CMA Final · Strategic Cost Management

Game Theory: formula sheet

Full chapter guide

Key formulas

Maximin value (row player): Maximin = maximum of the row minimums; Player A (maximiser, rows) finds the minimum of each row and picks the largest of these. It is A's lower value of the game.
Minimax value (column player): Minimax = minimum of the column maximums; Player B (minimiser, columns) finds the maximum of each column and picks the smallest of these. It is B's upper value of the game.
Saddle point condition: Maximin = Minimax = Value of the game; If they are equal, the game has a saddle point and is strictly determinable. Pure strategies are then optimal.
Zero-sum condition: Gain of one player + Gain of other player = 0; A payoff matrix shows only the payoff to A. B's payoff is the negative of each entry.
Fair game: Value of the game = 0; A game with value above 0 favours A. A game with value below 0 favours B.
Maximin value: Maximin = max over rows of (minimum of each row); Player A (maximiser) writes the row minima, then picks the largest of them.
Minimax value: Minimax = min over columns of (maximum of each column); Player B (minimiser) writes the column maxima, then picks the smallest of them.
Saddle point condition: Maximin = Minimax = V; If equal, the game has a saddle point and V is the value of the game. Always maximin ≤ minimax.
Fair game: V = 0; A game is fair when the value is zero. It is strictly determinable when a saddle point exists.
Saddle point test: Maximin = max of row minima; Minimax = min of column maxima; saddle point exists if Maximin = Minimax; If they are not equal, use a mixed strategy.
2x2 game with payoffs a b / c d (to the row player): p1 (probability of row 1) = |d − c| ÷ (|a − c| + |d − b|), p2 = 1 − p1; The matrix is row 1 = a, b and row 2 = c, d. Applies only when no saddle point exists.
Column player's probability of column 1: q1 = |b − d| ÷ (|a − b| + |c − d|), q2 = 1 − q1; Same condition: no saddle point.
Value of the game: V = (ad − bc) ÷ [(a + d) − (b + c)]; Valid for a 2x2 game with no saddle point. Check by finding the expected payoff of either row using the q values.
Oddments check: Row probabilities sum to 1; column probabilities sum to 1; Each probability is an oddment divided by the sum of the oddments.
Row dominance: Row i is dominated by row k if aᵢⱼ ≤ aₖⱼ for every column j; Row player maximises, so the smaller row is deleted. Delete row i.
Column dominance: Column j is dominated by column l if aᵢⱼ ≥ aᵢₗ for every row i; Column player minimises, so the larger column is deleted. Delete column j.
Dominance by average: Row i is dominated if aᵢⱼ ≤ p·aₖⱼ + (1 − p)·aₘⱼ for every column j, where 0 < p < 1; For columns, reverse the inequality: column j is dominated if its entries are ≥ the weighted average of two other columns in every row.
Order of reduction: Reduce → check saddle point → else mixed strategy or graphical method; Dominance only simplifies the game. It does not give the value by itself.
Expected payoff line, 2xn game: E(Bj) = a1j × p + a2j × (1 − p) = a2j + (a1j − a2j) × p; A plays row 1 with probability p and row 2 with 1 − p. Payoffs are to A. Draw one line for each column j.
Expected payoff line, mx2 game: F(Ai) = ai2 + (ai1 − ai2) × q; B plays column 1 with probability q and column 2 with 1 − q. Draw one line for each row i.
Optimal point rule: 2xn: maximise the lower envelope. mx2: minimise the upper envelope.; Use the envelope itself, not just any intersection of two lines.
Solving the 2x2 sub-game (no saddle point): For rows (a b / c d): p(row 1) = (d − c) ÷ (a − b − c + d); V = (ad − bc) ÷ (a − b − c + d); Valid when the 2x2 game has no saddle point. Use the same idea for B's probabilities.
Probability check: p + (1 − p) = 1 and every probability ≥ 0; Strategies not in the final 2x2 sub-game get probability 0.
Maximin (row player): Maximin = maximum of the row minimums; Row player is the maximiser. Write the minimum of each row, then take the largest.
Minimax (column player): Minimax = minimum of the column maximums; Column player is the minimiser. Write the maximum of each column, then take the smallest.
Saddle point condition: Maximin = Minimax = value of the game; If equal, the game has a pure strategy solution. If maximin < minimax, use mixed strategy.
2x2 mixed strategy, row probabilities: p1 = (d − c) ÷ (a − b − c + d); p2 = 1 − p1; Matrix is [a b; c d], with a, b in row 1 and c, d in row 2. Use only when there is no saddle point.
2x2 mixed strategy, column probabilities: q1 = (d − b) ÷ (a − b − c + d); q2 = 1 − q1; q1 is the probability of column 1 for the column player.
Value of 2x2 game: V = (ad − bc) ÷ (a + d − b − c); Use when there is no saddle point. Check that V lies between maximin and minimax.

Quick revision

Game theory models decisions where the outcome depends on a competitor's choice.
Two-person zero-sum game: one player's gain equals the other's loss.
Row player seeks the maximin: the maximum of the row minima.
Column player seeks the minimax: the minimum of the column maxima.
Saddle point exists when maximin = minimax; that common value is the value of the game.
A game with value zero is a fair game.
With no saddle point, players use mixed strategies: probabilities over their choices.
Dominance: a row can be dropped if another row is at least as good in every column; a column can be dropped if another column is at least as good for the column player, meaning smaller or equal payoffs.
For a 2x2 game with payoffs a, b / c, d, and no saddle point, the row player's probability of the first row = (d − c) ÷ ((a + d) − (b + c)).
Value of the game for 2x2 = (ad − bc) ÷ ((a + d) − (b + c)).
Graphical method suits games where one player has only two strategies.
Limitations: assumes rational players, known payoffs and a zero-sum setting, which real markets rarely meet.

Common mistakes

Calling every competitive situation a zero-sum game. Fix: Check that the payoffs of both players add to zero. If both can gain or lose together, it is non-zero-sum.
Mixing up maximin and minimax. Fix: A (rows) takes row minimums then the maximum of them. B (columns) takes column maximums then the minimum of them.
Taking the row maxima and column minima instead of the reverse. Fix: Remember: A is the maximiser, so A looks at the worst case in each row, the minimum. B looks at the worst case in each column, the maximum.
Declaring a saddle point when maximin and minimax differ. Fix: Always compute both and compare. A saddle point exists only when they are equal.
Applying oddments without checking for a saddle point. Fix: Always compute maximin and minimax first. Use mixed strategy only when they differ.
Not swapping the differences. Fix: The probability of row 1 uses the row-2 difference. Test it: the larger payoff gap gets the smaller weight in the other row.
Deleting the larger column because it has bigger numbers Fix: For columns, delete the column with larger entries in every row. Write 'Column player minimises' beside the matrix.
Deleting a row that is better in some columns but worse in others Fix: Dominance needs the inequality to hold in every column (or every row). One exception means no dominance.
Picking the highest intersection of any two lines instead of the highest point of the lower envelope. Fix: Put the p value into every line. If any line is below the crossing, that crossing is not on the envelope. Use the pair that actually forms the envelope peak.
Using the lower envelope for an mx2 game, or the upper envelope for a 2xn game. Fix: In 2xn, lines show A's gain, and B pushes it down, so A maximises the lower envelope. In mx2, lines show B's loss, and A pushes it up, so B minimises the upper envelope.

Exam tips

Expect 2-mark MCQs on definitions: zero-sum, fair game, saddle point, pure strategy. Learn each in one line.
Always write row minimums and column maximums beside the matrix. Examiners give marks for visible working.
State the conclusion in words with units, such as the value in ₹ lakh and who it favours.
For theory questions, list assumptions and types of games in short numbered points. Add a business example for each type.
Check first whether a saddle point exists. If not, the question needs the mixed strategy method from the next topics.
Show the row minima and column maxima beside the matrix. Examiners award marks for these workings.
Always state the strategies of both players and the value, with the sign explained.
If a question gives a matrix with unknown x and asks for a range that keeps a saddle point, set maximin = minimax conditions and solve the inequalities.