Finite Extensive Games Research Articles

On equilibria in games with imperfect recall

We generalize the modified multiselves approach of Piccione and Rubinstein to (multiplayer) games of imperfect recall. Four solution concepts are introduced: the multiselves agent equilibrium, the multiselves Nash equilibrium, the multiselves sequential equilibrium, and the multiselves perfect equilibrium. These modified equilibrium notions satisfy two important properties not fulfilled by the original ones. First, they always exist: every finite extensive game has at least one multiselves equilibrium of each type. Second, they form a strict hierarchy: every multiselves perfect equilibrium is a multiselves sequential equilibrium, every multiselves sequential equilibrium is a multiselves Nash equilibrium, and every multiselves Nash equilibrium is a multiselves agent equilibrium—but not conversely. Finally, in games of perfect recall, the multiselves equilibrium notions reduce to their original counterparts.

Uncertain extensive game with application to resource allocation of national security

Uncertain theory is a powerful tool for describing human uncertainty, especially in social economic studies. In this paper, we consider finite extensive games with payoffs being uncertain variables. We introduce three different criteria: the expected value criterion, the optimistic value criterion and the pessimistic value criterion to describe players’ behaviors in different decision-making situations. Accordingly, three types of uncertain equilibria and uncertain subgame perfect equilibria for the uncertain extensive game are proposed. Moreover, we provide theorems to prove the existence of uncertain equilibria and uncertain subgame perfect equilibria. Finally, the resource allocation problem for national security is formulated and analyzed using the uncertain extensive game. Results show that different decision criteria may lead to different subgame perfect equilibria, demonstrating the significance of introducing different uncertain subgame perfect equilibria.

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 (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 (1975), by allowing for ambiguous trembles characterized by sets of distributions. We prove existence for two of our equilibrium notions, and relate our de finitions 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.

Credibilistic extensive game with fuzzy payoffs

This paper considers a finite extensive game with fuzzy payoffs. Within the framework of credibility theory, expected value criterion, optimistic value criterion and pessimistic value criterion are introduced to define the behaviors of players in different decision situations. Accordingly, three types of credibilistic equilibria and credibilistic subgame perfect equilibria for the fuzzy extensive game are proposed, respectively. Moreover, theorems are given to confirm the existence of these credibilistic equilibria in the fuzzy extensive game. Finally, three examples are given to demonstrate the significance of these new concepts.

QUANTUM INFORMATION APPROACH TO NORMAL REPRESENTATION OF EXTENSIVE GAMES

We modify the concept of quantum strategic game to make it useful for extensive form games. We prove that our modification allows us to consider the normal representation of any finite extensive game using the fundamental concepts of quantum information. The Selten's Horse game and the general form of two-stage extensive game with perfect information are studied to illustrate a potential application of our idea. In both examples we use the Eisert–Wilkens–Lewenstein approach as well as the Marinatto–Weber approach to quantization of games.

Rational Rules of Thumb in Finite Dynamic Games N-person Backward Induction with Inconsistently Aligned Beliefs and Full Rationality

Recent work has cast considerable doubt on the plau sibility of specific assumptions about how rational agents form out-of-equilibrium beliefs in finite extensive games in which beliefs are induced backwards. The point is that the resulting consiste ntly aligned beliefs are incoherent in view of the counterfactuals they rely on. This study asks: How will the possibility of inconsistently aligned beli efs affect the manner in which rational players play such games? It shows that, provided beliefs are aligned monotonically, the more interesting qualitative fea tures of the conventional approach remain unchanged.

Proper rationalizability and backward induction

This paper introduces a new normal form rationalizability concept, which in reduced normal form games corresponding to generic finite extensive games of perfect information yields the unique backward induction outcome. The basic assumption is that every player trembles “more or less rationally” as in the definition of a e-proper equilibrium by Myerson (1978). In the same way that proper equilibrium refines Nash and perfect equilibrium, our model strengthens the normal form rationalizability concepts by Bernheim (1984), Borgers (1994) and Pearce (1984). Common knowledge of trembling implies the iterated elimination of strategies that are strictly dominated at an information set. The elimination process starts at the end of the game tree and goes backwards to the beginning.

Inconsistencies in extensive games

In certain finite extensive games with perfect information, Cristina Bicchieri (1989) derives a logical contradiction from the assumptions that players are rational and that they have common knowledge of the theory of the game. She argues that this may account for play outside the Nash equilibrium. She also claims that no inconsistency arises if the players have the minimal beliefs necessary to perform backward induction. We here show that another contradiction can be derived even with minimal beliefs, so there is no paradox of common knowledge specifically. These inconsistencies do not make play outside Nash equilibrium plausible, but rather indicate that the epistemic specification must incorporate a system for belief revision. Whether rationality is common knowledge is not the issue.

Structural consistency and strategic independence in extensive games

Summary In this note I provide a formulation of the joint principle of structural consistency and strategic independence, which is used to model players' expectations in finite extensive games. I compare updating systems of conjectures and conditional probability systems, showing that they represent equivalent formalizations of structural consistency. The notion of strategic independence cannot be adequately formalized by properties of updating systems of conjectures. However, it can be naturally translated in an intuitive stochastic independence property for conditional probability systems.

Demand commitment bargaining in three-person quota game experiments

The paper reports results of experiments on three-person quota games without the grand coalition and with zero values for the one-person coalitions. The experimental procedure used is the demand commitment model. This model generates finite extensive games with perfect information whose neutral equilibrium point predicts quota agreements. The termneutral means that at every decision point all locally optimal choices are taken with equal probability. Quota agreements do not perform better than equal division payoff bounds, but nevertheless the experiments show that quotas have some behavioral relevance. The demand commitment model seems to favor learning of the quota concept if subjects have the opportunity to gain experience by frequently playing the same game.