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.

Players typically select a pure strategy where they choose a specific action with certainty. However, not all games allow for pure strategies, leading players to adopt mixed strategies by randomly selecting from available actions.

Consider a penalty kick scenario. The kicker can kick either left or right.

The goalie can dive left or right. If the kicker and the goalie choose the same direction, the goalie wins. Otherwise, the kicker wins.

The payoff matrix shows the outcomes.

This scenario results in no pure-strategy Nash equilibrium. This is because there is no set of strategies where both players simultaneously choose the best response to the other's choice.

Here, the game has a mixed-strategies Nash equilibrium, when each player chooses left or right 50% of the time.

This randomness ensures that neither player can predict or gain an advantage, as both choices are equally likely.

By choosing randomly, both the kicker and the goalie keep their moves unpredictable, which means no player has a reason to switch from this strategy since it doesn't give the opponent an edge.