Further Explanation and Illustration of Why
Net Benefit Calculations in the War of Attrition Can Be Complex

Lets say that our focal supporting pure strategy (e.g., strat. A in the earlier equations) is prepared to pay a maximum cost of 0.4 fitness units. So, we'll write it as fix(x=0.4). Now, recall that our focal strategy is essentially opposing the entire mix (i.e., a mixed strategist or an entire population at the equilibrial mix). Since the mix consists of an infinite number of possible maximum costs, the focal pure strategy will usually not play an opponent who is willing to pay the same maximum cost in each contest.

For example:

• Sometimes the opponent plays 0.2 and fix(x = 0.4) wins
• Other times the opponent plays x = 1.0 and fix(x=0.4) loses.

So, the overall net benefit to fix(x=0.4) in wins against against the mix will be the sum of each possible net gain times the chance of getting that win. Here again is a simplified equation for net gain:

eq. 3b: net Benefit (B) = Sum{(V - x) * (Prob of facing x)}

Eq. 3b gives us lifetime expected net benefit by summing this value for each cost where the focal strategy wins -- i.e. for each x < m.