Is it trivial to assume a version of Hamilton's rule that applies for numerous generations?

Is it trivial to assume that a version of Hamilton's rule that applies to numerous generations is:

C > rB

C = lineage fitness lost by an actor,

B = lineage fitness gained from the act, and

r = relatedness between actor and those that gained lineage fitness?

Or does this rely upon a particular derivation of Hamilton's rule? If so, does anyone know of a citation in which this derivation is made?

Thank you very much.

I am not sure what you mean by "applies to numerous generations". If the following does not help, can you please clarify what you meant?

Hamilton's rule expresses the condition for which, under a prisoner's dilemna game (see game theory), the stable equilibrium that will be reached is the one where everyone is cooperating. Hamilton's rule is, therefore, a condition to determine the directionality of a dynamic system (the system is dynamic because the state of the system varies over generations).

EDIT

what if the values of B and C differ depending on [the generation]

In other words, you are asking

What if the (social or not) environment is changing over time?

Well, under such circumstance, of course, the simplistic rule that consider B and C as constant won't hold.

You could not simply replace B and C by the the arithmetic mean (or by the geometric or harmonic or any other mean) as the system as two stable equilibriums and upon reaching one there would be no way to get out of it (at least as long as the game remains a prisoner's dilemna).

• Hamilton's rule states (according to Wikipedia, for one): A gene responsible for an altruistic act should increase in frequency if rB > C, where: B = additional reproductive benefit gained by a recipient C = reproductive cost to the actor r = the genetic relatedness of the actor and recipient My question is, what if the values of B and C differ depending on how many generations have elapsed since the act because of a disparity in the number of, for example, grandoffspring or greatgrandoffspring? Commented Mar 22, 2019 at 10:18
• For example, what if after 1 generation, B is F offspring and C is H offspring and after 5 generations, B is G descendants and C is J descendants? And what if H > rF but J < rG? Is it trivial to assume the act will gain frequency after a long number of (> 5) generations because J < rG? Is it still Hamilton's rule even if it deals with C and B in the form of descendants rather than offspring? Commented Mar 22, 2019 at 10:18
• @sterid Thanks for clarification. Please see edit. Commented Mar 22, 2019 at 20:25
• I am not asking about the environment changing. Here's an example. Consider Jim relinquishes resources and they go 10% to Bob and 90% equally to all individuals (so, r = .1). Consider that Jim would have 2 offspring from the resources and Bob would get 9 offspring from them. But after 5 gen, Jim would have 8 descendants from the resources and Bob would get 100 descendants from them. 2 > .1*9 but 8 < .1*100 Then after 1 gen, there is less of the variant responsible for the act than at gen 0, but after 5 gen, there is more of the variant than at gen 0 (the variant has evolved). Commented Mar 22, 2019 at 22:10