Game Theory Question

[Pudge714]

There is an auction for a $500 prize. The auction works in the following way.

Each individual can only bid once.
None of the individuals are aware of the other bids.
The winner of the $500 prize will be the person who bids the most.
The winner will pay the second highest amount submitted.

IE three people are playing
Person 1 $400
Person 2 $350
Person 3 $1000

Person 3 would win and pay $400 for the prize.

If there are only two people playing this game who are rational what number would you write down? How does this change as more people are added? How does this change if there are more people, but it is not guaranteed that they are rational?

[diddle]

Hi,
Since the value of the prize is known beforehand and both of the players are rational, no one wins any money.

Both bid $500.

[Thythe]

Scenario 1: 2 players, the nash equilibrium (NE) would be both players bid 500. Also, one player bidding 499 and one bidding 500 would also be a NE since neither would have a rational reason to leave this particular point.

Scenario 2: 3 players - N Players. A variety of (500, 500, 500, 499, 500, 500) etc pairs would exist that are NE. I'm pretty sure that only one player can bid 499 and still have it be NE. So that would mean that the number of NE for any particular game of this type and N players would be N+1 solutions. The overall idea is the same, though, and at best, the winners makes $(1/(N-1)) or $0.

Scenario 3: Non rational players. Anything goes I guess. I would make some guess as to the players non rationality. The less rational they seemed, the lower I would bid. I would suspect that the very least rational might try to bid something like $50, so maybe I would bid $100. In real life, though, I think I would bid around $400 (against 1 opponent), suspecting that they would bid lower than that and not be rational enough to raise their bid.

[JaredL]

In any 2nd price auction, it is a dominant strategy for each player to bid her value. In this case every player has a value of 500 so they all should, but it generalises as well to for example auctions for a painting (or whatever) where some value it higher than others.

Here's the argument:
Suppose you value the object at $x.

If you bid x + k there are 3 cases:
1. you lose the auction
2. you win the auction and the second price (what you have to pay) is some y < x
3. you win the auction and the second price is some y such that x < y < x+k

In case 1 you get the same payoff, zero, if you bid x - either way you lose the auction and get nothing and pay nothing. In case 2 you would get the same payoff if you had bid x. You would win the object and pay less than what you value the object at. In the third case you win the auction but must pay more than you value it at. Therefore, you would have been better off not winning the auction and hence in this case bidding x is better.

I have just shown above that bidding your value is better than bidding above your value, no matter what the other bidders do.

Similarly suppose you bid x - k. There are 3 cases here as well:
a) You win the auction - second price is some y < x - k
b) You lose the auction, highest bid is some y such that x - k < y < x
c) You lose the auction, highest bid is some y > x

In case a had you bid x you would have the same outcome - winning the object and paying y for it. In case b if you had bid x you would pay y < x so you would be better off (you get the object and pay less than you think it's worth). In case c you would get the same outcome if you had also bid x.

So you are better off bidding your value than anything lower.

Hence you are better off bidding your value than anything higher or lower.

[Thythe]

[ QUOTE ]
Hence you are better off bidding your value than anything higher or lower.

[/ QUOTE ]

agreed

[soon2bepro]

[ QUOTE ]
Scenario 3: Non rational players. Anything goes I guess. I would make some guess as to the players non rationality. The less rational they seemed, the lower I would bid. I would suspect that the very least rational might try to bid something like $50, so maybe I would bid $100. In real life, though, I think I would bid around $400 (against 1 opponent), suspecting that they would bid lower than that and not be rational enough to raise their bid.

[/ QUOTE ]

No, no no. You bid $500 and still pay only what they bid, no matter if it's $50 or $499 or what.

[prosellis]

A slightly different approach:
The value of the prize is "x" and the minimum bidding increment is "i"
Bid= x+i
Using this strategy there are only two possible outcomes:
1)You win the auction and obtain the prize at a price that is equal to or less than the value, in which case you have gained

2)You lose the auction but the other bidding party has paid a price that is equal to x+i or greater, in which case he has experienced a loss

This assumes that there is something to be gained in the other bidder having a negative expectation.

[soon2bepro]

[ QUOTE ]
This assumes that there is something to be gained in the other bidder having a negative expectation.

[/ QUOTE ]

Why would you care. He loses more, the house wins more. He loses money on his bad play and he's less likely to play again, and less likely to keep making bad plays. All purely hypotetical here.

[Thythe]

[ QUOTE ]
[ QUOTE ]
Scenario 3: Non rational players. Anything goes I guess. I would make some guess as to the players non rationality. The less rational they seemed, the lower I would bid. I would suspect that the very least rational might try to bid something like $50, so maybe I would bid $100. In real life, though, I think I would bid around $400 (against 1 opponent), suspecting that they would bid lower than that and not be rational enough to raise their bid.

[/ QUOTE ]

No, no no. You bid $500 and still pay only what they bid, no matter if it's $50 or $499 or what.

[/ QUOTE ]

Whoops, yeah, you're totally right. I momentarily forgot that you pay the 2nd highest price.

[prosellis]

Whether or not you stand to gain from an opponents loss really depends on a lot of circumstantial factors. If this were a one time event then you would gain nothing by your opponent having negative expectation, but if you are in a long term economic competition (like two competing companies) then your position is strengthened by the weakening of your opponent's.