The Tragedy of Optimality

I have children.  And because I have had them in the era of DVDs and iPods, I have watched all of the Pixar feature films, many of them well over a hundred times.  It's an occupational hazard, although there are some countervailing benefits; for one, my impression of Shark, doing his impression of Woody, is spot on.

Some time ago, as I was watching Finding Nemo for probably about the 267th time, I thought: Now isn't it too bad that Nemo just didn't stay away from the boat (butt)?  All this hassle could have been avoided.  But on the other hand, if they avoided the hassle, Nemo would have spent the rest of his childhood being helicopter-parented around by Marlin.  And it occurred to me that a lot of the other Pixar films had similar moments.  If only Woody hadn't sent Buzz flying out the window.  But then they would never have bonded or saved the neighborhood from a bully.  If only Bob hadn't sent his boss Mr Huph flying through four office walls and gotten fired.  But then he never would have stumbled onto a plot on the lives of the former supers and regained the trust of the people.  And so on.  I think this pattern, as formulaic as it might seem in retrospect, is part of the strength of Pixar stories.

I was reminded of all of this a couple of days ago as I was making my way through James Gleick's Genius, a biography of Richard Feynman that, for a wonder, wasn't written by the man himself.  As a boy, Gleick writes, possibly in reaction to not having been blessed with extraordinary or even ordinary athleticism, Feynman disdained the fine arts—music, drawing, poetry, and so forth—as not being masculine enough, for being too impractical.  You almost want to go back through the decades and slap some sense into the boy...and yet, if he had not disdained those things, would the world have been deprived of the great genius of Feynman?  As the mathematician G.H. Hardy said of the Indian prodigy Ramanujan,

He would probably have been a greater mathematician if he had been caught and tamed a little in his youth; he would have discovered more that was new, and that, no doubt of greater importance.  On the other hand he would have been less of a Ramanujan, and more of a European professor and the loss might have been greater than the gain.

The general notion is one of the trade-off as yet unseen.  We hear all the time about the value of being willing to fail, of being ready to risk substantial loss in search of almost inestimable gain.  But it's easier to be brave, I think, when you know what you might gain.  Few will wholly fault you then.  As both fiction and fact tell us, though, there are plenty of moments when things of value are risked, and seemingly without even the hope of gain, simply through recklessness or stubbornness, and yet things of value too are gained nevertheless.  Under the circumstances, without willful insistence, it seems an error even to call these risks.  More accurate to call it routine imperfection (although certainly also more of a mouthful).

In light of my posts on game theory and the like, it may sound as though I'm advocating for occasionally suboptimal behavior as a way to obtain optimal results.  That's not exactly right; as far as they go, the results of game theory are inviolate.  You can't get optimal results from suboptimal choices.  But what you can do is discover that your measure of what's optimal wasn't quite right.  You can optimize perfectly for dollars (or regular-season wins, or family time, or whatever), and yet thereby miss greatness.

Or, you might miss nothing.  In fact, most of the time, and for the vast majority of people, that's exactly what you miss.  And that's what makes routine imperfection so unappealing on an individual level, because it's regularly unrewarding.  But on a social level, with millions or billions of people operating in general autonomy, it's at once unavoidable and essential.

Speaking of the Electoral College

About four years ago, I made a somewhat long-winded post (not by the standards of this blog, I suppose, but generally) about the electoral college, prompted by Bernie visiting my office to ask me about it.  Of course, I had primed him by saying I had something nerdy to say about it, and he's unable to resist that kind of bait.  One of the best things about nerdy posts of this sort is that timeliness is not a big attribute, so here it is, four years on:

Bernie just came into my office because he wanted to hear my spiel on the electoral college. Put aside for the moment the question of whether this indicates he's some kind of pedagogical masochist; what started this was the question of whether voters in a big state like California suffer because their vote is diluted, or are favored because the state's electoral power is so huge. The short answer is that it's mostly the latter, but there are a few interesting wrinkles along the way.

One way to approach the question is to consider how many Missouris (11 electoral votes) it would take to match the electoral power of California (55 electoral votes). The obvious answer is five Missouris, but this assumes that the Missouris all vote as a bloc, as California would (in a presidential election). In general, assuming independent Missouris, this is unlikely. Because some of the Missouris would be likely to cancel others out, the swing power they hold would not be 11 times 5, but 11 times the square root of 5, or about 24 electoral votes. (Remember random walks and square roots?) If all Missouris were independent—and why shouldn't they be?!—it would take 25 Missouris to match one California (in terms of the states' electoral power).

Now, each individual voter in California has less power to impact the state's overall direction, just because there are oodles of people in California—roughly five times as many as in Missouri, to match the disparity in electoral votes. (It's actually a little more than that, but we'll deal with that in a bit.) That means that it takes roughly five Californians to make the same percentage impact on their state's result as one Missourian. Again, that only happens if the Californians vote as a bloc; assuming they vote independently, it would take 25 Californians to equal one Missourian.

So at first blush, it seems that these two effects cancel each other out: California has 25 times as much electoral power as Missouri, but each Missouri voter has 25 times as much individual impact on the Missouri result as a California voter has on the California result. However, there is one additional effect of California's large population: The required swing in close votes in California is smaller, percentage-wise, than it is in Missouri. It's basically the law of large numbers: In any evenly contested election, the outcome probably won't ever be exactly 50-50, but the more populous the state, the closer it will be to 50-50, and the smaller the percentage swing required to change that outcome. This factor is again equal to the square root of 5, and it's what drops out in the final result—that a California voter has a larger impact on the national electoral result than a Missouri voter.

One complicating factor is that the number of electoral votes belonging to a state is not quite proportional to that state's population, not even when rounding is taken into account. The reason is that only the number of Representatives belonging to the state is proportional to the state's population; there are also the Senators, which are two a state. Since there is one electoral vote per Congress member (Representatives and Senators combined), small states have a much higher representation per capita than large states.

The upshot is that the most overall power is held by voters in the largest states, like California, Texas, and New York. Intermediate are voters in states with moderately large populations, such as Ohio or Illinois, as well as the smallest states. The weakest are voters in states like Arizona and Colorado, which are too large to gain much advantage from the "bonus" two electoral votes corresponding to Senators, but are too small to gain advantage from the enormous impact of a large population (and large electoral college representation).

It should be pointed out that the foregoing discussion only applies to votes where each state is contested—a "battleground" or "swing state," in recent election parlance. In practice, the impact a California voter has in the 2008 presidential election is nearly nil, since the state is almost guaranteed to go to Obama. (We'll see if I eat those words. [Obviously, I didn't. —brian]) The necessary swing is way too large for a reasonable number of California voters to overcome. That the predisposition of a state's voters is more than enough to swamp the effect of the largest population in the Union is, in my opinion, an indication that those trying to "fix" the election system (typically by replacing it with direct popular vote) are barking up the wrong tree, often in an irrational attempt to right a wrong—the 2000 Bush victory, say, which went against the popular vote—that ultimately had to do with factors distinctly different from the structure of the electoral college.