Here is a description of the Hawk and Dove:

HAWK: very aggressive, always fights for some resource.

• Fights between hawks are brutal affairs with the loser being the one who first sustains injury. The winner takes sole possession of the resource.
• Although hawks that lose a contest are injured, the mathematics of the game requires that they not die and in fact are fully mended before their next expected contest
• For simplicity, we will assume that all hawks are equal in fighting ability; that is, each hawk has a 50% chance of winning a hawk - hawk conflict. Another way of saying this is that Hawk vs. Hawk contests are symmetrical.

DOVE: never fights for a resource -- it displays in any conflict and if it is attacked it immediately withdraws before it gets injured.

• Thus, in any conflict situation, dove will always lose the resource to a hawk, but it never gets hurt (never sustains a decrease in fitness) when confronting a hawk and therefore the interactions are neutral with respect to the dove's fitness.
• A corollary to this rule is that doves do not display for very long against hawks. After starting their displays, they immediately recognize that their opponent is a hawk and they withdraw without paying a meaningful display cost. (press here to review the concept of display costs)
• On the other hand, if a dove meets a dove there will be a period of displaying with some cost (time, energy for display) but no injury. We assume that all doves are equally good at displaying and and they adapt a strategy of waiting for a random time period (see "War of Attrition Game") therefore when two doves face off, each has a 50% chance of winning.
• Notice that both doves will pay essentially the same display cost in any contest. The winner is the individual willing to pay more. However, note that the winner quits displaying essentially at the same time as the loser withdraws (see war of attrition game)

Equation #1:

Eq. 1. Payoff(to Strat., when vs. a Strat.) = [(chance of win) * (resource value - cost of win)] + [(chance of loss) * cost of loss]

Action	Benefit or Cost (arbitrary units)
Gain Resource	+ 50
Lose Resource	0
Injury to Self	- 100
Cost of Display to Self	- 10

Payoff to Hawk when vs. Hawk:

Eq. 2: E(H,H) = (0.5 * 50) + (0.5 * -100) = 25 - 50 = -25

Payoff to Hawk when vs. Dove:

Eq. 3: E(H,D) = 1.0 * 50 - 0 = +50

Payoff to Dove when vs. Hawk:

Eq. 4: E(D,H) = 0 * 50 + 1.0 * 0 = 0

Payoff to Dove when vs. Dove:

Eq. 5: E(D,D) = (0.5) * (50 - 10) - (0.5) * (-10) = +15

Here is the payoff matrix for this particular version of the Hawk vs. Dove game:

	Opponent
Focal Strategy	Hawk	Dove
Hawk	-25	+50
Dove	0	+15

the fitness of Hawk, W(H), is:

Eq. 8: W(H) = h * E(H,H) + (1-h) * E(H,D)

the fitness of Dove, W(D) is:

Eq. 9: W(D) = h * E(D,H) + (1-h) * E(D,D)