# Does multi-player games which is characterized a global concave function has unique Nash Equilibrium

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

$$0,1,2,...N$$

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 $$F(s_0,S_N)$$.

Question:

1. 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?

2. 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.

Further clarification:

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.

• "this game can be characterized by a global function " what does this mean? Please, clarify. Nov 19 '19 at 18:42
• "any initial setting/strategy of the game, the game always converges to that Nash Equilibrium" you do not mention which dynamic process is being used here. Please clarify. Nov 19 '19 at 18:43
• @SergioParreiras Sorry, I don't have much background on Game theory. I am not sure what you mean by dynamic process. Could you elaborate on that a little bit? What information you would need for the dynamic process? Nov 19 '19 at 19:36