Cooperation vs. Defection: Invasion Resistance

The Logic of Invasion and Stability

Before we dive into specific models, let's refine our intuition about what makes a strategy evolutionarily stable. The core idea, as your background in dynamic systems would suggest, is about reaching a state that is robust to perturbations. In evolution, these perturbations are "mutants"—individuals playing a different strategy.

Two short videos offer excellent conceptual framing for this. The first, featuring Richard Dawkins, uses the "fishers and pirates" analogy to explain the idea.

The second video formalizes this by breaking down the characteristics of a successful, stable strategy.

This two-part condition is the essence of analyzing invasion resistance. For a strategy to become an ESS, it must first be a successful invader, and then it must become a successful resistor.

Case Study 1: Invasion Conditions for "Dads vs. Cads"

Let's apply this invasion logic to a concrete model. In the previous lesson, we briefly met the "Dads vs. Cads" model. Here, "Dads" are cooperators who provide parental investment, while "Cads" are defectors who eschew investment to seek more mating opportunities. This provides a clean analogy for committed, investing partners versus opportunistic, non-investing ones.

We will now examine the mathematical conditions under which one strategy can invade the other.

The key insight from this analysis is that invasion is not a foregone conclusion. The ability of cooperators (Dads) to invade a population of defectors (Cads) depends on a clear cost-benefit calculation: the fitness benefit from cooperation ($f$) must be sufficiently large to outweigh the mating opportunities lost by not defecting ($c$).

Conversely, defectors can invade cooperators when the benefits of cooperation are spread too thinly (large group size $N$), or when the temptation to defect is high (large $c$). When the conditions for invasion are met for both strategies (i.e., Dads can invade Cads, and Cads can invade Dads), neither can achieve full dominance. The result is a stable mixture of both strategies in the population—a polymorphism, just as Dawkins suggested with the 80/20 fisher/pirate ratio.

Case Study 2: Hidden Strategies, Policing, and Cyclical Invasions

The "Dads vs. Cads" model assumes that strategies are known. But what about a core feature of your area of interest—cuckoldry and extra-pair mating (EPM)? Here, the defection is, by design, a hidden strategy. This secrecy fundamentally changes the game.

A fascinating paper by Damle et al. models exactly this scenario. It defines cooperation as genetic monogamy and defection as engaging in EPM. It then introduces more complex, realistic strategies like mate guarding, social policing, and even opportunistic blackmail.

This model reveals a much more complex and dynamic reality. When the cost of being punished for defection is greater than the benefit of defecting, no single strategy is stable. Instead, the population is subject to constant cyclical invasions, like a game of Rock-Paper-Scissors:

1. Defectors invade Cooperators: In a population of simple, trusting cooperators (Monogamous, Non-Guarding, Non-Policing: M+G-P-), a defector (M-) can easily prosper.
2. Policers invade Defectors: Once defectors are common, the population becomes a ripe environment for individuals who police this behavior. A strategy that defects but also guards and polices (M-G+P+) will do well, as it can continue to defect while punishing others.
3. Cooperative Policers invade Defecting Policers: A population full of policing defectors is a high-risk environment with frequent costly punishments. A cooperative policer (M+G+P+) can now invade. This strategy benefits from the policed environment but avoids the risk of being caught and punished for its own defections.
4. Simple Cooperators invade Cooperative Policers: Finally, once the population is dominated by cooperative policers, defection is rare. The costly machinery of guarding and policing is no longer necessary. A simple cooperator (M+G-P-) who doesn't pay those costs can now invade, bringing the cycle back to the beginning.

The result is not a single ESS, but an oscillating polymorphism where cooperators, defectors, and various policing strategies coexist in a dynamic, fluctuating balance. This provides a powerful theoretical explanation for the persistent diversity of mating strategies observed in human populations.

Visualizing Evolutionary Trajectories

These invasion dynamics can be visualized. The plots below, from a study on the co-evolution of strategies and payoffs, show how the frequencies of different strategies change over time.

These graphs illustrate how the proportion of different strategies (A, C) and the nature of the game itself (B, D) can change over evolutionary time. The first phase shows evolution with fixed payoffs, while the second phase allows payoffs and strategies to co-evolve. Source: Stewart & Plotkin (2014), PNAS.

Look at graph A. You can see an initial phase where one strategy ("self-defect," in red) rises to prominence, invading and replacing the others. However, after time $1\times 10^5$, the conditions change. The dominant "self-defect" strategy begins to decline as it is invaded by the "self-alternate" strategy (purple), which then rises toward fixation. This mirrors the kind of cyclical or sequential invasion we discussed in the Rock-Paper-Scissors dynamic. As one strategy becomes common, it changes the environment, potentially making it vulnerable to invasion by another.

Graph B shows the consequence of these strategy shifts. As the "self-defect" strategy rises, the game being played in the population is a Prisoner's Dilemma. However, as the "self-alternate" strategy successfully invades, the game itself is transformed into a Snowdrift Game (a version of Hawk-Dove/Chicken), which has entirely different strategic properties. This illustrates a profound point: the process of invasion doesn't just change who wins; it can change the rules of the game itself.

Conclusion

Today we have moved from a static view of stability to a dynamic one centered on the process of invasion. Analyzing the invasion resistance of strategies is the key to understanding which behaviors are likely to persist in a population.

• Key Takeaways:
  • An Evolutionarily Stable Strategy must be able to resist invasion by any alternative "mutant" strategy when it is common in the population.
  • The ability of a strategy (e.g., cooperation) to invade another (e.g., defection) depends on a cost-benefit analysis determined by the game's parameters, such as payoffs and group size.
  • In complex scenarios with hidden information and countermeasures like policing, a single ESS may not exist. Instead, we can observe cyclical invasions in a Rock-Paper-Scissors dynamic.
  • This dynamic leads to polymorphism: a stable, but often oscillating, mix of different strategies coexisting within the population. This provides a robust explanation for the diversity of behaviors we see in the real world.

Preview of the next lesson: So far, we have analyzed the evolution of strategies used by individuals within a given population structure. But what about the stability of the structure itself? In our next and final lesson of this module, we will assess the evolutionary stability of the vee configuration by analyzing conditions under which the structure persists or dissolves. We will ask: can a vee relationship structure itself resist "invasion" from pressures that would break it apart or transform it into a different form, like a triad?