I was wondering if the uniqueness of equilibrium conditions in n-person games as published in Rosen's 1965 paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 1965) are for single-dimensional strategy spaces only? It says in the paper that the strategy of each player can be a vector, but the rest of the conditions seem to be derived for one-dimensional strategy spaces. Can the results be generalized to multi-dimensional spaces as well? Thank you.
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These results can indeed be extended to a more general setting. In this case, define the $\nu$-th player's strategy vector $x^\nu \in R^{n_\nu}$ and so $x=(x^1,\dots,x^N)^T$ of a size $n_1+\dots+n_N$ for a $N$ player game.
In this context, $\nabla_\nu \varphi_\nu(x)$ is a vector of size $n_\nu$.
The pseudo-gradient (denoted $g(x,r)$ in the paper) is still the vector concatenated by all the gradients of $\varphi$ i.e. $g(x,r)=(\nabla_1 \varphi_1(x),\dots,\nabla_N \varphi_N(x))^T$. In this general case, the positivity constraint in the definition of diagonally strictly concave means that each component of a vector is positive.
Now, the reasoning of Rosen can be followed in the exact same way with the notations above.
