Case study in the politics of mathematics: optimality of Nash equilibria

580 words

Back in the Soviet era, Leonid Vitaliyevich Kantorovich got himself in trouble. He developed a theory of optimization problems, originally for the sake of optimizing plywood production, but he saw a potential for using his techniques to optimize the entire Soviet economy. So he sent letters to the central planning bureau to convince them to make use of his ideas.

He has been involved in advanced mathematical research since the age of 15; in 1939 he invented linear programming, one of the most significant contributions to economic management in the twentieth century. Kantorovich has spent most of his adult life battling to win acceptance for his revolutionary concept from Soviet academic and economic bureaucracies; the value of linear programming to Soviet economic practices was not really recognized by his country’s authorities until 1965, when Kantorovich was awarded a Lenin prize for his work.

Excerpt from original CIA file on Kantorovich.

There was just one problem, arising from the theory of linear programming duality. For any linear optimization problem, we can derive a dual problem. If you solve both the primal and dual problems, they turn out to have the same solution value, and moreover the optimal solution to the one certifies optimality of the solution to the other. These “dual solutions” can have clear interpretations. In the case of optimizing resource allocations, the dual solution can be interpreted as market prices.

LP duality theory connects the notion of optimal resource allocation as used for central planning with the efficient market hypothesis and Nash equilibria. This connection can be interpreted as “capitalist markets find the optimal allocation of goods and services”. Obviously, the communists did not like that.

This interpretation has been popular in the US as well. They’d see this interpretation in light of the slightly weaker “first welfare theorem” and Smith’s invisible hand.

Of course there are numerous solid arguments for why these interpretations are bogus. To name just a few: markets are not convex, barriers to market entry are non-zero, humans are not perfectly rational nor omniscient, computation and communication are neither free nor instantaneous, negative externalities are common, the dual LP only takes prices into account and not net cash flow of individuals, and the “welfare function” doesn’t necessarily exist.

All of this was known 50+ years ago, but apart from the 1950 cutesy example of the prisoner’s dilemma, game theorists didn’t do much of note to dispute the notion that Nash equilibria are typically good. Nothing really got formalized until 20 or so years go with the introduction of the price of anarchy. The price of anarchy (stability) of a game is the ratio of social welfare in the optimal solution versus the social welfare in the worst (best) Nash equilibrium. The PoA and PoS can be unboundedly bad, and recent decades have seen a lot of exciting work happening here.

Mathematics is political. When we want to apply mathematical theorems to the real world, we need to make certain simplifying assumptions. Or actually, if we want to think any thoughts about the real world, but whatever. The simplifications we make determine the theorems we can prove. We can work on theorems that “prove” that capitalism is good or we can work on theorems that “prove” that capitalism is bad. Both would be valid mathematics, but both individuals and communities can decide to value one over the other, resulting in undergrads being taught either “10/10 scientists agree: capitalism is good” or “beware: capitalism can be arbitrarily bad”.

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