During a lesson, my professor showed that the following 2-player game has a unique Nash equilibrium. I was curious to figure out if it also has a unique correlated equilibrium or not, but I failed miserably so far.
Here is a streamlined version of the game, which should be enough to capture the issues: simultaneously, Player 1 (the merchant) decides a price $p\in[0,1]$ and Player 2 (the customer) commits to buying a quantity $q \in [0,1]$ of the good or service on sale. The utility of the merchant is $u_1(p,q) = pq$ and that of the customer is $u_2(p,q) = \frac 1 6 (2-q)q - pq$. The constants are not particularly meaningful, the gist is that some money $pq$ is exchanged and the customer gets some (polynomial) utility from the purchase.
Now, the problem I am facing is that the game is infinite, and I do not know any way to reasonably attack the problem, other than trying to apply the definition of correlated equilibrium, i.e. (see https://arxiv.org/pdf/0812.4279.pdf), a joint probability measure $\pi$ on $[0,1]^2$ such that $$ \int_{[0,1]^2} \Big[ u_1 \big( f(p), q \big) - u_1 (p,q) \Big] \, \mathrm{d} \pi(p,q) \le 0 \quad \text{and} \quad \int_{[0,1]^2} \Big[ u_2 \big( p, g(q) \big) - u_2 (p,q) \Big] \, \mathrm{d} \pi(p,q) \le 0 $$ for any measurable functions $f,g\colon [0,1] \to [0,1]$.
This is leading nowhere, though, because it is essentially a variational problem in a space that is way too big for my brain to comprehend. There must be some clever way to solve this... or maybe for people used to maximize integrals over spaces of distribution, it's an easy task. Not sure but I sure as heck would appreciate any insight!