Perfect equilibrium in games with compact action spaces

Abstract

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.

Behavioral perfect equilibrium in Bayesian games

“…i is a best response against the completely mixed profile β k , for every k, it follows from Lemma A.2 in Bajoori et al (2013) that σ k i is undominated, for every k. 7 Hence β i is limit undominated. 2…”

“…While perfect equilibrium satisfies invariance in the setting of finite strategic form games-see for example Mertens (2003) and Vermeulen and Jansen (1997)-perfection violates invariance even in games with complete information if the action spaces are infinite, as is pointed out in Bajoori et al (2013).…”

“…They show existence of these notions of perfect equilibrium, and investigate the properties of and relations between the various resulting solution concepts. Bajoori et al (2013) examine the two approaches proposed in Simon and Stinchcombe (1995) in further detail and provide an improved definition of the finitistic approach. Their results seem to imply a critique on the finitistic approach.…”

Uniform Sampling of the Infinite Noncooperative Game on Unit Hypercube and Reshaping Ultimately Multidimensional Matrices of Player’s Payoff Values

“…However, NE-solutions render a lot of the refined or modified principles of optimality, allowing to smooth differences in utility and equity [2], [10], [11]. Mainly, they are principles of Pareto equilibrium [2], [6], [8], [10], [13], [14], Mertens-stable equilibrium [15], trembling hand perfect equilibrium [16], proper equilibrium [17], [18], correlated equilibrium [19], sequential equilibrium [20], [21], quasi-perfect equilibrium [18], [22], [23], perfect Bayesian equilibrium [18], [20], [24], [25], quantal response equilibrium [26], [27], self-confirming equilibrium [28], [29], strong Nash equilibrium [30], [31], Markov perfect equilibrium [32], [33]. The question is only to find NEsolutions as fast as possible.…”

“…Moreover, often an admissible player's action is described with a series of its continuous parameters, constituting thus an infinite (continuous) set of pure strategies [1], [6], [7], [12], [39], [40]. If this continuous set is compact then it is easy to find an isomorphic game to the initial one, that the set of every player's pure strategies would be Euclidean finite-dimensional subspace [6], [10], [12], [23], [41]. Normally, the spoken subspace may be a unit cube of the appropriate dimension [6], [10], [11].…”