Rationalizable strategies in random games

Pei, Ting; Takahashi, Satoru

doi:10.1016/j.geb.2019.08.011

Cited by 12 publications

(10 citation statements)

References 14 publications

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“…Concerning the use of best-response structures as a tool to study convergence frequencies, Pangallo et al [ 17 ] and Pei and Takahaski [ 18 ] both obtained exact results for the frequency of one or more PSNEs in the 2-player case. The authors in [ 12 ] use different methods to bound the convergence frequency in multi-player games.…”

Section: Introductionmentioning

confidence: 99%

The Frequency of Convergent Games under Best-Response Dynamics

Wiese

Heinrich

2021

Dyn Games Appl

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We calculate the frequency of games with a unique pure strategy Nash equilibrium in the ensemble of n-player, m-strategy normal-form games. To obtain the ensemble, we generate payoff matrices at random. Games with a unique pure strategy Nash equilibrium converge to the Nash equilibrium. We then consider a wider class of games that converge under a best-response dynamic, in which each player chooses their optimal pure strategy successively. We show that the frequency of convergent games with a given number of pure Nash equilibria goes to zero as the number of players or the number of strategies goes to infinity. In the 2-player case, we show that for large games with at least 10 strategies, convergent games with multiple pure strategy Nash equilibria are more likely than games with a unique Nash equilibrium. Our novel approach uses an n-partite graph to describe games.

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Section: Introductionmentioning

confidence: 99%

The Frequency of Convergent Games under Best-Response Dynamics

Wiese

Heinrich

2021

Dyn Games Appl

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“…For any given Column's action, there are n − 1 other actions that may strictly dominate it, each with probability 1 2 m . Using a union bound, 17 A variant of this observation for m = n is also stated as Proposition 5 in Pei and Takahashi (2019). 18 We can use the same union bound to show that for m = 2 log 2 n + ω(1), Column cannot delete any action at all.…”

Section: Arbitrary Action Sets: General Analysismentioning

confidence: 98%

“…The random games we study are, to some extent, more complex in that respect, structures described here. Pei and Takahashi (2019) consider point-rationalizability in similar classes of games, see also our discussion in Section 6. 32 The conditional distribution can be stated in terms of the unconditional one.…”

Section: Alternative Classes Of Gamesmentioning

confidence: 99%

“…In addition, we draw on experimental literature that suggests most individuals cannot perform many iterations, not without substantial experience (see, for instance, Nagel, 1995;Costa-Gomes, Crawford, and Broseta, 2001;Camerer, Ho, and Chong, 2004). Powers (1990) and McLennan (2005) also consider random games and analyze the number of Nash equilibria, pure and mixed, while Pei and Takahashi (2019) study the distribution of the number of point-rationalizable actions in such games. Our paper is related in spirit to these predecessors, though we address different questions and use different methodologies.…”

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confidence: 99%

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Dominance Solvability in Random Games

Alon¹,

Rudov²,

Yariv³

2021

Preprint

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We study the effectiveness of iterated elimination of strictly-dominated actions in random games. We show that dominance solvability of games is vanishingly small as the number of at least one player's actions grows. Furthermore, conditional on dominance solvability, the number of iterations required to converge to Nash equilibrium grows rapidly as action sets grow. Nonetheless, when games are highly imbalanced, iterated elimination simplifies the game substantially by ruling out a sizable fraction of actions. Technically, we illustrate the usefulness of recent combinatorial methods for the analysis of general games.

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“…Researchers examined the probability that a game possesses a pure Nash equilibrium, the distribution of the number of pure Nash equilibria, as well as questions related to convergence of learning processes. Rather than reviewing an extensive literature on random one-shot games, we refer to the two recent studies (Amiet, Collevecchio, Scarsini, and Zhong [1] and Pei and Takahashi [22]) that contain a survey of the literature.…”

Section: Introductionmentioning

confidence: 99%

Random perfect information games

Flesch¹,

Predtetchinski²,

Суомала³

2021

Preprint

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The paper proposes a natural measure space of zero-sum perfect information games with upper semicontinuous payoffs. Each game is specified by the game tree, and by the assignment of the active player and of the capacity to each node of the tree. The payoff in a game is defined as the infimum of the capacity over the nodes that have been visited during the play. The active player, the number of children, and the capacity are drawn from a given joint distribution independently across the nodes. We characterize the cumulative distribution function of the value v using the fixed points of the so-called value generating function. The characterization leads to a necessary and sufficient condition for the event v ≥ k to occur with positive probability. We also study probabilistic properties of the set of Player I's k-optimal strategies and the corresponding plays.

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Rationalizable strategies in random games

Cited by 12 publications

References 14 publications

The Frequency of Convergent Games under Best-Response Dynamics

The Frequency of Convergent Games under Best-Response Dynamics

Dominance Solvability in Random Games

Random perfect information games

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