I have continuous strategy game that consists two type of players, core player, and normal players. The number of core player in the game is $1$. The number of normal player in the game is $N$.
Let's index these players as
Where player $0$ is the core player and player $i$ is normal player
Let $s_i$ denote the strategy of player $i$ Let $S_N$ denote the set of strategy of normal players
We know that
the utility function of core player $u_c(s_0, S_N)$ depends on on the strategy of the set of normal players' strategy and his own strategy.
the utility function of normal player $i$ $u_i(s_i,s_0)$ depends on the core player's strategy and his own strategy.
Both $u_c(\cdot)$ and $u_i(\cdot)$ are strictly concave.
We also know this game can be characterized by a global function $F(s_0,S_N)$.
If we can show the global function $F(s_0,S_N)$ is strictly concave, does that mean the game exist an unique Nash Equilibrium?
If a game has unique pure Nash Equilibrium, does that mean given any initial setting/strategy of the game, the game always converge to that Nash Equilibrium?
I hope I have provided enough information to make the question clear and happy to give information if needed.
When I say the the game can be characterized by a global function $F(s_0, S_N)$, what I mean is that to find the optimal strategy of core and normal players is equivalent to find $s_0$ and $S_N$ that optimize the function F.