Monday, April 14, 2014
Billion Dollar Brackets and Fine Tuning
For this year's NCAA Men's Basketball Tournament, Warren Buffet offered to pay $1 billion to anyone who could correctly predict the outcome of every game. Unsurprisingly, no one won. Many stories and commentaries have cited numbers about just how improbable it is to pick every game correctly. What none of these stories and commentaries seem to have noticed is that the question is meaningless.
The usual comparison is with a lottery. Lotteries are run so that every possible combination of numbers is equally likely, which means that no one strategy is any better than another. Choosing "1 2 3 4 5" for Powerball works just as well as "2 3 5 7 11" (the first 5 primes) or "8 15 16 23 42" (from the Lost sequence) or any other sequence of 5 natural numbers less than 60. (Don't play any of those sequences, though. Because there is a "reason" for each of them, several other people are probably already playing them, which means that even if you won the prize, it is much more likely that you'd have to split the winnings!)
Most estimates of the "odds" of getting all the games right in a bracket challenge probably assume that the odds getting each individual game right are 50/50. Since there are 32+16+8+4+2+1 = 63 games in a bracket (not counting the play-in games), this would make the odds of a random bracket winning 1 in 263, or 1 in 9.223 x 1018. However, no 16 seed has ever beaten a 1 seed in the NCAA basketball tournament, but 93.75% of brackets chosen completely at random will have at least one 16 seed beat a 1 seed. Clearly, there are better strategies for filling out a bracket than just flipping a coin.
So what is the best strategy for filling out a bracket? No one knows! My strategy is to choose the winner of each matchup so that the odds of the team with the lower seed winning are (15 + difference in seed numbers)/30; that way, the odds of a 1 seed beating a 16 seed are 100% but the odds of one 1 seed beating another 1 seed (in the Final Four or Championship Game) would be 50%. This assumes that the people on the committee that sets the seeding really know what they are doing, which is debatable -- but what isn't debatable is that they know more about college basketball than I do.
Of course, there is a lot more information available -- the outcomes of games through the season and the order in which they came, the stats for individual players, the general strategies of the coaches, and the announced injury situations. No one really knows how to put all this together! Then there is information which is not publicly available (exactly how the players feel, physically and emotionally) and there are unpredictable things that will happen over the course of the tournament.
In the case of Powerball, any strategy is the ideal strategy, but in the case of a basketbal bracket, the ideal strategy is unknown. If the strategy is unknown, how can we know how likely it is to win?
At any rate, let me pretend that someone who really knows basketball can choose the 1 vs 16 matchups correctly 100% of the time and the other ones 70% of the time. His odds of getting all 63 games right are 0.759, or 1 in 1.378 x 109. Probably no one is really that good, and he would still face long odds, but he is a billion times more likely to fill out his bracket correctly than the crude estimate had indicated!
Let's go a little further. Suppose someone is a super-genius at college basketball and can pick games correctly 99% of the time (and I'll give him the 1 vs 16 matchups for free). Someone like that would already have his own billion dollars, but he's going to play anyhow. His odds of filling out the whole bracket correctly are 0.9959, or 55.27%. To me, that is the best measure of just how hard this is!
Unfortunately, it's not hard to find people quoting probabilities as though they were meaningful when they really are not. What are the odds that your marriage will end in divorce? Sorry, national stats are really useless in this case, because you are not a typical person. No one is. Your background and personality (and that of your spouse) are unique, and they have everything to do with whether you two will decide to remain together.
One particularly frustrating (to me) use of bad statistics comes in the fine-tuning argument for a Creator. This is based on the fact that there are 25 free parameters in the Standard Model; these are physical constants with values that can only be determined by experiments, not on the basis of some principle we know. In several cases, a change of a few percent to one of these parameters might have caused the Big Bang to collapse on itself again, or for stellar nucleosynthesis to create a world in which no atoms are more complex than hydrogen or in which all stars are neutron stars. In such a universe life as we know it would be impossible. What are the odds that this happened by chance?
Once again, we do not have enough information to answer the question, and it is dishonest to pretend we do. Since these are "free parameters", we obviously need additional information to determine their values -- but we also need additional information to know anything about the distribution from which they might be randomly chosen.
Think of it this way: Maybe the universe is the way it is because, Einstein notwithstanding, God did throw dice to determine the answer to the Ultimate Question of Life, the Universe, and Everything, and 42 was the number that came up. We can further specify that all the dice were fair and identical, and that the most likely value for the throw was 63. What are the odds that 42 came up? The number will depend on whether it was 6 20-sided dice (6 x 10.5 = 63) or 18 6-sided dice (18 x 3.5 = 63). Even in this simple example, if we do not know what kind of dice were used, we cannot answer the quesion.