Fictitious Play in Extensive Form Games

This article analyzes the fictitious play process originally proposed for strategic form games by Brown (1951) and Robinson (1951). We interpret the process as a model of preplay thinking performed by players before acting in a one-shot game. This model is one of bounded rationality. We discuss how fictitious play should then be defined for extensive form games and conclude that this is somewhat problematic. We therefore study two alternative definitions. For either of these, under a weak condition of initial uncertainty, a convergence point of a fictitious play sequence is a sequential equilibrium. For generic games of perfect information initial uncertainty also implies convergence of fictitious play.Journal of Economic LiteratureClassification Number: C72.     

Quantal Response Equilibria for Extensive Form Games

This article investigates the use of standard econometric models for quantal choice to study equilibria of extensive form games. Players make choices based on a quantal-choice model and assume other players do so as well. We define an agent quantal response equilibrium (AQRE), which applies QRE to the agent normal form of an extensive form game and imposes a statistical version of sequential rationality. We also define a parametric specification, called logit-AQRE, in which quantal-choice probabilities are given by logit response functions. AQRE makes predictions that contradict the invariance principle in systematic ways. We show that these predictions match up with some experimental findings by Schotter et al. (1994) about the play of games that differ only with respect to inessential transformations of the extensive form. The logit-AQRE also implies a unique selection from the set of sequential equilibria in generic extensive form games. We examine data from signaling game experiments by Banks et al. (1994) and Brandts and Holt (1993). We find that the logit-AQRE selection applied to these games succeeds in predicting patterns of behavior observed in these experiments, even when our prediction conflicts with more standard equilibrium refinements, such as the intuitive criterion. We also reexamine data from the McKelvey and Palfrey (1992) centipede experiment and find that the AQRE model can account for behavior that had previously been explained in terms of altruistic behavior. This revised version was published online in August 2006 with corrections to the Cover Date.     

Subgame Monotonicity in Extensive Form Evolutionary Games

A theory of subgame monotonicity is developed for finite extensive form games to model evolutionary processes that respect the subgame structure of the game. It is shown how these evolutionary processes can be decomposed into their evolution on smaller games and also recombined to simplify the analysis of the original game. When applied to the replicator dynamic, a generalization of the Wright manifold from population genetics is a natural consequence of monotonicity. Subgame monotonicity and the Wright manifold are also shown to arise naturally in imitation models based on the extensive form. Journal of Economic Literature Classification Numbers: C70, C72.     

An Example of Dynamic (In)Consistency in Symmetric Extensive Form Evolutionary Games

An example is developed to show that equilibrium selection via dynamic stability, when applied to the normal form of extensive form games, is inconsistent with the subgame structure. Alternative dynamic approaches are proposed and discussed to circumvent this inconsistency. Journal of Economic Literature Classification Numbers: C70, C72.     

Finitely Many Players with Bounded Recall in Infinitely Repeated Games

We study the set of limit points of equilibrium payoffs in n-player repeated games, with bounded recall, when the memory capacities of all the players grow to infinity. Two main issues are explored: (i) whether differential information enables players to play correlatively, and (ii) the extent to which boundedly rational players can learn others′ behavior patterns and conceal their own. Journal of Economic Literature Classification Number: 026.     

Bayesian Estimation of Entry Games with Multiple Players and Multiple Equilibria

Entry game models are often used to study the nature of firms' profits and the nature of competition among firms in empirical studies. However, when there are multiple players in an oligopoly market, the resulting multiple equilibria have made it difficult in previous studies to estimate the payoff functions of players in complete information, static and discrete games without using unreasonable assumptions. To overcome this difficulty, the present paper proposes a practical estimation method for an entry game with three players using a Bayesian approach. Some mild assumptions are imposed on the payoff function, and the average competitive effect is used to capture the entry effect of the number of firms. Our proposed methodology is applied to Japanese airline data from the year 2000, when there were three major airline companies, ANA, JAL and JAS. The model comparison is conducted to investigate the nature of strategic interaction among these Japanese airline companies.     

Extensive Form Implementation in Incomplete Information Environments

We study the implementation of social choice rules in incomplete information environments. A sufficiency condition calledposterior reversalis given for extensive form implementation. The condition has a natural interpretation in signaling terms: Consistent posterior distributions under truth-telling are different from consistent posteriors under deception. This variation in the distribution over player types leads to variation in the distribution over actions and outcomes (comparing truth-telling and deception). We exploit this feature to implement social choice rules.Journal of Economic LiteratureClassification Numbers: D78, D82.     

Measuring Strategic Uncertainty in Coordination Games

This paper proposes a method to measure strategic uncertainty by eliciting certainty equivalents analogous to measuring risk attitudes in lotteries. We apply this method by conducting experiments on a class of one-shot coordination games with strategic complementarities and choices between simple lotteries and sure payoff alternatives, both framed in a similar way. Despite the multiplicity of equilibria in the coordination games, aggregate behaviour is fairly predictable. The pure or mixed Nash equilibria cannot describe subjects' behaviour. We present two global games with private information about monetary payoffs and about risk aversion. While previous literature treats the parameters of a global game as given, we estimate them and show that both models describe observed behaviour well. The global-game selection for vanishing noise of private signals offers a good recommendation for actual players, given the observed distribution of actions. We also deduce subjective beliefs and compare them with objective probabilities.     

Endogenous Games and Mechanisms: Side Payments Among Players

We characterize the outcomes of games when players may make binding offers of strategy contingent side payments before the game is played. This does not always lead to efficient outcomes, despite complete information and costless contracting. The characterizations are illustrated in a series of examples, including voluntary contribution public good games, Cournot and Bertrand oligopoly, principal–agent problems, and commons games, among others.     

Working and Shirking: Equilibrium in Public-Goods Games with Overlapping Generations of Players

In overlapping-generations models of public-goods provision,in which the contribution decision is binary and lifetimes arefinite, the set of symmetric subgame-perfect equilibria canbe categorized into three types: seniority equilibria, in whichplayers contribute (effort) until a predetermined age and thenshirk thereafter; dependency equilibria, in which players initiallyshirk, then contribute for a set number of periods, then shirkfor the remainder of their lives; and sabbatical equilibria,in which players alternately contribute and shirk for periodsof varying length before entering a final stage of shirking.In a world without discounting we establish conditions for equilibriumand demonstrate that for any dependency equilibrium there isa seniority equilibrium that Pareto dominates it ex ante. Weproceed to characterize generational preferences over alternativeseniority equilibria. We explore the aggregation of these preferencesby embedding the public-goods provision game in a voting frameworkand solving for the majority-rule equilibria. In this way wecan think of political processes as providing one natural frameworkfor equilibrium selection in the original public-goods provisiongame.     

Socially Subjective Equilibrium in Strategic Form Games

This study provides a new framework and a new equilibrium concept, which are able to describe the situation where people have various images of the society and have various solution concepts for social outcomes, and where people accept the social outcomes. In socially subjective equilibrium, people have a coherence of their own norms in two senses. One is the consistency of the norm itself. Imagined outcomes should satisfy a certain (subjective) solution concept. The other is the consistency between the imagined outcomes and realized one. These are the main features of our equilibrium concept. This paper forms a part of my doctoral thesis, which is titled “On socially subjective equilibrium”. The first person I would like to thank is my direct supervisor Professor Ken Urai (Osaka University). I obtained a basic idea of the main concept of my doctoral thesis, that is, the solution concept scheme, from Professor Urai. I am grateful to Professor Hiroaki Nagatani (Osaka University) and Professor Ken-Ichi Shimomura (Kobe University), who monitored my works and took efforts in providing me with valuable comments on earlier versions of my thesis. I also obtained a fruitful advice from Professor Kenichi Amaya (Kobe University). Lastly, I specially thank to Kozo Shiraishi (Osaka University).     

Extensive and strategic forms: Games and models for games

This paper aims to make precise, in the context of epistemic models for games, some relations between the normal or strategic form representation of a game and the extensive or dynamic form representation. It is argued, first, that epistemic models defined for strategic form representations provide all the materials necessary for defining models for corresponding extensive form representations of the game, models that provide information about the way the game is played that is sufficient to evaluate the rationality of the choices that the players make, and are disposed to make, in the course of playing the dynamic game. Second, two definitions of rationality are compared — one for strategy choices in the normal form representation, and one for the individual choices that the player is disposed to make in the course of playing the dynamic game. It is shown that they are essentially equivalent in games with perfect recall. The main focus is on the intuitive foundational assumptions about rationality and dynamic choice that are needed to motivate the definitions. It is argued that to evaluate the rationality of a player's choices in a dynamic context, it is essential to distinguish passive knowledge (knowledge about nature and about the prior beliefs and strategy choices of other players that is based on observation and inference) fromactive knowledge (knowledge about one's own choices and future choices that is grounded in one's decisions).     

Equilibrium Tariff Schedules In Commercial Policy Games Under Uncertainty

Abstract A partial two country equilibrium model is built in which two different exogenous random shocks may occur. the governments simultaneously choose tariff functions relating their specific tariff to the level of an observable variable (volume of trade or international price). In the case of a “volume of trade shock” the Nash equilibria of this game are more protectionist the larger the possible trade swings and autarky is always an equilibrium outcome. In the case of a “terms of trade shock”, constant tariffs, at their Nash equilibrium in specific tariff levels are the only sensible equilibrium outcome.     

Reciprocity, Trust, and Payoff Privacy in Extensive Form Bargaining

We report decision making in two-person extensive form game trees, using six treatments that vary matching protocol, payoffs, and payoff information. Our objective is to examine game theoretic hypotheses of decision making based on dominance and backward induction in comparison with the culturally or biologically derived hypothesis that reciprocity supports more cooperation than predicted by game theory. We find strong support for cooperation under complete information, even in single-play treatments and in games of trust, unreinforced by the prospect of punishment for defection from reciprocity. Only under private information do we observe strong support for noncooperative game theory.Journal of Economic LiteratureClassification Numbers: C78, C92.     

A Payoff Uncertainty Explanation of Results in Experimental Centipede Games

This paper investigates how well a simple two-sided incomplete information game with a full support prior can be used to explain non-Nash equilibrium outcomes observed in the centipede game experiments. I estimate the variance of the uncertainty about preferences in several versions of the model, select two models, and compare these models to the estimation results of the altruism model and the quantal response models. It is shown that the two selected models have a better fit than the one-parameter altruism and quantal response models and that the estimated variance can explain all qualitative features of these experimental results.Journal of Economic LiteratureClassification Numbers: C72, C19, C 44, C91.     

A Logic for Suspicious Players: Epistemic Actions and Belief–Updates in Games

This paper introduces a notion of 'epistemic action' to describe changes in the information states of the players in a game. For this, ideas are developed from earlier contributions. The ideas are enriched to cover not just purely epistemic actions, but also fact–changing actions ('real moves', e.g. choosing a card, exchanging cards etc.) and nondeterministic actions and strategies (conditional actions having knowledge tests as conditions). The author considers natural operations with epistemic actions and uses them to describe significant aspects of the interaction between beliefs and actions in a game. A logic is used that combines in a specific way a multiagent epistemic logic with a dynamic logic of 'epistemic actions'. The author presents a complete and decidable proof system for this logic. As an application, the author analyses a specific example of a dialogue game (a version of the Muddy Children Puzzle, in which some of the children can 'cheat' by engaging in secret communication moves, while others may be punished for their credulity). Also presented is a sketch of a 'rule–based' approach to games with imperfect information (allowing 'sneaky' possibilities, such as cheating, being deceived and suspecting the others to be cheating).     

On the Existence of Undominated Nash Equilibria in Normal Form Games

In this paper the existence problem of undominated Nash equilibrium in normal form games is analyzed. It is shown that an undominated Nash equilibrium exists, if (a) strategy sets are convex polytopes inRⁿand (b) utility functions are affine with respect to each player's own strategy. It is shown by counterexamples that, first, it is not sufficient to have concave utility functions instead of affine under condition (b) even when condition (a) is satisfied, and, second, it is not sufficient to have just compact and convex strategy sets instead of polytopes in condition (a) even when condition (b) is satisfied.