# Recursive definition of fitness

Evolutionary theory predicts fitness maximization by organisms over a large enough number of generations. Fitness is described as the ability to survive and reproduce given the environment, but I have not seen a formal definition of fitness even in mathematical models of adaptation.

A direct definition of fitness is difficult to give. Fitness is not the number of offspring, because then a fitness-maximizing organism would accept the following trade: add one child and remove the reproductive ability of all your children. Similarly, fitness is not the number of grandchildren or grand-to-the-n-children, because if it was, the reproductive ability of the grandchildren would be traded away for one more grandchild. If fitness was the number of fertile offspring surviving to adulthood, this trade could be shifted by one generation: add one adult fertile child and remove the reproductive ability of grandchildren or all descendants in a more distant generation. Clearly one has to take into account all descendants in the infinite future in some way.

Fitness can be defined recursively: it is the sum of the fitnesses of the offspring (multiplied by some positive constant). The fitness of each child in turn is the sum of the fitnesses of that child’s offspring, their fitnesses are the sums of the fitnesses of their offspring, and so on to infinity.

Under this definition, the trade described initially would not be made: changing the fitnesses of all descendants in some generation to zero would not be accepted, no matter what is offered in return (increasing the fitness of more distant descendants cannot be part of the same trade). Increasing the summary fitness of descendants in some more distant generation, other things equal, is inconsistent with reducing the fitness of descendants in a nearer generation. This is because if the fitness of an organism increases, other things equal, then the fitness of all that organism’s ancestors increases.

Under multilevel selection, as in Reeve and Holldöbler 2007, there is more than one fitness concept. There is individual fitness and group fitness, if both individuals and groups reproduce. The definitions of these fitnesses are also recursive, but more complicated, since fitness at one level of reproduction will interact with the fitness at another.