27 May 2011

Who'm I Friday - Some Perspectives on Probability

So... what were the odds that I'd write another blog about probability? In retrospect, 1.0 (that's a math-y way of saying 100%), and given that blog content is not represented by an agreed-upon categorical model, that's about as close as we can get to an accurate answer.

But fear not, this one will be very light on the math. But fear, because it will be heavy on pedantic philosophizing.

Sports Predictions
Brian Burke has created a website that combines highly intelligent mathematics with the mass appeal of professional football (American football, for my global friends). That, in itself, is an improbable (tee hee) accomplishment. Burke evaluates tons of historical data to build various statistical models of player value and game outcome. These outcome predictions are presented, naturally, as probabilities.

One question Burke is frequently asked (and one that he asks of himself) is this: How is it that "luck" seems to play such a big role in what ought to be a game of pure skill?

Intelligently, Burke is quick to point out that the statistical model cannot account for all of the factors in a game like football. Brilliantly, he then goes on to analyze his own analysis to determine the limits of the model.

Now, I can't match Burke's skill with probability math, but I can give an epistemological explanation for the phenomenon we call "luck" in regard to sports predictions. Here it is:

First-- and this is something that Burke himself has described quite well-- an accurate prediction of performance doesn't necessarily produce an accurate prediction of outcome. This can be easily demonstrated by imagining a baseball game where each team gets nine singles over the course of nine innings. But the Davistown Dawgs got exactly one single per inning, and thus scored no runs; whereas the Haverford Hawgs got nine hits in a single inning, scoring seven runs. The batting, pitching and fielding performances were identical, but the Hawgs came away with a big win. This sort of thing makes analysts chuckle, and causes sports writers to rip out their hair in search of a plausible story to appease the limited imagination of their readers (and themselves). They usually come up with a bunch of nonsense about "superior clutch hitting", which they promptly ignore the following day when the tables are turned and the Dawgs blow out the Hawgs 13-1.

Second is perhaps a corollary to the first. Imagine, if you will, that Dirk Nowitzki has made his last seven free throws and then-- inexplicably-- missed. How is this possible? Shooting free throws is obviously a matter of skill, and individual differences are predictable. The answer is that Dirk himself is a microcosm of performance & outcome variability: Each time he shoots a basket, he must judge the distance to the hoop, select an angle of entry, determine the necessary force and therefore how much to contract his arm muscles, predict the necessary release timing and decide how severely to contract his wrist muscles, etc. etc. etc. Because Dirk is a living organism, his mind and body are in a constant state of change, so that not even two consecutive free throws are truly identical. If he makes several small errors in judgment, or if the errors cancel out (e.g., too much arm and not enough wrist), he can make the basket; but if one error is large, or several errors align (e.g., not enough arc and a late release) the shot will miss. Simple, right?

Third: Players and coaches employ deliberate random factors. Your team might perform best passing the ball, but if you pass on every play you will be at a disadvantage. Ergo, the better strategy is to pass often, occasionally run left, and occasionally run right. So if you run left and the opposing defense happens to blitz left (playing their own randomization strategy), that's just bad luck. Those sports writers will say otherwise, but they're probably wrong.

Fourth-- and here's the tricky one-- we have to realize that the concept of "luck" applies to the predictor, not the predictee. In other words, if I say that the Colts will beat the Saints based on my statistical model, but then the Saints beat the Colts, the Saints did not have good luck-- I had bad luck. Some would call this an inaccuracy or flaw in the statistical model, but I couldn't disagree more. If a statistical model claims to predict winners with 75% accuracy and is correct three times out of four, it is statistically perfect, just like my prediction that heads will come up 50% of the time on a fair coin flip. And similarly, if the coin comes up tails several times in a row, it is not the coin that was lucky or unlucky-- it was me. New Orleans fans should take comfort in knowing that their Saints won based on skill.

State Lotteries
Once upon a time, a person could use drugs like cocaine and heroin without becoming a social pariah, whilst gamblers were considered the heathen scourge that we cheered for superpowered movie cops to beat up on [citation needed].

Alas, for those days.

I've nothing against a friendly game of poker, mind you, but gambling against the house (or state) for a superjackpot is sheer lunacy.

Let me put it this way: Would you gamble $3,875.97 of your own money on a 258:1 shot at winning $1,000,000? No? Then quit buying lottery tickets.

Those numbers come from the Washington state lotto, which uses a 6 of 49 ticket (two tickets per $1) and offers the following prizes for picking the correct numbers:
3 / 6 = $3
4 / 6 = $30
5 / 6 = $1000
6 / 6 = $1,000,000+

There are 13,983,816 combinations possible, so for every $6,991,908 (again, two sets per $1) collected they pay out an aggregate of $749,133 to 3-number winners, $406,320 to 4-number winners, $258,000 to 5-number winners, and about $1,000,000 to 6-number winners (it's incremented when there is no winner).

Ergo, if they eliminated the super jackpot and split that $1,000,000 amongst the 258 5-number winners, they would each get $3,875.97. The current format does not do this. The current format, paying the same amount, takes away 258 chances (out of every 13,983,816) to win $3,875.97 for that one chance (out of the same 13,983,816) at a million. If you play the lottery, you are therefore accepting a gamble of $3,875.97 at 258:1 to win a million.

My proposal is that they eliminate the cognitive barriers, keep the exact same net payout ratio, and allow players to really decide what sort of stakes they want to play for. The normal drawing would therefore pay everything out to anyone who got at least 3 numbers (56:1 per 50-cent ticket) -- hence that would be $9.66 each.
Then a second drawing would be held, for an entry of $6.66 (the 3/6 winnings less the original $3 prize), that pays $122.90 at 18.44:1 odds.
Then a third drawing would charge $92.90 (the previous winnings less the original $30 prize) and pay $4,877.37 at 52.5:1 odds.
Finally, a fourth drawing would charge $3,877.37 to win $1,000,000 at 258:1 odds.

If you make the initial bet and just let it ride, your chance of winning the jackpot is the same. Some legislator might argue that $3,877.37 is too much for people to be gambling, but that's the point, dunderhead. If you like, we can limit the fourth drawing to those who are replaying previous winners, but then the legislator would sulk and whine about people not wanting to risk that kind of money once they've got it, and it takes all the fun out of having a chance to win a super jackpot right off the bat-- which is to say, the chance to fantasize about winning a super jackpot without actually considering the probabilities involved.

And that's the point.

The worst damage from these state lotteries is not the $1-$5 a week that people waste gambling, but the mental and emotional investment given to a future that is both extremely unlikely and completely beyond their control. This investment is stolen directly from financial dreams that are actionable-- like applying for a better job, improving one's education, starting a business on the side, etc. Some would argue that the amount of lost emotional investment per person is negligible. If so, I find you guilty of the continuum fallacy and sentence you to shut the hell up until your brain starts working (possibly never). One in three people in the U.S. now believe that winning a lottery jackpot is the only way to become financially secure.


Long ago, when other eight-year-olds were dreaming about being pirates or whatever, I was wondering what it would be like to meet myself.

Specifically, an exact copy of myself.

More precisely, I imagined an exact copy of myself who'd been flipped through a fourth dimension so that his left- and right-hand sides were now switched up (or perhaps mine were, neither of us would be able to tell) and how was now standing facing me on some road or path.

Most explicitly, I wondered how the heck we'd get around each other. "Let's both go left--" wouldn't work, because my left would be his right. Perhaps I'd try to climb over him while he stooped down, but then he'd do exactly the same thing; so then I'd switch-- but so would he.

Maybe we could talk things over and sort it out-- or maybe not. Whatever I decide to say, my mirror-duplicate is going to be saying the exact same thing at the exact same time. The cool thing here would be that we understood each other perfectly, but the obvious drawback is that there would be no actual conversation, and if I couldn't work it out in my head all by myself there wouldn't be any point in talking, would there?

Could I/we flip a coin? My duplicate will try the same thing (with his own coin) of course, but we'd both agree that "heads goes that way" (I point left, he points to his right, which is the same direction) and keep flipping until only one of us got heads. However... if the road is as perfectly symmetrical as us, our identical muscles are going to produce a series of identical flips.

I never resolved that, but it came back to me recently while I was mulling over the Surprise Quiz Paradox (which is a variant of the unexpected hanging paradox). The paradox involves a professor who wants to give a surprise quiz and some students who don't want to study. I decided to approach the scenario as something of a "game" with the following rules:

Each Friday, Professor X has the option of giving a quiz.
Each Thursday night, his students have the option of studying.

If there is a surprise quiz (quiz but no studying), the Professor scores three points of satisfaction. If he gives a quiz that is not a surprise, he loses one point (because he worked for nothing-- he only likes surprise quizzes). The students, on the other hand, gain three points of satisfaction if there is no quiz and no studying; if they study when there is no quiz, they lose a point (because they worked for nothing).

Superficially, it would seem that the best strategy for the students is to never study. Regardless of what the Professor does, studying will never give them more satisfaction. But with that strategy, they will be taking a quiz every week, and never get a chance at the big payoff (no study + no quiz).

The Professor might be secure in always giving quizzes, but if the students are clever they will respond by always studying (after all, studying with a quiz gives them the same satisfaction-- zero points-- as having to take a surprise quiz without studying).

When I sat back to consider this little game, I imagined the possibility of a single iteration (i.e., this only happens once). And given that the Professor and students are perfectly clever and have mutual knowledge of rationality, I concluded that there was no certain conclusion as to how either party should behave. In other words, they were in a similar boat (on a similar path) to that of me and my duplicate self. And it struck me that the only rational solution in either scenario was for at least one party to engage in random behavior. If the Professor decides on a 50-50 chance of giving a quiz, the students won't have to study, no effort will be wasted, and both parties will get pretty much the best they could get out of the deal.

And then it struck me that maybe this is why electrons and quarks and stuff have wave functions. You know, quantum physics and stuff-- reality, as far as science can observe, is not deterministic. Perhaps, given that one electron is pretty much identical to another, and those sorts of things have to interact, randomization is a precondition of existence. And no, I can't back that up with good science or even a decent theory. This is a blog. I'm allowed to speculate based on intuition. But maybe, just maybe, someone with a little more knowledge of particle physics will see a useful idea in that.

I give it 3 to 1 odds.



ViolaNut said...

What about my 50-50-90 rule? You know, the one that says that whenever I have a 50-50 chance on something, I've got a 90% track record of choosing the wrong one. :-P

Cruella Collett said...

Or MY 50-50 rule. Either something happens (i.e. you win the lottery) or it doesn't (you don't) - two possible outcomes, fifty-fifty chance. Makes the lottery seem much more promising!

Cold As Heaven said...

If people knew more about statistics, they wouldn't waste their money on the Lotteries. I don't, never did. On the other hand, I enjoy playing French Roulette from time to time (the Vienna casino is my favorite). I start by playing on red/black and odd/even, to keep it going for a while, before I get rid of my money (with rare exceptions) by setting on single numbers, doubles and quads. And it's always entertaining to watch the skills of a good croupier >:)

Cold As Heaven