Fractal Geometry of Equilibrium Payoffs in Discounted Supergames

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: Introductionmentioning

confidence: 99%

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

Schoenmakers

2017

Games

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“…, δ n on the diagonal. Now, we are ready to present the fixed-point characterization of equilibrium payoffs (Abreu et al 1986(Abreu et al , 1990Cronshaw and Luenberger 1994;Mailath and Samuelson 2006;Berg and Kitti 2014).…”

Section: Monotonicity Of Equilibriamentioning

confidence: 99%

“…Recently, Berg and Kitti (2015) have shown that the equilibrium paths consist of repeating fragments called elementary subpaths. This has provided a new methodology for analyzing the set of equilibria, computing the pure-strategy payoffs (Berg and Kitti 2013) and identifying the equilibrium payoffs as particular fractals (Berg and Kitti 2014). These papers assume that the punishment payoffs are known but it has turned out that the punishment paths may be very complicated and difficult to find in some games.…”

mentioning

confidence: 99%

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.

“…Berg and Kitti [12] propose two measures to analyze the set of equilibria: the Hausdorff dimension measures the density of the payoffs, see also [14] for unequal discount factors, and the asymptotic growth rate measures the number of different equilibrium paths. These measures make it possible to compare equilibria for different discount factor values and between games, and they can be extended to stochastic games by modifying the graphs so that each node corresponds to playing one action profile.…”

Section: Propositionmentioning

confidence: 99%

“…Berg and Kitti [12] have developed this idea in repeated games, where it is shown that the equilibrium paths consist of repeating fragments called elementary subpaths. These subpaths offer a novel way of computing and analyzing the set of equilibria [13,14]. The pure-strategy payoffs can be identified as particular fractals called sub-self-affine sets, and their density can be measured by the Hausdorff dimension.…”

Section: Introductionmentioning

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.