18.12 : Mixed Strategies

In game theory, mixed strategies involve players choosing their actions randomly from a set of available options. This approach contrasts with pure strategies, where players select a specific action with certainty. Mixed strategies become relevant in scenarios where there is no pure strategy equilibrium.

A mixed-strategies Nash equilibrium occurs when players adopt strategies so that no one can benefit by unilaterally changing their own strategy, given the strategies of the others. In this equilibrium, every player's strategy is an optimal response to the others.

Consider the game of rock-paper-scissors, where players can choose between three options: rock, paper, or scissors. In this game, rock beats scissors, scissors beat paper, and paper beats rock. A tie occurs if both players select the same item. Since each choice can be countered, there is no dominant move, leading to the absence of a pure-strategy Nash equilibrium.

However, a mixed-strategies Nash equilibrium exists where each player selects either rock, paper, and scissors with equal probability (one-third each). This random strategy ensures unpredictability, maintaining a balance as no player can foresee the other's move.

This equilibrium demonstrates how mixed strategies can stabilize games by neutralizing direct counter-actions between players.