Modeling Conflict with Hawk-Dove in Vee Relationships

The Hawk-Dove Game: A Model of Conflict

You are likely already familiar with the Hawk-Dove game from your game theory studies, but a brief refresher will be useful. It's a classic model of a conflict over a shared resource, where two players can choose between an aggressive strategy (Hawk) and a passive one (Dove). It's also known as the "Game of Chicken," and it beautifully captures the tension between wanting to win and wanting to avoid mutual destruction.

To set the stage, let's watch a short video that explains the classic setup and payoffs.

The key insight from the classic game is that when the cost of a Hawk-Hawk fight is greater than the value of the resource, the stable outcomes involve one player acting as Hawk and the other as Dove. It's a game of anti-coordination. This basic structure is a useful starting point for modeling conflicts within a vee.

Imagine two non-hinge partners, P1 and P2, in a vee with a hinge partner, H. A conflict arises over a resource—for instance, who gets to spend a special holiday with H. We can map the strategies as follows:

• Hawk Strategy ("Compete"): Insist on getting your way. Escalate the conflict if necessary, creating stress for all three individuals.
• Dove Strategy ("Accommodate"): Concede to the other partner's preference to maintain peace and harmony within the vee.

The payoffs are:

• Value of the resource (V): The benefit of winning the conflict (e.g., getting the holiday time).
• Cost of fighting (C): The negative consequence of a mutual Hawk-Hawk escalation. This isn't just a lost holiday; it's emotional distress, damage to the relationships, and potential destabilization of the entire vee structure. We assume $C>V$.

The symmetric payoff matrix would look like this, where P1's payoffs are listed first:

Introducing Asymmetry: Resource Holding Potential

The classic model's biggest limitation is its symmetry. It assumes both players are identical. In reality, partners in any relationship, including a vee, are rarely on perfectly equal footing. One partner might have been with the hinge longer, have more financial influence, or simply possess more emotional leverage.

We can capture this asymmetry using the concept of Resource Holding Potential (RHP). RHP is a measure of an individual's ability to win a contested resource. In the context of a vee, a partner's RHP could be derived from:

• Relationship longevity or status (e.g., a pre-existing spouse).
• Financial contributions or dependence.
• Shared responsibilities, like children or a mortgage.
• The hinge's emotional or logistical dependence on them.
• A more dominant or assertive personality.

This asymmetry doesn't change the payoffs for Dove-Dove, Dove-Hawk, or Hawk-Dove interactions. However, it fundamentally changes the Hawk-Hawk outcome. Instead of an equal chance of winning, the partner with higher RHP is more likely to prevail. This transforms the game.

A paper by Hall, Porter, and Dawkins provides an excellent framework for analyzing this modified, asymmetric Hawk-Dove game.

As you've just read, the crucial insight is that the nature of the conflict is not fixed. It depends on the relative power of the participants.

For example, consider a scenario from the perspective of the weaker partner (low RHP, low $p_A$). If their RHP is so low that their expected payoff from fighting is worse than getting nothing (${\mu }_A<0$), but the stronger partner's expected payoff is positive (${\mu }_B>0$), the game might become a PD-Deadlock. The weaker partner faces a Prisoner's Dilemma-like choice, while the stronger partner is in a "Deadlock" situation where Hawk is their dominant strategy regardless of what the other does. This asymmetry is what drives the emergence of complex, stable strategies.

Finding the Evolutionarily Stable Strategy (ESS)

So, in this population of non-hinge partners with varying RHPs, what strategy is evolutionarily stable? Is it always "Compete" (Hawk), always "Accommodate" (Dove), or something else?

The Hall et al. paper investigates this by simulating tournaments between different strategies. A key finding is that a simple, unconditional strategy is rarely the winner. Instead, a conditional strategy proves to be remarkably robust and often emerges as the ESS. This strategy can be summarized as:

Play Hawk if you have a higher RHP than your opponent; play Dove if you have a lower RHP.

This is the strategic logic of a dominance hierarchy. The "stronger" individual asserts themselves, and the "weaker" individual yields. This outcome is stable because it allows both parties to avoid the mutually destructive cost of a Hawk-Hawk fight. The weaker partner avoids a costly, likely loss, and the stronger partner secures the resource without a fight.

Another paper, by Soltis and McElreath, explores a similar dynamic in the context of paternal investment. Their "Dads vs. Cads" model is analogous to our Dove vs. Hawk setup.

The "Dads vs. Cads" model reinforces the core idea: the ESS is not fixed but depends on the parameters of the environment. In their model, the Dad (Dove) strategy is more likely to be stable when the benefits of cooperation are high and the group size is small. In our vee model, this translates to the "Accommodate" strategy being more viable when the "cost of conflict" (C) is very high relative to the "value of the resource" (V).

Interpreting the ESS in Vee Relationships

What do these findings tell us about the long-term dynamics of vee relationships?

1. The Stability of Dominance Hierarchies: The model predicts that if there are significant, recognizable asymmetries in RHP between non-hinge partners, the most stable outcome is a dominance hierarchy. One partner consistently adopts the "Hawk" role, and the other adopts the "Dove" role. This explains why many polyamorous relationships, even those aspiring to be egalitarian, can settle into a stable pattern where one partner's needs and preferences are implicitly prioritized to avoid conflict. The structure is stable not because it's "fair," but because any other strategy is evolutionarily unstable—it leads to lower payoffs (fitness) for at least one player who would then be better off switching to the hierarchical strategy.

2. The Fragility of "Sharing": The Hall et al. paper also finds that resource sharing (mutual Dove-playing) can be a stable outcome, but only under limited conditions: when fighting costs are low and, crucially, when partners have very similar RHPs. In a vee context, this suggests that a truly egalitarian, cooperative dynamic between non-hinge partners is most likely to be stable when they perceive each other as being on equal footing and when the cost of escalating conflict is not seen as catastrophic.

3. Continual Fights: The model also allows for a third outcome: continual fighting. This occurs when the RHP difference is moderate. The stronger player still has an incentive to play Hawk, but the weaker player's expected payoff from fighting, while negative, might still be better than getting nothing (the Dove's payoff against a Hawk). This can lead to a stable but costly pattern of repeated conflict.

To check the stability of these outcomes, we can use the logic from the video on ESS you saw in the last lesson. Imagine a population at the "dominance hierarchy" ESS. If a "mutant" Dove appears who decides to play Hawk against a stronger opponent, they will suffer a costly loss, reducing their fitness and preventing the mutation from spreading. Similarly, if a "mutant" Hawk decides to play Dove against a weaker opponent, they give up a resource they could have easily won, again reducing their fitness relative to the population. The conditional strategy is therefore stable against invasion.

Conclusion

By adapting the simple Hawk-Dove game to include the crucial element of asymmetry (RHP), we have developed a powerful model for understanding conflict dynamics in vee relationships. This approach moves beyond a static analysis of a single interaction and provides an evolutionary perspective on which behavioral patterns are likely to persist over time.

• Key Takeaways:
  • The classic Hawk-Dove game can be adapted to model conflict between non-hinge partners by mapping Hawk to a "Compete" strategy and Dove to an "Accommodate" strategy.
  • Introducing asymmetry via Resource Holding Potential (RHP) makes the model more realistic, as the nature of the conflict changes depending on the power differential between partners.
  • A conditional strategy—"Play Hawk if you have higher RHP, Dove if you have lower"—often emerges as the Evolutionarily Stable Strategy (ESS).
  • This ESS corresponds to a dominance hierarchy, a stable but unequal relationship dynamic where one partner consistently defers to the other to avoid costly conflict. This provides a game-theoretic explanation for the emergence of implicit hierarchies in some vee relationships.
  • Other outcomes, like mutual accommodation ("sharing") or continual conflict, are also possible, but the model predicts they are stable under more limited conditions.

Preview of the next lesson: In this lesson, we've identified an ESS by analyzing its resistance to invasion. In the next lesson, we will formalize and generalize this idea. The goal will be to analyze the invasion resistance of cooperative versus defecting strategies in a population of potential partners, looking more broadly at the conditions required for cooperation to survive and thrive in an evolutionary context.