Game theory type question

[The Truth]

My girlfriends philo professor gave this one.

You are at an auction. The prize is 1 dollar. Bidding starts at .50 cents. Both the winner and the 2nd highest bidder pay their final bid, but only the winner gets the dollar.

Is there a way to win over time?

blake

[Copernicus]

It seems like something is missing from the question? Otherwise why not jump to $.99 immediately, preempting bids from the opponent and making $.01 every time.

[luckyme]

[ QUOTE ]
It seems like something is missing from the question? Otherwise why not jump to $.99 immediately, preempting bids from the opponent and making $.01 every time.

[/ QUOTE ]

We all know jerks who'll bid $1 just to watch us squirm.

luckyme

[madnak]

Because then it's in the opponent's interest to choose $1.00 and outbid you. Either he pays $0.50 or he breaks even. Then of course, you have to bid $1.01. Either you pay $0.99 or you pay $0.01. At each stage bidding an additional penny is the best approach.

I've heard sometimes this experiment is done and people will end up paying $20 or more for the dollar. Personally I try not to play games offered by game theorists. They won't get me with their sinister tricks!

[daryn]

wouldn't it just go on forever until someone cries uncle

[The Truth]

[ QUOTE ]
wouldn't it just go on forever until someone cries uncle

[/ QUOTE ]

You have to pay if you win.

blake

[madnak]

Pretty much. I thought you were a student of game theory?

I'd like to think the game theory response is to cut your losses, but game theory seems to recommend a lot of really awful strategies in situations like this.

[DougShrapnel]

[ QUOTE ]
Is there a way to win over time?

[/ QUOTE ]
Be the auctionier. You can break even by not bidding. That is the best you can do. Or if everyone is assumed to play optimaly be the 1st bidder. If there are only 2 bidders playing optimaly. They each make a quater an auction.

[Utah]

There is a very good discussion of Shubik's dollar auction dilemma starting on page 260 in "Prisoner's Dilemma" By Poundstone. There are also 2 papers by Barry O'Neill and Wolfgang Leininger on this.

I believe the answer depends on whether the players have finite bankrolls.

[Copernicus]

[ QUOTE ]
wouldn't it just go on forever until someone cries uncle

[/ QUOTE ]

Then whats missing is a definition of "win". I wouldnt call paying more than $.99 "winning" since if I pay more my position is worse off than not bidding at all.

[MathEconomist]

The optimal response is to not get involved at all, or to collude with the other bidder to split any winnings (but there are none in this case since the opening bid is .50).

[Go Blue]

[ QUOTE ]
My girlfriends philo professor gave this one.

You are at an auction. The prize is 1 dollar. Bidding starts at .50 cents. Both the winner and the 2nd highest bidder pay their final bid, but only the winner gets the dollar.

Is there a way to win over time?

blake

[/ QUOTE ]

What are the tie-breaking rules of this game?

[hmkpoker]

If there must be a second bidder, this game is strictly negative sum.

That being said, yes, there is a way to win: 1) don't be the first bidder, 2) bid $.99. It is the only incremental amount that is profitable and cannot be trumped by a higher bid (note the problem with bidding $.98, or any smaller amount)

Eventually, people will figure out the strategy and no one will bid first. Then there will be no more game.

[MathEconomist]

This isn't actually correct. If you bid .99, the other guy has an incentive to bid 1.00 and make zero profit as opposed to -.50 or whatever. You shouldn't ever get involved in this institution to begin with. I think, however, that the strategy p1-always outbid by .01 and p2-quit immediately (and vice versa) is a Nash Equilibrium. You can't lose less than .50 if the other player is always outbidding you, and you will make money by always outbidding if the other player will choose to quit immediately, so the strategies are mutual best responses. But since a rational person doesn't get involved to begin with, these equilibria don't make much sense, particularly because there isn't anything distinguishing the players.

[Copernicus]

[ QUOTE ]
This isn't actually correct. If you bid .99, the other guy has an incentive to bid 1.00 and make zero profit as opposed to -.50 or whatever. You shouldn't ever get involved in this institution to begin with. I think, however, that the strategy p1-always outbid by .01 and p2-quit immediately (and vice versa) is a Nash Equilibrium. You can't lose less than .50 if the other player is always outbidding you, and you will make money by always outbidding if the other player will choose to quit immediately, so the strategies are mutual best responses. But since a rational person doesn't get involved to begin with, these equilibria don't make much sense, particularly because there isn't anything distinguishing the players.

[/ QUOTE ]

Is cooperation between the bidders permitted?

[MathEconomist]

I don't know, but you can't make any money that way anyway since the rules obligate both to pay at least .50. But obviously in the general case if you end up in one of these situations your best bet is to try to collude with the other player.

[Utah]

The interesting thing is that if you cant collude and you begin - where do you quit?

Lets say your last bid is 5.00 and your opponent bets $5.01 - do you bet $5.02?

Game theory cannot get you to the answer at this point.

[bunny]

I would guess it depends on your knowledge of your opponent and probably his bankroll. If he is going to bid again - you should quit. If he is going to quit, you should bid again. You can probably use this information earlier - something like

"I've just bid .99 and he's now bid $1. If he has a "bankroll" of 2.00 then I should quit now as I can see where this is heading and he will eventually bid $2 - the best I can do after this (-1.01) is worse than I am getting now (-.99). I should only bid 1.01 if I think he is going to "chicken out" before the bidding gets to 2.00 as then I am guaranteed to improve my current loss."

[hmkpoker]

[ QUOTE ]
This isn't actually correct. If you bid .99, the other guy has an incentive to bid 1.00 and make zero profit as opposed to -.50 or whatever. You shouldn't ever get involved in this institution to begin with. I think, however, that the strategy p1-always outbid by .01 and p2-quit immediately (and vice versa) is a Nash Equilibrium. You can't lose less than .50 if the other player is always outbidding you, and you will make money by always outbidding if the other player will choose to quit immediately, so the strategies are mutual best responses. But since a rational person doesn't get involved to begin with, these equilibria don't make much sense, particularly because there isn't anything distinguishing the players.

[/ QUOTE ]

oof, excellent point. I didn't think about that. In theory, then, the bids would continue forever in an attempt to minimize losses. With no rational end in sight, the only correct strategy is not to play from the get-go.

[Our House]

[ QUOTE ]
My girlfriends philo professor gave this one.

You are at an auction. The prize is 1 dollar. Bidding starts at .50 cents. Both the winner and the 2nd highest bidder pay their final bid, but only the winner gets the dollar.

Is there a way to win over time?

blake

[/ QUOTE ]
If the bill doesn't sell for 50 cents, everyone else at the auction is retarded.

[Copernicus]

[ QUOTE ]
[ QUOTE ]
My girlfriends philo professor gave this one.

You are at an auction. The prize is 1 dollar. Bidding starts at .50 cents. Both the winner and the 2nd highest bidder pay their final bid, but only the winner gets the dollar.

Is there a way to win over time?

blake

[/ QUOTE ]
If the bill doesn't sell for 50 cents, everyone else at the auction is retarded.

[/ QUOTE ]

not retarded, just irrational

[etizzle]

theres no way to win, if both players act simultaneously they will both bid $1.