# Perfect equilibrium in games with compact action spaces

Working paper

Issue number:

RM/11/029

Publisher:

Maastricht University

Year:

2011

We investigate the relations between diﬀerent types of perfect equilibrium, introduced by Simon and Stinchcombe (1995) for games with compact ac- tion spaces and continuous payoﬀs. Simon and Stinchcombe distinguish two approaches to perfect equilibrium in this context, the classical “trem- bling hand” approach, and the so-called “ﬁnitistic” approach. We propose an improved deﬁnition of the ﬁnitistic approach, called global-limit-of- ﬁnite 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.