CSCE 475/875

Handout 4: Strategies in Normal-Form Games

September 8, 2011

(Based on Shoham and Leyton-Brown 2011)

Pure Strategy

Certainly, one kind of strategy is to select a single action and play it. We call such a strategy a pure strategy, and we will use the notation we have already developed for actions to represent it. We call a choice of pure strategy for each agent a pure-strategy profile.

Mixed Strategy

Players could also follow another strategy: randomizing over the set of available actions according to some probability distribution. Such a strategy is called a mixed strategy. (Although it may not be immediately obvious why a player should introduce randomness into his choice of action, in fact in a multiagent setting the role of mixed strategies is critical. WHY?)

Definition 3.2.4 (Mixed strategy) Let be a normal-form game, and for any set let be the set of all probability distributions over . Then the set of mixed strategies for player is

Definition 3.2.5 (Mixed-strategy profile) The set of mixed-strategy profiles is simply the Cartesian product of the individual mixed-strategy sets, .

By we denote the probability that an action will be played under mixed strategy The subset of actions that are assigned positive probability by the mixed strategy is called the support of .

Definition 3.2.6 (Support) The support of a mixed strategy for a player is the set of pure strategies .

The generalization to mixed strategies relies on the basic notion of decision theory—expected utility. Intuitively, we (1) calculate the probability of reaching each outcome given the strategy profile, and (2) we calculate the average of the payoffs of the outcomes, weighted by the probabilities of each outcome. Formally, we define the expected utility as (we use for both utility and expected utility):

Definition 3.2.7 (Expected utility of a mixed strategy) Given a normal-form game , the expected utility for player of the mixed-strategy profile is defined as