On refinements of subgame perfect $$\epsilon $$ ϵ -equilibrium

Flesch, János; Predtetchinski, Arkadi

doi:10.1007/s00182-015-0468-8

Cited by 22 publications

(13 citation statements)

References 24 publications

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“…If the set of actions is finite, the existence of a pure SPE follows by the truncation approach of Fudenberg and Levine (1983). When the set of actions is infinite, an SPE need not exist, but one could follow the approach in Flesch and Predtetchinski (2015) to obtain existence of a pure -SPE, for all > 0.…”

Section: Discussionmentioning

confidence: 99%

Subgame-perfect $$\epsilon $$ ϵ -equilibria in perfect information games with sigma-discrete discontinuities

Flesch

Predtetchinski

2015

Econ Theory

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Multi-player perfect information games are known to admit a subgameperfect -equilibrium, for every > 0, under the condition that every player's payoff function is bounded and continuous on the whole set of plays. In this paper, we address the question on which subsets of plays the condition of payoff continuity can be dropped without losing existence. Our main result is that if payoff continuity only fails on a sigma-discrete set (a countable union of discrete sets) of plays, then a subgame-perfect -equilibrium, for every > 0, still exists. For a partial converse, given any subset of plays that is not sigma-discrete, we construct a game in which the payoff functions are continuous outside this set but the game admits no subgameperfect -equilibrium for small > 0.

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Section: Discussionmentioning

confidence: 99%

Subgame-perfect $$\epsilon $$ ϵ -equilibria in perfect information games with sigma-discrete discontinuities

Flesch

Predtetchinski

2015

Econ Theory

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“…It is not difficult to show, by means of examples like the one above, that the ordinal ξ * can be arbitrarily large. Examples of this sort have been previously constructed in Flesch et al [7] and Flesch and Predtetchinski [11].…”

Section: Examplesmentioning

confidence: 99%

“…7 This technique runs into difficulties when the number of players is not finite, since then there might be no such subgame. However, whether the player set is finite or not, a slight modification of the algorithm in Flesch et al [7] lends itself to a proof of the existence of SPE (see Flesch and Predtetchinski [11]).…”

Section: Special Casesmentioning

confidence: 99%

“…We show by means of an example that this result cannot be strengthened: there might be plays that survive all stages of elimination and yet that are not supportable by any subgame-perfect equilibrium. We argue that such a discrepancy arises because Player II might not have an optimal strategy in the auxiliary zero-sum game.Our final result is motivated by the frequent use of the technique of discretizing payoff functions: approximating a given bounded payoff function by a function with finite range, as is done, for instance, by Mertens and Neyman (see Mertens [22]), Flesch et al [7], Purves and Sudderth [24], Flesch and Predtetchinski [10,11]. The technique allows one to reduce the problem of existence of (subgame-perfect) -equilibrium for a game with bounded payoff functions to the problem of existence of (subgame-perfect) 0-equilibrium for a game where the payoff functions have finite range.…”

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A Characterization of Subgame-Perfect Equilibrium Plays in Borel Games of Perfect Information

Flesch

Predtetchinski

2017

Mathematics of OR

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We provide a characterization of subgame-perfect equilibrium plays in a class of perfect information games where each player's payoff function is Borel measurable and has finite range. The set of subgame-perfect equilibrium plays is obtained through a process of iterative elimination of plays. Extensions to games with bounded Borel measurable payoff functions are discussed. As an application of our results, we show that if every player's payoff function is bounded and upper semicontinuous, then, for every positive epsilon, the game admits a subgame-perfect epsilon-equilibrium. As we do not assume that the number of players is finite, this result generalizes the corresponding result of Purves and Sudderth [24] [Purves RA, Sudderth WD (2011) Perfect information games with upper semicontinuous payoffs. Math. Oper. Res. 36(3):468-473].The game begins with Player I choosing an action. Player II then proposes a play that respects Player I's move. Player I then chooses to accept the proposed play or to deviate. In the former case, the game ends. In the latter case, Player I also announces a deviation, and Player II proposes a new play that accommodates Player I's deviation. This process continues ad infinitum. Player II is restricted to propose plays that have survived the elimination thus far. The value of this auxiliary zero-sum game is defined to be the active player's security level.We believe that the characterization established here could yield useful insights into the nature of subgame perfection. Our second contribution on upper semicontinuous games, obtained as an application of this characterization, serves as an illustration.Contribution II. As an application of the characterization result, we address an open question raised by Purves and Sudderth [24]. Purves and Sudderth work in a setup that is identical to ours, except for their assumption that the number of players is finite. They show that if each player's payoff function is bounded and upper semicontinuous, then for each positive , the game admits a subgame-perfect -equilibrium. The authors further pose the question of whether the assumption that the number of players be finite could be dispensed with.The technique that Purves and Sudderth use is an induction on the number of distinct payoff vectors in the game. Because of this feature of their proof, extending it beyond the case of finitely many players seems to be difficult. We take a different approach. We show that in games where each player's payoff function has a finite range and is upper semicontinuous, our algorithm returns a nonempty set of plays at each step of the iteration. This allows us to deduce that any such game has a subgame-perfect equilibrium. The result for games with bounded upper semicontinuous payoff functions then follows by discretizing the payoff functions, exactly as is done by Purves and Sudderth.Additional results. For games with bounded Borel measurable payoffs we obtain the following result: subgame-perfect equilibrium plays survive all stages of the elimination. Convers...

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“…For results on the existence of subgame perfect -equilibrium, see Solan and Vieille [16], Solan [17], Flesch et al [3], Laraki et al [10], and Mashiah-Yaakovi [11], and Flesch and Predtetchinski [6,5].…”

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Subgame-Perfect ϵ-Equilibria in Perfect Information Games with Common Preferences at the Limit

Flesch¹,

Predtetchinski²

2016

Mathematics of OR

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On refinements of subgame perfect $$\epsilon $$ ϵ -equilibrium

Cited by 22 publications

References 24 publications

Subgame-perfect $$\epsilon $$ ϵ -equilibria in perfect information games with sigma-discrete discontinuities

Subgame-perfect $$\epsilon $$ ϵ -equilibria in perfect information games with sigma-discrete discontinuities

A Characterization of Subgame-Perfect Equilibrium Plays in Borel Games of Perfect Information

Subgame-Perfect ϵ-Equilibria in Perfect Information Games with Common Preferences at the Limit

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