QUESTION 1

Consider a simultaneous two-player second-price auction concerning a single

indivisible good. The game-frame is as follows: 1 2 S S B where 1 2 { , ,..., } B p p p m is a finite

subset of [0, ) with k k 1 p p for every k m {1,..., 1}, the set of outcomes is the set of pairs

( , ) i p where i{1, 2} is the winner of the auction and p B is the price that winner has to pay

and the outcome function is as follows ( i b denotes the bid of player i):

2 1 2

1 2

1

(1, ) if ( , ) (2, ) otherwise

b b b

f b b b

. Let i v be the value of the object to Player i (that is, Player i

views getting the object as equivalent to getting $ i v ). We shall consider three kinds of

preferences. We state them in terms of Player 1, but the same definitions apply to Player 2. The

following apply to all three preferences (this is the "selfish" part):

for every p < 1 v and for every p, 1 (1, ) (2, ) p p ;

for every p and p, 1 (1, ) (1, ) p p if and only if p p .

Player 1 is selfish and uncaring if, in addition, his preferences are as follows:

for every p and p, 1 (2, ) (2, ) p p ;

for every p, 1 1 (2, ) (1, ) p v ;

and everything that follows from the above by transitivity.

Player 1 is selfish and benevolent if, in addition, his preferences are as follows:

for every p and p, 1 (2, ) (2, ) p p if and only if p p ;

1 1 (2, ) (1, ) mp v ;

and everything that follows from the above by transitivity.

Player 1 is selfish and spiteful if his preferences are as follows:

for every p and p, 1 (2, ) (2, ) p p if and only if p p ;

1 1 1 (2, ) (1, ) p v ;

and everything that follows from the above by transitivity.

In what follows assume that m > 3, 1 2 v v B , , 1 1 m p v p and 1 2 m p v p .

(a) Suppose that Player 1 is selfish and uncaring. Does she have a weakly or strictly dominant

strategy? If your answer is Yes, state whether it is weak or strict dominance; if your answer is

No, prove it.

(b) Suppose that Player 1 is selfish and benevolent. Is bidding 1 v a weakly dominant strategy?

Prove your claim.

(c) Suppose that Player 1 is selfish and spiteful. Is bidding 1 v a weakly dominant strategy? Prove

your claim.