AN EXPERIMENTAL STUDY OF UNCERTAINTY IN COORDINATION GAMES

• DOI: http://doi.org/10.1111/iere.12367
• Published date: 01 May 2019
• Author: Miltiadis Makris,Christos A. Ioannou
• Date: 01 May 2019

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 2, May 2019 DOI: 10.1111/iere.12367

AN EXPERIMENTAL STUDY OF UNCERTAINTY IN COORDINATION GAMES∗

BYCHRISTOS A. IOANNOU AND MILTIADIS MAKRIS1

Universit´

e Paris 1 Panth´

eon-Sorbonne, France; University of Kent, U.K.

Global games and Poisson games have been proposed to address equilibrium indeterminacy in Com-

mon Knowledge Coordination games. The present study investigates in a controlled setup, using as con-

trols Common Knowledge games, whether idiosyncratic uncertainty about economic fundamentals (Global

games) or uncertainty about the number of actual players (Poisson games) may influence subjects’ behavior.

We find that uncertainty about the number of actual players has more influence on subjects’ behavior than

idiosyncratic uncertainty about economic fundamentals. Furthermore, subjects’ behavior under Poisson

population-size uncertainty is closer to the respective theoretical prediction than subjects’ behavior under

idiosyncratic uncertainty about economic fundamentals.

1. INTRODUCTION

Coordination games with strategic complementarities have been widely used to capture

setups, such as speculative attacks, start-up investments, and new technology adoption under

network externalities (see, e.g., Milgrom and Roberts, 1990; Obstfeld, 1996). If the state of

the economy (i.e., profitability of the risky action) and the number of stakeholders/players is

common knowledge, then equilibrium cannot always be pinned down uniquely because beliefs

can be indeterminate. To escape a prediction of indeterminacy of equilibria, the received

theoretical literature has focused on uncertainty about fundamentals (see, e.g., Morris and

Shin, 1998; Herrendorf et al., 2000; Frankel and Pauzner, 2000; Burdzy et al., 2001; Makris,

2008).

Global Coordination games (see Morris and Shin, 1998) constitute the most popular approach

to escape the prediction of equilibrium indeterminacy by means of deploying uncertainty about

economic fundamentals (e.g., the profitability of a successful speculative attack). A more recent

approach, Poisson Coordination games, is motivated instead by the fact that, in the above

strategic environments, the number of economic agents is often very large. As Myerson (2000)

points out, in games with a very large number of players, “it is unrealistic to assume that every

player knows all the other players in the game; instead, a more realistic model should admit

some uncertainty about the number of players in the game” (p. 7). Following the suggestion

of Myerson (2000), this approach models the number of actual players as a Poisson random

variable (see Makris, 2008).2

∗Manuscript received May 2017; revised July 2018.

1The article has benefited greatly from the comments of Antonio Cabrales, Nobuyuki Hanaki, Valentin Patilea,

Anastasios Magdalinos, Miguel Fonseca, Alex Michaelides, Francesco De Sinopoli, and Frank Heinemann. We would

also like to thank the seminar participants at the University of Texas at Dallas, University of Pittsburgh, University of

Purdue, Florida State University, Universit´

eC

ˆ

ote d’ Azur, and University of Verona. Finally, we are indebted to the

editor, Masaki Aoyagi, and three anonymous referees for their insightful comments, which significantly improved the

article. The usual disclaimer applies. Please address correspondence to: Miltiadis Makris, School of Economics, Keynes

College, University of Kent, CT2 7NP, United Kingdom. E-mail: mmakris.econ@gmail.com.

2This modeling choice is driven, in part, by certain convenient properties of the Poisson distribution (see Myerson,

1998). As a complementary justification for this modeling choice, suppose that the identity of every stakeholder is

common knowledge and that binding individual orders for, say, short sales of a currency must arrive with the central

bank by a given time. Standard theory suggests that each agent will decide on her action by taking the number of orders

at the collector’s disposal as given. However, the probability that a phone call to a busy switchboard will go through

751

C

(2018) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

752 IOANNOU AND MAKRIS

Importantly, Global and Poisson Coordination games lead to different predictions. The

Global Coordination game prediction about, say, the onset of speculative attacks manifests

a threshold level of economic fundamentals that defines two areas in the region where Com-

mon Knowledge Coordination games predict multiplicity of equilibria: one in which a successful

attack takes place and another where a successful attack does not materialize. However, the

Poisson Coordination game prediction is that no speculative attack will take place as long as

the ratio of the short-selling cost per reward is greater than the probability of having suffi-

ciently many players in the game; otherwise, multiplicity of equilibrium outcomes still arises

(see Section 3 for more details).

Motivated by the aforementioned theoretical papers, the present study investigates in a con-

trolled setup, using as controls Common Knowledge Coordination games, whether idiosyncratic

uncertainty about economic fundamentals or uncertainty about the number of actual players

may influence subjects’ behavior. Specifically, we design a novel experiment to compare the

behavior of subjects in Poisson, Common Knowledge, and Global Coordination games (hence-

forth, for brevity, referred to as Poisson,Common Knowledge, and Global games, respectively,

unless there is a risk of confusion). The experimental design is formulated around asking sub-

jects to state their intent to buy a cash amount.3Registering to buy the cash amount entails

paying a nonrefundable fee, which is less than the cash amount. Additionally, in order to get

the cash amount, a threshold number of registrations has to be met. If fewer subjects than the

number dictated by the threshold register, then the cash amount is not awarded.

We make three key contributions in this article. First, to the best of our knowledge, we are the

first to provide an experimental investigation of Poisson Coordination games. Games assuming

Poisson population-size uncertainty have been studied theoretically in mostly Voting games

(see, e.g., Krishna and Morgan, 2011; Bouton and Castanheira, 2012; Medina, 2013; Bouton and

Gratton, 2015) and Discrete Public Goods games (Makris, 2009). The only other experimental

studies of Poisson games we know of are those of Ostling et al. (2011) and Herrera et al.

(2014). Ostling et al. (2011) study the Swedish Lowest Unique Positive Integer (LUPI) game,

and find that the behavioral patterns of the field and laboratory data are closely related with

the theoretical predictions. Herrera et al. (2014) investigate a voter turnout model, where they

compare the turnout in two electoral systems: a winner-takes-all system and a proportional

power sharing system. Their results from a laboratory experiment are broadly supportive of the

theoretical predictions.

The second key contribution of our study is methodological. Specifically, our experiments

are conducted over the Internet. Internet is ideal for Poisson games, as subjects cannot infer the

number of participants, which is typically the case in a laboratory experiment. Crucially, in order

to circumvent the difficulties that would arise given the (assumed) unfamiliarity of many subjects

with Poisson probabilities, we applied the specific probabilities onto a roulette wheel and noted

that the latter is not a standard wheel. To maintain consistency with the Poisson experiments,

the Global and the baseline (i.e., Common Knowledge) sessions were also conducted over the

Internet while accommodating the underlying assumptions of the theories. A value-added of this

approach is that it resembles how managers and investors commit to their decisions nowadays:

After contemplating the pros and cons of various alternatives, managers and investors will often

place their (short-selling, purchase, or investment) orders online.

Our third and most important contribution is substantive. We find that uncertainty about

the number of actual players has a more significant impact on subjects’ behavior than

or the webpage of an online site will be uploaded successfully at times of high traffic decreases with the number of

stakeholders. As a result, and under the assumption that the average number of successful phone calls or online visits

is known, in a large environment, stakeholders should actually view the number of actual players in the Coordination

game as a Poisson random variable.

3In the lingo of the speculative attack model of Morris and Shin (1998), registering to buy the cash amount is

analogous to short selling the currency. Alternatively, in the context of investors and technology adopters under

network externalities, registering to buy the cash amount is analogous to undertaking the investment opportunity and

adopting the new technology, respectively.

AN EXPERIMENTAL STUDY OF UNCERTAINTY IN COORDINATION GAMES 753

idiosyncratic uncertainty about economic fundamentals when we focus on parameters for which

both Poisson and Global games predict a unique equilibrium. Specifically, we find that, in their

vast majority, subjects in the Poisson games forgo registering to buy the cash amount (i.e.,

choose the “safe” action), whereas in both Global and Common Knowledge games, subjects

split almost evenly between forgoing registering to buy the cash amount and registering to buy

the cash amount. Therefore, the introduction of uncertainty regarding the number of actual

players may influence empirical behavior in large environments with strategic complementari-

ties, whereas the introduction of idiosyncratic uncertainty about economic fundamentals may

not. Finally, subjects’ behavior under Poisson population-size uncertainty is closer to the re-

spective theoretical prediction than subjects’ behavior under idiosyncratic uncertainty about

economic fundamentals.

The article adheres to the following plan. We present next other related experimental liter-

ature. In Section 3, we review the theoretical predictions of Common Knowledge, Global, and

Poisson games. In Section 4, the experimental design is presented. In Section 5, we report the

results of our experiments. In Section 6, we conduct a robustness analysis, and, in Section 7,

we discuss comparative statics and a possible explanation for the main results based on limited

depth of reasoning. Finally, in Section 8, we conclude and offer suggestions for future research.

2. OTHER RELATED EXPERIMENTAL LITERATURE

Common Knowledge Coordination games have been studied extensively experimentally (see,

e.g., Van Huyck et al., 1990, 1991, 1993; Brandts and Cooper, 2006; Cooper et al., 2018). Re-

garding experimental studies of Global Coordination games, we are aware only of the following

three studies. Heinemann et al. (2004) study an experiment that resembles the speculative at-

tack model of Morris and Shin (1998), but with repeated play. In comparing sessions between

Common Knowledge and Global Coordination games, they find that subjects use threshold

strategies in both informational protocols. In the Global games, they find that observed behav-

ior is closer to the Global game solution. In their setup, the relevant economic fundamental is

the profit from short selling the currency, which is drawn anew at the start of each repeated

interaction.4In the Common Knowledge games, the authors find that observed behavior lies

between the payoff-dominant equilibrium and the Global game solution.

Cabrales et al. (2007) study an experiment that resembles the 2 ×2 setup of Carlsson and van

Damme (1993). Analogous to Heinemann et al. (2004), Cabrales et al. (2007) also investigate

subjects’ behavior in Common Knowledge and Global Coordination games, but distinguish

between short-term and long-term play. The authors utilize a discrete state space with five

possible states and signals to make the theoretic reasoning simpler. Cabrales et al. (2007) find

that in the Global games with long-term play, subjects’ behavior converges toward the Global

game solution. The authors also find that in the Common Knowledge games with short-term

play, observed behavior of subjects can be anywhere (weakly) between the payoff-dominant

equilibrium and the Global game solution. Moreover, Cabrales et al. (2007) establish that

subjects’ behavior across the Common Knowledge and Global games with short-term play is

statistically similar. This is a departure from the findings of Heinemann et al. (2004). According

to Cabrales et al. (2007, p. 232), the difference in results may be driven by the absence of learning

effects.

Szkup and Trevino (2015) implement a different informational structure than the two previous

studies. Specifically, the authors develop a two-stage model, where each agent, in the first stage,

has the possibility to choose, at a cost, the precision of his private signal, and, in the second

stage, play the Coordination game, as in Morris and Shin (1998), using the information acquired

4The context of a subject’s decision differs in our setup compared to the one in Heinemann et al. (2004). In our

setup, a subject has to sacrifice an amount of money (pay a nonrefundable fee) from the initial endowment to buy the

cash amount. Otherwise, a subject gets to keep the endowed amount. In the study of Heinemann et al. (2004), subjects

are required to decide between the safe and the risky action; however, the risky action does not take away any money

from their total earnings.