Problem 15 Player A chooses a random integer between 1 and 100, with probability pj of choosingj {forj = 1,2, ..., 100). Player B guesses the number that player A picked, and receives that amount in dollars if the guess is correct (and 0 otherwise). a Suppose for this part that player B knows the values of pr. What is player B's optimal strategy to J maximize expected earnings)? (b) Show that if both players choose their numbers so that the probability of picking j is proportional to 1 / 3', then neither player has an incentive to change strategies, assuming the opponent's strategy is xed. (In game theory terminology, this says that we have found a Nash equilibrium.) c Find the expected earnin s of pla er B when followin the strate from b . Express our answer both g y g gy y as a sum of simple terms and as a numerical approximation. Does the value depend on what strategy player A uses? Solution (a) The expected earning of player B if she guesses j is jpj, thus she should choose the j with maximal jpj. (b) Since pj ~ l/j, there must exists a constant a such that pj : a/j. From the point of player B, the expected earning of player B if B guesses j is j -pj = (1. So there is no need for player B to change her strategy. From the point of player A. the expected earning of player B if A chooses j is j - - pj = (1. Thus there is no need for player A to change her strategy. (G) Since 231031 f = 1, the expected earning of player B is 100 Zj - P(A chooses j, B guesses j) {1) j=1 ' 3 {2) = - = a? (3) which does not depend on what strategy player A uses