# Non-cooperative Nash Equilibrium in political game

I have difficulties deriving the non-cooperative Nash Equilibrium of this problem.

The objective function is to maximize the expected total rent over the two periods, that is:

\begin{align} \max_{S_i} (1-e_i S_i + P_i(S_i, S_j) \delta(1-\underline{S})) \end{align}

with $P_i(S_i, S_j)=\frac {S_i}{S_i+S_j}$, and $i,j \in \{A,B\}$, and $e_A=1-d$ and $e_B=1+d$

$S_i$ denotes the quality of the service, $e_i$ is a parameter that captured the spending needs in $i$, $P_i(S_i, S_j)$ is the re-election probability, $\delta$ is the discount factor and $\underline{S}$ is the minimum service quality.

The solution of the Nash Equilibrium of the game is \begin{align} (S_A^*, S_B^*)=\Big(\frac{\delta (1+d)(1-\underline{S})}{4}, \frac{\delta (1-d)(1-\underline{S})}{4}\Big) \end{align}

I have tried but I don't get that same solution.

This is what I have done: I solve the maximization problem \begin{align} \frac {\partial{U_i}}{\partial{S_i}} = -e_i+\dfrac{\delta(1-\underline{S})S_j}{(S_i+S_j)^2}\stackrel{!}{=}0 \end{align}

and get the best respond function for $i$ and $j$ respectively as follows \begin{align} \delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\ \delta(1-\underline{S})S_i=e_j(S_i+S_j)^2 \end{align}

I impose symmetry $S_i=S_j=S^*$ to the reaction function and I get this: \begin{align} \delta(1-\underline{S})S^*&=e_i(2S^*)^2 \\ S^*&=\dfrac{\delta(1-\underline{S})}{4e_i} \end{align}

I am not sure though.

You could add up the equations \begin{align} \delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\ \delta(1-\underline{S})S_i=e_j(S_i+S_j)^2 \end{align} to get \begin{align} \delta(1-\underline{S})(S_i+S_j)=(e_i+e_j)(S_i+S_j)^2 \end{align} which simplifies to \begin{align} \delta(1-\underline{S})=(e_i+e_j)(S_i+S_j). \end{align} This gives you $(S_i+S_j)$. From the initial equations \begin{align} \delta(1-\underline{S})S_j=e_i(S_i+S_j)^2 \\ \delta(1-\underline{S})S_i=e_j(S_i+S_j)^2 \end{align} you can also get $$\frac{S_j}{e_i} = \frac{(S_i+S_j)^2}{\delta(1-\underline{S})} = \frac{S_i}{e_j},$$ which gives you $$\frac{S_j}{S_i} = \frac{e_i}{e_j}.$$ Now you have both the sum and ratio of the variables $S_j,S_i$. Calculating their individual values should be a straightforward matter.