Tuesday, January 27, 2015

Lee Smolin's take on Arrow-Debreu

I've been reading Lee Smolin's (of loop quantum gravity fame) take on Arrow-Debreu equilibrium (mentioned by Tyler Cowen awhile ago), and I was struck by how much we are looking at the economic problem the same way. It's probably just our shared backgrounds in particle physics. His take is a lot more sympathetic to the idea that gauge symmetries and other mathematics may be of use, something I would probably share if the information transfer model didn't seem to point to non-ideal information transfer or spontaneous drops in entropy from time to time [1]. Anyway, I've put together (mostly just for my own notes) a collection of quotes from Smolin's paper with something similar I've said on this blog. I follow the quotes with square brackets that get at the differences between what we are saying.

Let's go ...

Smolin: We are then interested in the simplifications that may happen in limits of large numbers, such as a large number of agents, or of goods. In these limits there may be universality classes that only depend on a few parameters that characterize the behaviors of the human actors that comprise the economy

Me: In reality, there may be a detailed balance that keeps the equilibria in an equivalence class described by e.g. a given NGDP growth rate. But that's the rub! Macroeconomics is the study of the behavior of those equivalence classes, not the instances of them!
[This is basically the same thing -- universality class is a particular kind of equivalence class.]
Smolin: The observables of economics are accounting and other records. One should then try to construct a theory of economics that involves only observables. The importance of this kind of operational principle in physics and other sciences have been paramount. Let’s see what it can do for economics. This restricts attention to what is actually measured by companies, individuals and governments. And it removes from consideration fictional elements that have nothing to do with how real economies work such as fixed spaces of products, fixed production plans, utility functions etc

Me: Game theory is a representation of economic microfoundations; the information transfer framework makes as few assumptions about microfoundations as possible. We are assuming we don't know any game theory. ... The following is a rough sketch, but one way to think about the information transfer model is as an ensemble of games with random payoff matrices (zero-sum and not) with players repeating games, switching between games and possessing some correct and/or potentially incorrect information about the payoff matrix (or its probability distribution). The only "constraint" is that all of the realized payoffs sum up to a macroeconomic observable like NGDP.
[Where Smolin says leave out unobservable things like utility, I say assume you know nothing about them -- in this case a payoff matrix in game theory is essentially a set of contingent utility functions. Expectations also seem unobservable in this sense.]
Smolin: An analogy to physics might be helpful here. Just like there is micro and macro economics, there is micro and macro physics. The former is atomic physics, the latter includes thermodynamics and the description of bulk matter in different phases. Macrophysics mostly deals with matter in equilibrium. The bridge between them is a subject called statistical mechanics, which is a general study of the behavior of large numbers of atoms, both in and out of equilibrium. Indeed, even though there is not a close analogy between the notions of equilibrium in economics and physics, there is clearly a need for a subject that might be called statistical economics. It would be based on a microscopic model of the basic agents and operations or processes that make up an economy and study the behavior of large numbers of them in interaction.

Me: ... assume the principle of indifference: given the macrostate information you know (NGDP, price level, MB, unemployment, etc), assume the system could be in any microstate consistent with that information with equal probability [2]. In Bayesian language, this is the simplest non-informative prior. This way lies statistical mechanics, thermodynamics and information theory.
[This is the same idea.]
Smolin: Furthermore, since equilibria are in general non-unique, there is no mechanism in the theory to explain why one rather than another of these equilibria could be chosen by the market mechanism. All we know about the market mechanism in the theory is that it looks for Pareto efficient states, but if there are many the market mechanism cannot choose among them.

Me: ... imagine a world where fuel was slightly more expensive and cars were slightly less expensive. Depending on the relative price this could still clear the market with the same amount of money being spent in aggregate on cars and fuel. ... However! If there is less fuel and more cars, then there might be fewer ways to associate gallons of fuel with cars (lower entropy since the fuel is fungible) and you could select the actual equilibrium based on maximum entropy production.
[Maximum entropy selects which Arrow-Debreu equilibrium; this quote represents a possible solution the problem in the quote from Smolin.]
Smolin: Time must be incorporated in a way that recognizes the irreversibility of most actions taken, as well as the asymmetry of the present, past and future.

Me: Basically, because the market is made of people, we can violate the second law of thermodynamics (ΔS > 0) by coordinating ourselves in a way that atoms or particles can't. There is no second law of econo-dynamics because of human behavior -- which is unfortunate because otherwise (if human nature didn't matter) ΔS > 0 would imply ΔNGDP > 0 -- the economy would always grow (absent real shocks like natural disasters or resources running out).
[The second law holds most of the time as there is usually economic growth; a recession represents moving backwards. My arrow of time in economics is essentially the thermodynamic arrow of time, with some exceptions during recessions. Entropy producing processes are irreversible processes. This is another case where what I am saying is a solution to a problem stated by Smolin.]
Smolin: Markets with large numbers of agents have very large approximate symmetries, expressing the fact that there are many individuals with similar educations, interests or aspirations and many firms competing to offer similar products and services. In the steady states reached by real economies these symmetries are usually broken. This leads to multiple equilibria or steady states. The representation theory of the broken symmetries is then relevant to the distribution of equilibria.

Me: If your macro system appears to be described by n << N degrees of freedom, then it seems highly likely that among the total number of microstates, large subsets of the microstates are going to be described by a given macro state -- i.e. the equilibrium (the microstate satisfying macro constraints) is not going to be unique. For example, in an ideal gas, you can reverse the direction of the particle velocities and obtain another equilibrium (actually, all spatial, rotational and time-reversal symmetries lead you to other equilibria).
[There is a slight difference here in that Smolin is being much more accurate from the physics perspective. Equilibria related to each other by a symmetry transformation are not actually distinct (e.g. reversing the particle velocities) in physics. The sense here is that you could exchange iPads for Nexus tablets in theory, but the actual dominance of iPads breaks the symmetry leading to two equilibria: one where the Nexus dominates and one where the iPad dominates.]
Footnotes:

[1] Smolin writes: ... we want to know far from ideal a real economy may be, and still count as evidence for the theory. For example, lets take the prediction that all markets clear in equilibrium. There are clearly lots of markets in the real world that do not perfectly clear. In the information transfer model, we have a theory that tells us the 'neoclassical' view holds if the system is in information equilibrium (we don't have episodes of non-ideal information transfer). Essentially, if the information transfer model holds assuming ideal information transfer, we have evidence for the ideal neoclassical theory -- except for the emergent aspects of macro like liquidity traps and nominal rigidity.

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