War of Attrition review materials

Winning and Costs in the War of Attrition

They simply wait or display up to a certain period of time or burn up to a certain amount of energy.

If a focal animal's opponent has not yet quit when it reaches its limit, it loses.
If their opponent quits before the limit is reached, the focal animal wins.
Since both animals quit at essentially the same time (determined by the loser since the winner is prepared to go on longer), both will pay the same cost in the contest (although for the winner it was less than it was willing to pay).
This means that the value of the resource is discounted to the winner by the cost of displaying for it. We have discussed this idea earlier when we developed a general formula for determining payoffs in a contest.

About Sign Conventions: Following Maynard Smith, we will consider costs as positive values and subtract them from the gross resource value. This is a different convention then we used in the Hawks, Doves, and Bourgeois games but the final mathematics are the same.

The Payoff Table for the War of Attrition:

Strategy and Outcome

Change in fitness for player A

Change in fitness for player B

m(A)> m(B), therefore
A wins

V - m(B)

- m(B)

m(A) < m(B), therefore
B wins

- m(A)

V - m(A)

m(A) = m(B),
Therefore, stalemate. Resource possession is decided at random so each wins half of the time.

0.5*V - m(B)

(equivalent expression
is 0.5*V- m(A))

0.5*V - m(B)

(equivalent expression
is 0.5*V - m(A))

Costs increase linearly with time displaying:

Eq. 3: m = Display Costs = k * t

Strategy and Outcome	Change in fitness for player A	Change in fitness for player B
m(A)> m(B), therefore A wins	V - m(B)	- m(B)
m(A) < m(B), therefore B wins	- m(A)	V - m(A)
m(A) = m(B), Therefore, stalemate. Resource possession is decided at random so each wins half of the time.	*0.5V - m*(B) (equivalent expression is 0.5V- m(A))	*0.5V - m*(B) (equivalent expression is 0.5V - m(A))