QBER DISCUSSION PAPER No. 6/2013. How Strongly Do Players React to Increased Risk Sharing in an Experimental Bank Run Game?

Transcription

1 QBER DISCUSSION PAPER No. 6/2013 How Strongly Do Players React to Increased Risk Sharing in an Experimental Bank Run Game? Alexander Klos and Norbert Sträter

2 How Strongly Do Players React to Increased Risk Sharing in an Experimental Bank Run Game? Alexander Klos University of Kiel Norbert Sträter Finance Center Münster March 2013 Abstract This paper investigates predictions of the global games approach in an experimental bank run game. Our main goal is to investigate how strongly players react to increased risk sharing, i.e., a higher payo in case of an early withdrawal, in order to learn more about the descriptive validity of dierent theoretical predictions in the laboratory. The global games predicts a strong reaction to increased risk sharing. We observe only a small reaction in the experiment. A level-k approach, in which level-1 types assume that level-0 types play a random threshold strategy, ts the data better. However, non-negligible features of the data are also inconsistent with a level-k approach. JEL classication: G21, C92, C72 Keywords: Experimental nance, bank runs, global games, level-k thinking, nancial intermediation We thank two anonymous referees, John Hey, Susanne Homölle, Rosemarie Nagel, Andreas Pngsten, Anika Schink, Andreas Trauten, as well as numerous conference and seminar participants. Timon Albers helped us running the experiments. Financial support from the Deutsche Forschungsgemeinschaft (German Science Foundation, grant KL 2365/1-1) is gratefully acknowledged. Institute for Quantitative Business and Economics Research (QBER), Christian-Albrechts-Universität zu Kiel, Heinrich-Hecht-Platz 9, Kiel, Germany, alexander.klos@qber.uni-kiel.de, phone: , fax: , corresponding author Finance Center Münster, Institut für Kreditwesen, University of Münster, Universitätsstraÿe 14 16, Münster, Germany, norbert.straeter@wiwi.uni-muenster.de 0

3 1 Introduction Economists typically model bank runs as a coordination game (Diamond and Dybvig (1983)). All agents act simultaneously and there are two equilibria. Everyone tries to withdraw money from a bank in the bank run equilibrium and everyone leaves money in the bank in the equilibrium without a bank run. 1 One problem with this modeling approach is that there are no probabilities attached to these possibilities, i.e., such models predict that panic-based bank runs are possible, but do not address the probability of a bank run. However, the probability of a bank run is of obvious interest for policy analysis. It is therefore desirable to have a model that removes the multiplicity of equilbria, or at least one that allows us to identify an economically plausible equilibrium from the dierent equilibria. The probability of a bank run can then be inferred from this equilibrium. The theory of global games (Carlsson and van Damme (1993)) provides such a framework (see Morris and Shin (2003) for an overview). The main dierence is that classic panic-based bank run models assume that everyone knows with certainty the return properties of the deposit (the fundamental state of the banking system). However, in reality, depositors typically do not know for certain which payos will result. Instead, they hold highly correlated but unidentical beliefs about the true payos. The global games approach models this situation by assuming that every agent receives a noisy private signal that is highly informative with respect to the true fundamental state of the economy (it often has the same expected value) but does not allow agents to determine the exact value. In such an environment, agents are expected to play so-called threshold (or synonymously cuto) strategies, i.e., they withdraw their deposits if their signal is below a certain threshold and leave their money in the bank if the signal is above a certain threshold. The intuition for this behavior is that agents form beliefs about the signals that the other players have received based on their own signal. For example, if an agent's signal suggests that the fundamental state of the banking 1 These models typically use one bank that represents the banking sector as a whole. We will follow this convention throughout the paper. 1

4 sector is bad, agents expect that other depositors have received an equally bad signal, and thus withdraw their money early. Given this belief, it is optimal to withdraw also. The opposite is true for good signals above the fundamental state. For some intermediate signal, leaving the money in the bank and withdrawing the money are equally attractive, given the beliefs about the other players' actions. This signal denes a threshold strategy equilibrium if the threshold for each player is the best response to the threshold of the other players. Goldstein and Pauzner (2005) provide a theoretical analysis of the global games approach applied to classic panic-based bank run models (see also Morris and Shin (2001), Dasgupta (2004), and Rochet and Vives (2004)). They show that the threshold equilibrium is the only equilibrium of the bank run game using reasonable assumptions. 2 Here, the probability of a bank run arises endogenously out of the model, which can be considered a substantial theoretical improvement. A further advantage of the global games approach is the theoretical integration of the panic-based and the fundamental view of bank runs, two related, but dierent, views found in the literature on nancial stability. The panic-based view stresses that bank runs can be pure sunspots, not necessarily related to changes in the real economy. Panic-based bank runs might even occur when the economic environment is suciently strong (see, e.g., Diamond and Dybvig (1983)). In contrast to this, the fundamental view assumes that bank runs are a natural outow of the business cycle and only occur in connection with negative real shocks (see, e.g., Allen and Gale (1998)). 3 In the global games approach, it is possible that fundamental runs occur if the fundamentals of the economy and therefore the signals which agents receive are very bad. However, in other situations, fundamentals may be good enough so that the prevention of a bank run is desirable from the viewpoint of the depositors, but runs still occur due to strategic uncertainty about others' beliefs. Moderate signals about the fundamental state of the economy may lead to the belief that there is a high probability that other 2 Even if these assumptions are not met, it can be argued that the threshold equilibrium is more plausible than an equilibrium that predicts that bank runs always or never occur for any signal (see Appendix B in Goldstein and Pauzner (2005), where the authors discuss several plausible equilibrium selection criteria for the case of multiple equilibria). 3 The literature on panic-based and fundamental bank runs is large. Allen and Gale (2007) provide a survey. 2

5 agents will withdraw. In such a situation it is rational to withdraw as well. In that sense, moderate expectations about the fundamental state of the nancial system can trigger panic-based bank runs. We consider a level-k approach as a behavioral alternative to the global games approach (see, e.g., Nagel (1995), Stahl and Wilson (1995), Camerer, Ho, and Chong (2004), Costa-Gomes and Crawford (2006), and Crawford and Iriberri (2007)). In these models, it is assumed that every player believes that she understands the game better than all other players. Agents rationally choose the best response given this belief. For example, a level-1 type best responds to a population of level-0 types, which are assumed to behave unstrategically. Level-2 types best respond to level-1 types, and so on. In empirical applications, the proportion of dierent types is estimated, typically resulting in a high proportion of level-1 and level-2 types. In complex incomplete information games, such as signaling games, high order thinking in the sense of more than one thinking step is uncommon (Camerer, Ho, and Chong (2004)). We consider, therefore, the behavior of a level-1 type as a reasonable alternative hypothesis. Crawford, Costa-Gomes, and Iriberri (2013) review level-k models and state that a level-k model is a richer and more plausible view of strategic thinking than the one that underlies the global-games approach (page 59) in their chapter on bank runs. However, this assessment is not accompanied by any data. This study adds evidence by investigating a situation where an experimental variation should cause a large increase in the used threshold values in a global games model, while a level-k model predicts only a small reaction. The data shows evidence of behavior that is more consistent with the level-k approach. However, non-negligible features of the data are also inconsistent with a level-k approach. Although any experimental study provides a stylized environment compared to a setting outside the lab, we believe that these insights can enrich the analysis of real-world bank runs. As described above, the global games approach has a very attractive feature: it allows the calculation of the probability of a bank run. It can, therefore, be used for policy analysis in the sense that a policy maker can change the parameters of the bank run model, such as increasing partial deposit insurance 3

6 or changing the expected return of a bank deposit, and calculate the predicted change in the probability of a bank run. However, such an inference should only be drawn if the global games approach provides a good approximation of behavior. We would be more optimistic about the descriptive validity of a modeling approach for real-world bank runs with many confused depositors if the modeling approach performs reasonably well in a stylized test in the lab with a student subject pool that is used to think in abstract terms. The same reasoning applies to a level-k approach that provides a similarly clear-cut prediction of the probability of a bank run. 4 In the following, we start with a short overview of literature. We then introduce the underlying bank run game, derive theoretical predictions, describe the experimental design, and present the results. The last section concludes with a short discussion of the main ndings in light of the previous literature. 2 Literature Overview Our paper is mainly related to two dierent strands of the literature: The investigation of the global games approach in the laboratory and the experimental investigation of bank runs. Heinemann, Nagel, and Ockenfels (2004) study the theory of global games in the context of currency attacks using a xed matching design. They nd evidence that players use threshold strategies, and that the observed behavior follows the comparative statics of the global games solution. Furthermore, observed thresholds are always below the global games prediction. Duy and Ochs (2012) nd that observed behavior is similar in static and in dynamic global games. They further conrm the comparative statics ndings of Heinemann, Nagel, and Ockenfels (2004) using a random rematching design. Further experimental studies of global games include Cornand (2006), Cabrales, Nagel, and 4 Duy (2008) provides a more detailed discussion of the potential merits of experimental macroeconomics. Another reason for an experimental approach is that an empirical test of these predictions with eld data is complicated because of the lack of suciently detailed data on individual depositors' information. In an experiment, we can control the information of agents. This aspect is dicult to control with eld data, but is essential for testing predictions of the global games approach. 4

7 Armenter (2007), and Shurchkov (2012). However, none of these papers study global games in a bank run context. Cornand and Heinemann (2013), Shapiro, Shi, and Zillante (2011), and Klos and Nöth (2012) also consider level-k and global games models. Cornand and Heinemann (2013) and Shapiro, Shi, and Zillante (2011) consider more complicated games with private and public signals. Both papers nd that subjects underweight the public information relative to the equilibrium prediction. This observation is consistent with a level-k model. However, Cornand and Heinemann (2013) show that level-k models are only able to explain part of the dierences between the global games approach and laboratory behavior. Shapiro, Shi, and Zillante (2011) nd that level-k models are successful in predicting subjects' behavior if public information is available and if there is a strong coordination motive (high payos if subjects predict the behavior of other players accurately). This is not the case if one of these conditions is not met. Klos and Nöth (2012) construct a series of simple (2x2)- games to show that behavior in global games depends on the set of possible threshold strategies. Although this eect is consistent with a level-k approach, the individual data shows deviations from level-k. Our paper diers from these papers in focusing on how strongly cuto values react to a payo change. Furthermore, our paper explicitly constructs a bank run frame, including the presence of one-sided strategic complementaries that are typical for bank run problems (see below). The paper is also related to the recently started experimental investigation of bank runs. Madiès (2006) tests the possibility and degree of self-fullling banking panics. His results indicate that partial runs occur frequently. Full deposit insurance reduces bank runs, but partial deposit insurance does not help much. Bank runs can also be curbed by a suspension of deposit availability (implemented as a break in the experiment) and a so-called narrow-banking solution. The latter expression refers to a relatively low expected payment in case of an early withdrawal, i.e., a low degree of risk sharing. In a policy-oriented study, Schotter and Yorulmazer (2009) investigate the factors that aect the severity of fundamental bank runs. In their experiments, subjects know a probability distribution 5

8 for the rate of return they get paid from the bank, but they do not know the exact value. They nd that bank runs appear to be less severe when subjects have more information about what other subjects have done. However, this is only true when the economy is in a relatively good state. They also nd that a higher repayment rate makes early withdrawals more likely, especially if all agents can observe previous actions of other player. Surprisingly, this does not happen with low information in a sequential decision task. If agents who know the rate of return are present, these insiders act on their private information and uniformed agents mimic their behavior. In contrast to Madiès (2006), Schotter and Yorulmazer (2009) nd that partial deposit insurance is enough to mitigate bank runs. Garratt and Keister (2009) study the conditions that lead to a self-fullling bank run and which factors aect its prevalence. They nd that the payo-dominant equilibrium is selected unless uncertainty about aggregated fundamental withdrawal demand is introduced, i.e., if players do not know how many subjects have to withdraw early. Garratt and Keister (2009) investigate static and dynamic decision situations. Players have the option to withdraw their money at dierent points of time in the latter case, while the static game corresponds to the classic one-period simultaneous move scenario. Subjects are more likely to withdraw when given multiple opportunities to do so. Kiss, Rodrìguez-Lara, and Rosa-Garcìa (2009) analyze the importance of observability to the appearance of bank runs both theoretically and experimentally. They nd in a sequential mover setup that bank runs do not occur if the depositors who decide late can observe the earlier actions. Kiss, Rodrìguez-Lara, and Rosa-Garcìa (2012) document an interaction eect between deposit insurance and observability. If choices are not observable, partial deposit insurance decreases the probability of a bank run. But if choices are observable, the behavior under partial and full deposit insurance is the same. Arifovic, Jiang, and Xu (2013) investigate the so-called 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 can earn higher prots than 6

9 the withdrawing depositors. The main result is that bank runs occur more frequently when the coordination task is more dicult. Arifovic, Jiang, and Xu (2013) explain their results with the logit evolutionary algorithm. Brown, Trautmann, and Vlahu (2012) is a unique study in the sense that they implement a pure stranger design, i.e., subjects never meet twice over the course of the experiment. They investigate contagion among dierent banks and nd that bank runs are only contagious if depositors know that the fundamentals of dierent banks are correlated. They are not contagious if subjects know that they are uncorrelated. None of these studies considers the case where agents have dierent but highly correlated opinions (they receive noisy private signals) about the fundamentals of the banking sector. Furthermore, none of these studies considers jointly global games and level-k models as a theoretical approach. 3 Underlying Bank Run Model Appendix A of the online appendix contains a simplied 2-player game that illustrates the main ideas of the global games and the level-k approach in an easier framework. Here, we start directly with the discussion of the underlying bank run model. Coordination games typically exhibit strategic complementarities. They arise when the optimal strategy of an agent depends positively upon the strategies of the other agents (Cooper and John (1988), page 441). In a bank run framework, this means that leaving money in the bank becomes more attractive the more other depositors choose this action. However, it is not generally true that the attractiveness of withdrawing funds increases with the number of other withdrawing depositors. Consider the situation of a full bank run, where everyone withdraws, and a partial bank run, where only a few depositors withdraw. A bank nances long-term credits with short-term deposits; therefore, it is not possible to pay all withdrawers in a bank run. Clearly, the probability of receiving (parts of) the deposited money is higher in a partial run than in a full run. Consequently, we have 7

10 strategic complementarities with respect to leaving money in the bank, but not with respect to withdrawing the money. This important feature of bank runs makes theoretical analysis more dicult: one-sided strategic complementarities (Goldstein and Pauzner (2005)), which may even change the behavior of agents in an experimental study. An example is the widespread use of threshold strategies in global games (Heinemann, Nagel, and Ockenfels (2004)). With one-sided strategic complementarities, it may be possible that players will leave their money in the bank given high and low signals and withdraw given an intermediate signal. The decision to leave the money in the bank for low signals may be driven by the expectation of having only a small chance to receive deposited money in a full bank run. If other agents have similar reasoning, such a strategy could be reasonable because bank runs may not occur for low signals. Our main framework is shown in Figure 1, which outlines the timing of events and actions in the underlying bank run model. There are three points in time. In t=0, all agents deposit 1 currency unit (CU) in a bank that represents the banking sector as a whole. Depositors can either be patient or impatient. Impatient savers only receive utility from consuming in t=1, while for patient savers consumption in t=1 and t=2 is of equal benet. Depositors do not know a priori whether they are patient or impatient. Each agent's ex-ante probability of being impatient is given by λ. There is no aggregate uncertainty about fundamental withdrawal demand. The bank invests all deposits collected from all depositors into a project expected to yield R CU per invested CU at the end of the investment horizon. The investment project can be liquidated before maturity, and then generates one unit of output for each unit of input. The success of the investment project depends on the fundamentals of the banking sector. The exact link is established below. At the beginning of t=1, all agents receive a private signal about the fundamentals. They are also informed whether they are an impatient saver and must therefore withdraw. Next, every patient depositor must decide whether to withdraw early, i.e., to withdraw immediately at t=1. The bank oers for every CU deposited at t=0 a payment of r 1 > 1 CU in the case of an early withdrawal. These payments are nanced by a partial liquidation of the investment project. The 8

11 Deposit 1 CU Receive private signal Decision about early withdrawal Payoff dependent on project success t=0 t=1 t=2 Bank invests N CU in project Partial or full liquidation Repay dependent on project success Figure 1: Timing of events and actions: The gure illustrates the timing of the dierent events and actions. The only decision is the binary decision whether or not to withdraw early at time t=1. Events and actions addressing subjects are single-framed, while events and actions addressing the bank are double framed. bank may run out of funds if too many subjects withdraw early. If the bank is still liquid in period 2, the remaining capital earns the project return R. The prot from the investment project is then divided equally among all subjects who have not withdrawn early. One important issue is how the fundamentals are dened. Two approaches can be distinguished. Morris and Shin (2001) model the project yield R as the uncertain fundamental variable, 5 while Goldstein and Pauzner (2005) x R and introduce a probability function p(θ) : [0, 1] [0, 1] instead. The probability function assigns every fundamental θ to a probability p(θ). p(θ) is the probability that the investment project is successful and earns the xed rate R. We consider both approaches and refer to the Morris-Shin approach as model A and the Goldstein-Pauzner approach as model B. In both models, we assume that fundamentals are uniformly distributed ex-ante with a minimal value θ min and a maximal value θ max. To simplify, we only consider the case where p(θ) = θ for model B, i.e., the uncertain fundamental is the probability of a successful project without further transformation. As mentioned above, the uncertain fundamental θ in model A is the project yield R. In Table 1, we summarize both models in terms of their dierent payment consequences for subjects. Without loss of generality, we normalize the total amount of deposits in the rst period to one in Table 1. A bank run occurs if n 1/r 1 where n is the proportion of early withdrawals relative to possible withdrawals. 5 See also Dasgupta (2004) and Rochet and Vives (2004) for bank run models in which R depends on a random fundamental variable. 9

12 Only some of the early withdrawers, who are determined randomly, 6 receive r 1 in a full bank run. Otherwise (n < 1/r 1 ), the bank is still liquid after period 1 and pays all late withdrawers (1 nr 1) 1 n R. (1 nr 1 ) is the investment left after early withdrawals and (1 nr 1 ) R is the nal outcome of the bank's project, which can be divided between the remaining depositors (given by (1 n)) who chose to wait. Note that model B closely resembles the Goldstein-Pauzner model, while there are several dierences between Morris and Shin (2001) and model A. Hence, model A could be regarded as a modication of the Goldstein-Pauzner approach with a dierent uncertain fundamental variable to check the robustness of this factor. Model A Ex-Post Payos Model B Withdraw n < 1/r 1 n 1/r 1 n < 1/r 1 n 1/r 1 { { 1 r 1 : Early r nr : 1 1 Late (1 nr 1 ) 1 n θ 0 { (1 nr1 ) 1 n R : θ 0 : 1 θ nr 1 r 1 r 1 : 1 nr 1 0 : 1 1 nr 1 Table 1: Payos: The table shows the ex-post payos for models A and B. θ denotes the fundamental variable. R is the project yield in model B and r 1 is the refund in the case of an early withdrawal. 0 4 Theoretical Approaches Consider the decision problem of an agent who is not forced to withdraw early in the rst period. Depending on her signal s i, the depositor can infer that the true fundamental value lies between s i ɛ and s i + ɛ. Because the a-priori probability distributions of the fundamental value and the error term are both uniformly distributed, the a-posteriori probability distribution of the true fundamental value given signal s i is also uniformly distributed. The optimal action depends now on the belief about other players' actions. Assume that all other agents have homogeneous preferences and follow the same threshold strategy which is dened by the threshold θ j. The distribution of the signal of one other player from the viewpoint of a single player with signal s i is a symmetric 6 Note that we use a form of suspension of convertibility. Banks suspend convertibility for randomly chosen savers if early withdrawals are too numerous. However, we do not implement a sequential service constraint (in contrast to Diamond and Dybvig (1983) or Goldstein and Pauzner (2005)) for practicability reasons. 10

13 triangular distribution with mean s i. The probability that another player j withdraws given the own signal s i is 0 for θ j < s i 2ɛ Prob GG (Withdraw s i, θ j ) = (θ j (s i 2ɛ)) 2 8ɛ 2 1 (s i+2ɛ θ j ) 2 8ɛ 2 for s i 2ɛ θ j < s i for s i θ j < s i + 2ɛ (1) 1 for s i + 2ɛ θ j The expected value of an early withdrawal is now the sum of the utilities given that none, one, two,..., or all (1 λ)n 1 of the other patient depositors withdraw 7, or more formally EV Withdraw (s i, θ j ) = (1 λ)n 1 n=0 1 2ɛ θ +ɛ θ ɛ [ ( ) B n, (1 λ)n 1, Prob GG (Withdraw s i, θ j ) ] EV (Payo Withdraw n) dθ (2) ( ) where B n, (1 λ)n 1, Prob GG (Withdraw s i, θ j ) is the probability that exactly n of the other (1 λ)n 1 patient depositors withdraw. B denotes the binominal distribution. The expected value given that n other patient depositors withdraw depends on the ability of the bank to pay o all early withdrawers. The number of withdrawers in that case is the sum of the n other patient depositors who withdraw, the λn people who have to withdraw early and our agent who decides to withdraw also: EV (Payo Withdraw n) = u(r 1 ) if RD(N/r 1 ) n + λn + 1 ( ) RD(N/r 1 ) n+λn+1 u(r 1) + 1 RD(N/r 1) n+λn+1 u(0) if RD(N/r 1 ) < n + λn + 1 (3) RD(x) depicits the round down of x. 7 We assume that λ is chosen in such a way that (1 λ)n 1 is an integer. 11

14 The expected value of waiting can be calculated analogously and is given by EV Wait (s i, θ j ) = (1 λ)n 1 n=0 [ 1 2ɛ θ +ɛ θ ɛ B ( ) n, (1 λ)n 1, Prob GG (Withdraw s i, θ j ) ] EV (Payo Wait n) dθ (4) The expected value of waiting given that n of the other patient depositors withdraw is now dierent for model A and for model B. For model A, we have EV (Payo A Wait n) = ( ) (1 n/n r θ u 1 ) (1 n/n) R + (1 θ) u(0) if RD(N/r 1 ) n + λn u(0) if RD(N/r 1 ) < n + λn (5) and for model B EV (Payo B Wait n) = ( ) (1 n/n r u 1 ) (1 n/n) θ u(0) if RD(N/r 1 ) n + λn if RD(N/r 1 ) < n + λn (6) The threshold θ, which constitutes the interior global games equilibrium, is dened as the solution of the equation EV Withdraw (s i = θ, θ ) = EV Wait (s i = θ, θ ). We will consider a level-k approach as a behavioral alternative in these types of games. In these models, dierent types of agents are considered. There exists a non-strategic level-0 type in the head of the players: this type chooses randomly. A level-1 type nds the best response to a population of level-0 types. A level-2 type best responds to the behavior of a level-1 type, and so on. Level-k models have at least in part emerged from the experimental responses in beauty contest games. In the best-known version of these games, participants are asked to pick a number between 0 and 100. The winner of the game is the person that comes closest to 2/3 of the average of the picks. 12

15 The Nash equilibrium of the outlined beauty contest is zero. However, responses are typically much larger, and many people choose 33 or 22. One way to explain these patterns is that people start with level-0 players in mind who behave unstrategically. A natural assumption is that these non-sophisticated players choose a random number between 0 and 100. Their average guess is 50. Two-thirds of 50 is 33. The number 33 is therefore the response of a level-1 type who optimizes against a population of level-0 types. 22 is the best response to the behavior of a level-1 type and therefore constitutes the level-2 type prediction. Camerer, Ho, and Chong (2004) show in an analysis of the experimental data from many dierent games that higher thinking types above level-3 are rare, while for more complicated incomplete information games even level-2 types are seldom. Our bank run game, which will be introduced in the next section, is a complicated game; we therefore consider the prediction of a level-1 type to be a reasonable behavioral prediction in this environment. An important issue in applying level-k thinking to models of the global games type is the question of what level-1 types think about the behavior of level-0 types. It is typically assumed that level-0 types randomly choose a pure strategy. In a global game this assumption could mean that level-0 types choose A or B with a probability of 1/2, or that they randomly choose one cuto strategy out of all possible cuto strategies. Carrillo and Palfrey (2009) nd that randomization over all possible threshold strategies is the appropriate specication in a two-player private information game (the so-called compromise game) using a cognitive hierarchy specication. We therefore utilize the same specication in our analysis. However, it is not clear how a subject aggregates her beliefs about whether or not one randomly chosen subject withdraws into a probability distribution for the number of withdrawers. We therefore consider two dierent level-1 models that vary on this dimension. In our rst variant level-1 types assume that level-0 types use a random threshold strategy. Level-1 types presume the play of threshold strategies by the lower type, but they do not assume that level-0 types choose their threshold strategically. Let θ min and θ max depict the minimal and the maximal 13

16 possible threshold strategy. The probability that one randomly chosen level-0 type withdraws for a given signal s i is equal to (1 s i θ min θ max θ min ). In a further step, the level-1 type has to calculate the probability distribution for the number of withdrawers under the assumption of independently choosing level-0 types. Formally, the level-1 type maximizes θl1 θ=θ min 1 θ max θ min θmax 1 + θ=θ L1 θ max θ min (1 λ)n 1 n=0 (1 λ)n 1 n=0 [ [ ( ( B n, (1 λ)n 1, 1 θ θ )) ] min EV (Payo θ max θ Withdraw n) dθ min ( ( B n, (1 λ)n 1, 1 θ θ )) ] min EV (Payo θ max θ Wait n) dθ min (7) with respect to θ L1. This level-1 variant is called L1 RT S. Ho, Camerer, and Weigelt (1999) and Costa-Gomes, Crawford, and Iriberri (2009) provide evidence that players in experimental games with large groups do not take the independence of others' responses into account. To investigate this issue, we additionally consider a further variant named L1 P C RT S. L1P C RT S equals L1 RT S except that the level-1 player assumes that the decisions of all other players are perfectly positively correlated. That is, the level-1 player calculates the probability that one randomly chosen level-0 type withdraws exactly as in L1 RT S, but assumes that all other patient agents withdraw with this probability. The state that none of the other patient agents withdraw occurs with one minus this probability. Formally, the level-1 type maximizes θ P C L1 θ max θ min θ=θ min 1 θmax + θ=θl1 P C 1 θ max θ min (( 1 θ θ ) min EV (Payo θ max θ Withdraw n = (1 λ)n 1) min (( 1 θ θ min θ max θ min + θ θ ) min EV (Payo θ max θ Withdraw n = 0) dθ min ) EV (Payo A Wait n = (1 λ)n 1) + θ θ min EV (Payo A Wait n = 0) θ max θ min ) dθ (8) with respect to θl1 P C. This level-1 variant is called L1P C RT S. Both level-1 variants dier with respect to the calculation of the probability distribution for the number of withdrawers. 14

17 5 Parametrization and Design We will consider a low and a high repayment rate (r low 1 and r high 1 ) in the case of an early withdrawal. A set of parameters (θ min, θ max, ɛ, N, r low 1, r high 1, λ) in model A and a set of parameters (θ min, θ max, ɛ, N, r low 1, r high 1, λ, R) in model B is called a parameterization. The parameters ɛ, N, r low 1, r high 1, and λ are the same in both models, while θ min, θ max, and R dier. One interesting theoretical feature of bank run models is the existence of one-sided strategic complementarities. One-sided strategic complementarities cannot be included in a 2-person game. We therefore decide to play our game with N=6 subjects, as in Schotter and Yorulmazer (2009). Furthermore, using six subjects seems to be a reasonable compromise between simplicity and the desire to implement a multi-player game. Exactly one depositor has an early liquidity need, i.e., each agent's ex-ante probability of being impatient is given by λ = 1/6. So, one subject in every group is forced to withdraw early, leaving ve analyzable decisions per group. There is no aggregated uncertainty about fundamental withdrawal demand. We consider a uniform distribution for the uncertain fundamentals, i.e., project yield R for model A and success probability p for model B. We choose a uniform distribution between θ min = R low = 1.3 and θ max = R high = 3.0 (θ min = p low = 0 and θ max = p high = 1) for the ex-ante distribution of the fundamental in model A (B). The noise variable ɛ is set to 0.1 in both models. Fundamentals and signals are rounded to two decimal places. In principle, the parameter choices are guided by the desire to avoid a ceiling eect for both repayment rates (e.g., the chosen thresholds are both close to the upper or lower bound of the uncertain payo parameter). In such a situation, a meaningful analysis of the response to a change in the repayment rate would not be possible. 8 8 To be more precise, the parametrizations are guided by the predictions of the theoretical approaches and by the results of some pre-tests. We started with r1 low = 1.05 and r high 1 = 1.25, but the low repayment rate of 1.05 resulted in a situation where every subject waited for any signal. The data for these tests are not included in the paper for several reasons. In the pre-tests, we use a xed matching design, do not oer monetary incentives, play fewer rounds for more parameters, and have a dierent subject pool. Subjects are student assistants, the majority of whom are familiar with the Diamond and Dybvig (1983) model. All these elements are dierent from the experiments reported in the paper. 15

18 The lack of global strategic complementarities among agents' actions plays a role for r 1 = 1.5 in our parameterization. In that case, a partial run with 4 or 5 withdrawers leads to a payo of zero if the agent waits. Accordingly, the agent who leaves his money at the bank expects zero payos in the cases that three or four out of the four other patient agents withdraw. The incentive to withdraw is greater if she expects that three other agents withdraw than if she expects that four other agents withdraw. Note that the lack of global strategic complementarities among agents' actions plays a smaller role for r 1 = 1.25, where a partial run leads only to a zero payo if all other patient agents withdraw. Therefore, if the lack of global strategic complementarities is important for the use of threshold strategies, we should expect a higher degree of deviations from threshold strategies for r 1 = 1.5. We implement a random rematching design. Every session consists of 30 participants. At the beginning of each round, subjects are randomly and anonymously divided into ve six-person groups for every decision. The participants can interact within each group, but there is no interaction between the groups. At the end of each round, participants receive information on the outcome of their group. Information about the other groups is not given. After each round, the groups are again randomly rematched. So, rematching with the same people is possible, but the large sessionto-group-size ratio should minimize repeated play eects (see also the discussion in footnote 9). After each round, each subject receives feedback about her signal, the realized fundamental value, how many participants decided to withdraw early, and the resulting payment. The experiments are framed as a bank run decision (see instructions in the online appendix). Another interesting methodological issue arises from the design of Heinemann, Nagel, and Ockenfels (2004). In their experimental test of the global games approach applied to currency attacks, subjects receive ten independent signals in every round. For every signal, participants must decide whether they should attack. Note that this is very dierent from a sequential decision situation. Each subject simultaneously chooses an action for ten dierent decision situations, while no connection 16

19 exists between these ten decisions. After each round, they also receive feedback about each of the ten decision situations together. A more realistic setting would be to confront subjects with only one signal in every round, because in applications, decision makers will seldom respond to ten dierent games simultaneously. If threshold strategies are still preferred, this result would support the robustness of their use. Additionally, an extension to dynamic global games is obvious if there is only one signal presented at a time. In summary, theory-guided considerations have led to our models A and B. Given that for our research questions, the most important variable is the repayment rate r 1, we need at least two dierent values for this variable. In each session, the participants are confronted with both values for r 1, but the ordering is varied. With regard to questions on strategic signaling in repeated games, we use a random rematching design with a large session-to-group-size ratio. Additionally, we vary the number of signals that subjects receive in one round to test whether separate decisions would produce dierent results than sessions with joint decisions. Participants in sessions A 1.25/ and B 1.25/ play eight independent rounds with ten signals for both repayment rates, while participants in sessions A 1.25/1.5 1 and B 1.25/1.5 1 play forty independent rounds with one signal for both repayment rates. These considerations lead to a (2x2x2)-design, shown in Table 2. One session per cell is conducted. There is a total of eight sessions consisting of thirty subjects each. 9 9 We have a total of eight sessions, which is relatively few in comparison with many experimental studies. In essence, the design considerations that led to this choice are based on a compromise between two lines of reasoning. The rst argues that the decision behavior of any group, no matter how large that group is, constitutes only one independent observation. Every group member inuences other (or all other) members. Data analysis can therefore only be done in a reliable way if session averages are treated as independent observations. Experiments with larger groups are therefore often done in a xed matching design, presumably to reach a sucient number of observations on the session level despite the limited supply of subjects. The other line of reasoning is concerned with repeated play eects. The theoretical predictions in game-theoretic experiments are typically based on the assumption of a one-shot gamble. However, in a laboratory experiment with xed matching, it is possible that some or even all subjects engage in strategic signaling. For example, players may choose to cooperate more often than they would in a one-shot coordination experiment, simply because they assume that others will recognize this behavior and cooperate also. In the context of public good games, Botelho, Harrison, Pinto, and Rutström (2009) show that behavior is considerably dierent in xed matching and random rematching designs. Furthermore, the behavior under a random rematching design approaches the behavior under a random matching design (where subjects never meet twice) as the session-to-group-size ratio increases. In the worst case, strategic signaling would lead to a situation where a researcher uses statistically independent session averages that come from a dynamic game with very dierent properties compared to the theoretical benchmark the researcher is interested in. Our design is an attempt to meet (as far as possible) the requirements of both outlined reasonings. We have a fairly high session-to-group-size ratio that should mitigate repeated play eects. Also note that forced withdrawals are present; they further complicate any kind of strategic signaling. However, we still have eight observations on the session level, and the most conservative test of our main hypothesis would be to compare the eight observed 17

20 Model Choices per Round Starts with r low 1 = 1.25 Starts with r high 1 = 1.5 A 10 Session A Session A A 1 Session A Session A B 10 Session B Session B B 1 Session B Session B1 1.5 Table 2: Experimental Design: The table shows the general outline of our design. We test two models each for dierent values of the repayment variable and dierent numbers of signals and choices per round. Each cell corresponds to one session. In total, 240 students participated. Each session included 30 people and lasted roughly 90 minutes, without a great deal of variation. The average payment in sessions using model A (B) was (14.10) euros. The aim was an average payment of 15 euros, but subjects were more (less) cooperative than we thought a priori in model A (B). We conducted a computer-controlled experiment. Sessions were run at the University of Münster. No subject could participate in more than one session. In total, 240 undergraduate business and economics students participated, leading to a total of 24,000 analyzable decisions where no early withdrawal was forced. The experiment was programmed and conducted with the software z-tree (Fischbacher (2007)). We used an experimental currency unit (ECU). After nishing the experiment, the subjects were paid by converting their total ECU into euros. The average payment was euros, which is approximately equivalent to the payout they would receive for two hours of work as a student assistant at the Finance Center (doing library work, computer support, etc.). 6 Predictions One of the main purposes of this paper is to assess the strength of the reaction to increased risk sharing, because the global games approach predicts a strong reaction, while the level-k approach predicts only a small change in behavior. Intuitively, the equilibrium reasoning of the global games approach generates feedback eects that are not present in the level-k approach. The exact predictions are given in Table We provide predictions for a continuous version (as in Section 4) changes with the predicted ones using a non-parametric test. Furthermore, we can apply additional tests that use two observations from each session for r high 1 and r high 1 (like, e.g., Heinemann, Nagel, and Ockenfels (2004)), maximum likelihood estimations of individual decisions (like, e.g., Stahl and Wilson (1995)), and panel data models (see Section 7 and the online appendix). These analyses have their own strengths and weaknesses, but if these tests and the non-parametric tests based on only a few observations point toward the same conclusions, we would be very condent in the statistical inferences, although we are using only a few sessions. 10 In light of the level-k literature, it is plausible that we should expect few (if any) higher types in the experiment; however, it is less clear how many level-0 types are present. Level-0 types choose a random threshold strategy. The 18

21 and a discrete version of the model with a precision of two decimal places (as in the experiments). Appendix B of the online appendix explicitly derives the theoretical predictions of the global games and the level-k approach for the discrete version. An interval of [1.67; 1.74] means that for every threshold strategy (dened by a cuto value with two digits after the decimal point that comes from this interval) the best response of a player to a situation where all other players are using this cuto value is to use the same cuto value also. 11 Global Games L1 RT S L1 P RT C S r1 low r high 1 r1 low r high 1 r1 low r high 1 Model A Continuous Dist Discrete Dist. [1.67; 1.74] [3.07; 3.10] Model B Continuous Dist Discrete Dist. [0.28; 0.29] [0.51; 0.58] Table 3: Theoretical Predictions: The table shows the theoretical predictions for all considered approaches. Another aspect of our experiment is that there is no upper dominance region. An upper dominance region is a parameter region where fundamentals are so strong that, independent of the beliefs about other players' actions, no patient depositor withdraws early. Such a region is needed to show theoretically that the interior threshold equilibria are the only equilibria. Therefore, it is also an equilibrium in our experiment that all players withdraw given any signal. The threshold equilibria in Table 3 are the only equilibria with one single interior cuto value. We cannot exclude the possibility that more complicated equilibria, for example, with multiple thresholds, exist. predicted cuto value for a population of level-0 types is therefore simply the average of all possible cuto values. This prediction holds for both repayment rates, i.e., the level-0 prediction is the same. If we are now considering a population of level-1 and level-0 types, then we end up with the level-1 prediction in Table 4 for 0% of level-0 types and with the average possible cuto value for 100% of level-0 types. The predicted dierence for proportions in between is linear, decreasing with the proportion of level-0 types in the population. Appendix G of the online appendix contains two graphs for both models that illustrate this relationship. The results of the maximum likelihood approach (presented below) contain empirical estimates. 11 Such a large interval of interior equilibria is a consequence of the discrete nature of the game. For example, the best response to a cuto value of in a game that allows for three digits after the decimal point would be a cuto value of

22 7 Results Result 1: Threshold strategies are consistently played in the presence of one-sided strategic complementarities. A strategy in the game maps a signal between 1.2 and 3.1 for model A (-0.1 and 1.1 for model B) into a binary decision whether to withdraw or leave the money in the bank. 12 The rst step is to determine a framework that allows the estimation of threshold strategies. We use a simple error model as an intuitive starting point. 13 We assume an arbitrary threshold ˆt i [1.2; 3.1] for subject i in model A. The threshold denes the strategy T S(ˆt i ). We then classify every decision that is inconsistent with strategy T S(ˆt i ), i.e., a withdrawal for a signal above ˆt i or leaving the money in the bank for a signal below ˆt i, as an error. The threshold t i that minimizes the percentage of errors denes the threshold strategy used by subject i. Typically, the optimal threshold t i is not unique. In this case, we take the average of the supremum and the inmum of all thresholds that lead to the minimal percentage of erroneous decisions e i. The percentage of erroneous decisions e i for strategy T S(t i ) is further used as a measure to judge how consistently subject i used the threshold strategy. The analysis can be done analogously for the aggregate data of a complete session, implicitly assuming that every subject uses the same threshold. The number of observations for an aggregate threshold in one experimental condition is 1,000 (2,000) in sessions with one (ten) decisions per round. 14 Figure 2 illustrates the estimation procedure for session A and r 1 = The smallest (highest) possible signal is the smallest (highest) possible payo parameter minus (plus) the smallest (highest) error term. 13 A similar model has been used by Palfrey and Prisbrey (1996) and Palfrey and Prisbrey (1997) in a public good context. 14 In sessions with 1 (10) decisions per round, we have 40 (80) decision situations for r1 low and r high 1. This makes 2,400 (4,800) individual decisions per session. In exactly 1/6 of all decisions, the subject is forced to withdraw. Therefore, the number of analyzable individual decisions is 2,000 (4,000) per session. 15 An alternative approach would be logit regressions of the individual decisions on the signal. The best tting threshold t i is dened as the signal for which subject i withdraws with a probability of 0.5 according to the estimated logit model. On an individual level, the residuals of the logit model represent the partial randomness of a subject's value evaluations according to the random utility model. On the session level, the residuals also reect the heterogeneity of preferences within the session. We ran probit and logit regressions, but they produced virtually the same results as the simple error model, so the results are not reported. 20

23 1.00 e i e i t i t i Figure 2: Estimation of Threshold Strategies: The graph shows the estimation of the threshold that best ts the data from session A for r 1 = The estimated threshold t for session A under the experimental condition r 1 = 1.25 is The use of threshold strategies is analyzed by utilizing the simple error model. For every subject, we apply the methodology outlined above to estimate personal thresholds t i and the percentage of decisions e i that contradict the personally optimal threshold strategy T S(t i ) of subject i. This method allows us to estimate the degree of usage of threshold strategies directly on an individual basis. Table 4 shows the results. The means and medians of error rates e i are surprisingly low, indicating that the vast majority of subjects use a threshold strategy right from the beginning. Mean of Individual Error Rates Simple Error Model A A A A r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 Mean 5.21% 5.02% 4.00% 6.91% 5.36% 3.27% 2.93% 4.36% Median 4.48% 4.77% 3.08% 5.20% 4.63% 3.03% 2.78% 3.13% B B B B1 1.5 r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 Mean 5.71% 3.66% 4.46% 7.17% 5.23% 2.51% 3.17% 4.55% Median 5.25% 3.10% 2.82% 4.49% 3.52% 1.39% 3.03% 3.13% Table 4: Result 1 (Use of Threshold Strategies): The table shows the average error rates that are estimated with the simple error model and provide a measure for how consistently subjects use threshold strategies. The presence of one-sided strategic complementarities seems to be of minor importance with respect to the wide use of threshold strategies. Recall that the lack of strategic complementarities is more important for r 1 = 1.5 than for r 1 = A Mann-Whitney-U-test reveals no dierence between 21

24 the eight average error rates in the low risk sharing condition and the eight average error rates in high risk sharing condition (p=0.9164). We also used a parametric analysis and estimate the following linear regression: ER = (.005) (.006) rlow DecP erround1.039 r1 lowxdecp erround1 (N=16) (9) (.009) (.007) ER is the average error rate in an experimental condition, r low 1 is a dummy variable for the low risk sharing condition, DecP erround1 is a dummy variable for conditions with one decision per round, and r1 low XDecP erround1 is the interaction term between these two dummy variables. In this specication, each session delivers two observations, one observation for each risk sharing condition. Standard errors are in brackets and clustered on the session level to take the positive correlation of observations within one session into account. Hypothesis testing with a few cluster tends to over-reject the null hypothesis. We therefore use the wild cluster bootstrap approach of Cameron, Gelbach, and Miller (2008) with 2,000 repetitions to make more accurate cluster-robust inference possible. 16 The p-values of a test of coecients equal to zero are.067 and.004 for the low risk sharing dummy and the interaction term, respectively. However, the coecients are economically small and both coecients have opposite signs. We conclude that the lack of strategic complementarities, which complicates theoretical analysis considerably, is of less importance in the lab. Result 2: The use of threshold strategies is only marginally aected by whether subjects make one or ten decisions in each round. Interestingly, the lowest error rates are observed on average for conditions A 1.25/1.5 1 and B 1.25/1.5 1, although threshold strategies are least obvious in these conditions. The null hypothesis of equal distributions for the eight error rates that come from conditions with one decision per round and 16 The general data structure (number of observations, two independent dummy variables, an interaction term between these dummies and a dependent metric variable) in our regressions 9 and 10 is identical to the Gruber and Poterba (1994) application in Cameron, Gelbach, and Miller (2008). We used the STATA command cgmwildboot for the calculation of the wild bootstrap p-values. We would like to thank Judson Caskey for making his STATA ado le (cgmwildboot.ado) publicly available. 22

25 for the eight error rates that come from conditions with ten decisions per round cannot be rejected (Mann-Whitney U test, p= ). The bootstrapped p-value of a test with the null hypothesis that the coecient of DecP erround1 is equal to zero in regression 9 is Again, the estimated eect is economically small. This result can be interpreted as further evidence of the intuitive appeal of threshold strategies in such games with incomplete information about payos. One possible criticism of previous experiments (Heinemann, Nagel, and Ockenfels (2004)), which also potentially applies to our sessions A 1.25/ and B 1.25/1.5 10, is that the joint decisions for ten given signals causally induce the use of threshold strategies. However, the low error rates for sessions A 1.25/1.5 1 and B 1.25/1.5 1 show that threshold strategies are also common when the experimental interface is less favorable for them. 17 Result 3: We observe higher thresholds (and therefore more bank runs) if the repayment rate for early withdrawers is increased (qualitative comparative statics). Observed reactions to an increase in the repayment are smaller than the predicted changes of the global games approach. There are mixed results for the level-k approaches (quantitative comparative statics). All theoretical approaches agree on the qualitative comparative statistics. A higher degree of risk sharing should lead to higher thresholds. Table 5 shows the estimated thresholds on the session level. We obtain higher session thresholds (higher average of the estimated individual thresholds) and therefore an increased probability of a bank run for the experimental conditions with a high repayment rate. The discussion of dierent matching protocols in Section 9 implies that the most conservative way to analyze the estimated change in the observed thresholds is to focus exclusively on the observations 17 Small error rates favor the use of threshold strategies, but other strategies, such as waiting for a signal above a certain threshold and randomizing below, cannot be ruled out. If such asymmetric strategies are employed, then one would expect an asymmetric distribution of errors. In the example, one would expect no large errors for signals above the best tting threshold, but a signicant number of large errors below the best tting threshold. However, if the sample of all errors is split into a subsample of errors that occur for signals above and a subsample of errors that occur below the best tting threshold, there is no such clear pattern in these distributions (see Appendix D). It is therefore unlikely that the error-free use of an asymmetric threshold strategy has led to the low error rates observed in the experiment. 23

26 Estimated Thresholds A A A A r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 Aggregated Data Means of Individual Thresholds Standard Deviation of Individual Thresholds B B B B1 1.5 r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 Aggregated Data Means of Individual Thresholds Standard Deviation of Individual Thresholds Table 5: Result 3: The thresholds for the pooled session data are shown. Also shown are the means and standard deviations of estimated individual thresholds for each session. at the session level. Each session generates one observation of the change in behavior in reaction to a change in the repayment rate r 1 under this paradigm. We apply a simple binominal test by counting the proportion of sessions with an observed change above zero. If the true proportion is equal to 0.5 (four out of eight sessions), then the observed frequency eight out of eight occurs by chance with probability The null hypothesis of a true proportion equal to 0.5 can therefore be rejected at a conservative 1% signicance level, suggesting that increased risk sharing leads indeed to higher thresholds. We take now a rst look on our main research question. How do subjects react quantitatively to the increase in the repayment rate? If we consider the estimated thresholds on the aggregate level based on the simple error model, the mean of the estimated threshold for model A (model B) increases from 1.74 (0.44) for r 1 = 1.25 to 2.22 (0.48) for r 1 = 1.5. The observed mean change is therefore = 0.48 in model A and = 0.04 in model B. Figure 3 summarizes the results of the experiment and compares them with some theoretical predictions given numerically in Table 3. Comparing the data estimates with the theoretical predictions of the global games approach reveals that a stronger eect of an increased repayment rate should have been expected. The predictions of this approach are not consistent with the rather small reactions. The level-1 variants L1 RT S 24

27 Model A Model B Figure 3: Data and Predictions: The gure shows the data and predictions for the global games approach, L1 RT S, and L1 P RT C S. We use the midpoints of the intervals that are relevant to the global games approach. and L1 P C RT S do a better job. L1 RT S roughly predicts the sluggish reaction to the repayment rate in model A, although the predicted thresholds are higher than the observed ones under both risk sharing conditions. L1 P C RT S provides a near-perfect t of the data in model B, whereas L1 RT S only slightly improves the predictions of the global games approach. We use binominal tests by counting the proportion of sessions with an observed change below the predicted change of a theoretical approach. The predicted dierence of the model is higher than the observed dierence in the experiment in 8 out of 8 cases for the global games approach, in 6 out of 8 cases for the level-k approach assuming independence of other players' actions, and in 4 out of 8 cases for the level-k approach assuming a perfect correlation among the other players' actions. Using a simple two-sided binomial test, the null hypothesis that the global games approach creates predictions that are half the times above and half the times below the observed changes can be rejected with a p-value of A corresponding hypothesis for the level-k models cannot be rejected at conventional signicance levels (p= for L1 RT S and p= for L1 P C RT S ). To investigate the comparative statics further, we estimate the regression IT = α+β 1 r low 1 +β 2 A+ β 3 r1 low XA, where IT is the average individual threshold in an experimental condition, r low 1 is a dummy variable indicating the low risk sharing condition, A is a dummy for model A, and r1 low XA is the interaction term between both dummies. 18 The following equation shows the estimated 18 Heinemann, Nagel, and Ockenfels (2004) use a similar approach to analyze their so-called Standard sessions, see their Table IV on page The ordering of risk sharing conditions does not aect average individual thresholds. The estimated eect on average individual thresholds is.01 in model A and -.01 in model B if we extend regression 25

28 results. Standard errors are in brackets and clustered on the session level. In this specication, the coecient β 1 (β 3 ) gives the estimated eect of lower risk sharing (r1 = 1.25 instead of r 1 = 1.5) on the average individual threshold in model B (A). IT =.487 (.009).041 (.009) rlow (.062) A.379 (.069) r1 lowxa (N=16) (10) Again, we use the wild cluster bootstrap approach of Cameron, Gelbach, and Miller (2008) with 2,000 repetitions for hypothesis testing. The null of a coecient of zero can be rejected for model B at the 10%-level (p=0.059) and for model A on the 1%-level (p=0.004). As predicted by both theoretical approaches, a higher degree of risk sharing leads to more bank runs. 19 Probably more interesting are the quantitative comparative statics. The global games approach predicts that the coecients for r low 1 and r1 low XA are equal to -.22 and respectively. 20 Both hypotheses can be rejected at the 1%-level (H 0 : β 1 =.22: p<.000; H 0 : β 3 = 1.33: p=.003). The observed reactions are smaller than the predicted reactions by the global games approach. The analysis of the level-k approach yields mixed results for both variants. While the null of a reaction equal to -.2 in model A can be rejected for L1 P C rts (H 0 : β 3 =.2: p=.041), the null of a reaction equal to cannot be rejected for model B (H 0 : β 1 =.06: p=.181). The pattern for the second level-k-variant L1 rts is exactly vice versa. The null can be rejected for model B (H 0 : β 1 =.18: p<.000), but not for model A (H 0 : β 3 =.4: p=.562). Result 4: Both level-k approaches t the data better than the global games approach in a maximum likelihood analysis. Results are mixed with respect to the comparison of both level-k approaches. 10 with dummy variables for experimental sessions that started with the low risk sharing condition. Both coecients are insignicant economically and statistically. 19 Probit and panel regressions in the online appendix further support this claim using individual responses instead of experimental condition averages as observations. 20 These numbers are the dierence between the smallest possible threshold equilibrium in the high risk sharing condition and the highest possible threshold equilibrium in the low risk sharing condition in the discrete version of the underlying bank run model. 26

29 An alternative way of analyzing the data of experimental games is the maximum likelihood estimation of dierent decision models (see Stahl and Wilson (1995) and Costa-Gomes, Crawford, and Iriberri (2009), among others). This approach makes full use of all decisions and does not aggregate all decisions in one experimental condition into one observation. We assume that there may be naive decision makers in the population who make random choices, i. e. the probability of withdrawing is 1/2 irrespective of the signal s i the agents receive (P W 0 (s i) = 1/2 s i ). In deriving the theoretical predictions, a level-1 type does not assume such a behavior, but we make the assumption for the maximum likelihood estimations in all variants (including equilibrium) to ensure the comparability of the results. These models are dubbed GG Rand, L1 RT S Rand, and L1 P RT C S Rand. We further estimate the level-k variants with level-0 types as naive decision makers, i.e., players who are expected to choose a random threshold strategy for each decision (P W 0 (s i) = (1 s i θ min θ max θ min )). These variants are called L1 RT S RandT S and L1 P RT C S RandT S. We only consider level-0 and level-1 types based on the assumption that higher types are uncommon in complex games (see e. g. Camerer, Ho, and Chong (2004)). A second reason is that considering higher types in our game simply means approaching the global games solution stepwise. For example, in model A under the high risk sharing condition, the best response of a level-2 type to the level-1 types is to increase the threshold a little bit. The same is true for a level-3 type and so on. In the limit, a level-n type nds that the best response to level-(n 1) types is to choose exactly the same threshold. This solution is in fact a global games solution. Therefore, the inclusion of higher types not only moves the theoretical predictions of the level-k variants away from the data, but it also makes it hard to distinguish higher types from each other as their strategies are only marginally dierent. For all level-1 variants, we proceed as follows. Denote with EV W (s i ) the expected value of withdrawing and with EV NW (s i ) the expected value of not withdrawing. We allow for errors by estimating 27

30 the probability that a level-1 type withdraws given signal s i via P W 1 (s i ) = exp(λ EV W (s i )) exp(λ EV W (s i )) + exp(λ EV NW (s i )) (11) λ gives the precision of the level-1 type calculations. If λ approaches zero, the player chooses randomly. If λ goes to, the agent chooses the alternative with the higher expected value. The way the expected values EV W (s i ) and EV NW (s i ) are calculated is dierent for the dierent variants of level-k models. The probability that a observed choice of player j who has received signal s i is generated by the proposed model is given by P j 1 (s i) = P W 1 (s i) 1 s j +(1 P1 W i (s i)) (1 1 s j), where 1 i s j is an indicator i function that is equal to one if player j has withdrawn and zero otherwise. A subject makes T choices in the experiment. The likelihood that a player j is of type 1 (0) is given by L 1 = ( T t=1 P j 1 (s t) L 0 = ) T t=1 P j 0 (s t). Let π 1 (π 0 ) be the proportion of level-1 thinkers (naive decision makers) in the population. The likelihood of observing T choices from a subject j unconditional of her type is given by T π 1 t=1 P j 1 (s t) + π 0 T t=1 P j 0 (s t) (12) Considering all N M subjects in a given population, the log-likelihood function is given by LL = N M j=1 ( log T π 1 t=1 P j 1 (s t) + π 0 T t=1 ) P j 0 (s t) (13) In addition to level-1 variants, we consider an equilibrium plus noise specication. Here, we assume that a player thinks that the group is in a threshold equilibrium. The used threshold from the interval of possible thresholds in the discrete version of the game is determined as follows. If the signal is below the smallest value of the relevant interval, we choose the smallest value. If the signal is above the highest value of the relevant interval, the maximum value in the interval is chosen. 28

31 If a signal within the interval is considered, we use exactly the signal as threshold. For example, the player under the low risk sharing condition (r 1 = 1.25) in model A that receives a signal of 1.4 plays a threshold strategy with θ = This agent assumes that all players use a threshold strategy with this cut-o value. If she observes a low signal like 1.4, she knows that nobody has observed a signal above 1.67 and that the probability of a full bank run is one. The opposite is true for good signals ( 1.95). If the player observes a signal of 1.7, then she assumes that everybody plays θ = 1.7. This procedure gives the equilibrium a good chance to perform well, because the thresholds adapt to the signals. For own signals that make it possible that some agents withdraw and some do not, the agent infers the probability distribution for the number of withdrawers based on the assumption that all players choose the equilibrium threshold strategy and that they choose independently of each other. Table 6 presents the results. There is only a tiny fraction of naive decision makers in GG Rand, L1 RT S Rand, and L1 P C RT S Rand, indicating that virtually nobody ignores his own signal. Note that all variants have the same number of free parameters, so we can compare the goodness of t directly via the log-likelihood. The global games plus noise specication ts relatively badly in all four subsamples. This result is especially pronounced in model A under r 1 = 1.5. Intuitively, the result is due to strict rules for the calculation of the probability distribution for the numbers of withdrawers. If an agent observes a signal below 2.87 in model A under r 1 = 1.5, she infers that all other patient agents withdraw with a probability of one. The beliefs are similarly strict in other models and risk sharing conditions. However, such beliefs are inconsistent with the behavior in the experiment. All other models assign a positive and often substantial probability to any number of withdrawers. The distinction between L1 RT S and L1 P C RT S is not clear-cut. L1 RT S performs better than L1 P C RT S under model A, but the opposite is true for model B. One disturbing aspect of the data is the relatively high number of level-0 types that choose a random threshold strategy for model L1 P C RT S. For example, the estimated proportion of level-0 types is roughly 60% in sessions A and A

32 Sessions A and A (obs.=8,000) GG Rand L1 RT S Rand L1 P C RT S Rand L1 RT S RandT S L1 P C RT S RandT S λ (0.026) (0.089) (0.077) (0.994) (0.155) π 0 <0.001 (<0.001) <0.001 (<0.001) <0.001 (<0.001) (0.391) (0.064) LL Sessions A and A (obs.=4,000) GG Rand L1 RT S Rand L1 P C RT S Rand L1 RT S RandT S L1 P C RT S RandT S λ (0.035) (0.100) (0.085) (0.173) (0.240) π 0 <0.001 (<0.001) (0.017) <0.001 (<0.001) (0.070) (0.080) LL Sessions B and B (obs.=8,000) GG Rand L1 RT S Rand L1 P C RT S Rand L1 RT S RandT S L1 P C RT S RandT S λ (0.038) (0.067) (0.092) (0.071) (0.102) π (0.032) (0.033) (0.036) (0.044) (0.042) LL Sessions B and B (obs.=4,000) GG Rand L1 RT S Rand L1 P C RT S Rand L1 RT S RandT S L1 P C RT S RandT S λ (0.056) (0.114) (0.130) (0.128) (0.198) π (0.017) (0.023) (0.017) (0.045) (0.044) LL Table 6: Maximum Likelihood Estimations: The table reports on the results of the maximum likelihood estimations. Shown are the coecients and the standard errors for the precision parameter λ and the proportion of naive decision makers π 0. In the models with the postx rand, a naive decision maker is an agent who choses action A with probability 1/2. In the models with the postx randts, a naive decision maker is an agent who choses a random threshold strategy. Also shown is the log-likelihood (LL) for every estimation. The ml estimations have been run separately for every model and number of decisions per round. Forced withdrawals do not enter the estimation process. `obs' is the number of observations. The number of observations per session is explained in footnote 14. The high percentage of level-0 types in specications L1 RT S RandT S and L1 P RT C S RandT S can be considered as evidence against level-k models in general. Result 5: Observed thresholds are constant over time. Learning in experimental games is a popular area of research (see, e.g., Camerer (2003) for a review). In our experimental game, learning might have changed the thresholds over time. To investigate this issue, we estimate thresholds with the logit model separately for rounds 12, 34, 56, and 78 for sessions A 1.25/ and B 1.25/1.5 10, and rounds 110, 1120, 2130, and 3140 for sessions A 1.25/1.5 1 and B 1.25/ Figure 4 reports the results. Thresholds for subperiods are remarkably stable over 30

33 time. No systematic change in played thresholds can be detected with the naked eye. Panel and probit regression reported in the online apendix conrms this impression. r 1 = 1.25 r 1 = 1.5 Figure 4: Thresholds over Time: The gure shows how the session thresholds evolve over time. For sessions A , A , B , and B10 1.5, we estimated thresholds using the logit model for rounds 12, 34, 56, and 78. For sessions A , A 1.5 1, B1 1.25, and B1 1.5, we estimated thresholds for rounds 110, 1120, 2130, and

34 8 Discussion Our results suggest that, applied to bank runs, the global games approach leads to experimentally valid predictions with respect to the qualitative comparative statics. Our subjects use threshold strategies, and increased risk sharing leads to increased thresholds. However, the rather small reaction of thresholds to changes in the repayment rates is at odds with the theory. This result is important with respect to the socially optimal repayment rate. One major advantage of the global games approach is the trade-o between the benets and the costs (increased probability of a bank run) of high repayment rates. Such a social optimization requires valid predictions about how people react to changes in the repayment rates. In our experimental setup, at least, this assumption is questionable. As a result, the optimal r 1 for the banking system as a whole is dicult to determine. A simple calibration based on past experience would not work, because depositors do not react quantitatively, as predicted by the theory, to a change in the repayment rate. This result is true notwithstanding if fundamentals are modeled through the project return R (model A) or through the success probability of the project (model B). Our results may be surprising in the light of the previous experimental literature on global games. Heinemann, Nagel, and Ockenfels (2004) and Duy and Ochs (2012) document that observed thresholds are always smaller than the global games prediction. We nd sometimes smaller and sometimes larger thresholds compared to the global games prediction. Heinemann, Nagel, and Ockenfels (2004) and Duy and Ochs (2012) further nd that the reaction of observed behavior to a change in the payo parameter is roughly equal to the prediction of the global games approach. In our experiment, the reactions are much smaller than the predicted ones. There are many dierences between our study and the previous literature on global games, but we believe that two aspects are especially important here. In Heinemann, Nagel, and Ockenfels (2004) and Duy and Ochs (2012), the theoretical prediction of the global games approach is always close 32

35 to the mean or smaller than the mean of the pre-specied interval for the fundamentals. 21 Klos and Nöth (2012) report that observed behavior is sensitive to the pre-specied interval for the fundamentals. Especially, they report results from treatments where there is no interior solution in the global games approach, i.e., where the unique prediction of the global games approach coincides with the upper or the lower bound of possible fundamentals. In these situations, subjects still choose interior thresholds on average. Such a situation also occurs in model A for r high 1. In general, the study of Klos and Nöth (2012) implies that results from studies with dierent pre-specied intervals for the fundamentals are dicult to compare. The second aspect that we want to stress is that we use a random rematching design instead of the xed matching design of Heinemann, Nagel, and Ockenfels (2004). Duy and Ochs (2012) are also concerned with the xed matching and rematched subjects in sessions with 20 participants randomly in two groups of ten respondents. While this certainly reduces strategic signal concerns, it is not clear if it is enough. Botelho, Harrison, Pinto, and Rutström (2009) show in public good games that the behavior under a random rematching design approaches the behavior under a pure stranger design (where subjects never meet twice) as the session-to-group-size ratio increases. As the session-to-group-size ratio is 5 in our study, 2 in Duy and Ochs (2012), and 1 in Heinemann, Nagel, and Ockenfels (2004), it is at least possible that repeated cooperation causes reactions to be smaller in xed matching designs than the reactions in a design where repeated play eects are less likely. Further research could address in more detail if both, one, or none of these two conjectures is correct. Our paper also adds to the ongoing comparison of the global games and the level-k approach. Our overall impression from the results in Cornand and Heinemann (2013), Shapiro, Shi, and Zillante (2011), and Klos and Nöth (2012) (briey surveyed in Section 2) is that level-k approaches help to organize the experimental data, but they fail in non-negligible respects. We observe a similar pattern in our study. While our experiment presents evidence that the global games approach predicts reactions to a change in the repayment rate that are too strong, our results are mixed with 21 Heinemann, Nagel, and Ockenfels (2004): Theoretical predictions are 29.73, 55.33, 41.83, and 57.04; Duy and Ochs (2012): Theoretical predictions for the static games are 42 and 62. Fundamentals in both studies are random draws from a uniform distribution over the interval [10, 90]. 33

36 respect to the question of whether players in a level-k model behave as if they believe that the other agents' actions are perfectly correlated. This result is unexpected because previous research has favored the perfect correlation assumption (Ho, Camerer, and Weigelt (1999) and Costa-Gomes, Crawford, and Iriberri (2009)), while our results are not clear-cut. We also identify a considerable estimated proportion of level-0 types in the maximum likelihood approach (up to 60%), again suggesting that level-k approaches lack at least some descriptive validity in global games. Our study further adds to the experimental investigation of bank runs. Madiès (2006) analyzes the interaction between narrow banking (low risk sharing) and deposit insurance or the suspension of deposit availability. We focus on narrow banking per se and analyze dierent degrees of risk sharing within respect to suitable theoretical benchmarks. This allows a quantitative assessment which may be informative for policy analysis. Furthermore, Madiès (2006) and Garratt and Keister (2009) show that pure panic-based runs are rare if aggregate fundamental withdrawal demand is known to subjects. We use the theoretical setup of Morris and Shin (2001) and Goldstein and Pauzner (2005) to show that this is no longer true if there exists ex-ante uncertainty about the protability of the banking sector. Although aggregate withdrawal demand is known in our experiments, subjects withdraw for many signals where coordination among the patient depositors would pay o. For example, the average threshold is 2.22 in model A for r 1 = 1.5. However, the expected outcome is 1.8 for a signal of 2 in the case that all patient depositors wait. We have these panic based bank runs in our experiment even without aggregated fundamental uncertainty if fundamentals are weak. We hope to see further studies that investigate the ndings of the current experimental bank studies (surveyed in Section 2) in a setup that allows for both, fundamental and panic-based runs. 34

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41 Online-Appendix How Strongly React Players to Increased Risk Sharing in an Experimental Bank Run Game? A Framework A.1 An Initial Example We start with a simple two-player example to illustrate the global games and the level-k approach used in this paper. The game is shown in Figure 1. If both players choose A (B), then each receives a payment of Θ (0). If one player chooses B and the other player chooses A, the A (B) choice leads to a payment of -1 (1). Player 1 A (Doesn't Run) B (Run) Player 2 A (Doesn't Run) Θ B (Run) 1 Θ Table 1: An example illustrating the basic structure of the studied coordination problem. This simple example captures some of the basic features of a bank run situation. The socially best outcome would be that all depositors choose A (Doesn't Run) and the bank oers a high repayment (Θ). The repayment depends on the investment success of the bank. If a depositor decides to withdraw his money, it is best for him if he is the only withdrawer. Otherwise, the bank may run out of funds. Clearly, B dominates A for Θ < 1. A.2 Global Games Approach The main idea of the global games approach is that the payo parameter Θ is not common knowledge, as it is in classic models. Instead, agents receive a noisy private signal about the 1

42 value of Θ. Let us assume that θ is a realization of Θ. Each player i now receives a signal s i = θ + ɛ before they simultaneously choose their actions. ɛ is a normally distributed error term with zero mean and standard deviation σ. Each player knows that the other agents receive such a signal, implying that each player knows after receiving a good (bad) signal that the other player also received a good (bad) signal (with high probability for suciently small σ). A natural assumption is that agents choose threshold strategies in such a situation, i.e., they choose A (B) if they receive a signal above (below) a certain threshold. An equilibrium is reached if the strategy of one player is the best response to the strategy of the other player, and vice versa. How can one determine such an equilibrium? Every agent knows that his signal is normally distributed with mean θ and standard deviation σ. Because the error term has zero mean, it also follows that the signal of the other player is normally distributed with mean s i and standard deviation 2 σ. If the other player also uses a threshold strategy, the probability that player j receives a signal below his cuto value c j is ( cj s P rob GG (B s i, c j ) = Φ i 2 σ ), where Φ is the cumulative standard normal probability density function. The expected payos for playing A or B are EV (A s i, c j ) = (1 P rob GG (B s i, c j )) s i + P rob GG (B s i, c j ) ( 1) and EV (B s i, c j ) = (1 P rob GG (B s i, c j )) 1, respectively. Our player chooses A if EV (A s i, c j ) > EV (B s i, c j ) and B otherwise. For some signal s i, it holds that EV (A s i, c j) = EV (B s i, c j). s i denes the cuto value of the threshold strategy that is the best response to the threshold strategy c j of the other player. Figure 1 shows the best response for every c j [1, 3]. An equilibrium in this symmetric game is reached for a cuto value of 2. In many applications, the interior cuto value equilibrium is the single equilibrium. However, it is important to note that this interior cuto equilibrium is not the only equilibrium in our example game. Another equilibrium is to play B for every signal because if a player believes that the other player always chooses B, the rst player's best response is to play B also. There are several ways to address this problem. Additional assumptions can be added to ensure the 2

43 Figure 1: Best response function for the game shown in Table 1: The gure illustrates the best responses of a player to every cuto value between one and three that the other player may choose. provableness of a single cuto equilibrium, or plausible equilibrium selection rules can be used to point toward the interior cuto equilibrium (Goldstein and Pauzner (2005)). In this paper, we will refer (somewhat loosely) to the interior cuto equilibrium as the prediction of the global games approach, although other equilibria (such as always playing B) cannot be ruled out theoretically. A.3 Level-k Approach If a level-1 type nds the best response to a level-0 type who chooses a random threshold strategy, it is straightforward to calculate the probability that the other player (the level-0 type) plays A or B for a given signal. Consider a signal of 1.5 in our example game and fundamentals between 1 and 3. If the level-0 type chooses a random threshold strategy, he will play A if the cuto value that denes his threshold strategy is less than or equal to 1.5. Because by assumption any cuto value between 1 and 3 is equally likely, the probability that the other player picks action A is 1/4. Figure 2 illustrates this situation. Given our assumptions, the probability that a level-0 type plays A is 1/4 and the expected value of a level-1 type playing A (B) is EV L1 (A s i = 1.5) = 1/ /4 1 =

44 Figure 2: Example (Level-k approach): The gure illustrates the expectations of a level-1 type with an own signal s i equal to 1.5 about the behavior of the other player. (EV L1 (B s i = 1.5) = 1/ /4 0 = 0.25). EV L1 (A s i ) is generally smaller than EV L1 (B s i ) if the signal is smaller than 2. A threshold strategy with cuto value 2 is therefore the prediction of a level-1 approach. Because a threshold strategy with a cuto value of 2 is an equilibrium of this game, the best responses of higher types coincide with the level-1 prediction. A.4 Responses to Changes in the Payo Structure The global games and the level-k approach make the same prediction (cuto value of 2) in our example. However, if the payo parameters of the models are changed, a dierence in the strength of the predicted responses to changes in the payo structure can be observed. In our initial example, an agent who chooses B while the other agent chooses A receives a payo of 1. Increasing this number obviously increases the attractiveness of action B using both models. However, the change in the predicted threshold strategy is much stronger in the global games than in the level-k approach. Increasing the payo from 1 to 1.5 yields a predicted threshold of 2.5 in the global games approach, but only in the level-k approach (considering only three digits after the decimal point). A payo of 2 in the case of a B-choice with the other agent playing A yields a solution of 3 in the global games approach, but only in the level-k approach. 1 We use this observation to design an experiment that assesses the strength of reactions to changed payo parameters empirically. 1 Note that the behavior of higher thinking types is not equal if the behavior of level-1 types does not coincide with an equilibrium prediction. In these cases, considering higher thinking types means stepwise approaching equilibrium because higher types best respond to lower thinking types until an equilibrium is reached. 4

45 B Theoretical Prediction of Thresholds B.1 Global Games Consider the decision problem of an agent who is not forced to withdraw early in the rst period. Depending on her signal s i, the depositor can infer that the true fundamental value lies between s i ɛ and s i + ɛ. Because the a-priori probability distributions of the fundamental value and the error term are both uniformly distributed, the a-posteriori probability distribution of the true fundamental value given signal s i is also uniformly distributed. The optimal action depends now on the belief about other players' actions. Assume that all agents have homogeneous preferences and follow the same threshold strategy which is dened by the threshold θ. Playing an equilibrium threshold strategy implies that the agent's best response to θ of the others is also θ. Signals were rounded to the next two decimal places in the experiment. If a signal s i is observed, the agent can infer that the true fundamental state is one of the 21 possible ones between s i 0.1 and s i +0.1 with an equal probability of 1/21. For each of these possibly-true fundamental states ˆθ, a randomly chosen agent receives one of the 21 signals between ˆθ 0.1 and ˆθ+0.1. For example, 1 a randomly chosen agent receives the signal s i 0.2 with a probability of 441 = , as there is only one true fundamental state (s i 0.1) and only one noise term of the other agent ( 0.1) that would lead to this situation. A signal (s i 0.19) would be observed with probability as there are two possibilities that lead to this situation (true fundamental state (s i 0.09), noise term 0.1 and true fundamental state (s i 0.1), noise term 0.09), and so on. The probability that a single randomly chosen agent withdraws if our agent observes signal s i is P (θ, s i ) = si +0.2 s=s i 0.2 ((21 s i s 100)/441) 1 s<θ, where 1 s<θ is equal to 1 if s < θ and 0 otherwise. More generally, the probability is P (θ, s i ) = s i +2ɛ s=s i 2ɛ ((nps s i s 1 step )/nps2 ) 1 s<θ, where 5

46 nps is the number of possible states in an interval of the length 2ɛ and step is the interval length between two discrete states. The expected value of an early withdrawing is now the sum of the utilities given that none, one, two,..., or all (1 λ)n 1 of the other patient depositors withdraw 2, or more formally EV WD (θ, s i ) = (1 λ)n 1 n=0 ( ) B n, (1 λ)n 1, P (θ, s i ) EV (Payo Withdraw n) ( ) where B n, (1 λ)n 1, P (θ, s i ) is the probability that exactly n of the other (1 λ)n 1 patient depositors withdraw. B denotes the binominal distribution. The expected value given that n other patient depositors withdraw depends on the ability of the bank to pay o all early withdrawers. The number of withdrawers in that case is the sum of the n other patient depositors who withdraw, the λn people who have to withdraw early, and our agent who decides to withdraw as well: EV (Payo Withdraw n) = u(r 1 ) if RD(N/r 1 ) n + λn + 1 ( ) RD(N/r 1 ) n+λn+1 u(r 1) + 1 RD(N/r 1) n+λn+1 u(0) if RD(N/r 1 ) < n + λn + 1 RD(x) depicts the round down of x. The expected value of waiting can be calculated analogously and is given by EV W ait (θ, s i ) = 1 nps s i +ɛ θ=s i ɛ (1 λ)n 1 n=0 ( ) B n, (1 λ)n 1, P (θ, s i ) EV (Payo Wait n, θ) The expected value of waiting given that n of the other four patient depositors withdraw is now dierent for model A and for model B, because the uncertain payo parameter is the project 2 We assume that λ is chosen in such a way that (1 λ)n 1 is an integer. 6

47 return for model A and the success probability for model B. For model A, we have EV (Payo A Wait n, θ) = ( ) (1 (n+λn)/n r u 1 ) (1 (n+λn)/n) θ u(0) if RD(N/r 1 ) n + λn if RD(N/r 1 ) < n + λn and for model B EV (Payo B Wait n, θ) = ( ) (1 (n+λn)/n r θ u 1 ) (1 (n+λn)/n) R + (1 θ) u(0) if RD(N/r 1 ) n + λn u(0) if RD(N/r 1 ) < n + λn The optimal threshold θ opt is dened as the smallest possible signal for which the expected value of waiting is greater that the expected value of withdrawing given the belief that all others play θ. The players are in equilibrium if θ opt = θ. The optimal threshold is not unique due to the discrete nature of the game. Therefore the outlined approach results in a range of optimal θ 's that are considered to be consistent with the global games approach. B.2 Level-1 Thinking In our rst variant level-1 types assume that level-0 types use a random threshold strategy. Level- 1 types presume the play threshold strategies by the lower type, but they do not assume that level-0 types choose their threshold strategically. Let θ min and θ max depict the minimal and the maximal possible threshold strategy, e. g. θ min = 1.2 and θ max = 3.1 in model A. It is tempting to say that the probability that one randomly chosen level-0 type withdraws for a given signal s i is equal to (1 s i θ min θ max θ min ), but this is only true for signals that are suciently far away from the boundaries of the strategy space. In that case, the probability that a randomly chosen level-0 type withdraws is given by s=s i +0.2 s=s i 0.2 ((21 s i s 100)/441) (1 s θ min θ max θ min ), an expression that reduces to (1 s i θ min θ max θ min ). However, in general the probability that a randomly chosen level-0 type withdraws is given by s=s i +0.2 s=s i 0.2 ((21 s i s 100)/441) Max(1 Min( (s θ min ) (θ max θ min ), 1), 0). 7

48 Because this only makes a dierence near the boundaries (for example if you observe a signal of 1.31 in model A), this approach is still behaviorally plausible as a heuristic use of the term (1 s i θ min θ max θ min ) works most of the time. In a further step, the level-1 type has to calculate the probability distribution for the number of withdrawers under the assumption of independently choosing level-0 types. This level-1 variant is called L1 RT S. We additionally consider a further variant named L1 P C RT S. L1P C RT S equals L1 RT S except that the level-1 player assumes that the decisions of all other players are perfectly positively correlated. That is, the level-1 player calculates the probability that one randomly chosen level-0 type withdraws exactly as in L1 RT S, but assumes that all four other patient agents withdraw with this probability. The state that none of the other patient agents withdraw occurs with one minus this probability. Level-1 types in L1 P RT C S assume that there is no chance that one, two, or three of the other four patient agents withdraw. Both level-1 variants dier with respect to the calculation of the probability distribution for the number of withdrawers. The optimal threshold θ is dened as the smallest possible signal for which the expected value of waiting is greater than the expected value of withdrawing. The following table (Table 2) summarizes the predictions of all theoretical approaches. Model A Model B Global Games L1 RT S L1 P RT C S Global Games L1 RT S L1 P RT C S r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 r1 low r high 1 [1.67; 1.74] [3.07; 3.10] [0.28; 0.29] [0.51; 0.58] Table 2: Theoretical Predictions: The table shows the theoretical predictions for all considered approaches. 8

49 C Instructions Instructions given to the participants varied according to dierent sessions. Here, we include an English translation of the instructions for session A , i.e. the session started with r 1 = 1.25 and shifted to r 1 = 1.5. For the other sessions instructions were adapted accordingly. Introduction Thank you for participating in this economic experiment. The principal aim of this experiment is to analyze your decision-making behavior in groups. Your prots depend on your decisions but also on the decisions of the other participants in the experiment. You will make all your decisions at your workstation in our lab and to assist you in your decision process you will receive some printouts with helpful tables. This experiment is nanced by the University. General Information You are one of 30 participants in this experiment and the rules are the same for all participants. The experiment consists of 8 independent rounds, which are played by ve 6-person groups. At the beginning of each new round ve new 6-person groups will be formed out of the 30 participants. Within each group the participants interact with each other, but there is no interaction between the groups; each group is completely separate. In order to conduct the experiment in a manageable time-frame we ask you to make your decisions in every round in the suggested processing time of 3 minutes. During each round a timer on your screen will inform you of the time remaining. A round is nished after every participant has made their decisions. In a one minute information phase each participant is informed about their results in the previous round. Following this, the next round will start. A quick summary of the rules 9

50 In every round each participant must make 10 decisions, all of which are similar in structure. Each participant will make a total of 80 decisions (8 rounds with 10 decisions). Each of you has one ECU (Experimental Currency Unit), which you placed into a bank in time period 0. The bank invests your and your group members' deposits in an investment project. The investment project will last for 2 periods. After the two periods (t=2) the bank can realize project earnings R, which vary between 1.30 and 3.00 ECU. If the bank has to partly liquidate the project in period 1, it receives no project earnings for the liquidated part. It only receives the invested capital back. However, the bank promises a xed payment of 1.25 ECU to all withdrawing participants who have already requested their money in period 1. Therefore, it has to liquidate parts of the investment project to settle these claims. Participants who leave their money until period 2 will receive a payment which depends on the money remaining in the project and the project earnings R. Your decision situation in detail In this experiment we ask you to decide at which point of time you would like to withdraw money; that means if you would like to withdraw money in period one or two. The chronological structure is summarized in the following illustration. 10

51 Actions of the participants 6 participants (including you) place 1 ECU with a bank Period 1: you withdraw your money in t=1 Period2 : you withdraw your money in t=2 t=0 t=1 t=2 Bank invest 6 ECU in the investment project Liquidation of the project for early withdrawals Investment project finished; project earnings R is paid to late withdrawals Actions of the bank You might ask yourself how you can determine your payment if you leave your money at the bank until period 2. The following rule may help you: If some of the participants withdraw their money in the rst period, the bank is forced to liquidate parts of the investment project to meet these expenses. If we call the number of participants who claim their money back in period one n, then n*1.25 ECU is claimed back. Due to this the bank retains (6-n*1.25) ECU of the originally invested amount (6 ECU), which realizes the project earnings of R ECU for each ECU remaining. In the second period the bank is liquidated and the generated proceeds are distributed among the remaining participants. Example: If two of the six participants in a group claim a repayment in period one, each receives 1.25 ECU. Thus the bank has to liquidate 2.50 (2*1.25) in period one and 3.50 ECU (6-2.50) remains in the investment project. Assuming the project earnings amount to R=3, the bank possesses ECU (3*3.50) at the end of the investment project. These ECU are shared equally among the remaining four participants, so that each receives 2.63 ECU. In the case that ve or six participants demand their deposits in period one, the bank goes bankrupt. We call this situation a bank run. For example, if ve participants claim their money back in period one, the bank has to pay out 6.25 ECU (5*1.25). Since this amount exceeds 6 11

52 ECU, the bank can no longer meet the withdrawal demands and the bank goes bankrupt. It can only pay out four participants with 1.25 ECU, for which it expends 5 ECU (4*1.25). Among the ve participants who wanted to withdraw their money four are selected by chance and they receive their payment of 1.25 ECU. The remaining amount of 1 ECU (6-4*1.25) is lost and the last (fth) participant receives no payment. If all six participants claim their money, again only four participants receive 1.25 ECU. Two participants receive no payment. To sum up, if there are more than four participants who claim back their deposits, it is decided randomly which four participants receive a payment and which participants receive no payment. The following illustration summarizes the payment scheme: n participants withdraw in total 1.25 * n ECU 6-(1.25*n) ECU remain at the bank Project earnings R*(6-1.25*n) ECU is paid to the late withdrawals t=1 t=2 Bank Runs hazard, if 5 or 6 participants withdraw early If you decide to withdraw your money in period one, your credit entry depends on how many participants would also like to withdraw their money in period one. If fewer than ve participants decide to claim back their money in period one then you will denitely have 1.25 ECU transferred into your account. If you decide to withdraw your money in period two, your credit entry depends on how many participants already claimed back their money in period 1 and how high the project earnings R are. The lower the number of participants who chose period one and the higher the project earnings R, the more credit you will receive in period two. If there are more than four participants 12

53 claiming their money back in period one, you will denitely receive zero ECU in period two for sure. Forced withdrawals In each decision situation there is exactly one participant in each 6-person group who has to claim back their money early. Thus this participant has to choose period one. The following message will appear on your screen: `As a result of an urgent need for money you are forced to withdraw your money. Hence you automatically have chosen period one.' In each decision the odds are six-to-one (one in each 6-person group) that you will be forced to withdraw. Calculation spreadsheets Mit einer Wahrscheinlichkeit von 1/6 (einer in jeder 6er Gruppe) sind Sie von dem Hinweis Below is a table that will help you to calculate the payos associated with your withdrawal decision assuming that the project earnings are R=3. The chart gives you payos for all the possible numbers of requests. Please take a moment to make sure that you completely understand the spreadsheet. betroffen und haben dann in dieser Entscheidungssituation keine Entscheidungsfreiheit. Berechnungstabellen Angenommen der Projektertrag sei R=3, so können Sie der folgenden Tabelle für jede mögliche Anzahl von Teilnehmern, die bereits in Periode 1 ihr Geld zurückfordern, Ihre Auszahlung entnehmen. Bitte nehmen Sie sich die Zeit, diese Tabelle zu verstehen. Number of participants wishing to withdraw Period 1 Period 2 In period 1 In period 2 Number of participants who can be satisfied Number of participants who are not satisfied Individual amount of the withdrawal Number of participants who can be satisfied Number of participants who are not satisfied Individual amount of the withdrawal or or Mit diesen Instruktionen haben Sie mehrere Tabellen enthalten, welche Ihre Auszahlungen für The handouts that you received as well as these instructions should aid your decision process. verschieden hohe Werte von R angeben. Während des Experiments erhalten Sie diese Tabellen als Hilfsmittel für Ihre Entscheidungen. They contain tables that show your payments for dierent values of R. 3 Der unbekannte Projektertrag R The unknown Bei jeder project Entscheidung earnings hängt der RErfolg von Periode 2 auch von der Höhe des Projektertrags 3 Subjects received R ab. Im (in Augenblick, addition in todem the Sie instructions) Ihre Entscheidung handouts fällen, withist payout Ihnen die tables Höhe with des Projektertrags numerous values for R. These handouts contained R nicht bekannt. 18 tables for any value of R between 1.3, 1.4, 1.5,..., and 3.0. R kann zwischen 1,30 und 3,00 variieren. Alle möglichen Zahlen im Intervall von 1,30 bis 3,00 haben die die gleiche Wahrscheinlichkeit gezogen zu werden, so dass beispielsweise der 13 Wert R=1,78 im Durchschnitt in jeder 171. Entscheidungssituation auftreten wird (von 1,30 bis 3,00 gibt es 171 verschiedene Werte auf zwei Dezimalwerte). Welche Konsequenzen unterschiedlich hohe Projekterträge R auf Ihre Auszahlungen haben, können Sie anhand der

54 As previously mentioned the success in period two also depends on the project earnings R. R can vary between 1.30 and Every number between 1.30 and 3.00 has the same probability to be drawn, so that for instance the value R=1.78 will appear on average in every 171st decision situation (from 1.30 to 3.00 there are 171 dierent values for R). When you make your decision you do not know the amount of the project earnings R. However, for each situation each participant will receive a hint for the unknown earnings R. This hint corresponds to a number which lies in a range between R and R All numbers in this interval have the same probability to be drawn. Participants' hints are independent of the hints which the other participants receive. If for example in one decision situation R will be 1.78, every participant will get a dierent signal lying between 1.68 and Thus within a complete round you get 10 independent signals. Please note Every decision is completely independent of other decisions. That means for each decision you start again with one ECU in your bank account and have to decide to withdraw in period one or two. The money you earned at the end of each round belongs to you and is credited to your account. In each round ve new 6-person groups will be randomly formed out of the 30 participants of the experiment. Information after each round After each round every participant will be informed about the outcomes of the previous round. For all ten decisions you will be informed about: (1) Your hint for R (2) The real value of R (3) Your chosen decision (4) How many participants decided to withdraw early 14

55 (5) The payment you received After a predetermined time limit the next round will start. If you wish you can leave the information round before the end by pushing the grey OK button. If you do this, there is no chance to return to the previous information. A new round starts when you get the following message: `Please make your decision now.' Quiz and test round To ensure that you understand these instructions and have a feeling for the decision situation, we will ask you to answer three easy questions correctly before you start with the real experiment. At the beginning of the experiment you will play one test round which should clarify the experiment to you. Variation of the experiment After you have made all your decisions (eight rounds each with ten decisions), we will repeat the experiment with a small variation. We will inform you about the variation after we have played the rst eight rounds. Questionnaire After completing the whole experiment we will ask you to ll out a questionnaire that allows you to give us feedback concerning the experiment. Payment At the end of the experiment the ECUs that you earned will be converted into euros and paid in cash. One ECU corresponds to 0.07 euro or seven euro-cents. Quiz before the start of the experiment Please answer the following questions to ensure that you understood the instructions. Please assume that the true value of R is equal to 3. 15

56 Two participants withdraw their money in the rst period. What is the payo of participants in the rst and the second period? How many participants are receiving a zero payo? a) The participants who withdraw in the rst period receive 1.25 ECU and the participants who wait until period 2 receive 2.85 ECU. Nobody receives a zero payo. b) The participants who withdraw in the rst period receive 1.25 ECU and the participants who wait until period 2 receive 2.63 ECU. Nobody receives a zero payo. c) The participants who withdraw in the rst period receive 1.25 ECU and the participants who wait until period 2 receive 2.85 ECU. One person receives a zero payo. Five participants withdraw their money in the rst period. What is the payo of participants in the rst and the second period? How many participants are receiving a zero payo? a) The participants who withdraw in the rst period receive 0 or 1.25 ECU and the participant who waits until period 2 receives 0 ECU. Two participants receive a zero payo. b) The participants who withdraw in the rst period receive 1.25 ECU and the participant who waits until period 2 receives 0 ECU. One participant receives a zero payo. c) The participants who withdraw in the rst period receive 1.25 ECU and the participant who waits until period 2 receives 1.50 ECU. Nobody receives a zero payo. Four participants withdraw their money in the rst period. What is the payo of participants in the rst and the second period? How many participants are receiving a zero payo? a) The participants who withdraw in the rst period receive 1.25 ECU and the participants who wait until period 2 receive 0 ECU. Two participants receive a zero payo. b) The participants who withdraw in the rst period receive 1.25 ECU and the participants who wait until period 2 receive 1.5 ECU. Nobody receives a zero payo. c) The participants who withdraw in the rst period receive 0 or 1.25 ECU and the participants who wait until period 2 receive 0 ECU. Three participants receive a zero payo. 16

57 Practice Round Questions about the instructions If you still have any questions about the instructions please ask now. Please do not ask afterwards and we ask you not to speak during the experiment. Thank you very much for following these instructions and for participating in this experiment. D Deviations from Threshold Strategies This appendix presents three gures on the distribution of error rates. The rst gure (Figure 3) shows the average error rates per round. Errors occur roughly equally often above and below the personally optimal threshold strategy in model A. For model B, we observe more errors below the best tting threshold. However, Figures 4 and 5 reveal that most errors are observed for intermediate signals. It is therefore likely that these errors are true errors and not simply the result of a strategy that randomizes below or above some cut-o point. Average Error Rates per Round Model A_10 Model A_ error_up error_down error_up error_down Model B_10 Model B_ error_up error_down error_up error_down Figure 3: Error Rates Per Round: The gure shows the error rates per round for both models and risk sharing conditions. For the sessions A 10 and B 10, we report aggregated error rates for the rounds 1 5, 5 10, 10 15, 15 20, 20 25, 25 30, 30 35, and

58 Figure 4: Histogram for Errors (r 1 = 1.25): The gure shows the distribution of errors under the low risk sharing condition. The strategy space has been split into twelve intervals of equal size. For example the label `1.2' in model A means that all errors are considered that fall in the interval [1.2; 1.35]. Figure 5: Histogram for Errors (r 1 = 1.5): The gure shows the distribution of errors under the high risk sharing condition. The strategy space has been split into twelve intervals of equal size. For example the label `-0.1' in model B means that all errors are considered that fall in the interval [ 0.1; 0.01]. 18