Hawk-Dove: The following game has been widely used in evolutionary biology to un

Question

Hawk-Dove: The following game has been widely used in evolutionary biology
to understand how fighting and display strategies by animals could
coexist in a population. For a typical Hawk-Dove game there are resources
to be gained (e.g., food, mates, territories), denoted as v. Each of two players
can choose to be aggressive, as Hawk (H), or compromising, as Dove (D).
If both players choose H then they split the resources but lose some payoff
from injuries, denoted as k. Assume that k > v/2
2 . If both choose D then they
split the resources but engage in some display of power that carries a display
cost d, with d < v/2
2 . Finally, if player i chooses H while j chooses D then i
gets all the resources while j leaves with no benefits and no costs.
a. Describe this game in a matrix.
b. Assume that v = 10, k = 6, and d = 4. What outcomes can be supported
as pure-strategy Nash equilibria?

please explain each step

Explanation / Answer

It is a prolem o game thoery. Two players are fighting for some resources. This resource may be food, mates or territories. the resource is indicated by v. I order to get this resource each of the player can go for two different strategies. Either they can be agrressive like Hawk. Alternatively they can adopt compromising strategies lke Dove. These two alternative strategies are indicated by H and D.

Now consider pay off of each player. Suppose both of them has decided to be agressive. In that case they will suffer injuries. For this some resources/pay off will be lost.This lost pay off is indicated by 'k'.Balance payof will be dostributed equally. Thus each of the player will get a payoff (v-k)/2.

Now suppose another combination where each player has decided to go for strategy D. In that case they have to display some power for which they have to incur cost of d. However rsource will be distibuted equally. Theefore pay off of each player is (v/2) -d.

Finally consider the situation when one player has adopted strategy H and other has adopted strategy D. In that case the player adopting strategy D will get the entire resources.v. Thus pay off matrix will be as follows.

Consider the above table. For player 1 best strategy is H. It will give full resource v to him. If he adopt H then player 2 will definitely go for strategy H also because he will not get anything if he chooses D. thus combination (H,D) is not possible. similarly combination (D,H) will not be possible as player 1 will get nothing.

So either both of them will go for H or for D. Which one will be better will depend upon the value of pay off. Here value v, k and d will decide the Nash equilibrium.

Now consider the conditions to be fulfilled.

1. k should be greater than v/2 and

2. d should be less than v/2.

Here v=10. So k should be more than 10/2 = 5. On the other hand d should be lass than v/2 i.e less than 5. The problem states that k=6 and d=4. Thus both conditions are satisfied. So substitute the value of v,d and k in the pay off table above. The table will be:

(10-6)/2, (10-6)/2

2,2

10,0

0,10

(10/2)-4,(10/2)-4

1,1

Now compare strategy H,H and D,D. here combination H,H will give higher pay off to both player. So they will settle at this strategy. It will Nash equilbtium as no one has an incentive to move to other option.

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