In a hawk-dove game, if the benefit of winning is 10 and the cost of losing is
7, what is the best strategy to play, hawk or dove?

In contests modeled after the hawk-dove game, when the benefit is greater than
the cost, the best strategy is to always play hawk.

Problem :

If the Prisoner's Dilemma game is repeated multiple times, why should you not
always cheat?

The Prisoner's Dilemma, when repeated, offers a maximum reward if you mix up
your strategy. It is true that cheating offers a better reward no matter what
your opponent does, however, by mimicking your partner's behavior, you will
obtain the maximum reward. When your partner cheats, you should cheat in the
next game. When your partner cooperates, you should cooperate in the next game.

Problem :

Consider two breeding strategies of the fictional Woozle. Dominator Woozles can
fight for a breeding territory, and if they win, will be able to rear 10
offspring. An alternative is to share territory with another Woozle which will
allow each to rear 5 offspring. Sharers who attempt to share with dominators
will be forced out of the territory, although they will be able to find a new
territory. Assume sharers become extra catious after encountering a dominator
and so will always find another territory to share the next time around, but due
to lost time will only be able to produce 3 offspring. Dominators are always
able to force sharers out of the territory and rear 10 young. Dominators who
meet dominators will win 50% of the time. When they lose, they are not able to
reproduce that season due to sustained injuries. Individual Woozles cannot
switch strategies. What proportion of the population would you expect to be
dominators, and what proportion would be sharers?

If a dominator meets another dominator, he has a 50% chance of winning and
producing 10 offspring, and a 50% chance of losing which means a cost of 10
offspring. Therefore the average payoff for a dominator/dominator encounter is
zero. Dominators who meet sharers will win the territory and produce 10
offspring. Sharers who meet sharers will produce 5 offspring each. Sharers who
meet dominators will always produce 3 offspring at their new territory. Since
these are the only two mating strategies Woozles can choose from, we know that
the proportion of dominators (p) and sharers (q) will equal the entire
population. We then use our equations to estimate the relative payoffs for the
dominator strategy and the sharer strategy. We end up with the following two
equations:
p + q = 1
p(0) + q(10) = p(3) + q(5)
Solving for p and q yields:
p=5/8
q=3/8

5/8 of the population will be dominators and 3/8 will be sharers.