vickrrey auction second price bidding

vickrey auction.txt

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.