# 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

If this is really a N+1 player game, it appears the classic Rosen paper applies: Rosen, J.B., Existence and uniquenss of equilibrium points for concave n-person games. Econometrica, 33,3, p.520-534, 1965. Usually there isn't a global objective, just player-specific objectives, and you can't just add them all up to get a global objective for ordinary optimization.

If each player's objective is concave in the player's own decision variables, then there is a Nash equilibrium. A sufficient condition is that the Hessian of the diagonal block of each player's objective w.r.t. that player's own variables is negative definite (ND).

For the N.E. to be unique, build a square matrix composed of blocks from each player's full Hessians (w.r.t. all the variables), selecting the rows associated with each player's own decision variables. If the symmetric part of this matrix is ND (for some positive weighting of each row group), then equilibrium is unique, a sufficient but not necessary condition. See also Bruce Hajek, 2018, An Introduction to Game Theory, pg 16.

This condition is easily demonstrated if all the players' Hessians are diagonally dominant. (I'm working on a problem in which the full Hessians are ND for all players, but haven't proven ND-ness for the pieced-together "Frankenstein" matrix.)