Abstract. Finding an equilibrium of an extensive form game of imperfect information is a fundamental problem in computational game theory, but current techniques do not scale to large games. To address this, we introduce the ordered game isomorphism and the related ordered game isomorphic abstraction transformation. For a multi-player sequential game of imperfect information with observable actions and an ordered signal space, we prove that any Nash equilibrium in an abstracted smaller game, obtained by one or more applications of the transformation, can be easily converted into a Nash equilibrium in the original game. We present an algorithm, GameShrink, for abstracting the game using our isomorphism exhaustively. Its complexity is Õ(n2), where n is the number of nodes in a structure we call the signal tree. It is no larger than the game tree, and on nontrivial games it is drastically smaller, so GameShrink has time and space complexity sublinear in the size of the game tree. Using GameShrink, we find an equilibrium to a poker game with 3.1 billion nodes—over four orders of magnitude more than in the largest poker game solved previously. To address even larger games, we introduce approximation methods that do not preserve equilibrium, but nevertheless yield (ex post) provably close-to-optimal strategies.

In 1952, Thompson defined four transformations on extensive games, and proved that they leave the reduced normal form intact (a so-- called "adequacy" theorem). Kohlberg&Mertens, in 1986, proposed two additional game transformations. In this paper, I will state and prove adequacy results for all six transformations in a uniform manner. Then, a number of well--known solution concepts will be surveyed. Finally, I will examine the (in)variance of the solution concepts under the game transformations, so as to determine whether they induced a plausible notion of game equivalence. 1 Introduction Game theory is often defined as the analysis of conflict---a description, however broad and misleading, that sets the stage for many of its applications in fields as diverse as the political sciences and evolutionary biology. More specifically, game theory is regarded as the study of solutions to conflicts. That is, a game theorist is expected to come up with a solution to particular games---this is...

Semantic games are an important evaluation method for a wide range of logical languages, and are frequently resorted to when traditional methods do not easily apply. A case in point is a family of independence-friendly (IF) logics which allow regulation over information flow in formulas, and thus perfect information fails in the games associated with such formulas. This mechanism of imperfect information is studied in this paper. It is noted that imperfect information of players often gives rise to the game-theoretic phenomenon of imperfect recall. Furthermore, independence-friendliness in epistemic logic is investigated. We also discuss a couple of misunderstandings that have occurred in the literature concerning IF first-order logics and gametheoretical semantics, related to such issues as intuitionism, constructivism, truth-definitions, mathematical prose, and the status of set theory. By straighten out these misunderstandings, we hope to show the importance of the role semantics ga...

We study partiality in propositional logics containing formulas with either unde ned or over-de ned truth-values. Unde ned values are created by adding a four-place connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally complete for all partial functions. Transjunction is seen to be motivated from a game-theoretic perspective, emerging from a two-stage extensive form semantic game of imperfect information between two players. This game-theoretic approach yields an interpretation where partiality is generated as a property of non-determinacy of games. Over-de ned values are produced by adding a weak, contradictory negation or, alternatively, by relaxing the assumption that games are strictly competitive. In general, particular forms of extensive imperfect information games give rise to a generalised propositional logic where various forms of informational dependencies and independencies of connectives can be studied.

ABSTRACT. As yet, no general agreement has been reached on whether the Bayesian or the frequentist (Neyman-Pearson, NP) approach to statistics is to be preferred. Whereas Bayesians adhere to coherence conditions of de Finetti, Savage, and others, frequentists do not consider these conditions normative and deliberately and knowingly violate them. Hence further arguments, bringing more clarity on the disagreements, are warranted. Providing such arguments, by refining the coherence conditions, is the purpose of this paper. It invokes recent arguments from the economic literature demonstrating that some seemingly self-evident principles for dynamic decision making have a surprising implication for static decisions: They imply Bayesianism. These principles are forgone-event independence (independence of past counterfactual events, often called consequentialism in decision theory and known as the conditionality principle in statistics), dynamic consistency (what is optimal at some given time point is independent of the time point at which that is decided), and two other conditions. Thus, a more sensitive diagnostic tool is obtained for identifying the disagreements between Bayesians and frequentists. If a frequentist does not mind violating Bayesian coherence, a Bayesian can now ask a follow-up question: Which of the dynamic principles will the frequentist give up? The debate may lead either to Bayesianism or to better implementations of non-Bayesian models in dynamic decision situations and to better non-Bayesian methods for updating information. The diagnostic tool sheds new light on NP hypothesis testing. NP theory requires that statistical procedures are laid down before data are observed. It adheres to dynamic consistency but violates forgone-event independence. Forgone-event independence, however, is so natural that NP practitioners adhere to it and observe the data before deciding on a statistical procedure. They are thus led into violations of dynamic consistency.

. Three aspects of structural information in extensive form of games are discussed. First, the way strategies are dened in extensive games, as coding the appropriate amount of information about histories, leaves the possibility of performing a transformation of imperfect information games into perfect information ones. Such a transformation is described here, and it is shown that subgame perfect equilibrium can be preserved. Second, the structure of extensive games with simultaneous moves is discussed, and it is argued that this class of games can be seen both as imperfect and perfect information games. A need for additional information partition is pointed out for these games, to capture the notion of imperfect recall. Third, it is suggested that extensive games in general benet from a wider and more dynamic concept of information set, generalising away from the requirement of players making decisions at the sets. In some cases, these dynamic sets are argued to be indisp...

Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably two-person zero-sum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexity-theory sense. However, in most interesting potential applications in artificial intelligence, the solutions are difficult to compute using current techniques due primarily to the extremely large state-spaces of the environments. In this thesis, we propose new algorithms for tackling these computational difficulties. In one stream of research, we introduce automated abstraction algorithms for sequential games of imperfect information. These algorithms take as input a description of a game and produce a description of a strategically similar, but smaller, game as output. We present algorithms that are lossless (i.e., equilibrium-preserving), as well as algorithms that are lossy, but which can yield much smaller games while still retaining the most important features of the original game. In a second stream of research, we develop specialized optimization algorithms for finding ɛ-equilibria in sequential games of imperfect information. The algorithms are based on recent advances in nonsmooth convex optimization (namely the excessive gap technique) and provide significant improvements

We study a weak-dominance analog to Pearce’s [28, 1984] fundamental solution concept of a best-response set. The concept, called a self-admissible set (SAS), arises from an epistemic analysis of weak dominance in Brandenburger-Friedenberg-Keisler [12, 2007]. Here, we ‘test’ the SAS concept by: (i) examining which of the Kohlberg-Mertens [22, 1986] axioms it satisfies; (ii) analyzing its behavior in the Finitely Repeated Prisoner’s Dilemma, Centipede, and the Chain Store Game; and (iii) characterizing it in perfect-information games.

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and Georgetown theory workshops, the Stanford SITE and Stonybrook theory conferences,