It really is. You can start to get a sense of just how involved the corrugation is with an image like this but it only hints at the surface features required.
Nash himself notes that the unique “user equilibrium” is similar to the Nash embedding theorem and it has pragmatic applications in areas such as fairness in traffic flow and such. It’s not an impossible problem but it can get VERY very complicated indeed, no matter HOW you map it out or utilize it.
Not in the original, no. But there are modifications that try to incorporate some aspect of direction.
I think the best place to find them is looking up “user equilibrium” and “directed acyclic graph”, as there’s a LOT of work done there for the purposes of optimizing traffic flow in networks (roads and computer networks alike) to calculate individual path costs.
As you can see by the introduction, “static user equilibrium traﬃc assignment (UETA)” is what’s in use, which is a similar problem to Nash embedding.
The difficulty they face is similar to the difficulty you face, which is for them, “A well-known limitation of this model has to do with its inability to produce a unique path ﬂow solution”.
I bring up the analogous problem because there’s simply not much work out there involving directed acyclic graphs in “nash embedding” but there is a LOT of DAG work done for “user equilibrium”.It’s where the $$$ is: traffic flow, and with funding comes many proposed solutions.I think it’s where you may find inspiration.
https://en.wikipedia.org/wiki/Probability_density_functionOh look, back to Dirac. “Link between discrete and continuous distributionsIt is possible to represent certain discrete random variables as well as random variables involving both a continuous and a discrete part with a generalized probability density function, by using the Dirac delta function. For example, let us consider a binary discrete random variable having the Rademacher distribution—that is, taking −1 or 1 for values, with probability ½ each. The density of probability associated with this variable is: “
If you had a ball of clay and a pile of sticks, you could do it but each corrugation would take a bit of water and sculpting to make pretty.