Vol. 10 No 3 - Pi Mu Epsilon

PROBLEMS AND SOLUTIONS91, 92, a total of 32 numbers. Here the fair price is P3 = $(A + B +320190.11. Solution to part (a) by the Proposer.Continuing Solution I, suppose the player selects a number completelyat random without regard to the three cases considered. (Perhaps thenumber is assigned by a drawing.) Then the probabilities of picking case 1,2, or 3 are 9/90 = 1/10, 9/90 = 1/10, and 72/90 = 8/10, so the fair priceshould be111. Solution to part (b) by Richard I. Hess, Rancho Palos Verdes,California.Assume you follow a mixed strategy with probabilities p, q, and r = 1- p - q of choosing numbers from cases 1, 2, and 3 (from Solution 1above) respectively. We shall solve for p and q so that your expectation isindifferent to whatever strategy the remaining population chooses. Let thepopulation consist of n people including yourself. We suppose the othersall pick pure strategies from cases 1, 2, and 3.Suppose the remaining population all pick from case 1, say they pick thenumber 10. The table shows your expectation for each choice you make.You pick tone of) With probability Your expectation23, 24, ..., 98(no 0, 1, or repeat)Hence the expectation E(l) is given bySimilar counting for the cases where the remaining population choosespurely from sets 2 and 3 givesandTo make the expectations equal we set0 = E(3) - E(2) =andwhich, since r = 1 - p - q, reduce to