The n-person war of attrition
type: Item
instance of: scholarly article
title: The n-person war of attrition
zbMATH Open document ID: 0669.92022
DOI: 10.1007/BF00046674
author: John Haigh
author: Chris Cannings
published in: Acta Applicandae Mathematicae
publication date: 1989-01-01T00:00:00Z
review text: A two-person war of attrition (WA) is a game in which contestants compete for a prize worth V \((V>0)\), and the one who is prepared to wait longer collects the prize; both contestants incur a cost equal to the length of time taken to resolve the contest. Symbolically, if E(x,y) denotes the amount gained by a contestant prepared to wait time x when the opponent is prepared to wait y, we have \[ E(x,y)=V-x\quad if\quad x>y,\quad and\quad E(x,y)=-x\quad if\quad x<y. \] Such a game has precisely one evolutionarily stable strategy (ESS), viz. to wait for a time x drawn at random from the exponential distribution with mean V, i.e. density \(V^{-1}\exp (-x/V)(x>0).\) The authors analyze several different models for the generalization of this WA to n-person conflicts. In Model A, n players compete for one reward, value V, and each player independently selects a time he is prepared to wait, in the hope of outlasting the (n-1) opponents. Once chosen, this time is fixed; as some players drop out, those remaining are not allowed to alter their initial ``bids''. If \(X_ i\) is the waiting time chosen by player i, the reward to player 1 (say) is \[ E(X_ 1;X_ 2,...,X_ n)=V-W_ 1\quad if\quad X_ 1>W_ 1 \] and \[ E(X_ 1;X_ 2,...,X_ n)=-X_ 1\quad if\quad X_ 1<W_ 1 \] where \(W_ 1=Max(X_ 2,...,X_ n)\). Theorem 1 says that this model has a unique ESS, when each player chooses a value \(X_ i\), independently, from the distribution functions \(G(x)=(1-\exp (-x/V))^{1/(n-1)}(x\geq 0).\) Other models, in which the players may alter their bids or in which several prizes are available, are also discussed and the ESS investigated (there may be none).
Mathematics Subject Classification ID: 92D50
Mathematics Subject Classification ID: 92D15
zbMATH DE Number: 4097185
zbMATH Keywords: war of attrition
zbMATH Keywords: evolutionarily stable strategy
zbMATH Keywords: exponential distribution
zbMATH Keywords: n-person conflicts
zbMATH Keywords: ESS
reviewed by: Manfred Nermuth
MaRDI profile type: Publication
cites work: Models of animal conflict
cites work: Q5332396
cites work: Evolutionary stable strategies and game dynamics for \(n\)-person games
full work available at URL: https://doi.org/10.1007/bf00046674
OpenAlex ID: W4236878521
description: scientific article; zbMATH DE number 4097185