Equilibrium existence and topology in some repeated games with incomplete information

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

doi:10.1090/s0002-9947-02-03098-2

Cited by 16 publications

(19 citation statements)

References 8 publications

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“…Claim a) above follows from Fact 1b), and for the other ones see (Simon, Spież, Toruńczyk 2002) (again, with M a sphere).…”

Section: The Spanning Property Of Correspondencesmentioning

confidence: 91%

“…A proof with M a sphere is given in (Simon, Spież, Toruńczyk 2002); using the orientablity of M over Z 2 a generalization causes no difficulty. Of course, [W ] depends on the ambient manifold M, e.g.…”

Section: The Spanning Property Of Correspondencesmentioning

confidence: 99%

“…In the present paper, we consider parametrized relationships between the mixed strategies and the payoffs, especially when that relationship satisfies the spanning property (Simon, Spież, Toruńczyk 2002). We show that if the payoffs relative to a parametrization have the spanning property, then the resulting parametrized myopic equilibria also have the spanning property.…”

Section: Introductionmentioning

confidence: 95%

“…First the spanning property is more general. With repeated games of incomplete information on one side the equilibrium payoff correspondence as a function of the probability distributions has the spanning property (Simon, Spież, Toruńczyk 2002), but there is no reason to believe that it has the homotopic property. Second, there is no indication yet that the homotopic property is robust with respect to composition, though we prove that the spanning property has such a robustness.…”

Section: Introductionmentioning

confidence: 99%

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Myopic equilibria, the spanning property, and subgame bundles

Simon,

Spiez,

Torunczyk

2020

Preprint

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For a set-valued function F on a compact subset W of a manifold, spanning is a topological property that implies that F (x) = ∅ for interior points x of W . A myopic equilibrium applies when for each action there is a payoff whose functional value is not necessarily affine in the strategy space. We show that if the payoffs satisfy the spanning property, then there exist a myopic equilibrium (though not necessarily a Nash equilibrium). Furthermore, given a parametrized collection of games and the spanning property to the structure of payoffs in that collection, the resulting myopic equilibria and their payoffs have the spanning property with respect to that parametrization. This is a far reaching extension of the Kohberg-Mertens Structure Theorem. There are at least four useful applications, when payoffs are exogenous to a finite game tree (for example a finitely repeated game followed by an infinitely repeated game), when one wants to understand a game strategically entirely with behaviour strategies, when one wants to extends the subgame concept to subsets of a game tree that are known in common, and for evolutionary game theory. The proofs involve new topological results asserting that spanning is preserved by relevant operations on set-valued functions.

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“…Claim a) above follows from Fact 1b), and for the other ones see (Simon, Spież, Toruńczyk 2002) (again, with M a sphere).…”

Section: The Spanning Property Of Correspondencesmentioning

confidence: 91%

Section: The Spanning Property Of Correspondencesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 95%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Myopic equilibria, the spanning property, and subgame bundles

Simon,

Spiez,

Torunczyk

2020

Preprint

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“…In non zero-sum repeated games with lack of information on one side, the existence of "joint plan" equilibria have been generalized to the case of state independent signalling ( [59]), and more generally to the case where "player 1 can send non revealing signals to player 2" ( [74]). The existence of a uniform equilibrium in the general signalling case is still an open question (see [75]).…”

Section: Vi3 Biconvexity and Bimartingalesmentioning

confidence: 99%

Repeated Games with Incomplete Information

Renault¹

2018

Encyclopedia of Complexity and Systems Science

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Article Outline Glossary 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.The 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 |.

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Repeated Games with Incomplete Information

Renault¹

2009

Encyclopedia of Complexity and Systems Science

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Article Outline Glossary 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 Notation Repeated 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. The 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 |.

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Equilibrium existence and topology in some repeated games with incomplete information

Cited by 16 publications

References 8 publications

Myopic equilibria, the spanning property, and subgame bundles

Myopic equilibria, the spanning property, and subgame bundles

Repeated Games with Incomplete Information

Repeated Games with Incomplete Information

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