# Question

Two college students, Al and Bob, very confident about their mid-term exam performance, decided to attend a party the weekend before the final exam. The party was so good that they overslept the whole Sunday. Instead of taking the final exam unprepared on Monday, they pleaded to the professor to give them a make-up exam. Their excuse was a flat tire without a spare and any help. The professor, surprisingly, agreed. On Tuesday morning, the professor placed them in separate rooms and handed them the test. The test had just one question: which tire?

(a) Suppose the final score is entirely based on the make-up test. If Al and Bob give the same answer on the flat tire question, both pass the exam, yielding a payoff of 10 for each of them. If Al and Bob, however, give different answers, both fail the exam, getting payoff zero. Write down the game table of this simultaneous-move game between Al and Bob.

(b) Find all pure-strategy Nash equilibria of this game.

(c) Find two mixed-strategy Nash equilibria of this game with the following requirements: in one of the mixed Nash equilibrium, each player randomizes using two pure strategies, and in the other mixed Nash equilibrium, each player randomizes using three or four pure strategies.

(a) Suppose the final score is entirely based on the make-up test. If Al and Bob give the same answer on the flat tire question, both pass the exam, yielding a payoff of 10 for each of them. If Al and Bob, however, give different answers, both fail the exam, getting payoff zero. Write down the game table of this simultaneous-move game between Al and Bob.

(b) Find all pure-strategy Nash equilibria of this game.

(c) Find two mixed-strategy Nash equilibria of this game with the following requirements: in one of the mixed Nash equilibrium, each player randomizes using two pure strategies, and in the other mixed Nash equilibrium, each player randomizes using three or four pure strategies.

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