Nash Equilbriums

 

I Am the Very Model of a Modern Major-General

I know our mythic history, King Arthur's and Sir Caradoc's,
I answer hard acrostics, I've a pretty taste for paradox,
I quote in elegiacs all the crimes of Heliogabalus,
In conics I can floor peculiarities parabolous.
I can tell undoubted Raphaels from Gerard Dows and Zoffanies,
I know the croaking chorus from the Frogs of Aristophanes,
Then I can hum a fugue of which I've heard the music's din afore,
And whistle all the airs from that infernal nonsense Pinafore.

Sometime a Paradox is hiding a deeper truth.

Governments in the US are individuals acting as if they were a system. Individuals CAN function as a system. The “wave” at many sports events is an example of individuals acting together as if they were a system.

Individuals might function as if they are a system, but they are NOT a system. If they were truly individuals, they should adopt  User Optimal solutions. If they were a system, they should adopt a System Optimal solution. Since they are not a system, it  is not surprising that the sum of their User Optimal solutions for a system is more than the System Optimal solution. But to act as if they are a system, the users can adopt a Nash Equilibrium, often called a User Equilibrium, because in that solution no individual can chose a solution that is better for themselves. It is a Nash Equilibrium that is observed. It can be summed, and it will be found to be greater than the System Optimal solution. However the Nash Equilibrium is unique to each system. If you change the system, then you change the Nash/User Equilibrium.

This is the basis for the Braess Paradox. https://en.wikipedia.org/wiki/Braess’s_paradox. When examining a traffic network, “Dietrich Braess, a mathematician at Ruhr University, Germany, noticed the flow in a road network could be impeded by adding a new road, when he was working on traffic modelling. His idea was that if each driver is making the optimal self-interested decision as to which route is quickest, a shortcut could be chosen too often for drivers to have the shortest travel times possible. More formally, the idea behind Braes' discovery is that the Nash equilibrium may not equate with the best overall flow through a network.”  Thus it can be argued that Braess’s Paradox is because people were confusing a Nash/User Equilibrium with a System Optimal.

If you’re unfamiliar with a Nash Equilibrium, Adam Smith was correct that everyone should do what is best for themselves, and while this is a User Optimal, it is incomplete.  Karl Marx was correct that everyone should do what is best for the common good and while that is a System Optimal, it is also incomplete. John Nash appears to be correct AND complete in that everyone should do what is best for them AND the common good, which is a Nash Equilibrium.    https://www.youtube.com/watch?v=vCyZvfRHkC4

Dafermos and Saprrow (Dafermos & Sparrow, 1969) developed what came to be the basis for what is known as an User (Nash) Equilibrium in Travel Demand Modeling. It solved the problem that the impedance on a link of a network depends on how it reponds to the volume on that link, but how the link responds to volume is not known, and thus an iterative algorithm is required to solve for the volume. Nagurney, (Nagurney, 1984), while a post doctoral student of  Dafermos, showed that while the exact response may not be known, the most efficient response could be found, and that a fourth power function, such as the Bureau of Public Roads, BPR, curve, is an efficient solution.  Azizi and Beagan, (Azizi & Beagan, 2022) showed that a discontinuous function, where the discontinuity happens at the link capacity, is an even more efficient solution than the BPR curve. Arguably the most efficient response is the correct response.

It is hardly surprising that the Nash/User Equilibrium is specific to each system. And therefore  if you change the system, you change the Nash Equilibrium. However the Nash Equilibrium is also only possible if some members of the System forego their own User Optimal and all Users block others from pursing their own User Optimal. The farther each User Optimal is from that Nash Equilibrium, the more likely it is for some users to leave that system and seek their own User Equilibrium.

Beagan (Beagan, 2016) appeared to shown that the equation which is used in User Equilibrium is itself a function of the Standard Deviation, σ aka SD, of the system, which is the reliability time that is used is a function of the 95^th percentile time, the mean time of a normal system plus two Standard Deviations.  Reducing the error of a system depends on increasing the number of individuals, n, in the system, i.e. Standard Error = SD/√n. Losing individuals in a system by straying too far from their UO solution can led to increasing error, or even to competing Systems, despite what Braaess's Paradox seems to suggest.

It is suggested that the Nash/User Equilibrium should not be reduced so far from the UO solution that individuals are tempted to leave the system. Thus making the sum of NE, UE, closer to a SO solution may not be desirable if it leads to fewer users in the system. What is instead desired is not NO government, and only UO solutions, but a government which is for ALL individuals. To parrot Lincoln "government of the people, by the people, and for the people". But fooling people, by convincing them to leave the system, is an example Lincoln's “You can fool all of the people some of the time, and some of the people all of the time, but you can’t fool all of the people all of the time.” Vote for the People, not just as a Republican or a Democrat, when you are choosing a government.

Works Cited

Azizi, L., & Beagan, D. (2022, January). Inclusion of Reliability in the Volume Delay Function. Poster Presented at Annual TRB Meeting.

Beagan, D. (2016). Including Reliability in VDF Curves. Prestentation to the 6th TRB Conference on Innovations in Transportation Modeling. Denver, Colorado.

Dafermos, S., & Sparrow, F. (1969). The Traffic Assignment Problem for a General Network. Journal of Research of the National Bureau of Standards, 73B, 91-118.

Nagurney, A. B. (1984). Comparative Tests of Muitimodal Equilibium. Transportation Research Part B: Methodological, 18B(No 6), 469-486.