DF: Game theory for zero-sum games — AQA A-Level Further Mathematics

Game theory in AQA Further Mathematics focuses on two-person zero-sum games, where the total sum of gains and losses across both participants is always zero. The topic explores decision-making under competition, starting with pure strategies, play-safe criteria (maximin and minimax), and identifying stable equilibrium points known as saddle points.

When a pure strategy equilibrium does not exist, the topic expands to probabilistic decision-making via mixed strategies. Students master reduction techniques via dominance, solve 2 x n and n x 2 games geometrically using boundary envelopes, and formulate larger, complex pay-off matrices into linear programming models solvable via the Simplex algorithm.

• →Zero-Sum Principle: The algebraic sum of payoffs to both players equals zero for every combination of strategies, meaning one player's gain is identically the other player's loss.
• →Play-Safe Criterion and Saddle Points: The row player chooses the maximum of row minimums (maximin), while the column player chooses the minimum of column maximums (minimax). If maximin equals minimax, the game has a stable solution (saddle point).
• →Dominance Rule: A strategy can be permanently deleted if another available strategy yields an equal or strictly better payoff under all opposing moves (higher payoffs for Row, lower payoffs for Column).
• →Mixed Strategy and Envelope Method: When no saddle point exists, players randomise choices with specific probabilities to maximise their minimum expected payoff, solved graphically using upper or lower boundary envelopes.
• →Linear Programming Conversion: Larger m x n games are converted into linear programming problems by defining non-negative scaled variables x_i = p_i / V after adding a constant k to ensure all payoffs are strictly positive.

• 💡Always check for a saddle point first before undertaking algebraic or graphical mixed-strategy methods; if maximin equals minimax, you can state the stable solution and game value immediately.
• 💡Clearly label axes (probability p from 0 to 1 on the horizontal, expected payoff on the vertical) and state line equations explicitly when sketching graphical solutions to guarantee full method marks.
• 💡When defining variables in Linear Programming formulations, always state the relationship between probabilities, game value, and variables clearly (e.g., x_i = p_i / V with p_i >= 0, sum p_i = 1).

• Assuming Column dominance follows the same rule as Row dominance: Students often forget that a pay-off matrix shows gains for Row and losses for Column. Therefore, Column B1 dominates Column B2 if all entries in B1 are less than or equal to B2.
• Selecting the upper envelope for the Row player in graphical methods: The Row player wants to maximise their worst-case outcome, so students must locate the highest point on the LOWER envelope, not the upper envelope.
• Forgetting to adjust the final game value after an LP transformation: If a constant k was added to the matrix to guarantee all entries are positive, the true value of the original game must be calculated as V = (1 / sum(x_i)) - k.

1. 1Day 1-2: Master pure strategies, calculate row minimums and column maximums, verify saddle points, and interpret stable solutions.
2. 2Day 3-4: Practise dominance reductions for both row and column players, ensuring clear justification for each eliminated row or column.
3. 3Day 5-7: Solve 2x2 and 2xn games using expected payoff equations and graphical lower/upper envelope analysis.
4. 4Day 8-10: Work through Linear Programming formulations of 3x3 games, focusing on adding constant k, setting up constraints, and finding the value of the game.
5. 5Day 11-14: Complete full AQA past-paper questions under timed conditions, checking against official mark schemes for correct terminology.

• 📋Pay-off Matrix Analysis & Saddle Point: Find row minimums and column maximums, deduce play-safe strategies, and determine whether a stable solution exists.
• 📋Dominance and Reduction: Reduce an m x n matrix by sequentially eliminating dominated rows and columns, giving clear justifications.
• 📋Graphical Solution (2 x n or m x 2): Formulate expected payoffs in terms of probability p, plot lines, highlight the envelope, and calculate optimal p and game value V.
• 📋Linear Programming Formulation: Formulate a game as an LP problem for the Simplex method, including shifting payoffs by constant k and defining variables.