On the invariance of solutions of finite games

Games and Economic Behavior

2016

JEL classification: C72Keywords: Trembling hand perfect equilibrium Bayesian game with infinite type spaces Behavior strategy Second-price auction with incomplete information We develop the notion of perfect Bayesian Nash Equilibrium-perfect BNE-in general Bayesian games. We test perfect BNE against the criteria laid out by Kohlberg and Mertens (1986). We show that, for a focal class of Bayesian games, perfect BNE exists. Moreover, when payoffs are continuous, perfect BNE is limit undominated for almost every type. We illustrate the use of perfect BNE in the context of a second-price auction with interdependent values. Perfect BNE selects the unique pure strategy equilibrium in continuous strategies that separates types. Moreover, when valuations become independent, the equilibrium converges to the classical truthful dominant strategy equilibrium. We also show that less intuitive equilibria in which types are pooled are ruled out by our selection criterion. We further argue that standard selection criteria for second-price auctions have no bite here. Bidders have no dominant strategies, and the separating equilibrium is not sincere.

Section: Ordinalitymentioning

confidence: 96%

Behavioral perfect equilibrium in Bayesian games

Bajoori

Flesch

Games and Economic Behavior

2016

“…This is surprising in the sense that ∞ is payoff equivalent to 1 for player 2. Thus, both strong and weak perfection violate invariance in this setting, while perfect equilibrium satisfies invariance in the setting of finite strategic form games (see for example Mertens [18] and Vermeulen and Jansen [25]). …”

Section: Invariancementioning

confidence: 99%

Perfect equilibrium in games with compact action spaces

Bajoori

Flesch

Games and Economic Behavior

2013

We investigate the relations between different types of perfect equilibrium, introduced by Simon and Stinchcombe (1995) for games with compact action spaces and continuous payoffs. Simon and Stinchcombe distinguish two approaches to perfect equilibrium in this context, the classical "trembling hand" approach, and the so-called "finitistic" approach. We propose an improved definition of the finitistic approach, called global-limit-offinite perfection, and prove its existence.Despite the fact that the finitistic approach appeals to basic intuition, our results-specifically examples (1) and (2)-seem to imply a severe critique on this approach. In the first example any version of finitistic perfect equilibrium admits a Nash equilibrium strategy profile that is not limit admissible. The second example gives a completely mixed (and hence trembling hand perfect) Nash equilibrium that is not finitistically perfect.Further examples illustrate the relations between the two approaches to perfect equilibrium and the relation to admissibility and undominatedness of strategies.JEL Codes. C72.

“…Now let (Γ * ) be the game with the same strategy space as Γ for which the payoff function of a player equals the composition of the payoff function of that player in the game Γ and f . Since each f i preserves best replies (by Lemma 1 of (Vermeulen and Jansen, 1997)), the games (Γ * ) and Γ are abr-equivalent. Since, moreover, f is a reduction map from (Γ * ) to Γ * , f −1 (S) is a BR-set of the game (Γ * ) .…”

Section: Theorem 2 the Solution τ Is Abr-invariantmentioning

confidence: 99%

“…Let Γ and Γ * be two abr-equivalent games and let S be a solution set of the game Γ * . Then, by Theorem 3 of (Vermeulen and Jansen, 1997), it must be a closed and connected set of perfect equilibria of the game Γ . In order to show that S is also an extensible set of the game Γ , take a reduction map f from a game Γ to Γ .…”

Section: Theorem 2 the Solution τ Is Abr-invariantmentioning

confidence: 99%

An ordinal selection of stable sets in the sense of Hillas

Journal of Mathematical Economics

Jansen

2001

Self Cite

In this paper, it is shown in an example that the original definition of stable sets in Hillas [Econometrica 58 (1990) 1365-1391] does not satisfy (an even slightly weakenened version of) the invariance condition proposed in Mertens [Ordinality in noncooperative games, Core Discussion Paper 8728, CORE Louvain de la Neuve, Belgium, 1987]. However, it is also shown that the basic stability condition of Hillas underlying his definition of stable sets does admit a selection that is invariant in the strong sense, and even ordinal.