A minimum effort coordination game experiment in continuous time

We conduct an experiment on a minimum effort coordination game in a (quasi-)continuous time-frame, where effort choices can be switched freely during a 60-s period. The cooperation levels of the continuous time treatments are not significantly different from the discrete time treatments. Providing subjects with the information on the effort choices of all group members increases the average effort level in continuous time only. The minimum effort level in continuous time with full information feedback is also substantially higher than that with limited information feedback, but the difference is statistically insignificant. With limited information feedback, subjects rarely coordinate to increase their efforts simultaneously to change the group minimum within a period. Our findings imply that continuous time games are not behaviorally equivalent to infinitely repeated discrete time games.     

Coordination,timing and common knowledge

There is tight link between coordination and common knowledge. The role of higher order beliefs in static incomplete information games has been widely studied. In particular, information frictions break down common knowledge. A large body of literature in economics examine dynamic coordination problems when there are timing frictions, in the sense that players do not all move at once. Timing frictions in dynamic coordination games play a role that is closely analogous to information frictions in static coordination games.This paper makes explicit the role of higher order beliefs about timing in dynamic coordination games with timing frictions. An event is said to be effectively known if a player knew the event when he last had an option to change his behavior. The lack of effective common knowledge of the time drives results of dynamic coordination games.     

Leadership and overcoming coordination failure with asymmetric costs

We study how the heterogeneity of agents affects the extent to which changes in financial incentives can pull a group out of a situation of coordination failure. We focus on the connections between cost asymmetries and leadership. Experimental subjects interact in groups of four in a series of weak-link games. The treatment variable is the distribution of high and low effort cost across subjects. We present data for one, two and three low-cost subjects as well as control sessions with symmetric costs. The overall pattern of coordination improvement is common across treatments. Early coordination improvements depend on the distribution of high and low effort costs across subjects, but these differences disappear with time. We find that initial leadership in overcoming coordination failure is not driven by low-cost subjects but by subjects with the most common cost type. This conformity effect may be due to a kind of group identity or to the cognitive simplicity of acting with identical others.      

Which hat to wear? Impact of natural identities on coordination and cooperation

As the workforce becomes increasingly diverse, motivating individuals from different backgrounds to work together effectively is a major challenge facing organizations. In an experiment conducted at a large public university in the United States, we manipulate the salience of participants' multidimensional natural identities and investigate the effects of identity on coordination and cooperation in a series of minimum-effort and prisoner's dilemma games. By priming a fragmenting (ethnic) identity, we find that, compared to the control, participants are significantly less likely to choose high effort in the minimum-effort games, leading to less efficient coordination. In comparison, priming a common organization (school) identity significantly increases the choice of a rational joint payoff maximizing strategy in a prisoner's dilemma game.     

Minimum-Effort Coordination Games: Stochastic Potential and Logit Equilibrium

This paper revisits the minimum-effort coordination game with a continuum of Pareto-ranked Nash equilibria. Noise is introduced via a logit probabilistic choice function. The resulting logit equilibrium distribution of decisions is unique and maximizes a stochastic potential function. In the limit as the noise vanishes, the distribution converges to an outcome that is analogous to the risk-dominant outcome for 2 × 2 games. In accordance with experimental evidence, logit equilibrium efforts decrease with increases in effort costs and the number of players, even though these parameters do not affect the Nash equilibria. Journal of Economic Literature Classification Numbers: C72, C92.     

Playing safe in coordination games:: the roles of risk dominance, payoff dominance, and history of play

This paper presents results from a series of experiments designed to test the impact on subject behavior of changes in the risk dominance and payoff dominance characteristics of two player coordination games. The main finding is that changes in risk dominance significantly affect play of the subjects, whereas changes in the level of payoff dominance do not. Observed history of play also has an important influence on subject behavior, both when subjects are randomly rematched after each game and when they remain matched with the same individual for a sequence of games.     

Imitation with asymmetric memory

Models of learning in games based on imitation have provided fundamental insights as the relevance of risk-dominance equilibria or perfectly competitive outcomes. It has also been shown, however, that the introduction of nontrivial memory in those models fundamentally alters the results. This paper further considers the effect of asymmetric memory length in the population. We focus on two classical results and find that, while asymmetric memory crucially affects equilibrium selection in coordination games, it reinforces the stability of perfectly competitive outcomes in oligopoly games. The latter result is generalized to aggregative games and to finite population ESS in symmetric games.     

Behavioral spillovers in coordination games

Motivated by problems of coordination failure observed in weak-link games, we experimentally investigate behavioral spillovers for minimum- and median-effort coordination games. Subjects play these coordination games simultaneously and sequentially. The results show that successful coordination on the Pareto optimal equilibrium in the median game influences behavior in the minimum game when the games are played sequentially. Moreover, this positive, Pareto-improving spillover is present even when group composition changes across games, although the effect is not as strong. We also find that the precedent for uncooperative behavior in the minimum game does not influence play in the median game. These findings suggest guidelines for increasing cooperative behavior within organizations.     

Difference-Form Contests and the Robustness of All-Pay Auctions

In much of the existing literature on rent-seeking games, the outcome of the contest is either infinitely sensitive or relatively insensitive to contestants' efforts. The current paper presents a family of contest games that permit characterization of equilibrium for all levels of sensitivity of the outcome to contestants' efforts. Specifically, the outcome of the contest depends on the difference between efforts, which encompass the lottery and the all-pay auction as polar cases. The equilibrium converges to that of the all-pay auction as the probability of winning the prize grows infinitely sensitive to one's effort, and the main qualitative features of equilibrium persist over a large parameter region. Journal of Economic Literature Classification Numbers: D44, D72.     

How Much Common Belief Is Necessary for a Convention?

We study Bayesian coordination games in which players choose actions conditional on the realization of their respective signals. Due to differential information, the players do not have common knowledge that a particular game is being played. However, they do have common beliefs with specified probabilities concerning their environment. In our framework, any equilibrium set of rules must be simple enough so that the actions of all players are common belief with probability 1 at every state. Common belief with probability close to 1 will not do.Journal of Economic LiteratureClassification Numbers: C72, D82.     

When and why? A critical survey on coordination failure in the laboratory

Coordination games with Pareto-ranked equilibria have attracted major attention over the past two decades. Two early path-breaking sets of experimental studies were widely interpreted as suggesting that coordination failure is a common phenomenon in the laboratory. We identify the major determinants that seem to affect the incidence, and/or emergence, of coordination failure in the lab and review critically the existing experimental studies on coordination games with Pareto-ranked equilibria since that early evidence emerged. We conclude that there are many ways to engineer coordination successes.      

The Impact of Social Comparisons on Reciprocity*

In this paper, we investigate how social comparison information about referent others (i.e., learning what similar others do, and how they are treated) affects reciprocal relationships. Using three‐person gift‐exchange games, we study how employees’ reciprocity towards an employer is affected by pay comparison information (what co‐workers earn) and effort comparison information (how co‐workers perform). We find strong evidence of the effects of effort comparison: employees are more willing to reciprocate by choosing high effort in response to a high wage if they observe others doing so. In our setting, we find somewhat weaker evidence of the effects of pay comparison.     

The Statistical Mechanics of Best-Response Strategy Revision

This paper compares local and global strategic interaction when players update using the (myopic) best-response rule. I show that randomizing the order in which players update their strategic choice suffices to achieve coordination on the risk-dominant strategy in symmetric 2 × 2 coordination games. The "persistant randomness" which is necessary to achieve similar coordination with global interaction is replaced under local interaction by spatial variation in the initial condition. An extension of the risk-dominance idea gives the same convergence result for K × K games with strategic complementarities. Similar results for K × K pure coordination games and potential games are also presented. Journal of Economic Literature Classification Number: C78.     

How noise matters

Recent advances in evolutionary game theory have introduced noise into decisionmaking to select in favor of certain equilibria in coordination games. Noisy decisionmaking is justified on bounded rationality grounds, and consequently the sources of noise are left unmodelled. This methodological approach can only be successful if the results do not depend too much on the nature of the noise process. This paper investigates invariance to noise of these results, both for the random matching paradigm that has characterized much of the recent literature and for a larger class of two-strategy population games where payoffs may vary non-linearly with the distribution of strategies among the population. Several parametrizations of noise reduction are investigated. The results show that a symmetry property of the noise process and, in the case of non-linear payoffs, bounds on the asymmetry of the payoff functions suffice to preserve the known stochastic stability results.     

The importance of higher-order beliefs to successful coordination

Beliefs about other players’ strategies are crucial in determining outcomes for coordination games. If players are to coordinate on an efficient equilibrium, they must believe that others will coordinate with them. In many settings there is uncertainty about beliefs as well as strategies. Do people consider these “higher-order” beliefs (beliefs about beliefs) when making coordination decisions? I design a modified stag hunt experiment that allows me to identify how these higher-order beliefs and uncertainty about higher-order beliefs matter for coordination. Players prefer to invest especially when they believe that others are “optimistic” that they will invest; but knowledge that others think them unlikely to invest does not cause players to behave differently than when they do not know what their partners think about them. Thus resolving uncertainty about beliefs can result in marked efficiency gains.     

Modelling International Policy Games: Lessons from European Monetary Coordination

Most international monetary policy games are modelled as prisoners' dilemma games. Political scientists suggest however that other game-theoretic structures (chicken, stag-hunt and deadlock games) could be more appropriate to describe international monetary coordination. This paper provides some empirical evidence on this issue, by studying the case of European monetary coordination from 1979 to 1989. First, central banks' and governments' preferences are revealed through an analysis of their actual behaviour. Then, the dynamic game describing policymakers' interactions is simulated under alternative institutional arrangements: coordination, defection, sub-group coalitions, etc. The main conclusion is that the stylised facts derived from those experiments seem not to be consistent with the implications of the prisoners' dilemma framework. Consistency is instead found with the features of a stag-hunt game. CEPR and FEEM This revised version was published online in July 2006 with corrections to the Cover Date.     

Equilibrium Selection inn-Person Coordination Games

This paper investigates several approaches to equilibrium selection and the relationships between them. The class of games we study aren-person generalized coordination games with multiple Pareto rankable strict Nash equilibria. The main result is that all selection criteria select the same outcome (namely the risk dominant equilibrium) in two-person games, and that most equivalences break for games with more than two players. All criteria select the Pareto efficient equilibrium in voting games, of which pure coordination games are special cases.Journal of Economic LiteratureClassification Numbers: C70, C72, D82.     

Individually selecting among conventions - an evolutionary and experimental analysis

Conventions can be narrowly interpreted as coordinated ways of equilibrium play, telling all players which of possibly several equilibria to play or more broadly how to choose in a game without imposing the equilibrium property. Since coordination often takes place before learning about the game, one has to coordinate on a prescribing principle. For the subclass of 2×2-bimatrix games with two strict equilibria, we analyze the evolutionary stability of various such principles. In our experiment, we allow participants to coordinate on principles before playing various games. Based on between-subjects treatments, participants do so being completely (they know neither their role nor the game parameters), partially (they know either their role or the game parameters) ignorant, or with no veil of ignorance (they know their role and the game parameters).     

The evolution of conventions in minimum effort games

The literature on minimum effort game has been concerned with a symmetric game with linear payoff functions. The main aim of the present paper is to study the coordination problem arising in a not necessarily symmetric minimum effort game with two players. The sources of asymmetry can be twofold: the productivity of effort and the distribution of the join output. To select among the Pareto ranked equilibria we use the stochastic stability criterion. We show that, for any configuration parameters, the set of stochastically stable equilibria coincides with the set of potential maximizers. We also show that when the disutility of effort is linear, the Pareto dominant equilibrium is stochastically stable provided that the distributive parameter belongs to a well defined range. When the disutility of effort is nonlinear no distributive arguments can be used to successfully affect the selection process. Lastly we prove that the connection between stochastic stability and maximum potential can fail when more than two agents are considered.     

Dynamic coordination with individual learning

We study coordination in dynamic global games with private learning. Players choose whether and when to invest irreversibly in a project whose success depends on its quality and the timing of investment. Players gradually learn about project quality. We identify conditions on temporal incentives under which, in sufficiently long games, players coordinate on investing whenever doing so is not dominated. Roughly speaking, this outcome occurs whenever players? payoffs are sufficiently tolerant of non-simultaneous coordination. We also identify conditions under which players coordinate on the risk-dominant action. We provide foundations for these results in terms of higher order beliefs.