<p>So you want to find a maximum of $\sigma(s,z)$. If $\sigma$ is diagonally strictly concave you can do so by starting at any point and just following the gradient $g(s,z)$ until you find the maximum and no matter where you start, you will always end up at the same point (Start at the lower black points and follow the direction of the gradient (the direction of the steepest ascent).).
<a href="https://i.sstatic.net/pLQRe.png" rel="noreferrer"><img src="https://i.sstatic.net/pLQRe.png" alt="Following the gradient in a diagonally strictly concave function"></a></p>

<p>However, if $\sigma$ isn't diagonally strictly concave, you could end up at different maxima by starting at an arbitrary point and following the gradient (Follow the direction of the steepest ascent starting from the two lower black dots; you'll end up at two different points.).
<a href="https://i.sstatic.net/wsE3S.png" rel="noreferrer"><img src="https://i.sstatic.net/wsE3S.png" alt="Following the gradient in a non-diagonally strictly concave function"></a></p>

CC BY-SA 4.0
2018-10-17T07:29:00.867Z
Thanks for your answer! What you write is essentially one of the results in the original Rosen's paper.  When I say intuition I mean what  property of the strategic interaction in the game is captured by the strict concavity condition? For example,  does this condition say something about how other players'  actions affect player i's payoff,  or how player i's action affects other players' payoff in the game.   Sorry if I wasn't clear enough in the question.
Nidjsi
CC BY-SA 3.0
2017-10-25T16:11:44.580Z
2017-10-25T16:11:44.580Z
muxamilian
<p>The diagonally strict concavity property is better known as the <strong>strict monotonicity</strong> property of the pseudo-gradient.</p>
<p>An operator <span class="math-container">$\Psi:\mathbb{R}^n \to \mathbb{R}^n$</span> is strictly monotone if the following holds true:
<span class="math-container">$$\forall x_0, x_1 \in \mathbb{R}^n: \left&lt; x_0 - x_1, \Psi(x_0) - \Psi(x_1)  \right&gt; &gt;0. $$</span></p>
<p>In your case <span class="math-container">$\Psi$</span> would be <span class="math-container">$g$</span>.
The intuition behind it is that for every line of <span class="math-container">$R^n$</span>, the projection of <span class="math-container">$\Psi$</span> on that line is strictly monotone.</p>
<p>I join here an example: the payoff function are given by <span class="math-container">$\varphi_i$</span> for both players <span class="math-container">$i \in {1,2}$</span>.</p>
<p><a href="https://i.sstatic.net/dlzGW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/dlzGW.png" alt="enter image description here" /></a></p>
<p><a href="https://i.sstatic.net/jKEHo.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/jKEHo.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2021-03-24T23:21:27.543Z
2021-05-12T12:55:28.650Z
2021-05-12T12:55:28.650Z
Julien V
<p>Consider a game with $n$ players, with strategy space $S \subset \mathbb{R}$, where $S$ is bounded set, and player's $i$ payoff function  $\pi_i:S^n \rightarrow \mathbb{R}$. 
Rosen's condition (<a href="http://www-inst.eecs.berkeley.edu/~ee228a/fa03/228A03/papers/rosen.pdf" rel="noreferrer">J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965</a>) for uniqueness of the Nash Equilibrium in n players game states that the  equlibrium will be unique when</p>

<ol>
<li>payoff function $\pi_i(\textbf{s}) \; i \in N$  is concave in own strategy</li>
<li>There exists vector $\textbf{z}$ ($(\forall i \in N)(z_i \geq 0)\ \wedge (\exists i \in N) (z_i &gt;0)$ such that function $\sigma(\mathbf{s}, \mathbf{z})=\sum_{i=1}^{n}z_i\pi_i({\textbf{s}})$ is diagonally strictly concave</li>
</ol>

<p>$N$ denotes the set of players. </p>

<p>To define the concept of diagonal strict concavity, fist introduce 'pseudogradient' of function $\sigma$, defined with:
\begin{align}
g(\mathbf{s},\mathbf{z}) = \begin{pmatrix}
z_1\frac{\partial \pi_1(\mathbf{s})}{\partial s_1} \\ 
z_2\frac{\partial \pi_2(\mathbf{s})}{\partial s_2} \\ 
... \\ 
z_n\frac{\partial \pi_n(\mathbf{s})}{\partial s_n}%
\end{pmatrix}
\end{align}
Then, function $\sigma$ is said to be <strong>diagonally strictly dominant</strong> in $\mathbf{s} \in S$ for fixed $\mathbf{z} \geq 0$ if for every $\mathbf{s}^0, \mathbf{s}^1 \in S$ the following holds:
\begin{align}
(\mathbf{s}^1 - \mathbf{s}^0)'g(\mathbf{s}^{0}, \mathbf{z}) + (\mathbf{s}^0 - \mathbf{s}^1)'g(\mathbf{s}^{1}, \mathbf{z})&gt;0
\end{align}</p>

<p>It is shown, in the paper I cite in the beginning, that a sufficient condition for $\sigma$ to be diagonally striclty concave is that matrix $\left[G(\mathbf{x}, \mathbf{z}) +G(\mathbf{x}, \mathbf{z})' \right]$ is negative defite for $\mathbf{s} \in S$, where $G(\mathbf{x}, \mathbf{z})$ is Jacobian of pseudogradient $g$ with respect to $\mathbf{s}$. I use ' to denote transpose of a matrix. 
What is the intuition behind diagonal strict concavity condition?</p>

<game-theory><nash-equilibrium>
Rosen's Diagonal Strict Concavity condition
Nidjsi
# Rosen's Diagonal Strict Concavity condition

(Asked by: username_0 on 2016-03-24)

Consider a game with $n$ players, with strategy space $S \subset \mathbb{R}$, where $S$ is bounded set, and player's $i$ payoff function $\pi\_i:S^n \rightarrow \mathbb{R}$.
Rosen's condition ([J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 33(3):520–534, 1965](http://www-inst.eecs.berkeley.edu/~ee228a/fa03/228A03/papers/rosen.pdf)) for uniqueness of the Nash Equilibrium in n players game states that the equlibrium will be unique when

1. payoff function $\pi\_i(\textbf{s}) \; i \in N$ is concave in own strategy
2. There exists vector $\textbf{z}$ ($(\forall i \in N)(z\_i \geq 0)\ \wedge (\exists i \in N) (z\_i >0)$ such that function $\sigma(\mathbf{s}, \mathbf{z})=\sum\_{i=1}^{n}z\_i\pi\_i({\textbf{s}})$ is diagonally strictly concave

$N$ denotes the set of players.

To define the concept of diagonal strict concavity, fist introduce 'pseudogradient' of function $\sigma$, defined with:
\begin{align}
g(\mathbf{s},\mathbf{z}) = \begin{pmatrix}
z\_1\frac{\partial \pi\_1(\mathbf{s})}{\partial s\_1} \\
z\_2\frac{\partial \pi\_2(\mathbf{s})}{\partial s\_2} \\
... \\
z\_n\frac{\partial \pi\_n(\mathbf{s})}{\partial s\_n}%
\end{pmatrix}
\end{align}
Then, function $\sigma$ is said to be **diagonally strictly dominant** in $\mathbf{s} \in S$ for fixed $\mathbf{z} \geq 0$ if for every $\mathbf{s}^0, \mathbf{s}^1 \in S$ the following holds:
\begin{align}
(\mathbf{s}^1 - \mathbf{s}^0)'g(\mathbf{s}^{0}, \mathbf{z}) + (\mathbf{s}^0 - \mathbf{s}^1)'g(\mathbf{s}^{1}, \mathbf{z})>0
\end{align}

It is shown, in the paper I cite in the beginning, that a sufficient condition for $\sigma$ to be diagonally striclty concave is that matrix $\left[G(\mathbf{x}, \mathbf{z}) +G(\mathbf{x}, \mathbf{z})' \right]$ is negative defite for $\mathbf{s} \in S$, where $G(\mathbf{x}, \mathbf{z})$ is Jacobian of pseudogradient $g$ with respect to $\mathbf{s}$. I use ' to denote transpose of a matrix.
What is the intuition behind diagonal strict concavity condition?

**Tags:** game-theory, nash-equilibrium

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## 2 answers

### Answer by username_2 on 2021-03-24. Score: 6 | Accepted answer

The diagonally strict concavity property is better known as the **strict monotonicity** property of the pseudo-gradient.

An operator $\Psi:\mathbb{R}^n \to \mathbb{R}^n$ is strictly monotone if the following holds true:
$$\forall x\_0, x\_1 \in \mathbb{R}^n: \left< x\_0 - x\_1, \Psi(x\_0) - \Psi(x\_1) \right> >0. $$

In your case $\Psi$ would be $g$.
The intuition behind it is that for every line of $R^n$, the projection of $\Psi$ on that line is strictly monotone.

I join here an example: the payoff function are given by $\varphi\_i$ for both players $i \in {1,2}$.

[![enter image description here](https://i.sstatic.net/dlzGW.png)](https://i.sstatic.net/dlzGW.png)

[![enter image description here](https://i.sstatic.net/jKEHo.png)](https://i.sstatic.net/jKEHo.png)


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### Answer by username_1 on 2017-10-25. Score: 6

So you want to find a maximum of $\sigma(s,z)$. If $\sigma$ is diagonally strictly concave you can do so by starting at any point and just following the gradient $g(s,z)$ until you find the maximum and no matter where you start, you will always end up at the same point (Start at the lower black points and follow the direction of the gradient (the direction of the steepest ascent).).
[![Following the gradient in a diagonally strictly concave function](https://i.sstatic.net/pLQRe.png)](https://i.sstatic.net/pLQRe.png)

However, if $\sigma$ isn't diagonally strictly concave, you could end up at different maxima by starting at an arbitrary point and following the gradient (Follow the direction of the steepest ascent starting from the two lower black dots; you'll end up at two different points.).
[![Following the gradient in a non-diagonally strictly concave function](https://i.sstatic.net/wsE3S.png)](https://i.sstatic.net/wsE3S.png)

> **Comments:**
> **username_0**: Thanks for your answer! What you write is essentially one of the results in the original Rosen's paper.  When I say intuition I mean what  property of the strategic interaction in the game is captured by the strict concavity condition? For example,  does this condition say something about how other players'  actions affect player i's payoff,  or how player i's action affects other players' payoff in the game.   Sorry if I wasn't clear enough in the question.  

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