Dynamic coordination games

Summary Gains from coordination provide incentives for delay. In this paper, the extent of delay is studied in a dynamic,N-person, coordination game. There is no social gain from delay, so an equilibrium with delay is always inefficient. For fixedN, there is no coordination failure when the period length is short: all equilibrium outcomes converge to the Pareto efficient outcome as the period length converges to zero. On the other hand, holding period length fixed, there exist equilibria in which delay is proportional toN, for arbitrarily large values ofN. In addition, it can be shown that the possibility of delay depends on the timing of strategic complementarities. However, under certain conditions, delay is shown to be a robust phenomenon, in the sense that well-behaved equilibria exhibit infinite delay forN sufficiently large.This paper grew out of discussions with Christophe Chamley. While writing it I benefited from discussions with Ken Binmore, Russell Cooper, Bob Rosenthal and Michael Manove. Joe Farrell, Drew Fudenberg, Martin Hellwig and Sawoong Kang made very useful comments on an earlier version that led to substantial improvements. Helpful comments were also made by seminar participants at the London School of Economics, the SUNY at Stoney Brook, the NBER Summer Institute, Northwestern University, and the University of Chicago. I would like to thank Nick Yannelis and an anonymous referee for their editorial advice. Financial support for this research was provided by the National Science Foundation under Grant No. SES 9196061.     

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.     

Stochastic strategy adjustment in coordination games

Summary. The paper explores a model of equilibrium selection in coordination games, where agents from an infinite population stochastically adjust their strategies to changes in their local environment. Instead of playing perturbed best-response, it is assumed that agents follow a rule of ‘switching to better strategies with higher probability’. This behavioral rule is related to bounded-rationality models of Rosenthal (1989) and Schlag (1998). Moreover, agents stay with their strategy in case they successfully coordinate with their local neighbors. Our main results show that both strict Nash equilibria of the coordination game correspond to invariant distributions of the process, hence evolution of play is not ergodic but instead depends on initial conditions. However, coordination on the risk-dominant equilibrium occurs with probability one whenever the initial fraction contains infinitely many agents, independent of the spatial distribution of these agents. Received: March 14, 2000; revised version: June 21, 2001     

Evidence on learning in coordination games

This paper reports an experiment designed to detect the influence of strategic uncertainty on behavior in order statistic coordination games, which arise when a player’s best response is an order statistic of the cohort’s action combination. Unlike previous experiments using order statistic coordination games, the new experiment holds the payoff function constant and only changes cohort size and order statistic.

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Electronic Supplementary Material The online version of this article () contains supplementary material, which is available to authorized users. Related research available at     

Learning and sophistication in coordination games

This paper studies the role of strategic teaching in coordination games and whether changing the incentives of players to teach leads to more efficient coordination. We ran experiments where subjects played one of four coordination games in constant pairings, where the incentives to teach were varied along two dimensions—the short run cost of teaching and the long run benefit to teaching. We show which aspects of the game lead subjects to adopt long run teaching strategies, and show that subjects try to manipulate their opponent’s actions to pull them out of a situation of coordination failure. We also show that extending a model of decision making by introducing a forward-looking component helps to track teachers’ behaviour more accurately, and describes the way players behave in a more unified way across both teachers and learners.     

Competition and coordination in experimental minority games

This work presents experimental results on a coordination game in which agents must repeatedly choose between two sides, and a positive fixed payoff is assigned only to agents who pick the minoritarian side. We conduct laboratory experiments in which stationary groups of five players play the game for 100 periods, and manipulate two treatment variables: the amount of information about other players’ past choices and the salience of information regarding the game history (i.e., the length of the string of past outcomes that players can see on the screen while choosing). Our main findings can be summarized as follows: aggregate efficiency in the game is in most cases significantly higher than the level corresponding to the symmetric mixed strategy Nash equilibrium. In addition, providing players with information about individual choices in the group does not improve aggregate efficiency with respect to when such information is absent. Displaying information about more rounds than just the previous one, on the other hand, seems to have a positive effect on aggregate efficiency. At the individual level, we find a stronger statistical relation between players’ current choices and their own past choices than between players’ choices and previous aggregate outcomes. In addition, the depth of the relation between present and past choices seems to be affected by the prompt availability of information about the game history. Finally, we detect evidence of a mutual co-adaptation between players’ choices over time that is partly responsible for the high level of efficiency observed.      

Equilibrium selection in coordination games with simultaneous play

Summary. We apply the dynamic stochastic framework proposed in recent evolutionary literature to a class of coordination games played simultaneously by the entire population. In these games payoffs, and hence best replies, are determined by a summary statistic of the population strategy profile. We demonstrate that with simultaneous play, the equilibrium selection depends crucially on how best responses to the summary statistic remain piece-wise constant. In fact, all the strict Nash equilibria in the underlying stage game can be made stochastically stable depending on how the best response mapping generates piece-wise constant best responses. Received: February 12, 2001; revised version: October 29, 2001     

Asymmetric and endogenous within-group communication in competitive coordination games

Within-group communication in competitive coordination games has been shown to increase competition between groups and lower efficiency. This study further explores potentially harmful effects of communication, by addressing the questions of (1) asymmetric communication and (2) the endogenous emergence of communication. Our theoretical analysis provides testable hypotheses regarding the effect of communication on competitive behavior and efficiency. We test these predictions using a laboratory experiment. The experiment shows that although asymmetric communication is not as harmful as symmetric communication, it leads to more aggressive competition and lower efficiency relative to the case when neither group can communicate. Moreover, groups vote to endogenously establish communication channels even though they would earn higher payoffs if jointly they chose to restrict within-group communication.     

Coordination failure in repeated games with private monitoring

Players coordinate continuation play in repeated games with public monitoring. We investigate the robustness of such equilibrium behavior with respect to ex-ante small private-monitoring perturbations. We show that with full support of public signals, no perfect public equilibrium is robust if it induces a “regular” 2×2$2\times 2$ coordination game in the continuation play. This regularity condition is violated in all belief-free equilibria. Indeed, with an individual full rank condition, every interior belief-free equilibrium is robust. We also analyze block belief-free equilibria and point out that the notion of robustness is sensitive to whether we allow for uninterpretable signals.     

Money illusion and coordination failure

Economists long considered money illusion to be largely irrelevant. Here we show, however, that money illusion has powerful effects on equilibrium selection. If we represent payoffs in nominal terms, choices converge to the Pareto inferior equilibrium; however, if we lift the veil of money by representing payoffs in real terms, the Pareto efficient equilibrium is selected. We also show that strategic uncertainty about the other players' behavior is key for the equilibrium selection effects of money illusion: even though money illusion vanishes over time if subjects are given learning opportunities in the context of an individual optimization problem, powerful and persistent effects of money illusion are found when strategic uncertainty prevails.     

Social learning in coordination games: does status matter?

We report the results of experiments designed to test the impact of social status on learning in a coordination game. In the experiment, all subjects observe the play of an agent who either has high status or low status. In one treatment the agent is another player in the game; in the other the agent is a simulated player. Status is assigned within the experiment based on answers to a trivia quiz. The coordination game has two equilibria: one is payoff-dominant but risky, and the other is risk-dominant. The latter is most commonly chosen in experiments where there is no coordination device. We find that a commonly observed agent enhances coordination on the payoff-dominant equilibrium more often when the agent has high status.      

Equilibrium selection through incomplete information in coordination games: an experimental study

We perform an experiment on a pure coordination game with uncertainty about the payoffs. Our game is closely related to models that have been used in many macroeconomic and financial applications to solve problems of equilibrium indeterminacy. In our experiment, each subject receives a noisy signal about the true payoffs. This game (inspired by the “global” games of Carlsson and van Damme, Econometrica, 61, 989–1018, 1993) has a unique strategy profile that survives the iterative deletion of strictly dominated strategies (thus a unique Nash equilibrium). The equilibrium outcome coincides, on average, with the risk-dominant equilibrium outcome of the underlying coordination game. In the baseline game, the behavior of the subjects converges to the theoretical prediction after enough experience has been gained. The data (and the comments) suggest that this behavior can be explained by learning. To test this hypothesis, we use a different game with incomplete information, related to a complete information game where learning and prior experiments suggest a different behavior. Indeed, in the second treatment, the behavior did not converge to equilibrium within 50 periods in some of the sessions. We also run both games under complete information. The results are sufficiently similar between complete and incomplete information to suggest that risk-dominance is also an important part of the explanation.      

An experimental study of coordination and learning in iterated two-market entry games

Summary. Tacit coordination in large groups is studied in an iterated market entry game with complete information and multiple market capacities that are varied randomly from period to period. On each period, each player must decide independently whether to enter any of the markets, and if entering, which of the two markets to enter. Across symmetric and asymmetric markets, we find remarkable coordination on the aggregate level, which is accounted for by the Nash equilibrium, together with considerable individual differences in frequency of entry and decision rules. With experience, the decisions of most players converge to decision rules with cutoff values on the combined market capacity that determine whether or not to enter but not which of the two markets to enter. This latter decision is determined probabilistically by the differential market capacities. The aggregate and individual results are accounted for quite well by a reinforcement-based learning model that combines deterministic and probabilistic elements.     

Investigating the failure to best respond in experimental games

Experimental Economics - We examine strategic sophistication using eight two-person 3?×?3 one-shot games. To facilitate strategic thinking, we design a ‘structured’...     

Evolutionary selection of ‘chivalrous’ conventions in coordination games without common expectations

We study an evolutionary game-theoretic model where players have to choose within a predetermined set of mixed strategies in a coordination game. Players are of two different kinds, male and female. No common expectations assumption is made; players tend therefore to adopt the strategy that yields larger than average expected payoffs for their kind. In this framework, every stable stationary point of the population dynamics can be interpreted as the emergence of a particular convention. A classification of the possible conventions is provided; conditions for their emergence are determined.     

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.      

The international business cycle as intertemporal coordination failure

Where investments are irreversible and the future is uncertain, people in two countries can make investment decisions that turn out to be mutually inconsistent. I argue that this intertemporal coordination failure explains international business cycles in a two-currency-area setting with a floating foreign exchange rate. The sequence of events starts with an expansionary domestic monetary shock, which decreases the domestic real interest rate. Facing low transactions costs, people spend the new money relatively early in the foreign exchange market and in the foreign market for loanable funds. Domestic monetary expansion thereby changes the relative prices of domestic and foreign goods and also of goods of earlier and later stages of production. The relative price changes lead to intertemporal and international coordination failures once the monetary expansion ends and relative prices change. Domestic monetary policy thereby causes the comovement across different currency areas we observe of business cycles.     

Solving coordination failure with “all-or-none” group-level incentives

Coordinating activity among members is an important problem faced by organizations. When firms, or units within firms, are stuck in bad equilibria, managers may turn to the temporary use of simple incentives—flat punishments or rewards—in an attempt to transition the firm or unit to a more efficient equilibrium. We investigate the use of incentives in the context of the “minimum-effort,” or “weak-link,” coordination game. We allow groups to reach the inefficient equilibrium and then implement temporary, flat, “all-or-none” incentives to encourage coordination on more efficient equilibria. We vary whether incentives are positive (rewards) or negative (penalties), whether they have substantial or nominal monetary value, and whether they are targeted to a specific outcome (the efficient equilibrium) or untargeted (apply to more than one outcome). Overall, incentives of all kinds are effective at improving coordination while they are in place, but there is little long-term persistent benefit of incentives—once incentives are removed, groups tend to return to the inefficient outcome. We find some differences between different kinds of incentives. Finally, we contrast our results to other recent work demonstrating greater long-term effectiveness of temporary incentives.      

Observability and overcoming coordination failure in organizations: An experimental study

Motivated by problems of coordination failure in organizations, we examine how overcoming coordination failure and maintaining coordination depend on the ability of individuals to observe others’ choices. Subjects’ payoffs depend on coordinating at high effort levels in a weak-link game. Treatments vary along two dimensions. First, subjects either start with low financial incentives for coordination, which typically leads to coordination failure, and then are switched to higher incentives or start with high incentives, which usually yield effective coordination, and are switched to low incentives. Second, as the key treatment variable, subjects either observe the effort levels chosen by all individuals in their experimental group (full feedback) or observe only the minimum effort (limited feedback). We find three primary results: (1) When starting from coordination failure the use of full feedback improves subjects’ ability to overcome coordination failure, (2) When starting with good coordination the use of full feedback has no effect on subjects’ ability to avoid slipping into coordination failure, and (3) History-dependence, defined as dependence of current effort levels on past incentives, is strengthened by the use of full feedback. Electronic Supplementary Material Supplementary material is available in the online version of this article at . JEL Classification C92, D23, J31, L23, M52     

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.