# Is there results about the existence of a Nash equilibrium in continuous games with non-compact strategy spaces?

Consider a $$n$$-player continuous game $$G=(P,S,U)$$ where:

• $$P=\{1,2,\dots,n\}$$ is the set of $$n$$ players.
• $$S=\{S_1,S_2,\dots,S_n\}$$ where $$S_i=\mathbb{R}^n_+$$ is the $$i$$- th player's set of pure strategies, and $$\mathbb{R}^n_+$$ is the set of non-negative $$n$$-tuples.
• $$U=\{u_1,u_2,\dots,u_n\}$$ where $$u_i:S\rightarrow\mathbb{R}$$ is the utility function of player $$i$$.

Let $$\sigma_i\in S_i=\mathbb{R}^n_+$$ denote a single strategy for player $$i$$, $$\sigma=\{\sigma_1,\sigma_2,\dots,\sigma _n\}\in S$$ denote a strategy profile, and $$\sigma_{-i}$$ denote a strategy profile of all players except for player $$i$$.

A strategy profile $$\sigma^*=\{\sigma_1^*,\sigma_2^*,\dots,\sigma_n^*\}\in S$$ is said to be a Nash equilibrium if the strategy $$\sigma_i^*$$ is a local maximum for the utility function $$u_i(\sigma_i; \sigma_{-i}^*)$$ for all the players $$i$$.

## Question:

Is there any property for the utility functions $$u_i$$ that could guarantee the existence of at least one Nash equilibrium? for example, concavity or quasi-concavity.

I know that if all the utility functions are continuous and the sets $$S_i$$ are compact, then, the existence of a Nash equilibrium is guaranteed, but $$\mathbb{R}^n_+$$ is not bounded.

Concavity or quasi-concavity is not enough, since it can be linear (for example take $$u_i(\sigma)=\sigma_i\ \forall i$$).
If $$\{-u_i\}$$ are continuous and coercive (meaning $$\lim_{|\sigma|\rightarrow\infty} u_i(\sigma)=-\infty\ \forall i$$) then you can restrict the problem to a compact rectangle subset of $$S$$ on which the functions $$u_i$$ are not to small and then use the result with compact strategy sets.
In particular, if $$\{u_i\}$$ are continuous and strongly concave, one can show $$\{-u_i\}$$ are also coercive.