Choose your bias

What this game-theory probability question tests

This is a medium-difficulty problem that combines probability, expected value, and strategic commitment. It appears in quant interviews because it requires you to reason about sequential decision-making under asymmetric information: Player 1 must choose a coin bias knowing that Player 2 will then optimize against it.

To solve it, you need to set up the payoff matrix from Player 1's perspective, express her expected payoff as a function of both players' bias choices, then determine what bias she should commit to in order to maximize her worst-case outcome (a maximin strategy). The key insight is that Player 2, moving second, will always choose the bias that minimizes Player 1's payoff for any bias Player 1 picks—so Player 1 must find the bias that makes Player 2 indifferent between his options.

• Expected value under mixed strategies
• Maximin and minimax reasoning
• Indifference conditions in two-player games
• Reduced fractions and exact arithmetic