Extremal Pure Strategies and Monotonicity in Repeated Games

Schoenmakers

2017

Games

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This paper examines how to construct subgame-perfect mixed-strategy equilibria in discounted repeated games with perfect monitoring. We introduce a relatively simple class of strategy profiles that are easy to compute and may give rise to a large set of equilibrium payoffs. These sets are called self-supporting sets, since the set itself provides the continuation payoffs that are required to support the equilibrium strategies. Moreover, the corresponding strategies are simple as the players face the same augmented game on each round but they play different mixed actions after each realized pure-action profile. We find that certain payoffs can be obtained in equilibrium with much lower discount factor values compared to pure strategies. The theory and the concepts are illustrated in 2 × 2 games.

Section: From This Condition We Getmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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

Schoenmakers

2017

Games

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“…The punishment payoffs are v − (V (T 1 )) and v − (V (T 2 )). Complementing the following result, it is shown in [7] that the equilibrium paths may not be monotone in the discount factor if the punishment payoffs increase. Note also that the monotonicity of paths holds even though the equilibrium payoffs may fail to satisfy the monotone comparative statics.…”

Section: Properties Of Elementary Subpathsmentioning

confidence: 67%

“…We shall derive a characterization for the equilibrium subpaths by assuming that the players' smallest equilibrium payoffs are known. Finding these payoffs is discussed in Section 4.1, see also [7,11] on the computation of the smallest equilibrium payoffs.…”

Section: Notation and Definitionsmentioning

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...

“…We leave for future research how the minimum payoffs should be found. It is not an easy task even in repeated games [10,11], but these methods can be generalized to stochastic games. For an upper bound, we can solvẽ…”

Section: Elementary Subpathsmentioning

confidence: 99%

Elementary Subpaths in Discounted Stochastic 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.