Paradox - Individuals don't make a difference

A certain type of paradox is actually relevant to real life. There are a few versions that are really bringing up the same problem.

The most common is the "grains of sand" paradox. If you have a heap of sand and remove one grain from it, it will still be a heap of sand. It logically follows that if you continue to remove grains, one by one, it will never stop being a heap of sand, even when there are no grains left.

Consider another: If a person is a child one day, the next day she will still be a child. From this it follows that she will never become an adult.

These all involve very gradual changes which are virtually undetectable taken alone, but summed up are.

When I was younger, I used to sing a lot and joined a choir. I told my mom that I didn't like being in it because, since there were so many people in it, I wasn't actually making a difference to how it sounded, so I felt there was no point in being there. She argued that everyone makes a difference and if enough people had the same mindset and quit, there would be no singing at all. But this didn't settle right with me. I mean it's true that if everyone stopped singing there would be no choir. But this doesn't negate the fact that if I were the only one who quit, there would be no difference.

The paradox applies to about any situation in which an individual is part of any large group that is making a difference. One of the more common real-life issues is the concept of voting. Unless the result happen to be one point off of a tie (which is extremely unlikely) one person's vote really isn't going to make a difference. That is a fact. So the conclusion should be that there is no need for me, an individual person, to vote. However, if enough people realized this (and many do) the results would be affected significantly. But even after realizing this, it is still true that the individual person you are now will not make a difference.

Any ideas on how this might be made sense of?

In all of the situations you proposed the issue is about perception. It seems like a grain of sand removed from a heap doesn't change it, but that's only because grains of sand seem so small an insignificant to us individually. Personality changes like growing up seem slow just because they really do take a long time to happen, but that doesn't mean there won't be shifts. One person lacking from a choir may not seem much different to us, but there is a difference involved. The volume will be a little lower and little narrower, even if it's below what we'd perceive as scaled to our regular sizes. Just like a grain of sand. Same with a vote, and one vote won't make a difference normally but that doesn't mean it can't.

Basically what I'm getting at is, these paradoxes are actually just generalizations, and they're not really paradoxes because they make complete logical sense, it's just in our subjective views that they arise. Every single change you've described actually does make a statistically significant difference regardless of how we perceive it, so it only follows that after enough changes we will begin to really notice.

Really, for using statements like that what it comes down to is when people need to realize that they don't apply anymore, like when something can no longer be considered a heap or when someone can be considered to have matured. There are no set rules to these things saying they must always be true, it's always an in-the-moment judgment.

I agree with Alzarin. This is not really a paradox so much as it is a problem with semantics or how things are observed. That one grain of sand DOES make a difference, whether you can see that difference or not.

Thinking about 3rd party candidates?

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Originally Posted by Indeed

I agree with Alzarin. This is not really a paradox so much as it is a problem with semantics or how things are observed. That one grain of sand DOES make a difference, whether you can see that difference or not.

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Ah, but when does it cease to be a heap of sand?

I guess I do agree with Aly and Indeed; it's the same conclusion I came to when I heard the problem. The changes are gradual and the problem (as with so many things) is over definition.

Not that this is an exception but I find it interesting. The problem becomes even harder to deal with once you ask when someone will judge it to be a heap, instead of when it becomes a heap. Let's say there's an 8 year old girl and we ask someone to label her either a child or an adult. She'll obviously be labeled a child. But what if we asked the same person to assign her one of the labels every day of her life? Imagine that this is a perfect experiment, in which each time he's asked the question, it's like time has gone back, erased his memory and duplicated all of the circumstances except that the girl looks one day older.

At some point, the person would have to start labeling her a child one day and then an adult from then on. That's the only thing that can really happen. You might say the judgment would very gradually become more and more uncertain. But if he really were forced to choose between 'child' and 'adult' the shift would need to happen eventually and suddenly.

So the real problem, I think, is that the premise which goes something like "if someone is labeled a child one day, she'll also be labeled a child the next day" is actually false, even though it really seems to be true. Similarly, there will eventually be one grain of sand which, when lost, will cause a person to judge that he's no longer looking at a heap.

But it's still a problem for things like voting, and I'm still not sure what to think of this. It's an indisputable fact that one vote is not going to make a difference. Or at least that it's so unlikely to make a difference (for there to be a point off from a tie) that it isn't worth the effort of voting. Just as lottery tickets aren't even worth spending $1 to buy no matter how big the prize is if the probability of winning is like 1/10^7. It seems the only way to get people to vote is to trick them so that they think they're actually making a difference. And the purpose of tricking them is so that they collectively do vote and do make a difference, even though they actually don't individually. When someone admits they aren't going to vote because it won't make a difference, people usually get really upset at the person, even though it's true. I'm just not sure where to stand morally here.

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Originally Posted by Dianeva

I guess I do agree with Aly and Indeed; it's the same conclusion I came to when I heard the problem. The changes are gradual and the problem (as with so many things) is over definition.

Not that this is an exception but I find it interesting. The problem becomes even harder to deal with once you ask when someone will judge it to be a heap, instead of when it becomes a heap. Let's say there's an 8 year old girl and we ask someone to label her either a child or an adult. She'll obviously be labeled a child. But what if we asked the same person to assign her one of the labels every day of her life? Imagine that this is a perfect experiment, in which each time he's asked the question, it's like time has gone back, erased his memory and duplicated all of the circumstances except that the girl looks one day older.

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That's not necessarily true. In between there would likely be a significant period in which he flip-flopped on that judgement based on various psychological factors changing on a day-to-day basis (because it would be close enough to go either way).

I figured out a satisfying method, for settling my thoughts on this subject some time ago, when I voted for primeminister the first time in my life. Basically remember, that while we are all inviduals, and we all have our reasons for voting, a lot of us will still vote for the same reasons, and some will consider not voting, for certain reasons. I simply made the judgement call, that these people are probably thinking somewhat like me, and for that reason, they understand that if they themselves don't vote, then people similar to themselves won't vote either, thus concluding that voting is the best option. Indeed, this requires a certain train of thought, but given a big enough group of people, it's likely that some think like yourself, and thus your vote means a lot more, than you initially thought.

This might not be a perfect solution for you, but to me it's satisfying enough.

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Originally Posted by Dianeva

But it's still a problem for things like voting, and I'm still not sure what to think of this. It's an indisputable fact that one vote is not going to make a difference. Or at least that it's so unlikely to make a difference (for there to be a point off from a tie) that it isn't worth the effort of voting. Just as lottery tickets aren't even worth spending $1 to buy no matter how big the prize is if the probability of winning is like 1/10^7. It seems the only way to get people to vote is to trick them so that they think they're actually making a difference. And the purpose of tricking them is so that they collectively do vote and do make a difference, even though they actually don't individually. When someone admits they aren't going to vote because it won't make a difference, people usually get really upset at the person, even though it's true. I'm just not sure where to stand morally here.

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You're getting a lot of chocolate in your peanut butter here. Voting is by nature a collectivist enterprise. The fact that an individual vote doesn't decide the issue is the whole point of voting. Every voter makes an equal difference, but that difference is only 'activated' by participation in the collective. The fact that it's difficult for people to see past the end of their noses doesn't make collective efforts paradoxical or a trick. The trick is on the other end, when we convince ourselves that we will gain the greatest benefit by acting in shortsighted self-interest despite the fact that we obtain most of the comforts and rewards in our lives from participation in the social collective.

The position that voting doesn't make a difference unless you cast the mythical deciding vote is the opposite of "indisputable fact;" it comes out looking paradoxical because it's riddled with faulty premises.

For purposes of understanding an election, Marvo has a pretty good approach of investing his identity in the collective of "people who think like me:"

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Originally Posted by Marvo

I figured out a satisfying method, for settling my thoughts on this subject some time ago, when I voted for primeminister the first time in my life. Basically remember, that while we are all inviduals, and we all have our reasons for voting, a lot of us will still vote for the same reasons, and some will consider not voting, for certain reasons. I simply made the judgement call, that these people are probably thinking somewhat like me, and for that reason, they understand that if they themselves don't vote, then people similar to themselves won't vote either, thus concluding that voting is the best option. Indeed, this requires a certain train of thought, but given a big enough group of people, it's likely that some think like yourself, and thus your vote means a lot more, than you initially thought.

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In other words, the solution to this "paradox" is to get over yourself :cheeky:

I agree it is more an issue of vague definitions than an actual paradox. If you were to define a heap of sand as being 534 grains of sand or more, then the problem would be fixed. However, as it stands a heap is just an arbitrary term. The definition of a child or adult can be pretty arbitrary as well, especially since some adults seem like children. Though if you define an adult as anyone has been through puberty, then there is a distinct cut off point and one day there is an actual physical difference between a child and an adult.

On a related note, I bet there are some people who are really into music who can tell the difference between a choir with a person missing or not. In fact some people are very picky about the number of people in situations like that.

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Originally Posted by Supernova

Ah, but when does it cease to be a heap of sand?

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That's an issue with semantics, honestly. What you call it is up to you.

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Originally Posted by Alric

On a related note, I bet there are some people who are really into music who can tell the difference between a choir with a person missing or not. In fact some people are very picky about the number of people in situations like that.

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One of the old music teachers at my high school who usually conducted the full band could supposedly pick out a single wrong note in a whole room, and point out who it as and what note they played.

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Originally Posted by Dianeva

Or at least that it's so unlikely to make a difference (for there to be a point off from a tie) that it isn't worth the effort of voting. Just as lottery tickets aren't even worth spending $1 to buy no matter how big the prize is if the probability of winning is like 1/10^7.

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This seems to be the crux of the voting question. As in the case of lotteries, the decision to participate should ideally be based not just on the probability of there being some kind of payoff that follows from your participation, but also on the size of that payoff. If the lottery prize is large enough relative to the probability of that payoff being realized, it might make good sense to enter the lottery. This notion is traditionally formalized in terms of the expected value (EV) of choosing to participate. Letting u and p be the size of the payoff and its probability of attainment, respectively, EV is defined as u*p. In other words, if a lottery offers a 1 in a million chance of winning a million dollars, and the price of a ticket is greater than $1, then entering the lottery is probably not a smart idea; but if the price of a ticket is less than $1, then I might want to buy a ticket (or two).

We can use these basic concepts to do a (very) rough calculation of a person's EV of voting, restricting ourselves to the particular case of an American voting in a presidential election.

As above, let u = the amount of money such that you are indifferent between being given u and getting to choose who wins the election. So u is an index of how much you would value being able to choose the candidate of your choice to win, in American dollars.

Let p = the probability that your vote will be decisive. In a recent paper (LINK), some statisticians (including Nate Silver of FiveThirtyEight fame) estimated this to be between 1 in 10 million and 1 in 10 billion, depending on the state you live in, with most Americans being around the 1 in 60 million range. So let's let p = 1/60000000.

At this point I should perhaps mention that using the probability of a decisive vote as an index of individual vote influence is not considered controversial by academics and scholars in this area. While there is substantive disagreement on whether it makes good sense or not to vote, nearly all of the disagreement stems from these broader issues of the expected value of voting, not from disagreements about whether it is appropriate to consider probability of a decisive vote as a good indication of individual influence. There is no way of deciding which individual vote was decisive (in the event that a single vote was decisive), but it is fairly straightforward to determine which groups of people had more influence on the outcome than others.

With that aside, let's say that in order for us to consider voting to be worth it for us, we need EV = u*p > $5 (for the non-Americans, this is roughly the price of an average McDonald's lunch ticket).

For this to be true, we need u > $300 million. In other words, we need the relative value of our preferred election outcome, compared to the other possible election outcomes, to be worth more than $300 milllion to us.

The first impression of this number (at least for me) is that it is awfully high. Our estimate is subject to some variation based on, e.g., what you consider to be a sensible minimum EV to make it worth voting, where you live, etc. But within any reasonable tweaking of these parameters, you're always going to end up with a Big Number.

There are two basic insights we can begin to draw from this. First, if you are a purely self-interested voter, there is virtually no conceivable way that voting in an American presidential election makes sense for you.

However (and this is the second point), if you start to view voting as an altruistic act, akin to donating to a charity, it starts to make more sense. Andrew Gelman has a nice way of framing this HERE (I adopt his framing although he illustrates using different figures; see the point above about variation). As Gelman points out, if your vote is decisive, it will make a difference for 300 million people. So, using our figures from above, if you believe that the difference between outcomes of your preferred candidate winning and the other candidate winning is equivalent to, on average, every American's quality of life being improved by > $1, and this is what you care about rather than how much it will benefit you personally, then voting could make sense. Of course, reasonable people could disagree on whether this belief about the anticipated outcome is realistic. (The "on average" clause from above is important because, realistically, the difference in outcomes will mean huge losses for some people, huge gains for other people, and relatively little for many others in between -- the consideration here is whether, after averaging across all those gains and losses, we have a net gain of > $1 for all, or > $300 million in total.)

Whether a similar line of reasoning would lead you to quitting your job and becoming a full-time lobbyist is left as an exercise to the reader...

Formal analysis requires conventions, but I doubt many voters are reckoning their "expected value" on the likelihood of influencing the outcome of the presidential race. Voting for most people is a team sport, and they turn out to maximize the influence of collectives to which they belong (not necessarily anything as formal as a political party) and reassert their membership in those collectives, including membership in the nation itself. The action is neither strictly self-interested nor altruistic, but collective. It takes place to maintain and renew the institutions in which it occurs, and the communities of interest which seek to guide those institutions. The continuation of those institutions establishes a base value for EV on top of which any influence on the outcome of a specific race is just gravy.

I have no doubt that, if taken as a description of what most people actually do mentally when considering voting, EV is essentially worthless. Its utility is as a prescriptive tool for those of us who don't see the answer to the voting question as so clear-cut and obvious. I think you are right that, on a descriptive level, voting is for many people an opportunity to reaffirm, publicly and/or privately, their membership in social collectives at various levels. The team sport analogy is apt: voting is a chance to cheer for the home team. To the extent that people find this cheering to be rewarding, it makes perfect sense for them to do it. Maybe some people even get sexual excitement from voting. I don't know. The thing is, not everyone places much value in these other motives (or at least, voting doesn't address these motives for them :wink:). If you are a person who views voting as, first and foremost, an opportunity to potentially effect beneficial national outcomes, I hope you will agree that it makes sense to start wondering about the details of what the likely impact of your vote will actually be. If you are such a person, it might of course still make sense to vote, possibly for reasons like those previously described, but this is at least non-obvious. This is of course why the question comes up so often.

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Originally Posted by DuB

If you are a person who views voting as, first and foremost, an opportunity to potentially effect beneficial national outcomes, I hope you will agree that it makes sense to start wondering about the details of what the likely impact of your vote will actually be.

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Well, yes and no. Obviously it's worth crunching the numbers and going into it with your eyes open; the data is valuable. If you have enough interest in effecting "beneficial national outcomes" to investigate in that much detail, though, maybe the question to ask is not whether voting in a presidential election is worthwhile, but whether it's sufficient. It's definitional that voting in the contest with the most voters and the fewest options is the least effective political action one can take. With any additional political involvement, however, your personal influence climbs and the cost of casting that one ballot falls. Even if your interest is not sufficient to inspire, say, consistent voting on all offices even between general elections, it doesn't take a terribly high level of interest to warrant doing something rather than nothing, i.e. accepting any EV greater than zero.