Constructing the Hawk-Dove Payoff Matrix

The Story: From 'Chicken' to Hawks and Doves

The Hawk-Dove game is a stylized model of conflict over a shared resource. It's often introduced with a more visceral story: the "Game of Chicken."

To get an intuitive feel for the setup, please watch the first two minutes of this video. It describes the classic "Game of Chicken" and then connects it to the Hawk-Dove model used in biology.

As the video explains, this scenario isn't just about reckless teenagers; it's a powerful model for any situation involving a contest where one party can choose to be aggressive (Hawk) or back down (Dove).

To build our model, we need to formalize these behaviors. We'll use a resource from Holy Cross College that provides a detailed description.

From this reading, we can summarize the core rules of engagement:

• Hawk: Always fights. Will escalate until it is injured or the opponent retreats.
• Dove: Never fights. Displays, but retreats immediately if attacked.

Now, let's translate these behaviors into mathematical payoffs.

Formulating the Payoffs

To create a payoff matrix, we need to assign quantitative values to the outcomes. In evolutionary game theory, these "payoffs" represent changes in an individual's fitness.

The two most important parameters are:

• V (Value): The value of the resource being contested. Winning the resource increases fitness by $V$.
• C (Cost): The cost of injury from a fight. Being injured decreases fitness by $C$.

A critical assumption in the standard Hawk-Dove game is that the cost of injury is greater than the value of the resource ($C>V$). This is what makes the game interesting; a fight is a net loss in expected value.

The General Payoff Formula

We can calculate the expected payoff for any given contest using a general formula. The text you just read presents it as follows:

Expected Payoff = (chance of win) × (value of win) + (chance of loss) × (cost of loss)

Let's use this logic to calculate the expected payoff for our "focal" player in each of the four possible encounters.

1. Hawk vs. Hawk (E(H,H))

• Narrative: Two Hawks meet. They fight until one is injured.
• Assumptions: They are equally matched, so the chance of winning is 50% (0.5). The winner gets the resource ($V$), and the loser sustains an injury ($-C$).
• Calculation:
  $E(H,H)=(0.5\times V)+(0.5\times -C)=\frac{V-C}{2}$
  This is the expected outcome for a Hawk fighting another Hawk.

2. Hawk vs. Dove (E(H,D))

• Narrative: A Hawk meets a Dove. The Dove immediately retreats.
• Assumptions: The Hawk wins the resource with 100% certainty. No fight occurs, so there is no cost of injury.
• Calculation:
  $E(H,D)=1.0\times V=V$
  The Hawk gets the full value of the resource.

3. Dove vs. Hawk (E(D,H))

• Narrative: A Dove meets a Hawk. The Dove retreats.
• Assumptions: The Dove wins with 0% certainty. Since it retreats before a fight, it avoids injury.
• Calculation:
  $E(D,H)=0.0\times V=0$
  The Dove gets nothing but also loses nothing.

4. Dove vs. Dove (E(D,D))

• Narrative: Two Doves meet. They engage in a display, but do not fight.
• Assumptions: They are equally matched, so they effectively split the resource. Each has a 50% chance of the other backing down first after a brief, low-cost display. We'll follow the simplest model where they share the resource equally without cost.
• Calculation:
  $E(D,D)=V/2$
  Each Dove gets half the value of the resource.
  (Note: More complex models, like the one in the Holy Cross resource, might also include a small cost for the display itself, but we will use this simpler, standard formulation).

The Complete Payoff Matrix

By combining these four calculations, we can construct the general payoff matrix for the Hawk-Dove game. The payoffs shown are for the row player.

This algebraic matrix is the fundamental model of the Hawk-Dove game. It's the same structure you'll find in many biophysics and economics textbooks.

A Numerical Example

To make this concrete, let's plug in the numerical values used in the Holy Cross resource:

• Value of resource, $V=50$
• Cost of injury, $C=100$

Let's calculate the payoffs:

• $E(H,H)=(50-100)/2=-25$
• $E(H,D)=50$
• $E(D,H)=0$
• $E(D,D)=50/2=25$ (The Holy Cross resource gets a value of +15 here because it includes a display cost of -10. Our value of +25 comes from the simplified model without display costs).

This gives us the following numerical payoff matrix:

You can now see the strategic dilemma clearly. Playing Hawk is tempting because of the high payoff against a Dove (50), but it's disastrous against another Hawk (-25). Playing Dove is safe, but it yields a lower payoff.

Conclusion

In this lesson, we deconstructed the narrative of the Hawk-Dove game and translated it into a formal, mathematical model. This process of moving from a qualitative story to a quantitative matrix is a cornerstone of game-theoretic analysis.

Key Takeaways:

• The Hawk-Dove game models a conflict over a resource where players can be aggressive (Hawk) or passive (Dove).
• The payoffs are determined by two key parameters: the value of the resource ($V$) and the cost of injury ($C$).
• By systematically considering each possible interaction (H-H, H-D, D-H, D-D), we can derive the expected payoff for each outcome.
• The resulting payoff matrix, $\left(\begin{array}{cc}(V-C)/2& V\\ 0& V/2\end{array}\right)$, captures the fundamental tension of the game, especially when the cost of fighting is high ($C>V$).

In our next lesson, we will use this very matrix to analyze the Nash Equilibria and the Evolutionarily Stable Strategy (ESS) of the Hawk-Dove game. As we've hinted, the solution will not be a pure strategy but a stable mix of Hawks and Doves in the population.