Deterministic graphical games revisited (original) (raw)
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Journal of Mathematical Analysis and Applications, 1990
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Zero-Sum Games: The Finite Case
Universitext, 2019
Zero-sum games are two-person games where the players have opposite evaluations of outcomes, hence the sum of the payoff functions is 0. In this kind of strategic interaction the players are antagonists, hence this induces pure conflict and there is no room for cooperation. It is thus enough to consider the payoff of player 1 (which player 2 wants to minimize). A finite zero sum game is represented by a real-valued matrix A. The first important result in game theory is called the minmax theorem and was proved by von Neumann [224] in 1928. It states that one can associate a number v(A) to this matrix and a way of playing for each player such that each can guarantee this amount. This corresponds to the notions of "value" and "optimal strategies". This chapter introduces some general notations and concepts that apply to any zero-sum game, and provides various proofs and extensions of the minmax theorem. Some consequences of this result are then given. Finally, a famous learning procedure (fictitious play) is defined and we show that the empirical average of the stage strategies of each player converges to the set of optimal strategies. 2.2 Value and Optimal Strategies Definition 2.2.1 A zero-sum game G in strategic form is defined by a triple (I, J, g), where I (resp. J) is the non-empty set of strategies of player 1 (resp. player 2) and g : I × J −→ R is the payoff function of player 1. The interpretation is as follows: player 1 chooses i in I and player 2 chooses j in J , in an independent way (for instance simultaneously). The payoff of player 1 is then g(i, j) and that of player 2 is −g(i, j): this means that the evaluations of the outcome induced by the joint choice (i, j) are opposite for the two players. Player 1
On Nash-solvability of n-person graphical games under Markov's and a priori realizations
ArXiv, 2021
Weconsider graphical n-person games with perfect information that have no Nashequilibria in pure stationary strategies. Solving these games in mixed strategies, we introduce probabilistic distributions in all non-terminal positions. The corresponding plays can be analyzed under two different basic assumptions: Markov’s and a priori realizations. The former one guarantees existence of a uniformly best response of each player in every situation. Neyertheless, Nash equilibrium may fail to exist even in mixed strategies. The classical Nash theoremis not applicable, since Markov’s realizations mayresult in the limit distributions and effective payoff functions that are not continuous. The a priori realization does not share many nice properties of the Markov one (for example, existence of the uniformly best response) but in return, Nash’s theoremis applicable. Weillustrate both realizations in details bytwo examples with 2 and 3 players and also provide some general results.
A three-person deterministic graphical game without Nash equilibria
Discrete Applied Mathematics
We give an example of a three-person deterministic graphical game that has no Nash equilibrium in pure stationary strategies. The game has seven positions, four outcomes (a unique cycle and three terminal positions), and its normal form is of size 2 × 2 × 4 only. Thus, our example strengthens significantly the one obtained in 2014 by Gurvich and Oudalov; the latter has four players, five terminals, and a 2 × 4 × 6 × 8 normal form. Furthermore, our example is minimal with respect to the number of players. Both examples are tight but not Nash-solvable. Such examples were known since 1975, but they were not related to deterministic graphical games. Moreover, due to the small size of our example, we can strengthen it further by showing that it has no Nash equilibrium not only in pure but also in independently mixed strategies, for both Markovian and a priori evaluations.
Games with Imperfect Information
The MIT Press eBooks, 2014
Games have been extensively studied, either in computer science, mathematics or even economy. Nevertheless, each discipline has its own interest in using this formalism. Computer science for instance is attached to calculability issues. These results have some direct applications in model checking or compilation. Recently, a new type of game has been introduced: games with imperfect information. They allow the modeling of more sophisticated systems, but bring also new calculability problems. In this document, we introduce a general method to prove the determinacy of any type of game. This method is used several times, and allow us to solve some open problems. This document introduces also several examples of important games stating for important properties. Then, a new type of game unifying the concepts of concurrency and imperfect information is presented. Finally, we discuss of the extension on infinite arenas.
Finding equilibria in games of no chance
2007
We consider finding maximin strategies and equilibria of explicitly given extensive form games with imperfect information but with no moves of chance. We show that a maximin pure strategy for a two-player game with perfect recall and no moves of chance can be found in time linear in the size of the game tree and that all pure Nash equilibrium outcomes of a two-player general-sum game with perfect recall and no moves of chance can be enumerated in time linear in the size of the game tree.
Why Chess and Backgammon can be solved in pure positional uniformly optimal strategies
2009
We study existence of (subgame perfect) Nash equilibria (NE) in pure positional strategies in finite positional n-person games with perfect information and terminal payoffs. However, these games may have moves of chance and cycles. Yet, we assume that All Infinite Plays Form One Outcome a∞, in addition to the set of Terminal outcomes VT . This assumption will be called the AIPFOOT condition. For example, Chess or Backgammon are AIPFOOT games, since each infinite play is a draw, by definition. All terminals and a∞ are ranked arbitrarily by the n players. It is well-known that in each finite acyclic positional game, a subgame perfect NE exists and it can be easily found by backward induction, suggested by Kuhn and Gale in early 50s. In contrast, there is a two-person game with only one cycle, one random position, and without NE in pure positional strategies. In 1912, Zermelo proved that each two-person zero-sum AIPFOOT game without random moves (for example, Chess) has a saddle point ...
Correlated equilibria in graphical games
2003
We examine correlated equilibria in the recently introduced formalism of graphical games, a succinct representation for multiplayer games. We establish a natural and powerful relationship between the graphical structure of a multiplayer game and a certain Markov network representing distributions over joint actions. Our first main result establishes that this Markov network succinctly represents all correlated equilibria of the graphical game up to expected payoff equivalence. Our second main result provides a general algorithm for computing correlated equilibria in a graphical game based on its associated Markov network. For a special class of graphical games that includes trees, this algorithm runs in time polynomial in the graphical game representation (which is polynomial in the number of players and exponential in the graph degree).
Graphical Models for Game Theory
2001
We introduce a compact graph-theoretic representation for multi-party game theory. Our main result is a provably correct and efficient algorithm for computing approximate Nash equilibria in one-stage games represented by trees or sparse graphs.