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


1 Answer 1


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


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

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