Economics 122 Homework 1 Examples

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Department of Economics, University of California, Davis Ecn 122  Game Theory  Professor Giacomo Bonanno 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: S1  S 2  {$3, $4, $5, $6, $7} (these are the possible bids), the set of outcomes is the set of pairs (i, p ) where i  {1, 2} is the winner of the auction and p is the price that winner has to pay and the outcome function is as follows ( bi denotes  (1, b ) if b1  b2 . the bid of Player i): f (b1 , b2 )   2 (2, b1 ) 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 $5, 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 , (1, p ) 1 (2, p) , for every p and p , (1, p ) 1 (1, p) if and only if p  p , for every p and p , (2, p) 1 (2, p) , for every p, (2, p) 1 (1,$5) , 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? Case 2. Player 1 is selfish and benevolent. 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 [this is the selfish part], (3) if he does not win the auction then he prefers if Player 2 pays less rather than more [this is the benevolent part] and (4) he is indifferent between winning the auction by paying $5 and not winning the auction and letting Player 2 pay the lowest amount. That is, for every p < $5 and for every p , (1, p ) 1 (2, p) , for every p and p , (1, p ) 1 (1, p) if and only if p  p , (2, p) 1 (2, p) if and only if p  p , (2,$3) 1 (1,$5) , for every p > $5, (2, p ) 1 (1, p) . (e) Use a utility function with values from the set {0,1, 2,3, 4,5, 6, 7} to represent the preferences of Player 1 and re-write the table of part (a) replacing outcomes with utilities for Player 1. (f) Is bidding his true value (namely $5) a weakly dominant strategy for Player 1? (g) How many weakly dominant strategies does Player 1 have? Page 1 of 1
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Economics 122 homework 1

Considering a simultaneous two-player second price auction concerning a single individual
good:

a. The game is represen...


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