Group play in games and the role of consent in network formation

We study games played between groups of players, where a given group decides which strategy it will play through a vote by its members. When groups consist of two voting players, our games can also be interpreted as network-formation games. In experiments on Stag Hunt games, we find a stark contrast between how groups and individuals play, with payoffs playing a primary role in equilibrium selection when individuals play, but the structure of the voting rule playing the primary role when groups play. We develop a new solution concept, robust-belief equilibrium, which explains the data that we observe. We provide results showing that this solution concept has application beyond the particular games in our experiments.     

Conditional behavior and learning in similar stag hunt games

This paper reports an experiment using Stag Hunt games with different payoff ranges. Specifically, in one treatment for half of the games the payoff dominant and the risk dominant equilibrium coincide and for the other half of the games they conflict; in the second treatment they always conflict. The experiment provides evidence that the payoff range experienced by the participant influences the likelihood of efficient conventions emerging. In particular, experiencing games where payoff dominance and risk dominance coincide appears to make payoff dominance more attractive in games in which they conflict. In the experiment, we also observe conditional behavior emerging with experience. We develop a model of conditional expectations to explain these stylized facts that depends crucially on the assumption that after a brief learning period participants categorize their experience using the same relative bandwidth in both treatments even though the range of experience is twice as large in treatment 1 as it is in treatment 2. The assumption cannot be rejected by the data. The analysis provides a formal example in which increasing experienced diversity by changing the way similar experiences are categorized increases the likelihood of efficient conventions emerging in communities playing similar Stag Hunt 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.     

Voluntarily separable repeated Prisonerʼs Dilemma with reference letters

We consider the voluntarily separable repeated Prisoner?s Dilemma model in which players randomly meet and form pairs to repeatedly play Prisoner?s Dilemma only by mutual agreement. While the literature has dealt with the case of no information flow across partnerships, we consider the case in which players can issue a “reference letter” to verify at least that the partnership entered the cooperation phase. We show that such reference letters can be voluntarily provided by the partners even at some cost, and that the sheer existence of a letter shortens the trust-building periods and thus improves social efficiency.     

囚徒困境的均衡辨析

传统的 《博弈论》 分析最终结果是两个囚徒均会坦白，但这并不是两个囚徒的最好结局，如何才能达到最好结局？本文从 《博弈论》 中“理性人”的假设出发，对传统的 （坦白，坦白） 均衡提出质疑。本文指出理性人不会满足于传统的均衡，理性人会积极寻找双方利益最大化的均衡，并达到此均衡。本文给出了新的均衡：广义均衡，并给出广义均衡的求解过程。通过对引入公共支付函数的概念，公共支付函数表达了两个人的共同利益。通过公共支付函数的最大化，使两个囚徒公共利益最大化，以求得囚徒困境的新的均衡，让两个囚徒的结果达到最好———即两人均抵赖。本文还将公共支付函数运用于其他 《博弈论》 案例，并指出广义均衡比纳什均衡更具有社会意义，因为前者体现了公共利益。文中还对囚徒困境的机制设计问题做了分析，指出囚徒困境中机制的设计会诱导囚徒作出不正确的选择，从而没有达到双方利益的最大化。     

Learning to alternate

The Individual Evolutionary Learning (IEL) model explains human subjects’ behavior in a wide range of repeated games which have unique Nash equilibria. Using a variation of ‘better response’ strategies, IEL agents quickly learn to play Nash equilibrium strategies and their dynamic behavior is like that of humans subjects. In this paper we study whether IEL can also explain behavior in games with gains from coordination. We focus on the simplest such game: the 2 person repeated Battle of Sexes game. In laboratory experiments, two patterns of behavior often emerge: players either converge rapidly to one of the stage game Nash equilibria and stay there or learn to coordinate their actions and alternate between the two Nash equilibria every other round. We show that IEL explains this behavior if the human subjects are truly in the dark and do not know or believe they know their opponent’s payoffs. To explain the behavior when agents are not in the dark, we need to modify the basic IEL model and allow some agents to begin with a good idea about how to play. We show that if the proportion of inspired agents with good ideas is chosen judiciously, the behavior of IEL agents looks remarkably similar to that of human subjects in laboratory experiments.     

Restricted feedback in long term relationships

This paper studies long term relationships, modeled as repeated games, with restricted feedback. Players condition current play on summary statistics of past play rather than the entire history, as may be the case in online markets. Our state strategy equilibrium framework allows for arbitrary restrictions on strategies. We derive a recursive characterization for the set of equilibrium payoffs similar to that of Abreu, Pearce, and Stacchetti (1986, 1990) [2], [3] for perfect public equilibria and show that the set of equilibrium payoffs is the largest fixed point of a monotone operator. We use our characterization to derive necessary and sufficient conditions for efficient trade in a repeated product choice game where costumers condition their purchase decisions only on the last performance signal.     

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.     

Cooperative behavior and the frequency of social interaction

We report results from an experiment that examines play in an indefinitely repeated, two-player Prisoner's Dilemma game. Each experimental session involves N subjects and a sequence of indefinitely repeated games. The main treatment consists of whether agents are matched in fixed pairings or matched randomly in each indefinitely repeated game. Within the random matching treatment, we elicit player's strategies and beliefs or vary the information that players have about their opponents. Contrary to a theoretical possibility suggested by Kandori [1992. Social norms and community enforcement. Rev. Econ. Stud. 59, 63–80], a cooperative norm does not emerge in the treatments where players are matched randomly. On the other hand, in the fixed pairings treatment, the evidence suggests that a cooperative norm does emerge as players gain more experience.     

On the Irrelevance of Risk Attitudes in Repeated Two-Outcome Games

We study equilibrium and maximin play in supergames consisting of the sequential play of a finite collection of stage games, where each stage game has two outcomes for each player. We show that for two-player supergames in which each stage game is strictly competitive, in any Nash equilibrium of the supergame, play at each stage is a Nash equilibrium of the stage game provided preferences over certain supergame outcomes satisfy a natural monotonicity condition. In particular, equilibrium play does not depend on risk attitudes. We establish an invariance result for games with more than two players when the solution concept is subgame perfection. Journal of Economic Literature Classification Numbers: C72, C9.     

Batch queues with choice of arrivals: Equilibrium analysis and experimental study

We study the decisions agents make in two queueing games with endogenously determined arrivals and batch service. In both games, agents are asked to independently decide when to join a queue, or they may simply choose not to join it at all. The symmetric mixed-strategy equilibrium of two games in discrete time where balking is prohibited and where it is allowed are tested experimentally in a study that varies the game type (balking vs. no balking) and information structure (private vs. public information). With repeated iterations of the stage game, all four experimental conditions result in aggregate, but not individual, behavior approaching mixed-strategy equilibrium play. Individual behavior can be accounted for by relatively simple heuristics.     

Why Any Seatbelt Mandate is an Infinitely Iterated Prisoner’s Dilemma: A Health Economics Perspective

Abstract

Among the most controversial of public health measures are those that restrict people’s freedom of choice presumably for the purpose of protecting their own health and safety. Mandatory use of automobile seatbelts can inspire allusions to the “tyranny of health” and the ‘health police.” This study revisits state seatbelt legislation, which is in force in all but one state of the USA. In modeling seatbelt mandates as indefinitely repeated Prisoner’s Dilemma (PD) games, the study qualifies and overcomes the limitations of conventional interest group and expressive voting theories. It indicates why and how utility-maximizing players, particularly carmakers, insurance companies, consumer advocates and regulators, may choose to cooperate, even in instances where mutual defection would bring each of them higher payoffs. The study suggests why state seatbelt mandates endure, despite ongoing concerns that they undermine individual autonomy or that their social costs outweigh the benefits of law-enforcement. The evolution and mediating effects of reputation and reciprocity in otherwise highly contestable healthcare decisions are consequently explored. Finally, the study identifies the implications of indefinitely repeated games on the continuing nature of strategic relationships and for mitigating conflict over traffic safety regulation, regardless of time and place.     

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.     

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.     

Cooperation,imitation and partial rematching

We study a setting in which imitative players are matched into pairs to play a Prisonerʼs Dilemma game. A well-known result in such setting is that under random matching cooperation vanishes for any interior initial condition. The novelty of this paper is that we consider partial rematching: players that belong to a pair in which both parties cooperate repeat partner next period, while all other players are randomly matched into pairs. Under certain conditions, this rematching mechanism makes cooperation the unique outcome in the long run. Furthermore, we show that if imitation happens infrequently enough then cooperative behavior is always present in the population.     

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.     

产业内贸易中的企业行为与贸易政策——基于古诺双寡头垄断模型的分析

20世纪60年代以后,学者研究发现,传统的贸易理论无法解释大量工业化国家之间的贸易,从而使产业内贸易的研究成为国际贸易的核心问题之一。本文建立古诺双寡头垄断模型对同质产品产业内贸易中的企业行为与贸易政策进行了博弈分析,通过研究发现：寡头垄断企业之间即使在比较优势和报酬递增都不存在的条件下也会互相向对方市场销售商品;追求社会福利最大化的政府很容易陷入贸易保护的囚徒困境,从而导致双方社会福利恶化。本文认为,政府之间应该加强交往,深化合作,通过谈判解决贸易争端,避免陷入贸易保护的囚徒困境。     

Characterising equilibrium selection in global games with strategic complementarities

Global games are widely used to predict behaviour in games with strategic complementarities and multiple equilibria. We establish two results on the global game selection. First, we show that, for any supermodular complete information game, the global game selection is independent of the payoff functions chosen for the game?s global game embedding. Second, we give a simple sufficient criterion to derive the selection and establish noise independence in many-action games by decomposing them into games with smaller action sets, to which we may often apply simple criteria. We also report in which small games noise independence may be established by counting the number of players or actions.     

Modeling finitely-repeated games with uncertain termination

A supergame is modeled in which two agents play a game repeatedly and while the finiteness of the repetitions is common knowledge, the number of repetitions is not. Such games can yield non-standard results. This is illustrated with the Prisoner's Dilemma where a cooperative outcome is shown to be possible.     

Voluntarily Separable Repeated Prisoner's Dilemma

Ordinary repeated games do not apply to real societies where one can cheat and escape from partners. We formulate a model of endogenous relationships that a player can unilaterally end and start with a randomly assigned new partner with no information flow. Focusing on two-person, two-action Prisoner's Dilemma, we show that the endogenous duration of partnerships generates a significantly different evolutionary stability structure from ordinary random matching games. Monomorphic equilibria require initial trust building, while a polymorphic equilibrium includes earlier cooperators than any strategy in monomorphic equilibria and is thus more efficient. This is due to the non-linearity of average payoffs.