War of Attrition (game) - Dynamic Formulation and Evolutionarily Stable Strategy

Dynamic Formulation and Evolutionarily Stable Strategy

Another popular formulation of the war of attrition is as follows: Two players are involved in a dispute. The value of the object to each player is . Time is modeled as a continuous variable which starts at zero and runs indefinitely. Each player chooses when to concede the object to the other player. In the case of a tie, each player receives utility. Time is valuable, each player uses one unit of utility per period of time. This formulation is slightly more complex since it allows each player to assign a different value to the object. Its equilibria are not as obvious as the other formulation.The evolutionarily stable strategy is a mixed ESS, in which the probability of persisting for a length of time t is:

The evolutionarily stable strategy below represents the most probable value of a. The value p(t) for a contest with a resource of value V over time t, is the probability that t = a. This strategy does not guarantee the win; rather it is the optimal balance of risk and reward. The outcome of any particular game cannot be predicted as the random factor of the opponent's bid is too unpredictable.

That no pure persistence time is an ESS can be demonstrated simply by considering a putative ESS bid of x, which will be beaten by a bid of x+.

It has also been shown that even if the individuals can only play pure strategies, the time average of the strategy value of all individuals converges precisely to the calculated ESS. In such a setting, one can observe a cyclic behavior of the competing individuals.