Results 1  10
of
22
Lossless abstraction of imperfect information games
 Journal of the ACM
, 2007
"... 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 abstractio ..."
Abstract

Cited by 33 (14 self)
 Add to MetaCart
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 multiplayer 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 closetooptimal strategies.
The power of paradox: some recent developments in interactive epistemology
 International Journal of Game Theory
, 2007
"... Abstract Paradoxes of gametheoretic reasoning have played an important role in spurring developments in interactive epistemology, the area in game theory that studies the role of the players ’ beliefs, knowledge, etc. This paper describes two such paradoxes – one concerning backward induction, the ..."
Abstract

Cited by 29 (2 self)
 Add to MetaCart
(Show Context)
Abstract Paradoxes of gametheoretic reasoning have played an important role in spurring developments in interactive epistemology, the area in game theory that studies the role of the players ’ beliefs, knowledge, etc. This paper describes two such paradoxes – one concerning backward induction, the other iterated weak dominance. We start with the basic epistemic condition of “rationality and common belief of rationality ” in a game, describe various ‘refinements ’ of this condition that have been proposed, and explain how these refinements resolve the two paradoxes. We will see that a unified epistemic picture of game theory emerges. We end with some new foundational questions uncovered by the epistemic program. 1
SelfAdmissible Sets
, 2004
"... We study a weakdominance analog to Pearce’s [28, 1984] fundamental solution concept of a bestresponse set. The concept, called a selfadmissible set (SAS), arises from an epistemic analysis of weak dominance in BrandenburgerFriedenbergKeisler [12, 2007]. Here, we ‘test’ the SAS concept by: (i) e ..."
Abstract

Cited by 7 (3 self)
 Add to MetaCart
We study a weakdominance analog to Pearce’s [28, 1984] fundamental solution concept of a bestresponse set. The concept, called a selfadmissible set (SAS), arises from an epistemic analysis of weak dominance in BrandenburgerFriedenbergKeisler [12, 2007]. Here, we ‘test’ the SAS concept by: (i) examining which of the KohlbergMertens [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 perfectinformation games.
Algorithms for abstracting and solving imperfect information games
, 2007
"... Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory s ..."
Abstract

Cited by 6 (2 self)
 Add to MetaCart
(Show Context)
Game theory is the mathematical study of rational behavior in strategic environments. In many settings, most notably twoperson zerosum games, game theory provides particularly strong and appealing solution concepts. Furthermore, these solutions are efficiently computable in the complexitytheory 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 statespaces 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., equilibriumpreserving), 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 ɛequilibria 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
Games in Philosophical Logic
, 1999
"... 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 independencefriendly (IF) logics which allow regulation over information flow in formulas, and thus pe ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
(Show Context)
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 independencefriendly (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 gametheoretic phenomenon of imperfect recall. Furthermore, independencefriendliness in epistemic logic is investigated. We also discuss a couple of misunderstandings that have occurred in the literature concerning IF firstorder logics and gametheoretical semantics, related to such issues as intuitionism, constructivism, truthdefinitions, mathematical prose, and the status of set theory. By straighten out these misunderstandings, we hope to show the importance of the role semantics ga...
National Academy of SciencesNational Research Committee on Vision (1980) Report of working group 39: Recommended standard procedures for the clinical measurement and specification of visual acuity
 Advances in Ophthalmology
, 1999
"... ABSTRACT. As yet, no general agreement has been reached on whether the Bayesian or the frequentist (NeymanPearson, 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 normati ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
ABSTRACT. As yet, no general agreement has been reached on whether the Bayesian or the frequentist (NeymanPearson, 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 selfevident principles for dynamic decision making have a surprising implication for static decisions: They imply Bayesianism. These principles are forgoneevent 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 followup question: Which of the dynamic principles will the frequentist give up? The debate may lead either to Bayesianism or to better implementations of nonBayesian models in dynamic decision situations and to better nonBayesian 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 forgoneevent independence. Forgoneevent 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.
Partiality and Games: Propositional Logic
, 2001
"... We study partiality in propositional logics containing formulas with either unde ned or overde ned truthvalues. Unde ned values are created by adding a fourplace connective W termed transjunction to complete models which, together with the usual Boolean connectives is shown to be functionally ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
We study partiality in propositional logics containing formulas with either unde ned or overde ned truthvalues. Unde ned values are created by adding a fourplace 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 gametheoretic perspective, emerging from a twostage extensive form semantic game of imperfect information between two players. This gametheoretic approach yields an interpretation where partiality is generated as a property of nondeterminacy of games. Overde 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.
Game Transformations and Game Equivalence
, 1999
"... 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 resul ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
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 wellknown 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 conflicta 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 gamesthis is...
FORWARD INDUCTION EQUILIBRIUM
"... Abstract. Forward induction is the notion that players in a game assume, even when confronted with an unexpected event, that their opponents chose rationally in the past and will choose rationally in the future. This paper modifies Govindan and Wilson’s (2009, Econometrica 77(1), 128) definition ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Abstract. Forward induction is the notion that players in a game assume, even when confronted with an unexpected event, that their opponents chose rationally in the past and will choose rationally in the future. This paper modifies Govindan and Wilson’s (2009, Econometrica 77(1), 128) definition of forward induction and constructs an admissible, invariant forward induction equilibrium concept for general games using normal form perfect equilibrium. Forward induction equilibrium according to this new definition exists for all finite, generic extensive form games with perfect recall. It does not satisfy backward induction. Yet for generic extensive form games the set of forward induction outcomes contains an invariant sequential equilibrium outcome. Forward induction is not equivalent to iterative elimination of strategies dominated at the equilibrium value. In signaling games, a forward induction equilibrium survives most existing equilibrium refinements. 1.
Three Short Plays: Structural Information In Extensive Games
, 1999
"... . 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 pe ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
. 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...