<p>For anyone else who is interested, I found a cool note online and apparently the following ugly monstrosity is a sufficient condition for strict diagonal concavity for my simplified case:</p>
<p><span class="math-container">$  2s^2_1 \frac{\partial^2 \pi_1}{\partial s_1^2} + 2s_1s_2 (\frac{\partial^2 \pi_1}{\partial s_1 \partial s_2} + \frac{\partial^2 \pi_2}{\partial s_2 \partial s_1}) + 2s^2_2 \frac{\partial^2 \pi_2}{\partial s_2^2} &lt; 0 \hspace{2mm} \forall [s_1 \hspace{2mm} s_2]^T \neq 0 $</span></p>
<p>Still no luck on the intuition...
Reference: <a href="https://ocw.mit.edu/courses/6-254-game-theory-with-engineering-applications-spring-2010/48cb7ae4210825b7de3d7fc0bcc8553f_MIT6_254S10_lec06b.pdf" rel="nofollow noreferrer">https://ocw.mit.edu/courses/6-254-game-theory-with-engineering-applications-spring-2010/48cb7ae4210825b7de3d7fc0bcc8553f_MIT6_254S10_lec06b.pdf</a></p>

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2023-05-28T16:15:24.603Z
2023-05-30T11:24:07.847Z
2023-05-30T11:24:07.847Z
Mwazr A
<p>I am working on a game theory exercise and I want to prove existence and uniqueness of pure strategy NE for a 2 person game with continuous strategy spaces. Lets call the payoff functions <span class="math-container">$\pi_1$</span> and <span class="math-container">$\pi_2$</span> for players 1 and 2. The strategy spaces are simple one dimensional intervals in Euclidean space that are common across players (<span class="math-container">$s_1,s_2 \in S = [a,b]$</span>). What do I need to do to show that the payoff functions are diagonally strictly concave? I am having a hard time wrapping my head around Rosen's paper and I don't need many dimensional strategy spaces and n players for now. I was wondering if there was a simpler, easy to understand version of the result for my simplified game. If someone could also provide some intuition for that condition, it would also be a massive help! Thanks</p>

<game-theory>
Proving existence of a unique pure strategy Nash Equilibrium in two-person continuous games
Mwazr A
# Proving existence of a unique pure strategy Nash Equilibrium in two-person continuous games

(Asked by: username_0 on 2023-05-26)

I am working on a game theory exercise and I want to prove existence and uniqueness of pure strategy NE for a 2 person game with continuous strategy spaces. Lets call the payoff functions $\pi\_1$ and $\pi\_2$ for players 1 and 2. The strategy spaces are simple one dimensional intervals in Euclidean space that are common across players ($s\_1,s\_2 \in S = [a,b]$). What do I need to do to show that the payoff functions are diagonally strictly concave? I am having a hard time wrapping my head around Rosen's paper and I don't need many dimensional strategy spaces and n players for now. I was wondering if there was a simpler, easy to understand version of the result for my simplified game. If someone could also provide some intuition for that condition, it would also be a massive help! Thanks

**Tags:** game-theory

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## 1 answers

### Answer by username_0 on 2023-05-28. Score: 0 | Accepted answer

For anyone else who is interested, I found a cool note online and apparently the following ugly monstrosity is a sufficient condition for strict diagonal concavity for my simplified case:

$ 2s^2\_1 \frac{\partial^2 \pi\_1}{\partial s\_1^2} + 2s\_1s\_2 (\frac{\partial^2 \pi\_1}{\partial s\_1 \partial s\_2} + \frac{\partial^2 \pi\_2}{\partial s\_2 \partial s\_1}) + 2s^2\_2 \frac{\partial^2 \pi\_2}{\partial s\_2^2} < 0 \hspace{2mm} \forall [s\_1 \hspace{2mm} s\_2]^T \neq 0 $

Still no luck on the intuition...
Reference: <https://ocw.mit.edu/courses/6-254-game-theory-with-engineering-applications-spring-2010/48cb7ae4210825b7de3d7fc0bcc8553f_MIT6_254S10_lec06b.pdf>


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