Larry Summers apparently doesn’t understand Arrow’s Impossibility Theorem

My earlier weblog text on Brexit and voting theory was republished by the Royal Economic Society (RES) Newsletter. One reason for the editor to take the piece (and give it a fine edit) was that Kenneth Arrow (1921-2017) has recently passed away, and that the piece highlights Arrow’s Impossibility Theorem in its relevance for the Brexit referendum question. The April Newsletter also contains an obituary of Arrow by Larry Summers, originally published in the Wall St. Journal.

It feels rather awkward to refer to an obituary, yet, as these events happen to coincide, it might serve a purpose.

Summers of course mentions Arrow’s Impossibility Theorem too. His statement indicates that he apparently doesn’t understand it. It might be that Summers does understand it actually, and that only his statement for the obituary was less thoughtful. For now, let us take the quote at face value.

“Drawing upon mathematical logic, it shows that there is no possible voting scheme that can consistently and sensibly reflect the preferences of a set of individuals with diverse views. Any scheme that could ever be invented will be at risk of perverse outcomes, where, for example, the choice between options A and B is affected by the presence or absence of option C; or where a vote switch by one person toward option A makes it less likely to prevail. Mathematical and abstruse it was. But it also explained why committees have so much trouble coming to consistent conclusions and why, with an increasingly polarized electorate, democracy can become increasingly dysfunctional.”

It is false that voting schemes (i.e. decision mechanisms) cannot consistently and sensibly reflect the preferences of a set of individuals with diverse views. It is only true when you confuse voting outcomes and decisions on those outcomes.

To understand the situation, let us take a closer look.

The distinction between voting and deciding

(This section has been adapted a bit from this paper, p3.)

Consider three chess players A, B and C. They are pairwise confronted in a tournament with the result A > B > C > A, meaning that A beats B, B beats C and C beats A. These results happen to be intransitive. The objective of the tournament may only have been to allow the players to play against each other. There need not be a notion to find the “best overall player”.

Even if the result had been A > B > C and also A > C so that the outcome happens to be transitive, then it need not be an issue that A would be the “best overall player”. The fact that A beats the two others need not be associated with a notion that this would be “best”. The question does not have to arise simply because it is not considered to be a relevant question, neither to the players nor the organisers of the tournament. (Indeed, A would be the best under the Condorcet rule but not necessarily under a Borda rule.)

In voting we start out with a similar situation like with chess. The voting scores are like the game scores. If A gets more votes than B then this doesn’t necessarily mean much for the relation to C or the overall situation.

This situation will be called a “voting field”.

There can be a drastic change in objectives. Namely, if the tournament wants to identify an “overall winner”. Then this becomes the issue of “direct single seat elections” (to distinguish the situation from the election of for example a multiple seat parliament or the indirect selection of the prime minister via such a parliament).

The notion of an overall winner amounts to using a “social decision function” (SDF). The SDF selects the winner from a list of candidates. It is the definition of the SDF that it does so.

For decisions we require transitivity. Above voting field doesn’t have to be transitive but for decisions we require this. The SDF always implies a ranking. For example, if A = SDF[A, B, C] then the second might be B = SDF[B, C] and then the third would be C. The ranking arises by stepwise dropping the best of the remainder. The ranking means a transitive order of the candidates.

Hence, we distinguish between the voting field and deciding. In everyday parlance we tend to associate voting with deciding. Voting thus tends to mean: using both a voting field and a decision. Hence there is a distinction. Sometimes “voting” can be used in the sense of a “voting field” where the “field” is dropped. “Voting” thus is a somewhat ambiguous term, with some ambiguity about what it is ambiguous about. If one keeps track of the context the meaning however will be clear.

Kenneth Arrow’s confusion

Kenneth Arrow’s Impossibility Theorem comes about by confusing voting fields and deciding.

When we have an intransitive voting result A > B > C > A, then Arrow requires this intransivity to be transitive, because he wants to see a decision. He however assumes something that is inconsistent, whence the impossibility.

Larry Summers isn’t aware that Arrow had this confusion, and copies it.

See my note in the RES Newsletter to see what this confusion means for the Brexit referendum.

The metaphor of a gavel

In some meetings, it is the convention that the chairman bangs the gravel when a decision is made. For example, in a pairwise vote between A and B, A gets more votes but there is no bang of the gavel since it is not a decision but only a mere count. Similarly for the pairwise vote between B and C, when B gets more votes. Similarly for the pairwise vote between C and A, when C gets more votes. Then it is observed that the voting field has A > B > C > A. Now the chairman can decide that the cycle indicates a deadlock, and then bangs the gavel for the decision that there is deadlock. The subsequent step is to search for the rule book and select a tie-breaking rule.

Court gavel (By Jonathunder – Own work, GFDL, wikimedia commons)

The crucial role of rules

The crucial question is how one handles deadlocks (indifferences). Theorists of axiomatics for example don’t like randomisation. Imagine Euclid with an axiom on something that is a point or line at random. Yet, to resolve voting deadlocks, people might flip a coin. Would you call it “inconsistent” when a coin shows different outcomes Head or Tail ?

It is also true that a vote switch by one person towards option A might make it less likely to prevail (in the collective outcome). It all depends upon your axioms.

A supposed axiom that isn’t an axiom

Arrow posed some axioms that caused an inconsistency. Thus these axioms cannot be simultaneously true for description of real world events. Democracy is something that we want to work for the real world. Thus democracy must eliminate at least one of Arrow’s axioms. If something is to be dropped, then one should not call it an axiom. The key axiom to drop is the one on pairwise decision making (a.k.a. independence of irrelevant alternatives). One can have pairwise voting results, but these need to be integrated to arrive at a decision. For a pairwise vote it is incorrect to say that a third option would be irrelevant, for it can be quite relevant for the final decision. Option A might get more votes than option B, but when we include option C, then there might be a cycle, A > B > C > A, which amounts to a deadlock or indifference in terms of decision making. In that case the focus shifts to the mechanisms to resolve deadlocks.

Summers on dysfunctional democracy

If democracy is getting dysfunctional, then this is e.g. because of district voting instead of proportional representation (see this paper), and the use of referenda with misleading questions (and the educational system, and the media, and so on).

Dale Jorgenson, once president of the AEA, once referred to Arrow’s theorem as if it implied the need for a dictatorship. This causes me to wonder whether misunderstandings about the theorem would support autocratic thoughts. Larry Summers had an exchange with Elizabeth Warren on the positions of insiders and outsiders. From her autobiography:

“[Summers] teed it up this way: I had a choice. I could be an insider or I could be an outsider. Outsiders can say whatever they want. But people on the inside don’t listen to them. Insiders, however, get lots of access and a chance to push their ideas. People — powerful people — listen to what they have to say. But insiders also understand one unbreakable rule: They don’t criticize other insiders. I had been warned.”

A situation in which insiders don’t listen to outsiders, and don’t criticise each other, reminds of an oligarchy and not an open society and democracy. Perhaps Summers only describes it factually but he also seems to support it. Perhaps his misunderstanding of Arrow’s theorem had misguided him into thinking that he was only taking the scientific point of view, that democracy was dysfunctional by definition to start with.

The latter is pure speculation. If Summers still thinks in terms of insiders and outsiders we might never discover the truth on this.

Arrow, Summers, Warren (Wikipedia Commons, Stanford News Service)


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