2
$\begingroup$

Rosen's paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965) presents a condition for the uniqueness of the Nash Equilibrium in $n$ players game. The setup is described in simple details by this other economics exchange post. My curiosity is why is the Jacobian of $g(\mathbf{s},\mathbf{z})$, where $g(\mathbf{s},\mathbf{z})$ is the pseudogradient of the function $\sigma$, a square matrix? If $s \in \mathbb{R}^m$ and there are $n$ players, it seems to me that the matrix would be $n \times m$ instead of $m \times m$. I have left out many of the details of the problem set up as they are covered in the other post; however, I can restate the set up or provide additional information if necessary.

$\endgroup$
3
$\begingroup$

While there are $n$ players, a player's strategy is not single dimensional. The total dimension of the strategies is $m$. So the vector of strategies $s$ is $m$ dimensional. $g(s,z)$ is also $m$ dimensional, which is why the Jacobian is $m \times m$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.