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:
Consider a game of coexistence where everyone exhibits two kinds of behaviour: ‘Hawk’ and ‘Dove’. The possible set of payoffs is given as follows:
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In mixed strategy Nash equilibrium, equate the expected payoffs of all pure strategies to find the equilibrium probability.
Updated On: Jun 5, 2026
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Correct Answer: 0.5
Solution and Explanation
Step 1: Understand the concept of mixed strategy Nash equilibrium.
In a mixed strategy Nash equilibrium, players randomize between available strategies in such a way that the opponent becomes indifferent between choosing either strategy.
Here, the two strategies are
Hawk and
Dove
Let the fraction of the population choosing Hawk be
\[ p \] Then, the fraction choosing Dove is
\[ 1-p \]
At equilibrium, the expected payoff from choosing Hawk must be equal to the expected payoff from choosing Dove.
Step 2: Compute the expected payoff from choosing Hawk.
If a player chooses Hawk:
- Against Hawk (probability \(p\)), payoff is \(-2\).
- Against Dove (probability \(1-p\)), payoff is \(4\).
Therefore, expected payoff from Hawk is
\[ E(H)=p(-2)+(1-p)(4) \]
Simplifying,
\[ E(H)=-2p+4-4p \] \[ E(H)=4-6p \]
Step 3: Compute the expected payoff from choosing Dove.
If a player chooses Dove:
- Against Hawk (probability \(p\)), payoff is \(0\).
- Against Dove (probability \(1-p\)), payoff is \(2\).
Therefore, expected payoff from Dove is
\[ E(D)=p(0)+(1-p)(2) \]
Simplifying,
\[ E(D)=2-2p \]
Step 4: Apply the Nash equilibrium condition.
At mixed strategy Nash equilibrium,
\[ E(H)=E(D) \]
Therefore,
\[ 4-6p=2-2p \]
Bring all \(p\)-terms to one side:
\[ 4-2=6p-2p \] \[ 2=4p \]
Hence,
\[ p=\frac{2}{4} \] \[ p=\frac{1}{2} \] \[ p=0.5 \]
Step 5: Verify the equilibrium interpretation.
At
\[ p=0.5 \] the expected payoff from playing Hawk and Dove becomes identical.
Thus, no player has an incentive to unilaterally change behaviour.
Hence, this value satisfies the mixed strategy Nash equilibrium condition.
Step 6: Final conclusion.
Therefore, the population fraction exhibiting Hawkish behaviour at Nash equilibrium is
\[ \boxed{0.5} \]
In a mixed strategy Nash equilibrium, players randomize between available strategies in such a way that the opponent becomes indifferent between choosing either strategy.
Here, the two strategies are
Hawk and
Dove
Let the fraction of the population choosing Hawk be
\[ p \] Then, the fraction choosing Dove is
\[ 1-p \]
At equilibrium, the expected payoff from choosing Hawk must be equal to the expected payoff from choosing Dove.
Step 2: Compute the expected payoff from choosing Hawk.
If a player chooses Hawk:
- Against Hawk (probability \(p\)), payoff is \(-2\).
- Against Dove (probability \(1-p\)), payoff is \(4\).
Therefore, expected payoff from Hawk is
\[ E(H)=p(-2)+(1-p)(4) \]
Simplifying,
\[ E(H)=-2p+4-4p \] \[ E(H)=4-6p \]
Step 3: Compute the expected payoff from choosing Dove.
If a player chooses Dove:
- Against Hawk (probability \(p\)), payoff is \(0\).
- Against Dove (probability \(1-p\)), payoff is \(2\).
Therefore, expected payoff from Dove is
\[ E(D)=p(0)+(1-p)(2) \]
Simplifying,
\[ E(D)=2-2p \]
Step 4: Apply the Nash equilibrium condition.
At mixed strategy Nash equilibrium,
\[ E(H)=E(D) \]
Therefore,
\[ 4-6p=2-2p \]
Bring all \(p\)-terms to one side:
\[ 4-2=6p-2p \] \[ 2=4p \]
Hence,
\[ p=\frac{2}{4} \] \[ p=\frac{1}{2} \] \[ p=0.5 \]
Step 5: Verify the equilibrium interpretation.
At
\[ p=0.5 \] the expected payoff from playing Hawk and Dove becomes identical.
Thus, no player has an incentive to unilaterally change behaviour.
Hence, this value satisfies the mixed strategy Nash equilibrium condition.
Step 6: Final conclusion.
Therefore, the population fraction exhibiting Hawkish behaviour at Nash equilibrium is
\[ \boxed{0.5} \]
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