Price Equation

The Price equation is a mathematical identity, derived by George Price in 1970, that describes how any trait changes in frequency across a generational transition. It is not a theory with empirical content but a formal decomposition: it partitions the total change in trait frequency into a covariance term (selection) and an expectation term (transmission bias, drift, or developmental change). It is exact and universal — it applies to any population of replicating entities, whether biological, cultural, or abstract.

The equation's canonical form is:

<math>w\Delta\bar{z} = \text{Cov}(w_i, z_i) + E(w_i \Delta z_i)</math>

where <math>w</math> is mean fitness, <math>\bar{z}</math> is mean trait value, <math>w_i</math> is individual fitness, <math>z_i</math> is individual trait value, and <math>\Delta z_i</math> is the change in trait value within an individual across the transition.

The first term captures selection: positive covariance between fitness and trait means the trait is being selected for. The second term captures everything else: mutation, recombination, developmental noise, and within-individual change.

The multi-level selection extension applies the Price equation recursively. If individuals are grouped, the covariance term can be further decomposed into within-group covariance (individual selection) and between-group covariance (group selection). This decomposition is exact — it does not assume that one level is more 'fundamental' than another. The debate about whether group selection or kin selection is the correct framework for explaining altruism is, from the Price equation's standpoint, a debate about bookkeeping rather than causation.

George Price himself found the implications of his equation disturbing. A converted atheist who used the mathematics of altruism to seek a proof of Christian self-sacrifice, he gave away his possessions to homeless people in London, became destitute, and died by suicide in 1975. His equation outlived him; his papers were recovered from a squat. The gap between the precision of a formal result and the life of the person who derived it is rarely so brutal.