The existence of equilibria in certain games, separation for families of convex functions and a theorem of Borsuk-Ulam type

Simon, Robert Samuel; Spież, Stanisław; Toruńczyk, H.

doi:10.1007/bf02762067

Cited by 47 publications

(38 citation statements)

References 13 publications

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“…In full generality, it is still open. Special cases of increasing generality have been established by Simon, Spież, and Toruńczyk [14], Renault [11] and again Simon, Spież, and Toruńczyk [15]. By a suitable construction, to the most general game considered in the present paper one can assign a game as treated by [15] with an essential preservation of the payoff structure.…”

Section: Conjecture For All Signalling Functions λ the Correspondingmentioning

confidence: 92%

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A parametrized version of the Borsuk-Ulam theorem

Schick¹,

Simon²,

Spież³

et al. 2011

Bulletin of the London Mathematical Society

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We show that for a "continuous" family of Borsuk-Ulam situations, parameterized by points of a compact manifold W , its solution set also depends "continuously" on the parameter space W . "Continuity" here means that the solution set supports a homology class which maps onto the fundamental class of W . When W ⊂ R m+1 we also show how to construct such a "continuous" family starting from a family depending in the same way continuously on the points of ∂W . This solves a problem related to a conjecture which is relevant for the construction of equilibrium strategies in repeated two-player games with incomplete information.A new method (of independent interest) used in this context is a canonical symmetric squaring construction inČech homology with Z/2-coefficients.MSC: 91A05, 55N45, 55N05, 55N91

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Section: Conjecture For All Signalling Functions λ the Correspondingmentioning

confidence: 92%

“…Assume that X ⊂ E. We say that X has property S for (W, W 0 ) if the projection p| X : X → W is H-essential for (W, W 0 ). Here, S stands for "spanning", compare with [14,15].…”

Section: A Parameterized Borsuk-ulam Theoremmentioning

confidence: 99%

A parametrized version of the Borsuk-Ulam theorem

Schick¹,

Simon²,

Spież³

et al. 2011

Bulletin of the London Mathematical Society

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“…The proof of proposition 4 relies, as explained in [59] or [72], on a fixed point theorem of Borsuk-Ulam type proved by Simon, Spież and Toruńczyk ( [73]) via tools from algebraic geometry. A simplified version of this fixed point theorem can be written as follows: Theorem 8 ( [73]): Let C be a compact subset of an n-dimensional Euclidean space, x ∈ C and Y be a finite union of affine subspaces of dimension n − 1 of an Euclidean space.…”

Section: Lemma 7 ([76])mentioning

confidence: 99%

“…A simplified version of this fixed point theorem can be written as follows: Theorem 8 ( [73]): Let C be a compact subset of an n-dimensional Euclidean space, x ∈ C and Y be a finite union of affine subspaces of dimension n − 1 of an Euclidean space. Let F be a correspondence from C to Y with compact graph and non empty convex values.…”

Section: Lemma 7 ([76])mentioning

confidence: 99%

Repeated Games with Incomplete Information

Renault¹

2012

Computational Complexity

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Article OutlineGlossary and Notation I. Definition of the Subject and its Importance II. Strategies, Payoffs, Value and Equilibria III. The standard model of Aumann and Maschler IV. Vector Payoffs and Approachability V. Zero-sum games with lack of information on both sides VI. Non zero-sum games with lack of information on one side VII. Non-observable actions VIII. Miscellaneous IX. Future directions X. Bibliography Glossary and NotationRepeated game with incomplete information: a situation where several players repeat the same stage game, the players having different knowledge of the stage game which is repeated. Strategy of a player: a rule, or program, describing the action taken by the player in any possible case which may happen. Strategy profile: a vector containing a strategy for each player. Lack of information on one side: particular case where all the players but one perfectly know the stage game which is repeated. Zero-sum games: 2-player games where the players have opposite payoffs. Value: Solution (or price) of a zero-sum game, in the sense of the fair amount that player 1 should give to player 2 to be entitled to play the game. Equilibrium: Strategy profile where each player's strategy is in best reply against the strategy of the other players. Completely revealing strategy: strategy of a player which eventually reveals to the other players everything known by this player on the selected state. Non revealing strategy: strategy of a player which reveals nothing on the selected state. 1The simplex of probabilities over a finite set: for a finite set S, we denote by ∆(S) the set of probabilities over S, and we identify ∆(S) to {p = (p s ) s∈S ∈ IR S , ∀s ∈ S p s ≥ 0 and s∈S p s = 1}. Given s in S, the Dirac measure on s will be denoted by δ s . For p = (p s ) s∈S and q = (q s ) s∈S in IR S , we will use, unless otherwise specified, p − q = s∈S |p s − q s |. I. Definition of the Subject and its Importance IntroductionIn a repeated game with incomplete information, there is a basic interaction called stage game which is repeated over and over by several participants called players. The point is that the players do not perfectly know the stage game which is repeated, but rather have different knowledge about it. As illustrative examples, one may think of the following situations: an oligopolistic competition where firms don't know the production costs of their opponents, a financial market where traders bargain over units of an asset which terminal value is imperfectly known, a cryptographic model where some participants want to transmit some information (e.g., a credit card number) without being understood by other participants, a conflict when a particular side may be able to understand the communications inside the opponent side (or might have a particular type of weapons),... Natural questions arising in this context are as follows. What is the optimal behavior of a player with a perfect knowledge of the stage game ? Can we determine which part of the information such a player should use ? Can we p...

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“…For two person games with lack of information on one side, existence has been recently proved by Simon, Spiez and Torunczyk (1995).…”

Section: Comments and Open Problemsmentioning

confidence: 99%