Sep 17, JDN 2458014

At a reader’s suggestion, I am expanding upon the argument I made a few weeks ago that political coalitions are strongest when they are willing to accept some disagreement. I made that argument with numbers, which is likely to convince someone like me; but I know that many other people don’t really think that way, so it may help to provide some visuals as well.

60% of this rectangle is filled in red.

This represents the proportion of the population that agrees with you on some issue. For concreteness but to avoid making this any more political than it already is, I’m going to pick silly issues. So let’s have this first issue be about which side of the road we should drive on. Let’s say your view is that we should drive on the right. 60% of people agree that we should drive on the right. The other 40% think we should drive on the left.

Now let’s consider another issue. Let’s say this one is about putting pineapples on pizza. You, and 60% of people, agree that pineapples should not be put on pizza. The other 40% think we should put pineapples on pizza.

For now, let’s assume those two issues are independent, that someone’s opinions on driving and pizza are unrelated. Then we can fill 60% of the rectangle in blue, but it should be a perpendicular portion because the two issues aren’t related:

Those who agree with you on driving but not pizza (that would include me, by the way) are in red, those who agree with you on pizza but not driving are in blue, those who agree with you on both are in purple, and those who disagree with you on both are in white. You should already be able to see that less than half the population agrees with you on both issues, even though more than half agrees on each.

Let’s add a third issue, which we will color in green. This one can be the question of whether Star Trek is better than Star Wars. Let’s say that 60% of the population agrees with you that Star Trek is better, while 40% think that Star Wars is better. Let’s also assume that this is independent of opinions on both driving and pizza.

This is already starting to get unwieldy; there are now eight distinct regions. The white region (8) is comprised of people who disagree with you on everything. The red (6), blue (4), and green (7) regions each have people agree with you on exactly one issue. The blue-green (3), purple (2), and brown (5) regions have people agree with you on two issues. Only those in the dark-green region (1) agree with you on everything.

As you can see, the proportion of people who agree with you on all issues is fairly small, even though the majority of the population agrees with you on any given issue.

If we keep adding issues, this effect gets even stronger. I’m going to change the color-coding now to simplify things. Now, blue will indicate the people who agree with you on all issues, green the people who agree on all but one issue, yellow the people who agree on all but two issues, and red the people who disagree with you on three or more issues.

For three issues, that looks like this, which you can compare to the previous diagram:

Now let’s add a fourth issue. Let’s say 60% of people agree with you that socks should not be worn with sandals, but 40% think that socks should be worn with sandals. The blue region gets smaller:

How about a fifth issue? Let’s say 60% of people agree with you that cats are better than dogs, while 40% think that dogs are better than cats. The blue region continues to shrink:

How about a sixth issue?

And finally, a seventh issue?

Now the majority of the space is covered by red, meaning that most of the population disagrees with you on at least three issues.

To recap:

By the time there were two issues, the majority of the population disagreed with you on at least one issue.

By the time there were four issues, the majority of the population disagreed with you on at least two issues.

By the time there were seven issues, the majority of the population disagreed with you on at least three issues.

This despite the fact that the majority of the population always agrees with you on any given issue!

If you only welcomed people into your coalition who agree on every single issue (the blue region), you wouldn’t win election if there were even two issues. If you only welcomed those who disagree on at most one (blue or green), you’d stop winning if there were at least four issues. And if there were at least seven issues, you couldn’t even win by allowing those who disagree on at most two issues (blue, green, yellow).

Now, this argument very much does rely upon the different opinions being independent, which in real politics is not the case. So let’s introduce some correlations and see how this changes the result.

Suppose that once someone agrees with you about driving on the right side of the road, they are 90% likely to agree on pizza, Star Trek, sandals, and cats.

That makes things look a lot better for you; by including one level of disagreement, you could dominate every election. But notice that even in this case, if you exclude all disagreement, you will continue to lose elections.

With enough issues, even with very strong correlations you can get the same effect. Suppose there are 20 issues, and if you agree on the first one, there is a 99% chance you’ll agree on each of the others. You are still only getting about half the electorate if you don’t allow any disagreement! Due to the very high correlation, if someone disagrees with you on a few things, they usually disagree with you on many things; yet you’re still better off including some disagreement in your coalition.

Obviously, you shouldn’t include people in your coalition who actively oppose its core mission. Even if they aren’t actively trying to undermine you, at some point, the disagreement becomes so large that you’ve got to cut them loose. But in a pluralistic democracy, ideological purism is a surefire recipe for electoral failure. You need to allow at least some disagreement.

This isn’t even getting into the possibility that you might be wrong about some issues, and by including those who disagree with you, you may broaden your horizons and correct your mistakes. I’ve thus far assumed you are completely correct and in the majority on every single issue, and yet you still can’t win elections with complex policy mixes unless you include people who disagree with you.