Questions: Mixed Strategies and Probabilistic Play

Question 1

In a mixed strategy Nash equilibrium of Matching Pennies, Player 1 plays Heads 50% of the time. What happens if Player 1 deviates to playing Heads 70% of the time?

• A. Nothing — Player 1 is still mixing, so it remains an equilibrium
• B. Player 2 will exploit the predictability by best-responding with a pure strategy, breaking the equilibrium
• C. Player 1's expected payoff increases because Heads is now favored
• D. Player 2 must also adjust to maintain equal mixing probabilities

Answer: (B)

In a mixed strategy Nash equilibrium, each player's mixing probabilities are chosen specifically to make the other player indifferent between their strategies. If Player 1 plays Heads 70%, Player 2 can now identify which pure strategy gives a higher expected payoff and will deviate to it — breaking the equilibrium. The 50/50 split is the unique profile where Player 2 has no profitable deviation, precisely because 50/50 leaves Player 2 indifferent.

Question 2

In a mixed strategy Nash equilibrium, what determines Player 1's equilibrium mixing probabilities?

• A. Player 1's own payoffs — Player 1 mixes in the proportions that maximize their expected payoff
• B. Player 2's payoffs — Player 1's mix must make Player 2 indifferent between their strategies
• C. The total number of strategies available to both players
• D. Player 1's risk aversion — more risk-averse players mix closer to 50/50

Answer: (B)

This is the counterintuitive core of mixed strategy equilibrium: your mixing probabilities are determined by the OTHER player's payoff structure, not your own. Player 1 must choose probabilities that equate Player 2's expected payoffs across the strategies in Player 2's support — because if Player 2 strictly preferred one strategy, Player 2 would not be willing to mix. Player 1's own mixing probabilities are pinned down by Player 2's indifference condition.

Question 3

In a mixed strategy Nash equilibrium, each player randomizes in order to maximize their own expected payoff.

• T. True
• F. False

Answer: False

Players do not mix to maximize their own payoff — in fact, in a mixed strategy NE, a player is INDIFFERENT between all strategies in their support (they all yield the same expected payoff). The purpose of mixing is to prevent opponents from exploiting predictability. A player randomizes to make opponents indifferent, which sustains the equilibrium. If a player were mixing to maximize their own payoff, they would play a pure strategy (the best-responding pure strategy).

Question 4

Nash's theorem guarantees that every finite strategic-form game has at least one Nash equilibrium.

• T. True
• F. False

Answer: True

Nash's theorem (proved using Kakutani's fixed-point theorem) states that every finite game — finitely many players, each with finitely many pure strategies — has at least one Nash equilibrium, possibly in mixed strategies. This is why mixed strategies matter: they ensure the equilibrium concept is never vacuous. Games like Matching Pennies have no pure strategy NE but always have a mixed strategy NE, so the solution concept remains well-defined.

Question 5

Why does predictability undermine equilibrium in games like Matching Pennies, and why does mixing solve this problem?

Model answer: In Matching Pennies, any deterministic pure strategy is exploitable: if Player 1 always plays Heads, Player 2 best-responds by always playing Tails, but then Player 1 wants to switch to Tails, and so on — there is no stable resting point. Mixing solves this by making Player 1 genuinely unpredictable: if Player 1 plays Heads with probability 1/2, Player 2 cannot improve by changing strategy because both Heads and Tails yield the same expected payoff. Unpredictability removes the opponent's ability to exploit any systematic pattern.

This captures the strategic logic of randomization: it is not about literally flipping a coin for its own sake, but about creating genuine uncertainty that neutralizes the opponent's ability to best-respond. The equilibrium mixing probabilities are exactly those that accomplish this — making the other player indifferent — and no other probabilities can sustain equilibrium.