Game Theory
Game theory analyses competitive situations with two or more players whose outcomes depend on each other. STPM 950 concentrates on two-person zero-sum games, saddle points, dominance, and 2×2 mixed strategies.
Pure strategies
Write the payoff matrix from the row player's viewpoint. For each row find the minimum; the maximin is the largest of these. For each column find the maximum; the minimax is the smallest of these. If maximin = minimax, that entry is a saddle point and both players have a pure strategy. The value of the game lies between the row player's maximin and the column player's minimax.
Mixed strategies (2×2)
If there is no saddle point, let the row player use row 1 with probability p and row 2 with probability 1−p. Set the expected payoff against column 1 equal to that against column 2, and solve for p. The value is then found by substitution. A dominated row or column can be deleted first.
Worked example: payoff matrix [[4,2],[1,5]]. Row minima are 2 and 1, maximin = 2; column maxima are 4 and 5, minimax = 4; no saddle. Row: 4p+1(1−p)=2p+5(1−p) gives 3p+1=5−3p, so p=2/3. Value = 3(2/3)+1=3. Column: 4q+2(1−q)=1q+5(1−q) gives q=1/2.
Traps: assuming every game has a saddle point; mixing up the row player's and column player's probabilities; and forgetting that dominance must be applied consistently.