Beyond classical Hamilton's Rule. State distribution asymmetry and the dynamics of altruism

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• github.com/ReplicatorDynamics/Evolutionary-Games-and-Population-Dynamics/blob/67ab24dcc36bf85782726513b65bcb9647cc1c8e/HamiltonArxiv.sci
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Abstract

This paper analyzes the relationships between demographic and state-based evolutionary games and Hamilton's rule. It is shown that the classical Hamilton's rule (counterfactual method), combined with demographic payoffs, leads to easily testable models. It works well when the roles of donor and receiver are randomly drawn during each interaction event. This is illustrated by the alarm call example. However, we can imagine situations in which role-switching results from external mechanism, such as, fluxes of individuals between the border and the interior of the habitat, when only border individuals may spot the threat and warn their neighbors. To cover these cases, a new model is extended to the case with explicit dynamics of the role distributions among carriers of different strategies, driven by some general mechanisms. It is shown that even in the case when fluxes between roles are driven by neutral mechanisms (acting in the same way on all strategies), differences in mortality in the focal interaction lead to different distributions of roles for different strategies. This leads to a more complex rule for cooperation than the classical Hamilton's rule. In addition to the cost and benefit components, the new rule contains a third component weighted by the difference in proportions of the donors among carriers of both strategies. Depending on the sign, this component can be termed the "survival surplus", when the donors survival have greater survival than receivers, or the "sacrifice cost" (when it decreases the benefit), when the receiver's survival exceeds that of the helping donor. When we allow different role-switching rates for different strategies, cooperators can win even in the case when the assortment mechanism is inefficient (i.e., the probability of receiving help for noncooperators is slightly greater than for cooperators), which is impossible in classical Hamilton's rule.

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