We consider the following public good provision game. There are 2 players, each choosing the amount of money $x_i$ ($i$ denotes 1 or 2) they will give to build a public good. We assume that each player has a maximum of 1 unit of money that he can give, so that $x_i$ denotes $[0,1]$ for both players. Once the good is built, they receive a utility $h(G)$ from using it, where $G = x_1 + x_2$ is the total amount that was invested in the public good. We assume that $h(G) = kG^{0.5}$, where $k \geq 0$ is a constant. Each player's utility is therefore
$$U_i(x_1, x_2) = k(x_1+x_2)^{0.5} - x_i$$ where $i$ denotes $(1, 2)$.
For each value of $k \geq 0$, find all Nash equilibria in pure strategies.
Can anyone explain how this question is done? I'm confused how to start. They are moving simultaneously.
To be a Nash equilibrium, a pair of pure strategies $(x_1,x_2)$ must be such that neither player can improve his payoff by shifting his strategy. That is, $$ U_1(x_1, x_2) = \sup_{x_1'\in[0,1]}U_1(x_1', x_2) $$ and $$ U_2(x_1, x_2) = \sup_{x_2'\in[0,1]} U_2(x_1, x_2'). $$ Note that $$ \frac{\partial}{\partial x_1}U_1(x_1,x_2)=\frac{\partial}{\partial x_1}\left(k\sqrt{x_1+x_2}-x_1\right)=\frac{k}{2\sqrt{x_1+x_2}}-1 $$ is zero on the line $x_1+x_2=\frac{1}{4}k^2$, negative for larger $x_1+x_2$, and positive for smaller $x_1+x_2$; and $\partial U_2(x_1,x_2)/\partial x_2$ is exactly the same. Since each player's utility is reduced when he moves away from this line, any pair $(x_1,x_2)$ with $x_1+x_2=\frac{1}{4}k^2$ is a Nash equilibrium in pure strategies. (Or, if $\frac{1}{4}k^2>2$, then $x_1=x_2=1$ is the only Nash equilibrium.)