In an earlier post, I started this 3-part series expanding on some of the key points raised in episode #116 of the InFi podcast, which was my two-hour-plus conversation with Eric Weinstein. If you didn’t catch that earlier post, you should definitely review it to understand the context. In the succeeding post (i.e. Part 2 of 3 in this series), I elaborated on the Weinstein/Malaney application to price indices.
And finally, in today’s post (i.e. Part 3 of 3), I’ll respond to the claim that Eric Weinstein couldn’t possibly have discovered what he claims to have discovered, because other papers have literally proved that it would be impossible to do so. Specifically, I’ll show why Eric’s work does not fell prey to Kenneth Arrow’s celebrated Impossibility Theorem on social welfare rankings, nor is it in the crosshairs of Van Veelen’s more recent Impossibility Theorem on currency comparisons.
The point of my defenses here is the broader claim I am making, to wit: The mathematical economics community is not giving Weinstein and Malaney the attention they deserve. That doesn’t mean their results are as important as Eric claims, but it certainly does mean the glib dismissals (such as coming from this video) miss their mark.
Kenneth Arrow versus Eric Weinstein
Eric told the actual anecdote during the original interview (starting at the 81-minute mark), but here’s the quick summary: When he first learned of Eric’s alleged results, Kenneth Arrow thought they couldn’t be right, because it would seem to contradict his own celebrated work. Therefore Eric had to be wrong.
Here’s what Arrow was thinking: His own Impossibility Theorem had shown that if you try to find a procedure that will take the ordinal rankings (meaning 1st, 2nd, 3rd, etc.) of various “choices” or “options” available to society—it could be political candidates but it doesn’t have to be—and somehow aggregate them into a “social ranking,” then it is impossible for that procedure to simultaneously satisfy certain axioms that seem quite reasonable.
For example, Arrow requires that if every single person in the community thinks Option A is better than Option B, then the “social ranking” had better also think that A is better than B. For a different requirement, Arrow says that the social ordering should be transitive: If A is preferred to B and B is preferred to C, then A had better be preferred to C as well.
Arrow’s Theorem also insists that the procedure does not simply take one individual’s ordinal rankings and do a copy-paste onto the “social” ranking. In Arrow’s framework, this person would be a dictator, and one of Arrow’s axioms is that there be no dictator.
With quite reasonable conditions like these, Arrow showed that there does not exist a formal procedure that takes any combination of individual rankings and spits out a “social” ranking, while satisfying all of Arrow’s criteria.
Now, in that mindset, Arrow encountered the claim from some guy Eric Weinstein that he and Malaney had found a logical, economically meaningful way to put the tastes of Jim Smith in the year 1960 on a non-arbitrarily “equivalent” footing as the tastes of Jim Smith in the year 1970. Weinstein/Malaney claimed that even though prices would have changed and even Jim Smith’s preferences could have evolved over the decade, nonetheless their use of gauge theory allowed them to handle the situation in a coherent fashion.
But, Arrow reasoned, this seemed to be formally equivalent to the case of comparing Jim Smith’s preferences in 1960 with Bill Johnson’s preferences also in 1960. And Arrow’s Theorem showed that that was impossible. So how could Weinstein claim that he had solved it in the case of Jim Smith in 1960 versus Jim Smith in 1970?
Weinstein Responds to Arrow
As Eric explained in the video, he pointed out two key differences: First, Arrow’s results were in the context of “social choice theory.” But Eric argued that an agent embedded in a market with various prices is a different setup.
The other difference—and the one I find decisive—is that in Eric’s framework, Jim Smith in 1960 evolves over time into Jim Smith in 1970. There’s a connection between past-and-future Jim Smith that isn’t present in Arrow’s framework, with discrete individuals who have subjective preference rankings that might have no consistency whatsoever.
Thus, even though Arrow had shown there is no coherent way to compare various individuals’ rankings at a single point in time, his proof does not rule out Eric’s claim to have discovered a coherent way to compare a single individual’s rankings over various points of time.
The Challenge from Van Veelen
In a 2002 article in a leading economics journal, Mathijs van Veelen published a theorem showing it was impossible to compare the volume of real (i.e. inflation-adjusted) economic output in three or more countries with a measure that obeyed certain desirable properties. Van Veelen himself in a podcast interview declared that Weinstein and Malaney couldn’t be right, because their approach was supposed to be a coherent procedure for measuring the change in price level over time—something van Veelen’s paper showed to be impossible.
Van Veelen’s paper listed four reasonable-sounding axioms or requirements for a coherent multilateral index of a country’s real output. One of them was the independence of irrelevant alternatives. In Van Veelen’s paper, the idea was that if you’re trying to see how much higher per capita output is in Germany compared to India, all that matters are the prices (in euros) and quantities sold in Germany, versus the prices (in rupees) and quantities sold in India—it doesn’t matter what real output is in Canada.
But here is Weinstein’s escape hatch: In his framework, van Veelen’s insistence on the independence of irrelevant alternatives would mean that when assessing the prices faced and quantities purchased by Jim Smith in 1960, versus Jim Smith in 1970, it shouldn’t matter what prices and quantities Jim Smith experienced in 1965. The only information we are allowed to consider are the prices and quantities in the starting and ending year.
But as we spelled out in Part 2 of this series, Eric argues that path dependence in this setting is a feature, not a bug. So yes, his approach doesn’t satisfy van Veelen’s axioms, but Eric would argue that it’s because van Veelen has unjustifiably restricted the set of viable procedures for assessing subjective human welfare over time.
Dr. Robert P. Murphy is the Chief Economist at infineo, bridging together the dependability of Whole Life insurance policies with the benefits of blockchain-based finance.
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