Finding best response function for both players?

Question

Two players, A, and B, are bidding on an apartment. The their bids be $x_1$, and $x_2$, respectively, with $x_1,\,x_2\geq 0$. Then, the probability that A wins the apartment is given by, $$p_1=\frac{\phi x_1}{\phi x_1+x_2},$$ where $\phi$ is some parameter. The probability that B wins the apartment is, $1-p_1$. I'm asked to graph the best response functions for both players. Naturally, the goal of A is to maximize the probability of winning the apartment regardless of B's action. Hence, we wish to find $x_1$, for which $\frac{dp_1}{dx_1}$ is a maximum. That is, find $x_1$, such that $\frac{dp_1}{x_1}=0$. Similarly, for B, we wish to find $x_2$, such that $\frac{dp_2}{dx_2}=0$. However, I find $\frac{dp_1}{dx_1}$ to be, $\frac{\phi x_2}{(\phi x_1+x_2)^2}$, and $\frac{dp_2}{dx_2}$ to be $\frac{\phi x_1}{(\phi x_1 +x_2)^2}$, such that there are no extrema. In such a case, I am not sure exactly what the best response functions would be, if they exist at all.

Answer 1

Note that in order to calculate a player's expected utility, you have to multiply the probability of winning by the benefit from winning, in this case $(R-x_i)$, for player $i$.

For player $1$: maximize w.r.t $x_1$: $$ p_1(R-x_1) = \frac{\phi x_1}{\phi x_1+x_2} (R-x_1) $$

The first order condition is:

$$ 0 = \frac{\phi x_2}{(\phi x_1+x_2)^2}(R-x_1) - \frac{\phi x_1}{\phi x_1+x_2} \implies 0 = \phi x_2(R-x_1) - \phi x_1(\phi x_1+x_2) $$

This is a quadratic polynomial in $x_1$. The non-negative solution is $$ x_1 = \frac{\sqrt{x_2 (R \phi + x_2)} - x_2}{\phi} $$

This is the best response funciton for the first player. You can obtain the second player's best response function similarly.

Hope that helps.