Free Trial

Safari Books Online is a digital library providing on-demand subscription access to thousands of learning resources.

Share this Page URL

3.2.2 Fundamental Notions of Repeated Ga... > 3.2.2 Fundamental Notions of Repeate... - Pg. 74

74 CHAPTER 3 Moving from Static to Dynamic Games After Kuhn's theorem was formulated (Kuhn, 1953), it was realized that, in finite extensive-form games with perfect recall, there is an equivalence in terms of game outcome between mixed and behavior strategies. More specifically, a mixed strat- egy i is said to be (realization) equivalent to a behavior strategy i when, for given mixed strategies chosen by the other players (-i), both strategies lead to the same outcome (every player obtains the same utility). Kuhn's result has been generalized by Aumann (1961) to infinitely repeated games with a finite number of players and action sets. This is the reason why mixed strategies do not need to be considered in standard repeated games with perfect recall. When the perfect recall assumption does not hold, this result needs to be reconsidered. It can turn out that mixed strategies and even general strategies 2 are needed. These aspects will not be here considered and the reader is referred to papers like Kaneko and Kline (1995) and Wichardt (2008) for further details about perfect recall refinements or imperfect recall. In repeated games, a player does not care about what he gets at a given stage, but about what his gains over the whole duration of the game. This is why utility func- tions resulting from averaging over the instantaneous utility have to be considered. There are three dominant models used in the literature. The corresponding expres- sions for the utility are provided by the following three definitions; pure strategies