Question

Consider a game of coexistence where everyone exhibits two kinds of behaviour: ‘Hawk’ and ‘Dove’. The possible set of payoffs is given as follows:

Hint:

In mixed strategy Nash equilibrium, equate the expected payoffs of all pure strategies to find the equilibrium probability.

Answer 1

Correct Answer: 0.5

Step 1: Set up the mixed strategy idea.
Let a fraction $p$ of the population play Hawk and $1-p$ play Dove. At a mixed equilibrium a player must be indifferent between the two, so the expected payoff from Hawk equals that from Dove.

Step 2: Expected payoff from Hawk.
Against Hawk the payoff is $-2$, against Dove it is $4$, so
\[ E(H)=-2p+4(1-p)=4-6p \]

Step 3: Expected payoff from Dove.
Against Hawk it is $0$, against Dove it is $2$, so
\[ E(D)=2(1-p)=2-2p \]

Step 4: Make them equal.
\[ 4-6p=2-2p\;\Rightarrow\;2=4p\;\Rightarrow\;p=0.5 \]

Step 5: Read the result.
So half the population plays Hawk at equilibrium.
\[ \boxed{0.5} \]