I cannot find for this Simple shared effort level game

Question

Each player can contribute to the project with non-negative effort.

Player 1's utility is $u_1=e_1(1+e_2-s\cdot e_1)$ where $s\in [0,1]$.

Player 2's utility is $u_2=e_2(1+e_1-e_2)$

For case 1, each player contributes to the project simultaneously.

For case 2, player 1 moves first then play 2 moves next

I cannot find a Nash equilibria (SPNE for case 2) if $s$ is less than $1/4$.

If I draw best a response diagram, the two best response functions do not intersect. But I don't think that $e_1=0$, $e_2=0$ is a Nash equilibrium, since they have would have an incentive to exert more than $0$ effort.

Can $e_1=\infty$, $e_2=\infty$ be a nash equilibrium?

Answer 1

You are right, there is no equilibrium for $s\leq \frac{1}{4}$. Note that Nash's theorem only guarantees the existence of an equilibrium in finite games (which this game is not). (Also note that $0$ is not in the best-response set of any of the two players, so we know that, if there were an equilibrium, it would not contain zero effort by any player.)

To get an intuition of what is going on, consider the plot below, which shows the best response functions for the two players, with $s=\frac{1}{4}$. As you point out, the lines do not intersect. Suppose we start at point $(0,0)$, and let each player revise their action in turn, holding the other player's action constant (dashed line). Player 1 will update their action to $e_1=2$, the best response to $e_2=0$. But then, player 2 will update their action to $e_2=\frac{3}{2}$. This goes on ad infinitum, never reaching an equilibrium.