  We consider game theory from the perspective of quantum algorithms.
Strategies in classical game theory are either pure (deterministic) or mixed
(probabilistic). We introduce these basic ideas in the context of a simple
example, closely related to the traditional Matching Pennies game. While not
every two-person zero-sum finite game has an equilibrium in the set of pure
strategies, von Neumann showed that there is always an equilibrium at which
each player follows a mixed strategy. A mixed strategy deviating from the
equilibrium strategy cannot increase a player's expected payoff. We show,
however, that in our example a player who implements a quantum strategy can
increase his expected payoff, and explain the relation to efficient quantum
algorithms. We prove that in general a quantum strategy is always at least as
good as a classical one, and furthermore that when both players use quantum
strategies there need not be any equilibrium, but if both are allowed mixed
quantum strategies there must be.
