Why is this the equation for the fitness of cooperators?

Question

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?

Connor McCormick · Accepted Answer

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).
$$
f_D = b frac{i}{N-1}
$$

user438383 · Answer

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

Why is this the equation for the fitness of cooperators?

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