 Solution

ID: past-exam-of-the-msc-in-advanced-computer-science-course-of-the-university-of-oxford/2021/computational-game-theory/2/a/solution

Put

p = Pr (T)

and

q = Pr (L)

, so the complementary probabilities are

1 - p

and

1 - q

. All four expressions are the row player's expected payoff, including those indexed by a column action. Direct expected value calculations give

π_{1} (p, L) π_{1} (p, R) π_{1} (T, q) π_{1} (B, q) = 2 p - (1 - p) = 3 p - 1, = - 3 p + (1 - p) = 1 - 4 p, = 2 q - 3 (1 - q) = 5 q - 3, = - q + (1 - q) = 1 - 2 q .

The column player's expected payoff is the negative of the row player's because this is a zero-sum game.

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