# Hamilton's Rule in practice

I came across this question in USABO 2013 open exam:

A mother antelope and its child are galloping along the plains, when they encounter a group of hungry lions. If the two antelopes try to escape the lions together, there is a 75% chance that both will be consumed and eaten, and a 25% chance that both will escape alive. If, however, the mother sacrifices herself to the lions, she may be able to buy her baby additional time to escape.

What is the minimum chance that the baby can have to escape from the lions following such a sacrifice such that the mother’s action will be evolutionarily favored?

You may assume that the baby antelope, once escaped, will be guaranteed to survive into a reproducing adult, and that the mother antelope is at the beginning of reproductive age.

I suppose one should use Hamilton's Rule ($$r \cdot B > C$$) to solve this issue. I'm not familiar with the working of this rule to put it into the question's context. To my understanding $$r=0.5$$ but this is pretty much how far I managed to go.

• I'm voting to close this question because this style of homework question doesn't lend itself to the format of SE. It's unclear exactly what about the topic you don't know. This question may involve someone answering correctly, but only after long commentary about the topic. Other users might disagree and with a few edits this might be a very decent question. Jun 1 '16 at 6:34
• I'm new to Bio SE, so sorry for the formatting. What can I do to improve my question? Jun 1 '16 at 17:55
• I disagree with the close votes that say the question is "unclear" or "too broad". I agree it is a homework question but it is hard to suggest what kind of efforts the OP might do to make the question perfectly on-topic. I edited the post hoping to slightly improve it. Jun 2 '16 at 16:49
• @James I agree that this question is a homework question, but what's the problem in answering it. It's quite visibile that this question is a good one. Iwould not vote for closing the question. Jan 18 '19 at 15:58

It doesn't have to be that complicated.

The evolutionary value of the offspring to the mother is half that of herself (0.5). Then the relative value of both mother and offspring is 1 + 0.5 = 1.5. So the value of preserving both lives is about 3 times the value of preserving the offspring alone, given the simplistic assumptions from the question. Therefore, to choose to make this sacrifice, the probability of survival of the offspring alone must be at least 3 times the probability of both surviving otherwise. 3 times 25% = 75%.

Edit:
In terms of B and C as requested (though I think it's simpler to step away from this):

B is the benefit to the relative. This is not the probability of survival given the action of sacrifice, but the increased probability of survival. The baby already had a 25% chance of survival, but this action will increase that chance to some p, which we are looking for. B is the improvement in the situation:
B = p - 0.25

C is the cost to only the individual (the mother). This is not 1 (loss of survival), but the difference between the situation without making the sacrifice and the situation when she makes the sacrifice. She's only giving up a 25% chance of survival. So:
C = 0.25.

r is the relation between mother and offspring.
r = 0.5.

So the equation is:
0.5(p - 0.25) = 0.25
0.5 p - 0.125 = 0.25
0.5 p = 0.375
p = 0.75

Someone feel free to format if you want...

• That's be great if you could put that in the $rB>C$ equation! I will then discretely delete my answer as if I never wrote it :) Jun 2 '16 at 17:17
• Hmm, this is an interesting way to view the question! Jun 2 '16 at 19:36
• @Remi.b If I'm reading your math correctly, you made two mistakes. First, it seems that you are looking for the minimum probability of survival of the baby after the incident, independent of the mother's choice at this moment, given an assumption that immediate survival is guaranteed by the sacrifice. This is a much harder question, and not what was asked. I believe the correct answer to that harder question though, would have been 0.67, because in your scenario B=p(1-0.25) => p = C/(r*(1-0.25)) = 0.25/0.375. Does it make sense why B is p(1-0.25) in that scenario? Jun 2 '16 at 21:22
• @Remi.b I suspect you also forgot the parent's contribution to fitness, which is assumed to be 1. This is added to the offspring's fitness, giving a total of 1.5 fitness if they escape the lions, and 0 fitness if they do not. Jun 3 '16 at 0:51
• Of course, I can't help but point out the place where the math ceases to support the reality around us, thus begging a greater explanation: The mother has another choice, which, in higher mammals, is almost impossible to even consider. She could sacrifice the baby, probably as easily as simply outrunning him. All else being equal (and it's not - she has a better chance of survival alone than he does), this choice would have double the value - the obvious choice. So there's your take-away ponderable... Jun 3 '16 at 18:58

There is another way to look at this question, which I suspect is even quicker (it is certainly more intuitive for me). Mathematically, it is identical to Mark's answer.

If the parent chooses to not sacrifice herself to save the child, they have a net fitness of 1.5 (1 parent + 0.5 child) if they escape, and 0 fitness if they do not. Therefore, their average fitness is 1.5*0.25 = 0.375.

If the parent sacrifices herself to save the child, they have a net fitness of 0.5 if the child escapes, and 0 if the child does not. Therefore, in order to have an average fitness >= 0.375, the probability of the child escaping must be >= 75%.