Perfectly almost strict equilibria for finite games in strategic form

https://doi.org/10.1016/S0165-4896(96)00836-0Get rights and content

Abstract

In this paper we introduce the perfectly almost strict equilibrium concept for finite games in strategic form. Such a new concept is a strict refinement of the perfect equilibrium which rejects completely mixed Nash equilibria which are not self-enforcing. We prove that the mixed extension of a finite game in strategic form has at least one perfectly almost strict equilibrium.

Introduction

One of the main targets of non-cooperative game theory is to propose concepts which select self-enforcing combinations of strategies in a wide class of games. That is a difficult task; in fact, many concepts have been introduced in game theoretical literature but none can be universally accepted. However, some have proved to be more valuable than others.

Perhaps one of the most relevant concepts is perfect equilibrium, introduced in Selten (1975). The perfect equilibrium concept has a number of desirable properties, but it also has some drawbacks. The drawback we are concerned with in this paper is that it is too “benevolent” with completely mixed strategy combinations, in the sense that all completely mixed Nash equilibria are also perfect. This is not very reasonable. For instance, in the game given by(e1, f1), (e2, f2) and (12e1+12e2,12f1+12f2) are perfect equilibria. However, the latter is less self-enforcing than the two former examples (these two are immune against small trembles; the completely mixed perfect equilibrium is not).

In this paper we present a refinement of the perfect equilibrium concept which corrects the drawback pointed out above. The organization of the paper is as follows. In Section 2we set up the notation and recall the framework of random games, introduced in Borm et al. (1995), because it will be important in this paper. In Section 3we introduce our new concept and study some of its properties.

Section snippets

Background

Let us establish the notation that we use further on in this paper. A finite game in strategic form Γ with player set M={1,...,m} is represented by Γ=〈{Si}iM, {Ki}iM〉 where, for each iM, Si denotes player i's finite set of pure strategies and Ki: S=∏j=1m SjR denotes player i's payoff function. We represent by Δi the set of player i's mixed strategies (for all iM), i.e.Δi={σiRSii≥0Si∈Si,si∈Siσi(si)=1}.Payoff functions {Ki}iM are extended to the set Δ=∏jM Δj of all mixed strategy

Perfectly almost strict equilibria

In this section we define the almost strict and the perfectly almost strict equilibrium concepts. The later is a modification of perfect equilibrium which is able to reject unsensible completely mixed Nash equilibria. First we prove an interesting property of the function Nε,P.
Lemma 3.1. Let Γ be a finite game in strategic form and, take σΔ and P=(Ω, A, π) a probabilistic space satisfying C.1–C.3. Then, there exists a real number εσ>0 such that Nεσ,P(σ)=Nε,P(σ) for all εεσ.
Proof. Take iM, ε

Acknowledgements

We thank DGICYT and Xunta de Galicia for financial support through projects PB94-0648-C02-02 and XUGA20704B95. We also acknowledge the suggestions of one anonymous Associate Editor which helped us to improve the paper.

References (3)

  • P.E.M. Borm, I. Garcı́a-Jurado, R. Cao-Abad and L. Méndez-Naya, Weakly strict equilibria in finite normal form games,...
There are more references available in the full text version of this article.

Cited by (1)

View full text