Mad as March

Note: This post has been updated to correct some of the probability figures, and to mention UMBC's defeat of Virginia in the 2018 tournament.

 

Hey, it's March, it's mad, it's March Madness!

Which means that there's math, too.

I was inspired to write math this time (as opposed to all those other times) by a short video in which some mathematics guy explained why filling out a perfect bracket (ignoring the "First Four" and focusing only on the 64 teams in the "real" tournament) is so hard.  According to him, it's because there are 63 games, each one eliminating one of the 64 teams, each of which has to be prognosticated correctly in a perfect bracket.  The number of possible brackets is therefore 2 to the 63rd power, or about 9 times 10 to the 18th power.  And so the odds of filling out a perfect bracket is 1 in that enormous number.  Even if everyone in the whole world filled out a bracket, the odds are still a billion to one against anyone getting it all right.

As viewers too numerous to list pointed out (correctly), this line of reasoning is entirely bogus because it assumes that each of the 63-game sequences is equally likely.  Of course they aren't.  Higher seeds are more likely to win their games than lower seeds.  In particular, in the 28 years they've had 64 teams in the tournament, no 16 seed has ever beaten a 1 seed.  That doesn't mean it's impossible, or that it'll never happen, only that it's very unlikely on a game to game basis.  Eventually, though, it's inevitable, provided the tournament goes on year after year. (ETA 2024-03-29: Sure enough, it's happened twice in the last several years. In 2018, the 16th seed University of Maryland Baltimore County Retrievers beat the top seed Virginia Cavaliers by twenty, 74–54, and in 2023, the 16th seed Fairleigh Dickinson Knights beat the top seed Purdue Boilermakers by the more sedate score of 63–58. Both Cinderellas went on to lose their second-round matchups, though.)

The upshot is that certain brackets are more likely to be correct than others, and I've even heard tell that people have filled out perfect brackets in the past.  So I wondered to myself: What are the odds of someone filling out a perfect bracket?

To estimate that (because this really isn't something you can determine empirically), I had to construct a model for simulating tournaments.  This has to be done because although there are plenty of statistics for how often a 5 seed beats a 12 seed (because that pairing always meets four times in the first round, once for each of the four regions), there aren't going to be statistics for how often a 5 seed beats a 14 seed, because that could only happen in the regional finals, after both teams have defeated three teams (mostly teams better than they are), which is highly unlikely.  In fact, I'm not sure that it's ever happened.  I needed a model that would be reasonably simple to calibrate, quick to evaluate, and was generally applicable to any pair of seeds.

The model I decided upon, fairly quickly, works as follows.  Each team has a certain "strength," which depends only on its seeding.  Then, if teams A and B meet, one with strength SA and the other one strength SB, the probabilities of each of them winning are given by

P(A wins) = SA / (SA+SB)
P(B wins) = SB / (SA+SB)

Fairly straightforward. (ETA 2024-01-21: This turns out to be the Bradley-Terry model. I'm sure it's been independently reinvented many times, since it's such a natural idea.) I calibrated it by finding statistics on how often teams of different seeds made it to various stages of the tournament.  I was a bit surprised, actually, by some of the statistics. I had imagined that the chances of winning the first-round game would decrease very little from the 1 seed down to about the 4 seed or so, and then accelerate quickly down to about the 13 seed, and then decrease very slowly again.  (I was aware that there were some seeds that historically have won more often than you might expect, such as the 12 seed, but I assumed those were statistical anomalies that one could reasonably expect to show up in only thirty years of history.  In particular, I did not want to assume that 12 seeds, for instance, were magically better than 11 seeds, or even 5 seeds.  I assumed seeds properly reflected relative talent.)

But no such pattern appeared.  Instead, the probability of winning the first round seems to decrease fairly steadily from the 1 seed down to the 16 seed.  The 12 seed teams do seem to win slightly more than you might expect, but they win no more often than do 11 seeds.  So here, at least, the 12 seed bump wasn't great.  (EDIT: Ha!  Both 12 seeds playing on the first day of the 2013 tournament won: the Oregon Ducks, and my California Golden Bears.  Further EDIT: And now 12th-seeded Ole Miss.  And 13th-seeded La Salle, for that matter.)

While I was trawling for these statistics, by the way, I also came upon the assertion that the way teams were seeded (and in particular, not re-seeded after each round) placed a penalty on the middle seeds, so that the 12 and 13 seeds, it was claimed, were in some cases likely to advance further than the 8 and 9 seeds, for instance.  I thought it might be interesting to see if that came out of the model.

Anyway, as a result of these statistics, I came up with the following strengths for the 16 seeds:

1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, 1/10, 1/11, 1/13, 1/17, 1/25, 1/41, 1/73

You may notice that these strengths imply that a 16 seed will win the first round 1/74 of the time.  (But they never win, I can hear you saying.  Nonsense.  You're only saying that because no one ever has. ETA: It happened! In 2018, the 16 seed University of Maryland, Baltimore County defeated the 1 seed Virginia. As of this writing, before the 2023 tournament, I think that makes the 16 seed one for 148, or exactly half as often as predicted by my wholly ex recto model.)  As you can see, the top 11 seeds have strengths that decrease harmonically—that is, as the reciprocal of the seed.  After that...well, maybe if you're dorky enough, you'll see what the pattern is.

To be sure, I don't really know how accurate this model is, largely because (as I mentioned previously) statistics for many of the match-ups just don't exist, or at least don't exist in sufficient quantity.  But since I'm taking the results with a grain of salt (I only need rough order-of-magnitude estimates), it only has to be moderately accurate for it to do what I want.

Anyway, my simulation engine runs through each of the 32,768 possible regional brackets (each of the regions is identically seeded) and determines how likely each bracket is, and how far each seed got in that bracket.

The results were sort of interesting.  In the first round, there are no notable oddities, which makes sense because the better teams always play against worse teams, so the higher a team is seeded, the more likely it is to win the first round.

1 seed wins First Round with probability 0.986486
2 seed wins First Round with probability 0.953488
3 seed wins First Round with probability 0.892857
4 seed wins First Round with probability 0.809524
5 seed wins First Round with probability 0.722222
6 seed wins First Round with probability 0.647059
7 seed wins First Round with probability 0.588235
8 seed wins First Round with probability 0.529412
9 seed wins First Round with probability 0.470588
10 seed wins First Round with probability 0.411765
11 seed wins First Round with probability 0.352941
12 seed wins First Round with probability 0.277778
13 seed wins First Round with probability 0.190476
14 seed wins First Round with probability 0.107143
15 seed wins First Round with probability 0.046512
16 seed wins First Round with probability 0.013514

(Remember to take all those digits with a sizable grain of salt.)  The second round is where it gets interesting.  Consider a moderately lower seed, like the 12 seed.  If it happens to win its first round, its second round game will be against either the 4 seed or the 13 seed.  It's likely to be against the 4 seed, but not overwhelmingly so; it will be the 13 seed about 19 percent of the time.  In that latter case, the 12 seed will actually be a slight (57 percent) favorite in the second round.  But even if it meets up against the 4 seed, it will be an underdog, but not a prohibitive one.  In such match-ups, the 12 seed beats the 4 seed about 24 percent of the time.

Compare that to one of the middle seeds—say, the 8 seed.  If it reaches the second round, it has to play against either the 1 seed or the 16 seed.  Not only is the 1 seed overwhelmingly likely to win its first round match-up, but it is also a much stronger opponent for the 8 seed than the 4 seed was for the 12 seed.  As I mentioned above, this happens because the teams are not re-seeded after each round, with the best of the surviving teams playing the worst of the surviving teams, the second best playing the second worst, and so on.  The moral of this story is that as far as the Sweet Sixteen is concerned, the pundits are right: The middle seeds are (slightly) cursed!

1 seed reaches Sweet Sixteen with probability 0.882035
2 seed reaches Sweet Sixteen with probability 0.763414
3 seed reaches Sweet Sixteen with probability 0.632753
4 seed reaches Sweet Sixteen with probability 0.496767
5 seed reaches Sweet Sixteen with probability 0.366148
6 seed reaches Sweet Sixteen with probability 0.248486
7 seed reaches Sweet Sixteen with probability 0.148009
8 seed reaches Sweet Sixteen with probability 0.064476
9 seed reaches Sweet Sixteen with probability 0.052084
10 seed reaches Sweet Sixteen with probability 0.080832
11 seed reaches Sweet Sixteen with probability 0.093788
12 seed reaches Sweet Sixteen with probability 0.082892
13 seed reaches Sweet Sixteen with probability 0.054193
14 seed reaches Sweet Sixteen with probability 0.024973
15 seed reaches Sweet Sixteen with probability 0.007745
16 seed reaches Sweet Sixteen with probability 0.001405

Now that I think about it, I think it's this bump for the 10 through 13 seeds that accounts for their "upset" reputation, more than their success in the first round.

The trend continues, though with decreased intensity, in the third round, in order to reach the Elite Eight (although notice that there's a bump at both the 6 seed and the 10/11 seed)...

1 seed reaches Elite Eight with probability 0.732698
2 seed reaches Elite Eight with probability 0.510341
3 seed reaches Elite Eight with probability 0.302688
4 seed reaches Elite Eight with probability 0.127560
5 seed reaches Elite Eight with probability 0.081095
6 seed reaches Elite Eight with probability 0.081461
7 seed reaches Elite Eight with probability 0.056441
8 seed reaches Elite Eight with probability 0.025441
9 seed reaches Elite Eight with probability 0.019169
10 seed reaches Elite Eight with probability 0.024748
11 seed reaches Elite Eight with probability 0.020596
12 seed reaches Elite Eight with probability 0.009123
13 seed reaches Elite Eight with probability 0.004812
14 seed reaches Elite Eight with probability 0.002918
15 seed reaches Elite Eight with probability 0.000807
16 seed reaches Elite Eight with probability 0.000101

...but interestingly, it's essentially gone by the time one reaches the Final Four.

1 seed reaches Final Four with probability 0.535913
2 seed reaches Final Four with probability 0.222277
3 seed reaches Final Four with probability 0.106313
4 seed reaches Final Four with probability 0.053660
5 seed reaches Final Four with probability 0.030044
6 seed reaches Final Four with probability 0.018613
7 seed reaches Final Four with probability 0.011601
8 seed reaches Final Four with probability 0.006982
9 seed reaches Final Four with probability 0.004848
10 seed reaches Final Four with probability 0.003929
11 seed reaches Final Four with probability 0.003042
12 seed reaches Final Four with probability 0.001761
13 seed reaches Final Four with probability 0.000753
14 seed reaches Final Four with probability 0.000221
15 seed reaches Final Four with probability 0.000039
16 seed reaches Final Four with probability 0.000004

There is a bit of an inflection in the curve at about the 10 seed, but it's still monotonically decreasing across all seeds.

Now, one might think that this is just an artifact of the strengths I chose, somewhat arbitrarily, and that either the trend itself, or its vanishing by the Final Four, might not arise with a different set of strengths.  For what it's worth, I thought that myself, and tried running the simulations with different sets of strengths.

As it turns out, with any reasonable set of strengths that I chose, the trend of the middle seeds being somewhat worse off at the Sweet Sixteen than the moderately lower seeds—that trend might arise or not, but if it did, it always disappeared by the Final Four.  I didn't find any set of strengths which retained the trend all the way through to the Final Four.  Heuristically, I think this is because by the time you get to the regional finals, you have to play the best teams in the entire bracket—you can't avoid anyone for sure.  So if your goal is to reach the Final Four, then have no fear: I don't think the lack of re-seeding significantly hurts you if you're in the middle seeds.  Your answer may be different, of course, if your goal is simply to make it to the Sweet Sixteen.

Oh yes, the question that first stimulated this exploration: What are the odds of filling out a perfect bracket?  Well, given the strengths as I initially had them, the most likely bracket is the one where all the favorites win every game.  For each region, this happens about one time in 123; since there are four regions in the entire bracket, the odds on the ultimate "chalk" bracket are that raised to the fourth power, or about one time in 225 million. That leaves the Final Four. Since any seeds can meet there, we'll simplify things and just assume that all three games can go any way, leading to an eight-fold increase in the number of overall tournament brackets, or 1.8 billion.

Some people, not accustomed to thinking through probability, will point out that no bracket has ever turned out perfectly chalk (which is true) and suggest that a slightly off-chalk bracket is individually more likely (which isn't).  One must not confuse "most likely" with "likely."  The reason that each year the tournament turns out slightly off-chalk is that there are enormously more off-chalk brackets than chalk ones.  (Strictly speaking, of course, there is only one perfectly chalk bracket.)  The fact that these off-chalk brackets are slightly less likely than the pure chalk bracket is more than compensated for by their superior numbers.

Anyway, there are a lot of brackets filled out each year, so given that the odds on the chalkier ones are probably not too much worse than 1 in 1.8 billion, it wouldn't be surprising if somewhere along the way, someone did end up filling out a perfect bracket. As far as we know, however, it hasn't happened.

Note: By the way, if you've only a passing familiarity with sports, you may wonder what I mean by "chalk."  Chalk is a sports betting term that refers to favorites winning; the more favorites win, the chalkier the outcome.  I've heard that it came from betting lines on horse races, which were written up in chalk, but I've no good idea if that's actually so.  Anyone?

EDIT: Here's a good write-up on how the term came to be.

A Harry Potter Puzzler

I recently had occasion to put together a bit of a puzzle for a fellow math fiend on the occasion of his birthday.  If you enjoy Harry Potter and recreational math—I know that's a limited bunch, but it occurred to me this blog readership might just select a little for that—you may well like this small offering:

Gryffindor and Slytherin played a Quidditch match.

1. Alicia Spinnet scored the first goal of the match, giving Gryffindor its one and only lead of the match until the moment that Harry caught the Snitch, winning the match for Gryffindor. The match was, however, tied three times (other than at 0-0).
2. Gryffindor never scored two goals in succession; Slytherin always scored at least one goal in between. Furthermore, each pair of Gryffindor goals was separated by a different number of Slytherin goals.
3. Ginny Weasley and Katie Bell each scored a hat trick (three goals) for Gryffindor.
4. Each Gryffindor score was assisted by the Chaser who scored the previous Gryffindor goal. Each Slytherin score was also assisted by the Chaser who scored the previous Slytherin goal.
5. Katie Bell's second goal came one hour into the match. Slytherin scored twice as many goals after that point than they did before it. Interestingly, Graham Montague scored all of his goals before that point, Cassius Warrington scored all of his after that point, and Adrian Pucey scored more than the other two combined.

Who scored the last goal of the match, and what was the final score after Harry ended it?

(Muggles—non-magical folk—may find it helpful to know that in Quidditch, goals are worth 10 points apiece.  Play continues until a player catches the Golden Snitch, which is worth 150 points and ends the match.  There are no other ways to score points.  The team with more points at the end of the match wins.  Assists are passes from one player to a different player leading directly to that second player scoring a goal.  Finally, each team has three and only three Chasers, who score all the goals; the other players are two Beaters, a Keeper, and a Seeker.)

Going Whole Ballhog

If you're one of the tens of readers who follow me, then unless the bottom of your rock doesn't carry ESPN, you've probably heard something about this kid from Grinnell who dropped 138 on a hapless Faith Baptist Bible College basketball team.  Now, granted, this was a Division III basketball game—hardly the acme of organized basketball.  Still, as Kobe Bryant said, "I mean, I don't care what level you're at, scoring 138 points is pretty insane."  Jack Taylor is a household name now, people.

Rather predictably, there was some backlash, with some people claiming that it was rigged, or that it was selfish basketball, or at least not The Way That Basketball Should Be Played (because anything that portentous has to be written upstyle).   I can't say anything as to whether it was rigged, although it didn't look like it to me, and as with any conspiracy theories, it's easy to say something like that when you don't have to offer any proof.  All you have to do is throw out your hands and say, "It's common sense!"

But we can say something about whether it was selfish or bad basketball.  Some folks have taken it upon themselves to make a virtue out of evenly distributed teamwork.  That's fine as a matter of personal opinion, but they make a mistake, I think, who believe that it's an intrinsic virtue of basketball.  It wasn't an intrinsic virtue of basketball when Naismith put up the first peach baskets, and until someone invents a game that makes teamwork an explicit scoring feature, there won't be a sport where it's an intrinsic virtue.  (I also think that some of these folks could benefit from playing with a scoring phenom, just to see what it's like, but that's neither here nor there.)

What makes it a virtue—when it is a virtue—is that it makes a team more efficient, by and large.  On the occasions when a player goes out and consciously attempts to score a bunch, it quite frequently turns out that the other players on the team are more efficient, and thus the team as a whole would have been more efficient if the offense had been more evenly distributed.  This is a basic result from game theory.

But that didn't turn out to be the case here.  Taylor scored 138 out of his team's 179 points.  That's 77 percent.  To get those points, of course, he used up a lot of his team's possessions: 69 percent, according to ESPN.  It is a lot, but it shouldn't overshadow the fact that the rest of his team used up the remaining 31 percent of the possessions and ended up scoring only 23 percent of the points.


Let's see how that stacks up against two other phenomenal scoring performances of the past: Wilt Chamberlain's mythic 100-point night in Hershey, and Kobe's own 81-point barrage at home against the Toronto Raptors.  (Taylor nearly had 81 just in the second half.)  I'm going to ignore claims that the Warriors game was a farce in the second half, or that the Toronto Raptors were a defensive sieve; I'm only interested in the efficiency figures.

Chamberlain's Warriors scored 169 points that night, so Chamberlain scored 59 percent of his team's points, using (again according to ESPN) 47 percent of his team's possessions.  Kobe's Lakers scored 122 points, so he contributed 66 percent of his team's points, while using (ESPN again) just 51 percent of the team's possessions.


One way to look at these feats is to consider how much more efficient the individual players were than the rest of the team.  So, on a percentage basis, Taylor scored 77 percent of the points on 69 percent of the possessions, whereas the rest of the team scored 23 percent of the points on 31 percent of the possessions.  Taylor, therefore, was (77/69) / (23/31) = 1.50 times as efficient as his teammates.  Similarly, Chamberlain was (59/47) / (41/53) = 1.62 times as efficient, and Kobe was (66/51) / (34/49) = 1.87 times as efficient.

However, such a measure can easily be misleading.  If someone plays a single minute, puts up a single three-pointer, and makes it, they might (as a normal example) have 3 percent of the team's points with only 1 percent of its possessions.  By the same metric, such a player would be (3/1) / (97/99) = 3.06 times as efficient as his teammates.  What's missing is some measure of the magnitude of the player's impact.

A more representative measure of the player's efficiency impact can be obtained by considering how efficient the team would have been if the other players had managed to use up all of their team's possessions, at the same efficiency they had been exhibiting.  For instance, Taylor's teammates used up 31 percent of the possessions, scoring 23 percent of the points they eventually scored.  If they had continued at that same clip, but used up 100 percent of the possessions, they would have eventually scored 133 points—about 74 percent as much as they actually did.  To put it another way, the team with Taylor was 31/23 = 1.35 times as efficient as they would have been without him.

Using that as our guideline, the Warriors with Chamberlain were 53/41 = 1.29 times as efficient as they would have been without him, and Kobe's Lakers were 1.44 times as efficient as they would have been without him.

Just as a demonstration of how amazing all of these numbers are, if a team averages a true shooting percentage of 50 percent amongst four players, and the remaining player uses up half the possessions with a true shooting percentage of 70 percent, that team is only 1.20 times as efficient as they would be without that player.  To increase their teams' efficiency as much as they did, these three athletes had to be remarkably efficient and prolific.

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.

Sealing Up Time

"But I thought—what about changing your own past?  What about the paradoxes?"

Dr. Vanner pursed her lips.  "Yes, I wondered about that too."

"So what happens if I shoot my grandfather?  Not that I would, but I could."

"Well, Jason, it turns out that's a bit of an interesting question, whether you could or not.  But grandfathers are very large, complicated things.  People are always trying to figure out how time travel could possibly work with grandfathers, and bullets, and messy macroscopic objects like that.  It's easier just to deal with simple particles first.  You figure out the particles, the grandfathers take care of themselves."

"OK..."

"Well, grandfathers are made out of particles, aren't they?"

"I guess that's one way to think of them."

"You know, I had a grandfather too."  Dr. Vanner smiled warmly.  "Anyway, I think the best way to answer your question is by way of example.  Say you're a particle.  An electron.  You didn't come from nowhere, you started out as a muon.  But muons don't live very long; they decay in a few microseconds to yield a couple of neutrinos, and you."  She sketched on the board as she spoke.

"Me—oh right, I'm an electron."

"That's right.  Now you, as an electron, can live essentially forever.  You step into the time machine (or a smaller version of it), and you go back in time.  Just a little bit: say, a microsecond."

"Ahh, I think I see where you're going.  I'm going to bump the muon just enough so that it decays somewhere else, and even if it decays into me, I'm nowhere near the time machine to go back in time.  Paradox."

"Exactly.  So what's the resolution?  The resolution is that particles aren't billiard balls.  As an electron, you don't really bump into the muon.  You 'interact' with it."

"What difference does that make?"

"The difference is that the interaction has a random element.  If I hit a cue ball into another billiard ball in the same exact way, over and over again, both balls will go off in the same directions, over and over again.  It's predictable, deterministic.  That's why you can have expert billiard players.  But subatomic particles aren't the same way.  They can hit in exactly the same way, as far as we can tell, but the results may be completely different from one time to the next.  There are no expert electron players.

"And that's the key.  There's going to be one way or another that you could end up hitting that muon that will end up with it decaying into you at the right place at the right time.  Maybe you give it an extra nudge, and it goes a bit faster in the same direction, but it decays sooner than it would have.  Maybe it goes off in a different direction, but when it decays into you, you still end up heading toward the time machine."

"But if there's so many different ways it can happen, which one actually does happen?"

"That's a complicated question.  The simplest way I can think of to understand it is to imagine the universe as a kind of simulation.  If we conduct an ordinary quantum-mechanical experiment, there's a certain probability that the experiment will end up one way, and the rest of the time, it ends up another way.  It can do that because the experiment is anchored on only one end: the start.

"But in the time travel case, it's actually anchored on both ends.  When you the electron exist at a particular time and place, there's an anchor at that point.  The universe is in a more or less definitive state at that point.  Normally, that's the only anchor.  But in this case, when you travel back in time, there's a second anchor, in that we know you have to end up back (or should I say 'forward'?) in the time machine.  In between, nearly anything can happen—subject to the laws of physics.

"So imagine that the universe runs a simulation.  How many different ways can you start at the first anchor point, and end up at the second anchor point?  Which ones are most likely, when you adhere to the laws of physics?  We don't even know which ones are most likely beforehand, except in the very simplest of cases."

"So as an electron, I end up taking the most likely path back to the time machine?"

"No, not quite.  If the chances of you taking that path are three in five, then three times out of five, that's the path you'll take.  Or you could end up taking a once-in-a-million path (like bouncing off of three other particles before entering the time machine); it's just that you only have a one-in-a-million chance of doing that."

"But I always end up back in the time machine."

"That's right."

"But then it sounds like I can't ever change anything.  If the universe is anchored on both ends, what point is there in going back in time?"

"Very good question!  The point is that the second anchor point is not a 'complete' anchor point.  The first point is.  It covers the whole universe.  But the second anchor point only consists of you.  The only thing that's required is that you—the original you, remember, not the one that goes back in time—you have to end up in the time machine.  Everything else can change."

"Wait a minute.  So forget about me being an electron and everything.  I'm me, Jason Sawyer.  I enter the time machine, and I go back a day or so.  I could see anything.  I could see me—the original me.  And anything might happen, but in a day or so, that original me has to end up getting into that same time machine.  But everything else could change.  I might forget to do yardwork that I actually did earlier today.  When I—the time-traveller me—catches back up to this present, I would know that the yardwork didn't get done.  But if you were watching, you'd all of a sudden see the leaves suddenly strewn across the yard, instead of put away in the yard trash?"

"Not quite.  Remember, when you rake the leaves, it affects more than just the leaves.  If I were watching and the leaves never got done, I'd never have any memory of the leaves being in the trash, would I?  So in fact, yes, the leaves would still be on the ground, but it wouldn't look to me like they suddenly appeared on the lawn.  In my brain, there would be lost any impression that the leaves were ever raked in the first place.  At least, I assume that's the most probable outcome.  There are all sorts of other outlandish possibilities that are more drastic but which I probably wouldn't have to worry about."

"All right, forget about the trash.  What about if I went back, really far back, far back enough to...well, let's not say I shot my grandfather, but I somehow set him up with someone other than my grandmother.  How can I possibly end up in the time machine then?"

"Hmm, let's think..."  Dr. Vanner considered this.  "OK, well, how well did you know your grandfather?"

"Huh?  Uhh, kind of?  I—he died when I was eight.  How does that matter?"

"It matters because it's not sufficient that you end up in the time machine.  You also have to end up going into the time machine in a state that's sufficiently 'consistent' (in a technical sense) with the way you ended up going into it 'the first time.'  And that state includes your brain: everything you remember and know about yourself and your experiences.

"So what happens?  You obviously have to go into the time machine.  So somehow, some collection of particles comes together to form you.  The way it actually happened is just one possibility: Your parents conceive you, and you start out as a small number of particles.  Over time, you take on some particles, and lose some other particles, and eventually, grow up to be who you are today.

"Now one other possibility is that somewhere, near here, just a few moments ago, a collection of particles just randomly happened to show up to make...well, you, but including all the memories you currently have (which would, in that case, be completely fictional).  In a classical world, that's impossible.  In a quantum-mechanical world, it's merely improbable—although we're talking really, really improbable.  Like you could run the universe a googol times and it wouldn't even come close to happening.  Still, it's possible.  And because by setting your grandfather up with someone other than your grandmother, you've already eliminated a bunch of probable outcomes that end up with you in the time machine, all the improbable options get a boost, so to speak.  As Sherlock Holmes said, 'When you have eliminated the impossible, whatever remains, however improbable, must be the truth.'  In this case, by setting up your grandfather on a date, you've made certain options impossible, and they're eliminated.  More than one path remains—all of them improbable a priori, perhaps—but one of them has to happen, in order for you to get back in that time machine.  The option of just a bunch of particles coming together to make a fully-formed you is always available.  If nothing is left besides that, then that's what'll happen.  In this case, though, I bet something else is more likely than that."

"Like what?"

"Like...suppose that after you set up your grandfather with someone else, you go off to visit the world.  You're not going to hang out with him forever, do you?  So after you leave him, suppose he breaks it off with the other person, and gets back with your grandmother.  And everything else happens more or less the same, so far as producing you is concerned.  His life history would be a bit different, but not in ways that are really all that critical.  That's why I asked you how well you knew him.  Do you know who he was with before he met your grandmother?"

"Hmm...not really."

"Exactly.  Remember, what happens to everyone else can change, but you have to stay more or less the same.  So my guess is that the most probable outcome is that his life would change in ways that you never knew about in the first place, so that when you go into the time machine 'the second time,' whatever you knew about your grandfather the first time remains true."

"Whoa.  Wait.  That means that I have...I can't really change too much about the people that I know really well.  Like Mom, Dad, my sister, and even you a little bit.  But, everything else and everyone else could change drastically?  Maybe I'd go back, and as a result of my getting my grandfather to go out on even one date with someone different, I still come back with the rest of my family, but—oh, let's just say—no World War II, ever?"

"Well, that's possible, but still unlikely, even after you eliminate the impossible.  Remember that World War II wasn't started by just one thing.  There were triggers, but there were broad forces too that were behind it.  The second anchor, and the fact that what happens is likely the most probable thing, means that the whole 'butterfly effect' thing is not as chaotic as one might think.  Just setting up your grandfather on one date is unlikely to reset all of world history.  In order to change that, you'd probably have to go back quite a bit further in time.

"Also, by the way, keep in mind that your time-traveled self is still aging.  If you go back far enough to set up your grandfather on a date, by the time you get back here to 2027, you'll probably be about as old as your grandfather would have been today.  That's assuming you make it all the way back.  There's no guarantee of that."

"Wait, I thought I had to get back.  To get into the time machine."

"No, remember, that's the original you who has to do that, not the time-traveled you."

Jason hesitated for a second.  "Right," he said finally.  "It's very confusing.  And it's weird to think that the results of going back in time are so intimately connected with me.  I can change really distant things a lot, but everything I know well is saved—at least to the degree that I know them.  How is that possible?  I mean, this machine doesn't know a thing about me."

"It doesn't know it in the usual sense, Jason, but when you enter it and go back in time, it knows everything about your current state, at the moment you go back.  It fixes it.  That's the second anchor.  The thing that makes all the other changes possible is that it is only you who is anchored in time and space.  Everything else is floating partly free—and the less you know about them, the freer they are.  Even me, to a point.  Although I still have to be able to invent the machine.  So I feel pretty safe, especially if I know the person well who's traveling in it."

"Why don't you get into it?"

Dr. Vanner looked for a moment as though she were going to answer that.  "I—I can't explain that to you yet," she said finally.  "You'll have to take my word for it that there's a good reason for me not to get into it."

"Hmm, OK."  Jason looked thoughtful again.  "All right, one more question.  Suppose I do something really drastic.  I shoot myself before I get into the machine.  Or I do something really memorable to myself, something that didn't happen the first time.  How can I—the original me—end up back in this time machine in a...what did you call it?"

"A consistent state."

"Yeah, that."

"Well, there's a limit to how reliable our senses are.  How do you know you didn't already go back in time and make a big noise in front of yourself?  Because you don't remember it.  But what makes you so sure that it didn't happen?  Is your memory that reliable?  There must have been some things that have happened that you don't remember.  Normally, it's because those things happened so far in the past that the memory has faded.  We have the notion that the memory is still there, locked inside us somewhere, that we just can't find it.  But what if the memory really went away?  My guess—although I really don't know, I haven't tested it—is that you'd just lose all memory that it happened."

"Weird.  But what if I shot myself?"

"That depends.  Maybe the shot misses, even though you think you hit yourself?  Maybe you miraculously heal in seconds?  Those sound ridiculously improbable, but you've already eliminated as impossible all the normal paths, so the truth must be something outlandish.  Even if you stay to watch yourself die, at some point, your body, reasonably healthy, must make its way into the time machine.  In that case, you might very well see your dead body vanish in front of your eyes, just in time to make it into the time machine at the right moment.  Again, impossible in the classical world, but possible in the quantum-mechanical world, and—if you've already eliminated everything else—even, in a sense, inevitable in that world."

"So the original me is immortal."

"The original you, yes," agreed Dr. Vanner.  "But only up to the point you get in that machine.  From that point on, that you vanishes, and the time-traveled you reappears at some point further back in time.  And that you is vulnerable.  Anything at all could happen to that you."

Excerpt from "Time Binder" copyright (c) 2012 Brian Tung

Review: Ready Player One

Ernest Cline's Ready Player One (Crown Publishing, 2011) features no grand, sweeping philosophical statements, no startling revelation about human nature, no moral judgments or object lessons.  Like the game that forms the backbone of its plot, it is an adventure with a beginning, a middle, and an end, and it rests its case on that straightforward simplicity.

Wade Watts, like the protagonist of many a science fiction novel, is a high-school student-cum-computer-geek in a dystopian society, but unlike many a science fiction novel, RP1 hardly dwells a second on the dystopia part.  The world of 2044 is in ruins, due to a catastrophic shortfall in fossil fuels, but this crisis is put in primarily to motivate the near-universal emotional investment in OASIS (Ontologically Anthropocentric Sensory Immersive Simulation), a sort of virtual-reality massively multiplayer online game that serves simultaneously as school, work, and escape for most citizens, including the continually impoverished Wade.

OASIS was the brainchild of James Halliday, a reclusive, Wozniakian genius with an intense penchant for 1980s pop culture, who, with his more affable business partner, Ogden Morrow, built a multi-billion dollar computer game empire starting in the 1990s.  Over the years, they gradually drifted apart, as Morrow focused on sustaining their company after the death of his wife (his and James's childhood friend) and Halliday seemed to fall deeper into mental illness.

The events of RP1 are set in motion when Halliday dies in 2039.  His death is announced not on the obituary page, but in a video will and testament shot by Halliday himself.  Halliday had no wife, no children, no surviving relatives at all, so in his video, he explains that he will bequeath his entire estate (valued at about a quarter of a trillion dollars) to the first person to solve a series of puzzles embedded into OASIS and its millions of fictional worlds.

Wade had a boundless admiration for Halliday even before his death, so he knows all about 1980s culture, an asset that will stand him in good stead in his quest for the billions.  The problem is, so do many of the other OASIS users, including his best friend Aech (pronounced "H"), geek-girl blogger Art3mis, and the obligatory bad guys, the faceless multi-national corporation Innovative Online Industries (or IOI)—none of whom Wade has actually met face-to-face.  Throughout RP1, Wade will have to contend with each of them and his other rivals, some of who are ready and willing to commit murder and worse in their race for the prize, as well as Halliday's own devilish imagination and his obsession with the 1980s. 

RP1 is written in a quick, breezy style with pulpish overtones.  (Of course, for those who grew up with the golden age of science fiction, the pulp might be a positive.)  In developing his story, Cline feels compelled to explain a bit of Halliday's world creation, and as a result occasionally gets caught up in his own world creation.  From time to time, we are treated with technical details on how his characters connect to OASIS—details that will abruptly jar many readers from his otherwise breathlessly scripted (and somewhat thin) plot.

The 1980s pop culture references are another matter.  They are dotted liberally throughout the book, sometimes merely for flavor, other times integral to the plot.  And Cline's novel features a slightly implausible ending, albeit one that mirrors that of popular movies in its time frame.  For those of us who grew up in the 1980s, reading RP1 is a bit like watching retrospective "clip" episodes of shows like Silver Spoons and Family Ties (one of Halliday's favorites); others will just be bemused by the constant parade of cultural touchstones they have no connection with. 

To a certain extent, Cline is trying to maintain his footing on a slippery slope.  The technical and pop-culture references are consistent enough to suggest that he had more in his back pocket, details that would have appealed to a very specific audience, but which he held back in aiming for a broader audience.  On the other hand, if he had held back even more, RP1 might have been more accessible, but it would have lost much of its childlike appeal.

Ultimately, RP1 spreads it on thick with its geek and pop culture details, thick enough to turn off readers who don't sympathize with its emphasis.  One gets the distinct impression, however, that it otherwise wouldn't be substantial enough to satisfy those of us who do, and while it teeters on the precipice from time to time, RP1 just does get the job done.

Brian's 0-10 score: 6.0