Game Theory Pure and Mixed Strategies: Concepts and Examples

Game theory studies how rational players choose strategies when their outcomes depend on others' choices. A pure strategy is a fixed plan of action; a mixed strategy randomises over several actions with chosen probabilities.

This matters far beyond games: companies set prices, armies plan defences, and athletes choose plays — all are strategic interactions. Pure and mixed strategies are the language used to analyse and solve them.

Key Concepts

• Payoff matrix: a table showing each player's outcome for every combination of choices.
• Pure strategy: always playing one specific action (e.g. "always advertise").
• Mixed strategy: playing actions with probabilities (e.g. "advertise with 60% chance").
• Saddle point: a cell that is the row minimum and column maximum — a stable pure-strategy solution.
• Value of the game: the expected payoff when both players play optimally.
• Dominance: a strategy that is always at least as good as another and can be eliminated.

Formulas

For a 2×2 game with row player payoffs [[a, b], [c, d]] and no saddle point:

• Row player's optimal mix: p = (d − c) / (a − b − c + d) for row 1 (plays row 1 with probability p).
• Column player's optimal mix: q = (d − b) / (a − b − c + d) for column 1.
• Value of the game: v = (ad − bc) / (a − b − c + d).

Assumptions: two players, zero-sum (one's gain is the other's loss), both rational and fully informed about payoffs.

Worked Example

Payoff matrix for player A: rows A1, A2; columns B1, B2: [[4, 1], [2, 3]].

1. Check saddle point: row minima are 1 and 2; column maxima are 4 and 3. Max of row minima = 2, min of column maxima = 3 → not equal, so no saddle point; use mixed strategies.
2. p = (3 − 2)/(4 − 1 − 2 + 3) = 1/4. So A plays A1 with 25%, A2 with 75%.
3. q = (3 − 1)/4 = 2/4 = 0.5. B plays B1 and B2 each with 50%.
4. Value v = (4·3 − 1·2)/4 = (12 − 2)/4 = 2.5.

Check: if B plays B1, A's expected payoff = 0.25·4 + 0.75·2 = 2.5; if B plays B2: 0.25·1 + 0.75·3 = 2.5. Equal — B cannot exploit A's mix. ✓

Common Misconceptions

• "Random play means playing badly." A mixed strategy is a calculated randomisation that keeps the opponent guessing.
• "You should always pick your best-looking move." If the opponent anticipates it, predictable play gets exploited — matching pennies proves this.
• "Game theory only applies to board games." Auctions, negotiations and pricing wars are games too.

Real-World Applications

• Advertisers randomise campaign timing so competitors cannot predict launches.
• Security agencies randomise patrol routes (game theory defends airports and wildlife reserves).
• Sports teams mix plays — a footballer randomises penalty-kick direction.

Pure strategies give clean answers when a saddle point exists; mixed strategies extend the theory to every zero-sum game. Together they explain why unpredictability itself can be the smartest strategy.