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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.
Posted by Tyler Cowen on August 1, 2006 at 04:26 AM in Current Affairs | Permalink
Comments
There is no current event that this is a good model for.
Posted by: joe o at Aug 1, 2006 1:51:07 AM
Joe: I assume the current event is meant to be the conflict between Israel and Hezbollah. Each side commits more and more resources in an attempt to win, because neither wants to be seen to have committed so many resources in vain.
The analogy is hardly perfect. It is not entirely clear that the two sides are contributing more resources than they originally planned. And there is a quite likely scenario in which each side "wins" in its own way: Israel wins the military battle, but Hezbollah wins increased power and influence in the Arab world.
Posted by: David Wright at Aug 1, 2006 3:28:42 AM
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.
Posted by: joan at Aug 1, 2006 5:26:46 AM
Isn't this broken if the first bidder bids exactly one dollar?
Posted by: Mo at Aug 1, 2006 7:45:16 AM
I think you are right Mo.
They say you can't cheat an honest man.
Posted by: Neil Craig at Aug 1, 2006 8:02:31 AM
I've heard this done as a radio show game with 2 callers and $100. The DJ goes up by $5 every few seconds, the first caller to call out their name, gets the last mentioned amount.
Every time I listen to it I think of game theory. ;-)
Posted by: Chris Meisenzahl at Aug 1, 2006 8:24:24 AM
If it is an open auction, then faced with an opening bid of $1, couldn't the auctioneer bid $1.01?
Posted by: Tom at Aug 1, 2006 8:28:39 AM
I think what Tyler means is the dollar auction with non-zero initial stakes. Suppose that for exogneous reasons, each player begins with an initial bid that is greater than zero... (consistent with the current event that is apparently being modeled.)
Posted by: jeff at Aug 1, 2006 8:55:53 AM
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?
Posted by: Paul Gowder at Aug 1, 2006 9:18:37 AM
Mike Munger uses that logic to explain the K Street phenomenon.
Posted by: Jessica Pickett at Aug 1, 2006 9:31:33 AM
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 ...)
Posted by: ryan at Aug 1, 2006 9:58:23 AM
I question the assumption of this game. It is stating that one will always bid higher to try and decrease their loss, however, there is a cost associated with bidding higher and that is the cost of being outbid again. For example:
Every time one makes a bid there is a chance that it will be the winning bid and a chance that it will be a losing bid. As such, every bid has an expected value that can be calculated. If we assume every bid has a 50% chance of being a winning bid and a 50% chance of not being the winning bid, then at 50 cents we have $0 expected return (50% chance of making 50 cents and a 50% chance of having to pay 50 cents).
All bids above $.50 will a negative expected value and thus no rational actor should make those bids. Perhaps 50/50 isn’t the correct percentages to use, however, at any percentage there should be a point where the expected return goes from positive to negative.
Anyone see where my logical is incorrect?
Posted by: Rob at Aug 1, 2006 9:59:59 AM
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.
Posted by: jim at Aug 1, 2006 10:25:35 AM
A model that some think better describes such current events is the "red queen hypothesis", which is used to model coevolution in predators and prey, ideas, and even state reactions to insurgencies. See Van Valen, L. 1973. A new evolutionary law. Evol. Theory 1: 1-30.
Posted by: Andy at Aug 1, 2006 10:35:02 AM
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
Posted by: Damien at Aug 1, 2006 11:53:45 AM
Also, has anyone ever tried that experiment permitting people to bid fractions of a penny? One would imagine that the bids in that case would converge on 0.99999...
Posted by: Paul Gowder at Aug 1, 2006 12:00:12 PM
Mo,
Why would you choose to bid $1?
Posted by: Maria at Aug 1, 2006 1:54:59 PM
I have thought about this game a little more than is healthy.
No one should ever really play this game.
Betting a buck, at the margin, appears the same as not betting; the expected return is o. Any rational opponent will not bid if Mo's bid is first. Mo however, having joined the game, now carries the risk of having an opponent who is irrational and bids a little more, say $1.01. Of course the opponent is worse off as they now have an expected return of 0.01 instead of 0, but then, they are irrational. If this is the case, Mo just lost his buck except if Mo keeps betting and loses some greater amount, x or x-1.
Given the route the game takes, why would one even place the dollar bet? It is only by not playing that one's expected return is strictly 0.
Posted by: disaggregated at Aug 1, 2006 2:18:31 PM
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.
Posted by: Keith at Aug 1, 2006 3:08:15 PM
"Given the route the game takes, why would one even place the dollar bet? It is only by not playing that one's expected return is strictly 0."
You would want to do that if you wanted to come out even and also didn't want your opponent to profit (which he would do if you refused to bid at all).
Posted by: Slocum at Aug 1, 2006 3:09:53 PM
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.
Posted by: Simon at Aug 1, 2006 3:33:59 PM
Is this what we see in reality? Not the experimental setting -- actual geopolitics (or, in Simon's example, lobbying)? Do states consistently commit far more resources than they could possibly gain? Do lobbyists actually spend more than they get? Even in the original Tullock model, where the higher bids only guarantee a higher chance of winning the rent rather than the entire prize, a game of two rent-seekers should conclude with each player consuming 1/4 of the rent. Is that happening?
Posted by: ryan at Aug 1, 2006 4:39:16 PM
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.
Posted by: will mcbride at Aug 1, 2006 4:51:40 PM
Keith:
But that can't be right. If we assume complete information and rationality on behalf of the bidders, any given bidder A will know the extent of the universe of other potential bidders. So A's utility calculuation while considering her first bid will be as follows. If she fails to bid, her expected gain is zero. If she does bid, then she must assume that everyone else has exactly the same utility calculation, and conclude that her expected gain is - all her money.
Can someone who knows more game theory than I chime in here? (Is anyone actually reading this blog yet?) I don't think I understand how nonparticipation is an equilibrium. The claim that there is no equilibrium seems to rely, sub rosa, on each actor's not having access to the utility calculations undertaken by the others.
Posted by: Johannes Climacus at Aug 1, 2006 4:54:18 PM
(ignore the "(Is anyone actually reading this blog yet?)" above -- I copied and pasted that paragraph in from my own new blog)
Posted by: Johannes Climacus at Aug 1, 2006 4:55:57 PM
