5.34

Consider the following lottery game, with n participants competing for a prize worth

$M (M > 1). Every player may purchase as many numbers as he wishes in the range

{1, 2,...,K}, at a cost of $1 per number. The set of all the numbers that have been

purchased by only one of the players is then identified, and the winning number is

the smallest number in that set. The (necessarily only) player who purchased that

number is the lottery winner, receiving the full prize. If no number is purchased by

only one player, no player receives a prize.

(a) Write down every player's set of pure strategies and payoff function.

(b) Show that a symmetric equilibrium exists, i.e., there exists an equilibrium in

which every player uses the same mixed strategy.

(c) For p1 (0, 1), consider the following mixed strategy i(p1) of player i: with

probability p1 purchase only the number 1, and with probability 1 p1 do not

purchase any number. What conditions must M, n, and p1 satisfy for the strategy

vector in which player i plays strategy i(p1) to be a symmetric equilibrium?

(d) Show that if at equilibrium there is a positive probability that player i will not

purchase any number, then his expected payoff is 0.

(e) Show that if M

value of the prize at equilibrium, there is a positive probability that no player

purchases a number. Conclude from this that at every symmetric equilibrium the

expected payoff of every player is 0. (Hint: Show that if with probability 1 every

player purchases at least one number, the expected number of natural numbers

purchased by all the players together is greater than the value of the prize M,

and hence there is a player whose expected payoff is negative.)