8. Reductions Op Game Matrices

Gale, David; Kuhn, Harold W.; Tucker, A. W.

doi:10.1515/9781400881727-009

Cited by 4 publications

(7 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In state h, B2 and S can increase their payoffs from (0, 5) to (2,6) by changing the tentative decision to T1, and thus, they should do so. On the other hand, in state l, no player attempts to change the tentative decision at first.…”

Section: An Exchange Economymentioning

confidence: 99%

See 1 more Smart Citation

Iterative information update and stability of strategies

Masuzawa

Hasebe

2010

Synthese

View full text Add to dashboard Cite

Abstract. In this paper, we investigate processes of iterative information update due to Benthem (International Game Theory Review, vol.9, pp.13-45, 2007), who characterized existent game-theoretic solution concepts by such processes in the framework of Plaza's public announcement logic. We refine this approach and make clearer the relationship between stable strategies and information update processes. We first extend Plaza's logic then demonstrate the conditions under which a stable outcome is determined independently of the order of iterative information update. This result gives an epistemic foundation for the order independence of iterated elimination of disadvantageous strategies.

show abstract

Section: An Exchange Economymentioning

confidence: 99%

“…The SIE of strictly or weakly dominated strategies can be traced back to Gale, et al [6], Gale [7], and Luce and Raiffa [13]. Bernheim [3] and Pearce [14] independently discussed IE of never best response strategies 4 .…”

Section: Definitionmentioning

confidence: 99%

Iterative information update and stability of strategies

Masuzawa

Hasebe

2010

Synthese

View full text Add to dashboard Cite

show abstract

“…Thus if we rearrange the rows and columns into their orbits, the resulting partition is as in Application (e), pages 94-95 of [Gale/Kuhn/Tucker (1950)]. Knowing that a group of symmetries is acting on a game makes it possible to tell what this partitioning is without writing down the matrix in full, and usually makes it unnecessary to write down more than a few of the columns (or rows) in order to calculate its value and find optimal strategies.…”

Section: Theorem 3 (Reduction With Respect To Player 1) Let B Be Obtmentioning

confidence: 99%

Reducing two person, zero sum games with underlying symmetry

Pearson

1982

J Aust Math Soc A

View full text Add to dashboard Cite

We consider two person, zero sum games with several symmetries. Where such symmetries are present there is a group acting on the strategies of the game. We show how to use this action to produce a reduced game with a smaller matrix, but having the same value as the original game, and how to obtain optimal strategies for the original game from optimal strategies of the reduced game. An analysis of a simplified version of the popular game Mastermind is given to illustrate the theory developed. The rules of many games give them a number of symmetries. When analysing such a game one keeps meeting cases which seem very much like ones already encountered and the natural approach is to try to link together somehow strategies that appear similar to each other.In this paper we formalize these feelings for two person, zero sum games (with imperfect information) with several symmetries. Where such symmetries are evident, there is in fact a group acting on the strategies of the game. This action yields homogeneous optimal strategies (that is, optimal strategies in which linked strategies are used with equal probability) for each player, and makes it possible to guarantee the existence of a smaller matrix game with the same value as the original one. Further, all homogeneous optimal strategies in the original game can be obtained from a knowlege of all optimal strategies in the smaller or reduced game. The greater the number of symmetries, the smaller will this reduced game be, in general.

show abstract

“…Reducing the size of a game without affecting its instrinsic properties such as the Nash equilibria, the value, and the optimal strategies, is desirable from the viewpoint of computation and modelling players' bounded rationality. In their seminal paper [1], Gale, Kuhn and Tucker (GKT) introduced two ways to reduce a zero-sum game by packaging some strategies with respect to some probability distribution on them. A reduction is desirable if optimal strategies of the original game are restorable from those of it.…”

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

A Note on Gale, Kuhn, and Tucker's Reductions of Zero-Sum Games

Liu¹

2017

Preprint

View full text Add to dashboard Cite

Gale, Kuhn and Tucker [1] introduced two ways to reduce a zero-sum game by packaging some strategies with respect to a probability distribution on them. In terms of value, they gave conditions for a desirable reduction. We show that a probability distribution for a desirable reduction relies on optimal strategies in the original game. Also, we correct an improper example given by them to show that the reverse of a theorem does not hold.

show abstract

8. Reductions Op Game Matrices

Cited by 4 publications

References 0 publications

Iterative information update and stability of strategies

Iterative information update and stability of strategies

Reducing two person, zero sum games with underlying symmetry

A Note on Gale, Kuhn, and Tucker's Reductions of Zero-Sum Games

Contact Info

Product

Resources

About