Consider the two-bidder auction example in class, where bidder i's valuation, Vi, is equal to 1 with probability q (0,1) and 0 with probability 1 q, for i = 1,2 (Note that we are using q in this question, rather than p as in the lecture). Assume V1 and V2 are mutually independent. The object is sold by a first-price auction. The bidders are allowed to bid any nonnegative price. Whenever there is a tie, each bidder gets the good with probability 1/2.

(a) Argue that, when Vi = 0, it is weakly dominant for player i to bid Bi(0) = 0 for i = 1,2.

(b) Show that there does not exist an equilibrium in which B1(1) = B2(1) = 0.

(c) Show that there does not exist an equilibrium in which B1(1) = B2(1) = 3/4.

(d) Verify that the mixed strategy profile given in class is indeed a Nash equilibrium.