Finite objects and more specifically finite games are formalized using induction, whereas infinite objects are formalized using coinduction. In this article, after an introduction to the concept of coinduction, we revisit on infinite (discrete) extensive games the basic notions of game theory. Among others, we introduce a definition of Nash equilibrium and a notion of subgame perfect equilibrium for infinite games. We use those concepts to analyze well known infinite games, like the dollar auction game and the centipede game and we show that human behaviors that are often considered as illogic are perfectly rational, if one admits that human agents reason coinductively.
Extensive games are not used in proving properties of automata and protocols @cite_10 @cite_19 . This work has only loose connection with @cite_34 @cite_1 , which is more interested by the complexity of the algorithms, especially those which compute equilibria, than by their correction, and does not deal with infinite games.
@cite_1 @cite_19 @cite_34 @cite_10
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Games are used to evaluate and advance Multiagent and Artificial Intelligence techniques. Most of these games are deterministic with perfect information (e.g. Chess and Checkers). A deterministic game has no chance element and in a perfect information game, all information is visible to all players. However, many real-world scenarios with competing agents are stochastic (non-deterministic) with imperfect information. For two-player zero-sum perfect recall games, a recent technique called Counterfactual Regret Minimization (CFR) computes strategies that are provably convergent to an e-Nash equilibrium. A Nash equilibrium strategy is useful in two-player games since it maximizes its utility against a worst-case opponent. However, for multiplayer (three or more player) games, we lose all theoretical guarantees for CFR. However, we believe that CFR-generated agents may perform well in multiplayer games. To test this hypothesis, we used this technique to create several 3-player limit Texas Hold'em poker agents and two of them placed first and second in the 3-player event of the 2009 AAAI IJCAI Computer Poker Competition. We also demonstrate that good strategies can be obtained by grafting sets of two-player subgame strategies to a 3-player base strategy after one of the players is eliminated. We analyse the following (solitaire) game: each node of a graph contains a pile of chips, and a move consists of selecting a node with at least as many chips on it as its degree, and letting it send one chip to each of its neighbors. The game terminates if there is no such node. We show that the finiteness of the game and the terminating configuration are independent of the moves made. If the number of chips is less than the number of edges, the game is always finite. If the number of chips is at least the number of edges, the game can be infinite for an appropriately chosen initial configuration. If the number of chips is more than twice the number of edges minus the number of nodes, then the game is always infinite. The independence of the finiteness and the terminating position follows from simple but powerful ‘exchange properties’ of the sequences of legal moves, and from some general results on ‘antimatroids with repetition’, i.e. languages having these exchange properties. We relate the number of steps in a finite game to the least positive eigenvalue of the Laplace operator of the graph. Game-playing is an approach to write security proofs that are easy to verify. In this approach, security definitions and intractable problems are written as programs called games and reductionist security proofs are sequences of game transformations. This bias towards programming languages suggests the implementation of a tool based on compiler techniques (syntactic program transformations) to build security proofs, but it also raises the question of the soundness of such a tool. In this paper, we advocate the formalization of game-playing in a proof assistant as a tool to build security proofs. In a proof assistant, starting from just the formal definition of a probabilistic programming language, all the properties required in game-based security proofs can be proved internally as lemmas whose soundness is ensured by proof theory. Concretely, we show how to formalize the game-playing framework of Bellare and Rogaway in the Coq proof assistant, how to prove formally reusable lemmas such as the fundamental lemma of game-playing, and how to use them to formally prove the PRP PRF Switching Lemma. This paper provides the first general technique for proving information lower bounds on two-party unbounded-rounds communication problems. We show that the discrepancy lower bound, which applies to randomized communication complexity, also applies to information complexity. More precisely, if the discrepancy of a two-party function @math with respect to a distribution @math is @math , then any two party randomized protocol computing @math must reveal at least @math bits of information to the participants. As a corollary, we obtain that any two-party protocol for computing a random function on @math must reveal @math bits of information to the participants. In addition, we prove that the discrepancy of the Greater-Than function is @math , which provides an alternative proof to the recent proof of Viola Viola11 of the @math lower bound on the communication complexity of this well-studied function and, combined with our main result, proves the tight @math lower bound on its information complexity. The proof of our main result develops a new simulation procedure that may be of an independent interest. In a very recent breakthrough work of kerenidis2012lower , this simulation procedure was the main building block for proving that almost all known lower bound techniques for communication complexity (and not just discrepancy) apply to information complexity.