Questions: Mixed Strategy Equilibrium and Equilibrium in Randomized Strategies

Question 1

In Matching Pennies, Player 2 mixes 50-50 between Heads and Tails. Why does Player 1 also play 50-50 in equilibrium?

• A. Player 1 randomizes to prevent Player 2 from predicting their action and exploiting it
• B. Player 1 randomizes because Player 2's 50-50 mix makes Player 1 exactly indifferent between Heads and Tails
• C. Player 1 randomizes to maximize expected payoff by averaging over all possible outcomes
• D. Player 1 randomizes because no pure strategy is a best response to any strategy Player 2 could play

Answer: (B)

This is the counterintuitive core of mixed-strategy equilibrium. Player 1 doesn't randomize to be unpredictable — they randomize because Player 2's mixing has made Player 1 exactly indifferent between Heads and Tails (equal expected payoffs). With equal expected payoffs, Player 1 has no incentive to deviate from mixing. If Player 2 instead played 60-40, Player 1 would have a strict best response (a pure strategy) and would stop mixing, collapsing the supposed equilibrium. Your mixing probabilities are determined by your opponent's indifference condition, not your own.

Question 2

An inspector and a firm play an inspection game. In equilibrium, the inspector audits with probability p* and the firm evades with probability q*. If the penalty for detected evasion doubles, what happens to the equilibrium audit probability p*?

• A. p* increases — a higher penalty requires more audits to maintain deterrence
• B. p* decreases — a higher penalty means fewer audits are needed to keep the firm indifferent between evading and complying
• C. p* is unchanged — the firm's behavior is what adjusts, not the inspector's mixing
• D. p* increases — the inspector's payoff from catching evasion is higher, so they audit more

Answer: (B)

The inspector's equilibrium probability p* is set by the firm's indifference condition: the firm must be exactly indifferent between evading and complying. With a higher penalty, evading becomes riskier at any given audit rate. To restore the firm's indifference (keep q* > 0), the inspector must audit less frequently. Counterintuitively, larger fines reduce equilibrium audit effort. This is the 'inspection game' result — a famous example of mixed-strategy logic generating non-obvious policy implications. Option A is the intuitive but wrong answer.

Question 3

In a mixed-strategy Nash equilibrium, a player who randomizes between two pure strategies earns a higher expected payoff than if they had played either pure strategy alone.

• T. True
• F. False

Answer: False

In a mixed-strategy equilibrium, the player is exactly indifferent among all pure strategies in their support — every one yields the same expected payoff as the mixture. Mixing achieves the same expected payoff as any of the supported pure strategies, not a higher one. The purpose of mixing is not to earn more but to make the opponent indifferent and thereby sustain an equilibrium. No player can improve by deviating (otherwise it wouldn't be an equilibrium), but they also don't gain from mixing per se.

Question 4

Nash's existence theorem guarantees that every finite strategic-form game has at least one Nash equilibrium, which may be in pure or mixed strategies.

• T. True
• F. False

Answer: True

Nash proved in 1950 that every finite game (finite players, finite strategy sets) has at least one Nash equilibrium when mixed strategies are allowed. Some games (like Prisoner's Dilemma) have pure-strategy equilibria; others (like Matching Pennies) have only mixed-strategy equilibria. Allowing randomization is the key that guarantees universal existence — without mixed strategies, many games would have no equilibrium at all. This result is foundational: it means the equilibrium concept is always applicable, not only in conveniently structured games.

Question 5

In a mixed-strategy equilibrium, a player's mixing probabilities are determined by the opponent's payoffs, not their own. Explain why.

Model answer: Your mixing probabilities are chosen to make your opponent exactly indifferent among their supported pure strategies. If your opponent were not indifferent, they would have a strict best response (a pure strategy they always prefer), and they would play it with certainty — making your mixing suboptimal and collapsing the equilibrium. So your mixing is constrained by your opponent's indifference condition, which depends on your opponent's payoffs. Your own indifference condition (which depends on your own payoffs) determines the opponent's mixing probabilities.

This mutual determination is the system of equations that characterizes a mixed-strategy equilibrium. Because each player's mixing is set by the opponent's indifference, changing your own payoffs changes the opponent's mixing but not your own — and vice versa. This also explains why making the penalty higher in the inspection game reduces the inspector's audit frequency: the firm's indifference condition changes (higher penalty → less evasion incentive at lower audit rates), so the inspector's equilibrium mixing adjusts downward.