Another "hot hand" false alarm

Here's a Deadspin article called "Gambling Hot Streaks are Actually Real." It's about a study by academic researchers in London who examined the win/loss patterns of online sports bettors. The more wagers in a row a client won, the more likely he was to also win his next bet. That is: gamblers appear to exhibit the proverbial "hot hand."  

It was a huge effect: bettors won only 48% of the time overall, but over 75% of the time after winning five in a row. Here's Deadspin's adaptation of the chart:

Keeping in mind the principle of "if it seems too good to be true, it probably is," you can probably think for a minute and come up with an idea of what might really be going on.

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The most important thing: the bets that won 75% didn't actually win more money than expected -- they were just at proportionately low odds. That is: the "streaking" bettors were more likely to back the favorites on their next bet. (The reverse was also true: bettors on a losing streak were more likely to subsequently bet on a longshot.)

As the authors note, bettors are not actually beating the bookies in their subsequent wagers -- it's just that they're choosing bets that are easier to win. 

What the authors find interesting, as psychologists, is the pattern. They conclude that after winning a few wagers in a row, the bettors become more conservative, and after losing a few in a row, they become more aggressive. They suggest that the bettors must believe in the "Gambler's Fallacy," that after a bunch of losses, they're due for a win, and after a bunch of wins, they're due for a loss. That is: they take fewer chances when they think the Fallacy is working against them.

But, why assume that the bettors are changing their behavior?  Shouldn't the obvious assumption be that it's selective sampling, that bettors on a winning streak had *always* been backing favorites?

Some bettors like long shots, and lose many in a row. Some bettors like favorites, and win many in a row. It's not that people bet on favorites because they're on winning streaks -- it's that they're on winning streaks because they bet on favorites! 

Imagine that there are only two types of bettors, aggressive and conservative. Aggressives bet longshots and win 20% of the time; conservatives bet on favorites and win 80% of the time.

Aggressives will win five in a row one time in 3,125. Conservatives will win five in a row around one time in 3. So, if you look at all bettors on a five-win hot streak, there are 1024 conservatives for every aggressive. (In fact, for every streak length, the factor increases by 4. 4:1 after one win, 16:1 after two wins, and so on, to 1024:1 after five wins.)

It seems pretty obvious that's what must be happening.

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But wait, it's even more obvious when you look closer at the study. It turns out the authors combined three different sports into a single database: horse racing, greyhound racing, and soccer.

A soccer game result has three possibilities -- win, lose, draw -- so the odds (before vigorish) have to average 2:1. On the other hand, if there are 11 horses in a race, the odds have to average 10:1. 

Well, there you go!  The results probably aren't even a difference between "aggressives" and "conservatives". It's probably that some bettors wager only on soccer, some wager only on racing, and it's the soccer bettors who are much more likely to win five in a row!

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There's strong evidence of that kind of "bimodality" in the data. The authors reported that, overall, bettors won 48% of their wagers -- but at average odds of 7:1. That doesn't make sense, right?  48% should be more like even money.

I suspect the authors just used a simple average of the odds numbers. They took a trifecta, with 500:1 odds, and a soccer match, with 1:1 odds, and came up with an simple average of 250:1. 

It doesn't work that way. You have to use the average of the probabilities of winning -- which, in this case, are 1/2 and 1/501. The average of those is 503/2004, which translates to average odds of 1501:503, or about 3:1. (Another way to put it: add 1 to all the odds, take the harmonic mean, and subtract 1 from the result. If you leave out the "add 1" and "subtract 1", you'll probably be close enough in most cases.)

The bigger the spread in the odds, the worse the simple average works. So, the fact that 48% is so far from 7:1 is an indication that they're mixing heavy favorites with extreme longshots. Well, actually, we don't need that indication -- they authors report that the SD of the odds was 38.

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Finally, if none of that made sense, here's an analogy for what's going on.

I study people who have a five year streak of eating lox and bagels. I discover that, in Year Six, they're much more likely to celebrate Hanukkah than people who don't have such a streak. Should I conclude that eating lox and bagels makes people convert to Judaism?