For a particular three player game the set of strategy profiles, it is the case $u_2(s)=3u_1(s),u_3(s)=[u_1(s)]^2,$ and $u_1(s) \in [0,1]$

Question

Question :

For a particular three player game the set of strategy profiles $S$ is finite. Also, for every $s \in S$, it is the case $u_2(s)=3u_1(s),u_3(s)=[u_1(s)]^2,$ and $u_1(s) \in [0,1]$

Then

a) does this game have a Nash Equilibrium?

b) does this game have an efficient Nash equilibrium?

I have no idea where to begin. Could somebody please give me a hint.

Answer 1

a) Yes. Since your set S is finite, and because $u_2(s)$ is linear and $u_3(s)$ concave on $u_1(s)$, and $u_1(s)$ is a constant assuming values from a compact interval $[0,1]$, then the game has a Nash Equilibrium.

b) It seems so. Since apparently there are no additional constraints worthy to mention and because we are dealing with strategic complementarities, then $1$ will choose an $s$ such that $u_1(s) = 1$, which is his best choice (and most efficient), which at the same time will maximize the utility of $2$ and $3$, considering the functional form defined.