Alice and Bob want to meet in one of three places, namely Aquarium (denoted by A

Question

Alice and Bob want to meet in one of three places, namely Aquarium (denoted by A), Botanical Gardens (denoted by B) and Centerville (denoted by C). Each of them has strategies A,B,C . If they both play the same strategy, then they meet at the corresponding place, and they end up at dierent places if their strategies do not match. You are asked to find a pair of utility functions to represent their preferences, assuming that they are expected utility maximizers.

Alice’s preferences: She prefers any meeting to not meeting, and she is indifference towards where they end up if they do not meet. She is indifferent between a situation in which she will meet Bob at A, or B, or C, each with probability 1/3, and a situation in which she meets Bob at A with probability 1/2 and does not meet Bob with probability 1/2. If she believes that Bob goes to Botanical Gardens with probability p and to the Centreville with probability 1-p, she weakly prefers to go to Botanical Gardens if and only if p 1/3 .

3

Bob’s preferences: If he goes to Centreville, he is indifferent where Alice goes. If he goes to Aquarium or Botanical Gardens, then he prefers any meeting to not meeting, and he is indifferent towards where they end up in the case they do not meet. He is indiifferent between playing,A, B or C if he believes that Alice may choose any of her strategies with equal probabilities.

(a) Assuming that they are expected utility maximizers, find a pair of utility functions ua : {A, B, C}^2 ->R and ub : {A, B, C}^2 ->R that represent the preferences of Alice and Bob on the lotteries over {A, B, C}^2

(b) Find another representation of the same preferences.

Explanation / Answer

a) ub:{A,B,C}= {01,01,0>1) ->outcome C -this is only valid outcome to win over the lotteries

    ua:{A,B,C}={(0,0,0),(0,>1/3,1/2),(1/3,1/3,1/3)=(1/2,0,0) }

after seeing possibilities in two cases A=1/3

                                                     B=1/3

                                                     C=1/3 is the feasible option to win the lottery

another representation could be

Ua=(A,B,C)1/2 ->R

Ub=(A,B,C)1/2 ->R

this representation is valid because as A,B,C (probability of A,B,C,)increases or decreases, the probabilities of square root of A,B,C also increases or decreases in order.

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