Current events

Remember that game where two people bid sucessively for a dollar bill?  The highest bidder takes the dollar home.  You also pay your highest bid whether or not you win the dollar.  The Nash equilibrium is an infinite bid from both players, or alternatively the equilibrium is undefined.

Comments

There is no current event that this is a good model for.

Hezbollah has said they expected Israel to respond with the usual tit for tat, which seem likely, as the cost for them has been very high. Israel military has has said they had not expect the fighting would be so difficult. Maybe one or both are not being truthful, and some clever plan is being executed which we will see only in hindsight.

I think you are right Mo.
They say you can't cheat an honest man.

If it is an open auction, then faced with an opening bid of $1, couldn't the auctioneer bid $1.01?

I confess to failing to understand the point of the game. Wouldn't a rational actor recognize the infinite-bidding possibility and refuse to participate in the first place, or if forced to participate refuse to bid more than one penny and take that loss?

Even with a non-zero bid, might a rational actor make a bid of 1.00 (or maybe 99 cents) more than the other bidder's initial bid and be done with the game? The other player has no possibility of being better off than his or her initial position.

Or, if this is a repeated game with many players, make a habit of bidding $5 if another player bids anything. Then you might win a lot. Say, Dr. Cowen, who was your dissertation adviser again? (Also, who was the last secretary of state? I seem to recall him saying something about the matter ...)

joe says "There is no current event that this is a good model for."

Probably true, but surely an experiment could be run that would permit this Nash equilibruim to be tested.

Rob: I think it's a matter not of logic and assumptions but of what people actually do in experiments.
http://erraticwisdom.com/2006/05/12/interesting-psychology-experiments

Mo,

Why would you choose to bid $1?

I think there's a little confusion about equilibrium here.

If there is no equilibrium, avoiding the game altogether is
not an equilibrium, either. After all, if nobody bids at
all, then you should a penny and win.

It's a paradox. Playing is a mistake. If nobody plays,
then not playing is a mistake, too.

This is about the Tullock Auction--where all bidders pay the amount of their bid, not just the winner. This podcast at econtalk.org
http://www.econtalk.org/archives/2006/06/giving_away_mon.html
provides an argument about why spending to get grants ('free money") is based on teh same principle.

Obviously, there are many equilibria in practice, because we have different
focal points. Some of us work on K street, some of choose not to. Some of
us vote, some of us don't.

(ignore the "(Is anyone actually reading this blog yet?)" above -- I copied and pasted that paragraph in from my own new blog)

Blar... I think you've solved it on the opening bid. Bids less than that amount will suffer from the same irrationality that the one penny opening bid will -- it requires the false assumption that the other players will give up before you do.

Actually, I believe two major results from economic theory apply here. By Nash, there MUST be an
equilibrium, although it is in mixed strategies. This is true even if the game has started and two
players have both bid. Secondly, the Revenue Equivalence Theorem tells us that this outcome will be
the same as if the auction were conducted using a more standard format (e.g. first-price auction).

I worked this out with some fellow economists and what you get is that each bidder must randomize their
probability of either raising their bid or dropping out in such a way that the other bidder is also
indifferent to staying in (and risking continued escalation) or dropping out (and taking a guaranteed
loss).

Of course, this isn't what we see empirically, either in classroom experiments or in current events.

The prediction is that, on average (i.e. in expectation), the auctioneer will receive the same revenue as in a 2nd-price sealed-bid auction, which is $1. But because the player are engaged in mixed strategies, sometimes in practice we should see lower bids, and sometimes we should see the bidding spiral very high. We didn't try to work it out for more than two bidders, but the revenue equivalence theorem should still hold in that case, also.

I stand corrected on the point of existence. I think we assumed 2 players making alternating offers. Bid increments are fixed (e.g. just enough to exceed the other bidder).

MITdude's model and the basic "dollar auction" model both assume that the size of the payoff is the same for both sides and is commonly known. Probably neither is true. Israel doesn't know how badly Hezbollah has to be beaten before they will abandon their cause, and Hezbollah doesn't know how many civilian casualties Israel is willing to cause in pursuit of its goal. By its current actions Israel is demonstrating that this number is a lot higher than some people thought.

MITdude: you rock. Your first iteration (and presumably subsequent ones too) is off by a penny though. Assume W_L = $100. Then by your first iteration of the reasoning, H should prefer to squash L when B_H reaches $99.01 rather than lose that amount. However, he doesn't prefer that. The squashing occurs at B_H = 100.01, where H's net loss is the same as it would be losing the $99.01 bid.

The original squishpoint thus should be B_H > W_L - $0.98.

God, that was embarassingly nitpickey. I'm glad I'm posting this under a fake name. :-)

Actually, on re-reading, I think there's a slightly larger flaw in the reasoning, one that actually counts.

Finally, player L could bid exactly B_H + $0.99, but I assume player H will then outbid him by one penny in this case (since he would be indifferent between this and losing with his current bid).

Why would H outbid him by one penny if H is indifferent between that and the loss? What's H's motivation in the face of indifference? (Perhaps to punish L for the bid? If so, that makes the model a bit more complicated.)

That's what I get for being nitpickey late at night. You're right, of course. I read > as >/=.

WRT your second point, I suppose this is where someone brings up the diminishing marginal utility of wealth or some such.

The proof above is valid, but it is fragile to trembling hands and relies heavily on both players having shared and common knowledge of each other's wealth. If you are H and W_L is a large sum of money for you to lose, then would you really sit there as the bidding escalated to play L out?

dsquared wrote:

> If you are H and W_L is a large sum
> of money for you to lose, then would
> you really sit there as the bidding
> escalated to play L out?

No, I would not. As with any game where the equilibrium depends on a long string of backward induction (e.g., the centipede game), it is not reasonable to assume that an opponent that has deviated from the equilibrium n times will then stick to the equilibrium behavior on play n+1, so subgame perfect nash equilibrium is not an appropriate solution concept for this game.

Climacus wrote:

> credibly makes it known that he has
> adopted a strategy of punishing anyone
> else who outbids him

Such a threat is not credible if all players are rational, but one way of operationalizing this idea is with a reputation model in which some players are irrational. For example, assume a small fraction of players are "crazy" and will always outbid their opponent until their wealth runs out. A rational player might initially want to pool with these crazy players by outbidding his opponent for a certain number of rounds. Therefore, we could have a situation where two rational players continue outbidding each other, with each trying to convince the other he is crazy enough to lose all his wealth in the game.

Yes, that would work too. The "doomsday device" in Dr. Strangelove is another good example!

This game is not so irrational as it seems. It might be explained like this:
I agree to commit 1 cent in order to win a dollar. If I am outbidden by 1 cent, the game starts again - I can commit 2 more cents in order to win a dollar - perfectly rational (the first cent is a sunk cost and it doesn't matter for the next bid).

In reality, we can see this in the lottery - you give some money every week because you expect to win the next time and you may continue doing it infinitely. However, not all people do it infinitely - some of them stop buying lottary tickets irrespective of the money they already gave up and irrespective of the possibility to win (others become addicted...).

For sure, you increase your chances for winning the lottery if you invest all your money in it. But most people just don't do it - because they have other needs also and these needs compete for the same money they have. So, irrespective of "anecdotal evidences" for any games played in classrooms, I'm not pessimistic on the results (surely, people are not giving up all money for lottery tickets).

I am surprised nobody mentioned yet a very old version of the same game, called blood feud.

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