Xavier, Yolanda, and Zachary are considering whether to pool their funds to buy into a lottery. There is a 20% chance that they will win big and make $8 million dollars, a 40% chance that they will win second prize and make $2 million, and a 40% chance that they will lose and win nothing. The entrance fee to participate in this lottery is $3 million. The partners decide whether or not to play by majority vote. Assume that Xavier has utility function u(x) = x². The other two partners have utility function u(x) = x, where x is the total amount of money won in the lottery. Will the partners buy in?

Question

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Xavier's utility from winning big is (8,000,000 - 3,000,000)² = 25,000,000,000. His utility from winning second prize is (2,000,000 - 3,000,000)² = 1,000,000,000. His utility from losing is (0 - 3,000,000)² = 9,000,000,000. The expected utility for Xavier is 0.2*25,000,000,000 + 0.4*1,000,000,000 + 0.4*9,000,000,000 = 8,200,000,000, which is greater than his utility from not playing, (0)² = 0. Therefore, Xavier will vote to buy in. Yolanda and Zachary's utility from winning big is 8,000,000 - 3,000,000 = 5,000,000. Their utility from winning second prize is 2,000,000 - 3,000,000 = -1,000,000. Their utility from losing is 0 - 3,000,000 = -3,000,000. The expected utility for Yolanda and Zachary is 0.2*5,000,000 + 0.4*(-1,000,000) + 0.4*(-3,000,000) = 400,000, which is less than the utility from not playing, 0. Therefore, Yolanda and Zachary will not vote to buy in.

Xavier, Yolanda, and Zachary Are Considering Whether to Pool Their