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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How long does it take until economic agents converge to an equilibrium? By studying the complexity of the problem of computing a mixed Nash equilibrium in a game, we provide evidence that there are games in which convergence to such an equilibrium takes prohibitively long. Traditionally, computational problems fall into two classes: those that have a polynomial-time algorithm and those that are NP-hard. However, the concept of NP-hardness cannot be applied to the rare problems where "every instance has a solution"---for example, in the case of games Nash's theorem asserts that every game has a mixed equilibrium (now known as the Nash equilibrium, in honor of that result). We show that finding a Nash equilibrium is complete for a class of problems called PPAD, containing several other known hard problems; all problems in PPAD share the same style of proof that every instance has a solution. 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. Thomas has presented a novel proof of the closure of ω-regular languages under complementation, using weak alternating automata. This note describes a formalization of this proof in the theorem prover Isabelle HOL. As an application we have developed a certified translation procedure for PTL formulas to weak alternating automata inside the theorem prover.