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I am looking for help on how to solve this game theory/manifolds question.

My thoughts:

Informally, E is the set of mixed strategy - utility profiles such that each player is indifferent between all of her strategies given the strategies of the other players (and given her utilities).

If I'm not mistaken, if I can show that E is locally the graph of a smooth function, then that implies it's a manifold. Intuitively, I think the idea is that if we perturb some of the variables e.g. the probability distribution of one of the players strategies than we should be able to make the other player(s) indifferent by perturbing some of the other variables e.g. her utilities.

(I would try to show more effort on the question, however it's very hard for me to type math right now--I type by voice because I'm injured; I was able to have someone copy the LaTeX here directly from Andrew McLennan's fixed point book. Please bear with me. Thanks!)

Below is the problem:

Let $n$ be a positive integer, and let $S_1,...,S_n$ be nonempty finite sets of pure strategies. For each $i=1,...,n$ let

$$H_i = \{\sigma_i : S_i \rightarrow \mathbb{R} : \sum_{s_i\in S_i} \sigma_i (s_i)=1\}$$

Let $S = S_1 \times \cdots \times S_n$ and $H=H_1\times \cdots \times H_n$. A game for $S_1,...,S_n$ is an $n$-tuple $u=(u_1,...,u_n)$ of functions $u_i : S \rightarrow \mathbb{R}$. Let $G$ be the space of such games. We extend $u_i$ to $H$ multilinearly: $$u_i(\sigma)=\sum_{s\in S}(\prod_j \sigma_j(s_j))u_i(s).$$

Let $$E=\{(u,\sigma)\in G\times H:u_i(s_i,\sigma_{-i})=u_i(t_i,\sigma_{-i}) \text{ for all $i$ and all $s_i,t_i\in S_i$}\}$$

Here $\sigma_{-i}$ means the vector without the $i$th component.

Prove that $E$ is a $n|S|$-dimensional $C^\infty$ manifold.

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