Order of Play, Forward Induction, and Presentation Effects in Two-Person Games

We investigate the effects of order-of-play (simultaneous, unobserved sequential and fully observed sequential play) and form of presentation (extensive vs. normal) in three simple two person games: battle-of-the-sexes with and without outside option and a three strategy game which differentiates between virtual observability (VO) and iterated elimination of dominated strategies as principles of equilibrium selection. VO predicts that knowledge of the order of play alone will affect the distribution of strategies chosen. We contrast this with the predictions of iterated elimination of dominated strategies. We report results from 1800 one-shot games conducted in 6 sessions with 120 subjects and analysed as panel data. The form of presentation strongly affects the distribution of outcomes and strategies. Information about order of play shifts the distribution of strategies away from the distribution in simultaneous play and towards the distribution in fully observed play, especially in the less complicated games presented in normal order. Order-of-play effects are less evident as complexity of the game increases. Extensive form presentation appears to induce sequential thinking even in simultaneously played games.     

A theory of sequential reciprocity

Many experimental studies indicate that people are motivated by reciprocity. Rabin [Amer. Econ. Rev. 83 (1993) 1281] develops techniques for incorporating such concerns into game theory and economics. His theory is developed for normal form games, and he abstracts from information about the sequential structure of a strategic situation. We develop a theory of reciprocity for extensive games in which the sequential structure of a strategic situation is made explicit, and propose a new solution concept—sequential reciprocity equilibrium—for which we prove an equilibrium existence result. The model is applied in several examples, and it is shown that it captures very well the intuitive meaning of reciprocity as well as certain qualitative features of experimental evidence.     

The Carrot vs. the Stick in Work Team Motivation

This paper reports on the use of carrot (positive) and stick (negative) incentives as methods of increasing effort among members of work teams. We study teams of four members in a laboratory environment in which giving effort towards the team goal is simulated by eliciting voluntary contributions towards the provision of a public good. We test the efficiency-improving properties of four distinct environments: monetary prizes given to high contributors versus monetary fines assessed to low contributors, where high/low contributor is defined first in terms of absolute contributions and then in terms of contributions relative to abilities—which we call handicapping. Our results show that both carrot and stick can increase efficiency (i.e., contributions) levels by 10–28%. We find that handicapped incentives promise the highest efficiency levels, and when handicapping is not used penalties may be more effective than prizes. The implications for work teams and suggestions for practical implementation are discussed.     

A Model of Behavior in Coordination Game Experiments

This paper constructs a structural model for behavior in expeiments where subjects play a simple coordination game repeatedly under a rotating partner scheme. The model assumes subjects' actions are stochastic best responses to beliefs about opponents' choices, and these beliefs update as subjects observe actual choices during the experiment. The model accounts for heterogeneity across subjects by regarding prior beliefs as random effects and estimating their distribution. Maximum likelihood estimates from experimental data suggest that distributions of initial beliefs vary across games, but in all games studied imply a convergence dynamic toward risk-dominant equilibrium.     

A Folk Theorem for Repeated Sequential Games

We study repeated sequential games where players may not move simultaneously in stage games. We introduce the concept of effective minimax for sequential games and establish a Folk theorem for repeated sequential games. The Folk theorem asserts that any feasible payoff vector where every player receives more than his effective minimax value in a sequential stage game can be supported by a subgame perfect equilibrium in the corresponding repeated sequential game when players are sufficiently patient. The results of this paper generalize those of Wen (1994), and of Fudenberg and Maskin (1986). The model of repeated sequential games and the concept of effective minimax provide an alternative view to the Anti–Folk theorem of Lagunoff and Matsui (1997) for asynchronously repeated pure coordination games.     

Experimental study of repeated team-games

We report an experiment in which the Intergroup Prisoner's Dilemma (IPD) game was contrasted with a structurally identical (single-group) Prisoner's Dilemma (PD). The games were played repeatedly for 40 rounds. We found that subjects were initially more likely to cooperate in the IPD game than in the PD game. However, cooperation rates decreased as the game progressed and, as a result, the differences between the two games disappeared. This pattern is consistent with the hypothesis that subjects learn the structure of the game and adapt their behavior accordingly. Computer simulations based on a simple learning model by Roth and Erev (Learning in extensive-form games: Experimental data and simple dynamic models in the intermediate term, Games and Economic Behavior 8, 164–212, 1995) support this interpretation.     

Cooperation in an Overlapping Generations Experiment

Recent theoretical work shows that folk theorems can be developed for infinite overlapping generations games. Cooperation in such games can be sustained as a Nash equilibrium. But, of course, there are other equilibria. This paper investigates experimentally whether cooperation actually occurs in a simple overlapping generations game. Subjects both play the game and formulate strategies. Our main finding is that subjects fail to exploit the intertemporal structure of the game. Even when we provided subjects with a recommendation to play the grim trigger strategy, most of the subjects still employed safe history-independent strategies. Journal of Economic Literature Classification Numbers: C72, C92, D90.     

Tacit Coordination in a Decentralized Market Entry Game with Fixed Capacity

We focus on a class of market entry games in which a newly emergent market opportunity may be fruitfully exploited by no more than a commonly known, exogenously determined number of firms. Our results show significant effects of the parameters manipulated in the study, namely, the market capacity, entry fee, and method of subject assignment to groups (fixed vs. random). In contrast to previous market entry games with linear payoff functions, we find no evidence of convergence to equilibrium play on the aggregate level. Shifting the focus of the analysis from the aggregate to the individual level, four clusters of subjects are identified. The patterns are: (1) choice of the same action that is independent of the parameters of the game or the outcome of previous presentations of the same game; (2) random choices with probabilities prescribed by the equilibrium solution for risk-neutral players; (3) random choices with probabilities equal to the individual observed overall proportion of entry; and (4) sequential dependencies that violate any model that assumes randomization. Subjects in the fourth and largest category are shown to adjust their choices in accordance with a simple principle of strategic reasoning.     

Reference Points and Negative Reciprocity in Simple Sequential Games

We investigate experimentally whether preferences over an outcome depend on what other possible outcomes of the situation under consideration are, i.e., whether choices are “menu dependent.” In simple sequential games we analyze whether reactions to a certain benchmark outcome are influenced by changes in the payoffs of another outcome, not attainable at that time, called the “reference point.” Our data provide evidence that is favorable to the notion of menu dependence. Alterations of the reference point can lead to quantitatively significant changes in behavior at the benchmark outcome. The behavior we observe can be interpreted in terms of negative reciprocity. Journal of Economic Literature Classification Numbers: C91, C92, C72.     

Rationalizability and equilibrium in N-person sequential bargaining

Summary. This paper deals with N-person sequential bargaining games with complete information. For N-person sequential bargaining games, uniqueness of the SPE has been obtained by allowing the players to exit with partial agreements. Adopting a non-equilibrium approach, we show that N-person sequential bargaining games with exit are solvable by a refinement of rationalizability for multi-stage games (trembling-hand rationalizability) whatever the impatience of the players. That is, once we adopt the non-equilibrium approach, the exit opportunity still fulfils its original aim: we achieve a unique solution by introducing the exit opportunity. Moreover, this unique solution is the unique SPE. Received: October 30, 1996; revised version: July 7, 1998     

Unobservable Contracts as Precommitments

It is well known that signing publicly observable contracts with third parties is a means of credibly committing to certain actions and hence may yield strategic advantages. Previous work on the commitment value of unobservable contracts has been limited to normal form games and extensive form games in which only one party has the option to sign a contract. In this paper, we extend the analysis to extensive form games in which both players can sign contracts, and characterize the set of sequential equilibria. We show that any Nash equilibrium outcome of the original game in which both players receive more than their individually rational payoffs can be supported as a sequential equilibrium outcome. Therefore, delegation acts not only as a commitment device to gain advantage over the opponent, but also as a cooperative device to attain Pareto improvements over the subgame perfect equilibrium outcome. I would like to thank Atila Abdulkadiroglu, Jean-Pierre Benoit, Alberto Bisin, Boyan Jovanovic, Ehud Kalai, Giuseppe Lopomo, George Mailath, Efe Ok, Ariel Rubinstein, Andy Schotter, seminar participants at various universities and conferences, and anonymous referees for helpful comments and suggestions. Support from the Alfred P. Sloan Foundation and Program for Economic Research at Columbia University is also gratefully acknowledged.     

Subgame-perfect equilibria of finite- and infinite-horizon games

We show that subgame-perfect equilibria of infinite-horizon games arise as limits, as the horizon grows long and epsilon small, of subgame-perfect epsilon-equilibria of games which are truncated after a finite horizon. A number of applications show that this result provides a useful technique for analyzing the existence and uniqueness of infinite-horizon equilibria. We extend our result to the sequential equilibrium concept.     

On the impossibility of achieving no regrets in repeated games

Regret-minimizing strategies for repeated games have been receiving increasing attention in the literature. These are simple adaptive behavior rules that lead to no regrets and, if followed by all players, exhibit nice convergence properties: the average play converges to correlated equilibrium, or even to Nash equilibrium in certain classes of games. However, the no-regret property relies on a strong assumption that each player treats her opponents as unresponsive and fully ignores the opponents’ possible reactions to her actions. We show that if at least one player is slightly responsive, it is impossible to achieve no regrets, and convergence results for regret minimization with responsive opponents are unknown.     

Sequential, nonzero-sum “Blotto”: Allocating defensive resources prior to attack

The strategic allocation of resources across multiple fronts has long been studied in the context of Blotto games in which two players simultaneously select their allocations. However many allocation problems are sequential. For example, a state trying to defend against a terrorist attack generally allocates some or all of its resources before the attacker decides where to strike. This paper studies the allocation problem confronting a defender who must decide how to distribute limited resources across multiple sites before an attacker chooses where to strike. Unlike many Blotto games which only have very complicated mixed-strategy equilibria, the sequential, nonzero-sum “Blotto” game always has a very simple pure-strategy subgame perfect equilibrium. Further, the defender always plays the same pure strategy in any equilibrium, and the attacker's equilibrium response is generically unique and entails no mixing. The defender minmaxes the attacker in equilibrium even though the game is nonzero-sum, and the attacker strikes the site among its best replies that minimizes the defender's expected losses.     

Sequential,nonzero-sum “Blotto”: Allocating defensive resources prior to attack

The strategic allocation of resources across multiple fronts has long been studied in the context of Blotto games in which two players simultaneously select their allocations. However many allocation problems are sequential. For example, a state trying to defend against a terrorist attack generally allocates some or all of its resources before the attacker decides where to strike. This paper studies the allocation problem confronting a defender who must decide how to distribute limited resources across multiple sites before an attacker chooses where to strike. Unlike many Blotto games which only have very complicated mixed-strategy equilibria, the sequential, nonzero-sum “Blotto” game always has a very simple pure-strategy subgame perfect equilibrium. Further, the defender always plays the same pure strategy in any equilibrium, and the attacker's equilibrium response is generically unique and entails no mixing. The defender minmaxes the attacker in equilibrium even though the game is nonzero-sum, and the attacker strikes the site among its best replies that minimizes the defender's expected losses.     

The value of commitment in contests and tournaments when observation is costly

We study the value of commitment in sequential contests when the follower faces small costs to observe the leader's effort. We show that the value of commitment vanishes entirely in this class of games. By contrast, in sequential tournaments—games where, at a cost, the follower can observe the effectiveness of the leader's effort—the value of commitment is preserved completely provided that the observation costs are sufficiently small.     

Preferences and beliefs in a sequential social dilemma: a within-subjects analysis

In empirical analyses of games, preferences and beliefs are typically treated as independent. However, if beliefs and preferences interact, this may have implications for the interpretation of observed behavior. Our sequential social dilemma experiment allows us to separate different interaction channels. When subjects play both roles in such experiments, a positive correlation between first- and second-mover behavior is frequently reported. We find that the observed correlation primarily originates via an indirect channel, where second-mover decisions influence beliefs through a consensus effect, and the first-mover decision is a best response to these beliefs. Specifically, beliefs about second-mover cooperation are biased toward own second-mover behavior, and most subjects best respond to stated beliefs. However, we also find evidence for a direct, preference-based channel. When first movers know the true probability of second-mover cooperation, subjects' own second moves still have predictive power regarding their first moves.     

REPEATED MORAL HAZARD AND CONTRACTS WITH MEMORY: THE CASE OF RISK‐NEUTRALITY*

We consider a repeated moral hazard problem where both the principal and the wealth‐constrained agent are risk‐neutral. In each of two periods, the agent can exert unobservable effort, leading to success or failure. Incentives provided in the second period act as carrot and stick for the first period, so that the effort level induced in the second period is higher after a first‐period success than after a failure. If renegotiation cannot be prevented, the principal may prefer a project with lower returns; i.e., a project may be “too good” to be financed or, similarly, an agent can be “overqualified.”     

Endogenous private information structures

We formally incorporate the option to gather information into a game and thus endogenize the information structure. We ask whether models with exogenous information structures are robust with respect to this endogenization. Any Nash equilibrium of the game with information acquisition induces a Nash equilibrium in the corresponding game with an exogenous structure. We provide sufficient conditions on the structure of the game for which this remains true when ‘Nash’ is replaced by ‘sequential’. We characterize the (sequential) Nash equilibria of games with exogenous information structures that can arise as a (sequential) Nash equilibrium of games with endogenous information acquisition.     

Coordination and transfer

We study the ability of subjects to transfer principles between related coordination games. Subjects play a class of order statistic coordination games closely related to the well-known minimum (or weak-link) and median games (Van Huyck et al. in Am Econ Rev 80:234–248, 1990, Q J Econ 106(3):885–910, 1991). When subjects play a random sequence of games with differing order statistics, play is less sensitive to the order statistic than when a fixed order statistic is used throughout. This is consistent with the prediction of a simple learning model with transfer. If subjects play a series of similar stag hunt games, play converges to the payoff dominant equilibrium when a convention emerges, replicating the main result of Rankin et al. (Games Econ Behav 32:315–337, 2000). When these subjects subsequently play a random sequence of order statistic games, play is shifted towards the payoff dominant equilibrium relative to subjects without previous experience. The data is consistent with subjects absorbing a general principle, play of the payoff dominant equilibrium, and applying it in a new related setting.