The n-person war of attrition
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).
scientific article; zbMATH DE number 4097185