First Order Condition for Profit Maximization in Gambling Industry
The expression in question is in footnote $11$ of the referenced article. Reading the paper, we see that the decision variable here is "the payout rate", which is the reciprocal of $P$. So equivalently, we can solve the maximization problem with respect to $P$ (and not w.r.t. $Q$). More over, "price elasticity of demand" involves the derivative of $Q$ with respect to $P$, and not the other way around: 

$$E_{PQ} = \frac {dQ/dP}{Q}P $$

and we expect it to be negative (higher price means lower payout rate leading to less demand for the quantity measure here, i.e. less "demand for prizes").

We can write the maximization problem as 
$$\max_{P}N = \max_{P}\left[P\cdot Q(P) - Q(P) - C(Q(P))\right]$$

The  first-order condition is

$$\frac{\partial N}{\partial P} = Q + P\cdot Q'  - Q' - C'\cdot Q' = 0 \tag{1}$$

Multiply throughout by $P/Q$:

$$Q\frac PQ + P\cdot Q'\frac PQ   - Q'\frac PQ  - C'\cdot Q'\frac PQ  = 0$$

$$\Rightarrow P +P\cdot E_{PQ} - E_{PQ} - C'\cdot E_{PQ}=0$$

$$\Rightarrow -E_{PQ} = \frac {P}{P-1-C'} \tag{2}$$

This makes sense. Plugging in the values presented in the reference, we have

$$-E_{PQ} = \frac {2}{2-1-.12} = \frac {2}{0.88} \approx  2.27 $$

which is very close to the value resulting from the equation presented by the authors. I haven't been able, by whatever algebraic manipulations I tried, to replicate their formula, but eq $(2)$ is correct in any case. If a reconciliation comes up, I will update.