# Why is this the equation for the fitness of cooperators?

This paper gives the equations as:

The fitness of cooperators and defectors is respectively given by fC = b(i − 1)/(N − 1) − c and fD = bi/(N − 1).

c - cost

b - benefit

i - number of cooperators

N - population size

For the equation fC, does this mean

(b(i − 1)/(N − 1)) − c, i.e.: $$\frac{b (i-1)}{N-1}-c$$

or b(i − 1)/((N − 1) − c), i.e.: $$\frac{b (i-1)}{{N-1}-c}$$

Guessing the former but I just want to check.

While we're at it, could someone explain why this is how fitness is modeled?

• Think about the units involved - only one of those options makes sense. – Bryan Krause Oct 20 '20 at 18:17

As Bryan Krause points out in the comments, looking at the units gives you the answer. The correct form of the equation is $$f_{c}=\frac{b(i-1)}{N-1}-c$$, where $$f_{c}$$ is the fitness of cooperators, $$N$$ is the total population size, $$i$$ cooperators, $$b$$ is the fitness benefit of cooperating and $$c$$ is the fitness cost of cooperating. It doesn't make sense to subtract $$c$$, which is a kind of relative cost in terms of offpsring production, from $$N$$, which is population size. It would be like subtracting the cost of a car from the speed of an aeroplane. It also makes sense from the viewpoint that we expect the the fitness of cooperators, $$f_{c}$$, to reduce as the cost of cooperating increases, but independently from the number of defectors in a population.

Also, in the paper, it gives the equation with square brackets:

then the fitness of cooperators and defectors, respectively,is given by fC=[b(i−1)/(N−1)]−c

Bryan Krause and user438383 are correct in pointing out that the second equation makes no sense since it would involve subtracting c, a cost, from N, a population.

As to why cooperation and defection is modeled this way, $$f_C$$ is just a linear equation where the fitness accrued to this cooperator is the benefit $$b$$ from every other cooperator $$i$$ except for themselves, thus $$i - 1$$, minus the cost of cooperating $$c$$:

$$f_C = b \frac{(i-1)}{N-1}-c$$

Meanwhile, $$f_D$$ is the fitness accrued by each defector. Since there is no cost to defecting there is no $$c$$ost to subtract, plus the defector also gets a slightly larger benefit because they get the cooperation $$b$$enefit from all the cooperators in the group (whereas each cooperator only gets the benefit from all other cooperators except themselves). $$f_D = b \frac{i}{N-1}$$