Mixed Strategy Games

When a game has no saddle point, players randomize over their strategies (a mixed strategy) instead of committing to one.

• Mixed strategy
  A probability distribution over a player’s pure strategies.

2x2 Game Without a Saddle Point

Solved algebraically: for optimal mixed strategy probabilities $p, 1-p$ (row) and $q, 1-q$ (column), the row player’s expected payoff is equalized across the column player’s 2 strategies, and vice versa. Solving the resulting linear equations gives $p$, $q$, and the game’s value.

Dominance

A strategy is dominated if another strategy gives the same player a payoff at least as good against every opposing strategy. Dominated strategies are removed before solving, reducing the payoff matrix.

2xn and mx2 Games

Solved graphically: plot each of the smaller player’s strategies as a line over the opponent’s mixing probability, take the upper envelope (row player) or lower envelope (column player), and read off the optimal point. Reduces the game to an equivalent 2x2 game at that point.