Voting Paradoxes

by on November 4, 2016 at 7:25 am in Philosophy, Political Science | Permalink

Here’s an excellent video on voting paradoxes from the San Francisco Exploratorium and here is my older post on Arrow’s Impossibility Theorem.

VotingParadoxes from Paul stepahin on Vimeo.

1 rayward November 4, 2016 at 8:23 am

Did Arrow consider different outcomes using different systems of voting? For example, proportional representation (as opposed to the majority system used here). Some corporations use a system of cumulative voting for directors: each voter (i.e., shareholder) gets the number of votes as board seats that are being voted on. For example, if there are three open seats, each voter gets three votes, which she may cast for one, two, or three candidates. In the video, the most irrational outcome arises when a bribe (i.e., gold) is offered to those who vote for a particular candidate. The real world equivalent occurs when one group collaborates with a sub-group from another group to guarantee the sub-group with representation but at the cost of the group to which the sub-group otherwise nominally belongs. [An aside, the lack of parallel construction (e.g., when an individual changes their vote) in both Tabarrok’s earlier blog post and in the video makes them very confusing – besides violating convention. The confusion is compounded because Arrow’s point was to highlight the difference between individual choice (singular) and group choice (plural). In my comment, I use the feminine “she” rather than the convention “he”. Indeed, I alternate between feminine and masculine in text to maintain parallel construction.]

2 rayward November 4, 2016 at 8:35 am

I should have mentioned that shareholders of corporations have weighted votes: a shareholder has the number of votes in proportion to his stock ownership percentage (e.g., if there are 100 shares outstanding and a shareholder owns 20 shares, she gets 20 votes). Some have proposed that American elections follow something similar, with the number of votes for each voter dependent on, for example, her net worth. Other criteria for apportioning votes include IQ, age, and ethnicity. A cynic might suggest that in practice America already apportions votes based on such criteria.

3 Dean November 4, 2016 at 10:00 am

Why would some propose something so insane?

4 NeedleFactory November 5, 2016 at 1:42 pm

Not insane! — the underlying reason is that voters with wealth will be those paying the costs of implementing the winning decision(s), whereas voters without wealth “have no skin in the game.” The method you decry as “insane” attempts to ensure that those who will do the work are those who get to choose the task. It may be a “bad” system, but it is not “insane.”

I think a crude example of this method appeared in early U.S. times, when only property holders could vote.

5 Thiago Ribeiro November 4, 2016 at 11:40 am

Some political scientists agree that the Brazilian voting system with two rounds (unless a candidate manages to gather 50% of the votes) for presidents and governors and proportional vote using the whole states as multiple-member districts for the Congress and State Assemblies is the best one. Founding Father James Wilson had predicted the good outcomes: “It may, I believe, be assumed as a general maxim, of no small importance in democratical governments, that the more extensive the district of election is, the choice will be the more wise and enlightened”. It prevents the systematic corruption that took hold of the American political system, allows real popular participation (Brazil has almost 40 registered polical parties and dozens more are seeking registration instead of America’s artificially sustained duopoly) and promotes good governance. The American voting system fosters apathy, cinism (the majority of American population hates or despises Mr. Trump and Mrs. Clinton and is de facto– if not de jure– disenfranchised by the fat cats in Washington), radicalism and hatred, the centre cannot not hold this way.

6 Troll me November 5, 2016 at 10:00 pm

I think any sort of runnoff mechanism in a winner-takes-all process, whether at the level of single constituents or national offices like the presidency, are superior to a simple plurality. In order to avoid the cost and inconvenience (especially politically, extended campaigns and the like) of a run-off vote, the candidates can be ranked and the run-off happens automatically. However, this system may not be appropriate where there might be reason for (or strong perception of) doubts about anything that happens in bits and bytes instead of paper, pencils and people, in which case the additional run-off process might be worth it.

7 Dan Lavatan-Jeltz November 4, 2016 at 6:30 pm

Arrow considered different systems. His result holds true for deterministic systems but not non-deterministic ones, for example under a random ballot voter’s should express their true preferences. Earlier sortition arrangements took advantage of this. With Condorcet systems and a lot of voters, it becomes difficult to game the system, although it is still possible.

I’d very much like to see a non-deterministic system that met Arrow’s criteria, but obviously entrenched interests want to keep things as bad as possible. Even the few bright spots such as Maine, exclude Presidential electors from choice-based voting.

8 Axa November 4, 2016 at 8:26 am

Very instructional video, thanks!

So, there’s no best method. Method A provides a solution but contains a problem…..I’ll look at the publications from Arrow. I suppose he demonstrated this with nice math, it should be interesting.

9 Ray Lopez November 4, 2016 at 8:29 am

Yawn. Another troll attempt by AlexT to usurp Arrow’s Impossibility theorem as his own. Rather than waste ten minutes of your life going through this video, which undoubtably moves too fast and would require a re-viewing, just spend ten minutes of your life going through the below example and pay attention to the key sentences I highlight. Move along, nothing to lern here…

key sentences: “Rationally we would expect that this would imply that vanilla would be preferred to strawberry” and “Thus we have the irrational result that socially vanilla is preferred to chocolate and chocolate is preferred to strawberry but strawberry is preferred to vanilla.”

10 Axa November 4, 2016 at 8:28 am
11 Lumen November 4, 2016 at 11:42 am

“Cuck stands for “cuckold,” which is defined as the husband of an adulterous wife. Often, it is a sexual fetish involving men who get off on watching their girlfriends/wives with other men, with their consent. Meaning that calling someone a “cuckold” is basically accusing them of enjoying sex in a very specific, totally legal, and perfectly fine way.”


12 Gary November 4, 2016 at 12:07 pm

I’m no fan of the alt-Right, but if you’re calling the author of that article a cuck, well, the shoe fits. I do wonder if the author is trying to bait his readers or if he really believes that garbage.

13 dearieme November 4, 2016 at 8:40 am

I’m looking forward to the TV pics of people holding their noses and voting for Trump.

Why not vote for Trump? It might give you some release, and the machine will transform it into a vote for Hillary anyway. And if that fails, the dead will outvote you.

14 anon November 4, 2016 at 9:12 am

I just left Twitter, where Noah Smith, Ezra Klein, Matt Yglesias, Justin Wolfers, and Kevin Fox all gave me a more reassuring view of reality. Fox had a graphic similar to those above. It matters.

Oh, and a cat playing a theramin.

15 Lumen November 4, 2016 at 11:36 am

“I just left Twitter, where Noah Smith, Ezra Klein, Matt Yglesias, Justin Wolfers, and Kevin Fox all gave me a more reassuring view of reality.”

But don’t forget you’re supposed to be a “center right” Republican!

16 anon November 4, 2016 at 1:13 pm

Independent ex-Republican. Where do you think we all went? Not right wing silos. Those are why we left.

I went to people who will explain their thinking, from fact to conclusions. The “wonk” sites do that. I also like the Economist and Financial Times for an external perspective.

Cue “no one believes the Economist or Financial Times!”

17 anon November 4, 2016 at 3:55 pm

This is a related and really amazing piece about where people get their news.

We probably all know Facebook people like that.

18 Thiago Ribeiro November 4, 2016 at 1:17 pm

I could play theramin, too, if I wanted.

19 Not A Communist November 4, 2016 at 9:31 am

If you have the voters allocate 100 points among their options, doesn’t that solve a lot of the problems here? IE for three options (Chocolate 40, Vanilla=30, Strawberry =30) etc? Then pick the option that gathers the most votes.

20 jb November 4, 2016 at 11:00 am

Only if everyone votes their actual preferences. The moment you allow strategic voting, everyone will allocate all 100 points to whichever of their 1st or 2nd choices has the best chance of beating their 3rd choice.

21 Sam The Sham November 4, 2016 at 11:26 am

Being able to allocate points like that is very vulnerable to devolving into single-vote FPTP. (still better than FPTP though) A twist on what you’re asking is square-root allocation. You have 100 points for your entire ballot. You could give Chocolate 10 votes (10^2= 100), or choco 8 votes and vanilla 6 (8^2=64 + 6^2=36 == 100). Or give choco and vanilla 1 vote and vote on Amendment 2 as well, etc.

I don’t like it for complexity reasons, but it more accurately measures degree of approval without being easily gamed.

My 10 votes for chocolate. Vanilla and strawberry are for commies.

22 Enrique November 4, 2016 at 9:40 am

Here is an example of Condorcet’s voting paradox applied to Puerto Rico:

In th paper I use the voting paradox known as “Condorcet’s paradox” to explain the impasse over Puerto Rico’s constitutional status. Although in reality Condorcet cycling is rare in politics, I explain how the impasse over Puerto Rico’s status can be modelled as a Condorcet cycle and I present a possible solution to this stalemate: a simple range voting procedure.

23 Dean November 4, 2016 at 10:03 am

Wikipedia has a handy chart for evaluating various systems against various errors:

24 Sam The Sham November 4, 2016 at 11:28 am

I love the chart. Why exactly do we still use plurality?

25 jb November 4, 2016 at 1:38 pm

It’s the simplest one, and we still can’t even get most people to understand it.

26 Just Saying November 4, 2016 at 11:37 am

I love how the Alt-right, about the weakest and most paranoid segment of males on the planet, refer to others as “beta”.

27 Tristan November 4, 2016 at 12:27 pm

No mention of range voting or quadratic voting? Those are two systems that don’t fall under the “certain assumptions” of Arrow’s theorem, right?

28 Spooky_Rationality November 4, 2016 at 3:56 pm

I wonder Arrow’s Impossibility Theorem has to do with Dutch Books problems and outcome pumps where you can rig the election to arbitrarily come up with any outcome you want. The reason the group can’t decide properly is because they’re behaving irrationally, they aren’t evaluating the ice cream on some consistent objective basis. Like if you made a list of objective properties and had people vote on which properties they like then you could make a rational rank ordering that would lead to the best group outcomes, rather than letting the people directly vote for the ice cream itself.

29 GoneWithTheWind November 4, 2016 at 8:19 pm

This is why third and fourth party candidates force the outcome to be the least satisfactory to the most people. One way to fix this would be to require a runoff vote whenever a candidate (or issue) failed to get a majority of votes. This would be a reasonable compromise.

30 Troll me November 5, 2016 at 10:10 pm

Extremely interesting. But it seems altogether too consistent with rejecting any change in the present system.

The paradoxes can be theoretically demonstrated with 10-15 people and a simple choice, but when you’ve got 30 million voting preferences I think the concerns are much reduced.

One thing the video failed to illustrate was the paradox of winner-takes all and median voter theory combining to predict a two-party system, and in such a situation that such an equilibrium may lead to a situation where ALL voters end up with something other than what they want for lack of a system which makes other choices available.

Consider the logistics of accessing funding, attention and support for a third party with 0% chance to take any seats in the very next election, and it becomes easy to understand the long-term result with respect to much reduced prospects for any flourishing of new ideas, interests, values, positions, etc., within new partisan and/or non-partisan electoral possibilities over time. If one defines political stability as over-arching socio-political stasis, this would be quite different from a situation that potentiates political evolution by virtue of a more open door to new parties.

31 Warren Meyer November 7, 2016 at 2:35 pm

To me, though, this video highlights the strong advantages of capitalism over socialism in at least three ways

+ Forcing one-size-fits-all socialist and authoritarian solutions sucks vs. allowing individuals to make choices based on their personal preferences regardless of other preferences in the group. While the video discusses a variety of voting approaches for forcing everyone into a single choice, all of these result in a lot of folks who don’t get their first preference. Obamacare is a great example, where product features have been standardized, essentially through a voting process (though indirectly) and huge numbers of people are unhappy.
+ The video fails to discuss one shortcoming of simple yes/no voting, and that is degree of preferance. In the real world, we both may prefer vanilla over chocolate, but your preference might be pretty close whereas I might be so allergic to chocolate that eating it will kill me. Socialist and authoritarian approaches don’t have a solution for this, but market capitalism does, as prices signal not only our preference but our degree of preference as well. The real market for ice cream is a preference expression process orders of magnitude more sophisticated than voting.
+ It is almost impossible for even an autocrat who legitimately wants to maximize well-being to do so, because the mass of individual preferences are impossible to encompass in any one mind. Towards the end of the video, it became harder and harder for a person to synthesize a best approach from the preference data, and this was just for 10 people. Imagine 300 million preferences.

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