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