Talk:War of attrition (game)

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It might be interesting to add notes about real-world decision makers who are or have used the "I'm insane and unpredictable" strategy. Consider Iran's nuclear program: It wouldnt bother us much if they were totally rational, and thus wouldnt be much good to them as a bargaining tool, so they foster the appearance of insanity. Also true of North Korea in some analyses. There must be other historical examples.

Kim Jong Il was the first one to come to my mind before I even that sentence. Picket's charge used that strategy- to capture a hill that was heavily defended in the American Civil War, Union soldiers charged it yelling theuir heads off and waving their guns over their heads through a hail of cannon fire, causing the Confederate artillery men to flee in panic, leaving their cannons behind. Ghengis Kahn, Ivan the Terrible, and John "Blackbeard" Teach also cutivated their horrible reputations for this reason: to make enemies surrender and save themselves a fight(and thus resources and men), although that almost borders on Hawk and dove, might depend on how one portrays it to how good of an example they would be.--Scorpion451 07:26, 19 June 2007 (UTC)

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[edit] does this game have a Nash equilibrium?

It seems to me this game doesn't have a NE, what would contradict the Nash Equilibrium article: "Nash equilibria must exist for all finite games with any number of players". Anyone can help? —Preceding unsigned comment added by Ezadarque (talk • contribs)

It does have one, it's a mixed ESS. It's proof should be here, it's on my list of things to do eventually. I can't remember off the top of my head which reference is the best to point you to. Or do you have reservations about the solution qualifying as a Nash? Pete.Hurd 14:06, 4 July 2006 (UTC)

Sorry, I see that I was confusing Nash equilibrium with ESS. I did some research and understood my mistake. However, maybe it would be interesting to point out that there is no symmetrical equilibrium in this game.

[edit] A decent example will do this article some good.

This article could do with some decent real life or simple fictional example. Kendirangu 06:46, 7 May 2007 (UTC)

It also needs the Grid table, that's fundamental to game theory game pages.--Scorpion451 07:18, 19 June 2007 (UTC)

[edit] Removing normal form matrices

I've moved the normal-form matrices here to the talk page. The game isn't a 2x2 matrix game, the players' strategies are continuous numbers. Pete.Hurd 03:18, 4 July 2007 (UTC)

Raise cost Accept cost
Raise Cost V/2 - c, V/2-c V-c, -c
Accept Cost -c, V-c V/2-c, V/2-c
War of Attrition: c starts at 0 and increases until either party accepts it(and the loss) or both accept it
Raise cost Accept cost
Raise Cost -c, -c V-c, -c
Accept Cost -c, V-c V/2-c, V/2-c
Waiting Game: This is an alternate version in which V is not awarded to the players for raising simultaniously. The concept here is that of two players waiting for the other to leave.

[edit] discrete time

I've removed the formulation in terms of discrete time, the WoA isn't a dynamic game, it's a bidding game with only one move per player, to chose a continuously valued bid. Pete.Hurd 03:17, 4 July 2007 (UTC)

One of the distinctive features of War of Attrition is that it is one of the few Game Theory games which is exclusivly a repeated game. Playing a single round, c equaling zero, the dominant strategy is to raise.

Raise cost Accept cost
Raise Cost V/2, V/2 V,0
Accept Cost 0, V V/2, V/2
at c=0 in the first round, there is only risk for accepting
Raise cost Accept cost
Raise Cost tie, tie win, lose
Accept Cost lose, win tie, tie
win-lose-tie form

However, as additional rounds are played, the continually rising cost of c cancels the tie situation, creating a fork in the game. This is the last opportunity for the players to cooperate before moving into the risk bearing portion of the game.

Raise cost Accept cost
Raise Cost 0,0 V-c,-c
Accept Cost -c, V-c 0, 0
if a player accepts now, they can escape penalty if their opponent also accepts, maening both benefit.


If the players choose to raise and continue, the game enters the lose-lose situation.The brief opportunity for cooperation has evaporated at this halfway point. The game is now into the risk-bearing stage of the game.

Raise cost Accept cost
Raise Cost lose,lose win, lose
Accept Cost lose, win lose, lose
the lose-lose scenerio

Ultimately, the game reaches it's endgame in a situation comperable to the Cold War: it is in each player's best interest to stop, as there is no possibility for gain in any scenario. Logic, however, indicates that stopping would cost that player an even higher cost if the other chooses to raise. They are locked into a mutually assured destruction situation. At this point the two players are in a game of Hawk and Dove.

Raise cost Accept cost
Raise Cost lose,lose lose less, lose more
Accept Cost lose more, lose less lose, lose
the lose-lose scenerio

[edit] more material removed from article

Moved material from article, not really on point. Pete.Hurd 05:43, 4 July 2007 (UTC)

[edit] Deception as a Strategic tool

The element of appearing unpredictible, however, to hide one's true intentions has long been military doctrine, mentioned as far back as Sun Tzu's The Art of War, in such advice as

"The greatest weapon a general has is a creative and innovative mind, for his foe will be unsure how he will strike next and thus be at unease"

and

"All warfare is based on deception. If your enemy is superior, evade him. If angry, irritate him. If equally matched, fight and if not: split and re-evaluate"

The attitude which Sun Tzu promotes is one of deception and misdirection, including threat displays such as marches along borders to display ones strength, and conserve resources in the long run.

In the mathematical sense, the idea of deception convinces the foe that you will accept a lower or higher cost "c" than you are prepared to. A foe believing that one will accept losses of 20% will not anticipate a fight when you will in reality accept 30% losses. They may accept losses of 25% believing they have already won, causing them to lose. A foe prepared to accept only 10% losses may surrender upon the idea that one will accept 20% losses, beliving that they have no chance to win, allowing one to suffer losses far less than 20%.

[edit] According to Source

Accoding to the game as presented in the sources, the game is an itterated game in which the two players compeate in an auction type gram in which the bids increase at a constant rate and may quit at any point. The players may bid any amount, but this is just as effectivly portrayed as above. For example, a "waiting game" style contest may take any amount of time, unknown at any one point in the game, leading to the unpredicatable nature. The contestants may choose to leave at at any one particular increment. Representing this as a single arbitrary bid confuses the insights given by this form. Yes the game can be considered, and for the total calculations is considered as a bid, but the individual decisions are still representable by the two by two grid in the context of "bid higher" or "walk away". The name of the game itself implies a contiuous game over a given time span.--Scorpion451 21:16, 4 July 2007 (UTC)

This is the souce upon which the 2x2 grids I created were based upon. As you can see here, as a tool for examining the individual cross sections of the integral, it is valid. While I default to Hurd as an established authority on this subject, I based my contributions on the work here, and so they were not simply pulled out of the aether.
http://www.holycross.edu/departments/biology/kprestwi/behavior/ESS/ESS_index_frmset.html
more specifically the section: "Part1: Introduction to Wars of Attrition: Fixed Cost Strategies"
--Scorpion451 15:24, 5 July 2007 (UTC)

[edit] Making some additions

I thought over Pete Hurd's comments, and reread the sections on attrition games in my game theory books. Having done that, I think I can find a way to present the War of attrition in a more explanatory way without making it appear to be turn based. As it stands, the article tells the reader little about the game itself other than than the equation for the solution. The reader needs to understand why there is no Nash equilibrium, or why a standard matrix does not describe the game in full.--scorpion 451 rant 20:33, 18 July 2007 (UTC)

Sounds good! It definitely needs more of what you describe. Cheers, Pete.Hurd 21:11, 18 July 2007 (UTC)