I need to find a nash equilibrium of a two-player game based on their best response function. The problem I have is that the functions I'm dealing with are somehow weird and hard to work with! So, let me start describing the problem for you:
There are two hosts in a city that are offering identical units, i.e., unit 1 (offered by host 1) and unit 2 (offered by host 2). The goal is to find the equilibrium price that both these hosts will have to offer in this two-players game. The profit function for each of them are described as follows:
$\begin{array}{l} {\pi _1} = {\rho _1}{P_1} - {k_1}\delta = \frac{{{P_1} - {P_1}^2 + b{P_1}{P_2}}}{{1 - {P_1} + b{P_2} + \mu }} - {k_1}\delta \\ {\pi _2} = {\rho _2}{P_2} - {k_2}\delta = \frac{{{P_2} - {P_2}^2 + b{P_2}{P_1}}}{{1 - {P_2} + b{P_1} + \mu }} - {k_2}\delta \end{array}$
Where:
${k_1} = \left\{ \begin{array}{l} 0,\, \, {\rho _1} \ge \theta \\ 1,\, \,{\rho _1} \le \theta \end{array} \right.$
${k_2} = \left\{ \begin{array}{l} 0,\, \, {\rho _2} \ge \theta \\ 1,\, \,{\rho _2} \le \theta \end{array} \right.$
And:
$\left\{ \begin{array}{l} {\rho _1} = \frac{{1 - {P_1} + b{P_2}}}{{1 - {P_1} + b{P_2} + \mu }}\\ {\rho _2} = \frac{{1 - {P_2} + b{P_1}}}{{1 - {P_2} + b{P_1} + \mu }} \end{array} \right.$
The parameters of the problem are: $0 \le \theta ,b,\mu \le 1$
If you are wondering about the logic behind these formulas, the term $-k_i\delta$ is about the time when the utilization of the unit $i$ is below a certain threshold, say $\theta$, and if this is true, the host of that unit should pay some amount of money as the vacancy tax (you can assume that the government sets the constant $\theta$ and announces that if a unit is going to be vacant more than $\theta$ percent of some period of time, its host needs to pay some extra money as the vacancy tax). That's why I defined the indicator $k_i$ which indicates that when the utilization, $\rho_i$, is going to be below the threshold and if this happens, in the profit equation $k_i$ turns into 1 and the amount of vacancy tax, $\delta$, should be deducted from the profit. (Utilization is defined as the percentage of the time that the unit is busy, e.g., there is a host there using the unit).
Now that you know the problem, I need to solve what I brought below:
$\left\{ \begin{array}{l} {B_1}({P_2}) = \arg {\max _{{P_1} \ge 0}}\left( {{\pi _1}} \right)\\ {B_2}({P_1}) = \arg {\max _{{P_2} \ge 0}}\left( {{\pi _2}} \right) \end{array} \right.$
In other words, I need to first take a partial derivative from the profit functions, and then reach to a pair of $(P_1^*,P_2^*)$ in a way that:
$\left\{ \begin{array}{l} {P_1^*} = {B_1}({P_2^*})\\ {P_2^*} = {B_2}({P_1^*}) \end{array} \right.$
Actually, if you're a bit familiar with game theory, you will notice that this is the time when neither of the hosts wants to deviate from their offered price (first-order condition). The problem, as you probably have noticed by now is that taking a derivative from the profit function is a bit complex mostly because of the indicator $k$ (at least I don't know how to deal with these functions while I want to find the optimal point).