QBUS2310 Chap.7 Zero-Sum Games as Linear Programs
Zero-Sum Games as Linear Programs
A finite two-player zero-sum game can be written as a linear program because one player chooses probabilities to maximise a guaranteed payoff against every opposing pure action. The opponent solves the dual minimisation problem. Equalising active responses is a useful shortcut in a small game, but endpoint and probability checks remain essential.
These terms let a reader confirm a proposed mixed strategy actually secures its claimed guaranteed payoff. Zero-Sum Game is a game in which one player's gain is the other player's equal loss. Mixed Strategy is a probability distribution over pure actions. Guaranteed Payoff is the worst payoff secured against all opposing actions. Maximin Value is the largest guaranteed payoff a player can secure.
Payoff Matrix is a table of outcomes indexed by the players' action choices. Game Value is the common guaranteed outcome when optimal strategies meet. The proposed equilibrium above is verified against both players' guarantees: The row player uses probabilities 0.6 and 0.4; the column player uses 0.5 and 0.5. Both secure value 1, so the strategies and value are optimal.
What this chapter covers
- 01
Payoff matrices
- 02
Pure and mixed strategies
- 03
Probability simplex
- 04
Guaranteed payoff
- 05
Maximin formulation
- 06
Opponent dual problem
- 07
Equalising active responses
- 08
Game value certificates
Zero-Sum Games as Linear Programs worked example
- +1Let p be the probability of row one. Against the first column the expected payoff is 5p-2; against the second it is 4-5p.
- +1The row player maximises the lower of these two lines. Equalise them to find 5p-2=4-5p.
- +1This gives p=0.6 and guaranteed payoff 1. Check p=0 and p=1 to ensure the intersection beats both endpoints.
- +1For the column player, probability q=0.5 on column one makes both row payoffs equal to 1, giving the matching upper certificate.
Key terms
- Zero-Sum Game
- A game in which one player's gain is the other player's equal loss.
- Mixed Strategy
- A probability distribution over pure actions.
- Guaranteed Payoff
- The worst payoff secured against all opposing actions.
- Maximin Value
- The largest guaranteed payoff a player can secure.
- Payoff Matrix
- A table of outcomes indexed by the players' action choices.
- Game Value
- The common guaranteed outcome when optimal strategies meet.
Zero-Sum Games as Linear Programs FAQ
Why does payoff matrices matter?
Zero-Sum Game: A game in which one player's gain is the other player's equal loss. Mixed Strategy: A probability distribution over pure actions. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How can I check guaranteed payoff?
Mixed Strategy: A probability distribution over pure actions. Guaranteed Payoff: The worst payoff secured against all opposing actions. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
What separates zero-sum game from mixed strategy?
Guaranteed Payoff: The worst payoff secured against all opposing actions. Maximin Value: The largest guaranteed payoff a player can secure. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Which error is most likely around opponent dual problem?
Maximin Value: The largest guaranteed payoff a player can secure. Payoff Matrix: A table of outcomes indexed by the players' action choices. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
How should I practise game value certificates?
Payoff Matrix: A table of outcomes indexed by the players' action choices. Game Value: The common guaranteed outcome when optimal strategies meet. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation.
Exam move
Write one expected-payoff line per opposing pure action. The player protects the worst line, so identify the lower envelope for a maximiser and verify probabilities stay between zero and one. Rehearse the chapter method in this order: Let p be the probability of row one. Against the first column the expected payoff is 5p-2; against the second it is 4-5p. The row player maximises the lower of these two lines.
Equalise them to find 5p-2=4-5p. This gives p=0.6 and guaranteed payoff 1. Check p=0 and p=1 to ensure the intersection beats both endpoints. For the column player, probability q=0.5 on column one makes both row payoffs equal to 1, giving the matching upper certificate.
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