# Game Theory: Mixed Strategies

## Introduction to Game Theory

• Game Theory is a branch of mathematics that studies strategic interaction, meaning the actions that different participants or ‘players’ take to maximise their own payoff.

• This theory provides a mathematical approach to decision-making in competitive situations, such as games, politics, and economics.

## Pure and Mixed Strategies

• In game theory, a player’s strategy is a complete action plan covering all possible moves that a player might need to make in the game.

• A pure strategy provides a complete definition of how a player will play a game. In particular, it determines the move a player will make for any situation he or she could face.

• A mixed strategy is a strategy in which a player makes a random choice among two or more possible actions, based on a set of chosen probabilities. A player using a mixed strategy is essentially saying, “I’m not sure what to do, so I will randomly select my action according to these probabilities.”

## Expected Payoff

• The expected payoff of a mixed strategy is the average payoff expected if the strategy is used over the long run.

• Expected payoff can be calculated by multiplying the probability of each outcome by the payoff of that outcome and then summing these figures.

## Strategies in Two-Person Games

• In a two-person, zero-sum game, each player seeks to maximise their own payoff at the expense of their opponent.

• When neither player has a dominant strategy (i.e., a strategy that always produces the best result regardless of the opponent’s actions), they might employ a mixed strategy, randomly selecting actions based on predetermined probabilities.

## Nash Equilibrium

• A Nash Equilibrium occurs when no player can benefit from changing their strategy while the other players keep theirs unchanged.

• A mixed strategy Nash Equilibrium occurs when each player’s strategy is the best response to the mixed strategy of the other player. This means no player has incentive to deviate from their strategy—there is no other strategy that will improve their payoff given what the others are doing.

## Solving Games with Mixed Strategies

• A common method to solve mixed strategy games is to set up and solve a system of equations: one for each player’s expected payoff.

• The idea is to find a probability distribution (i.e., a mixed strategy) for each player that makes the other player indifferent among their strategies.

• When each player adopts the mixed strategy that makes the opponent indifferent, a mixed strategy Nash Equilibrium is reached.