2nd bid also pays problem

[Paul2432]

In my business school class (CU-Boulder MBA evening program) the professor made the following offer to the class: I will sell this $10 bill to the highest bidder, with the catch that the second place bidder also pays.

You can probably guess what happened, the bidding quickly rose above $10 as the second place bidder made higher and higher bids to avoid paying the 2nd bid penalty and receiving nothing in return (e.g. a $14 2nd place bid loses $14 while a $15 winning bid only loses $5). The professor cut-off the bidding because his point had been made but it left me wondering what the proper strategy for this game would be (assuming collusion is not possible).

I figure that a bid must be $10 more than the previous bid for the second place bidder to prefer not to bid again. If this is true there would be no bids. Of course, this gives incentive for someone to make an opening bid with the hope that no other bidder will make a bid. Even that exposes you to a "spite" bid from another bidder who could force you to lose money. The higher you bid, the more money you could be forced to lose.

Has anyone seen this problem before? What is the proper strategy?

Paul

[AaronBrown]

There's no good answer to the question in theory. In practice, don't bid.

The trouble is if everyone looks ahead enough steps, no one will bid. That makes it rational to shout out the first bid of $0.01. But it also makes it rational for someone else to bid $0.02, figuring it's irrational for you or anyone else to compete.

There are lots of games like this where, as they said in War Games, the only way to win is not to play.

[KurtSF]

Nice game theory question!

I see three possible strategies:

(1) Assuming rational opponents and zero transaction costs, open bidding at $10. No rational oponent can beat the bid. Result: no one wins anything.

(2) Conspire with opponents against the seller. Put in one bid of $0.01 and everyone shares $9.99/n. I'm guessing the seller would disallow this in the rules of the game. Result: everyone wins equally.

(3) Don't play. Result: An irrational opponent may benefit. If you have several irrational opponents, one will also lose. Very possibly both will lose.

Since the assumptions of (1) are likely no good, and (2) is probably against the rules of the game, clearly (3) is the best solution. This is the best solution in any circumstance too, because not only does it prevent losses (a win in a non-competitive environment), but also the average expectation of you oponents is less than yours (a win in a competitive environment)!

[AaronBrown]

I did think of one other possibilty. Bid negative ten billion dollars. If no one else bids, you get rich. If one other person bids, you get rich. If two other people bid, you get into the usual problem and end up with nothing (but at least you don't lose money).

A bid of zero works if everyone else refuses to play the irrational part of the game.

[KurtSF]

[ QUOTE ]
Bid negative ten billion dollars.

[/ QUOTE ]

That's brilliant!

I'm going to try that on ebay. Easy money here I come!!!

[alThor]

This is a great example to show how people don't think ahead (or as your professor will say, use backwards induction).

Against rational opponents, you want to be the first to bid 0.01. If your opponents are rational, they'll realize that they can't make a profit by competing with you, if you plan to fight back. The equilibrium outcome is that you win a profit of 9.99.

Now you are talking about an extensive form game, which creates some technical problems I won't go into (e.g., infinite bidding paths). But in the sealed-bid version of this kind of game, equilibria can be asymmetric.

answer to this question, in (economic) theory

[Siegmund]

Yes, a very famous problem. Inflation has taken its toll: it's traditionally called the "dollar auction," introduced by Martin Shubik about 30 years ago.

A google search on words like shubik dollar auction will get you a good array of hits describing the paradox and different people's less-than-satisfactory attempts to explain it away.

[Paul2432]

Thanks for all the replies.

Paul

[f97tosc]

The game is somewhat undefined because you haven't specified exactly in which order the players may bid and at what point the game will end. The procedure "whoever yells first and loudest" isn't very well defined mathematically. And in your case the professor interrupted but what would be the "normal" termination protocol? Typically these "small" things have a huge impact on the optimal strategy for these types of games.