Risk attitudes and the stag-hunt game

A stag-hunt game (with the risky and safe actions) has two pure Nash equilibria that are Pareto-rankable. The risky action leads either to the Pareto-superior equilibrium (high payoff) or to out of equilibrium (low payoff) depending on the opponent’s action. Both players may want to obtain high payoff but uncertainty about the opponent’s action may prevent them to take such strategic risk. This paper investigates how information about the risk attitude of an opponent affects a player’s action choice in the stag-hunt game. I find that although a subject’s propensity to choose the risky action depends on her opponent’s risk attitude, it does not depend on her own risk attitude.     

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.      

Coordination and Information in Critical Mass Games: An Experimental Study

We present experimental results on a repeated coordination game with Pareto-ranked equilibria in which a payoff from choosing an action is positive only if a critical mass of players choose that action. We design a baseline version of the game in which payoffs remain constant for values above the critical mass, and an increasing returns version in which payoffs keep increasing for values above the critical mass. We test the predictive power of security and payoff-dominance under different information treatments. Our results show that convergence to the payoff-dominant equilibrium is the modal limit outcome when players have full information about others' previous round choices, while this outcome never occurs in the remaining treatments. The paths of play in some groups reveal a tacit dynamic coordination by which groups converge to the efficient equilibrium in a step-like manner. Moreover, the frequency and speed of convergence to the payoff-dominant equilibrium are higher, ceteris paribus, when increasing returns are present. Finally, successful coordination seems to crucially depend on players' willingness to signal to others the choice of the action supporting the efficient equilibrium.     

Dynamic common agency

A general model of dynamic common agency with symmetric information is considered. The set of truthful Markov perfect equilibrium payoffs is characterized and the efficiency properties of the equilibria are established. A condition for the uniqueness of equilibrium payoffs is derived for the static and the dynamic game. The payoff is unique if and only if the payoff of each principal coincides with his marginal contribution to the social value of the game. The dynamic model is applied to a game of agenda setting.     

Binary Games with State Dependent Stochastic Choice

This paper studies a stochastic equilibrium selection model for binary coordination games. Players switch strategies stochastically so that the mistake probabilities are fully dependent on the population states. A probabilistic behavior is said to be aspiration (imitation, resp.) oriented if strategy switches are mainly driven by the aspiration (imitation, resp.) effect. In general, a strategy switch by one player generates externalities on others. Strategies in a coordination game can be classified according to the relative magnitude of their externality effects. It is shown that the selection outcome for a linear coordination game is determined in a specific way by the balance of the risk dominance, the aspiration/imitation, and the externality effects. It is also shown that an aspiration (imitation, resp.) oriented behavior tends to select payoff dominant (maxmin, resp.) equilibrium and that risk dominant equilibrium is always selected if and only if the aspiration and the imitation effects exactly cancel each other out, which in turn makes the selection process insensitive to externality effects. Journal of Economic Literature Classification Number: C72.     

Coordination and delay in global games

What is the effect of offering agents an option to delay their choices in a global coordination game? We address this question by considering a canonical binary action global game, and allowing players to delay their irreversible decisions. Those that delay have access to accurate private information at the second stage, but receive lower payoffs. We show that, as noise vanishes, as long as the benefit to taking the risky action early is greater than the benefit of taking the risky action late, the introduction of the option to delay reduces the incidence of coordination failure in equilibrium relative to the standard case where all agents must choose their actions at the same time. We outline the welfare implications of this finding, and probe the robustness of our results from a variety of angles.     

Repeated games with present-biased preferences

We study infinitely repeated games with perfect monitoring, where players have β-δ preferences. We compute the continuation payoff set using recursive techniques and then characterize equilibrium payoffs. We then explore the cost of the present-time bias, producing comparative statics. Unless the minimax outcome is a Nash equilibrium of the stage game, the equilibrium payoff set is not monotonic in β or δ. Finally, we show how the equilibrium payoff set is contained in that of a repeated game with smaller discount factor.     

Technology adoption in markets with network effects: Theory and experimental evidence

We examine a technology-adoption game with network effects in which coordination on either technology A or technology B constitutes a Nash equilibrium. Coordination on technology B is assumed to be payoff dominant. We define a technology’s critical mass as the minimal share of users, which is necessary to make the choice of this technology the best response for any remaining user. We show that the technology with the lower critical mass implies risk dominance and selection by the maximin criterion. We present experimental evidence that both payoff dominance and risk dominance explain participants’ choices in the technology-adoption game. The relative riskiness of a technology can be proxied using either technologies’ critical masses or stand-alone values absent any network effects.     

UNIQUENESS OF EQUILIBRIUM PAYOFFS IN FINITELY REPEATED GAMES WITH IMPERFECT MONITORING*

This paper examines when a finitely repeated game with imperfect monitoring has a unique equilibrium payoff vector. This problem is nontrivial under imperfect monitoring, because uniqueness of equilibrium (outcome) in the stage game does not extend to finitely repeated games. A (correlated) equilibrium is equilibrium minimaxing if any player's equilibrium payoff is her minimax value when the other players choose a correlated action profile from the actions played in the equilibrium. The uniqueness result holds if all stage game correlated equilibria are equilibrium minimaxing and have the same payoffs. The uniqueness result does not hold under weaker conditions.     

Coordination of strategic responses to security threats: Laboratory evidence

Laboratory experiments are used to evaluate the extent to which players in games can coordinate investments that diminish the probability of losses due to security breaches or terrorist attacks. In this environment, economically sensible investments may be foregone if their potential benefits are negated by failures to invest in security at other sites. The result is a coordination game with a desirable high-payoff, high-security equilibrium and an undesirable low-security equilibrium that may result if players do not expect others to invest in security. One unique feature of this coordination situation is that investment in security by one player generates a positive externality such that all other players’ expected payoffs are increased, regardless of those other players’ investment decisions. Coordination failures are pervasive in a baseline experiment with simultaneous decisions, but coordination is improved if players are allowed to move in an endogenously determined sequence. In addition, coordinated security investments are observed more often when the largest single security threat to individuals is preventable by their own decisions to invest in security. The security coordination game is a “potential game,” and the success of coordination on the more secure equilibrium is related to the notion of potential function maximization and basin of attraction.      

Bargaining and markets: complexity and the competitive outcome

Rubinstein and Wolinsky (Rev. Econ. Stud. 57 (1990) 63-78) consider a simple decentralised market game in which agents meet randomly or voluntarily and bargain over the terms of trade. They show that any individually rational price can be sustained as a sequential equilibrium even though the model has a unique competitive outcome. Here, I consider Rubinstein and Wolinsky's model and show that if complexity costs of implementing strategies enter players’ preferences, together with the standard payoff in the game, then every equilibrium is stationary/Markov and induces the unique competitive price. Thus, I demonstrate that aversion to complexity may provide a justification for the competitive outcome.     

When less information is good enough: experiments with global stag hunt games

There is mixed evidence on whether subjects coordinate on the efficient equilibrium in experimental stag hunt games under complete information. A design that generates an anomalously high level of coordination, Rankin et al. (Games Econo Behav 32(2):315–337, 2000), varies payoffs each period in repeated play rather than holding them constant. These payoff “perturbations” are eerily similar to those used to motivate the theory of global games, except the theory operates under incomplete information. Interestingly, that equilibrium selection concept is known to coincide with risk dominance, rather than payoff dominance. Thus, in theory, a small change in experimental design should produce a different equilibrium outcome. We examine this prediction in two treatments. In one, we use public signals to match Rankin et al. (2000)’s design; in the other, we use private signals to match the canonical example of global games theory. We find little difference between treatments, in both cases, subject play approaches payoff dominance. Our literature review reveals this result may have more to do with the idiosyncrasies of our complete information framework than the superiority of payoff dominance as an equilibrium selection principle.     

Would Excess Capacity in Public Firms Be Socially Optimal?

We analyse oligopolistic interactions between a welfare-maximizing public firm and a profit-maximizing private firm in a repeated game. We find that the public firm can hold excess capacity as a strategic punishment device to sustain a subgame perfect equilibrium which is welfare-superior to the static Nash equilibrium. Basically, potential punishment from the public firm in the dynamic game can make the self-interested private firm behave in the public interest. Furthermore, if capacity is endogenous, public excess capacity can occur in a welfare efficient equilibrium when the cost of public capacity investment is higher than that of private investment.     

A consumer-based model of competitive diffusion: the multiplicative effects of global and local network externalities

Competitive diffusion of two incompatible technologies, such as PC vs. Macintosh, VHS vs. Betamax and so on, is studied under the framework of a spatial game in which consumers are distributed on a two-dimensional square lattice network. The consumers play coordination-like games with their nearest neighbors and imitate the most successful strategy in their neighborhood in terms of aggregated payoffs after each round. The effects of global network externality are realized in the dynamic payoff matrix of the game, and the framework of spatial game provides the model with the effects of local network externality. These two types of externalities are set as multiplicative, that is, as nonlinear. Both simulations and mean-field approximation show that not only total but also partial standardization (robust polymorphic equilibrium) occurs depending upon the parameters and initial configurations, even when there are positive effects of both global and local network externalities. Moreover, effects of innovation factors that alter paths toward a lock-in situation are studied. It is shown that both the timing and the size of the innovation factors matter for a disadvantaged technology in order to overwhelm a market.JEL Classification: C7, D8, O3We are obliged to Professor John Paul Boyd at the University of California, Irvine and our anonymous referees for their constructive comments.     

Risk and evolution

Summary. I examine a Knightian (1921) model of risk using a general equilibrium model of investment and trade. A population of agents with various preference types can choose between a safe production technology and a risky production technology. In addition, the distribution of types of agents changes through a standard evolutionary dynamic. For a given population distribution, the equilibrium is in general inefficient, however, by allowing the population distribution to change in response to market generated rewards, the population will converge to one where the equilibrium is efficient and where the population as a whole behaves as if all agents were risk neutral. Received: November 7, 1996; revised version: October 20, 1997     

Finitely repeated games with monitoring options

We study finitely repeated games where players can decide whether to monitor the other players? actions or not every period. Monitoring is assumed to be costless and private. We compare our model with the standard one where the players automatically monitor each other. Since monitoring other players never hurts, any equilibrium payoff vector of a standard finitely repeated game is an equilibrium payoff vector of the same game with monitoring options. We show that some finitely repeated games with monitoring options have sequential equilibrium outcomes which cannot be sustained under the standard model, even if the stage game has a unique Nash equilibrium. We also present sufficient conditions for a folk theorem, when the players have a long horizon.     

Institutional reform paths

This paper develops a dynamic population game in which agents play a simple anonymous-exchange game of cooperating or defecting. Agents switch to the strategy with a higher expected payoff. Reformers can affect the payoff structure of the stage game to maximize the number of cooperators in the population by either enacting legal reform (institutional quality of contract law) or focusing on the macro outlook of the economy. Based on the theoretical model, the paper predicts which types the reformer should enact first and under which conditions reform will not be successful.     

Satisficing and optimality in 2×2 common interest games

Summary. The paper studies the evolution of cooperation when satisficing players repeatedly play a symmetric two-by-two game of common interest. We show that if initial aspiration levels are sufficiently close to the efficient payoff and aspiration adjusts at a sufficiently slow speed then the unique long run state will be the efficient outcome. In the special case of coordination games, the more tension there is between payoff dominance and risk dominance, the longer it takes for the system to lock into the payoff dominant outcome. Received: June 23, 1997; revised version: November 19, 1997     

Coordination and learning in dynamic global games: experimental evidence

Coordination problems are ubiquitous in social and economic life. Political mass demonstrations, the decision whether to join a speculative currency attack, investment in a risky venture, and capital flight from a particular country are all characterized by coordination problems. Furthermore, all these events have a dynamic nature which has been largely omitted from previous experimental studies. Here I use a two-stage variant of a dynamic global game to study experimentally how the arrival of information in a dynamic setting affects the relative aggressiveness of speculators. In the first stage, subjects exhibit excess aggressiveness, which appears to be driven by beliefs about others’ actions rather than an intrinsic taste for attacking. However, following a failed first-stage attack, subjects learn to be less aggressive in the second stage. On the other hand, the arrival of new, more precise information after a failed attack leads to an increase in subjects’ aggressiveness. Beliefs, again, play a crucial role in explaining how the arrival of information affects attacking behavior.     

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.