How to Solve Hamilton's Rule Problems 100% Accuracy Every Time

I promise this isn't clickbait. I've never gotten a Hamilton's question wrong using this strategy. Trust.

To skip the basics of Hamilton's rule and get straight to the 100% success rate problem-solving strategy, go below the minion.

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Inclusive fitness is the concept that one's evolutionary fitness (the degree to which they passed their genes onto the next generation) not only depends on their offspring, but also the offspring of their relatives. Because relatives share DNA, it makes sense that the reproductive success of an individual partially depends on the reproductive success of their relatives.

A quantitative model for this concept is Hamilton's rule, which is frequently tested in USABO. The equation is:

rB>C,

where r is the coefficient of relatedness between the two individuals being compared,

B is the benefit incurred by saving the relative,

and C is the cost of saving the relative.

This equation applies to situations where one individual needs to make a decision to either save (risking their own lives) or ignore a relative in danger. The coefficient of relatedness (r) describes how much genetic material the individual and their relative share. This is important to see how important the relative should be (genetically) to the individual. For example, all else being equal, the individual should care more about their biological parent over a biological cousin, because the parent and the individual have more DNA in common than the cousin and the individual.

The benefit (B) and cost (C) are situation-dependent. It's important to know what each variable corresponds to: B is the benefit to the recipient of the altruism, and C is the cost to the altruist. This is why we multiply r by B instead of C; we are trying to see the benefit to the individual and compare it to the cost for the individual. The benefit to the individual equals the benefit to the relative times the relationship between the individual and relative. The cost to the individual is just C. Now we can compare them, since they're relative to the same person.

Let's try a simple problem.

The first step to solving these problems is to identify the coefficient of relatedness (r) between the individual (in this case, you) and the relative(s). In this problem, you are trying to save all relatives at once, so you can treat your grandma, uncle, and cousin as one entity.

The total r = 1/4 + 1/4 + 1/8 = 5/8

Next, we need to weigh the benefits to you vs. the costs to you. The most foolproof way to do this is to solely evaluate the benefits of saving versus the benefits of not saving the relatives. This automatically takes care of any costs, and makes the calculations much more straightforward.

Blue denotes coefficient of relatedness. Pink denotes likelihood of survival. Because reproduction is a product of survival (you need to be alive to have babies), the likelihood of survival can be viewed as the "B" term in "rB." Pink x blue = rB.

Green denotes the rB terms for the relatives. Orange is the rB terms for you. The rBs for you and your relatives add up to the total benefit for you.

Let's now find which situation is more favorable. These are the benefits for each situation:

If we want to find the maximum likelihood of your death (drowning), we need to set the benefit of saving (risky) to be greater than the benefit of ignoring, and solve for p.

So the answer is C!

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This problem-solving strategy doesn't resemble the straightforward rB>C equation we saw earlier. However, this equation is implicit in our calculations. By subtracting the benefits of each situation (risking or ignoring) and accounting for p (likelihood of your death) and using chances of survival in the calculations (note that 1-(chance of survival)=chance of death which is a risk), we essentially factored in cost. The importance of rB>C in our case is that we need to remember the concept that if the benefits to you outweigh the costs to you, it is evolutionarily favorable to you to perform the sacrifice.

Here's a template for these problems:

where green = relatives' benefit to you

orange = your benefit to you

blue = r (coefficient of relatedness)

pink = B (benefit) to whoever it concerns

Note: to apply this model to a wider range of situations, we can replace "benefit of saving (risky)" and "benefit of ignoring" with "benefit of risk" and "benefit of no risk," respectively.

Try to fill out this template for the below problem (answer below):