Trembles in extensive games with ambiguity averse players

We introduce and analyze three definitions of equilibrium for finite extensive games with imperfect information and ambiguity averse players. In a setting where players’ preferences are represented by maxmin expected utility, as characterized in Gilboa and Schmeidler (J Math Econ 18(2):141–153, 1989), our definitions capture the intuition that players may consider the possibility of slight arbitrary mistakes. This generalizes the idea leading to trembling-hand perfect equilibrium as introduced in Selten (Int J Game Theory 4(1):25–55, 1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions and relate our definitions to standard equilibrium concepts with expected utility maximizing players. Our analysis shows that ambiguity aversion can lead to behavioral implications that are distinct from those attained under expected utility maximization, even if ambiguous beliefs only arise from the possibility of slight mistakes in the implementation of unambiguous strategies.     

INCOMPLETE INFORMATION GAMES WITH MULTIPLE PRIORS*

We present a model of incomplete information games, where each player is endowed with a set of priors. Upon arrival of private information, it is assumed that each player “updates” his set of priors to a set of posterior beliefs, and then evaluates his actions by the most pessimistic posterior beliefs. So each player's preferences may exhibit aversion to ambiguity or uncertainty. We define a couple of equilibrium concepts, establish existence results for them, and demonstrate by examples how players’ views on uncertainty about the environment affect the strategic outcomes.     

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.     

Existence of valuation equilibria when equilibrium strategies cannot differentiate between equal ties

This paper reconsiders the valuation equilibrium concept (Jehiel and Samet, 2007) and proposes an additional regularity condition concerning the players? equilibrium strategies. The condition, which requires equilibrium strategies to induce the same local behaviour at all nodes with “similar” optimal actions, increases both the predictive power and the internal consistency of the concept — especially when used as a tool to study boundedly rational behaviour in games with imperfect information and/or imperfect recall. It is shown not to conflict with existence.     

Are the treasures of game theory ambiguous?

Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001) experimentally study a number of games. In each case, they initially find strong support for Nash equilibrium; however, by changing an apparently irrelevant parameter, they find results which contradict Nash equilibrium. In this paper, we study the five normal form games from Goeree and Holt (Am Econ Rev 91:1402?C1422, 2001). We argue that their results may be explained by the hypothesis that subjects view their opponents?? behaviour as ambiguous. Ambiguity-aversion causes players to avoid strategies, which give low out of equilibrium payoffs. Similarly, ambiguity preference can make strategies with high payoffs more attractive.     

Decompositions and potentials for normal form games

A basic model of commitment is to convert a two-player game in strategic form to a “leadership game” with the same payoffs, where one player, the leader, commits to a strategy, to which the second player always chooses a best reply. This paper studies such leadership games for games with convex strategy sets. We apply them to mixed extensions of finite games, which we analyze completely, including nongeneric games. The main result is that leadership is advantageous in the sense that, as a set, the leader's payoffs in equilibrium are at least as high as his Nash and correlated equilibrium payoffs in the simultaneous game. We also consider leadership games with three or more players, where most conclusions no longer hold.     

OPTIMISM AND PESSIMISM IN GAMES

This article considers the impact of ambiguity in strategic situations. It extends the existing literature on games with ambiguity‐averse players by allowing for optimistic responses to ambiguity. We use the CEU model of ambiguity with a class of capacities introduced by Jaffrray and Philippe (Mathematics of Operations Research 22 (1997), 165–85), which allows us to distinguish ambiguity from ambiguity‐attitude, and propose a new solution concept, equilibrium under ambiguity (EUA), for players who may be characterized by ambiguity‐preference. Applying EUA, we study comparative statics of changes in ambiguity‐attitude in games with strategic complements. This extends work in Eichberger and Kelsey (Journal of Economic Theory 106 (2002), 436–66) on the effects of increasing ambiguity if players are ambiguity averse.     

The dual theory of the smooth ambiguity model

This paper studies the “dual” theory of the smooth ambiguity model introduced by Klibanoff et al. (Econometrica 73:1849–1892, 2005). Unlike the original model, we characterize attitudes toward ambiguity captured by second-order probabilities. First, we give a set of axioms to derive a dual representation of the smooth ambiguity model. Second, we present a characterization of ambiguity aversion. Last, as an application of our dual model to a portfolio problem, we conduct comparative static predictions which give sufficient conditions to guarantee that an increase in smooth ambiguity aversion decreases the optimal portfolio.     

The single deviation property in games with discontinuous payoffs

We study equilibrium existence in normal form games in which it is possible to associate with each nonequilibrium point an open neighborhood, a set of players, and a collection of deviation strategies, such that at any nonequilibrium point of the neighborhood, a player from the set can increase her payoff by switching to the deviation strategy designated for her. An equilibrium existence theorem for compact, quasiconcave games with two players is established as an application of a general equilibrium existence result for qualitative games. A new form of the better-reply security condition, called the strong single deviation property, is proposed.     

An Experiment to Evaluate Bayesian Learning of Nash Equilibrium Play

We report on an experiment designed to evaluate the empirical implications of Jordan's model of Bayesian learning in games of incomplete information. A finite example is constructed in which the model generates unique predictions of subjects' choices in nearly all periods. When the “true” game defined by players' private information was one with a unique equilibrium in pure strategies, the experimental subjects' play converged to the equilibrium, as Jordan's theory predicts, even when the subjects had not attained complete information about one another. But when there were two pure strategy equilibria, the theory's predictions were not consistent with observed behavior. Journal of Economic Literature Classification numbers: D83, C72, C92.     

Group efforts when performance is determined by the “best shot”

We investigate the private provision of a public good whose level is determined by the maximum effort made by a group member. Costs of effort are either commonly known or privately known. For symmetric perfect-information games, any number of players may be active and we characterize the unique (mixed-strategy) equilibrium in which active contributors use the same strategy. Increasing the number of active players leads to stochastically lower individual efforts and level of the public good. When information is private, the symmetric equilibrium is in pure strategies. Increasing the number of players yields a pointwise reduction in the equilibrium contribution strategy but an increase in equilibrium payoffs. Comparative statics with respect to costs and levels of risk aversion are derived. Finally, whether information is public or private, equilibria are inefficient—we provide mechanisms that improve efficiency.     

Repeated games with one-memory

We study the extent to which equilibrium payoffs of discounted repeated games can be obtained by 1-memory strategies. We establish the following in games with perfect (rich) action spaces: First, when the players are sufficiently patient, the subgame perfect Folk Theorem holds with 1-memory. Second, for arbitrary level of discounting, all strictly enforceable subgame perfect equilibrium payoffs can be approximately supported with 1-memory if the number of players exceeds two. Furthermore, in this case all subgame perfect equilibrium payoffs can be approximately supported by an ε-equilibrium with 1-memory. In two-player games, the same set of results hold if an additional restriction is assumed: Players must have common punishments. Finally, to illustrate the role of our assumptions, we present robust examples of games in which there is a subgame perfect equilibrium payoff profile that cannot be obtained with 1-memory. Thus, our results are the best that can be hoped for.     

Equilibrium play and best response to (stated) beliefs in normal form games

We report experimental results on a series of ten one-shot two-person 3×3 normal form games with unique equilibrium in pure strategies played by non-economists. In contrast to previous experiments in which game theory predictions fail dramatically, a majority of actions taken coincided with the equilibrium prediction (70.2%) and were best-responses to subjects' stated beliefs (67.2%). In constant-sum games, 78% of actions taken were predicted by the equilibrium model, outperforming simple K-level reasoning models. We discuss how non-trivial game characteristics related to risk aversion, efficiency concerns and social preferences may affect the predictive value of different models in simple normal form games.     

Dragon Slaying with Ambiguity: Theory and Experiments

This paper studies the impact of ambiguity in the best‐shot and weakest‐link models of public good provision. The models are first analyzed theoretically. Then, we conduct experiments to study how ambiguity affects behavior in these games. We test whether subjects' perception of ambiguity differs between a local opponent and a foreign one. We find that an ambiguity‐safe strategy is often chosen by subjects. This is compatible with the hypothesis that ambiguity aversion influences behavior in games. Subjects tend to choose contributions above (respectively, below) the Nash equilibrium in the best‐shot (respectively, weakest‐link) model.     

Equilibrium selection in signaling games with teams: Forward induction or faster adaptive learning?

Teams are shown to violate the most basic of equilibrium refinements in signaling games: single-round deletion of dominated strategies (part of the Cho–Kreps intuitive criteria). This is important because, to the extent that teams can be easily induced to violate the most basic of equilibrium refinements even under a “best case” scenario (teams that rapidly develop strategic play in games of this sort), it implies that one must rely on learning models, and past empirical research with these models, when predicting equilibrium outcomes.     

Equilibrium play in matches: Binary Markov games

We study two-person extensive form games, or “matches,” in which the only possible outcomes (if the game terminates) are that one player or the other is declared the winner. The winner of the match is determined by the winning of points, in “point games.” We call these matches binary Markov games. We show that if a simple monotonicity condition is satisfied, then (a) it is a Nash equilibrium of the match for the players, at each point, to play a Nash equilibrium of the point game; (b) it is a minimax behavior strategy in the match for a player to play minimax in each point game; and (c) when the point games all have unique Nash equilibria, the only Nash equilibrium of the binary Markov game consists of minimax play at each point. An application to tennis is provided.     

Kindness,confusion, or … ambiguity?

Other-regarding preferences or decision errors are the main explanations put forward to justify contributions exceeding the non-cooperative optimum in VCM games. An alternative rationale relies on ambiguity aversion. Ambiguity aversion increases the perceived marginal benefit of own contributions, which in equilibrium will exceed the Nash level. We present a series of experiments testing this hypothesis. To control for other-regarding preferences, we run a two-player game in which a human player plays with a virtual agent. Players are assigned either to a risky setting (known probabilities of opponent’s choices) or to an ambiguity setting (probabilities of opponent’s contribution are vague). Results show that ambiguity affects contributions. However, attitude to ambiguity appears to be affected by the location of the aggregate Nash optimum inside the decision space.     

Fault tolerance in large games

A Nash equilibrium is an optimal strategy for each player under the assumption that others play according to their respective Nash strategies, but it provides no guarantees in the presence of irrational players or coalitions of colluding players. In fact, no such guarantees exist in general. However, in this paper we show that large games are innately fault tolerant. We quantify the ways in which two subclasses of large games – λ-continuous games and anonymous games – are resilient against Byzantine faults (i.e. irrational behavior), coalitions, and asynchronous play. We also show that general large games have some non-trivial resilience against faults.     

知识表达、知识互补性、知识产权均衡

基于知识的“代数格”表达方式和知识的互补性原理 ,这篇文章讨论知识产权的博弈均衡问题。每个人都拥有“局部知识” ,显然 ,如果法律对每个人的局部知识设立产权并由政府全面加以保护的话 ,人类知识将无法获得任何发展。另一方面 ,如果法律不承认任何知识的产权且政府对任何知识产权都不加以保护的话 ,人类知识也难以获得发展的激励。这里将证明 ,N个具有局部知识的社会成员在社会整体知识构成的代数格上关于各自的“局部知识”的价格的博弈存在至少一个纳什均衡———此即该社会内部自恰的和最优的知识产权配置。由周林 ( 1 994) ,这类均衡点构成一完备格。又由MilgromandShannon( 1 994) ,博弈的每一参与者最佳对应集内的最大策略和最小策略在社会知识的代数格上 (按集合强序 )单调非减 ,这意味着当社会成员的初始知识水平有所提高时 ,由产权博弈均衡所决定的各个局部知识的知识产权价格将相应有所上升。所有上述结果都不需要通常的凸性假设 ,因此在以“知识”为核心要素的经济分析中 (包括收益递增经济学 )具有独特重要的意义     

Asymmetric evolutionary games with non-linear pure strategy payoffs

In this paper, I extend the analytical framework of evolutionary game theory to games with two distinct types of players where the type-specific payoff functions are nonlinear on the right—bi-player-type nonlinear (BNL) evolutionary games. That is, asymmetric games where the payoffs for pairwise interactions are influenced by the ambient frequencies of the pertinent strategies of both types of players.While this paper is motivated by my interest in inter- and intra-gender conflicts of interest in sexual reproduction, BNL games also arise just as naturally in most two-species interactions. Thus, the methodology may provide a novel approach to investigating such classical ecological phenomena as prey–predator/host–parasite interactions and interspecies competition for resources.Finally, given the growing tendency to utilize evolutionary games in economic modeling, a case can be made for the current paper being of interest to traditional game-theoretic modelers in connection with situations such as buyer/seller or employer/employee interaction.