Was Arrow’s Impossibility Theorem an Endorsement of Free Market Capitalism?
By John Lawrence
Kenneth Arrow, Nobel prize winning economist, passed away February 21, 2017 at the age of 95 at his home near Stanford University where he was Professor Emeritus. He was the youngest person ever to win the Nobel prize. Dr. Arrow made many academic contributions, but chief among them was his Impossibility Theorem.
The theorem, in my opinion, is an example of the over-mathematization of the social sciences. As such, it is absolutely elegant and correct but it’s implications on a practical level — that capitalist economics and representative government are the best a society can do — Arrow himself could not defend in his later years. In fact he suggested at the age of 90 that further research would come up with ways to work around his system if not discredit it altogether.
The damage, however, had already been done. Capitalism, as evidenced by his nephew, Larry Summers, was exalted while welfare economists and political reformers were stopped in their tracks. Larry Summers is an economist and President Emeritus of Harvard University. He served as the 71st Secretary of the Treasury for President Clinton and the Director of the National Economic Council for President Obama.
Summers pushed Clinton for deregulation of the Glass-Steagall Act which led directly to the 2008 financial meltdown. When he was President of Harvard, he gained notoriety for a sexist remark in a 2005 speech in which he suggested that the under-representation of women in science and engineering could be due to a “different availability of aptitude at the high end,” and less to patterns of discrimination and socialization. Summers evidently took his uncle’s Impossibility Theorem to heart and championed free market capitalism. An encomium to his uncle can be found here.
Plurality Voting Holds in Most US Elections
Dr. Arrow said in a 2012 interview:
“I think the answer is you have to ask, in effect, which ones get closest to this combination? And we have to then begin to look at what actual votes are. The real way we do this is to apply some rule and to take elections and apply different methods and see what violates these conditions as little as possible. Remember all we’re saying is there could be a [violation]. We’re not saying you’re always getting a violation of these rules.”
What Arrow was saying is that there are other political and economic systems that, while not perfect according to his criteria, were nevertheless possibly better than what we have today. In fact Arrow decried the pitfalls of winner-take-all (plurality) voting which does satisfy his criteria. It chooses one winner out of a number of candidates and is the method used in almost all US elections, including the election for President.
Arrow had this to say about plurality voting:
“I think plurality voting leads to very unsatisfactory solutions in a number of cases. Now, there is a problem which certainly my social choice theorem didn’t address and a lot of discussions don’t address about elections, which is the long-run implications of an electoral system on parties.
“Partly, if you have a plurality system, you’re kind of driven to a two-party system. If a party splits, both factions lose because they’re less likely to get a plurality. So there’s some pressure to create two-party systems.
“Now some people have argued that two-party systems are a good idea. And others say no. It stifles innovation. It stifles real contest. New ideas are suppressed in a two-party system. I’m inclined to the latter view that if there really are a number of political positions, it’s better they be raised explicitly.
“The plurality system chokes off free entry. In other words, in the economic world we’re accustomed to the virtues of free entry. We don’t want a small number of corporations to be dominant. We favor the idea of new firms entering in order to compete to bring in new ideas, to bring in new products.
“Well, the same way in the political field. We should be encouraging free entry, I think, in order to have new political ideas come in. And they may flourish. They may fade. That’s what you want, them to be available.
“So I’m inclined that the plurality system will choke off by encouraging, the two-party system will choke off new entry. So I’m really inclined to feel that we don’t want plurality as a voting system. It’s likely to be very stifling.”
Social Choice: Mathematics Meets the Social Sciences
Social choice is an esoteric, abstruse and specialized subject generally only of interest to the academic community and then only to those interested in political or economic science. In this article I will try to simplify the concepts and point out how they could be applicable in developing more democratic political and economic systems.
In 1951, Kenneth Arrow published his book, Social Choice and Individual Values, in which he claimed that the amalgamation of individual choices whether in a political system (voting, for example) or in an economic system (consumer choice, for example) to form an overall social choice (such as the winner of an election) was impossible if you postulated a simple set of (supposedly) rational and ethical criteria.
As Arrow stated in the first sentence of his book, “In a capitalist democracy there are essentially two methods by which social choices can be made: voting, typically used to make ‘political’ decisions, and the market mechanism, typically used to make ‘economic’ decisions.”
He goes on to say, “The methods of voting and the market … are methods of amalgamating the tastes of many individuals in the making of social choices.” They are similar in that both involve collective choice among a limited range of alternatives.
Arrow then went on to make a lengthy mathematical analysis, at the culmination of which he asserted that social choice was impossible. This has come to be known as Arrow’s Impossibility Theorem. For this and other work in a distinguished career, Arrow won the Nobel prize in economics in 1972.
The fact that social choice was apparently impossible put the kibosh on any further work in this field and led to much despair among voting theorists and welfare economists. If there is no rational way to make social decisions based on the amalgamation of individual ones, not only welfare economics but any hope for economic democracy is ruled out. Also democratic voting systems other than majority rule in first-past-the-post (plurality), one member districts are ruled out.
The dichotomy between political and economic systems remains with the implication being that representative democracy and capitalist economics are the best systems that can be attained. Amartya Sen in his Nobel Prize acceptance speech said, “Two centuries after the flowering of the ambitions of social rationality, in Enlightenment thinking and in the writings of the theorists of the French Revolution, the subject seemed to be inescapably doomed.”
Kenneth Arrow and his Theory: Getting Down to the Nitty-Gritty
Let’s take a look at Arrow’s analysis. As a simple example let’s take a voting system in which there are three candidates: A, B and C and three voters: 1, 2 and 3. Arrow postulates that each voter will rank the candidates in order of preference: A is preferred to B is preferred to C would be indicated ABC. One of Arrow’s criteria is that for every possible arrangement of voter preferences there should be a unique social choice consisting of one and only one ranking, for example, ABC.
So what about the case when voter 1 votes ABC, voter 2 votes BCA and voter 3 votes CAB?
If we let ABC be the social choice, we notice that two out of three voters preferred C to A so that can’t be it. Likewise for BCA and CAB. The phenomena is called the paradox of voting and was discovered by the Marquis de Condorcet in the 1700s.
Arrow could have stopped right there and said that social choice was impossible according to his terms, but he didn’t. He went on to mathematicize the heck out of the problem reaping much credit in the process. The stopped progress in social theory is at least partly due to the fact that no one has been able to come up with a more elegant mathematical analysis!
But wait a minute. Is Arrow’s set-up of the problem even realistic? I say it’s not. Arrow demands that a social choice be a complete ordering such as ABC. That’s neither realistic nor necessary. For multiple candidates and multiple winners, all that’s necessary is to know which ones are in the winning set, not their order. For instance, if A, B, C and D are candidates, and the outcome produces two winners rather than one, the only necessary information is which two winners, not their ordering. If A and B are both winners such as in a two-member district, for example, we don’t need to know if A is preferred to B or vice versa.
Proportional Representation is a voting system for choosing multiple winners and has been around for years. Whereas first-past-the-post (plurality), single member districts lead to a two party system, Proportional Representation induces a multi-party Congress or Parliament in accordance with the percentage of votes received by the party. While a big improvement over majority rule, proportional representation has no theoretically optimal underpinning.
Arrow states: “In the theory of consumer’s choice, each alternative would be a commodity bundle; in welfare economics, each alternative would be a distribution of commodities and labor requirements.” It is clear that in consumer’s choice, the collective choice mechanism would narrow down the possible number of commodity bundles to a “limited range of alternatives” in which case there would be no need to order each alternative in the “limited range.” Consumers would just choose the commodity bundle that represented their highest preference.
So Arrow’s fundamental assumptions were somewhat arbitrary to say the least. But other academics were so impressed with his mathematical prowess that they gave him the Nobel Prize. And evidently, rather than continuing work which might have shown the limitations of his Impossibility Theorem, Arrow was content to sit on his laurels and be a respected professor at Stanford University, a conservative enclave if there ever was one. (Truth in advertising: I’m a graduate of Stanford University.)
Economic Democracy, Co-ops and Shift Scheduling
As for welfare economics, there are also implications of social choice for economic democracy.
We envision cooperative enterprises in which worker/owners freely choose their shift and pay preferences from the totality of possibilities in accordance with a social choice mechanism which maximizes collective satisfaction among all worker/owners.
Today already shift scheduling software is available for nurses on a first come, first served basis. In a profit making environment, the scheduling software is more about reducing overtime costs and costs in general than it is about maximizing shift satisfaction among employees, but that would change in an employee-owned enterprise.
Arrow assumes that all information will be gathered from the voters or worker/consumers in the form of rankings. (A is preferred to B is preferred to C, etc.) However, there is another form of indicating preferences over candidates and that is ratings. A citizen can indicate preferences by rating candidates on a scale consisting of real numbers between zero and one or minus one and plus one. The scale itself is arbitrary. Then the social choice is made by adding up the numbers for each candidate over all individual choosers. This is called utilitarian voting.
Clearly, this method will yield consistent and rational results. It is also called range voting or score voting which was reinvented by Warren D. Smith. Utilitarianism as a philosophy was invented by Jeremy Bentham and John Stuart Mill, 19th-century English philosophers, who defined their system as “the greatest good for the greatest number.”
Two guys, Brams and Fishburn, came up with a method of voting called approval voting. They said just give every candidate you approve of a one and all others a zero and then tally up the votes. The candidate with the highest number of votes was declared the winner. The problem is how do you choose which candidates to approve in an optimal manner?
I have had a lot of animosity towards Arrow in my lifetime because of his Impossibility Theorem. I have a website dedicated to proving him wrong. Why didn’t he go on to study what was possible even if it wasn’t perfect according to his seemingly rational (but not really) criteria?
There must be a workaround or a compromise for surely nothing is really impossible. But how dare I or anyone else criticize such a formidable mathematical analysis? Well, I do.
Did the conditions he set up doom his theoretical outcome to failure? I think so.
Did Arrow stop there because he was an apologist for the capitalist system and the traditional US method of voting? Did he bask in the approval of free market capitalists who liked nothing better than for someone to prove theoretically that any system other than capitalism was fatally flawed? I think so.
Did he gain brownie points because he supposedly proved that the American political system rested on a firm theoretical footing while parliamentary democracy did not? You bet.
Ah well, may he rest in peace and may other economists and political scientists not be stymied by his overly mathematical but soulless result. As it turns out Arrow’s conditions don’t have much to do with the real world. His analysis represents a theoretical result which amounts to proving how many angels can dance on the head of a pin.
I guess the Nobel committee was impressed by such mathematical legerdemain.