Perfect equilibrium in games with compact action spaces

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.