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May 1, 2017 22:12
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vickrrey auction second price bidding
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Vickrey Auction | |
System Description: There are n bidders | |
each bidder has a bid b(i) and a true value v(i) that the item is truly worth | |
payoff to bidder = v(i) - b(i) | |
payoff too bidder = 0, if I lose. | |
The winner of the auction has the highest bid. | |
However, the winner doesn't pay the highest bid, he pays the second highest bid. | |
What does the winer pay? | |
He pays max(b(j)), j!= i | |
Suppose the item is worth $100 to me. | |
ie. v(i) = 100 | |
Case 1. Suppose I overbid ie. b(i) = 105 | |
1a. suppose max(b(j), j!= i) is 95. | |
ie. 95(second price)....100(true worth)....105(my bid) | |
I win, but I pay 95. | |
Note I would have won even if I just paid the true value 100! | |
Thus, truth wins. | |
1b. suppose max(b(j), j!= i) is 103 | |
ie. 100(true worth)....103(second price)...105(my bid) | |
I win, but I pay 103 | |
Note I would NOT have won if I had bid the true value of 100. | |
Truth does not win. | |
1c. suppose max(b(j), j!=i) is 107 | |
I lose because I don't have the highest bid. My payoff is 0. | |
I would have lost if I had bid the true value of 100. My payoff would still be 0. | |
So bidding truthfully & overbidding has the same payoff of 0. | |
Truth wins. | |
So out of 3 cases 1a,1b,1c, truth wins in two of them! | |
Truth dominates. | |
Case 2. Suppose I underbid ie. b(i) = 95 | |
2a. suppose max(b(j), j!= i) is 93 | |
ie. 93(second price)....95(my bid)...100(true worth) | |
I win, but I pay 93. | |
Note I would have won even if I just paid the true value 100! | |
Thus, truth wins. | |
2b. suppose max(b(j), j!= i) is 97 | |
ie. 95(my bid)....97(second price)....100(true worth) | |
I lose. My payoff is 0. | |
If I had bid truthfully I would have won. | |
So truth wins. | |
2c. suppose max(b(j), j!=i) is 103 | |
ie. 95(my bid)....100(true worth)....103(second price) | |
I lose because I don't have the highest bid. My payoff is 0. | |
I would have lost if I had bid the true value of 100. My payoff would still be 0. | |
So bidding truthfully & underbidding has the same payoff of 0. | |
Truth wins. | |
So out of 3 cases 2a,2b,2c, truth wins in all three of them! | |
Truth dominates. | |
So overall, truth dominates. | |
It is best to bid one's true value, as truth dominates in 5/6 cases. | |
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