QBER DISCUSSION PAPER

No. 4/2013

How to Influence Behavior in Coordination

Games

Alexander Klos, Markus Nöth

How to In�uence Behavior in Coordination Games∗

Alexander Klosa Markus Nöthb

August 2012

Abstract

Using data from three experiments with 843 subjects, we show that the set of possible

cuto� strategies a�ects behavior in global games. The in�uence on behavior is stronger

in initial responses than in repeated situations. Thus, in one-shot situations, in�uencing

the perceived set of possible cut-o� strategies may be a viable and rather cheap policy

option.

JEL classi�cation: C92, D03, G28

Keywords: coordination games, global games, level-k-models, experimental eco-

nomics

∗We would like to thank Markus Glaser, Oliver Gloede, Frank Heinemann, seminar participants at themicroeconomics workshop in Hamburg, and conference participants at the 2010 BDRM in Pittsburgh, 2010ESA World Meeting in Copenhagen, and 2010 DGF (German Finance Association) Conference in Ham-burg for helpful comments. Financial support from the Deutsche Forschungsgemeinschaft (German ScienceFoundation, grant KL 2365/1-1) is gratefully acknowledged.

aInstitute for Quantitative Business and Economics Research (QBER), Christian-Albrechts-Universitätzu Kiel, Heinrich-Hecht-Platz 9, 24118 Kiel, Germany, alexander.klos@qber.uni-kiel.de, phone: +49 431880-5551, fax: +49 431 880-5595

bLS Bankbetriebslehre & Behavioral Finance, Von-Melle-Park 5, University of Hamburg, 20146 Hamburg,Germany, markus.noeth@wiso.uni-hamburg.de

1 Introduction

Coordination games have many important applications, such as panic-based bank run models,

the investment in complementary goods, or currency attacks. This class of games typically

has multiple equilibria, and traditional models are not able to predict the probability that a

speci�c equilibrium is selected. Thus, they cannot directly be used for a policy decision.

However, there are approaches that remove the multiplicity of equilibria in coordination

games. One prominent example is the global games approach (Carlsson and van Damme,

1993). Allen and Gale (2007) recommend the global games approach as �the essential analyt-

ical tool for policy analysis� (page 93) from a theoretical point of view in potential bank runs.

Whereas the payo� structure in classic coordination games is common knowledge, e.g. players

know with certainty their payo�s if they both cooperate, the global games approach assumes

that this fundamental parameter of the game is not common knowledge. Instead agents re-

ceive a noisy private signal about the true value of the fundamental parameter. If one player

receives a very good signal about the true fundamental value, cooperation will be the opti-

mal strategy even if the other player defects (upper-dominance region). For very bad signals,

a similar argument applies and leads to defection irrespective of the other player's action

(lower-dominance regions). There exists a signal in-between at which players are indi�erent

between cooperation and defection given the beliefs about the other player's signal and his

decision rule. This signal de�nes the equilibrium cuto� value of their strategies, i.e. players

cooperate for a signal above the cuto� value and defect for a signal below the cuto� value.

This interior cuto� equilibrium is the single equilibrium in many applications. Even in

situations when other equilibria exist, e.g. if a lower or upper dominance region is missing,

the interior cuto� equilibrium is generally a more plausible prediction and can therefore be

used for comparative statics.1

The potential bene�t of the global games approach can be seen by considering bank run

models such as the classic Diamond and Dybvig (1983) model. In this model, there is a bank

run equilibrium and an equilibrium where no bank run occurs. However, the model provides

1Goldstein and Pauzner (2005) discuss this point in a setting in which the existence of an upper dominanceregion is not obvious, see especially their appendix B.

1

no information on the probability of a run. Goldstein and Pauzner (2005) show that in a

global games setting the probability of a bank run emerges endogenously out of the model.

The probability can be calculated on the basis of the ex-ante distribution of fundamentals

and the resulting optimal cuto� strategies of agents. The global games approach allows a

trade-o� between the bene�ts of risk sharing of a demand-deposit contract and the danger

of a bank run that emerges from these contracts.

Theories providing an alternative framework are non-equilibrium theories such as level-k

models (see e.g. Costa-Gomes et al. (2009) and the references therein) or cognitive hierarchy

models (Camerer et al., 2004). These models are especially successful in explaining initial

responses to new games and may be used to derive policy implications, too. As a key

assumption, players best respond to the belief that other players play a strategy that is less

sophisticated. A level-1 type �nds the best response to a level-0 type who does not behave

strategically. A level-2 type �nds the best response to a level-1 thinker and so on.2 In general,

if level-0 type players use a random cuto� strategy, the set of possible cuto� strategies will

in�uence the predicted behavior of these non-equilibrium theories, i.e. the predicted cuto�

value is positively correlated with the set of possible cuto� values in a game.

This paper tests this prediction and �nds strong support in two one-shot online experiments

and in a repeated laboratory experiment with 843 subjects. While a level-k approach captures

the e�ect of the perceived set of possible cuto� values and organizes the aggregated data fairly

well, the patterns of the individual data are inconsistent with the behavioral approach. For

example, we do not �nd pronounced mass points at the level-1 prediction in our one-shot

games, as it is often but not always the case in beauty contest games (see e.g. Camerer (2003)

for a short survey).

The results have important implications. If a policy maker or, more generally, an actor in

a coordination game is able to in�uence the set of perceived possible cuto� strategies, the

aggregated economic outcome is in�uenced as well.

Although the primary goal of this paper is to test the relationship between the set of possible

cuto� values and observed behavior, our study also allows some insights into the descriptive

2We will use the terms level-k types and level-k thinkers synonymously throughout the paper.

2

performance of the global games approach and level-k models. The related experimental

literature on global games has recently received increased attention. In experiments inspired

by the model of Morris and Shin (1998), Heinemann et al. (2004) show that the observed

cuto� values follow the comparative statics of the global games approach, although observed

cuto� values are typically lower than the theoretical predictions. Du�y and Ochs (2012)

present evidence that behavior observed in static and in dynamic games are virtually indis-

tinguishable. Moreover, these authors show that the comparative statics of the global games

approach are also con�rmed in dynamic games. Heinemann et al. (2009) estimate the pa-

rameters of a global game and recommend the global games approach as a solution concept

for one-shot games. However, initial responses in two di�erent experimental scenarios seem

to be inconsistent with the global games solution in Cabrales et al. (2007). In one of their

two scenarios, the observed behavior approaches equilibrium, while such a convergence is not

observed in the second scenario.

To the best of our knowledge, there are three recent papers, Cornand and Heinemann (2011),

Shapiro et al. (2011), and Klos and Sträter (2012) that address both level-k and global games

type of models. These authors compare level-k models to the global games approach using

considerably more di�cult games than we do. Cornand and Heinemann (2011) and Shapiro

et al. (2011) use an experimental setup with private and public information, while Klos and

Sträter (2012) investigate both approaches in a bank run framework, including many features

that are speci�c for bank runs. None of these studies provide data on pure one-shot decision

situations, an important design feature of our study that allows us to provide additional

insights on the predictive power of both theoretical approaches. Furthermore, the focus of

our study, a variation of the set of possible cuto� value strategies, has not been previously

investigated; neither by studies that consider only the global games approach, nor by the

experiments that additionally consider applications of level-k models.

The remaining paper is structured as follows. Section 2 introduces a simple coordination game

and shows how the global games approach removes the multiplicity of equilibria. Section 3

presents a pure one-shot study that implements a variation of this game. Section 4 discusses

a further one-shot experiment to address an alternative explanation. The following Section

3

5 contains the results of a laboratory study that allows for learning over time. Based on

aggregated pattern of our data, we take a closer look at the individual data in Section 6. The

last section provides a short summary of the main results and implications.

2 The Basic Coordination Problem

Coordination games are characterized by multiple equilibria due to strategic complements.

Consider the following two-player example in Table 1 (Carlsson and van Damme, 1993). For

T ≥ 1 both players choose A, while for T ≤ 0 both players choose B. But if 0 < T < 1 is

common knowledge, the game has two equilibria in pure strategies.

Player 2A B

Player 1A

T T−1T 0

B0 0

T−1 0

Table 1: Basic structure of the studied coordination problem.

In most real-world applications of coordination games, e.g. a bank run situation, the common

knowledge assumption does not hold since not everybody will know the true value of T ,

e.g. the bank and its deposits, and not everybody will assume that all other involved agents

know this true value. At best, all depositors have highly correlated beliefs about the true

value of T , as in our experimental setup.

The theory of global games formalizes this intuition. It is assumed that both players know

the a-priori probability distribution of parameter T . In our laboratory experiment, the a-

priori distribution is uniform with Tmin denoting the minimal and Tmax denoting the maximal

possible value of T . Both players observe a noisy private signal about the true fundamental

value (e.g. the perception of news on the economic strength of a bank). The error term of

the signal is uniformly distributed such that every player receives a signal from the interval

[U − ε;U + ε] with U as the true fundamental value. These assumptions imply that it is

possible to observe a signal that is smaller than Tmin and larger than Tmax.

4

In general, let player i observe signal si and let him assume that the other player uses the

cuto� strategy T ∗. He can infer that the true fundamental value is in [si − ε; si + ε]. The

signal of the other player lies in [si − 2 · ε; si + 2 · ε]. The other player can only observe a

signal of si− 2 · ε if the true fundamental value T is equal to si− ε and the noise term of the

other player is equal to −ε. Thus, the probability that the other player chooses action B is

given by

ProbGG(B|si, T ∗) =

si+2ε∑s=si−2ε

a− |si − s| · 1ba2

· 1s<T ∗ , (1)

where 1s<T ∗ is equal to one if the signal s is smaller than T ∗, and zero otherwise. a is

the number of possible discrete states in [−ε;ε], and b is the distance between two adjacent

discrete states (0.01 in our lab experiment). The expected value of choosing action A for our

player is then given by3

EV(A|si, T ∗) =1

a

si+ε∑T=si−ε

ProbGG(B|si, T ∗) · (V − 1) + (1− ProbGG(B|si, T ∗)) · V (2)

with V = Tmax if T > Tmax,V = Tmin if T < Tmin,V = T otherwise

Using the parameters of our lab experiment (see Section 5; Lab2: T ∈ [−0.2; 1.2] and ε =

0.05), there are eleven possible signals in the range [si−0.05; si+0.05]. The probability that

the other player observes si − 0.1 is therefore 1121

(= 111∗11).

4

The expected value of action B is always zero. The best response to a cuto� strategy T ∗ is

therefore action B if EV(A|si, T ∗) is negative and action A otherwise. This best response is

again a cuto� strategy. An equilibrium is reached when the best response to the assumed

cuto� strategy cj of player j is the cuto� strategy with cuto� value cj. Figure 1 plots the

3Since the relationship Tmin ≤ T ≤ Tmax must hold for each possible T in the sum, we introduce V asde�ned in the equation.

4A signal equal to si − 0.09 is observed by the other player if the true fundamental value T is equal tosi − 0.05 and the noise term is equal to -0.04 or if the true fundamental vale T is equal to si − 0.04 and thenoise term is equal to -0.05. The probability that the other player observes this signal is therefore 2

121 , andso on.

5

best response to an assumed threshold strategy of the other player in experiment Lab2. A

cuto� strategy with a cuto� value of ci = 0.50 is an equilibrium.

0Assumed Cutoff Strategyof Other Player

Best Response

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Bisecting Line

Best Response

Abbildung 1: inputGG

1

Figure 1: Best responses to assumed cuto� strategies of the other player in our lab experiment Lab2 withT ∈ [−0.2; 1.2] and ε = 0.05.

To enrich our analysis with a further theoretical approach, we will now also look at the

predictions of level-k models in the described situation. The probably best known application

of level-k models are so-called beauty contests. In their easiest form, subjects have to guess

a number between zero and one-hundred. The winner of the contest is the person guessing

a number that is closest to p times the average guess. A common choice is p = 23. The

experimental data of such games has peaks at 33 and 22 that can be explained by a level-k

model (see for example Nagel (1995)). A natural assumption for level-0 behavior in this

game is choosing a random number from the interval [0; 100]. The expected average guess

is therefore 50 and 2/3 of 50 are roughly 33, i.e. the guess of a level-1 type. A level-2 type

optimizes against a level-1 type, i.e. choosing 22, and so on.

Level-k models imply a di�erent assumption about other players' beliefs than the assumption

used in the global games approach: A level-1 type assumes that all other agents play a random

cuto� strategy, i.e. every possible cuto� strategy is equally likely. The probability that the

other player chooses action B is therefore

6

ProbL1(B|si) =

si+2ε∑s=si−2ε

a− |si − s| · 1ba2

·(

1− w − (Tmin − ε) + b

(Tmax + ε)− (Tmin − ε) + b

)(3)

with w = Tmax + ε if s > Tmax + ε,w = Tmin − ε if s < Tmin − ε,w = s otherwise.

The relationship Tmin−ε ≤ s ≤ Tmax+εmust hold for every possible s in

(1− s−(Tmin−ε)+b

(Tmax+ε)−(Tmin−ε)+b

),

but not for every possible s in

(a−|si−s|· 1b

a2

). If the signal is su�ciently far away from the

boundaries, ProbL1(B|si) is reduced to

(1− si−(Tmin−ε)+b

(Tmax+ε)−(Tmin−ε)+b

). Including b in the numer-

ator and denominator is necessary as we assume that a level-1 type believes that the other

player chooses A if he receives a signal that is exactly equal to his cuto� value. For example,

the probability that a level-0 type chooses B if he receives a signal equal to Tmax + ε is zero,

but if the level-0 thinker receives Tmin− ε, it is possible that his randomly chosen cuto� value

is Tmin− ε, which would imply an A choice. The probability of choosing B in the latter case

is therefore close to, but not equal to, one. The best response to this belief is the behavioral

prediction.5 Higher thinking types would �nd best responses to the lower types. The major

di�erence between cognitive hierarchy models and level-k models is that higher types in the

former models �nd best responses to a distribution of lower types, while the higher types in

level-k models best respond to the behavior of the next lower type.

3 Study I: A One-Shot Online Experiment With Con-

stant Global Games-Predictions

We �rst present the design and data of a pure one-shot study: subjects made exactly one

decision without any repetitions or between-design features, i.e. learning is not possible and

repeated play e�ects are ruled out by design. This design closely resembles the theoretical

framework and avoids repeated play e�ects.

5Alternative speci�cations of level-k models and their consequences are discussed in Appendix B.

7

3.1 Design

Our main goal is to test whether behavior in coordination games depends on the set of possible

cuto� values, or not. Thus, we use the a-priori distribution of T as our main treatment

variable. This study's experimental game is shown in Table 2 with S = 100. The fundamental

value T is drawn from a uniform distribution between 0 and 60 (treatment On1), 20 and 80

(treatment On2), or 40 and 100 (treatment On3). In contrast to the game presented in the

previous section, we do not use noisy private signals. Such a setup without an additional

noise term is much easier to understand and therefore more suitable for a short internet

experiment. We assume as in Du�y and Ochs (2012) that players �e�ectively transform the

game in their own minds to a related game of incomplete information� (page 98). The global

games prediction is then the interior equilibrium of the corresponding incomplete game with

a tiny error rate. The parameters of the game were chosen such that the expected payout

conditional on being selected for payment was slightly above e 100 for a player.

Player 2A B

Player 1A

T+100 TT+100 S

BS S

T S

Table 2: Basic structure of the game used in the �rst one-shot online experiment.

The a-priori distribution of T is uniform and the experiment was conducted via the internet.

Undergraduate students taking their �rst �nance course at a large German university were

invited to visit a web site that contained the instructions (see Appendix A.1). We used a

method similar to the static games of Du�y and Ochs (2012), but we implemented a pure

one-shot decision. Participants were asked to enter a number that must be an integer between

the minimum (Tmin) and the maximum (Tmax) of the possible values of T , i.e. the cuto� value

of participants' strategies is exactly speci�ed.

After the data collection phase, we randomly matched all participants into groups of two

persons. Every participant received an email with feedback about the behavior of his matched

8

experiment participant, the randomly determined value of T , and the resulting payo�s. We

paid 14 out of 299 participants and they received on average about e 111.

3.2 Theoretical Predictions

The theoretical prediction is derived as in Section 2. The main di�erence is that we are now

using a normally distributed error term, which makes the calculations easier. While applying

the global games approach, we follow the assumptions in the introductory example of Morris

and Shin (2003) and assume that the realization of T (T r) is randomly drawn from the real

line. Each player i observes a signal si = T r + εi, where εi is normally distributed with mean

0 and standard deviation σ. The signals of the players are generated independently of each

other.

From the viewpoint of player i after receiving si, the realization of T is normally distributed

with mean si and standard deviation σ. The signal of the other player, sj, is normally

distributed with mean si and standard deviation√

2 · σ.

The expected value of choosing action B is always 100, while the expected value of choosing

action A under the assumption that the other player uses a cuto� strategy with value Tj is

given by

EV(A|si, Tj) = ProbGG(B|si, Tj) · si + (1− ProbGG(B|si, Tj)) · (si + 100) (4)

= (si + 100)− 100 · ProbGG(B|si, Tj)

The probability that the other player chooses action B is given by ProbGG(B|si, Tj) =

Φ(Tj−si√

2·σ

)with Φ being the cumulative probability density function of a standard normal

distribution. Player i chooses A if EV(A|si, Tj) > 100, and B otherwise, a strategy that

is itself a cuto� strategy. An equilibrium is reached if the best response to Tj is Ti = Tj.

The interior equilibrium of the global games approach is 50 for a tiny variance σ in all three

treatments.

9

An upper and a lower dominance region are necessary to prove that the interior cuto� equi-

librium is the only equilibrium in the game. An upper (lower) dominance region of possible

values for T is a region where playing A (playing B) is the optimal decision irrespectively of

the behavior of the other player's action. If we assume that the realizations 0 and 100 do

not constitute such regions, there are further equilibria in which agents play A or play B for

every signal.

Assuming level-1-thinking, we obtain

EV(A|si) =

(1− si − Tmin + 1

Tmax − Tmin + 1

)· si +

si − Tmin + 1

Tmax − Tmin + 1· (si + 100), (5)

where si is the realization of T . The player chooses now the cuto� value T l1 that maximizes

EV(T l1) =1

61·

Tmax∑T=Tmin

1T≥T l1 · E(A|T ) + 1T<Tl1 · 100. (6)

Level-1-thinking predicts 38, 50, and 63 for treatment On1, On2, and On3, respectively.

Note that higher level types do not change the prediction of a level-k model in our one-shot

study: An agent always wants to match the other agent's choice. If a level-2 type believes

that a level-1 type uses a cuto� value of 38, the best response would be to use a cuto� value

38 as well. As a result, the level-2 type would choose A (B) whenever the level-1 type plays

A (B). The strategy of every higher thinking type therefore coincides with the strategy of a

level-1 type.

3.3 Results

The observed cuto� values in this experiment increase from treatment On1 to On3 with mean

values close to the level-1 approach predictions. In addition, the cuto� values are consistently

higher than the predicted ones. Table 3 contains the descriptive results.

Previous experiments have suggested that cuto� values are biased towards cooperation,

i.e. they are typically smaller than the global games prediction. Since this is not the case

10

All Data

Treatment Mean Median N

On1 [0; 60], S = 100 40 45 114On2 [20; 80], S = 100 54 56 97On3 [40; 100], S = 100 71 70 88

Table 3: Descriptive results for the three treatments (On1, On2, On3) of the �rst one-shot online experi-ment.

here, the previous results may partly be due to repeated play e�ects, i.e. strategic signaling

to in�uence other subjects' behavior in subsequent rounds. An alternative or complementary

explanation could be that risk aversion leads to higher cuto� values. Although the expected

value of earnings is relatively small, the payo� of a player depends on a single decision, mak-

ing payo�s more volatile than in traditional laboratory experiments. We explore risk aversion

as a possible explanation in the next Section 4.

Due to data censoring, standard non-parametric tests for di�erences between the treatments

(such as Wilcoxon tests) cannot be used without modi�cations. For example, a cuto� value

of 15 in treatment On1 cannot be compared to one of 20 in treatment On2 because 20 is the

minimum in treatment On2 and thus the true cuto� value may be smaller than or equal to

20. However, we do not know whether it is smaller than, equal to, or greater than 15. As a

consequence, we use the smallest possible cuto� value that can be observed in both treatments

(cmin). Whenever we identify a cuto� value smaller than cmin, we adjust this cuto� value

to be equal to cmin. In our example, a cuto� value of 15 in treatment On1 would result in

a tie with a cuto� value of 20 in treatment On2 in the modi�ed Wilcoxon test. Similarly,

we de�ne cmax as the highest possible cuto� value that can be observed in both treatments

and adjust the observed cuto� values accordingly. Note that the modi�ed Wilcoxon test

is a conservative test that is biased towards non-rejection of the null hypothesis of equal

distributions. Nevertheless, we can reject the null hypothesis at the 1%-level (cmin = 20,

cmax = 60, p=0.0059) in treatments On1 and On2. The comparison of treatments On2 and

On3 leads to a rejection of the null hypothesis, too (cmin = 40, cmax = 80, p=0.0001).

11

In addition to this non-parametric analysis, we also performed a parametric analysis. Censored-

normal regressions are used that allow censored values to vary from observation to observa-

tion. The dependent variable consists of subjects' chosen cuto� values. Explanatory variables

are dummies for treatments On1 and On3. The reference category is treatment On2. Table

4 contains the results. The estimated models in Table 4 reveal that cuto� values in treatment

On1 are 10.53 units smaller than in treatment On2 whereas cuto� values in treatment On3

are 16.25 units higher as in treatment On2.

(1)Cuto� Value

Treatment On1 (d) -10.53***(3.899)

Treatment On3 (d) 16.25***(4.112)

Constant 55.41***(2.836)

Observations 299Pseudo R2 0.017

Standard errors in parentheses

* p < 0.10, ** p < 0.05, *** p < 0.01

Table 4: Results of censored normal regressions with subjects' cuto� values as dependent variables.

Summing up, the ex-ante distribution of T matters in the �rst one-shot experiment as hy-

pothesized.

4 Study II: A One-Shot Online Experiment With Con-

stant Level-k-Predictions

4.1 Design

Using a simple anchoring-and-adjustment heuristic would lead to an alternative explanation

of our Study I results: Subjects may anchor on the midpoint of the speci�ed interval and

12

then adjust their decision depending on their beliefs and preferences. Such an anchor would

introduce the reported correlation between cuto� values and the pre-speci�ed interval of T .

To address this concern, we modify the design such that even when we shift the a-priori

interval of T, the level-k prediction always predicts the same cuto� value. At the same

time, the interior equilibrium of the global games prediction decreases as the interval shifts

upwards. Thus, if the observed cuto� values still increase, the results in Studies I and

II could be explained with an anchoring and adjustment heuristic but not with level-k or

global games. We use the same game as in Study I (see Table 2) but now the safe option

S is varied as well (S = 100: study I). With level-k thinking, option B becomes more

attractive as we move the pre-speci�ed interval for T upwards. However, option B becomes

obviously less attractive as we decrease the safe payo� S. We use three combinations of

pre-speci�ed intervals and safe payo�s S that lead to a predicted cuto� value of 50 for the

level-k approach. These combinations are labeled treatment On4 (T ∈ [0; 70];S = 121),

treatment On5 (T ∈ [15; 85];S = 100), and treatment On6 (T ∈ [30; 100];S = 79). The

interval length has been increased slightly relative to the interval length in the �rst one-shot

experiment. This makes it more likely that a di�erence between treatments will be detected

by the modi�ed Wilcoxon test because more information enters the rank test. The global

games approach with a tiny σ, as in Section 3.2, predicts decreasing cuto� values (On4: 71,

On5: 50, On6: 29) if we consider only the interior cuto� equilibrium. Given that subjects

are constrained in some treatments, the predictions are 70 (On4), 50 (On5), and 30 (On6).

In addition to the above, we also address the conjecture that risk aversion is important in

explaining variation in individual cuto� values. A quick and easy way to build a proxy for

risk tolerance is to ask directly. We use the general risk attitude question that has been

used in the German Socio-Economic Panel (�How would you rate your willingness to take

risks in �nancial matters ?�). Subjects respond on a 11-point scale to this question. This

question can be easily and quickly answered in an online experiment, but subjects are not

paid in an incentive compatible manner. Dohmen et al. (2011) have shown that the answers

to this question correlate with behavior in economic experiments using incentive-compatible

payments.

13

Subjects were recruited from a mailing list of a large German university. All other aspects of

the design, including the payment procedure, were identical to the �rst one-shot experiment.

285 students participated and 14 randomly selected students were paid on average e 108.

4.2 Results

All observed cuto� values are greater than predicted by the level-k approach but are rea-

sonably close to the theoretical prediction. This result is consistent with the �rst one-shot

experiment. The mean (median) cuto� values for the three treatments are similar, ranging

from 52 to 58 (50 to 60). None of these di�erences is signi�cant using our modi�ed Wilcoxon

test. The p-values for a comparison of treatments On4 and On5, of treatments On4 and

On6, and of treatments On5 and On6 are 0.6511, 0.8143, and 0.8241, respectively. Table 5

contains the descriptive results of Study II.

All Data

Treatment Mean Median N

On4 [0; 70], S = 121 52 55 93On5 [15; 85], S = 100 54 50 93On6 [30; 100], S = 79 58 60 99

Table 5: Descriptive results of the second one-shot online experiment.

A parametric analysis leads to a similar conclusion, see Table 6. The variables in these

censored normal regression are generated as in Study I (see Section 3.3). In addition, we now

control for risk aversion using the answers to the risk attitude question. The risk tolerance

dummy is generated using the mean split of all responses, i.e. a subject is classi�ed as risk

tolerant if the self-assessment on the 11-point scale is higher than the sample mean (4.296).

As mentioned, level-k predictions are the same for all three treatments. If the results in the

previous experiments were driven primarily by an anchoring-and-adjustment heuristic, we

would again expect increasing cuto� values from treatment On4 to On6, in contrast to the

level-k prediction. Consistent with the non-parametric analysis, the treatment dummies are

14

(1) (2)Cuto� Value Cuto� Value

Treatment On4 (d) 3.254 3.010(4.301) (4.248)

Treatment On6 (d) 0.593 0.634(4.167) (4.116)

Above Average Risk Tolerance (d) -9.167***(3.431)

Constant 55.12*** 59.19***(2.991) (3.329)

p-value H0: TOn4 = TOn6 0.533 0.573

Observations 285 285Pseudo R2 0.000 0.004

Standard errors in parentheses

* p < 0.10, ** p < 0.05, *** p < 0.01

Table 6: Results of censored normal regressions with subjects' cuto� values as dependent variables instudy II. The row �p-value TOn4 = TOn6� shows the p-value of a test for equality of the treatment dummycoe�cients (On4 and On6).

not signi�cant, neither for treatment On4 nor for treatment On6, which rules out an impor-

tant anchoring-and-adjustment e�ect. Risk tolerance helps to explain variation in individual

cuto� values. The cuto� values of risk tolerant individuals are signi�cantly smaller.

5 Study III: A Repeated Laboratory Experiment

The two one-shot studies have demonstrated that the a-priori distribution of the state variable

T a�ects behavior and that this is not primarily caused by an anchoring-and-adjustment

heuristic. However, behavior might change if the same situation is faced repeatedly and/or

with similar but often not identical agents.

Thus, we use a random rematching design to evaluate the hypothesis that the set of possible

cuto� values in�uences behavior in a repeated game situation in the lab. An experimental

setup is used that is broadly comparable to previous lab investigations of the global games

approach (see especially Heinemann et al. (2004)). We assume that the a-priori distribution

of T is uniform, and we use the game shown in Table 1 of Section 2.

15

5.1 Design

Table 7 contains the prediction of both theoretical approaches for our experimental variations.

Note that level-k-thinking predicts increasing cuto� values whereas the interior equilibrium of

the global games approach predicts non-increasing cuto� values. For example, in treatment

Lab2, T is drawn from the interval [-0.2; 1.2]. At the beginning of each round, the true

fundamental value of T is randomly determined. Players receive private signals about the

true fundamental value T that are drawn independently from a uniformly distributed interval

[T − ε;T + ε]. In our experiment, ε is set to 0.05 and is common knowledge.6 We consider

only values for T and ε with two digits after the decimal point.

Treatment [Tmin; Tmax] level-1-thinking Global Games

Lab1 [-0.2; 0.6] 0.33 [0.48; 0.53]Lab2 [-0.2; 1.2] 0.5 [0.48; 0.53]Lab3 [0.4; 1.2] 0.67 [0.48; 0.53]

Table 7: Theoretical predictions of the level-1-approach and the global games approach for the three di�erenttreatments of the repeated lab experiment with ε = 0.05.

It is an equilibrium to always choose option B in treatment Lab1 (a cuto� value of 0.65) and

to always choose option A in treatment Lab3 (a cuto� value of 0.35) because there is either

no upper or no lower dominance region. The global games approach predicts no di�erence

if we restrict our analysis to situations where global games decision maker sometimes choose

A and sometimes choose B. There exists no other equilibrium with a single cuto� value.

A few aspects of our design require some explanations: First, the interior global games

prediction is not unique. This is a consequence of the discrete nature of the experimental

game. If we allowed for three digits after the decimal point, the best response to the cuto�

strategy 0.48 would be 0.483. Second, our game is a simple game compared to existing

studies on global games. The simplicity should make it di�cult for our treatment variation to

work. Third, we do not consider higher types of hierarchical thinking in deriving hypotheses.

The behavioral prediction in Table 7 is therefore dubbed level-1-thinking. For example,

6We chose a relatively small ε because the upper bound in treatment Lab1 (0.6) and the lower bound intreatment Lab3 (0.4) are close to the global games solution. A higher error term would have increased theimportance of the calculations at the boundaries.

16

we could think of a level-2 type who �nds the best response to the behavior of a level-

1 type. Considering higher types means stepwise approaching the global games solution

until the interior cuto� equilibrium is reached. Since the strategy of a level-n type is only

marginally di�erent from the strategy of a level-(n-1) type,7 it would be di�cult (and in some

cases impossible) to clearly distinguish types in an empirical analysis. As higher thinking

types choose cuto� values that approach stepwise the interior global games equilibrium, their

presence would make it again harder for our experimental variation to work.

In the lab experiment, subjects played the described situation repeatedly. They received a

private signal about the true fundamental value T at the beginning of each round and chose

either action A or B.8 The participants received feedback, i.e. the other player's decision, after

each round. All payo�s were accumulated, but the current sum of pro�ts was not provided.

In this experiment, we used ECUs (Experimental Currency Unit) that were converted at an

exchange rate of 1 ECU = e0.50.

We used a random rematching design to mitigate e�ects caused by the repeated nature of the

experimental game. Six persons participated in one session, and they were randomly divided

into three independent two-person groups at the beginning of each round. Every session

started with a publicly announced a-priori distribution for the value of T . After 40 rounds,

we changed the interval length for the value of T , and the participants in the sessions used the

changed interval length for another 40 rounds. Subjects were not informed at the beginning

of the experiment that the interval size would be changed. A comparison of only the �rst 40

rounds of each session is thus close to a pure between subject design (see Appendix A.2 for

the instructions).

The theoretical considerations in Section 2 suggest that the probability of the other player

choosing action A is the key parameter that di�ers between the global games and the level-1-

thinking approach. Therefore, we elicited this probability in about 50% of our experimental

7For example, the best response to a cuto� value of 0.33 (0.67) in treatment Lab1 (Lab3), i.e. the strategyof a level-1 type, is 0.35 (0.65), i.e. the strategy of a level-2 type.

8The fact that subjects made binary choices in the lab experiment resembles the elicitation in the majorityof previous experimental studies on global games. For obvious reasons, this elicitation procedure is notsuitable for an online experiment. However, although these di�erent elicitation modes appear to be a majordi�erence at the �rst glance, Du�y and Ochs (2012) show that the estimated cuto� values from binary choicedata are a good approximation of explicitly elicited cuto� values.

17

sessions. The results based on the belief data largely mirror the results of the thresholds in

the following subsection. Details can be found in Appendix D.

5.2 Results

258 students from two large German universities participated in our 43 sessions with and

without belief elicitation (see Table 8 for details) and they made 20,640 binary decisions.

Followed by Followed by Followed bytreatment Lab1 treatment Lab2 treatment Lab3

Started with treatment Lab1 4 (4) 4 (3)Started with treatment Lab2 5 (4) 4 (2)Started with treatment Lab3 4 (3) 4 (2)

Table 8: Overview of all 43 lab sessions. Each session consists of six persons who were randomly rematchedat the beginning of each round. The �rst number in each cell is the number of sessions in which we elicitedbeliefs explicitly about the other player's actions. The number of sessions without belief elicitation is givenin brackets.

We de�ne an experimental �condition� as the sum of 240 individual decisions that are made

when the six subjects of a session were exposed to one particular treatment. For example,

if six subjects participated in a session that started with treatment Lab1 and continued

with treatment Lab2, then these six subjects created two observations on the experimental

condition level.

First, we calculate the fraction of A choices. Figure 2 shows, for each treatment, the pro-

portion of observed A choices if signals fall within the intervals [-0.25; -0.16], [-0.15; -0.06],

[-0.05; 0.04], . . . , and [1.15; 1.25].

Figure 2 shows that the proportion of A choices tends to be highest in treatment Lab1,

followed by treatment Lab2 and treatment Lab3. This feature of the aggregated data is

consistent with the level-k approach.

The cuto� values are estimated using a simple error method as in Palfrey and Prisbrey

(1997). We include all decisions of all participants in one condition i and count all decisions

that are inconsistent with a given cuto� value ci. The cuto� value leading to the smallest

18

Figure 2: Proportion of A choices in 15 subintervals for each treatment using interval midpoints as labels.

percentage of decisions e∗i that are inconsistent with this cuto� value de�nes the best �tting

c∗i . If multiple cuto� values are associated with the same value of e∗i , c∗i is the average of the

smallest and the largest best �tting cuto� value.9

The analysis of the comparative statics con�rms that cuto� values increase when moving

from treatments Lab1 to Lab3, in line with the level-k hypothesis. The mean condition

cuto� values are 0.15 for treatment Lab1, 0.23 for treatment Lab2, and 0.44 for treatment

Lab3. The di�erence in the mean cuto� values is especially pronounced if we restrict the

analysis to the �rst 40 rounds. The mean cuto� values are 0.12, 0.24, and 0.46 for the

treatments Lab1 to Lab3, respectively. Table 9 reports the details of order e�ects. There is

only a small di�erence between the estimated cuto� value in treatments Lab1 and Lab2 if

participants have not started with this condition.

An OLS regression analysis using the experimental condition cuto� values supports the de-

scriptive statistics (see Table 10). We exclude treatment Lab3 data since the lower bound

9The mean (median) distance between the smallest and the largest best �tting cuto� value is 0.0136(0.01). The mean corresponding error rate is 6.4% (15.38 out of 240). The median error rate is 5.8% (14 outof 240). Given that these error rates partially re�ect individual heterogeneity and/or learning, we considerthese error rates to be low. Furthermore, we also estimated cuto� values based on the 40 individual responsesof one subject in an experimental condition. The mean (median) error rate of individual cuto� values thatleads to the best �tting cuto� value is 1.24 (1). Over 40% of all individual cuto� values are associated witha zero error rate, despite the fact that �errors� could result from learning. Overall, the evidence is consistentwith previous research: Subjects consistently play cuto� strategies in global games type coordination games(Heinemann et al., 2004).

19

Treatment Lab1 Treatment Lab2 Treatment Lab3

Start Start Start Start Start Start# sessions 15 16 15 14 13 13Mean CCV 0.12 0.18 0.24 0.21 0.46 0.41

Mann-Whitney z 2.174 -0.742 -1.103p-value 0.0297 0.4581 0.2701

Table 9: Descriptive statics on experimental condition cuto� values � Start : started with this treatment;Start: started not with this treatment; Mean CCV: mean of condition cuto� values.

of the fundamental value T is 0.4 and thus many participants would choose a lower cuto�

value if they were unconstrained. The dependent variable in Table 10 is the estimated cuto�

value of one experimental condition. Every session consists of two experimental conditions

and most sessions therefore enter the regressions twice. Because the descriptive evidence

suggests an interaction, we use three interaction dummies between the treatment and the

ordering within a session: Lab1/Start, Lab2/Start, and Lab2/Start. This leaves Lab1/Start

as the baseline case and all e�ects are reported relative to this baseline. Standard errors are

corrected for clusters on the session level (see Model (1) in Table 10).

(1) (2)Condition Cuto� Values Condition Cuto� Values

Robustness Check

TLab1 / Not Started With This Condition 0.0580** 0.0576**(0.0255) (0.0249)

TLab2 / Not Started With This Condition 0.0848** 0.0713*(0.0353) (0.0364)

TLab2 / Started With This Condition 0.121** 0.110**(0.0466) (0.0462)

Constant 0.122*** 0.128***(0.0130) (0.0122)

Observations 60 60R2 0.125 0.101

Standard errors in parentheses

* p < 0.10, ** p < 0.05, *** p < 0.01

Table 10: Results of OLS regressions with the experimental condition cuto� values as dependent variables.Standard errors are clustered on the session level because many sessions enter the regressions twice, leadingto 60 observations from treatments Lab1 and Lab2. There are 43 clusters, one cluster for each session.Observations from treatment Lab3 are not included.

20

Condition cuto� values are smaller in treatment Lab1 and there also exists a signi�cant

order e�ect. While the di�erence between estimated cuto� values is a signi�cant 0.121 if

the sessions have started with the treatment, the di�erence is only 0.0268 (0.0848 - 0.0580)

and insigni�cant if the sessions have not started with the treatment. This observation is

consistent with the view that experimental variations that should not work according to the

global games approach only yield an e�ect if the experimental situation is relatively new for

participants.

As noted above, estimated cuto� values are not unique. Therefore, we compare the highest

best �tting cuto� value for treatment Lab1 with the lowest best �tting cuto� value from

treatment Lab2 as a further robustness check. All other speci�cations are identical to re-

gression model (1). The estimated di�erences between cuto� values of treatment Lab1 and

treatment Lab2 are smaller (0.110 and 0.0137 instead of 0.121 and 0.0268), but the overall

pattern does not change (see Model (2), Table 10).

The results in Table 9 indicate cuto�s values that are substantially di�erent from both the

global games and the level-k predictions. However, Figure 3 shows a slow, but steady increase

of cuto� values over time in treatments Lab1 and Lab2, i.e. they may eventually converge to

the global games prediction. Note, however, that learning in this experiment is rather slow

because of limited informative feedback. In treatment Lab3 cuto� values do not change over

time but are fairly close to the global games prediction.

If we disaggregate the data and compare the results between treatments before and after the

treatment change, subjects tend to start with similar cuto� values in treatments Lab1 and

Lab2 if the session has not started with the considered treatment. In treatment Lab2, the

cuto� value after the treatment change is lower in the �rst rounds and increases over time,

as in treatment Lab1.

In general, the sensitivity of early responses to the set of possible cuto� values is consistent

with a level-k approach. However, the observed values are always substantially smaller than

predicted by the level-k approach. Nevertheless, the level-k approach provides a better �t of

the data than the global games approach. If we classify our subjects based on a maximum-

likelihood analysis (see Appendix C for details), the estimated proportion of level-1-thinkers

21

Figure 3: Estimated session cuto� values for the subperiods 1�10, 11�20, 21�30, and 31�40.

22

is roughly 68%. The remaining 32% of our subjects are more likely to be of the global

games type. Moreover, the proportion of level-1-thinkers in the population varies across

treatments (between 99% in Lab1 and 18% in Lab3) suggesting that a level-k model has a

higher predictive power than the global games approach but is not a perfect model and it

fails in particular in treatment Lab3.

Furthermore, the e�ect of the experimental variation decreases after the treatment change.

This can be interpreted as evidence against an experimenter demand e�ect. Such an e�ect

would predict that subjects respond to the set of possible cuto� values because they think that

the experimenter is expecting such a behavior. However, when the experimental variation

was most salient to them, i.e. when the treatment was changed, behavior is virtually identical

under the sets of possible cuto� values [-0.2; 0.6] and [-0.2; 1.2].

6 Individual Behavior

So far, we have analyzed the aggregated pattern of the data that is largely consistent with

a level-k approach. In this section, we will look at the distribution of chosen cuto� values:

The upper part of Figure 4 contains the histograms of chosen cuto� values in the one-shot

online experiments.

In line with the evidence on level-k models from beauty contest games, we expect pronounced

mass points at the level-k predictions. However, with the exception of On2 and On5 where

level-k prediction, global games prediction, and a round number e�ect for 50 coincide, we

do not observe such mass points. Instead, the increase of the average and median cuto�

value in the �rst study (On1 to On3) is driven by a changing proportion of the extreme

cuto� strategies, i.e playing either always A or always B regardless of the signal. This

distributional feature of the data is inconsistent with the level-k approach.

Figure 4 also contains the corresponding histograms for the second study (On4 to On6). The

distribution of responses shows again a similar inconsistency with the level-k approach as in

the �rst one-shot experiment. As the aggregated behavior is predicted reasonably well, the

23

Figure 4: Histograms of cuto� values are shown. The chosen cuto� values in the online experimentsare shown in the upper part of this �gure. The estimated cuto� values based on the individual data fromlaboratory experiments are shown in the lower part of the �gure.

24

individual data is lacking the peaks that are expected in level-k models. The consistency of

level-k models on the aggregated level comes again mainly from the fact that the treatment

variation changes the proportion of subjects with extreme cuto� values.

The lower part of Figure 4 shows the histograms of individual cuto� values for the lab

experiment. In treatments Lab1 and Lab2, we observe only a few extreme cuto� values that

imply always action A or always action B, in contrast to the one-shot gambles. The majority

of individual cuto� values is now relatively close to the experimental condition means.

As can be seen in the aggregated analysis, many participants choose to cooperate almost

always in treatment Lab3. Furthermore, cuto� values are less disperse if the session has not

started with this condition, suggesting that experience leads to improved coordination among

agents.

Figure 4 suggests that extreme responses are much more likely in pure one-shot gambles

than in games that are repeatedly played with a similar player composition. However, there

are several di�erences between the lab and the online experiments. The lab experiment was

designed to provide an experimental test of the hypothesis that the set of cuto� strategies

in�uences behavior within a framework that has been previously used to investigate the

theory of global games. The main purpose was not to compare one-shot and repeatedly

played scenarios in a rigorous way. We hope that further research will investigate this issue

in more depth using a design that minimizes di�erences between one-shot and repeated

studies.

7 Summary, Conclusions, and Further Research

In this paper, we demonstrate how the economic outcomes of global games type coordination

games can be in�uenced. Based on the prediction of a level-k model, we �nd that the

set of possible cuto� values has a strong in�uence on behavior in initial responses but not

if su�ciently large learning possibilities are provided. Our results are not driven by an

25

anchoring-and-adjustment heuristic. From a political point of view, the results suggest that

trying to in�uence the perceived set of cuto� values is a policy option.

Even though behavioral models are a valuable alternative for predicting behavior in pure

one-shot situations, potential applications such as bank runs, currency attacks, or investment

decisions in complementary goods exhibit speci�c features that may change our conclusions.

For example, the number of players are either only a few (investment decisions) or many

(currency attacks or bank runs). Such features should be investigated in further research.

Finally, while the level-k approach can explain the aggregate pattern of the data better

than the global games approach, we do not observe pronounced mass points at the level-k

predictions. If the goal is to organize the individual data perfectly, either the level-k approach

needs some re-speci�cation or a completely di�erent theoretical approach is needed. We leave

this for further research.

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28

Online-Appendix

A Instructions

A.1 One-Shot Online Experiment

The following instructions were used in treatment On2.

Instructions

Welcome to this web experiment that is being conducted by and . We

are interested in studying your behavior in a series of decision-making situations. At the end

of the experiment, we will randomly select 14 participants who will receive a payment that

depends on the decisions they have made in the experiment.

Your decision situation

Your decision is a strategic one, i.e. the result depends on your decision as well as on the

decision of another player. After collecting all decision data, we will randomly split up all

participants into groups of two players. For example, if there are 200 participants, there will

be 100 groups of 2 participants each. You will thus be playing with a randomly chosen person

whom you do not know and with whom you cannot communicate. After you and the other

player have submitted your decisions, we will randomly choose an integer T between 20 and

80 (from, up to, and including). Each integer in this interval is equally likely to occur. As

the interval contains a total of 61 integers, the probability of guessing the drawn number in

this interval is exactly 1/61.

The number T is the same for you and the other player. Neither you nor the other player know

the actual value of the number T . Only the distributional information is public knowledge

(any integer between 20 and 80 is equally likely to occur).

We ask you for a unique threshold value that has to be an integer between 20 and 80. You

can enter this number in the text box at the end of these instructions. If the randomly chosen

29

number T is larger or equal to your threshold value, you choose action A. If the randomly

chosen number T is smaller than your chosen threshold value, you choose action B.

If your threshold value implies action A, you will receive a payment of T + 100 e if the

other player has also chosen A. If the other player has chosen action B, you will receive a

payment of T e. If your threshold value implies action B, you will receive a payment of 100

e independently of the other player's action.

• Example 1: You have entered 35 as your threshold value. The randomly determined

number T is 40. As the number T is greater than your threshold value, your chosen

action is A. Your co-player chose action B. You will receive a payment of 40 e.

• Example 2: You entered 35 as your threshold value. The randomly determined number

T is 40. As the number T is greater than your threshold value, your selected action is

A. Your co-player has also chosen action A. Your payment is 140 e.

• Example 3: You have entered 55 as your threshold value. The randomly determined

number T is 60. As the number T is greater than your threshold value, your chosen

action is A. Your co-player has also chosen action A. You will receive a payment of

160 e.

• Example 4: You have entered 65 as your threshold value. The randomly determined

number T is 60. As the number T is less than your threshold value, your chosen action

is B. Your co-player has chosen action A. You will receive a payment of 100 e.

Payment

After completing the web experiment, we will randomly choose 14 participants who will be

paid based on the above stated rules.

Short quiz

In the following, we would like to ask you three short questions: Answering them will improve

your understanding of the decision making scenario.

1. Let's assume T = 50 and you have chosen a threshold value of 40. The other player

chooses action A. What will be your payment ?

30

• 50

• 100

• 150

2. Let's assume T = 50 and you have chosen a threshold value of 60. The other player

chooses action B. What will be your payment ?

• 50

• 100

• 150

3. Let's assume T = 50 and you have chosen a threshold value of 35. The other player

chooses action B. What will be your payment ?

• 50

• 100

• 150

Enter your threshold value

Which threshold value (between 20 and 80) would you like to choose?

A.2 Repeated Laboratory Experiment

The following instructions were used in treatment Lab3 with elicitation of the probability that

the other player chooses A.

Instructions

Welcome to this experiment that is conducted by and . We

are interested in studying your behavior in a series of decision-making situations. At the end

of the experiment, all participants will receive a payment based on their decisions during the

experiment. Thus, you should consider your decisions carefully as the payment that you will

receive at the end of the experiment depends on it. You will only use the computer in front of

you. The experiment is �nanced by the Deutsche Forschungsgemeinschaft (German Science

Foundation).

31

General information

Your decision situation

Your decision is a strategic one, i.e. the result depends on your decision as well as on the

other player's decision. As mentioned before, at the beginning of each round, new groups

of two are randomly drawn among the six participants. Thus, you play with a randomly

selected person in this room with whom you cannot communicate and who cannot identify

you.

At the beginning of each round, a number �T� is randomly picked. The number �T� is

between 0.40 and 1.20 (from, up to, and including). All numbers, rounded to two decimals

in the interval of 0.40 to 1.20 have the same probability of being selected, such that the value

T = 0.50, for example, can occur on average in every 81st decision situation (from 0.40 to

1.20 there are 81 di�erent values with two decimals).

The T value is the same for you and the other player. Every participant receives an indepen-

dently drawn signal about this value T . These signals are drawn from the interval between

T−0.05 and T+0.05. All numbers rounded to two decimals within this interval have the same

probability of being picked. Note, that one player's signal is drawn independently from the

other player's signal. If, for example T = 0.50, both participants will receive independently

drawn signals (with replacement) between 0.45 and 0.55.

After you have received your signal, you must choose between actions A and B. If you

choose action A, you will receive a payment of T ECU (experimental currency unit) if the

other player has also selected A. If the other player has selected action B, you will receive

T − 1 ECU.

If you select action B, you will always receive a payment of 0 ECU regardless of the action

of the other player.

• Example 1: The randomly selected number T is 0.70. Based on your signal, you selected

action A. The other player selected action B. Your payment is -0.30 ECU.

32

• Example 2: The randomly selected number T is 0.70. Based on your signal, you selected

action A. The other player selected action A. You are paid 0.70 ECU.

• Example 3: The randomly selected number T is 0.90. Based on your signal, you selected

action A. The other player also selected action A. You are paid 0.90 ECU.

• Example 4: The randomly selected number T is 0.90. Based on your signal, you selected

action B. The other player selected action A. You are paid 0 ECU.

During each round you will also be asked to estimate the probability that the other player

will select action A.

Payment

At the end of the experiment the ECU you earned will be converted into Euro and paid out.

The exchange rate is 1 ECU = 0.50 e.

Experiment variation

After you have made 40 decisions (40 rounds, each with 1 decision), a variation of the

experiment will be introduced. We will inform you about the variation after the �rst 40

rounds.

We will start the experiment with two test rounds so that you can familiarize yourself with

the program and the procedures. If you have any questions, please ask now or after the test

rounds. Once the experiment has been started, no questions will be answered!

B Alternative Model Speci�cations

In this paper, an application of level-k models is presented in an attempt to organize the

experimental data of more than 800 participants. In general, all level-k models depend on

the speci�cation of the level-0 types. In this section, we discuss alternative speci�cations and

their main consequences.

33

A common choice for level-0 types is that they choose each possible action with the same

probability.10 In our games, this would imply that level-0 types would choose action A and

action B with equal probability independent of further information. The best response given

this belief can be easily seen by calculating the expected value of action A if the signal is

su�ciently far away from the boundaries:

EV(A|si) =1

a

si+ε∑T=si−ε

1

2· (T − 1) +

1

2· T =

1

2· (si − 1) +

1

2· si = si −

1

2(7)

A level-1 type should play a cuto� value of 12in this speci�cation. Furthermore, the predicted

cuto� value does not depend on the prior distribution of T . Higher types in a level-k model

would also choose a cuto� value of 12because 1

2is the best response to a cuto� value of 1

2.

Thus, the predictions coincide with the global games approach. In our speci�cation, higher

thinking types approach the global games equilibrium stepwise. This convergence is slow if

ε is small. Figure 1 (on page 6) illustrates this process. Starting with the level-1 behavior

in treatment 3, a cuto� value of 0.67, the best response is 0.65, which is also the level-2

prediction. The best response to 0.65 de�nes the level-3 behavior and so on.

In our speci�cation, a level-1 type assumes that all possible cuto� values are equally likely.

However, it is not clear if level-1 types consider dominated strategies of level-0 types in their

optimization. For example, in our laboratory experiment, it is always optimal to play B for

signals lower than -0.05, independent of the other players' decision. If level-1 types assume

that level-0 types do not play dominated strategies, the behavioral predictions will change

only a bit or not at all. In the lab experiment, the level-1 predictions change to 0.38 in

treatment Lab1 and 0.63 in treatment Lab3. The prediction for treatment Lab2 does not

change. No predictions are changed for the �rst online experiment since the possible cuto�

values are all included in the pre-speci�ed interval for T . The single prediction that changes

in the second one-shot experiment is the prediction for treatment On4 (54 instead of 50).

Given that these di�erences are fairly small, we cannot answer the question whether subjects

behave as if dominated cuto� values enter their level-1 calculations, or not.

10However, deviating from this assumption is not uncommon in the recent literature on level-k models, seefor example Crawford and Iriberri (2007).

34

Next, we analyze the dispersion of signals ε. Instead of using the ε given in the experiment

(0.05), we can estimate the ε from the data that gives the best description of the data. If

this estimated value is large, agents act as if there is a high degree of uncertainty about the

strategic behavior of the other player. In a global games framework, the dispersion of signals

needed to �t a global game is large (Heinemann et al., 2009). If we restrict our analysis

to the global games approach, we can replicate this �nding. We use maximum-likelihood

estimations to determine the best �tting value of ε. It is assumed that all decision makers

are of the global games type (or in the terminology of the maximum likelihood estimations

in Appendix C: π = 0 is assumed). A high value of 0.34 �ts the behavior best.

In general, increasing εmeans approaching a level-1 model where the level-1-thinker optimizes

against non-dominated random threshold strategies. Figure 5 shows theoretical beliefs for

di�erent values of ε and for the level-1 models that exclude or include dominated threshold

strategies in their calculations. Moreover, the �gure illustrates the interesting connection

between level-k models, in which level-0 types optimize against random cuto� values, and

the global games approach.

Figure 5: Predicted beliefs of di�erent models under di�erent assumptions and parameters. It is assumedthat 0.5 is the single global games cuto� equilibrium of the discrete game.

35

However, higher values of ε lead only to marginally smaller values of π in a mixed maximum

likelihood model, using the level-1 model in which the level-1 thinker optimizes against ran-

dom cuto� values coming from the interval [TMin − ε;TMax + ε]. The minimal π = 0.659 is

reached for ε = 0.28.

C Maximum Likelihood Estimation

In this section, we use a maximum likelihood estimation to classify subjects based on their

behavior as level-1 or global games types. Subject j makes n choices in the experiment.

Given a signal si in decision situation i, the probability that the other player chooses action

A is Prob(si)L1 (level-1 approach) or Prob(si)

GG (global games approach). The expected

value of playing A is then

EV(si)L1 =

1

a

si+ε∑T=si−ε

Prob(si)L1 · V + (1− Prob(si)

L1) · (V − 1) (level-1-approach) or

EV(si)GG =

1

a

si+ε∑T=si−ε

Prob(si)GG · V + (1− Prob(si)

GG) · (V − 1) (global games approach)

with V = Tmax if T > Tmax,V = Tmin if T < Tmin,V = T otherwise.

The expected value of playing B is always zero. Assuming a multinominal logit model, the

probability that subject j plays A is given by

ChoiceProbA(si)L1 =

exp(λ · EV(si)L1)

exp(λ · EV(si)L1) + exp(λ · 0)(level-1-approach) or

ChoiceProbA(si)GG =

exp(λ · EV(si)GG)

exp(λ · EV(si)GG) + exp(λ · 0)(global games approach).

Random choices are predicted as λ, the error term, approaches zero while a high λ indicates

a low error component. The probability that a player chooses B is (1− ChoiceProbA(si)L1)

or (1 − ChoiceProbA(si)GG). Let ChoiceProb(si)

L1 (ChoiceProb(si)GG) be the probability

that a choice for signal si is generated by the level-1 (global games) model. The proba-

36

bility of a speci�c choice history of subject j is∏n

i=1ChoiceProb(si)L1 (level-1 approach)

or∏n

i=1ChoiceProb(si)GG (global games approach). Let π be the share of level-1-thinkers

in the population. The log-likelihood that we observe the choice history in the experiment

with N participants under this set of assumption is∑N

j=1 ln(π ·∏n

i=1ChoiceProb(si)L1 +

(1− π) ·∏n

i=1ChoiceProb(si)GG). Note that each participant enters this estimation in both

treatments she was exposed to.

Full Sample Full Sample Lab1 Lab2 Lab3≤ 3 Rounds

π 0.68 (0.02) 0.92 (0.06) 0.99 (0.01) 0.81 (0.04) 0.18 (0.03)λ 3.24 (0.04) 2.44 (0.13) 2.94 (0.07) 2.87 (0.07) 4.32 (0.11)

Observations 20,640 1,548 7,440 6,960 6,240Log Likelihood -8,412.97 -788.54 -3,640.64 -2,910.72 -1,611.85

Table 11: The results of maximum likelihood estimations. Standard errors are in parentheses. π is theestimated proportion of level-1 types and λ captures the error component. A high (low) λ implies a low(high) error rate.

The estimated proportion of level-1 types is 68% (see Table 11).11 The maximum likelihood

of this model is -8,412.97, a value that can be compared to models that allow only level-

1-thinkers or global games decision makers. The log likelihoods for π = 1 and π = 0 are

-9,565.70 (λ = 2.96) and -10,898.89 (λ = 1.58), respectively. Not surprisingly, level-1 thinking

is especially pronounced if we restrict the analysis to the �rst three rounds of the game since

this theory was originally developed to describe �rst response behavior.

There are substantial di�erences between the treatments. While virtually all participants can

be classi�ed as level-1 types in treatment Lab1, the vast majority of participants in treatment

Lab3 is classi�ed as global games decision makers. Note that responses in both treatments

may come from the same persons. The main reason for this seemingly contradictory result

is that virtually nobody, i.e. less than one percent, expects action B for signals above 0.75.

These high cooperation rates in the upper part of the interval are more consistent with global

11If we allow for di�erent error terms λ for both approaches, the estimated proportion of level-1-thinkersincreases marginally from 0.684 to 0.686. The estimated error rate for the global games types λGG is 4.18while the estimated error rate for the level-1 types λL1 is only 2.95, i.e. global games types follow their assignedtheory more closely. However, this extended model �ts only marginally better. The log-likelihood increasesfrom -8,412.97 to -8,351.42, despite the fact that the extended model has one additional free parameter.

37

games than with level-1-thinking. A similarly strong prediction of this approach is also the

main reason for the failure of the global games approach in treatment Lab1. For most signals,

action B is much more likely according to the incomplete information approach, in contrast

to the observed behavior.

To evaluate the reliability of our estimated proportion of level-1 types in the population,

we use a standard bootstrap approach. We randomly drew 40 decisions of a subject in one

experimental condition with replacements from our original sample to generate a sample

that is as large as our original sample. In such a generated sample, some subjects enter the

estimation procedure multiple times, while other subjects are not considered at all. This

procedure is repeated 100 times to calculate an approximately correct con�dence interval

for π. The 95% interval is [0.6268; 0.7114]. The estimated proportion of level-1 types

in the population (about 68%) is therefore reliable given our speci�cation. Since the two

models make substantially di�erent predictions for the majority of signals, we can di�erentiate

precisely between the two theoretical approaches.

The varying treatment behavior is also consistent with previous global games approach ex-

periments. Heinemann et al. (2004) have shown that cuto� values tend to be smaller than

the global games prediction. Such a behavior is interpreted within the maximum likelihood

procedure for treatment Lab1 as evidence in favor of level-1. In treatment Lab3, however,

the tendency to choose lower cuto� values in the repeated laboratory experiment favors the

global games approach because the lower bound of T is close to the global games but not

to the level-1 prediction. Furthermore, individual heterogeneity favors level-1 since expected

payo�s from picking the �wrong� action are not reduced as much for level-1 types as for global

games types.

38

D Elicitation of Beliefs

D.1 Design Issues and Results

Beliefs were elicited in roughly half of all lab sessions. This section contains a description

and an analysis of this data.

We refrained from paying subjects based on the stated probabilities for several reasons.

First (and most important), our primary interest is in behavior, and the rather complicated

mechanism for eliciting beliefs in the absence of risk neutrality (see e.g. O�erman et al.

(2009)) may shift participants' attention to the crucial importance of beliefs. Second, there

is the possibility that subjects try to hedge belief and action responses, a behavior that is

likely among students with a solid knowledge of �nance. Since many participants of the lab

experiment were recruited in a �nance course, it it possible that belief elicitation causally

a�ects behavior. Furthermore, there is evidence that di�erent methods of eliciting beliefs lead

to systematically di�erent results (Palfrey and Wang, 2009), although they are all incentive

compatible according to some theoretical considerations. Last, participants had also no

incentive to lie since beliefs were not public information. The most likely consequence of not

paying beliefs is that noise in the responses increases. Given the fact that we used a relatively

easy (2x2)-game, we expect high consistency rates, e.g. that most actions are indeed based

on stated beliefs.

Since there are no systematic di�erences between experimental conditions with and without

belief elicitation,12 we do not distinguish between data from conditions with or without belief

elicitation in the paper.

The elicited beliefs of the other player's action allow us to investigate more closely the causes

of di�erent behavior. Figure 6 shows the predicted distribution of the level-1 and the global

games approach as well as the empirical distribution. The elicited values are displayed as

12The average condition cuto� values based on 240 individual responses in the respective experimentalconditions are 0.161 (0.182/0.444) if probabilities were not elicited and 0.145 (0.257/0.432) if probabilitieswere elicited in treatment Lab1 (Lab2/Lab3). None of these di�erences is signi�cant at the 10%-level if weassume independent observations and use a Wilcoxon test.

39

the result of a locally weighted regression using Stata's lowess command.13 Elicited beliefs

are very close to the level-1 prediction in treatment Lab1. However, in treatment Lab2, and

especially in treatment Lab3, participants attach a higher probability to the other player

choosing A for each signal.

The results for treatment Lab3 shed further light on the success of the global games approach

in this treatment. Subjects believed for each signal that the other agent would cooperate

with a high probability. Especially for higher signals, this belief is consistent with the global

games probabilities. In global games in which cooperation is very attractive, the equilibrium

approach is valuable because it predicts the observed high cooperation rates while the behav-

ioral alternative does not.The beliefs in the �rst 40 rounds of a session are not systematically

di�erent from the beliefs in the second 40 rounds of a session. The corresponding �gures are

reproduced below. The elicitation of beliefs therefore generates no additional insights with

respect to the order e�ect.

Figure 3 suggests that condition cuto� values are slowly approaching the global games pre-

diction. A natural follow-up question is if elicited beliefs also approach global games. To

investigate this issue, we calculate for each subject in each experimental condition the ab-

solute distance between the stated belief and the global games belief for the �rst ten and

the last ten rounds of each 40 rounds. In the last ten rounds, the beliefs are closer to the

global games predictions in 61.8% (treatment Lab1), 76.5% (treatment Lab2), and 58.3%

(treatment Lab3) of all cases. The corresponding p-values of a two-sided binomial test with

the null hypothesis that the proportion of subjects with beliefs closer to global games in the

last ten rounds is equal to 12are 0.0223 (treatment Lab1), <0.0000 (treatment Lab2), and

0.1253 (treatment Lab3), respectively.

The elicitation of beliefs without paying participants in an incentive compatible way is often

criticized. In addition to the short discussion at the beginning of this section, Appendix D.2

provides three ex-post arguments why our belief elicitation is reliable.

13Simply calculating means in subintervals for di�erent signals would result in a very similar graph.

40

Figure 6: Subjects' beliefs about the other players' actions. Also shown are the predictions of level-1thinking and global games.

41

Figure 7: The stated probabilities of the �rst and the second 40 rounds from the repeated lab experiment.

D.2 Validation of Beliefs

First, the percentage of choices that are consistent with the stated beliefs is high. A choice

is classi�ed as belief-consistent if the expected value for the chosen action is higher under

the stated probability than the expected value of the alternative action. The percentages

are 80.4%, 87.8%, and 92.4% in treatments Lab1, Lab2, and Lab3, respectively. These num-

42

bers are higher than similar percentages reported in papers that used incentive compatible

payment like Costa-Gomes and Weizsäcker (2008) with about 60% (8.41 out of 14 games).

However, our consistency rates are probably higher because our (2x2)-game is easier than

the (3x3)-games in Costa-Gomes and Weizsäcker (2008).

Second, using the stated probabilities in the maximum likelihood estimation task outlined

in Appendix C increases the �t of the model substantially. The log-likelihood of the mixed

model is −4775.57 if we only use the data from the sessions with belief elicitation (λ = 3.321;

π = 0.676). However, if we calculate the expected value of action A by

EV(si)ST = Prob(si)

ST · si + (1− Prob(si)ST ) · (si − 1)

with Prob(si)ST equal to the stated probability for signal si, the log likelihood increases to

−3700.93 (λ = 4.43). Note further that the model based on stated probabilities has only one

free parameter (λ) while the mixed model in Appendix C has two (λ and π). The stated

probabilities are therefore informative and not only cheap-talk or noise.

Third, we used the mixed model from Appendix C to classify participants based on their be-

havior. Recall that the probability that a given choice history of subject j would have been

observed is given by∏n

i=1ChoiceProb(si)L1 (level-1 approach) or

∏ni=1ChoiceProb(si)

GG

(global games approach). If we assume Bayes' law and the estimated π as a-priori proba-

bility that a randomly chosen subject is a level-1-thinker, we can calculated the a-posteriori

probability that a subject j is a level-1-thinker by

∏ni=1ChoiceProb(si)

L1 · π∏ni=1ChoiceProb(si)L1 · π +

∏ni=1ChoiceProb(si)GG · (1− π)

. (8)

If this probability is greater that 12, the decision maker is classi�ed as a level-1 type. Oth-

erwise, it is assumed that the observed behavior is more consistent with the global games

approach.

All participants in treatment Lab1 are classi�ed as level-1 types and a distinction of types

is therefore not possible. Figure 8 shows the beliefs in treatments Lab2 and Lab3 for level-

43

1 types and global games decision makers. The estimated line of beliefs of global games

decision makers in treatment Lab2 crosses the bisecting line close to 12as predicted by the

global games approach. The beliefs of level-1 types (global games decision makers) are closer

to the predictions of the level-1 approach (global games approach) in treatment Lab3. This

is strong evidence that the belief elicitation is reliable, i.e. participants are classi�ed based on

behavior in an incentive-compatible experiment and their unpaid beliefs are consistent with

the belief predictions of the decision models.

Figure 8: Beliefs of global games decision makers, level-1-thinkers, and the empirical observations fromthe repeated lab experiment.

44