Appendix A: Optimal prices for sigmoidal acceptance
This appendix describes the steps to obtain the optimal prices described in the paper.
A.1 Non-saturated equilibrium price
Here is the derivation of the optimal price value (16) described in Section 4.1.2 in the hypothesis that infinite rooms are available.
We want to find the stationary point of u ⋅ pa(u), where u is the price and pa(u) is the corresponding acceptance probability (15):
$$ \frac d{du}\bigl(u\cdot p_{a}(u)\bigr) = 1 - \sigma\left( \frac{u-\mu}\sigma\right) - \frac1\eta u\sigma^{\prime}\left( \frac{u-\mu}\eta\right); $$
(24)
by applying the substitutions
$$ s = \frac{u-\mu}\eta,\quad \beta=\frac\mu\eta, $$
and reminding the identity \(\sigma ^{\prime }(s)=\sigma (s)\bigl (1-\sigma (s)\bigr )\), we obtain
$$ (\beta+s)\sigma(s)^{2} - (\beta+s+1)\sigma(s) + 1 = 0. $$
(25)
After replacing the sigmoid function definition, multiplying by (1 + e−s)2 and simplifying, we are left with
$$ (s+\beta-1)e^{s} = 1 $$
and, multiplying by eβ− 1,
$$ (s+\beta-1)e^{s+\beta-1} = e^{\beta-1}. $$
(26)
Let us observe that this equation is in the form xex = a for a > −π/2, whose solution can be analytically expressed as x = W0(a), where W0(⋅) is the main branch of Lambert’s function. We get
$$ s+\beta-1 = W_{0}(e^{\beta-1}). $$
By replacing the original variables, we finally obtain (16).
A.2 Dynamic programming optimal price
The derivation of the optimal price policy (23) described in Section 4.1.3 for the dynamic programming technique follows the same steps outlined above.
After replacing (15) into (22), let us perform the following variable substitutions and quantity replacements:
$$ s = \frac{u-\mu}\eta,\quad \beta=\frac{V_{1}-V_{2}+\mu}\eta, $$
(27)
after which we obtain (25), whose solution is, again, (26).
By replacing the original variables from (27), we obtain (23).