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In a 2nd price option, if all the player choice their bids independently, the optimal strategy is to bid their valuation. However, in some situations, their decisions can effect each other.

For example, an over bidder could announce that they will bid \$1000000 for a painting worth \$1000. This would cause all the bidders to not bid (since now there is no point), and the over bidder pays $0 since there were no other bids.

A solution to the over bidder is a spiter, who predicts the bid of and underbids the highest bidder by 1 cent, forcing them to pay their bid. However, the spiter presents a new problem. If the players know their is a spiter, they will bid like they would in a first price auction, since they know they will end up paying whatever they bid. Even worse is a shill, someone who colludes with auctioneer to underbid the highest bidder by a cent. If they accidentally win (if they overestimated the highest bid), the auctioneer can secretly refund the purchase. If they succeed, however, they can share the extra revenue with the auctioneer, meaning that they have an incentive to succeed.

My question is, are there any modifications you can make to the auction to solve this? Of course, you could just say "no over bidders, spiters, or shills allowed", but I would prefer a mathematical solution.

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  • $\begingroup$ @afreelunch Essentially, but also over bidders. It should also be similar to the second price auction (having truthiness, for example). $\endgroup$ – PyRulez Feb 12 at 11:44
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    $\begingroup$ Why would I believe your \$10000000 announcement? Even if I do, why not bid \$1000 if the picture is worth that much? I cannot possible lose, and in a 2nd price setting I might gain, if you backtrack or make a mistake. So no, the overbidder's decision does not affect my strategy. $\endgroup$ – Giskard Feb 12 at 12:25
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    $\begingroup$ @densep You would believe my \$10000000 announcement because it is an equilibrium (albeit one in weakly dominated strategies) for me to bid \$10000000 and you to bid zero. If participation in the auction has some (e.g., time/resource) cost then this equilibrium doesn't even rely on play of dominated strategies. This is not purely hypothetical either. A big challenge, for example, in spectrum auctions is convincing new entrants to show up to bid when everyone expects the big incumbents will win all of the licenses. $\endgroup$ – Ubiquitous Feb 12 at 17:07
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Overbidding

The second price auction has an equilibrium in which player 1 bids a very large amount and all other players bid zero. None of the losing bidders have a profitable deviation because they would have to pay more than their value to win. People usually ignore this equilibrium because not bidding is weakly dominated. But we should acknowledge that it is indeed an equilibrium because it is relevant for some practical applications. A big challenge, for example, in spectrum auctions is convincing new entrants to show up to bid when everyone expects the big incumbents will win all of the licenses.


One way to eliminate this equilibrium is to make it strictly dominant for players to submit a bid. This can be done, for example, by modifying the auction in the following way:

1) run a second price auction as normal.

2) with probability $1-\epsilon$, determine the allocation and payments as usual. With probability $\epsilon$, ignore the bid of the highest bidder and give the item to the second highest bidder (who pays the third highest bid).

Even very small $\epsilon$ will suffice to make is strictly dominant for everyone to bid. Thus, the efficiency loss from this scheme can be kept relatively small.

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Ubiquitous has a good answer.

To add: a shill works the same way as a hidden reserve price. So any literature on one should be applicable to the other.

Shills are especially damaging, if they can act after seeing every one else's bids. You can get around that with some cryptography that has all bidders commit to their bid before anyone reveals.

(Or a alternatively a third party trusted to offer this kind of commitment service, but not needed to be trusted with eg handling money.)

You can probably extent Ubiquitous' idea of ignoring the highest bid with low probability and going for the second highest to giving any of the top $k$ bids a small chance of winning, so that more than $k-1$ shills would be needed to make not bidding your true value a better strategy.

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