Construction of Subgame-Perfect Mixed-Strategy Equilibria in Repeated Games

2015

Dyn Games Appl

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This paper examines the subgame-perfect equilibria in discounted stochastic games with finite state and action spaces. The fixed-point characterization of equilibria is generalized to unobservable mixed strategies. It is also shown that the pure-strategy equilibria consist of elementary subpaths, which are repeating fragments that give the acceptable action plans in the game. The developed methodology offers a novel way of computing and analyzing equilibrium strategies that need not be stationary nor Markovian.

Section: Introductionsupporting

confidence: 67%

Elementary Subpaths in Discounted Stochastic Games

2015

Dyn Games Appl

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“…The Hausdorff dimension, on the other hand, measures how the payoff set fills the space and hence serves as a measure for the complexity of the equilibrium payoff set. This paper lays foundations for further research, e.g., on extending the methodology to stochastic games [8] or mixed strategies [12], on designing algorithms for computing equilibria and minimum payoffs [9,11], and on the analyses of the fractal properties of payoff sets [10].…”

Section: Discussionmentioning

confidence: 99%

Equilibrium paths in discounted supergames

Discrete Applied Mathematics

Kitti

2019

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This paper examines the subgame-perfect pure-strategy equilibria in discounted supergames with perfect monitoring. It is shown that all the equilibrium paths are composed of fragments called elementary subpaths. This characterization result makes it possible to compute and analyze the equilibrium paths and payoffs by using a collection of elementary subpaths. It is also shown that all the equilibrium paths can be compactly represented by a directed graph when there are finitely many elementary subpaths. In general, there may be infinitely many elementary subpaths, but it is always possible to construct finite approximations. When the subpaths are allowed to be approximatively incentive compatible, it is possible to compute in a finite number of steps a graph that represents all the equilibrium paths. The directed graphs can be used in analyzing the complexity of equilibrium outcomes. In particular, it is shown that the size and the density of the equilibrium set can be measured by the asymptotic growth rate of equilibrium paths and the Hausdorff dimension of the payoff set. JEL Classification: C72, C73players' available moves at different positions, we use graphs to describe the variety of equilibrium behavior. Moreover, we emphasize that this characterization concerns all the equilibrium paths simultaneously rather than gives a condition for individual paths as in [1,2].We propose two complexity measures to analyze the equilibrium paths and payoffs based on the graph presentation. The first one is the asymptotic growth rate of the equilibrium paths. This measure tells us the rate at which the number of finitely long equilibrium paths grows when their length is increased. Because the paths of action profiles are given by strategies, the growth rate reflects the size of the equilibrium set and the increase of strategies producing the finitely long equilibrium paths.The second complexity measure is the Hausdorff dimension of the payoff set. This measure reflects the density of the equilibrium payoff set, which is a fractal in general [10]. The phenomenon that the payoff set behaves in a rather complex manner, as fractals do, is not completely new, see [31] and [34]. We offer a more comprehensive view to the structure of equilibria: when the discount factors vary, the elementary subpaths change, which affects the graph that generates the payoffs. The proposed complexity measures make it possible, e.g., to compare different repeated games in terms of equilibrium behavior.Our approach to the complexity of repeated game equilibria is new, and it differs from the previous literature on strategic complexity [15,28] and computational complexity [16,21]. We analyze the complexity of all the equilibrium outcomes without relying on the complexity of individual strategies nor their computation. It has been shown that computing even an approximate equilibrium in a stage game is difficult [20], and the task is not any easier in repeated games [13,33]. However, there are efficient algorithms that work for a class of repeated gam...

“…Note that the mapping B goes through all the action profiles a ∈ A and all the possible continuations payoffs w that can follow a, i.e., the set C a (W ). See Berg and Schoenmakers (2014) for the corresponding characterization in mixed strategies in a model where the players only observe the realized pure actions. There are many algorithms that use the iteration of B in computing approximations of the set of equilibrium payoffs (Cronshaw 1997;Judd et al 2003;Burkov and Chaib-draa 2010;Salcedo and Sultanum 2012;Abreu and Sannikov 2014).…”

Section: Theorem 1 the Payoff Set V Is The Largest Fixed Point Of Bmentioning

confidence: 99%

“…For example, Berg and Kärki (2014b) examine the lowest discount factor values when the payoff set covers all the reasonable payoffs in the symmetric 2 × 2 games under pure, mixed and correlated strategies, and this also gives a bound when the payoff set becomes convex and monotone. We give the result for the prisoner's dilemma; see Berg and Kärki (2014b), Berg and Schoenmakers (2014), Stahl (1991) for more details.…”

Section: Thus V C (δ) Is Convex and Monotone In δmentioning

confidence: 99%

See 1 more Smart Citation

Extremal Pure Strategies and Monotonicity in Repeated Games

2016

Comput Econ

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The recent development of computational methods in repeated games has made it possible to study the properties of subgame-perfect equilibria in more detail. This paper shows that the lowest equilibrium payoffs may increase in pure strategies when the players become more patient and this may cause the set of equilibrium paths to be non-monotonic. A numerical example is constructed such that a path is no longer equilibrium when the players' discount factors increase. This property can be more easily seen when the players have different time preferences, since in these games the punishment strategies may rely on the differences between the players' discount factors. A sufficient condition for the monotonicity of equilibrium paths is that the lowest equilibrium payoffs do not increase, i.e., the punishments should not become milder.