Subgame-perfection in recursive perfect information games, where each player controls one state

Kuipers, Jeroen; Flesch, János; Schoenmakers, Gijs; Vrieze, Koos

doi:10.1007/s00182-015-0502-x

Cited by 11 publications

(10 citation statements)

References 18 publications

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“…We have v i (q) u − i (q). Indeed, if q ∈ P\S, this follows automatically by the definition of v i , and if q ∈ S, this follows by (16) since v i (p) < v i (q) implies that v i (p) < u + i (q). Furthermore, v i (p) u + i (p).…”

Section: Claim 3 the Strategy Profile σ Is An Spe Of The Game ω(V)mentioning

confidence: 98%

“…This is a very mild assumption. Games with semicontinuous payoffs (Purves and Sudderth [24], Flesch et al [7]), perfect information stopping games (e.g., Solan [25], Mashiah-Yaakovi [21]), recursive stochastic games with perfect information (e.g., Flesch et al [8], Kuipers et al [16]) are all examples of games with Borel measurable payoffs.…”

Section: Introductionmentioning

confidence: 99%

“…In the construction of subgame-perfect -equilibria, the primary play is to be followed with a high probability, and the secondary play, which acts as a threat, is to be followed with a small probability. Kuipers et al [16] refine this algorithm to establish the result for perfect information stochastic games with the crucial feature that every player controls only one state. Alós-Ferrer and Ritzberger [1] consider an algorithm very different from ours to obtain an existence of subgame-perfect equilibrium in a class of continuous games.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: Claim 3 the Strategy Profile σ Is An Spe Of The Game ω(V)mentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Flesch

Predtetchinski

2017

Mathematics of OR

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“…Compatible plans: Consider that a player i ∈ N can influence play by choosing a specific action if play visits one of his states, say t ∈ S. Now, if every α-viable plan after the selected action yields a strictly higher reward for player i than α t , then α t can be increased without eliminating any plan that may occur in equilibrium. This idea formed the basis for the iterative procedure in Flesch et al (2010b) and Kuipers et al (2016). In those papers, it was sufficient to consider only one state at a time per iteration to eventually eliminate all non-equilibrium plans.…”

Section: Strategic Concepts and An Update Proceduresmentioning

confidence: 99%

“…Our existence result extends several earlier results, where further restrictions were imposed on either the transition structure or the reward structure of the game. A subgame perfect ε-equilibrium, for every ε > 0, was previously established in games with only nonnegative rewards (Flesch et al 2010a), in free transition games (Kuipers et al 2013), and in games where each player only controls one state (Kuipers et al 2016). In the literature, we can also find sufficient conditions for other classes of games, such as in the classical papers by Fudenberg and Levine (1983) and Harris (1985), and more recently, in the papers by Solan and Vieille (2003), Flesch et al (2010b), Purves and Sudderth (2011), Brihaye et al (2013), Roux and Pauly (2014), Flesch and Predtetchinski (2016), Roux (2016), Mashiah-Yaakovi (2014), Cingiz et al (2019) and Flesch et al (2019).…”

Section: Introductionmentioning

confidence: 99%

Subgame perfection in recursive perfect information games

et al. 2020

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We consider sequential multi-player games with perfect information and with deterministic transitions. The players receive a reward upon termination of the game, which depends on the state where the game was terminated. If the game does not terminate, then the rewards of the players are equal to zero. We prove that, for every game in this class, a subgame perfect ε-equilibrium exists, for all ε > 0. The proof is constructive and suggests a finite algorithm to calculate such an equilibrium.

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Non-Zero-Sum Stochastic Games

Jaśkiewicz¹,

Nowak²

2016

Handbook of Dynamic Game Theory

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Abstract. This paper treats stochastic games. A nonzero-sum average payoff stochastic games with arbitrary state spaces and the stopping games are considered. Such models of games very well fit in some studies in economic theory and operations research. A correlation of strategies of the players, involving "public signals", is allowed in the nonzero-sum average payoff stochastic games. The main result is an extension of the correlated equilibrium theorem proved recently by Nowak and Raghavan for dynamic games with discounting to the average payoff stochastic games. The stopping games are special model of stochastic games. The version of Dynkin's game related to observation of Markov process with random priority assignment mechanism of states is presented in the paper. The zero-sum and nonzero-sum games are considered. The paper also provides a brief overview of the theory of nonzero-sum stochastic games and stopping games which are very far from being complete.

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Subgame-perfection in recursive perfect information games, where each player controls one state

Cited by 11 publications

References 18 publications

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

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

Subgame perfection in recursive perfect information games

Non-Zero-Sum Stochastic Games

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