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proud sycamore and a young weeping willow. Too weak or too ill or too heartbroken to follow this path peacefully, perhaps it was his ultimate and enduring message to the world he left behind.

Appendix 1: Covariance and Kin Selection 1
     
    Price did not himself show how Hamilton’s coefficient of relatedness in the narrow sense of common descent could be replaced with association. The first to do that was Hamilton in his 1970 paper. Later Queller expanded on this in a 1985 paper, “Kinship, Reciprocity and Synergism in the Evolution of Social Behaviour,” clarifying further (if it had yet to be entirely clear) that association need not mean genetic relatedness in the narrow sense of descent via direct replication. 2
    To see how the covariance equation translates into Hamilton’s kin-selection equation, we begin with the equation wg = Cov ( w, g ) where g is the breeding value that determines the level of altruism. The least-squares multiple regression that predicts fitness, w, can be written as

     
    where g´ is the average g value of an individual’s social neighbors,is a constant, andis the residual which is uncorrelated with g and g´. Theare partial regression coefficients that summarize costs and benefits:is the effect an individual’s breeding value has on its own fitness in the presence of neighbors’ g´ (that is, the cost of altruism), andis the effect of an individual’s breeding value on the fitness of its neighbors (that is, the benefit of altruism). Substituting into the covariance equation and solving for the condition under which wg > 0 gives us Hamilton’s inclusive fitness equation ( r B > C) in the following form

     
    whereis the regression coefficient of relatedness.
    This derivation, performed by Hamilton in 1970 3 following Price, clearly shows that it is natural to use statistical association instead of common descent. It is considered the first modern theoretical treatment of inclusive fitness.
    Among other things, it allows us to see that spiteful behavior, where an organism acts in such a way as to harm itself in order to harm another organism even more, can evolve since the product of a negative relatedness and a negative benefit to the recipient (harm) is positive, meaning that benefit multiplied by relatedness can outweigh the cost.

Appendix 2: The Full Price Equation and Levels of Selection 1
     
    Unbeknownst to George Price in 1968, the simple covariance relationship
     
     
    (1) w z = Cov (w, z)
     
     
    had already been worked out and published independently by two other men, Robertson and Li, in 1966 and 1967 respectively. 2 But the uniqueness of the full Price equation stems from the inclusion of the further, expectation term. Here is the derivation:
    Let there be a population in which each element is labeled by an index i . The frequency of elements with index i is q i , and each element with index i has some character, z i . Elements with a common index form a subpopulation that comprises a fraction q i of the total population, and no restrictions are placed on how the elements are grouped.
    Now imagine a second, descendant, population with frequencies q´ i and characters z´ i . The change in the average character value,between the mother and descendant populations is

     
    This equation applies to anything that evolves because z can be defined however one likes. For this reason, the equation applies not only to genetics, but to any selection process whatsoever.
    What is special about the Price equation is the way in which it associates statistically between entities in groups, a “mother” and “daughter” population. Instead of the value of q i obtaining from the frequency of elements with index i in the daughter population, it obtains from the proportion of the daughter population derived from the elements in with index i in the mother population. If we define the fitness of the element i as w i , the contribution to the daughter population from type i in the mother