# Condition for a Nash equilibrium

Consider two people have a mutually advantageous relationship. That is, if both dedicate more effort to the relationship both improve. Specifically, each individual chooses a level of effort $$x_{i}\in [0,1]$$ and the payoff of the person $$i$$ is

$$\pi_{i}(x_{i},x_{j})=cx_{i}+x_{i}x_{j}-x^2_{i}$$. where $$c$$ is constan $$c\in [0,1]$$

Verify conditions that guarantee the existence of Nash equilibrium in pure strategies and determine such equilibrium.

QUESTIONS:

Q1: for the equilibrium.

$$\pi_{i}(x_{j})=cx_{j}+x_{i}x_{j}-x^2_{i}$$

$$\max_{\mathbf{x}_{i}\in [0,1]} \pi_{i}(x_{j})=\max_{\mathbf{x}_{i}\in [0,1]}(cx_{i}+x_{i}x_{j}-x^2_{i})$$

First order conditions $$c+x_{j}-2x_{i}=0$$ $$x_{i}=\dfrac{c+x_{j}}{2}$$ $$\max_{\mathbf{x}_{j}\in [0,1]} \pi_{j}(x_{i})=\max_{\mathbf{x}_{j}\in [0,1]}(cx_{j}+x_{j}x_{i}-x^2_{j})$$

First order conditions $$c+x_{i}-2x_{j}=0$$ $$x_{j}=\dfrac{c+x_{i}}{2}$$ So then I can say $$x_{i}=x_{j}=c$$ is nash equilibrium.

Is correct?

Q2: (Conditions for existence). I think that I must verify the conditions of the theorem(Debreu. 1952;Fan, 1952;Glicksberg, 1952): Let $$G$$ be a game in strategic form such that, for each $$i=1,2,...,n, S_{i}\subset \mathbb{R^{m}}$$ is compact and convex, and the function $$\pi_{i}:S_{1}\times...\times S_{n}\rightarrow \mathbb{R}$$ is continuous in $$s=(s_{1},s_{2},...,s_{n})$$ and quasi-concave in $$s_{i}$$.Then, the game $$G$$ has a Nash equilirbium in pure strategies.

$$\rho:[0,1]\rightrightarrows [0,1]$$ where $$\rho_{i}(x_{j})=arg \max_{x_{i}\in [0,1]}\pi_{i}(x_{j})$$

*$$S\neq \phi$$

*$$S=[0,1]$$ is compact.

*$$\rho_{i}(x_{j}) \neq \phi$$ and convex...

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But The teacher indicates that it is not the theorem that must be analyzed. But I think he can be wrong. Do you recommend that I continue with this?

Or who will be asking me to apply?