23506 Chap.5 Mixed-Strategy Nash Equilibrium: Expected Payoffs and the Indifference Condition
Mixed-Strategy Nash Equilibrium: Expected Payoffs and the Indifference Condition
A mixed strategy is a probability distribution over pure strategies chosen before the realised action. It can make a player strategically unpredictable when no pure profile is stable, and it can coexist with pure equilibria in coordination games. Each probability needs an owner and an action order. A player mixing with positive probability across several actions must be indifferent among those actions, and every supported action must attain the maximum expected payoff.
The probability ownership rule crosses players: Row’s probability determines Column’s expected payoffs, so equating Column’s supported actions solves Row’s probability. Column’s probability similarly solves Row’s indifference. With two actions, p and 1−p automatically sum to one. With larger supports, add the probability-sum equation and non-negativity, then compare every excluded action. An excluded action earning more invalidates the candidate; a zero calculated probability reduces the true support.
Best-response diagrams plot a player’s optimal action against the opponent’s mixing probability. A crossing is an indifference point; endpoint tests reveal which response applies on each side. In Battle of the Sexes, the mixed equilibrium supplements two coordinated pure equilibria rather than replacing them. Randomisation can be interpreted as private one-shot uncertainty, population proportions or long-run frequencies, but these interpretations share expected-payoff algebra without being behaviourally identical.
What this chapter covers
- 01Probability distributions and support
- 02Expected payoff and probability complements
- 03Cross-player indifference equations
- 04Best-response crossings
- 05Mixed equilibrium in penalty and coordination games
- 06Three-action support checks and interpretations
Practice-final 2b mixed equilibrium
- Mark pure best responses and establish no pure equilibrium.
- Define p as Row’s probability of U and make Column indifferent.
- Define q as Column’s probability of L and make Row indifferent.
- Solve exact vectors in the printed action order.
- Substitute back to verify supported payoffs and completeness.
Key terms
- Mixed strategy
- A probability distribution over pure strategies.
- Support
- Pure strategies assigned strictly positive probability.
- Indifference condition
- Equality of maximum expected payoffs across supported actions.
- Interior mix
- A mix giving each proposed supported action probability strictly between zero and one.
- Expected payoff
- The probability-weighted average payoff before the action realisation.
- Best-response crossing
- An opponent probability at which two expected-payoff lines tie.
Mixed-Strategy Nash Equilibrium: Expected Payoffs and the Indifference Condition FAQ
Why must supported actions tie?
If one paid less, shifting its probability to the higher action would improve payoff.
Whose probability am I solving?
Equate Column’s payoffs to solve Row’s mix and Row’s payoffs to solve Column’s mix.
Can a mixed equilibrium coexist with pure equilibria?
Yes. Battle of the Sexes has two pure equilibria and one interior mixed equilibrium.
What if a probability is negative?
Reject the proposed support; a negative value is not a feasible strategy.
What if a solved probability is zero?
Treat the candidate as a smaller-support profile and verify that support anew.
Must unused actions earn less?
They may tie, but none may earn strictly more than supported actions.
Why check endpoints in a response diagram?
The crossing locates equality; an endpoint shows which payoff line is higher on each side.
Does a player literally choose a fractional action?
No. A pure action is realised; the fractions describe ex ante probabilities.
Do mixed probabilities maximise total payoff?
No. They make individual best-response conditions hold.
How do I report a vector?
Use the action order printed by the matrix and include zero probabilities for excluded actions when returning from a reduced game.
What is a reliable support-enumeration procedure?
Begin with every pure best-response intersection because a game can contain pure and mixed equilibria together. For a candidate mixed support, assign probabilities in the printed action order and choose one supported action as a payoff reference. Equate each other supported action to that reference, using the opponent’s probabilities, and add the probability-sum equation. Solve exactly. Reject the candidate if any probability is negative or greater than one; if a probability is zero, reclassify it as a smaller support. Compute the expected payoff of every excluded action. No excluded action may earn more than the supported value, though a tie may indicate another equilibrium or larger support. Substitute the final probabilities independently into every supported action to verify equality. Restore zero coordinates when reporting the strategy in the original game. Finally list the pure equilibria alongside the mixed result and state which vectors belong to which players. This routine prevents a correct algebraic fraction from being attached to the wrong action or an invalid support.
Exam move
Start every mixed problem with a pure best-response scan; this identifies missing pure equilibria and plausible supports. Label p and q with player and action before forming expectations. Write endpoint-expanded expected payoffs, solve exact fractions and substitute them back into both supported actions. For a 3×3 game, choose a support, solve k−1 equalities plus the probability sum, check the domain and test every excluded action. Rehearse the practice-final 2b vector until you can explain why each fraction belongs to the opposite player’s indifference. Add one interpretation sentence: equilibrium frequencies prevent exploitation, but a realised play remains a pure action profile.
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