Pure vs. mixed strategies

Try it yourself

1. For Matching Pennies (payoffs $(1,-1)$ if they match, $(-1,1)$ if not), show that no pure strategy is a best response to any other pure strategy.
2. Compute the mixed strategy that makes your opponent indifferent between H and T. (Spoiler: 50/50.)
3. For Rock-Paper-Scissors, write down the unique symmetric mixed-strategy equilibrium. Why is any deviation exploitable?
4. Bias RPS so you get $+2$ (not $+1$) for winning with Rock. Recompute the equilibrium mix — does the mix shift toward or away from Rock? Sit with the answer until it feels right.

Prompt your AI

Use these three in order. Each builds on the one before.

1. Basics & terminology

In one paragraph, explain what a 'mixed strategy' is and when a rational player would actually want to randomize.

2. Why it works (the mechanism)

Walk me step-by-step through how to find a mixed-strategy equilibrium in a 2x2 game using the indifference condition. Use the Matching Pennies payoffs (±1) as the worked example.

3. Advanced — application & what's next

In a penalty kick, the striker and goalkeeper both randomize. Empirically, the mix *isn't* 50/50 — it reflects the striker's stronger foot. Explain using indifference how the payoff asymmetry pins down the empirical mix, and how you'd estimate the mix from game tape.

References

• Chiappori, Levitt & Groseclose — Testing Mixed-Strategy Equilibria (AER, 2002)
• SEP — Game Theory (mixed-strategy section)