Wednesday, March 13, 2013

From the Top

Cross-posted to my lovely wife's Web Librarian blog. The American Library Association is currently having a discussion about how they elect their Council, and she figured "Hey, I know a guy who loves to talk about election stuff!" This is a good back-to-basics introduction to our topic:

It has been said that democracy is the worst form of government, except all the others that have been tried. And the essential component for democracy is voting. So it seems reasonable that a better way of voting would lead to better democracy. That’s the motivation behind the Center for Election Science, the book Gaming the Vote: Why Elections Aren’t Fair (and What We Can Do About It), and this post.

Let’s break voting down. You have 1) a group of mutually-exclusive options to choose from. We don’t have to be talking about candidates for office; we could be deciding between different pizza toppings, or names for your new band. You have 2) a group of people with opinions about those options. Finally, you have 3) a system which takes those opinions and uses them to select one of the options. Actually, you have a whole lot of different systems, and you get to choose one. Your system could be “pick one of the options at random.” That’s not a good system, since you could end up with an option that everyone hates. Or, you could pick one of the people at random, and let them choose. That’s a little better—you know you’ve picked an option that at least one person thinks is the best—but what if everyone else thinks they’re wrong? Shouldn’t we consider everyone’s opinions?

Okay, so what if we let each person name their favorite option? And then, whichever option the most people choose, that’s the one we go with. This is the system you’re probably most familiar with. It’s called “plurality” or sometimes “first past the post.” And since you’re familiar with it, you’re probably also familiar with some of its shortcomings. For example, sometimes you might really like one of the options, but you also know that there are two other options where almost everyone else thinks one of them is the best. If you also have a strong opinion between those two, you need to decide if you’re actually going to support your real, no-hope favorite, at the risk of getting something popular that you don’t like. Or perhaps, there are two options which both seem equally good to you, and another popular one that you think is terrible. It can be difficult to determine which of the two good ones to get behind, and if people split evenly between them, it’s quite likely both will lose. You shouldn’t have to distort your opinions in order to vote effectively. Instead, you could use a voting system which collects more information about the voters’ opinions on the options, and incorporates as much of that as possible into the process.

It turns out, there are many other systems to choose from that are better than plurality. Some of them ask you to rank the options in order from your favorite on down. However, every single one of those methods suffers from at least one of the same shortcomings we identified for plurality: either there will be situations where you have a compelling reason not to vote for your real favorite first, or multiple too-similar options will stomp on each others’ support. Or both! This problem is a consequence of a very famous (and Nobel-prize-winning) theory from economics called Arrow’s Impossibility Theorem. Arrow’s theorem is usually pithily summed up as “There is no perfect voting system.” The whole truth is, of course, much more complicated than that. And while actual perfection probably is impossible, there has recently been a push in favor of voting systems which achieve what Arrow proved to be unachievable. Arrow wasn’t wrong; rather, his proof simply—and intentionally—ignored another class of voting systems, a class that doesn’t ask voters to rank the options, but rather, asks them to rate them.

A (sort of weird) example approval voting ballot

Think of it like giving “stars” to products on Amazon, or “liking” comments on social media. You don’t consider each product by directly comparing it against all other products you’ve ever purchased, or each comment by directly comparing it against every other comment you’ve read today. Instead, the process is more indirect. This book was about a 4. That banana slicer was definitely a 1. You like your aunt’s comment enough to say so, or you don’t. What are the best products? Probably the ones with the highest scores. What are the best comments? Probably the ones where more of the people who saw it, liked it.

Approval voting is the simplest of this class of voting methods. It’s exactly like “liking.” For each option, each voter can either approve of it, or not. The option with the most approvals wins. You can expand on the idea by using an idea like “stars.” Each voter gives each option as many points as they like, up to some limit, often 5 or 9 or 99. This method is called score voting. (And you can think of approval voting as score voting with a limit of 1.) Because you rate each option completely independently, similar options don’t ruin each others’ chances, and you can always give top mark to the choice that’s your true favorite. By fixing the problems that encouraged people to distort their opinions, by collecting more information from people about their opinions, and by incorporating it into the process in an intelligent way, a group is better-able to select the option that is the best choice for it. Approval voting is better democracy.

Further reading: What Do You Mean By "Best"?


  1. But wasn't Arrow's decision to disregard these rules based on a well accepted belief that utility tends to be ordinal rather than cardinal?

    If ordinality is more true than cardinality then wdn't there be an indeterminacy in how many candidates a voter gives their approval to?


  2. Not quite.

    It was based on the well-accepted belief that people's actual cardinal utilities were non-comensurable. My "10" is not your "10", and my "6, on a scale from 1 to 10" is not the same as your "6, on a scale from 1 to 10."

    The thing is though, they're close; close enough, at least. And one-person-one-vote is probably more of handicap in the translation anyway.

    Also, Arrow was concerned with (full!) societal orderings, which aren't exactly the same as voting either. And also, in a rigorous proof, which is easier to do with ordinals than cardinals.

  3. Did you mean "easier to do with cardinals than ordinals?"

    I know that in Economics, there's long been a debate about cardinal vs ordinal utilities. I believe Arrow initially presumed utility was ordinal.

    Here's from the wikipedia entry on Arrow's impossibility theorem. "cardinal" utilities that are non-commensurable n

    Arrow's framework assumes that individual and social preferences are "orderings" (i.e., satisfy completeness and transitivity) on the set of alternatives. This means that if the preferences are represented by a utility function, its value is an ordinal utility in the sense that it is meaningful so far as the greater value indicates the better alternative. For instance, having ordinal utilities of 4, 3, 2, 1 for alternatives a, b, c, d, respectively, is the same as having 1000, 100.01, 100, 0, which in turn is the same as having 99, 98, 1, .997. They all represent the ordering in which a is preferred to b to c to d. The assumption of ordinal preferences, which precludes interpersonal comparisons of utility, is an integral part of Arrow's theorem.
    Not all voting methods use, as input, only an ordering of all candidates.[24] Methods which don't, often called "rated" or "cardinal" (as opposed to "ranked", "ordinal", or "preferential") voting systems, can be viewed as using information that only cardinal utility can convey. In that case, it is not surprising if some of them satisfy all of Arrow's conditions that are reformulated.[25] Range voting is such a method.[26][27] Whether such a claim is correct depends on how each condition is reformulated. [29] Other rated voting systems which pass certain generalizations of Arrow's criteria include Approval voting and Majority Judgment. Note that although Arrow's theorem does not apply to such methods, the Gibbard–Satterthwaite theorem still does: no system is fully strategy-free, so the informal dictum that "no voting system is perfect" still has a mathematical basis.

    dlw: It seems specious to say scalings are "close" when such is not measureable. But you'll concede that if utility were ordinal that the number of approvals would be indeterminate? Such as wd be the case if in an algorithm if, in addition to generating a utility Xi for candidate i from between 0 and 10, one generate a number Y from a standard normal distribution and made the effective utilities for all candidates become z=x^y*10^1-y.

    Then if you averaged the effective utilities across all candidates for an individual the number of candidates an individual would give their approval to would be a random function.

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  5. (Or rather I should say "would be trivial with cardinals," but less meaningful useful due to the non-comensurability.)

  6. So you'll concede that some of the diffs are maybe philosophical or based on different conceptual understandings of voters/"voting in political elections"?


  7. So, Arrow wanted rigor. He wanted to make some "In all cases" type claim. Since cardinal utilities are non-comensurable, he couldn't use them to make that kind of claim.

    But that doesn't mean utilities are necessarily ordinal, nor does it mean that cardinal utilities aren't useful. For instance, most humanities research asks participants to rate their responses on a 0 to 6 scale ("disagree strongly", "disagree", "disagree somewhat", "no opinion", "agree somewhat", "agree", "agree strongly").

    Trying to force people into using ordinal utilities conceals some important aspects of their beliefs. Trying to force people into using a fixed cardinal scale conceals import aspects of their beliefs. You always lose something. Arrow chose to keep comensurability, and was able to show--despite the hopes of his times--that it leads to an impossibility. Without comensurability, it's hard to say anything with absolute certainty because, as you point out, 4.00 > 3.99 is just as true as 4 > 3.

    But, by analogy to the humanities research, we can still get useful information, and quite possibly even BETTER information, from asking people to shoehorn themselves into the cardinal state of mind rather than the ordinal one, when trying to make group decisions. And that's what it's about: extracting information from the voters, to determine what's best for the group. And until we get mind-reading machines, effective voting systems are the best we can do.

  8. Maybe political preferences transcend the ordinal/cardinal dichotomy...

    Methinks, perhaps the true implication of Arrow's impossibility theorem is that no one election rule works the best for all situations, since the possible ways to value election rules will have varying import in different types of elections.

    I don't think we shd use the same rule for judging wine, for which cardinal preferences seem appropos, or Olympics performances to judge political elections.

    In concrete decisions, like should we build bridge at location X, then getting folks educated and then to shoehorn their preferences into Cardinal form makes a lot of sense. Many political elections often involve many issues, on which most voters are quite rationally ignorant, and ineffable notions of trust. It's not as easy to argue for a cardinal state of mine in that setting.

    And IRV is an effective rule. It reliably lowers the diff between the true "center" and the de facto center and changes the dynamics to give dissenters more voice in elections. It only leads to strategic voting or coercion when a major party refuses to move towards the true center. But It's faced a strong opposition from those who benefit from the status quo and they've been more than apt to use whatever works to subvert its use, including thru the old divide and conquer tactic that is quite effective in our current system at reinforcing the status quo.

  9. By the way, if you want to know what Dr. Arrow thinks, you should see this:

    <a href=">CES interviews Dr. Kenneth Arrow</a>


    CES: Do you have any particular preferences or ideas as far as how voting methods should be evaluated in the future? Or, do you think there are certain things we should look at in trying to figure out what voting methods we should push?

    Dr. Arrow: Well, I’m a little inclined to think that score systems where you categorize in maybe three or four classes probably (in spite of what I said about manipulation) is probably the best.

  10. If it's an empirical question, as seems to be the implic from the combo of Arrow/Gibbard–Satterthwaite theorems, then Arrow's preference as a theorist should not be unduly ranked.

    electoral analytics is the hand-maiden, not the queen of electoral reform...