Strategy (game Theory) - Mixed Strategy - A Disputed Meaning

A Disputed Meaning

During the 1980s, the concept of mixed strategies came under heavy fire for being "intuitively problematic". Randomization, central in mixed strategies, lacks behavioral support. Seldom do people make their choices following a lottery. This behavioral problem is compounded by the cognitive difficulty that people are unable to generate random outcomes without the aid of a random or pseudo-random generator.

In 1991, game theorist Ariel Rubinstein described alternative ways of understanding the concept. The first, due to Harsanyi (1973), is called purification, and supposes that the mixed strategies interpretation merely reflects our lack of knowledge of the players' information and decision-making process. Apparently random choices are then seen as consequences of non-specified, payoff-irrelevant exogeneous factors. However, it is unsatisfying to have results that hang on unspecified factors.

A second interpretation imagines the game players standing for a large population of agents. Each of the agents chooses a pure strategy, and the payoff depends on the fraction of agents choosing each strategy. The mixed strategy hence represents the distribution of pure strategies chosen by each population. However, this does not provide any justification for the case when players are individual agents.

Later, Aumann and Brandenburger (1995), re-interpreted Nash equilibrium as an equilibrium in beliefs, rather than actions. For instance, in Rock-paper-scissors an equilibrium in beliefs would have each player believing the other was equally likely to play each strategy. This interpretation weakens the predictive power of Nash equilibrium, however, since it is possible in such an equilibrium for each player to actually play a pure strategy of Rock.

Ever since, game theorists' attitude towards mixed strategies-based results have been ambivalent. Mixed strategies are still widely used for their capacity to provide Nash equilibria in games where no equilibrium in pure strategies exists, but the model does not specify why and how players randomize their decisions.

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