#arrow's theorem

[publiusmaximum]

This is called "single transferable vote" balloting, and it's my preferred method of voting in any election with a single winner, and widespread attention.

(It kinda fails in city council races where people don't have opinions on more than 1 or 2 candidates.)

I love that the Hollywood Reporter is explaining balloting methods to a popular audience- because it's weirdly relevant to the Oscars.

However- their "surprise winner" point is a bit odd- Arrow's Theorem is pretty clear that there can never be a perfect balloting system- just ones with different trade-offs.

[politicalsciencebitches]

43. Arrow’s Theorem Attempts to Solve Everything

Today we are going to get into Arrow’s Theorem further and see if we can learn anything about our f*cked up political system.  I have no idea if we will or if this is even relevant.  The fun we have!

As I introduced yesterday, Arrow’s theorem tells us that there is no system of voting that satisfies basic requirements of fairness and also gives results that are rational.  This isn’t just to say that there is no system currently in place that can do it -- it’s deeper.  It says we can’t do it.  In principle.  In logic.  Oh, goodness, he’s good.

Let’s break this down.  First, we’ll talk about these fairness conditions.  There are five.  I’ll break them down and apply them to us.  Then we’ll return to rationality.

1. Universal Admissibility: Every individual voter has a set of rational preferences.

Assessment: So far so good.  The idea is that as long as you have a rational set of preferences over the things you’re choosing between, then your choices are valid.  Your therapist may tell you that your non-rational preferences are valid, but he/she is wrong.

Applied: Rationality is a tricky thing to apply to real life.  At first glance it would seem voters follow this.  All we are really saying is you can’t have transitive preferences, or any that aren’t logical.  E.g., you can’t prefer Repeal ACA>Keep ACA but at the same time also prefer Keep ACA>Repeal ACA.  But, look, humans ARE irrational, and Americans famously have all kinds of conflicting views about politics.  So already this assumption may break down in real life.

2. Positive Association of Individual and Social Values: If individuals start to like one choice more, that will be reflected in the group preference.

Assessment: I previously knew this one to mean something broader, which is that generally speaking the group preferences should map the individuals’. But I read Arrow’s original paper for this post and it seems he only cares about the group preferences for one option INCREASING if the individuals’ preferences for that option increase.  He says it’s because they only care about social “welfare” not “illfare”.  He could have been a rapper.

Applied: I’m ok with applying this to us.  If most individual Americans prefer Do Nothing>Repeal ACA>GOP Health Plan, but then most individuals change their preferences to Repeal>Do Nothing>GOP Plan, then the social preference should reflect that promotion of “repeal”.  I don’t think that’s problematic.  I accept counterarguments.

3. Independence of Irrelevant Alternatives: This one is my favorite!  If you’re choosing between A and B and you prefer A>B, adding choice C doesn’t make you suddenly prefer B>A.  

Assessment: This sounds like it would never happen, but an example from my advisor makes it seem plausible: Suppose you’re on a plane and the flight attendant asks if you want chicken or beef and you say beef.  Then you find out there’s a vegetarian option, which reminds you that you were going to try to be healthier.  So you change your vote to chicken.  Applying this to politics, suppose you prefer Sanders>Clinton.  But then you add choice Trump.  Now you prefer Clinton>Sanders.  This would violate IIA.  Arrow demands this of the social choice, too.

Application: Now I’m wondering what makes something “irrelevant”.  I think Arrow is saying that the group choice should always reflect the true preference of the body, regardless of the order or range of choices given.  That’s a tall order, actually, because humans are irrational and weird.  We for sure are violating this.

4. The Social Welfare Function is not to be Imposed: Stick that on your t-shirt!  This one says you can’t limit the social choice set (every option should be allowed at the group level).

Assessment: This seems reasonable, though Arrow says it’s actually stronger a condition than is needed.

Application: I’m really going out on a limb here, but wouldn’t the DNC throwing its support behind Clinton > Sanders be a form of social welfare constraining?

5. No dictators!  You can’t have someone sweep in and make the choice for everyone; that is, the group choice shouldn’t just reflect the preference of one man.

Assessment: Makes sense for what we’re trying to do here.

Application: We violate this.  Arrow’s theorem applies to cases where there are three or more options to choose from.  In our presidential elections we only choose between two candidates.  This is because we have a whole primary and party system that rules some of this stuff out.  In a way, it’s a kind of dictatorship.  Also, I’m wondering now if the median voter theorem violates this -- the idea that in some conditions the tie-breaker voter is the one who gets his way.  Isn’t that a kind of dictator?

Rationality: Arrow says you cannot have all of these conditions and guarantee against cycling, or irrational, group preferences (the whole A>B>C>A thing).  This doesn’t mean you definitely will get cycling.  It just means you can’t protect against it.

UNLESS

You are willing to violate some of these.  You could impose a dictator.  You could constrain the social choice set.  You could restrict the preferences individuals are allowed to have.

And in fact: ALL VOTING SYSTEMS DO THIS!  ALREADY!

That’s why there are so many different kinds of democracy out there.  Just like snowflakes, no democratic regimes are alike.  And just like snowflakes, if there are too many of them they will eventually kill you.

Back to my original reason for wanting to do this: I am frustrated by our country’s seemingly worse ability to make choices that reflect the preferences of the group.  I was curious whether it’s individual preferences or something in the structure of our decision-making that’s making it seem more extreme.  I think from this exercise I now lean more to the side of our preferences just being so extreme that if about half of us VASTLY prefer A>B and the other half VASTLY prefer B>A, then (1) even a fair choice that reflects the will of the group won’t feel fair, and (2) the possibility of cycling preferences means that it will seem arbitrary which side is winning when.

That feels consistent with my experience as an American voter.

Solutions?  Adjust our preferences.  Think about restricting them in different ways.  Choose a better dictator.

Send help.

Fig. 1. Carrot Therapy.

Main Work

Still our gorgeous paper from Arrow!

Read Further

“Arrow’s Theorem Proves No Voting System is Perfect,” The Tech, Feb. 28, 2003.

Clark, William R., Matt Golder, and Sona Nadenichek Golder.  2017.  “Chapter 11: Problems with Group Decision-Making,” in Principles of Comparative Politics, 2nd Ed.  CQ Press.

[politicalsciencebitches]

42. Arrow’s Theorem Explains Everything

The purpose of government is to aggregate preferences.  All nations are just groups of people who live together under some agreed-upon set of decision-making rules.  We call these groups nations or countries, but we might as well call them clubs or societies or cults.  

Aggregating individual preferences for a group in a way that reflects what everyone actually wants is really hard.  Think about the last time your group of friends tried to decide on a restaurant.  It was probably infuriating as you cycled through options until someone took charge and just picked a place. And half of you probably hated it.  Is this how friendship works?  I have no idea.

Authoritarian governments solve this problem by having one guy in charge who makes all the choices for everyone (or to be more precise: makes them in a way that satisfies the small body of people who keep him in power (Bueno de Mesquita et al. 2004)).  Democratic governments do this by having people vote on things or vote for people who decide things.  

Here in the US we love our democracy and think it’s amazing.  But according to a very smart man named Kenneth J. Arrow, there’s actually no way to aggregate group preferences through voting in a way that guarantees results that are both fair and rational.  Oh, Christ.

Ken Arrow was a Nobel Prize laureate, brilliant social scientist, and, according to my advisor who was friends with him, a kind man.  He passed away recently, so I want to use Political Science Bitches to talk about his contributions.

Arrow is behind something we now call Arrow’s Theorem.  He originally called it a Possibility Theorem.  But social scientists (pessimists) somehow warped it over the decades to call it his Impossibility Theorem.  

The theorem is from a beautifully written, math-heavy paper (two qualities that rarely go together; when they do we must celebrate) called “A Difficulty in the Concept of Social Welfare.”  Hell, yes.

Arrow says that when a group of people in a capitalist democracy want to decide on something, they have two ways of doing it: market mechanisms or voting.  For political things like policy choices, markets don’t make a lot of sense, so we vote.  The problem is, just adding up votes and picking the one with the most doesn’t always work.  Why not?

If you have two or more voters trying to choose over three or more things, there’s a chance that adding up their votes will get you something called cyclical preferences.  Each individual may prefer a>b>c, but the group could add up to prefer something that doesn’t make any sense, like a>b>a.  And, no clever way of adding up votes (majority rules, ranked preferences, etc.) gets us around this problem.  And so if we have a vote, the result may not really reflect what anyone wants (sound familiar?).

If you want to avoid this irrationality, this means we need to impose rules that restrict the fairness of the election.  We’d need to do something, for example, like impose a dictator (someone to make the choice for us; a.k.a. our bossy friend in the group or Kim Jong-un) or restrict the choices people can make (hello DNC and RNC!).

I’ve been thinking about Arrow’s Theorem a lot lately -- not just because Arrow has been on my mind -- but because when I look at the fury that is the US political system, I cannot help but worry that we are doing an extra bad job compared to usual at aggregating our individual preferences.  I can’t decide if that’s because our choice sets have gotten harder, or because we are less informed, or because the people we elected to make choices for us can’t add up to anything.  I think I need to dig through Arrow’s Theorem some more and see if I can learn anything.

And so I will.

Thanks for reading -- let’s call this the introductory essay.

Hang in there, everyone.

Fig. 1. How do you like them arrows?  (Irrationally.)

Main Work

Arrow, Kenneth J.  1950.  “A Difficulty in the Concept of Social Welfare.”  Journal of Political Economy 58(4): 328-346.

Other References

Bueno de Mesquita, Bruce, Alastair Smith, Randolph M. Siverson, and James D. Morrow.  2004.  The Logic of Political Survival.  Cambridge: MIT Press.

[shitzido]

I find this so fascinating.

[isomorphismes]

You can put more than one order on a set. For example, regarding salsas from the set {mild, medium, hot}:

mild < medium < hot, where < means the predicate "is less spicy than"

medium ≻ hot ≻ mild, where x≻y means the predicate "She prefers x to y".

One could launch off into defining ≻ᵢ's for every ᵢ of the population. (This is the kind of thing one needs to do in talking about Arrow's Theorem.) That's when notation begins to suck.