The dark horse and a 2012 interview with Kenneth Arrow

Given Kenneth Arrow’s impossibility theorem, it is a fair question to ask what voting system he himself would advise. There is a 2012 interview with him, with a phone recording and transcript, by Aaron Hamlin of the Center for Election Science. Arrow’s advice is:

  • Not plurality and no US Electoral College, with its winner-take-all selection of the US President
  • Not approval voting, since this uses too little information
  • A system that uses more information:

“Dr. Arrow: Well, I’m a little inclined to think that score systems [range voting] where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best. (…) In France, [Michel] Balinski has done some studies of this kind which seem to give some support to these scoring methods.”

His statement about strategic voting – or manipulation:

“Dr. Arrow: There’s only one problem that bothers me about that. And that’s something my theorem really doesn’t cover. In my theorem I was assuming people vote sincerely. The trouble with methods where you have three or four classes, I think if people vote sincerely they may well be very satisfactory. The problem is the incentive to misrepresent your vote may be high. In other words, a classic view is that there’s a candidate I really like, but I know is hopeless. I may put him down at the bottom and vote for the next candidate simply because I feel there’s a chance. Now, if you have a very large electorate you might say no individual has much of an incentive to misrepresent. But I’m not sure. You probably need experience rather than theory.”

Observe that Arrow cautiously states “a little inclined to think (…) probably the best”. His advice to have more empirical research can be supported. The interview touches on some points that call for a closer discussion, also in the light of this earlier weblog text.

Definitions

In plurality, voters only can vote for their best candidate. In a district, often the one with the highest score wins, which is the “first past the post” (FPTP) system. If there are only two candidates, then the winner will also have more than 50%. If there are more candidates, the winner may have less than 50%. There may be ways to assure that a final vote only concerns two candidates. A Putin hack that eliminates a particular candidate will not quickly be accepted, yet, voting theorists still wonder what method would be reasonable. A current example is that Donald Trump got elected with 46% of the popular vote, while Hillary Clinton got 48%. With a turnout of 60% Trump has only 28% support in the electorate, while the House of Congress depends upon district results too. A prime minister who is elected by a coalition in a parliament that has proportional representation (PR) generally has more than 50% support in parliament, and by representation also in the electorate.

In approval voting, voters mention which candidates they approve. The candidate with the highest total approvement is selected.

  • In economics this links up with satisficing (Herbert Simon).
  • Strategic voters will tend not to approve of candidates that might harm their best candidate (even the second best), so that this system devolves into plurality. Steven Brams claims that such fears are overrated but are they ? Brams declines to look into non-satisficing alternatives like the Borda Fixed Point method.

In Borda ranking, each voter puts the candidates in order of preference, and assigns rank numbers.

  • In economics this reflects the notion of ordinal utility.
  • Strategic voters will give a low score to candidates that harm their best candidate (even the second best), which means that “dark horses” (of mediocre approval) might win. See the discussion below.

In range voting, the voters grade the candidates like on a report card, and the candidate with the highest grade point average (GPA) wins. There is the tantalizing but empirically perhaps small complexity of the distinction between a 0 grade (included in the GPA) and a blank vote (not included in the GPA).

  • In economics this reflects the notion of cardinal utility (with voters restricted to the same range).
  • Strategic voters will give a low score to candidates that harm their best candidate (even the second best), which means that the system devolves into plurality. (The use of ordinal preferences and Borda explicitly intends to resolve this again.)

(See also the distinction in levels of measurement.)

Beware of the distinction between cardinal and ordinal preferences

Arrow’s impossibility theorem is about aggregating individual rank orders into a collective rank order. The theorem uses rank orders, or ordinal preferences. Arrow does more than only use rankings. He also defends the “axiom of pairwise decision making” (APDM) a.k.a. the “axiom of independence of irrelevant alternatives” (AIIA) as reasonable and morally desirable (Palgrave Dictionary of Economics).

Range voting allows more information than just ordinal preferences, and it is similar to cardinal preferences (but limiting people to the same range). Cardinal preferences imply ordinal preferences. Yet rank voting doesn’t satisfy the requirements of Arrow’s impossibility theorem, for cardinality violates APDM or AIIA.

One might say that Arrow’s theorem is not about voting systems in general, since it only looks at ordinal and not at cardinal preferences. Instead, Arrow’s position is that he looks at voting theory in general and only proposes axioms that are “reasonable” and “morally desirable”  When cardinality and range voting are excluded from his axioms, then it is because they would be unreasonable or morally undesirable.

These distinctions are discussed – and Arrow’s notions are debunked – in my “Voting theory for democracy” (VTFD). (See especially chapter 9.2 on page 239.)

Arrow’s theorem is only about those voting systems that satisfy his axioms. Since his axioms cause an inconsistency, there is actually no system that matches his conditions. Something that doesn’t exist cannot be reasonable and morally desirable. Arrow’s theorem confuses voting results with decisions, see this earlier weblog discussion.

However, there still remains an issue for voting theory. Range voting allows more scope for strategic voting or manipulation. The reason to restrict votes to rank orders is to reduce the scope for strategic voting.

Gerry Mackie’s “Democracy Defended

A reader alerted me to Gerry Mackie’s thesis with Jon Elster, now commercially available as “Democracy Defended“. I haven’t read this but the blurb seems to confirm what I have been arguing since 1990 on Arrow (but not on Riker).

“Is there a public good? A prevalent view in political science is that democracy is unavoidably chaotic, arbitrary, meaningless, and impossible. Such scepticism began with Condorcet in the eighteenth century, and continued most notably with Arrow and Riker in the twentieth century. In this powerful book, Gerry Mackie confronts and subdues these long-standing doubts about democratic governance. Problems of cycling, agenda control, strategic voting, and dimensional manipulation are not sufficiently harmful, frequent, or irremediable, he argues, to be of normative concern. Mackie also examines every serious empirical illustration of cycling and instability, including Riker’s famous argument that the US Civil War was due to arbitrary dimensional manipulation. Almost every empirical claim is erroneous, and none is normatively troubling, Mackie says. This spirited defence of democratic institutions should prove both provocative and influential.” (Cover text of “Democracy Defended“)

My point however would be that issues of cycling are of concern, like we see with the Brexit referendum question. The concern causes support for representative democracy with proportional representation, rather than populism with referenda.

The key context is switching to parliaments with PR

Discussions about voting theory best be seen in the context of the switch towards parliaments that are elected with PR and that select the prime minister. The president may have a cerimonial role and be elected by parliament too (like in Germany).

It is most democratic when there is proportional representation (PR) of the electorate in the elected body. The more complex voting methods can then be used by the professionals in the elected body itself only. A prime minister is best elected by a parliament with PR, instead of a president by direct elections.

The interview with Arrow contains a criticism on plurality and FPTP compared to PR.

“Dr. Arrow: Yes. I think definitely. I think there’s no question about that. 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 dominate. 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.”

“(…) proportional representation [PR] plays very little role in The United States, but they do play a role in a number of countries. And the question of whether single-member districts are appropriate or not. The Germans, for example, have some kind of compromise between single-member and broader districts. (…)”

See my comparison between the Dutch PR and the UK district system.

Proposals that assume that the voters themselves would use the complexer voting systems – perhaps an enlightened form of populism – are complicating election reform, because these methods put too high demands upon the voters and the electoral process.

In the interview, Arrow referred to the proportional systems, but still expressed the idea that voters themselves would use the three or four categories. In this manner Arrow contributed to this confusion on context.

“CES: If you could, just sort of dictatorially, change something about the way that we do voting in the US, something that would make the biggest impact in your mind, what do you think you would do?

Dr. Arrow: The first thing that I’d certainly do is go to a system where people ranked all the candidates, or as many as they wish, and not just two. And that these data are used in some form or another to choose the candidate, say by eliminating the lowest, or some method of that kind. I’d be interested in experimenting with the idea of categorization and creating interpersonal comparisons by that. And those are the things that I would argue for, and certainly the abolition of the Electoral College. It goes without saying.”

In my experience Arrow is often more confused than one would expect. (1) His original theorem confused voting outcomes and decisions. (2) If he really assumed that people would vote sincerely, then he might as well have assumed cardinality, but he didn’t, for then he wouldn’t have had a theorem. (3) He made a theorem on ordinal preferences but now is inclined to cardinality, even though he defends his theorem that cardinality would be unreasonable and morally undesirable since it doesn’t satisfy APDM a.k.a. AIIA. (4) He now mentions PR but doesn’t draw the conclusion of the selection of the prime minister by parliament, and apparently still thinks in terms of a direct election of the president.

Arrow’s contributions to economics derive from the application of mathematics to economics in the 1950s, and not because he was exceptionally smart in economics itself. Paul Samuelson expressed this idea about himself once too, as a physicist entering into economics. If Arrow had been real smart then he also would have had the common sense to see that his theorem confuses voting results and decisions, and that it amounts to intellectual fraud to pretend that it is more than that.

A major issue is that abstract thinking mathematicians can get lost about reality. In VTFD I show that Amartya Sen is confused about his theorem about a Paretian liberal. Sen’s article with Eric Maskin in the NY Book Review about electoral reform also neglects the switch to a parliamentarian system with PR. A major problem in society is that many intellectuals have insufficient background in mathematics and follow such lost mathematicians without sufficient criticism, even when common sense would warn them.

Warren Smith’s parable of the bees

Warren Smith suggests that bees also use range voting to select the next location for their hive. My problem is that bees aren’t known for strategic voting. My VTFD already suggested – as Jan Tinbergen – that aggregation of cardinal utility would be best indeed. Thus I don’t feel the need to check how bees are doing it.

The problem in voting theory is that humans can vote strategically, also guarded by secrecy in the ballot box. Potentially this strategic vote might be less of a problem when votes for the prime minister in parliament are made public, so that people can wonder why a party has a particular vote. But transparency of the vote might not be the key issue.

Smith on Bayesian regret

Smith has a notion of Bayesian regret, as a more objective criterion to judge voting systems. I am amazed by the existence of such a notion for social optimality and haven’t looked into this yet.

Smith is too enthousiastic about Arrow’s support

Smith interpretes Arrow’s “a little inclined to think” as an endorsement for range voting.  Smith provides full quotes properly – and I must thank him for directing me to this interview with Arrow. But I would advise Smith to be more critical. Arrow mainly indicates an inclination, he is also confused and doesn’t repeal his interpretation of his theorem. Also Smith is advised to grow aware and alert readers of his website that the real improvement in democracy lies not in range voting but in a switch to a prime minister selected by a PR parliament. It is another issue how voting mechanisms operate in other situations, like the Eurovision Song Contest.

Smith’s discussion of the dark horse and the war of the clones

To reduce the options for strategic voting, the voters can be restricted to the use of rankings, and then we get systems like Borda, Condorcet, or my suggestion of the Borda Fixed Point method (BordaFP). The latter wasn’t designed to be a compromise between Borda and Condorcet but still can be seen as one. For example, in the 2010 general elections in the UK, with David Cameron, Gordon Brown and Nick Clegg, it appears that Clegg would be a Borda Choice, but Cameron would still be the BordaFP choice because he would beat Clegg in a pairwise contest.

The reader would enjoy Smith’s discussion of the dark horse and the war of the clones, in his criticism of the Borda method. There is no need for me to repeat his short statement, and I simply refer to here. While you are reading, there is also a picture of Frisian horse Fokke of 2013, and we continue the discussion below it. This discussion is not in VTFD since I mainly pointed to strategic voting but didn’t develop the argument, and thus I thank Smith for his succinct criticism.

 

Frisian Fokke 2013

War of the clones

This assumes the Borda system. Smith (point 8) compares the election between Mush (51%) and Bore (49%) with the election between Mush and some clones Bore1, Bore2, Bore3 (leaving unclear who the real Bore is). Supposedly it is publically known that Mush selects Bore1 in second place, so that the Bores can collect all their votes on Bore1 too. Now Mush loses. This criticism is accurate.  With Condorcet’s rule, Mush would beat all Bores, but the idea of Borda is to mitigate Condorcet. With enough Bores, the BordFP method is not immune to this either.

In above key context, the method would not be applied by the whole electorate but only by parliament. The number of parties would be limited, and each party would only mention one candidate. In the current Dutch parliament there are 13 parties, see Bloomberg with a graphical display of the political spectrum and my analysis on an application of BordaFP. Here the problem doesn’t really arise.

In general people might feel that parties and their candidates differ. If not, then this would require attention. For applications of Borda or BordaFP to smaller committees, it would be sensible to be aware of this. Committees might devise rules about when candidates are too much alike, bunch their votes as if they were one (and rerank), and only call for a decision vote between the clones when they would actually be chosen.

The dark horse

Smith (point 2) considers candidates A, B, C and various nonentities. Kenneth Arrow used the more polite term “irrelevant alternatives”. Let me settle for Dark Horse D. Let me also distinguish truthful voting and strategic voting. In a truthful vote there is no difference between the true preference and the ranking submitted to the ballot box. In a strategic vote there is the strategy provided by the truth and the tactic vote submitted to the box. (Potentially one might design a voting system in which a voter submits those two rank orders simultaneously, but then we must relabel between truth and those two submissions.)

A member of parliament (MP) faces a dilemma. If the MP prefers A > B > C > D then giving the ranks 4, 3, 2, 1 will give 3 points to B, which might cause that B is chosen instead of A. This MP has the incentive to shift points to the Dark Horse, as in 4, 1, 2, 3, hoping that nobody else will vote for this dark horse anyway. If all MPs think in this manner, then the Dark Horse will be elected with an impressive score.

Smith provides an anecdote how such an event happened in the selection of a job application, where there was disagreement about an excellent macro-economist and an excellent micro-economist, whereupon a mediocre candidate got the job.

This is the prisoners’ dilemma. (1) If everyone votes truthfully then they all benefit from the true selection. (2) If everyone votes strategically then they all suffer the worst outcome. (3) Each has an incentive to deflect from the true vote.

The BordaFP method is sturdier than Borda but is not immune to this situation.

A prime answer to Smith is that in parliament the rankings for the selection of the prime minister might be public, so that voters and the press can question party tactics. A party that gives so much points to a Dark Horse might be criticised for not appreciating a better candidate.

Looking for balance

For now, I find Smith’s discussion a bit unbalanced. He emphasizes the disadvantages of Borda, but these have the answers above, for the proper context, while the disadvantages of range voting don’t get as much attention. Range voting stimulates the strategy of giving zero points to alternative candidates, whence it reduces to plurality with all its drawbacks. A candidate with 51% of the vote in plurality might not be better, since more extremist, than a candidate with a higher Borda score who is more moderate. The main point remains that the key issue is that countries with district voting like the USA, UK and France better switch to PR.

By way of conclusion

It remains true that Borda has the risk of a Dark Horse, and that the search for better algorithms is open. How can we elicit information from voters about their true preferences ? In the ballot box we might numb their brains so that they vote like bees (perhaps also with the dance) ?

An idea that I already mentioned at another place: MPs might submit two inputs, one with the strategy (supposed to be true) and one with the intended tactic. (One would design a test whether these better be rankings or ranges.) The intermediate result would be based upon the tactics. A random selection of the true preferences then is used to revise the tactics to improve the results for those MPs who have the luck to be selected. This prospect encourages MPs to be truthful about the strategy.

Another possibility for such double submissions: One might first determine the outcome according to the submitted strategies (supposedly true) and then use a random selection to use the allowed tactics, and only uses these if they indeed cause an improvement in the eyes of the MP. This sanctions a moderate degree of unavoidable strategic voting, but reduces the chaos when all do it without information about others.

Such calculations are simple for a partial outcome for a single MP. The problem lies in the aggregation of all MPs. Perhaps money helps in solving this too. Voters in the electorate aren’t allowed to sell their vote directly, with the obvious horror stories, also involving the distribution of income. But in parliament there is coalition bargaining which involves money, i.e. budget allocations. Potentially this helps in designing better algorithms. Perhaps the Bayesian Regret comes into play here, but I haven’t checked this. In Holland there is professor Frans Stokman who studies coalition bargaining with his “Decide” model.

Thus the search for better voting schemes hasn’t ended. Yet the main step for the USA, UK and France would be to accept the choice of a prime minister by parliament selected by PR.

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