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Her payoff from Bus when Bob plays Bus, multiplied by the probability that Bob will play Bus, or 3 times q

plus

Her payoff from Bus when Bob plays Walk times the probability that Bob plays Walk, or 6 times (1 – q)

Her expected payoff from Walk is the sum of:

Her payoff from Walk when Bob plays Bus times the probability that Bob plays Bus, or 5 times q

plus

Her payoff from Walk when Bob plays Walk times the probability that Bob plays Walk, or 4 times (1 – q)

Summarizing,

Alice’s expected payoff for Bus = 3q + 6(1 – q)

Alice’s expected payoff for Walk = 5q +4(1 – q)

Applying similar reasoning to calculating Bob’s expected payoffs yields:

Bob expected payoff for Bus = –3p + –5(1 – p)

Bob expected payoff for Walk = –6p + –4(1 – p)

Now, Alice’s total expected payoff for the game will be her probability of choosing Bus times her Bus expected payoff, plus her probability of choosing Walk times her Walk expected payoff. Similarly for Bob. To achieve a Nash equilibrium, their probabilities for the two choices must be such that neither would gain any advantage by changing those probabilities. In other words, the expected payoff for each choice (Bus or Walk) must be equal. (If the expected payoff was greater for one than the other, then it would be better to play that choice more often, that is, increasing the probability of playing it.)

For Bob, his strategy should not change if

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