Suppose that the preferences of two players satisfy the von Neumann-Morgenstern axioms. Player I is indifferent between receiving $600 with certainty and participating in a lottery in which he receives $300 with probability 1 4 and $1,500 with probability 3 4 . He is also indifferent between receiving $800 with certainty and participating in a lottery in which he receives $600 with probability 1/2 and $1,500 with probability 1/2. Player II is indifferent between losing $600 with certainty and participating in a lottery in which he loses $300 with probability 1/7 and $800 with probability 6/7. He is also indifferent between losing $800 with certainty and participating in a lottery in which he loses $300 with probability 1/8 and $1,500 with probability 7/8. The players play the game whose payoff matrix is as follows, where the payoffs are dollars that
Player II pays to Player I.
(a) Find linear utility functions for the two players representing the preference relations of the players over the possible outcomes.
The players play a game whose outcomes, in dollars paid by Player II to Player I, are given by the following matrix.
(b) Determine whether the game is zero sum.
(c) If you answered yes to the last question, find optimal strategies for each of the players. If not, find an equilibrium.