# Altruism

Altruism is selfish.

"The work of Kruuk (1964) has been reinforced by further studies, summarized by Lack (1968). The latter seem to have shown that all marginal nests failed to rear young, mainly due to predation. Perhaps, nevertheless, the gulls that could not get places in the centre of the colony were right in nesting on the edge rather than in isolation where, for a conspicuous bird like a gull, the chances would have been even worse. " Hamilton, William D. "Geometry for the selfish herd." Journal of theoretical Biology 31.2 (1971): 295-311.

## Example: Evolution of altruism

To study the evolution of a genetic predisposition to altruism, altruism will be defined as the genetic predisposition to behavior which decreases individual fitness while increasing the average fitness of the group to which the individual belongs. First specifying a simple model, which will only require the simple Price equation. Specify a fitness wi by a model equation:

$n'_i}{n_i$

where zi is a measure of altruism, the azi term is the decrease in fitness of an individual due to altruism towards the group and bz is the increase in fitness of an individual due to the altruism of the group towards an individual. Assume that a and b are both greater than zero. From the Price equation:

$var$

where var(zi) is the variance of zi which is just the covariance of zi with itself:

$var}(z_i) \;\stackrel{\mathrm{def}}{=}\; \operatorname{E}(z_i^2) - \operatorname{E$

It can be seen that, by this model, in order for altruism to persist it must be uniform throughout the group. If there are two altruist types the average altruism of the group will decrease, the more altruistic will lose out to the less altruistic.

Now assuming a hierarchy of groups which will require the full Price equation. The population will be divided into groups, labelled with index i and then each group will have a set of subgroups labelled by index j. Individuals will thus be identified by two indices, i and j, specifying which group and subgroup they belong to. nij will specify the number of individuals of type ij. Let zij be the degree of altruism expressed by individual j of group i towards the members of group i. Let's specify the fitness wij by a model equation:

$ij} = \frac{n'_{ij}}{n_{ij}} = k - a z_{ij$

The a zij term is the fitness the organism loses by being altruistic and is proportional to the degree of altruism zij that it expresses towards members of its own group. The b zi term is the fitness that the organism gains from the altruism of the members of its group, and is proportional to the average altruism zi expressed by the group towards its members. Again, in studying altruistic (rather than spiteful) behavior, it is expected that a and b are positive numbers. Note that the above behavior is altruistic only when azij >bzi. Defining the group averages:

align} n_i &= \sum_j n_{ij} \\ z_i &= \frac{1}{n_i}\sum_j z_{ij}n_{ij} \\ w_i &= \frac{1}{n_i}\sum_j w_{ij}n_{ij} = k + (b - a)z_i \\ n_i'&= \sum_j n_{ij}' = n_i[k + (b - a)z_i] \\ z_i'&= \frac{1}{n_i'}\sum_j z_{ij}n_{ij}' \end{align

and global averages:

align} n &= \sum_{ij} n_{ij} = \sum_i n_i \\ z &= \frac{1}{n}\sum_{ij} z_{ij}n_{ij} = \frac{1}{n}\sum_i z_in_i \\ w &= \frac{1}{n}\sum_{ij} w_{ij}n_{ij} = \frac{1}{n}\sum_i w_in_i \\ n'&= \sum_{ij} n_{ij}' = \sum_i n_i' \\ z'&= \frac{1}{n'}\sum_{ij} z_{ij}n_{ij}' = \frac{1}{n'}\sum_i z_i'n_i' \end{align

It can be seen that since the zi and zi are now averages over a particular group, and since these groups are subject to selection, the value of Δzi = zizi will not necessarily be zero, and the full Price equation will be needed.

$cov}\left(\frac{w_i}{w}, z_i\right) + \operatorname{E}\left(w_i\,\Delta \frac{z_i}{w$

In this case, the first term isolates the advantage to each group conferred by having altruistic members. The second term isolates the loss of altruistic members from their group due to their altruistic behavior. The second term will be negative. In other words, there will be an average loss of altruism due to the in-group loss of altruists, assuming that the altruism is not uniform across the group. The first term is:

$cov}\left(\frac{w_i}{w}, z_i\right) = \left(b - a\right)\operatorname{var$

In other words, for b>a there may be a positive contribution to the average altruism as a result of a group growing due to its high number of altruists and this growth can offset in-group losses, especially if the variance of the in-group altruism is low. In order for this effect to be significant, there must be a spread in the average altruism of the groups.