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?

  • 5
    $\begingroup$ Think about the units involved - only one of those options makes sense. $\endgroup$ – 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). $$ f_D = b \frac{i}{N-1} $$


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