The asymmetric war of attrition

The paper, which has an informal discussion at the end, provides a game theoretical analysis of the asymmetric “war of attrition” with incomplete information. This is a contest where animals adopt different roles like “owner” and “intruder” in a territorial conflict, and where the winner is the individual prepared to persist longer. The term incomplete information refers to mistakes in the identification of roles. The idea by Parker & Rubenstein (1981) is mathematically worked out and confirmed that there exists only a single evolutionarily stable strategy (ESS) for the model with a continuum of possible levels of persistence and no discontinuities in the increase of cost during attrition. The ESS prescribes to settle the conflict according to “who has more to gain or less to pay for persistence”. The only evolutionarily stable convention is thus to give the player access to the resource who has the role which is favoured with respect to payoffs. By contrast, it was shown earlier (Hammerstein, 1981) for various asymmetric versions of the “Hawks-Doves” model that an ESS can exist which appears paradoxical with respect to payoffs. The nature of this contrast is further analyzed by introducing elements of discreteness in the asymmetric war of attrition. It turns out that some conditions must be satisfied in order to have the possibility of an alternative ESS which is not of the above simple commonsense type. First, a decision to persist (or escalate) further in a contest must typically commit a contestant to go on fighting for a full “round”, before he can give up without danger. Second, such a “discontinuity” must occur at a level of persistence where the contest is still cheap, and, finally, errors in the identification of roles must be rare.