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Decision Mathematics Applications

Key Terms

A two-person zero-sum game: one player’s gain equals the other’s loss
The payoff matrix shows what the row player (A) wins from the column player (B)
Row player (A)
wants to maximise the payoff; Column player (B) wants to minimise it
Maximin strategy
(A): find the minimum in each row; choose the row with the largest minimum
Minimax strategy
(B): find the maximum in each column; choose the column with the smallest maximum
Saddle point
: the entry where maximin = minimax — both players have a pure strategy optimal solution
Dominant strategy
: a row (or column) that is always at least as good as another; dominated strategies can be eliminated
Mixed strategy
: when no saddle point exists, players randomise between strategies with optimal probabilities
Finding a saddle point:
Row player: find max(row minima) — this is the maximin value
Column player: find min(column maxima) — this is the minimax value
If maximin = minimax, a saddle point exists at that entry. Value of the game = saddle point.

Mixed strategy (2×2 matrix only):
For matrix  
ab
cd
 where D = (a+d) − (b+c):

Row player plays Row 1 with probability: p = (d−c) / D
Column player plays Col 1 with probability: q = (d−b) / D
Value of the game: V = (ad−bc) / D
Worked Example: Find the optimal strategy and value for the game:
B1B2
A152
A216
Saddle point check: Row minima: A1=2, A2=1. Maximin=2.
Column maxima: B1=5, B2=6. Minimax=5. 2 ≠ 5 → no saddle point.

Mixed strategy: a=5, b=2, c=1, d=6. D = (5+6)−(2+1) = 11−3 = 8.
Row player: p = (6−1)/8 = 5/8 (play A1 with prob 5/8, A2 with prob 3/8).
Column player: q = (6−2)/8 = 4/8 = 1/2 (play B1 with prob 1/2, B2 with prob 1/2).
Value: V = (5×6−2×1)/8 = (30−2)/8 = 28/8 = 3.5
Hot Tip: Always check for a saddle point before applying mixed strategy formulas — if maximin = minimax, the pure strategy solution is simpler and exact. Only use the mixed strategy formula for 2×2 matrices when no saddle point exists.

What Is a Two-Person Zero-Sum Game?

In a two-person zero-sum game, two players make decisions simultaneously, and one player’s gain exactly equals the other player’s loss. Examples include: competing businesses choosing advertising strategies, sports teams choosing tactics, or players choosing moves in a game.

The situation is represented by a payoff matrix, where each entry is the amount the row player (Player A) receives from the column player (Player B). Negative entries mean A pays B.

Pure Strategies and Saddle Points

If each player commits to a single strategy regardless of what the other does, this is a pure strategy.

  • Row player (A) uses the maximin strategy: find the minimum payoff in each row (the worst outcome for A in each row), then choose the row with the largest minimum. This guarantees A at least the maximin value, no matter what B does.
  • Column player (B) uses the minimax strategy: find the maximum payoff in each column (the worst outcome for B in each column), then choose the column with the smallest maximum. This guarantees B pays at most the minimax value.

If the maximin value equals the minimax value, this entry is called a saddle point. Both players have found their optimal pure strategy, and the payoff at the saddle point is the value of the game. A value > 0 favours A; a value < 0 favours B; value = 0 is a fair game.

Dominant Strategies

A strategy is dominant if it is at least as good as another strategy in every situation (and strictly better in at least one). Dominated strategies can be eliminated because a rational player would never choose them.

  • For the row player: a row that is always ≥ another row (entry by entry) means the other row is dominated — eliminate the smaller row.
  • For the column player: a column that always gives a lower or equal payoff (entry by entry) dominates a higher-payoff column — eliminate the larger column.

After eliminating dominated strategies, you may reach a smaller matrix that has a saddle point, or a 2×2 matrix suitable for mixed strategy analysis.

Mixed Strategies (No Saddle Point)

When no saddle point exists, neither player has a single best pure strategy. Instead, each player should randomise over their strategies with specific probabilities — this is a mixed strategy.

For a 2×2 payoff matrix with entries a (top-left), b (top-right), c (bottom-left), d (bottom-right), let D = (a+d) − (b+c):

  • Row player plays Row 1 with probability p = (d−c)/D
  • Column player plays Column 1 with probability q = (d−b)/D
  • Value of the game: V = (ad−bc)/D

These probabilities make the game fair in the sense that neither player can benefit by changing their strategy — the expected payoff is V regardless of what the other player does.

Strategy: Step 1 — check for saddle point. Step 2 — if not found, eliminate dominated strategies. Step 3 — if a 2×2 remains with no saddle point, apply the mixed strategy formulas.

Mastery Practice

See Answers ➔
  1. Fluency A payoff matrix for a two-person zero-sum game is shown below (payoffs to the row player).
    B1B2
    A13−1
    A2−24
    State the maximin value and minimax value. Does a saddle point exist?
  2. Fluency Find the saddle point, if it exists, for the payoff matrix below. State the value of the game and the optimal pure strategies.
    B1B2B3
    A1243
    A2564
    A3132
  3. Fluency In the payoff matrix below, identify any dominated strategies for the row player (A). Which row(s) should be eliminated?
    B1B2
    A135
    A212
    A343
  4. Fluency Apply the mixed strategy formula to the 2×2 payoff matrix below (no saddle point exists). Find the optimal probabilities for each player and the value of the game.
    B1B2
    A114
    A232
  5. Understanding Two players compete in a game with the following payoff matrix (payoff to A). Perform a complete analysis: check for a saddle point; if none, find the optimal mixed strategies and the value of the game.
    B1B2
    A15−1
    A223
  6. Understanding Two rival cafés (A and B) each independently choose a daily special: pasta or salad. The payoff matrix below shows the number of extra customers A gains from B (negative = A loses customers to B).
    B: PastaB: Salad
    A: Pasta64
    A: Salad28
    (a) Does a saddle point exist? (b) Find the optimal mixed strategy for each café. (c) What is the expected gain for Café A per day in the long run?
  7. Understanding The 3×3 payoff matrix below has at least one dominated strategy. Use dominance to reduce the matrix, then solve the resulting 2×2 game (find optimal strategies and game value).
    B1B2B3
    A1253
    A2461
    A3132
  8. Understanding A game has value V = −2. Explain what this means for each player. Is this a fair game? Which player has the advantage in the long run?
  9. Problem Solving Two companies simultaneously choose a pricing strategy: Low or High. The payoff matrix (market share gained by Company A) is:
    B: LowB: High
    A: Low42
    A: High16
    (a) Check for a saddle point.
    (b) Find the optimal mixed strategy for each company.
    (c) In the long run, what is the expected market share gain for Company A?
    (d) If Company B always plays “Low”, what is Company A’s best response?
  10. Problem Solving A 3×3 decision game has the payoff matrix below.
    B1B2B3
    A1253
    A2462
    A3132
    (a) Identify and eliminate any dominated strategies (row or column) step by step.
    (b) Solve the reduced game: find optimal strategies for both players and the value of the game.
    (c) Interpret the value of the game in context: who benefits, and by how much on average?

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