Game Theory Basics: Saddle Points and Mixed Strategies

Game theory studies strategic decisions between competing players. The classic starting point is the two-person zero-sum game, where one player's gain is exactly the other's loss. This guide shows how to find a pure-strategy solution (a saddle point) and, when none exists, how to compute optimal mixed strategies.

Solve any payoff matrix with the free Game Theory Solver.

The Payoff Matrix

The row player chooses a row, the column player chooses a column, and the entry is the row player's gain (the column player's loss). The row player wants large values; the column player wants small ones.

Step 1: Look for a Saddle Point (Pure Strategy)

Maximin (row player): for each row, find its minimum; take the largest of these.
Minimax (column player): for each column, find its maximum; take the smallest of these.

If maximin = minimax, that value is a saddle point and the game has a pure-strategy solution — each player has a single optimal move.

Example with a saddle point

📐 Formula

C1 C2 C3 Row min R1 1 2 3 1 R2 3 4 5 3 ← maximin R3 2 1 0 0 Col max 3 4 5 ↑ minimax = 3

Maximin = minimax = 3, so the value of the game is 3 (row player picks R2, column player picks C1).

Step 2: No Saddle Point → Mixed Strategies

When maximin ≠ minimax, players must randomize. For a 2×2 game, the optimal probabilities have closed-form solutions; larger games are solved with linear programming. Consider:

📐 Formula

C1 C2 R1 2 −1 R2 −1 1

Here maximin = −1 and minimax = 1, so there is no saddle point. Solving gives:

📐 Formula

Row player: play R1 with p = 0.4, R2 with 0.6 Column player: play C1 with 0.4, C2 with 0.6 Value of the game = 0.2

So the row player can guarantee an expected gain of 0.2 per play by mixing 40/60.

The Minimax Theorem

Von Neumann's minimax theorem guarantees that every finite two-person zero-sum game has a value and optimal (possibly mixed) strategies for both players — the maximin equals the minimax when mixed strategies are allowed.

Dominance to Simplify

Before solving, remove dominated strategies: a row that is never better than another can be deleted (and similarly for columns). This shrinks the matrix and can even reveal a saddle point.

Common Mistakes

Reading the matrix from the wrong player's view — entries are the row player's payoff.
Assuming a saddle point always exists — many games require mixed strategies.
Forgetting to check dominance — it simplifies the problem.

Try It Yourself

Enter your payoff matrix into the Game Theory Solver to detect saddle points, compute optimal mixed strategies, and find the value of the game with step-by-step working. For decisions against "nature" rather than an opponent, use the Decision Analysis Calculator.

Key Takeaways

Check maximin vs. minimax first; if equal, the game has a pure-strategy saddle point.
Otherwise, players randomize using mixed strategies with a well-defined game value.
The minimax theorem guarantees a solution for every finite zero-sum game.