Mixed Strategies

Players adopt mixed strategies when a game lacks a pure-strategy Nash equilibrium, meaning no set of strategies exists where both players simultaneously choose their best response. In such games, players randomly select from available actions to maintain unpredictability. The penalty kick scenario exemplifies this: when neither the kicker nor goalie can gain an advantage by choosing a specific direction, both must randomize their choices to achieve equilibrium.

A mixed-strategies Nash equilibrium occurs when players adopt randomized strategies such that no player benefits by unilaterally changing their own strategy. Each player's strategy becomes an optimal response to the others' strategies. In the penalty kick example, both players achieve equilibrium by choosing left or right 50% of the time, ensuring neither can predict the opponent's move or gain an advantage.

By choosing randomly, players keep their moves unpredictable, eliminating any pattern the opponent could exploit. When both the kicker and goalie randomize equally between options, neither can anticipate the other's choice. This unpredictability ensures both choices are equally likely, meaning no player has incentive to deviate from the mixed strategy since switching provides no competitive edge.

Rock-paper-scissors has no pure-strategy equilibrium because each choice can be countered by another: rock beats scissors, scissors beats paper, and paper beats rock. Since no dominant move exists, players cannot select a single strategy that guarantees success. A mixed-strategies Nash equilibrium emerges when each player selects rock, paper, and scissors with equal probability (one-third each), maintaining balance through unpredictability.

Mixed strategies stabilize games by neutralizing direct counter-actions between players. When players randomize their choices equally across available options, no single strategy can consistently counter another. This randomness ensures that each player's expected payoff remains constant regardless of which action they choose, eliminating the incentive to deviate and creating a stable equilibrium.

If a player deviates from a mixed-strategy equilibrium, they cannot improve their payoff because the opponent's randomized strategy already accounts for all possible deviations. Since the opponent chooses each action with equal probability, switching to any single pure strategy yields the same expected outcome. This mutual best-response property ensures both players remain committed to their mixed strategies.

In the penalty kick scenario, the kicker and goalie face a game with no pure-strategy equilibrium because matching choices favor the goalie while mismatches favor the kicker. By each randomizing left or right 50% of the time, both players achieve equilibrium. This randomness ensures neither player can predict or exploit the other's move, making the mixed strategy the optimal solution.