Patrick (patrickwonders) wrote,

Let's Talk About Coin Flips

There has been lots of talk in the last 24 hours about coin flips. If a precinct had, for example, five delegates to award and the caucus-goers were evenly divided between two candidates, they would award two delegates to each and toss a coin to see which of them got the fifth delegate.

Most of the social-media that I've seen in response has been either Hillary is the coin-flipping champion or Really? Coin flips?.

Coin-flipping champion

Hillary won six out of six coin-tosses! That the author of one article knew about.

In actuality, there were at least a dozen coin-tosses. Sanders won some of them, too.

Even if there were only six and Hillary won all six, there is no reason for adulation or alarm. The odds of six coin tosses out of six going in your favor are 1 in 64. Yes, that's less than 2%. Things with odds way less than 1 in 64 happen every single day. In fact, if there were six coin tosses in each of the 50 states, the odds that Hillary would win all six in some state are about 46%. In other words, it's basically a coin toss whether or not she'd win all six in one of the states.

This Six out of Six! reporting also smacks of confirmation bias. If you're a Sanders supporter, but Clinton won an extra delegate in your precinct by a coin toss, you're going to tell all of the other Sanders supporters you know about it. You aren't going to notice the precincts where Sanders won by a coin toss.


TL;DR: Yes, really.

To those that are skeptical, my first question is: What would you propose instead? How else can we deal with an odd number of delegates in an evenly divided electorate?

One possible answer would be, Make the caucus goers stay until one side concedes the extra delegate. I can't understand the appeal here. I don't know if anyone is actually even thinking it should be this way. This is just what it feels like someone would propose if they were pushed to say more than just, Really? A coin toss?. Trust me, you will not get people to attend the caucus at all in two years if this year's caucus degenerates into Who can keep one hand on the Trans Am for the longest?

Another possible answer would be, Throw away the extra delegate. There is some appeal to that. It is completely fair as far as that one caucus goes. On the other hand, it means that at the next level of caucuses, your precinct is under-represented. So, that's less than ideal.

Another possible answer would be, Give them each half a delegate. Again, there is some appeal to that. It is fair. On the other hand, it really makes things awkward as things roll up into the next level of caucusing. Excuse me, sir. Are you a full delegate or a half delegate?

Flip a coin is another alternative. It is fair. It is quick. It gives your precinct all of the representation it was entitled to have. It doesn't degenerate going into the next round of caucusing.

In case it isn't blatantly obvious, I firmly believe in the coin flip.

But, but.... The Will of the VotersTM!!!

There are implicit assumptions there like: each voter has an unwaivering (captial-W) Will (or at least unwaivering long enough to count), there really is some way to go from the Will of each individual voter to the Will of the voters collectively, there really is some way to measure that collective Will with such precision that there will never be a tie, and only those voters who physically attended the caucus that particular evening should count anyway.

The Will of a Single Voter

What is your favorite color? Has that always been your favorite color? Will it be your favorite color tomorrow? How certain are you that it is your favorite color at this very moment?

Right now, my favorite color is, I think, a sort of cornflower blue. That has not always been my favorite color. I don't have any confidence that it will still be my favorite color if you ask me again in twenty minutes let alone if you ask me tomorrow. I am not even really certain it is my favorite color at the moment. [Edit: during proofreading, I think emerald green has edged out cornflower blue.]

When I asked myself the question What is your favorite color?, cornflower blue was the fourth answer that came to mind. It was a better answer than the first three colors that came to mind. It was a better answer than the next five colors that came to mind. No other colors came to mind as viable candidates.

In an election, the question isn't quite as open-ended (if you ignore write-in candidates), but I don't think it is any more clear.

If I had a ballot in my hand right now with both Clinton and Sanders on it, whom would I pick? I'm not sure. I think I would pick Hillary. But, I answered this same question less than a week ago and was sure that I'd pick Sanders. If you ask me in an hour, I'm not sure which I would pick.

Why so fickle? When I envision life under a Clinton presidency and life under a Sanders presidency (or an election pitting Clinton vs. Cruz or Sanders vs. Trump), there are many, many facets to that picture. Is it better to have a President who won't make poor compromises with the legislature or to have a President who is already prominent on the world stage? Is it better to vote with my head or my heart? Is this? Is that?

The answers to each of those questions vary slightly from moment to moment in my head (and heart). How important each of those questions is to me varies greatly from moment to moment. How well I think I can predict the future behavior of either candidate in, as yet, hypothetical, straw-man circumstances varies greatly from moment to moment.

Once I have voted, does my cast vote actually reflect my Will as a voter? I don't think the question makes sense. I don't even think the question makes sense if I refine it to ask whether my vote reflected my Will as a voter at the moment at which I cast it.

I contend that my Will is neither binary nor final. My vote is necessarily both. My vote will never, ever reflect my Will.

My Will plus Your Will

If you're an ardent Sanders supporter and I am as on the fence as I just described, then what is our collective Will. Certainly, it seems that if we had to pick the closest binary answer to the question, then you and I together want Sanders to be elected.

If you're an ardent Sanders supporter and I am an ardent Clinton supporter, is our Will that they both be elected or that neither be elected?

I don't think it makes any sense to even try to define what one means by our combined Will. But, even if that were a meaningful concept, our combined Will cannot be binary. The election outcome is binary. The election outcome will never, ever reflect our combined Will.

I think this is where Nate Silver really knocks it out of the park (obligatory baseball metaphor just for Nate). You've got 30 polls. They've got different margins of error. The pollsters have different historical biases. Some of the polls are more recent than others. You weight them all and then you run simulations. You run hundreds and hundreds of simulations. You don't try to answer the question, Who will win the election? Instead, you answer the question, Given what we know at the moment (taking into account the sources we know it from), if we held the election a few thousand times, what percentage of those elections would each candidate win? My Will plus your Will is some percentage of elections in which I vote for Clinton and you for Sanders plus some percentage of elections where we both vote for Sanders.

Measurement Accuracy

I am running a Jellybean Guess contest. I count the jellybeans as I am putting them into the jar. I wrote down the number of jellybeans that I put into the jar on a piece of paper.

Now, the contest is over, and I have to figure out who came the closest. Uh-oh. I have misplaced the piece of paper with the real answer on it. So, I count the jellybeans as I am taking them out of the jar. I come up with 742 jellybeans. I award the prize to the closest guess.

Two days later, I find the piece of paper tucked inside a book that I am reading. The paper says there are 739 jellybeans. Oh-no!

Ugh! I counted wrong. If I counted wrong the second time and right the first time, then I just awarded the prize to the wrong guesser! If I counted wrong the first time but right the second time, then everything is okay. If I counted wrong both times, then I don't know if everything is okay or not.

Even if there is such a thing as the combined Will of the voters and even if such a thing can be counted as an integer number of votes, counting is actually really, really hard. I contend that the count of the number of votes cast for a given candidate will almost never reflect the actual number of votes cast for that candidate (unless the number of votes cast is very, very small).

In the 2008 senatorial election in Minnesota, the ballot count showed that Norm Coleman won by 215 votes over Al Franken. That margin was less than 0.008% of the vote.

Minnesota law requires that a recount be conducted for a statewide, federal election if the winning margin is less than a quarter of a percent. This winning margin was less than one thirty-third of a quarter of a percent.

After the recount, Al Franken won by 225 votes.

The recount cost the counties of Minnesota an estimated $460,000. Now, don't get me wrong, I think having Al Franken instead of Norm Coleman probably more than makes up for the cost, but still. We paid $460,000 for a second measurement. Even if one argues that the recount was conducted with more care than than initial count, it is ludicrous to think that the recount number was, in fact, an absolutely correct counting of the marks that the voters actually placed on their ballots on election day (even ignoring the fact that voters may have actually placed more or fewer marks than they were trying to place).

You Forgot to Vote

If, for any reason, you are unable to get to the polls on election day or to the caucus on caucus night, you are out of luck. Any Will you had as a voter is irrelevant. Any measurement of your Will as a voter is multiplied by a one if you show up and a zero if you don't.

If I walk out of the polling booth and get hit by a bus, I have way more Will as a voter than the lady who got hit by the same bus on her way into the polling booth.

Even if each voter's Will were constant and perfectly reflected in their ballot and even if counting votes were absolutely accurate, one would get a totally different result if the election were one day earlier or one day later or one hour earlier or one hour later.

I contend that even if one is able to define a collective Will of the voters and measure it with absolute precision, the election results will never, ever actually reflect the Will of the whole electorate (or even the likely voters) (unless the collective Will was defined as a binary thing).

What I Would Propose for Caucuses

Even if I grant that each voter had a Will, there is a way to add up the Will of each voter into a collective Will, that measuring that collective Will is possible, and only those voters who physically attended the caucus that evening should count, I am still stuck trying to measure that collective Will with such precision that it can never be tied. Every voting system that we've ever had (since the thirteenth amendment, mumble, mumble) has been based on whole numbers of votes. We have no framework for dealing with rounding errors.

In a caucus-like situation, we have some number of voters and some number of delegates to award. At the end of all of the debating and wrangling and changing of allegiances, each candidate has some number of supporters. Unless each such number is an integer multiple of the ratio of voters to delegates, then you've got to decide how to handle round-off errors.

In poker tournaments, as the tournament progresses, the tournament directors start phasing out the lower denominations of chips. When the minimum bid is $100, it is not useful to have a whole pile of $1 chips. When it is time to phase out a particular denomination, the tournament does a Chip-Up. If we are making the $5 chip the new minimum chip and you have $17 in $1 chips, then you get three $5 chips and still have two $1 chips left. Now, you trade your two $1 chips for cards. You now have two cards. If there were, in total, at your table, ten extra $1 chips, then the person with the highest card on the table gets a $5 chip (and discards any remaining cards he has) and the person with the next highest card on the table gets a $5 chip. (Note: most tournaments use the bridge ordering of the suits so that the Ace of spades is considered higher than an Ace of hearts so there is never a tie for the highest card on the table.)

In a one table tournament where everyone started with some number of chips that was a multiple of this new lowest denomination, the end result is that there is the same amount of money on the table as before. If the new lowest denomination is $5 chips, the most you could gain or lose in the chip-up is $4. If you were only faced with losing $1 in the chip-up, then you only got one random chance to win it; but if you were faced with losing $4, then you had four chances to win.

I think delegates should go the same way. If there are 100 voters and five delegates, then every candidate should get a delegate for every full batch of 20 supporters. After that, any left-over supporters should each get a ping-pong ball with their candidate's name on it. These should all go into a bingo or lotto mixer. First ball out, gets a delegate. Next ball out for a candidate who hasn't had a ball come out yet, gets a delegate. Repeat until all of the delegates are gone.

Yes, it's tough if you had 17 balls in the bin while your opponent won with one of their 3 balls. Suck it up. Win in the next precinct. At least you weren't cheated nor were your supporters strong-armed into supporting someone else.

What I Would Propose for Recounts

Recounts are slightly different than caucus situations. If you think there is any utility to a recount, then you are admitting the possibility that the count might be wrong. If you are admitting the possibility that the count might be wrong, you also have to admit the possibility that the recount might be wrong.

If you could accurately gauge the probability distribution of the counting error, then you could, with enough recounts, reach any level (less than 100%) of satisfaction that your counting favored the right winner. However, the right winner isn't the winner of the last recount, but instead some composite winner from the information provided by all of the recounts combined.

Would you be satisfied with the election results if, after five recounts, you've finally established that you have the correct winner with at least 99.5% confidence?

If the margin of victory is so small that it may well be counting error, then a coin toss is far cheaper than a recount and costs almost nothing.

Given the variance of the counting error (which, I admit we would have to estimate), one can answer this statistical question Given that this candidate's count was this and that candidate's count was that, what is the probability that an infallible count would show that the first candidate actually beat the second candidate? Then, I would flip a biased coin (biased with that probability of landing heads) and declare the first candidate the winner if the coin comes up heads.

There have probably been enough recounts in the last few decades that one could make a good stab at the variance of the counting error. Even if one can't come up with a very accurate estimate of that variance, I feel like a variance value could be legislated.

And, as creating biased coins is non-trivial, I would simulate this. I would either use a secure random number generator, or I would calculate some threshold such that the probability that the sum of tonight's Powerball numbers is greater than the threshold is the same as the probability that I have in mind. (Though, there aren't very many possible sums in Powerball so we'd be stuck picking the threshold closest to the desired probability or having a re-do if the Powerball sum hits one of the numbers adjacent to the threshold). There are probably better sources of random numbers than this. Oh, man, I bet I could write a viral article about how the most likely sum of the white Powerball balls is 175 and so you should make sure that whatever numbers you pick add up to 175.)

Wrapping It Up

Regardless, back to my original point in the TL;DR above. Yes, coin tosses.

Life is way too complex for binary decisions to have a single right answer. We should just admit that our elections are actually random events and make sure that when the outcome is too close to be obvious, that we have an outcome that transparently reflects the randomness involved.

Tags: elections, math
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