<p>These results can indeed be extended to a more general setting. In this case, define the <span class="math-container">$\nu$</span>-th player's strategy vector <span class="math-container">$x^\nu \in R^{n_\nu}$</span> and so <span class="math-container">$x=(x^1,\dots,x^N)^T$</span> of a size <span class="math-container">$n_1+\dots+n_N$</span> for a <span class="math-container">$N$</span> player game.</p>

<p>In this context, <span class="math-container">$\nabla_\nu \varphi_\nu(x)$</span> is a vector of size <span class="math-container">$n_\nu$</span>. </p>

<p>The pseudo-gradient (denoted <span class="math-container">$g(x,r)$</span> in the paper) is still the vector concatenated by all the gradients of <span class="math-container">$\varphi$</span> i.e. <span class="math-container">$g(x,r)=(\nabla_1 \varphi_1(x),\dots,\nabla_N \varphi_N(x))^T$</span>. In this general case, the positivity constraint in the definition of diagonally strictly concave means that each component of a vector is positive.</p>

<p>Now, the reasoning of Rosen can be followed in the exact same way with the notations above.</p>

CC BY-SA 4.0
2018-11-30T01:29:29.870Z
2018-11-30T01:29:29.870Z
Tanj
<p>I was wondering if the uniqueness of equilibrium conditions in n-person games as published in Rosen's 1965 paper (<a href="http://www.jstor.org/stable/pdf/1911749.pdf" rel="nofollow noreferrer">J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 1965</a>) 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.</p>

<nash-equilibrium><competitive-equilibrium>
Rosen's unique equilibrium conditions: Multi dimensional strategies?
SelinH
# Rosen's unique equilibrium conditions: Multi dimensional strategies?

(Asked by: username_0 on 2018-01-18)

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](http://www.jstor.org/stable/pdf/1911749.pdf)) 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.

**Tags:** nash-equilibrium, competitive-equilibrium


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## 1 answers

### Answer by username_1 on 2018-11-30. Score: 1

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.


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