Today’s topic comes from a reader’s suggestion (yes, I do have readers!) prompted by his observation that a friend of his bet, in his words, “waaaaay too many top 20s” in a golf tournament.

Some of my earliest childhood experiences with gambling were when my dad would take me to the track, either the local Woodbine racetrack in Toronto or the greyhound track on vacation in Florida (it was a long time ago, don’t call PETA). Funny story about my dad, he took my mother to the track on their first date so he could impress her with his “ability to make money”. Of course he won nothing, but I guess the whole thing was adorable enough to start a chain of events that led to me being brought into this world. Anyway, we’d go to the track once or twice a year, we’d study the statistics in the program and, high rollers that we were, we’d put one or two $2 bets down on each race. The whole outing probably cost my dad and I each $10 in -EV and $2 for the program. The education was priceless.

One day there was a race where I liked two of the horses and I wanted to bet them both to win. “Don’t do that”, my dad said, “you’re betting against yourself”. Meaning that if one of the bets won, the other was necessarily going to lose. Was my dad right when he said that was a bad strategy?

Today’s topic is correlation among a set of bets; specifically, negative correlation. How to spot it, what it means, when it helps and when it hurts. We are NOT talking about parlays here (at least not yet), we’re talking about multiple individual bets made by the same person on the same event.

You have negative correlation if, conditional on one of the bets winning, the likelihood of the other bets winning is diminished. The most extreme case is when they are diminished all the way down to zero, in the case of multiple “to win” bets on the same horse race or golf tournament, but that isn’t necessary. Multiple place or show bets on different horses and multiple “top 20” or “make the cut” in golf count too. Same with multiple “make the playoffs” bets on different teams in the same division or conference. Anything where you have a limited number of spots to fill means that each spot filled makes one fewer spot available for somebody else. Thus, you are literally betting against yourself.

What are the implications of betting against yourself? Let’s start with the most important measure of success: EV. In fancy statistical terms, EV is a “linear operator“. In regular terms,

The EV of a set of bets is the sum of the EV of each of the bets in the set.

This is true regardless of whether your bets have negative correlation, positive correlation or no correlation. In terms of your EV, the impact of betting against yourself is…absolutely nothing.

So why do we care? Let’s talk about variance.

All three of these “bell curves” have the same EV but different variance. The lower the variance, the more likely your actual outcome will be close to your EV. The higher the variance, the more likely your actual outcome could be significantly better OR worse than your EV (“good luck” / “bad luck”).

Ready for the punch line? OK, here it is:

Betting against yourself reduces your variance.

The more negative correlation you have in a set of bets, the lower your variance becomes.

Now, is lower variance a good thing or a bad thing? To answer that, let’s look at the consequence of lower variance…it tightens your range of probable outcomes around your EV. Is that a good thing? Well…

If your EV is positive, variance is your worst enemy and you want to lower it as much as possible. Betting against yourself is a good thing.

If your EV is negative, variance is your best friend and you want to raise it as much as possible. Betting against yourself is a bad thing.

During the pandemic with no sports, I’ve been catching up on some good TV shows I’ve missed over the years. One of them is “Billions”. (I’m only on season 3 so no spoilers please!) Cool show, from the guys who made Rounders, and it even had an out-of-nowhere Taleb reference! The reason hedge funds are called hedge funds is that they take a (presumably) +EV position, then reduce their variance by hedging with another position that’s negatively correlated. For example if they believe strongly in Apple stock they might take a long position in Apple and complement it with a short position in the S&P 500. When you have +EV, variance is the only way you can lose.

Back to my dad and me at the track. With a few small exceptions, the commissions that the tracks take out of the betting pools are enormous, usually 15-25%. As sharp as my dad and I liked to think we were as we pored through the stats in the program, there’s no way it was enough to climb that mountain. When you have -EV, variance is the only way you can win. The winning strategy for the hedge fund was a losing strategy for us. Whether he fully understood the theory or he was just going by intuition, my dad was right.

As I said up front, we are talking about a set of individual bets, NOT a parlay. All that stuff about correlation having no impact on your EV? That’s out the window when we’re talking about parlays. Putting negatively correlated outcomes together in a parlay reduces your EV, and you should never do it. A couple years ago, a well known sports betting media personality (doesn’t matter who, we’re not in the name-and-shame business here) tweeted out a parlay of 8 golfers to all make the cut at the British Open. Because there are a fixed number of spots above the cut line, each golfer who makes the cut reduces the likelihood of the next one making the cut. At the British Open, the top 70 players make the cut. To extend this into an extreme example, if this guy’s parlay had had 71 legs instead of 8 legs, it would have been a guaranteed loser as it would have been impossible for all 71 players to make the cut (yeah I know, ties, blah blah blah, you get the point). He would have been better off making 8 individual bets instead of parlaying them.

So to summarize,

Betting against yourself is a good idea if you have +EV (but never in a parlay).

Betting against yourself is a bad idea if you have -EV.

Until next time…may your variance be favourable!

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