## D2

any of strategies which survive iterated strict dominance. Since it is assumed that this set contains one element, this will prove the required result.

Let (s*»s2 be a (mixed) NE and suppose in negation that it does not survive iterated strict dominance. Let i be the player whose strategy is first ruled out in the iterative process (say in the k**1 round). Therefore, k-1

there exists <r. and a. such that u.(o\,s .) ✓ u.(a.,s .) V s .€ S . , and a.

i t li-ii i -l -I-I l is played with positive probability s?(a.). Since k is the first round at which any of the NE strategies, (s*,s*,...,s*), are ruled out, we must have k-1

that s*.€ S . . Hence, u.(cr.,s*.) > u.(a.,s*.). Let the strategy s'. be

-i-i 11-1 11 -i * i derived from s* execpt that any probability of playing a. is replaced by

u.(s'.,s* ) = u.(s?,s*) + s?(a.)-[u.(<r.,s*) - u.(a.,s*)l > u.(s?,s*,) li -l li-l 11 li-l ii -l ii-l which contradicts the assumption that (s*,s*,...,s*) is a NE.

8.D.3 First of all, notice that the first auction bid is a simultaneous move

game where a strategy for a player consists of a bid. Let b^ be the bid of Player 1, and b^ be the bid of Player 2.

(i) If b^ > b^, Player 1 gets the object and pays b^ for it;

Player 2 does not get the object. Thus, in this case: