Extensive Games Research Articles

Quantum signaling game

We present a quantum approach to a signaling game; a special kind of extensive game of incomplete information. Our model is based on quantum schemes for games in strategic form where players perform unitary operators on their own qubits of some fixed initial state and the payoff function is given by a measurement on the resulting final state. We show that the quantum game induced by our scheme coincides with a signaling game as a special case and outputs nonclassical results in general. As an example, we consider a quantum extension of the signaling game in which the chance move is a three-parameter unitary operator whereas the players' actions are equivalent to classical ones. In this case, we study the game in terms of Nash equilibria and refine the pure Nash equilibria adapting to the quantum game the notion of a weak perfect Bayesian equilibrium.

Trembles in extensive games with ambiguity averse players

We introduce and analyze three denitions of equilibrium for nite 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 denitions 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 denitions 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.

Bounding the Support Size in Extensive Form Games with Imperfect Information

It is a well known fact that in extensive form games with perfect information, there is a Nash equilibrium with support of size one. This doesn't hold for games with imperfect information, where the size of minimal support can be larger. We present a dependency between the level of uncertainty and the minimum support size. For many games, there is a big disproportion between the game uncertainty and the number of actions available. In Bayesian extensive games with perfect information, the only uncertainty is about the type of players. In card games, the uncertainty comes from dealing the deck. In these games, we can significantly reduce the support size. Our result applies to general-sum extensive form games with any finite number of players.

Games and Logic

The idea behind these games is to obtain an alternative characterization of logical notions cherished by logicians such as truth in a model, or provability (in a formal system). We offer a quick survey of Hintikka's evaluation games, which offer an alternative notion of truth in a model for first-order langauges. These are win-lose, extensive games of perfect information. We then consider a variation of these games, IF games, which are win-lose extensive games of imperfect information. Both games presuppose that the meaning of the basic vocabulary of the language is given. To give an account of the linguistic conventions which settle the meaning of the basic vocabulary, we consider signaling games, inspired by Lewis' work. We close with IF probabilistic games, a strategic variant of IF games which combines semantical games with von Neumann's minimax theorem.

Strategic complexity in repeated extensive games

This paper studies a machine (finite automaton) playing a two-player repeated game of a simple extensive-form game with perfect information. We introduce a new complexity measure called multiple complexity which incorporates a strategyʼs responsiveness to information in the stage game as well as the number of states of the machine. We completely characterize the Nash equilibrium of the machine game. In the sequential-move prisonerʼs dilemma, cooperation can be sustained as an equilibrium.

Observable implications of Nash and subgame-perfect behavior in extensive games

We provide necessary and sufficient conditions for observed outcomes in extensive game forms, in which preferences are unobserved, to be rationalized first, weakly, as a Nash equilibrium and then as the unique subgame-perfect equilibrium. Thus, one could use these conditions to find that play is (a) consistent with subgame-perfect equilibrium, or (b) not consistent with subgame-perfect behavior but is consistent with Nash equilibrium, or (c) consistent with neither.

Quantum Information Approach to the Ultimatum Game

The paper is devoted to quantization of extensive games with the use of both the Marinatto-Weber and the Eisert-Wilkens-Lewenstein concept of quantum game. We revise the current conception of quantum ultimatum game and we show why the proposal is unacceptable. To support our approach, we present a new idea of the quantum ultimatum game. Our scheme also makes a point of departure for a general protocol for quantizing extensive games.

Discovering the discovery of the hierarchy of formal languages

The hierarchy of formal languages is a mathematical representation of linguistic geometry (LG). LG is a type of game theory for a class of extensive discrete games called abstract board games (ABG), scalable to the level of real life defense systems. LG is a formal model of human reasoning about armed conflict, a mental reality “hard-wired” in the human brain. LG, an evolutionary product of millions of years of human warfare, must be a component of the primary language of the human brain (as introduced by Von Neumann). Experiences of development of LG must be instructive for solving another major puzzle, discovering the algorithm of discovery, yet another ancient component of the primary language. This paper reports results on discovering mental processes involved in the development of the hierarchy of formal languages. Those mental processes manifesting execution of the algorithm of discovery are called visual streams. This paper reveals the visual streams that were involved in the thought experiments led to the development of the formal theory of LG. Specifically, it demonstrates the streams involved in choosing the formal-linguistic representation of LG; the type of formal languages and grammars, the so-called controlled grammars; the construction of the grammars of shortest trajectories and the grammar of zones. This paper introduces a hypothesis of how we construct and focus visual streams.

Evolutionary stability in repeated extensive games played by finite automata

We discuss the emergence of cooperation in repeated Trust Mini-Games played by finite automata. Contrary to a previous result obtained by Piccione and Rubinstein (1993), we first prove that this repeated game admits two Nash equilibria, a cooperative and a non-cooperative one. Second, we show that the cooperative equilibrium is the only (cyclically) stable set under the so-called best response dynamics.

The order independence of iterated dominance in extensive games

Shimoji and Watson (1998) prove that a strategy of an extensive game is rationalizable in the sense of Pearce if and only if it survives the maximal elimination of conditionally dominated strategies. Briefly, this process iteratively eliminates conditionally dominated strategies according to a specific order, which is also the start of an order of elimination of weakly dominated strategies. Since the final set of possible payoff profiles, or terminal nodes, surviving iterated elimination of weakly dominated strategies may be order-dependent, one may suspect that the same holds for conditional dominance. We prove that, although the sets of strategy profiles surviving two arbitrary elimination orders of conditional dominance may be very different from each other, they are equivalent in the following sense: for each player i and each pair of elimination orders, there exists a function φi mapping each strategy of i surviving the first order to a strategy of i surviving the second order, such that, for every strategy profile s surviving the first order, the profile (φi(si))i induces the same terminal node as s does. To prove our results, we put forward a new notion of dominance and an elementary characterization of extensive-form rationalizability (EFR) that may be of independent interest. We also establish connections between EFR and other existing iterated dominance procedures, using our notion of dominance and our characterization of EFR.

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.

A Note on Kuhn's Theorem with Ambiguity Averse Players

Kuhn’s Theorem shows that extensive games with perfect recall can equivalently be analyzed using mixed or behavioral strategies, as long as players are expected utility maximizers. This note constructs an example that illustrates the limits of Kuhn’s Theorem in an environment with ambiguity averse players who use a maxmin decision rule and full Bayesian updating.

Normal Form of Some Infinite Extensive Games and Continuity Properties(<Special Issue>Bilevel Programming, Optimization Methods, and Applications to Economics)

Normal Form of Some Infinite Extensive Games and Continuity Properties(<Special Issue>Bilevel Programming, Optimization Methods, and Applications to Economics)

Existence of Nash Equilibrium in Infinite Extensive Games(<Special Issue>Bilevel Programming, Optimization Methods, and Applications to Economics)

Existence of Nash Equilibrium in Infinite Extensive Games(<Special Issue>Bilevel Programming, Optimization Methods, and Applications to Economics)

Discovering the discovery of the No-Search Approach

No-Search Approach is a component of Linguistic Geometry (LG), a type of game theory for extensive discrete games scalable to the level of real life defense systems. LG is a formal model of human reasoning about armed conflict, a mental reality “hard-wired” in the human brain. LG, an evolutionary product of millions of years of human warfare, must be a component of the Primary Language of the human brain (as introduced by J. Von Neumann). Experiences of development of LG must be instructive for solving another major puzzle, discovering the algorithm of discovery, yet another ancient component of the Primary Language. This paper reports results on discovering mental processes involved in the development of the major result in LG, the No-Search Approach. Those mental processes manifesting execution of the algorithm of discovery are called visual streams. It seems that those streams are the only interface of the algorithm of discovery. This paper demonstrates a typical situation in discoveries in mathematics and computer science when the original result (an idea) has been envisioned via one type of semi-formal visual streams and later proved via different streams that involve formal mathematical theories.

Extensive games with possibly unaware players

Standard game theory assumes that the structure of the game is common knowledge among players. We relax this assumption by considering extensive games where agents may be unaware of the complete structure of the game. In particular, they may not be aware of moves that they and other agents can make. We show how such games can be represented; the key idea is to describe the game from the point of view of every agent at every node of the game tree. We provide a generalization of Nash equilibrium and show that every game with awareness has a generalized Nash equilibrium. Finally, we extend these results to games with awareness of unawareness, where a player i may be aware that a player j can make moves that i is not aware of, and to subjective games, where players may have no common knowledge regarding the actual game and their beliefs are incompatible with a common prior.

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.

Discovering the discovery of Linguistic Geometry

Linguistic Geometry (LG) is a type of game theory for extensive discrete games scalable to the level of real life defense systems. This scalability is based on changing the paradigm for game solving—from search to construction. LG was developed by generalizing experiences of the advanced chess players. LG is a formal model of human reasoning about armed conflict, a mental reality “hard-wired” in the human brain. It appears that this algorithm, an evolutionary product of millions of years of human warfare, served in its turn as the principle mover for evolution of human intelligence. Its special role in human history and its relationship to the Primary Language of the human brain (as introduced by J. Von Neumann) suggest that the experiences of development of LG must be instructive for solving another major puzzle, discovering the Algorithm of Discovery. This algorithm should be an ancient component of the Primary Language as well. This paper refining our experiences of discovering LG reports the first results of this research.

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.

Existence of perfect equilibria: a direct proof

We provide a direct proof of the existence of perfect equilibria in finite normal form games and extensive games with perfect recall. It is done by constructing a correspondence whose fixed points are precisely the perfect equilibria of a given finite game. Existence of a fixed point is secured by a generalization of Kakutani theorem, which is proved in this paper. This work offers a new approach to perfect equilibria, which would hopefully facilitate further study on this topic. We also hope our direct proof would be the first step toward building an algorithm to find the set of all perfect equilibria of a strategic game.