Player A chooses a random integer between 1 and 100 with pro

Player A chooses a random integer between 1 and 100, with probability pj of choosing j (for j = 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 pj . What is player B’s optimal strategy (to maximize expected earnings)?

(b) Show that if both players choose their numbers so that the probability of picking j is proportional to 1/j, then neither player has an incentive to change strategies, assuming the opponent’s strategy is fixed. (In game theory terminology, this says that we have found a Nash equilibrium.)

(c) Find the expected earnings of player B when following the strategy from (b). Express your answer both as a sum of simple terms and as a numerical approximation. Does the value depend on what strategy player A uses?

Solution