Strategies, Payoffs, and the Matrix

    There are two basic strategies in the Hawk-Dove game as the name indicates. A hawk is an aggressive member of a population. It will fight for a resource until it loses. A Dove on the other hand is a non-aggressive member of the population. It will display, but never enter a physical confrontation. When a hawk meets a hawk, both hawks will fight until one of them loses and the other is left with the resource. The losing hawk incurs a fitness cost because of its wounds, energy loss, and so forth. When a hawk meets a dove, the dove will display, but when the it realizes it is facing a hawk that will attack, the dove retreats, not incurring any cost. The hawk is then left with the entire resource. When a dove meets a dove, both will display for a time, but neither will attack. Sooner or later, one will back down and the other will reap the rewards. However, both incur a display cost because of the energy needed to perform the display. Some game theorists do not factor in the display cost when analyzing this game (Cressman 1992, Weibull 1995).
    The resource can be any arbitrary value. In this case, we will assign it a value of 50. Many assumptions are made when we create the matrix game for these situations. 

Assumptions:

    1)  Resources and fitness points are the same at every encounter.

    2)  All hawks are equally strong.

    3)  All doves are equally good at displaying and waiting.

    4)  When a hawk meets a hawk, or a dove meets a dove, both have the same chance of gaining the resource.

    5)  When a dove displays to a hawk, and then retreats, it does not incur any cost.

    6)  The population is competing over the same limited resource.

The following section will list the different payoffs when a hawks and doves meet.

Hawk vs. Hawk - When two hawks meet, half of the time, one hawk wins, and the other half of the time, the other hawk wins.  The winning hawk will win the 50 fitness points and the other will incur a cost of 100 points due to damage from the battle.  Since both hawks have an equal chance of winning, we calculate the expected value of each encounter.  E(x) = 0.5(50) + 0.5(-100) = -25  Thus, each time a hawk meets a hawk, both are expected to lose 25 points on average.

Hawk vs. Dove - When a hawk meets a dove, the hawk will begin to attack and the dove will retreat.  The hawk will be left with the resource every time, winning 50 fitness points.  The dove will end up uninjured with no gain and no loss.  Thus, one will end up with 40 points, the the other with -10 points.  If we take the expected value, E(x) = 0.5(40) + 0.5(-10) = 15, we see that each will on average end up with 15 points.

Dove vs. Dove - When a dove meets a dove, both will display until one of them leaves.  There will be no conflict, and one of the doves will end up with the 50 points.  However, both doves incur a cost of 10 points because of the time and energy wasted displaying.

Thus, if we construct a payoff matrix, it will look like this:

    This is the basic hawk-dove game with the two basic strategies.  There are also other possible strategies that can be introduced into the matrix.  One of these strategies is called the Bully strategy.  In this situation, the animal will attempt to attack, and if it sees it is facing a hawk, it will retreat unharmed.  If it faces a dove, the dove will retreat and it will be left with the prize.   When a bully meets a bully, one will run away faster than the other, and one will be left unexpectedly with the resource.

   Another strategies is the Retaliator strategy.  The retaliator will remain passive like a dove unless the opponent attacks.  When an opponent attacks, it will attack with its full force until it wins or loses.  The following are the additional payoffs with these added encounters.

Bully vs. Hawk - When a bully faces a hawk, it will attempt to attack, but when it sees its opponent is a hawk, it will flee.  The hawk will be left with 50 fitness points every time, and the bully with 0.

Bully vs. Dove - When a bully faces a dove, it will attempt to attack and this time the dove will flee.  The bully will be left with 50 fitness points and the dove will lose nothing.

Bully vs. Bully - When a bully faces a bully, both attempt to attack, but then when they each see the other attack, both will attempt to flee.  One will escape faster than the other and the reward will be left unexpectedly to one.  Since each one is left with the resource half of the time, we can compute the expected value.  E(x) = 0.5(50) + 0.5(0) = 25  Thus, the average value for each bully is 25 points every encounter.

Retaliator vs. Hawk - When a retaliator faces a hawk, the hawk will attack first and then the retaliator will behave as a hawk and fight back.  The payoff is the same as a hawk vs. a hawk, -25 each.

Retaliator vs. Dove - When a retaliator faces a dove, both will remain passive.  The encounter will be exactly like the dove vs. dove scenario.  The payoff will be 15 points to each on average.

Retaliator vs. Bully - When a retaliator encounters a bully, the bully will start to attack, and when the retaliator responds, the bully will retreat.  The retaliator will be left with 50 points every time, and the bully with 0 points.

Retaliator vs. Retaliator - In this scenario, since both will not attack, the situation will be just like the situation between 2 doves.  The payoff will therefore be 15 points to each player.

The payoff matrix containing the additional strategies will look as follows.

We see that none of these strategies are dominant over any of the others meaning that no strategy has payoffs greater than the corresponding payoffs in all other strategies.

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