Imagine that at every U.S. presidential election, the system randomly picked one random U.S. voter and asked them to pay a fee to become a “kilo-voter.” Come election day, if there is a kilo-voter then the election system officially tosses sixteen fair coins. If all sixteen coins come up heads, the kilo-voter’s vote decides the election. If not, or if there is no kilo-voter, the election is decided as usual via ordinary votes. The kilo-voter only gets to pick between Democrat and Republican nominees, and no one ever learns that they were the kilo-voter that year.

More half-retarded commentary from the Savvy Humanist: One who does not know his/her asshole from a hole in the ground. Fortunately, not all suffer from "No balls" Ball-ter's affliction. Ball-ter: Perhaps your bicycle seat is riding your ball-ters too hard: cutting off the circulation to your already tiny brain.

You're making a key error that Gelman does not. When the polls say one candidate has a 10-point lead, the majority of the other candidate's chances don't come from winning way more coinflips than expected, they come from the poll having been wrong. And polls have standard error of a couple percentage points, not even counting sources of bias (nonresponse rates being the biggest).

And if you do this as a probability problem where you have a noisy estimate of the true coinflip probability and a lot of coins, you find that the chance of a pivotal vote is basically the value of the probability density function (for the coinflip probability) at 0.5, divided by the number of voters.

Ah, OK. That's another factor of 20 or so (since chance of being pivotal is weighted toward larger states). So for the 2012 election, my back-of-the-envelope calculation gives an average chance of 1 in 8 million, and for 2000 plausibly less than 1 in 1 million.

Let's say there's an election with 1,000 voters. Early polls say that 55% are voting one way. Assuming that each randomly selected person has a 55% chance to vote that way, the probability of the election being tied (i.e. one additional vote matters) is:

nchoosek(1000,500)*(0.55)^(500)*(0.45)^(500)

= 1.6574e-04

... or a 1/6000 chance.

It's difficult to compute higher N accurately, since nchoosek(1000,500) is already 2.7e299. There are approximations but I'm too lazy to do a full analysis.

Anyway, the link is doing some misleading math. You can't just divide by the number of votes and multiply the probabilities. It's completely pointless to try and figure out the probability of the election being decided by within 25,000 votes, because for 25,000 coinflips to come up at exactly 12,500 heads is extremely low (and computable).

Let's look at 2012, since there Nate Silver was nice enough to do a model incorporating the chance of uniform bias in polls (which is what would have needed to be very true in 2008 in order for a close race in a tipping-point state).

If you scroll down enough, he thinks that despite Obama having an advantage of ~2% in the tipping point states, there was still a 6.4% chance that at least one decisive state would be within 0.5%. Let's be conservative and say this scenario is dominated by cases where the decisive state is a reasonably big state like Ohio, with a voting total of 5 million between the top 2 candidates.

Then that's a 6.4% chance that 25,000 votes decide the election, or a 1-in-400,000 chance that your vote is decisive if you live in a swing state.

And that's with what seemed to be a clear edge unless the polls were all biased by a few percentage points. In 2000 or 2004, you could plausibly get another factor of ten back by a higher chance of states being decided by small margins, including smaller states.

People are not effective altruists and consider votes more important because voting affects people who are in your community and saving someone from malaria does not. Only to an EA are all people of equal value.

Anon, firstly, $10K is not remotely small compared to the average voters' total wealth! (about 50% of the population has essentially zero wealth). Secondly, there is a second relevant non-linearity here I didn't mention before - diminishing returns to charitable giving. For both reasons, if there were a charitable matching program where a person could pay $50 to give $1,000 to charity, we wouldn't say, "Therefore, because $50 is small as a fraction of your total wealth, you are irrational not to also pay $100 to give $2,000" to charity." That is what Robin's example amounts to saying. (As an aside, to perform the exercise you propose where you measure the marginal value of a dollar one would need a third good as a numeraire that could be compared to both dollars and votes). Also, I agree that signalling explanations likely play some roll in explaining why people actually vote, I just don't think Robin's reasoning demonstrates this.

You took the time to comment here, but didn't take the time to explain your reasoning or even highlight any of the points you disagreed with. I guess you're a busy guy with only enough time to get a nice, satisfying dig in before you get back to work.

Jason, you're correct in that the marginal value of each dollar you have decreases as your wealth grows. $10,000 is worth more to me than it is to Bill Gates. This is a non-linear relationship and doing back-of-the-handkerchief math with non-linear relationships is hard.

No need to throw up your hands and give up though, you can reasonably ignore these non-linear effects when the range of values you are dealing with is small (linearization). Is $10 to $10,000 a small enough interval? I'm not so sure, but Robin does openly hedge here by stating "As long as these numbers are both small compared to a voter’s wealth...". If you wanted to you could research the marginal value per dollar for the typical US voter and scale the $10,000 based on that. I think actuaries measure that sort of stuff.

Robin, your argument illegitimately assumes no income effects. Think of it this way - suppose voting in a Presidential election is like giving $10,000 to charity. I think this comparison is pretty reasonable - you have about a 1/10^8 chance of being pivotal, and it's plausible in many presidential elections that the difference in social welfare between the candidates is on the order of a trillion dollars (compare to cost of Iraq War). See, e.g. Andrew Gelman's work on voting as a rational decision.

Then your argument amounts to saying - suppose that I am willing to pay $500 in order to give $10,000 to charity. Then I must also be willing to pay $500,000 to give $10,000,000 to charity. But that is ridiculous. You might not even have $500K. There are income effects.

We know that while individual voters do not determine elections, large groups of voters certainly do. But large groups of voters do not exist without the individual voters who comprise them. So while your vote doesn't matter individually, your participation in the group phenomenon of voting does matter.

Using that new brain medication your doctor recommended? Colon-block?

Wow. I blocked this thing on twitter and now it's stalking me on Discus. Fortunately Discus has added a block feature, so away with it.

More half-retarded commentary from the Savvy Humanist: One who does not know his/her asshole from a hole in the ground. Fortunately, not all suffer from "No balls" Ball-ter's affliction. Ball-ter: Perhaps your bicycle seat is riding your ball-ters too hard: cutting off the circulation to your already tiny brain.

You're making a key error that Gelman does not. When the polls say one candidate has a 10-point lead, the majority of the other candidate's chances don't come from winning way more coinflips than expected, they come from the poll having been wrong. And polls have standard error of a couple percentage points, not even counting sources of bias (nonresponse rates being the biggest).

And if you do this as a probability problem where you have a noisy estimate of the true coinflip probability and a lot of coins, you find that the chance of a pivotal vote is basically the value of the probability density function (for the coinflip probability) at 0.5, divided by the number of voters.

Ah, OK. That's another factor of 20 or so (since chance of being pivotal is weighted toward larger states). So for the 2012 election, my back-of-the-envelope calculation gives an average chance of 1 in 8 million, and for 2000 plausibly less than 1 in 1 million.

Let's say there's an election with 1,000 voters. Early polls say that 55% are voting one way. Assuming that each randomly selected person has a 55% chance to vote that way, the probability of the election being tied (i.e. one additional vote matters) is:

nchoosek(1000,500)*(0.55)^(500)*(0.45)^(500)

= 1.6574e-04

... or a 1/6000 chance.

It's difficult to compute higher N accurately, since nchoosek(1000,500) is already 2.7e299. There are approximations but I'm too lazy to do a full analysis.

Anyway, the link is doing some misleading math. You can't just divide by the number of votes and multiply the probabilities. It's completely pointless to try and figure out the probability of the election being decided by within 25,000 votes, because for 25,000 coinflips to come up at exactly 12,500 heads is extremely low (and computable).

So what is your estimate of the number that Gelman estimated, of the AVERAGE chance of being pivotal in the US?

Let's look at 2012, since there Nate Silver was nice enough to do a model incorporating the chance of uniform bias in polls (which is what would have needed to be very true in 2008 in order for a close race in a tipping-point state).

http://fivethirtyeight.blog...

If you scroll down enough, he thinks that despite Obama having an advantage of ~2% in the tipping point states, there was still a 6.4% chance that at least one decisive state would be within 0.5%. Let's be conservative and say this scenario is dominated by cases where the decisive state is a reasonably big state like Ohio, with a voting total of 5 million between the top 2 candidates.

Then that's a 6.4% chance that 25,000 votes decide the election, or a 1-in-400,000 chance that your vote is decisive if you live in a swing state.

And that's with what seemed to be a clear edge unless the polls were all biased by a few percentage points. In 2000 or 2004, you could plausibly get another factor of ten back by a higher chance of states being decided by small margins, including smaller states.

How's that?

I wrote more extensive comments elsewhere on this page, you hypocritical jackass.

vvv I blocked this piece of garbage below on twitter and now it's stocking me here. Amazing.

People are not effective altruists and consider votes more important because voting affects people who are in your community and saving someone from malaria does not. Only to an EA are all people of equal value.

Thanks.

Anon, firstly, $10K is not remotely small compared to the average voters' total wealth! (about 50% of the population has essentially zero wealth). Secondly, there is a second relevant non-linearity here I didn't mention before - diminishing returns to charitable giving. For both reasons, if there were a charitable matching program where a person could pay $50 to give $1,000 to charity, we wouldn't say, "Therefore, because $50 is small as a fraction of your total wealth, you are irrational not to also pay $100 to give $2,000" to charity." That is what Robin's example amounts to saying. (As an aside, to perform the exercise you propose where you measure the marginal value of a dollar one would need a third good as a numeraire that could be compared to both dollars and votes). Also, I agree that signalling explanations likely play some roll in explaining why people actually vote, I just don't think Robin's reasoning demonstrates this.

You took the time to comment here, but didn't take the time to explain your reasoning or even highlight any of the points you disagreed with. I guess you're a busy guy with only enough time to get a nice, satisfying dig in before you get back to work.

Jason, you're correct in that the marginal value of each dollar you have decreases as your wealth grows. $10,000 is worth more to me than it is to Bill Gates. This is a non-linear relationship and doing back-of-the-handkerchief math with non-linear relationships is hard.

No need to throw up your hands and give up though, you can reasonably ignore these non-linear effects when the range of values you are dealing with is small (linearization). Is $10 to $10,000 a small enough interval? I'm not so sure, but Robin does openly hedge here by stating "As long as these numbers are both small compared to a voter’s wealth...". If you wanted to you could research the marginal value per dollar for the typical US voter and scale the $10,000 based on that. I think actuaries measure that sort of stuff.

Robin, your argument illegitimately assumes no income effects. Think of it this way - suppose voting in a Presidential election is like giving $10,000 to charity. I think this comparison is pretty reasonable - you have about a 1/10^8 chance of being pivotal, and it's plausible in many presidential elections that the difference in social welfare between the candidates is on the order of a trillion dollars (compare to cost of Iraq War). See, e.g. Andrew Gelman's work on voting as a rational decision.

Then your argument amounts to saying - suppose that I am willing to pay $500 in order to give $10,000 to charity. Then I must also be willing to pay $500,000 to give $10,000,000 to charity. But that is ridiculous. You might not even have $500K. There are income effects.

We know that while individual voters do not determine elections, large groups of voters certainly do. But large groups of voters do not exist without the individual voters who comprise them. So while your vote doesn't matter individually, your participation in the group phenomenon of voting does matter.