, we de…ne two more performance categories based on the starting and ending values of z. If
S < 5 and T
and T
8, we say the economy "switches to non-run equilibrium" (denoted as SN); if S > 5
2, we say the economy "switches to run equilibrium" (denoted as SR). The de…nitions of
the performance categories are summarized in table 4. Table 5 categorizes the performance of the
20 sessions of experiment.
[Insert Table 3: Starting, Terminal and Mean Value of the Number of Subjects
Choosing to Wait]
[Insert Table 4: De…nition of Performance Classi…cation]
[Insert Table 5: Classi…cation of Performance of Experimental Economies]
In the following, we …rst discuss the results from the …rst 8 sessions of experiment with the
coarse grid. For each of the three hypotheses, we show whether there is evidence to support it.
We then discuss the results from the 12 new sessions with the …ner grid and revisit all the main
…ndings.
4.1
Findings from Treatments with Coarse Grid
Let us …rst investigate what kind of evidence the …rst 8 sessions of experiment provide for the three
hypotheses raised in Section 3:
14
Finding I: There are less withdrawals when a lower amount of coordination is required. However, the results suggest an indeterminacy value higher than 0:5 as predicted in Temzelides (1997).
As shown in table 3, the average number of subjects choosing to wait tends to decrease with .
A higher coordination requirement means that there is a higher risk associated with waiting and
therefore, discourages subjects from choosing the action. As a result, one would expect the number
of subjects choosing to wait to decrease as coordination becomes more di¢ cult.
As for the occurrence of bank runs, the experimental data suggests a higher indeterminacy value
than 0:5. As shown in table 5, when
0:5 as the non-run region. When
0:8 the run region. When
0:5, all economies are marked by NN or CN. We call
0:8, all economies are marked by RR, FR or CR. We call
= 0:7, the experimental economies experience indeterminacy with
much less clear-cut outcomes. 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 …rst, 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 , sessions 5, 6 and 8 begin with very low coordination and
stay there; they are marked by "RR". Session 7 starts with 6 subjects who coordinate on "wait"
and the number rises to 8 by the end; this session is marked by "CN".
The result that a higher indeterminacy value is suggested can be explained as follows. Temzelides
(1997) derives 0:5 as the indeterminacy point under the implicit assumption that agents care about
only strategic uncertainty. However, the higher payo¤ associated with non-run equilibrium generates an additional force to draw people to that equilibrium and pushes up the indeterminacy value
above 0:5.
Finding II: There is some weak evidence of the order e¤ect as
takes the indeterminacy value
of 0:7.
The results from the …rst 8 sessions of experiment show that the ordering of
on the experimental economies when
0:5 (or
has little e¤ect
0:8) as all economies manage to stay or
converge to the non-run (or run) equilibrium. Given the great variation in performance of di¤erent
economies around the indeterminacy value of 0:7, we run the rank sum test to detect potential
order e¤ect for the phase with
= 0:7. To do the test, we separate the 8 sessions of experiment
into two groups of observations: the …rst group contains the 4 sessions with increasing
and the
other group includes the 4 sessions with decreasing . We conduct one-sided Mann-Whitney rank
sum tests on the starting, ending and mean value of z between the two groups. The null hypothesis
is that there is no di¤erence between the two groups. The alternative hypothesis is that the group
with increasing
has higher z or lower number of withdrawals than the other group with decreasing
. The test results are in the …rst part of table 6a, including the average values of S; T and M
15
in each group, the value of the test statistic Z and the one-sided p-value.11 The initial number of
people who choose to wait tends to be higher in sessions with increasing
with an average of 7
versus an average of 2 in the group with decreasing . The di¤erence is statistically signi…cant with
a p-value of 1:38% (and the di¤erence between group averages is 5) suggesting that the order of
a¤ects the performance of the economy at the start of the phase. This could be a result of inertia:
subjects tend to follow their old strategies immediately after a change of . However, the e¤ect
of history weakens after the initial periods. The mean and ending values of z are not signi…cantly
di¤erent with p-values of 12:41% and 37:05%, respectively. Overall, there is some, albeit weak,
evidence that the ordering of
a¤ects the performance of the economy.
[Insert Table 6: Rank Sum Tests of the Order and Grid E¤ect]
Finding III: There is evidence of intra-phase learning when the coordination parameter is
around the indeterminacy value.
As discussed earlier, previous studies …nd that the initial coordination behavior well predicts
the …nal outcomes in repeated coordination games. Here we …nd the same result when
= 0:1,
0:2, 0:3 or 0:9: most subjects keep their initial choices, and the experimental economies reach one
of the two equilibria instantly. However, we detect intra-phase learning for some sessions when
is in the neighborhood of the indeterminacy value 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
= 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
coordination at "wait" (S
= 0:7, three sessions (sessions 2; 3 and 4) start with a high level of
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
= 0:8 in session 7. In these 6 sessions, there is a signi…cant di¤erence
between the starting and ending values of the number of subjects choosing to wait as subjects learn
from previous rounds and change their actions.
The bank run game involves a tension between safety and e¢ ciency. Withdrawing ensures a
payo¤
1, but has a low return ceiling de…ned by r; waiting is risky with a possible low payo¤ of
0, but there is also the potential to achieve the e¢ cient allocation of R. The tension is stronger
in games with the coordination parameter lying around the indeterminacy region. Consequently,
switches are more likely to be observed in these games. In a switch from a low level of coordination
to the non-run equilibrium, the e¢ ciency consideration wins. However, a switch in the opposite
direction is also possible if the safety concern becomes dominant.
11
In the absence of strong argument (resulted from strong previous research, theoretical or logical considerations)
about the direction that the order e¤ect, two-sided tests should be used with the p-values doubling the values reported
in Tables 6 and 7. Here, one-sided tests are appropriate here because it is highly unlikely that the order e¤ect would
work in the opposite direction. As mentioned in Section 2, previous studies about history dependence (Weber, 2006;
Brandts and Cooper, 2006; Romero, 2011) all show that following a game with parameters that are conducive to
more coordination at the payo¤-dominant strategy, subjects are more likely to choose the strategy in a new game.
16
One may also notice a regularity concerning the direction of switches: for sessions with increasing
, the switches are always from a high level of coordination at "wait" toward the run equilibrium
(marked with "SR"), while for the treatment with decreasing , the switches are always in the
opposite direction (marked with "SN").12 This regularity is largely due to inertia: the starting
level of coordination in a new phase is often a¤ected by the level of coordination in the preceding
phase.
In the increasing treatment, this means subjects tend to start a new phase with di¢ cult coordination with a high level of coordination that cannot be sustained in the rest of the phase. For
example, in the treatment with increasing , three sessions experience a switch from high z to the
run equilibrium in the phase with
= 0:7. The initial high levels of coordination in the phase is
due to the high level of coordination exhibited in the preceding phase with
= 0:5. However, the
con…dence is not sustained throughout the whole phase and economies experience switches toward
the run equilibrium.
In contrast, as
decreases over time, the history of bad coordination in the past could carry
over to a new phase with easier coordination, resulting in switches in the reverse direction. For
example, sessions 5 and 8 (with decreasing ) experience switches in the phase with
= 0:5: The
low levels of coordination at the start of the phase is directly linked to the low levels of coordination
in the preceding phase where
4.2
= 0:7.
Findings Revisited after Treatments with Fine Grid
Given the observed richness of activities around the indeterminacy value of
= 0:7, we run a
new set of sessions with emphasized focus around the value. In particular, in contrast to the …rst
8 sessions of experiment which involve a coarse grid around 0:7, the new sessions feature a …ner
grid around the value and include
(non-monotonic) ordering of
= 0:4 and 0:6. In addition, we run experiment with random
to study whether the change of coordination behavior is induced by
the monotonic change of . Altogether, we have three new treatments, all with a …nder grid of
of 0:1 but in three di¤erent orderings of : increasing, decreasing and random.
The path of z for each session is presented in …gure 1b. The starting, ending and mean 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 …ndings are in general con…rmed by the new sessions with the …ner grid. In addition,
the new sessions generate stronger evidence for the order e¤ect. In the following, we revisit each of
the three main …ndings in details.
Finding I revisited. Finding I is robust to the adoption of a …ner grid and non-monotonic
ordering of . We still observe more withdrawals for phases with higher coordination parameters,
and the indeterminacy value is higher than 0:5. However, due to the …ner grid being used, the new
data captures another indeterminacy value 0:6 in addition to 0:7 as in the 8 old sessions.
12
We thank a referee for pointing out this regularity.
17
When
0:5, including
= 0:4 which was not included in the 8 old sessions, all economies stay
close or converge to the non-run equilibrium; when
0:8, all economies stay close or converge
to the run equilibrium with a small exception. In session 12 when
= 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
= 0:6 or 0:7, the performance of the experimental economies varies
greatly across di¤erent treatments and di¤erent sessions within the same treatment. When
all 4 economies with increasing
= 0:6,
stay close to the non-run equilibrium; 2 out of the 4 economies
with the decreasing and random treatments stay close or converge to the non-run equilibrium, while
the other 2 stay close or converge to the run equilibrium. When
= 0:7, all 4 economies with the
increasing treatment again stay close to the non-run equilibrium; all 4 economies with decreasing
stay close to the run equilibrium; 3 out of 4 economies with the random ordering stay close to
the run equilibrium, and 1 economy stays close to the non-run equilibrium.
In summary, we can characterize the occurrence of bank runs in three regions of the coordination
parameter. The new sessions identify the same non-run region
0:5 and run region
0:8. In
these two regions, the performance of the experimental economies is more uniform and is robust
to the ordering of : When
when
lies between 0:5 and 0:8 or the "indeterminacy" region, for example,
= 0:6 or 0:7, the economy is hard to predict and the outcome is sensitive to the ordering of
, the magnitude of the grid of
and the coordination result in the past.
In their experimental study of the global game theory, Heinemann et al. (2004) also run sessions
with common information, which in essence converts the global game with unique equilibrium into
a pure coordination game with multiple equilibria. They …nd individual subjects adopt a threshold
strategy that constitutes a best response to the belief that other subjects choose the payo¤-dominant
strategy ("wait" in the context of bank runs) with probability 2=3. In the context of the bank run
game, the strategy is to withdraw if
> ^, wait if
< ^ and randomize if
= ^ with ^ = 2=3.13
This implies that the indeterminacy region is located around 2=3.
Finding II revisited. The results from the new sessions lend more support to …nding II. With
the …ner grid, the order e¤ect becomes much stronger and we identify another indeterminacy value
13
Under the belief that each of the other 9 subjects chooses to wait with probability 2=3, the expected
payo¤s to "withdraw" is
!
9
n
9 n
X
9
1
2
10
min r;
e =
3
3
1
+n
n
n=0
and the expected payo¤ to "wait" is
z
=
9
X
n=0
The solution to
e
=
z
9
n
!
1
3
n
2
3
9 n
max 0;
2(10
10
nr)
n
is r = r^ = 1:5 and the implied value of the coordination parameter is
=^=
R(r
r(R
1)
= 2=3:
1)
The best response to the belief that everyone else chooses to wait with probability 2=3 is to withdraw
if > ^, wait if < ^ and randomize if = ^.
18
= 0:6 besides
= 0:7.
As in the sessions with the coarse grid, the ordering of
0:5 (or
has little e¤ect on the economy when
0:8) in the sense that all economies manage to stay close to or converge to the
non-run (or run) equilibrium. However, the new data shows a strong order e¤ect when
= 0:6 or
0:7.
As is obvious from table 5, when
= 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
coordination when
= 0:6 and no session achieves a high level of
= 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
manages to stay close to the run equilibrium when
= 0:6, and only a single session
= 0:7 (the other 3 stay close to the run
equilibrium).
The order e¤ect is also supported by rank sum tests on the starting, ending and mean value of
z (denoted as S; T and M; respectively). The test results for
= 0:7 are shown in table 6a. The
increasing treatment achieves signi…cantly better coordination than the decreasing treatment. The
average starting, terminal and mean value of the number of subjects choosing to wait are all much
higher when
increases. The p-values for S; T and M are 0:76%; 0:41% and 1:01%, respectively.
The p-values of the other two pairwise comparisons, increasing versus random and random versus
decreasing, are all below 7%. Now let us turn to the test results for
= 0:6 presented in table 6b.
There is again strong evidence that the increasing treatment exhibits a higher level of coordination
than the decreasing treatment. The p-value of M is highly signi…cant at 1:01%. The evidence is
less strong for the other two pairwise comparisons. Even with a small sample size of 4 for each
treatment, the data provides clear evidence that the performance of the economy is path-dependent.
Finally, a comparison of the new and old sessions with increasing
9-12) suggests that a more gradual increase in
indeterminacy value of 0:7. When
(sessions 1-4 versus sessions
is associated with better coordination at the
increases from 0:5 to 0:7 with a step of 0:2, only 1 out of 4
economies achieves a moderate high coordination when
= 0:7. In contrast, when
increases from
0:5 to 0:6 and then to 0:7 with a step of 0:1, all 4 out of 4 economies manage to stay close to the
non-run equilibrium when
= 0:7. A possible reason is that with a smaller grid, subjects perceive
the current game to be closer to the previous game with easier coordination. As a result they
are more likely to maintain the old strategies that were adopted in the previous game. The rank
sum tests also show that for
= 0:7, the economy tends to have higher starting, ending and mean
value of z with the …ner grid with p-values
1:3%. The …nding that a gradual change facilitates
coordination reminds us of the work by Steiner and Stewart (2008). The authors study learning
in a set of complete information normal form games where players continually face new strategic
situations and form beliefs by extrapolation from similar past situations. They show that the use
of extrapolations in learning may generate contagion of actions across games when players learn
from games with payo¤s very close to the current ones. Our experimental results show that the
19
contagion e¤ect is stronger when the change in the payo¤ structure is smaller and more gradual.
Finding III revisited. For …nding III, we still observe some learning e¤ect when
is around
the indeterminacy region. With the increasing treatment, 3 sessions (sessions 9-11) start with
relatively high levels of coordination (with S
when
= 0:8. When
8) and switch to the run equilibrium (with T
1)
= 0:9; session 9 starts with S = 6 and switches to the run equilibrium at
the end of the phase (with 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
random treatment, session 19 switches to the run equilibrium when
= 0:6: With the
= 0:8 and 0:9 (with S = 9
and T = 0). Session 20 follows a similar pattern with S = 6 and T = 1. As explained in the
discussion of results from the sessions with the coarse grid, the direction of the switches can be
largely accounted for by inertia.
5
The Logit Evolutionary Algorithm
The evolutionary algorithm developed by Young (1993) and Kandori et al. (1993) has been introduced as an equilibrium selection mechanism into many rational expectation models with multiple
equilibria and have greatly contributed to the understanding of these models. Temzelides (1997)
applies the evolutionary algorithm to a repeated version of the DD model. The algorithm has two
main components. The …rst is myopic best response, which means that when agents react to the
environment, they respond by choosing the strategy that performed the best in the previous period.
In the context of the DD model, the myopic best response is "withdraw" if the number of people
choosing to wait in the previous period, zt
1,
is
z , and "wait" otherwise. The second com-
ponent is experimentation, during which agents change their strategies at random with a certain
probability. In the context of the DD model, experimentation means to ‡ip 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
< 0:5. In other words,
= 0:5 is predicted to be
the watershed for the occurrence of bank runs.
Obviously, the limiting case does not describe the experimental results well. In particular, the
experimental results suggest an indeterminacy region located above 0:5. When
= 0:5, all 20
experimental economies stay close or converge to the non-run equilibrium. Only when
= 0:6 or
0:7 do we observe large variation across di¤erent treatments and sessions. One way to improve the
algorithm is to allow it to feed on the information that agents have about the economy. For examples, the coordination parameter
captures how di¢ cult the coordination task is, and the number
of subjects choosing to wait in the last period, zt
1,
measures how the group coordinated in the
past. These two variables may in‡uence a subject’s belief about what other subjects will do in the
next period and in turn the subject’s strategy choice. In this section, we show that the evolutionary algorithm combined with logit models to estimate the rate of experimentation as functions of
and/or zt
1;
can successfully explain the experimental data. Recently in a similar spirit, Hommes
20
(2011) shows that the evolutionary learning logit model …ts data of so-called learning-to-forecast
experiments well.
In the rest of this Section, we …rst describe the main features of the logit evolutionary algorithm.
We then run logit regressions to estimate the probability of experimentation from the experimental
data. Finally, we evaluate the performance of the algorithm.
5.1
Description of the Algorithm
The main feature of our algorithm is that it feeds on the available information about the past
period, so …rst we describe the amount of information subjects have in the experiment.
In the experiment, we inform subjects about r and subjects can always deduce
tables. We do not directly inform subjects about zt
payo¤ tables to deduce zt
1
1,
from the payo¤
but subjects can potentially refer to the
from their own payo¤s in the past period. For the new algorithm, we
assume that if a subject cannot deduce whether zt
1
> z , she skips the …rst 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
ically, if the subject can deduce the exact value of zt
on both zt
1
1;
and/or zt
1
Specif-
then the rate of experimentation depends
and ; otherwise, her experimentation rate depends only on . The value of the rate
of experimentation is to be estimated from the experimental data with logit models using
zt
1.
and/or
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 …rst
period are taken as given). Sessions 9 and 17 each have 9 phases and these two sessions generate
2x9x10x9 = 1; 620 observations. The remaining 18 sessions each have 7 phases and generate
18x7x10x9 = 11; 340 observations. The …rst 4 columns of table 7 summarize how a subject’s
information about the past period depends on her strategy and payo¤ 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 payo¤. If she got 0 payo¤, she can infer the bank was bankrupt, but she cannot deduce
the exact value of zt
exact value of zt
1
1.
If the subject chose to withdraw in the previous period, she can deduce the
if her payo¤ 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
93:19% of the
1
in
observations.14
14
As two referees point out, one may wonder how the results from the experiment and the evolutionary algorithm
depend on whether subjects have full information about the performance of the economy in the past. As shown
above, most of the time, subjects can derive the exact value of zt 1 . Our experimental economies are thus very close
to having full information. Nonetheless, we have also run three sessions of experiments with full information, one
21
[Insert Table 7: Information]
Temzelides (1997) assumes agents always update to best response and adopt an exogenously
…xed rate of experimentation (which approaches zero in the limiting case). Given that subjects can
deduce the best response most of the time, making whether to update to best response information
contingent has negligible e¤ect on the relative performance of the new algorithm. The critical
part of the new algorithm is to make the rate of experimentation depend on information about
and/or zt
1.
The success of the new algorithm is mainly due to the endogenization of the rate of
experimentation.
5.2
Estimation of Probability of Experimentation
To estimate the rate of experimentation, we construct each of the 12; 960 observations in …ve
steps. First, collect information about the individual’s last period’s strategy (st
payo¤ (
zt
1
t 1)
1 );
last period’s
and current period’s strategy (st ). Second, decide whether the individual can deduce
> 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 s0t . Fourth, decide whether
the individual ‡ips her strategy by comparing s0t and st and denote it by a binomial variable, f .
Count it as a ‡ip (f = 1) if s0t 6= st and a non-‡ip (f = 0) otherwise. Fifth, divide observations
into three groups according to s0t and whether the individual can deduce the exact value of zt
Group A includes observations with
exact value of zt
A)
1;
s0t
1.
="withdraw" and involves subjects who cannot infer the
these observations are used to estimate the experimentation rate (denoted as
for subjects who have information about zt
1
and consider ‡ipping from "withdraw" to "wait".
Group B includes data to estimate experimentation rate (denoted as
for subjects who cannot infer the exact value of zt
1.
to "withdraw". Note that subjects who have
in the same direction
Group C includes observations with s0t =
"wait" and are used to estimate the experimentation rate (denoted as
s0t
B)
C)
of ‡ipping from "wait"
="wait" can always infer zt
1
(see table 7).
We run three logit regressions to estimate the experimentation rate for each of the three groups
of data. The dependent variable is the binomial variable f (whether a ‡ip of strategy is observed).
For those who cannot infer the exact value of zt
expected sign of
1
(group A), we use
as the single regressor. The
is negative, meaning that subjects in groups A are less likely to ‡ip strategy
from "withdraw" to "wait" as coordination becomes more di¢ cult. For those who can infer the
exact value of zt
1
(groups B and C), we use the di¤erence between zt
1
and z as the single
regressor. According to feedback from subjects (after the sessions, we spent a few minutes talking
for each ordering of and all with the …ner grid. To investigate the e¤ect of information availability, we conduct a
two-sided rank sum test on the session mean of z between two groups. The …rst group includes the 12 sessions with
asymmetric information (sessions 9 to 20), and the second group includes the 3 sessions with full information. While
calculating the session mean of z, we omit the two phases with = 0:1 & 0:2 for sessions 9 and 17 to ensure the same
number of periods is included for each session. The p-value is 0:5637 showing that the performance of the economy
is not signi…cantly a¤ected by the adoption of full information.
22
with subjects about their experiences in the experiment), the di¤erence between zt
1
and z plays
an important role in determining the probability that subjects experiment with a strategy, even
though they know that the other strategy gave a higher payo¤ in the previous period. Note that
z = N so z contains information about
the case of
as zt
1
B,
as well. The expected sign of (zt
1
z ) is positive in
meaning that agents are more likely to change strategies from "withdraw" to "wait"
moves closer to z from below. The expected sign of (zt
z ) is negative for
1
that the probability of experimentation from "wait" to "withdraw" increases as zt
1
C,
meaning
moves closer
to z from above. The logit regression models are formulated as follows:
logit (
A)
= log
logit (
B)
= log
logit (
C)
= log
A
1
A)
B
1
=
A
+
A
=
B
+
B (zt 1
z );
=
C
+
C (zt 1
z ):
B
C
1
;
C
The results from the logit regression models are listed in table 8. All the coe¢ cients are highly
signi…cant 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
or zt
1
when they decide whether to ‡ip their
strategies. The estimated probability of experimentation is given by:
^A ( ) =
^B (zt
1;
) =
^C (zt
1;
) =
1
;
1+e
0:91+2:45
1+e
0:52 0:50(zt
(3)
1
1
1 + e2:83+0:31(zt
1
1
z )
z )
;
:
(4)
(5)
[Insert Table 8: Logit Regression of Rate of Experimentation-All Data]
5.3
Evaluation of the Algorithm
We evaluate the performance of the evolutionary learning logit model in two ways. The …rst is to
apply the algorithm and use the estimated probability of experimentation in Section 5:2 (equations
3 to 5) to simulate the path of the number of subjects choosing to wait, and compare the aggregate
performance of the simulated economies to that of the experimental economies. In particular,
we classify the performance of the simulated economies according to the de…nitions of the 10
performance categories in table 4 and check whether the distribution of the performance categories
is consistent with that of the actual experiments. The second is to make out-of-sample predictions
of individual strategies. Speci…cally, we use one set of data (the Chinese data) to estimate the rate
of experimentation, and apply the algorithm to predict the action choice in each of the Canadian
observations. The rate of successful predictions can then be used to evaluate the performance of
the algorithm.
23
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 , and 10 rounds in each phase. Each simulation adopts the starting values (S) in one of
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
…rst 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 …rst round. From round
2 on, each simulation agent updates its strategy according to the rules of the logit evolutionary
algorithm. In the …rst 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 ‡ipping strategies developed in the …rst step. The calculated probability
of experimentation depends on the direction of ‡ipping and whether the agent can deduce zt
1.
For
example, if the agent chooses "withdraw" as the strategy in the …rst step and can infer the value of
zt
1,
the probability of changing strategy to "wait" is given by
B (zt 1 ;
) and can be calculated
according to equation (4). After each agent’s strategy is determined, we calculate the number of
subjects who choose to wait. 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 de…ned in table 4. An investigation of table 9 shows
that the modi…ed evolutionary algorithm captures the main features of the experimental data. For
instance, the economies stay close or converge to the non-run equilibrium when
is small. For
0:4, all simulated economies stay close to the non-run equilibrium, spending 100% of the time
at "NN" or "FN". When
= 0:5, all economies spend more than 95% of the time at "NN", "FN" or
"CN", and very occasionally at "H", "L" or "CR" when the economy starts with low initial number
of agents choosing to wait (S = 4 or 5). When
time at "RR" or "FR". When
= 0:9, all simulated economies spend 100% of the
= 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
= 0:6 or 0:7, as in the actual experiment, the simulated
economies produce very di¤erent results, spending a positive amount of time in each of the …rst
8 performance categories. The evolutionary learning 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]
24
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
in table 10. Note that the estimated coe¢ cients from regressions with Chinese data are very close
to those from regressions with all data (compare tables 8 and 10). This can be taken as evidence
that the location of experiment does not a¤ect the experimental results in a signi…cant way.15 The
process to conduct out-of-sample prediction is as follows. For each Canadian observation, we apply
the algorithm using information about the individual’s strategy and payo¤ in the previous period
to determine his current action s0t . 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 ‡ips her
strategy if and only if the estimated rate of experimentation is > 0:5. If the predicted action, s^t ,
coincides with the actual action st , we count it as a successful prediction. The prediction success
rates are 63:66% for group A, 90:48% for group B, and 97:36% for group C. The overall rate of
successful prediction is 91:77%. We take this as another piece of evidence that the 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
a¤ects 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 de…ned by a demand deposit
contract. We …x the rate of return of the bank’s long-term investments to rule out the deterioration
of the bank’s assets as the source of bank runs. The only variation in the experiment is that the
coordination parameter, which measures the amount of coordination that is required for subjects
choosing to wait to receive a higher payo¤ than those who withdraw, changes every 10 periods.
We …nd that 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 coordination 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 …nd evidence of the order e¤ect: the sequences followed by the coordination parameter a¤ects
the performance of experimental economies, especially if the coordination parameter lies in the in15
We also divide the sessions into 2 groups according to location (China versus Canada), and run a two-sided
rank sum test of the mean of the number of subjects choosing to wait across the whole session (excluding the two
phases with = 0:1 and 0:2 for sessions 9 and 21). The p-value is 0:663, which suggests the location does not have a
signi…cant e¤ect on the experimental results.
25
determinacy region. 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
e¤ect in the sense that the initial coordination behavior may not predict the …nal outcomes in
repeated coordination games. In particular, we observe some experimental economies start a phase
(which 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 lies in or around the indeterminacy region.
In order to capture the learning e¤ect 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. Du¤y and Fisher (2005)
and Fehr et al. (2012) have provided direct evidence of sunspots in the laboratory. However, the
payo¤ structure is such that the multiple equilibria are not Pareto rankable or are Pareto equivalent.
Most recently, Arifovic et al. (2012) generate some initial experimental evidence of sunspot behavior
when the payo¤ structure is associated with multiple equilibria that can be Pareto ranked. One
important …nding is that training with respect to the sunspot variable during the practice period
is crucial for the observation of sunspot behavior in later periods.16 The experimental results in
our paper suggest that the level of coordination requirement may also be an important factor that
a¤ects the occurrence of sunspot behavior. 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 coordination
parameter 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.
16
The training is achieved by building a correlation between the realization of the sunspot variable and the aggregate
outcome. The process is determined by the experimenter and is not known to the subjects.
26
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29
Table 1: Values of r and η Used in Experiments
r
η
1.05 1.11 1.18 1.25 1.33 1.43 1.54 1.67 1.82
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
30
Table 2: Experimental Design
Session No.
1
2
3
4
5
6
7
8
Grid
17
18
19
20
η
Date
0.1;0.2;0.3;0.5;0.7;0.8;0.9
0.1;0.2;0.3;0.5;0.7;0.8;0.9
0.1;0.2;0.3;0.5;0.7;0.8;0.9
0.1;0.2;0.3;0.5;0.7;0.8;0.9
0.9;0.8;0.7;0.5;0.3;0.2;0.1
0.9;0.8;0.7;0.5;0.3;0.2;0.1
0.9;0.8;0.7;0.5;0.3;0.2;0.1
0.9;0.8;0.7;0.5;0.3;0.2;0.1
2006-11
2008-11
2006-11
2008-11
2007-06
2008-11
2008-11
2008-10
UIBE
UIBE
UIBE
SFU
0.1:0.1:0.9
0.3:0.1:0.9
0.3:0.1:0.9
0.3:0.1:0.9
2012-03
2012-03
2012-05
2012-05
Decrease
UIBE
UIBE
UIBE
SFU
0.9:-0.1:0.3
0.9:-0.1:0.3
0.9:-0.1:0.3
0.9:-0.1:0.3
2012-03
2012-04
2012-06
2012-05
Random
UIBE
UIBE
UIBE
SFU
0.4;0.9;0.1;0.8;0.2;0.7;0.3;0.6;0.5
0.7;0.3;0.9;0.6;0.4;0.8;0.5
0.8;0.4;0.7;0.3;0.6;0.9;0.5
0.8;0.4;0.7;0.3;0.6;0.9;0.5
2012-03
2012-04
2012-05
2012-05
Coarse
Decrease
Increase
Fine
Location
UIBE
UIBE
SFU
UofM
UIBE
UIBE
SFU
UofM
Increase
9
10
11
12
13
14
15
16
Order
31
Table 3: Starting, Terminal and Mean Value of Number of Subjects Choosing to Wait
Session
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
S
10
9
10
8
10
10
10
10
0.1
T
10
10
9
10
10
10
10
9
S
10
10
10
9
8
10
10
8
0.3
T
9
10
9
10
10
10
10
9
M S
9.7
10
9.7
9.8
9.7
10
10
9.6
10 9.9 10
10
9
7
10
10
10
10
10 9.6 10 10 10 9
9
9
9
10
10
9
8
10
10
10
10
10
9
10
9
10
10
9
8.5
10
10
10
9.9
9.8
9.5
9.9
9.7
M
10
9.8
9.8
9.7
10
10
10
9.7
S
9
8
9
10
10
10
10
10
10 10 9.9 9
7
0.2
T
10
10
9
10
10
10
10
10
M
9.9
9.5
9.7
9.9
9.9
10
10
10
10
10
10
9
10
10
10
10
10
9
10
10
0.4
T M
10
10
10
9
10
10
10
10
10
9
10
10
10
10
10
8.6
10
10
10
10
10
9.3
10
9.7
S
8
9
10
9
4
6
10
4
0.5
T
10
10
9
10
8
9
10
9
M S
9.3
9.6
9.6
9.9
7.2
8.7
9.8
7.8
10
10
10
10
5
10
9
7
7
9
10
8
10
10
10
10
10
10
10
10
9
9
10
10
10
9.9
10
9
9.3
9.9
9.9
8.2
8.5
9.3
10
9.7
32
9
10
10
9
3
4
3
3
3
2
10
7
0.6
T M
10
10
10
9
0
9
10
0
0
0
10
10
9.8
10
10
8.7
2.7
7.8
8.1
0.4
0.6
0.7
10
9.2
S
6
7
8
7
1
0
6
1
0.7
T
6
0
2
0
0
0
9
0
M
7.5
2.1
3.1
2.2
0.4
0.5
7.9
1
S
2
3
2
1
1
2
6
1
0.8
T
0
0
0
0
0
0
0
1
M
1.2
0.4
1.1
0.7
0.5
0.7
1.9
0.6
S
1
0
1
0
1
1
4
3
0.9
T
0
0
0
0
1
0
2
3
M
0.2
0
0.2
0.1
0.6
0.5
2.2
2
10
10
10
8
3
3
1
3
3
3
10
5
10
10
10
10
0
0
0
0
0
3
10
0
9.7
10
10
8.4
1
0.7
0.2
0.5
0.3
1.7
10
1.4
9
8
9
8
3
1
2
3
5
4
9
6
0
1
0
5
0
1
0
0
0
0
0
1
1.8
2.4
2.3
4.6
0.6
0.7
0.3
0.3
0.8
0.9
3
2.5
6
3
1
4
3
3
3
4
3
3
9
6
0
0
0
2
1
1
0
1
0
0
0
1
0.8
0.9
0.1
2
0.9
1.1
0.8
1.9
1.1
0.7
2.5
1.5
Table 4- Performance Classification
Performance Category
Label
Criterion
Very close to the non-run equilibrium
NN
M≥9
Fairly close to the non-run equilibrium
FN
8≤M<9
Converging to the non-run equilibrium
CN
5<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
Switch to run equilibrium
SN
SR
S<5 and T≥8
S>5 and T≤2
33
Table 5: Classification of Performance of Experimental Economies
η
Session
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
0.1
0.2
0.3
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
FN
NN
NN
NN
NN
NN
NN
NN
NN
0.4
0.5
0.6
NN
NN
NN
NN
CN; SN
FN
NN
CN; SN
NN
NN
NN
FN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
NN
FN
FN
NN
NN
NN
NN
NN
NN
FN
CR
CN; SN
FN; SN
RR
RR
RR
NN
NN
34
0.7
0.8
0.9
H
CR; SR
CR; SR
CR; SR
RR
RR
CN
RR
FR
RR
FR
RR
RR
RR
FR; SR
RR
RR
RR
RR
RR
RR
RR
CR
FR
NN
NN
NN
FN
RR
RR
RR
RR
RR
FR
NN
FR
FR; SR
CR; SR
CR; SR
L
RR
RR
RR
RR
RR
RR
CR; SR
CR; SR
RR;SR
RR
RR
FR
RR
FR
RR
FR
FR
RR
CR; SR
FR; SR
Table 6: Rank-sum Tests of the Order and Grid Effect
Table 6a: Rank Sum Tests for η=0.7
Order Effect - Old data
Sessions 1-4
Sessions 5-8
Z-value
p-value (1-sided)
Sample size
4
4
S
7.000
2.000
2.205
1.38%
T
2.000
2.250
0.331
37.05%
M
3.725
2.450
1.155
12.41%
Sample size
4
4
S
9.500
2.500
2.428
0.76%
T
9.750
0.000
2.646
0.41%
M
9.625
0.600
2.323
1.01%
Order
Increase
Random
Sample size
4
4
S
9.500
5.500
1.703
4.43%
T
9.750
5.000
2.000
2.28%
M
9.625
5.125
1.479
6.96%
Order
Random
Decrease
Sample size
4
4
S
5.500
2.500
1.654
4.91%
T
5.000
0.000
1.512
6.53%
M
5.125
0.600
1.443
7.45%
Sample size
4
4
S
9.500
7.000
2.247
1.24%
T
9.750
2.000
2.477
0.66%
M
9.625
3.725
2.323
1.01%
Order
Increase
Decrease
Order Effect - New data
Order
Sessions 9-12
Increase
Sessions 13-16
Decrease
Z-value
p-value (1-sided)
Sessions 9-12
Sessions 17-20
Z-value
p-value (1-sided)
Sessions 17-20
Sessions 13-16
Z-value
p-value (1-sided)
Grid Effect - New and old data
Order
Sessions 9-12
Increase
Sessions 1-4
Increase
Z-value
p-value (1-sided)
35
Table 6: Rank-sum Tests of the Order and Grid Effect Continued
Table 6b: Rank Sum Tests for η=0.6
Sessions 9-12
Sessions 13-16
Z-value
p-value (1-sided)
Sessions 9-12
Sessions 17-20
Z-value
p-value (1-sided)
Sessions 17-20
Sessions 13-16
Z-value
p-value (1-sided)
Order
Increase
Decrease
Sample size
4
4
S
9.500
3.250
2.397
0.83%
T
9.750
4.750
1.559
5.95%
M
9.625
4.750
2.323
1.01%
Order
Increase
Random
Sample size
4
4
S
9.500
5.500
1.488
6.84%
T
9.750
5.000
1
15.87%
M
9.625
5.125
1.183
11.84%
Order
Random
Decrease
Sample size
4
4
S
5.500
3.250
0.461
32.23%
T
5.000
4.750
0.316
37.91%
M
5.125
4.750
0.577
28.19%
36
Table 7: Information
Last period's Last period's Can deduce exact
payoff π t-1
value of z t-1 ?
choice s t-1
r
No
Withdraw
[1,r)
Yes
>r
Yes
Wait
(0,r)
Yes
0
No
Can deduce
Number of Percentage in
best response? observations all observations
No
474
3.66
Yes
4362
33.66
Yes
7449
57.48
Yes
267
2.06
Yes
408
3.15
37
Choice before experimentation s t '
Group
Withdraw (keep old strategy)
Withdraw (update to best response)
Wait (update to best response)
Withdraw (update to best response)
Withdraw (update to best response)
A
B
C
B
A
Table 8: Logit Regression of Rate of Experimentation - All Data
Table 8a
constant
η
coefficient
0.91
-2.45
std error
0.26
0.39
t-statistic
3.47
-6.34
p-value
0.00
0.00
coefficient
0.52
0.50
std error
0.13
0.02
t-statistic
3.87
21.76
p-value
0.00
0.00
coefficient
-2.83
-0.31
std error
0.26
0.05
t-statistic
-11.05
-6.05
p-value
0.00
0.00
Table 8b
constant
z t‐1 ‐z*
Table 8c
constant
z t‐1 ‐z*
Notes:
Table 8a: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who cannot infer z t-1
Table 8b: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who can infer z t-1
Table 8c: result from regression to esitmate the experimentation rate from "wait" to "withdraw"; subjects can infer z t-1 in this case.
38
Table 9: Performance of Simulated Economies
1
2
3
4
η
0.1 0.2 0.3 0.4
NN 100 100 100
FN 0
0
0
CN 0
0
0
H
0
0
0
RR 0
0
0
FR 0
0
0
CR 0
0
0
L
0
0
0
SN 0
0
0
SR 0
0
0
0.5
81
19
0
0
0
0
0
0
0
0
NN 100
FN 0
CN 0
H
0
RR 0
FR 0
CR 0
L
0
SN 0
SR 0
100
0
0
0
0
0
0
0
0
0
0.7
0
1
0
0
0
32
58
9
0
89
0.8
0
0
0
0
78
22
0
0
0
0
0.9
0
0
0
0
98
2
0
0
0
0
95
5
0
0
0
0
0
0
0
0
1
4
1
3
0
17
65
9
0
84
0
0
0
0
65
33
2
0
0
0
0
0
0
0
100
0
0
0
0
0
NN 100 100 100
FN 0
0
0
CN 0
0
0
H
0
0
0
RR 0
0
0
FR 0
0
0
CR 0
0
0
L
0
0
0
SN 0
0
0
SR 0
0
0
99
1
0
0
0
0
0
0
0
0
37
27
0
12
0
0
22
2
0
27
0
0
0
0
78
22
0
0
0
0
0
0
0
0
98
2
0
0
0
0
NN 100 100
FN 0
0
CN 0
0
H
0
0
RR 0
0
FR 0
0
CR 0
0
L
0
0
SN 0
0
SR 0
0
95
5
0
0
0
0
0
0
0
0
1
4
1
3
0
17
65
9
0
84
0
0
0
0
86
14
0
0
0
0
0
0
0
0
100
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
0.6
5
6
7
8
0.1 0.2 0.3
100 100 99
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.4
0.5
2
44
49
0
0
0
1
4
96
0
0.6
0.7
0
0
0
0
53
42
4
1
0
0
0.8
0
0
0
0
86
14
0
0
0
0
0.9
0
0
0
0
98
2
0
0
0
0
100 100 100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
39
53
8
0
0
0
0
0
0
0
0
0
0
0
68
28
3
1
0
0
0
0
0
0
78
22
0
0
0
0
0
0
0
0
98
2
0
0
0
0
100 100 100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
0
1
0
0
0
32
58
9
0
89
0
0
0
1
10
76
13
0
0
99
0
0
0
0
86
14
0
0
0
0
100 100
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
44
49
0
0
0
1
4
96
0
0
0
0
0
53
42
4
1
0
0
0
0
0
0
86
14
0
0
0
0
0
0
0
0
95
5
0
0
0
0
99
1
0
0
0
0
0
0
0
0
39
Table 9: Performance of Simulated Economies Continued
9
10
11
12
0.1 0.2 0.3 0.4
NN 100 100 100 98
FN 0
0
0
2
CN 0
0
0
0
H
0
0
0
0
RR 0
0
0
0
FR 0
0
0
0
CR 0
0
0
0
L
0
0
0
0
SN 0
0
0
0
SR 0
0
0
0
0.5
99
1
0
0
0
0
0
0
0
0
0.6
84
14
0
1
0
0
1
0
0
1
0.7
76
12
0
8
0
0
4
0
0
10
0.8 0.9
23
0
8
0
0
0
25
0
0
51
0
49
44
0
0
0
0
0
70 100
NN
FN
CN
H
RR
FR
CR
L
SN
SR
100
0
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
96
4
0
0
0
0
0
0
0
0
76
12
0
8
0
0
4
0
0
10
0
0
0
1
0
43
55
1
0
99
0
0
0
0
95
5
0
0
0
0
NN
FN
CN
H
RR
FR
CR
L
SN
SR
100
0
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
96
4
0
0
0
0
0
0
0
0
76
12
0
8
0
0
4
0
0
10
0
0
0
1
0
43
55
1
0
99
0
0
0
0
95
5
0
0
0
0
NN
FN
CN
H
RR
FR
CR
L
SN
SR
90
10
0
0
0
0
0
0
0
0
97
3
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
84
14
0
1
0
0
1
0
0
1
37
27
0
12
0
0
22
2
0
27
0
0
0
1
0
43
55
1
0
99
0
0
0
0
86
14
0
0
0
0
0.1
13
14
15
16
0.2
0.3 0.4
100 98
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.5
23
64
12
1
0
0
0
0
0
0
0.6
0
0
8
0
2
19
28
43
12
0
0.7
0
0
0
0
22
59
15
4
0
0
0.8
0
0
0
0
65
33
2
0
0
0
0.9
0
0
0
0
95
5
0
0
0
0
100
0
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
0
5
11
1
0
16
26
41
20
0
0
0
0
0
22
59
15
4
0
0
0
0
0
0
86
14
0
0
0
0
0
0
0
0
95
5
0
0
0
0
100
0
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
95
5
0
0
0
0
0
0
0
0
0
0
8
0
2
19
28
43
12
0
0
0
0
0
53
42
4
1
0
0
0
0
0
0
78
22
0
0
0
0
0
0
0
0
95
5
0
0
0
0
100
0
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
61
36
3
0
0
0
0
0
0
0
0
0
8
0
2
19
28
43
12
0
0
0
0
0
22
59
15
4
0
0
0
0
0
0
65
33
2
0
0
0
0
0
0
0
86
14
0
0
0
0
40
17
18
19
20
0.1 0.2 0.3
98 100 98
2
0
2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0.4
98
2
0
0
0
0
0
0
0
0
0.5
61
36
3
0
0
0
0
0
0
0
0.6
0
0
8
0
2
19
28
43
12
0
0.7
0
0
0
0
22
59
15
4
0
0
0.8
0
0
0
0
30
63
7
0
0
0
0.9
0
0
0
0
95
5
0
0
0
0
98
2
0
0
0
0
0
0
0
0
97
3
0
0
0
0
0
0
0
0
95
5
0
0
0
0
0
0
0
0
0
0
8
0
10
21
21
40
12
0
0
0
0
0
22
59
15
4
0
0
0
0
0
0
47
49
4
0
0
0
0
0
0
0
95
5
0
0
0
0
98
2
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
99
1
0
0
0
0
0
0
0
0
96
4
0
0
0
0
0
0
0
0
76
12
0
8
0
0
4
0
0
10
23
8
0
25
0
0
44
0
0
70
0
0
0
0
0
57
43
0
0
100
98
2
0
0
0
0
0
0
0
0
98
2
0
0
0
0
0
0
0
0
81
19
0
0
0
0
0
0
0
0
40
51
2
4
0
0
2
1
0
2
0
0
0
0
2
52
38
8
0
0
0
0
0
1
10
76
13
0
0
99
0
0
0
0
51
49
0
0
0
100
Table 10: Logit Regression of Rate of Experimentation - Chinese Data
Table 10a
constant
η
coefficient
0.62
-2.34
std error
0.37
0.55
t-statistic
1.69
-4.25
p-value
0.09
0.00
coefficient
0.50
0.54
std error
0.18
0.03
t-statistic
2.81
17.51
p-value
0.00
0.00
coefficient
-3.77
-0.31
std error
0.53
0.10
t-statistic
-7.17
-2.96
p-value
0.00
0.00
Table 10b
constant
z t‐1 ‐z*
Table 10c
constant
z t‐1 ‐z*
Notes:
Table 10a: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who cannot infer z t-1
Table 10b: result from regression to esitmate the experimentation rate from "withdraw" to "wait" for subjects who can infer z t-1
Table 10c: result from regression to esitmate the experimentation rate from "wait" to "withdraw"; subjects can infer z t-1 in this case.
41
Figure 1a: Experimental Results (Sessions 1-8)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
8
8
6
6
4
4
2
2
0
0
Session 1 10
10
8
8
6
6
4
4
2
2
0
0
Session 2 10
10
8
8
6
Session 6
4
2
2
0
Session 3 0
10
10
8
8
6
6
4
4
2
2
0
Session 5
6
4
Session 4 0
Session 7
Session 8
40 Figure 1b: Experimental Results (Sessions 9-20)
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 10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Session 9 0
Session 13
0
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Session 10 0
Session 14
0
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Session 11 0
Session 15
0
10
10
10
8
8
8
6
6
6
4
4
4
2
2
2
0
Session 12 0
Session 16
41 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0
Session 17
Session 18
Session 19
Session 20
Appendix: Instructions – Increasing Treatment with Fine Grid
This experiment has been designed to study decision-making behavior in groups. During
today's session, you will earn income in an experimental currency called “experimental dollars”
or for short ED. At the end of the session, the currency will be converted into dollars. 1 ED
corresponds to 0.2 dollars. You will be paid in cash. The participants may earn different amounts
of money in this experiment because each participant's earnings are based partly on his/her
decisions and partly on the decisions of the other group members. If you follow the instructions
carefully and make good decisions, you may earn a considerable amount of money. Therefore, it
is important that you do your best.
Please read these instructions very carefully and ask questions whenever you want to.
Description of the task
You and 9 other people (there are 10 of you altogether) have 1 ED deposited in an
experimental bank. You must decide whether to withdraw your 1 ED, or wait and leave it
deposited in the bank. The bank promises to pay r>1 EDs to each withdrawer. The money that
remains in the bank will earn interest rate R>r. At the end, the bank will divide what it has
evenly among people who choose to wait and leave money in the bank. Note that if too many
people desire to withdraw, the bank may not be able to fulfill the promise to pay r to each
withdrawer (remember that r>1). In that case, the bank will divide the 10 EDs evenly among the
withdrawers and those who choose to wait will get nothing.
Your payoffs depend on your own decision and the decisions of the other 9 people in the
group. Specifically, how much you receive if you make a withdrawal request or how much you
earn by leaving your money deposited depends on how many people in the group place
withdrawing requests.
Let e be the number of people who request withdrawals. The payoff (in ED) to those who
withdraw will be:
min{r,
10
10
} or the minimum of r and
.
e
e
The payoff (in ED) to those who leave money in the bank will be:
max{0,
10  e  r
10  e  r
R } or the maximum of 0 and
R.
10  e
10  e
In the experiment, R will be fixed at 2, and r will take 8 values: 1.38, 1.18, 1.25, 1.33, 1.43,
1.54, 1.67 and 1.82. To facilitate your decision, the payoff tables 0∼9 list the payoffs if n of the
other 9 subjects request to withdraw (n is unknown at the time when you make the withdrawing
decision - it can be any integer from 1 to 9 - and you have to guess it) for each value of r. Table 0
will be used for practice, and table 1∼7 will be used for the formal experiment. Let's look at two
examples:
Example 1. Use table 0 where r=1.38. Suppose that 3 other subjects request withdrawals.
Your payoff will be 1.38 if you request to withdraw: the number of withdrawing requests e will
10 10


 2.5 =1.38. If you choose to wait and
be 3+1=4, so your payoff will be min r  1.38, 
4
e


1
leave money in the bank, your payoff will be 1.67: the number of withdrawing requests will be
10  3  1.38
 10  e  r

R
 2  1.67 =1.67.
e=3, so your payoff is max 0,
10 - 3
 10  e

Example 2. Continue using table 0 where r=1.38. Suppose that 8 of other subjects request
10 10


 1.11 =1.11. Your
withdrawals. Your payoff for withdrawing will be min r  1.38, 
e
9


payoff
for
waiting
and
leaving
money
in
the
bank
will
be
e
r
10
10
8
1
.
38






max 0,
R
 2  - 1.04 = 0.
10 - 8
 10  e

Now let's take a closer look at the tables. Notice the following features of tables:

The payoff to withdrawing is more stable: it is fixed at r for most of the time and is
bounded below by 1.

The payoff to waiting and leaving money in the bank is more volatile. When n - the
number of other people requesting withdrawals - is small, waiting and leaving money in
the bank offers higher payoff than withdrawing. The opposite happens when n is large
enough. For your convenience, bold face is used to identify the threshold value of n at
which the payoffs to the two choices are equal or almost equal (the difference is ≤0.02
ED).

The threshold values of n vary from table to table. The general pattern is that it is smaller
when r is bigger.
Note you are not allowed to ask people what they will do and you will not be informed about
the other people's decisions. You must guess what other people will do - how many of the other 9
people will withdraw - and act accordingly.
Procedure
You will perform the task described above for 70 times. Each time is called a period. Each
period is completely separate. I.e., you start each period with 1ED in the experimental bank. You
will keep the money you earn in every period. At the end of each round, the computer screen will
show you your decision and payment for that period. Information for earlier periods and your
cumulative payment is also provided.
Note that the payment scheme changes every 10 periods, so please use the correct payoff
table:
 Table 1 for period 1-10 (r=1.18),
 Table 2 for period 11-20 (r=1.25),
 Table 3 for period 21-30 (r=1.33),
 Table 4 for period 31-40 (r=1.43),
 Table 5 for period 41-50 (r=1.54),
 Table 6 for period 51-60 (r=1.67), and
 Table 7 for period 61-70 (r=1.82).
You will be reminded when you need to change to a new table, pay attention to the message.
2
Beside the 70 paid periods, you will also be given 10 trials (periods -9 to 0) to practice and get
familiar with your task. You will not be paid for the trials. Please use table 0 (r=1.38) for the
trials. After the practice periods, you will have a chance to ask more questions before the
experiment formally start. You will be paid for each of the 70 formal periods.
Computer instructions
You will see three types of screens: the decision screen, the payoff screen and the waiting
screen.
Your withdrawing decisions will be made on the decision screen as shown in figure 1. You
can choose to withdraw money or leave money in the bank by pushing one of the two buttons.
Note that your decision will be final once you press the buttons, so be careful when you make the
move. The header provides information about what period you are in and the time remaining to
make a decision. After the time limit is reached, you will be given a flashing reminder “please
reach a decision!”. The screen also shows which table you should look at.
Figure 1: The decision screen
After all subjects input their decisions, a payoff screen will appear as shown in figure 2. You
will see your decision and payoff in the current period. The history of your decisions and payoffs
as well as your cumulative payoff is also provided. After finishing reading the information, click
on the “continue” button to go to the next period. You will have 30 seconds to review the
information before a new period starts.
3
Figure 2: The payoff screen
You might see a waiting screen (as shown in figure 3) following the decision or payoff screens
- this means that other people are still making decisions or reading the information, and you will
need to wait until they finish to go to the next step.
Figure 3: The waiting screen
Payment
At the end of the entire experiment, the supervisor will pay you in cash. Your earnings in
dollars will be:
total payoff (in ED)×0.2.
4
Table 0 (for practice): payoff if n of other 9 subjects withdraw
r=1.38 (period -9 to 0)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.38
2.00
1
1.38
1.92
2
1.38
1.81
3
1.38
1.67
4
1.38
1.49
5
1.38
1.24
6
1.38
0.86
7
1.25
0.23
8
1.11
0.00
9
1.00
0.00
Table 1: payoff if n of other 9 subjects withdraw
r=1.18 (period 1-10)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.18
2.00
1
1.18
1.96
2
1.18
1.91
3
1.18
1.85
4
1.18
1.76
5
1.18
1.64
6
1.18
1.46
7
1.18
1.16
8
1.11
0.56
9
1.00
0.00
5
Table 2: payoff if n of other 9 subjects withdraw
r=1.25 (period 11-20)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.25
2.00
1
1.25
1.94
2
1.25
1.88
3
1.25
1.79
4
1.25
1.67
5
1.25
1.50
6
1.25
1.25
7
1.25
0.83
8
1.11
0.00
9
1.00
0.00
Table 3: payoff if n of other 9 subjects withdraw
r=1.33 (period 21-30)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.33
2.00
1
1.33
1.93
2
1.33
1.84
3
1.33
1.72
4
1.33
1.56
5
1.33
1.34
6
1.33
1.01
7
1.25
0.46
8
1.11
0.00
9
1.00
0.00
6
Table 4: payoff if n of other 9 subjects withdraw
r=1.43 (period 31-40)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.43
2.00
1
1.43
1.90
2
1.43
1.79
3
1.43
1.63
4
1.43
1.43
5
1.43
1.14
6
1.43
0.71
7
1.25
0.00
8
1.11
0.00
9
1.00
0.00
Table 5: payoff if n of other 9 subjects withdraw
r=1.54 (period 41-50)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.54
2.00
1
1.54
1.88
2
1.54
1.73
3
1.54
1.54
4
1.54
1.28
5
1.54
0.92
6
1.43
0.38
7
1.25
0.00
8
1.11
0.00
9
1.00
0.00
7
Table 6: payoff if n of other 9 subjects withdraw
r=1.67 (period 51-60)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.67
2.00
1
1.67
1.85
2
1.67
1.67
3
1.67
1.43
4
1.67
1.11
5
1.67
0.66
6
1.43
0.00
7
1.25
0.00
8
1.11
0.00
9
1.00
0.00
Table 7: payoff if n of other 9 subjects withdraw
r=1.82 (period 61-70)
n
payoff if withdraw
payoff if wait & leave money in the bank
0
1.82
2.00
1
1.82
1.82
2
1.82
1.59
3
1.82
1.30
4
1.82
0.91
5
1.67
0.36
6
1.43
0.00
7
1.25
0.00
8
1.11
0.00
9
1.00
0.00
8