Question

HOMEWORK # 1 (for due date see web page) Consider a simultaneous two-player second-price auction concerning a single indivisible good. The game-frame is as follows: S S ($3, S4, $5, $6, $7 (these are the possible bids), the set of outcomes is the set of pairs (i, p) where ie1,2 is the winner of the auction and p is the price that winner has to pay and the outcome function is as follows (b denotes the bid of Player i): f(bb (2,b,) otherwise (a) Represent this game frame by means of a table (Player 1 chooses the row, Player 2 chooses the column and inside each cell you write the corresponding outcome) Player 1 values the object at S5, that is, he considers getting the object to be as good as getting $5. We shall consider two types preferences for Player 1 Case 1. Player 1 is selfish and uncaring. This means that (1) he prefers to get the object himself provided that he does not pay more than $4, (2) if he wins the auction then he prefers to pay less rather than more, (3) if he does not win the auction then he is indifferent as to how much Player 2 pays and (4) he is indifferent between winning the auction by paying $5 and not winning the auction. That is, for every p $5 and for every p',,p)(2, p'), for every p and p', (1, p, p) if and only if p< p', for every p and p',(2, p)(2, p), for every p, (2, p), (1,S5), and everything that follows from transitivity (b) Use a utility function with values from the set 0,1,2,3,4) to represent the preferences of Player 1 and re-write the table of part (a) replacing outcomes with utilities for Player 1. (c) Is bidding his true value (namely $5) a weakly dominant strategy for Player 1? (d) How many weakly dominant strategies does Player 1 have?

Answer 1

part a

	3	4	5	6	7
3	(1,3)	2,3	2,3	2,3	2,3
4	1,3	1,4	2,4	2,4	2,4
5	1,3	1,4	1,5	2,5	2,5
6	1,3	1,4	1,5	1,6	2,6
7	1,3	1,4	1,5	1,6	1,7

part b

	3	4	5	6	7
3	1	0	0	0	0
4	2	1	0	0	0
5	0	0	0	0	0
6	0	0	0	0	0
7	0	0	0	0	0

Partc-Yes, bidding the true value is a weakly dominant strategy for player 1 as it is a second price auction. for the proof, refer Fudenberg Tirole Chapter 1