The Price equation

The viability of cooperation under multilevel selection can be illustrated using the Price equation⁶⁷. The Price equation assumes selective migration where groups with higher average payoffs will grow in size compared to those with lower payoffs. Denote q_j as the fraction of the population that is in group j. Let w_j = ∑_iw_ij/n_j be the average payoff group j, and w = ∑_jq_jw_j be the average population payoff.

The sizes of the groups change from one period to the next proportionally to their relative payoff. Denote as the fraction of the population in group j in the next period, then selective migration prescribes

Now consider the altruistic trait A which prescribes full contribution to the public good, and the selfish trait S which prescribes zero contribution. The Price Equation can be used to study any trait that can potentially influence individual payoffs. Because a cooperative action by definition benefits others at a personal cost, here cooperation and altruism are used interchangeably and we can thus simply call individuals with trait A (unconditional) cooperators. We use p_ij = 1 to represent that individual i in group j has the trait A and p_ij = 0 otherwise. Denoting p_j as A’s frequency in group j and p its population frequency of A, we can derive the following Price equation [refer to Supplementary Information for details]

Here the expectation and covariance are weighted by q_j. Because w is always positive, trait A increases in frequency as long as the right hand side is positive. The change in its population frequency can hence be partitioned into two parts: the two terms on the right hand size characterize inter-group and intra-group selection respectively.

Substituting the payoff represented by Equation (1) into the above equation, we obtain the Price equation in the context of public goods provision:

Note that the second term is always negative. In order for evolution to favor altruists, we need inter-group selection measured by var(p_j) to be strong compared to intra-group selection measured by E[var(p_ij)]. Assuming individuals are drawn to reproduce with probability equal to their share of the total group payoff, we must have w₀ ≫ 1 in Equation (1) to make it possible for cooperation to emerge. With strong selection w ≈ 0, the payoff of a defector in a group of cooperators is much higher, making it impossible for trait A to persist in the population.

The Price equation has great limitations. It only applies to systems with a relatively small number of traits (strategies). As we include more strategies, it quickly becomes intractable to characterize the system analytically. It helps identify equilibria but cannot provide the dynamics that lead to the equilibria. We know that group conflicts caused by limited resources were common in early human society, but the Price equation is abstracted away from direct group conflicts – considering selective extinction (where groups become extinct due to conflicts or natural disasters) makes it difficult to derive closed-form solutions. To overcome these limitations, we use agent-based modeling to provide insight into systems with interactions between a broad set of conditional strategies. In order to add realism to our model, instead of using Equation (2) to model group-level selection, we introduce direct group conflict; we also incorporate migration between groups into the evolutionary model.