Answer First
A mixed strategy Nash equilibrium is found by identifying the set of actions each player randomizes over (the support), then making the opponent indifferent across those actions. The equilibrium probabilities are the ones that equalize expected payoffs within each player’s support.
Problem Setup
Let player 1 mix over actions A and B with probabilities p and 1−p. Let player 2 mix over actions C and D with probabilities q and 1−q. Mixed strategy equilibrium conditions:
- Each player must be indifferent across actions in their support.
- Actions outside the support must yield weakly lower payoffs.
- Probabilities must lie between 0 and 1.
Indifference condition example: \[ \text{EU}_1(A) = \text{EU}_1(B). \]
Step-by-Step Explanation
1. Identify pure strategy Nash equilibria
If none exist, a mixed strategy equilibrium is likely.
2. Choose candidate support sets
For a 2×2 game, the typical support is both actions for both players.
3. Write expected payoffs for each action
Compute expected payoffs as functions of the opponent’s mixing probability.
4. Impose indifference conditions
Set the expected payoffs of supported actions equal. This yields equations that solve for p and q.
5. Check feasibility
Probabilities must satisfy 0 ≤ p, q ≤ 1. If not, adjust the support set.
6. Verify best responses
Actions outside the support must not yield higher payoffs.
Intuition
In a mixed strategy equilibrium, each player randomizes just enough to keep the opponent indifferent. If one action gave strictly higher payoff, the opponent would never mix.
Common Exam Mistakes
- Forgetting to check that probabilities lie between 0 and 1.
- Not verifying that unsupported actions yield lower payoffs.
- Mixing up which player’s probabilities affect which payoffs.
- Solving only one player’s indifference condition.
Final Summary
To find a mixed strategy equilibrium, identify the support set, write expected payoffs for each supported action, impose indifference conditions, solve for probabilities, and verify that no unsupported action yields a higher payoff.
This explanation belongs to the broader Economics Tutoring pillar.
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