Play the Percentages: Go All-in

Why the optimal strategy is more aggressive than you think

In 2009 a mysterious figure known only by the screenname Isildur1 stormed through the ranks of high stakes online poker. In a series of head-up (one-on-one) matches with some of the best players in the world, he won over $5 million in the span of a few months.

Isildur specialized in a poker variant called no-limit Texas hold-em. “No limit” means you can bet any amount, up to everything you have on the table, up to and including going all-in. Most players, however, rarely leverage this rule to the fullest. Novice players often make very small bets, perhaps uncomfortable with risking more. Experienced players tend to think in fractions of the pot, betting 50%-100% of what’s already in the middle.

Isildur’s signature move was the overbet: betting many times the size of the pot. For example, if there was $10,000 in the pot, he might go all-in for $100,000. Intuitively, this move doesn’t seem to make sense. Why risk so much to win so little, relatively speaking?

But put yourself in his opponent’s shoes. Unless you have a really strong hand, it’s a tough call to make, and you’ll rarely have a really strong hand (weak to medium hands are far more common). This is especially true if you’re up against someone who’s good at recognizing situations where you’re vulnerable - which Isildur was.

The more times this happens, the more it gets in your head. Why does he keep betting so much? Can he really have such a strong hand every time? Is it time for me to take a stand with a mediocre hand?

There’s no question Isildur was a nightmare to play against, but there was still a sense that his style was mostly effective on psychological grounds, that it couldn’t be mathematically correct. This seemed to be born out when Isildur crashed back to earth as quickly as he had soared, losing all his winnings and then some.

But personal ups and downs notwithstanding, in terms of pure poker strategy, Isildur had the last laugh. In 2017, when a poker AI called Libratus defeated some of the best human players in the world, a core part of its strategy was overbetting.

“[The] AI was more likely than humans to make huge overbets — meaning that they would bet three, five or even 20 times the amount of chips in the pot,” according to one article summarizing the match.

In strategic terms, overbets make sense in situations where you can have the nuts - poker jargon for the best possible hand - and your opponent can’t, or is very unlikely to. It’s a way of leveraging your higher concentration of the very strongest hands to apply maximum pressure, either forcing the opponent to fold most of their hands, or getting them to pay off big when you actually have it.

There’s even a mathematical proof that if you know for sure whether or not you have the best hand, the optimal bet size is to go all-in no matter how much money you have on the table. Even if there was only $1 in the pot and you had $1 million in your stack, the optimal bet size would still be all-in!

Given the math behind it, why did it take the rest of the poker community so long to get onboard with overbets? Viewed from a short term perspective, an overbet is extremely risky. If you’re bluffing and get caught, you’ll lose a lot. Perhaps even scarier, everyone will be able to see that you made a huge bet with absolutely nothing and got called. It’s embarrassing. The psychological specter of this happening is hard to overcome. Yet viewed from a longer term perspective, it’s arguably riskier not to overbet. As the computer showed, overbetting is part of an optimal strategy. If you never overbet, that means you’re missing opportunities to make the most profitable bet.

In a 2002 paper with the esoteric title, “Do Firms Maximize? Evidence from Professional Football,” the economist David Romer analyzed what happened when football teams had the ball on fourth down. According to Romer, teams were punting far, far too often. The math supported a much more aggressive strategy of going for it on fourth down. Romer was largely mocked or ignored by NFL coaches, even as a consensus formed around his conclusions in the analytics community. Going for it was still seen as irrational, even though the numbers overwhelmingly supported it.

As 49ers head coach Steve Mariucci put it, "The crowd is going 'Go for it,' and they're just drinking beers and just going for it. Sometimes you get swayed a little bit. So you've got to block them out and you've got to make sense of it all.”

In other words, going for it on fourth down is the purview of beer-swilling yahoos, while cooler heads prefer to punt. In Mariucci’s analysis, the emotion is all on the side of going for it. However, it’s clear there are powerful emotions involved in playing it safe as well: the fear of embarrassment if you take an unconventional risk and lose; the regret of facing a big setback as a result of a risk you didn’t have to take.

Only in the last few years have teams really embraced going for it on fourth down. A key turning point was when Patriots head coach Bill Belichick went for it on fourth down against Peyton Manning and the Colts in a widely televised game. The attempt failed, but it sparked a wave of discussion. Even before that fateful attempt, Belichick was known for being more aggressive on fourth down than other coaches. One reason for that might be that he holds an economics degree and was one of few coaches who claimed to have actually read the Romer paper.

Another possible reason was job security. When punting on fourth down was the default play, a coach could hardly be fired for punting, but they could certainly be fired for making an unconventional call to go for it on fourth down and having it blow up in their face. As perhaps the greatest coach of all time, Belichick was not likely to be fired over one call, regardless of the result. Maybe this gave him more leeway to make risky calls if he thought it was the right play.

Like poker players, most football coaches resisted going for the aggressive play for a long time, even though the math said it made sense. For many people, there seems to be a built-in reluctance to take risks. One possible reason for that is known as the precautionary principle; basically, the idea that in uncertain situations it’s best to play it safe. From an evolutionary perspective this makes sense. A species that risked destruction on a regular basis would not survive for long. However, this reasoning depends on asymmetry in the payoffs: it only makes sense if the consequences of failing are much worse than the upside of succeeding.

In poker and football, while one play might feel riskier than another, it’s doubtful whether this kind of asymmetry is really present. When it’s fourth down, whether you punt or go for it, the stakes are the same: winning or losing one game of football. So it seems like some kind of evolutionary, built-in risk aversion is kicking in when it doesn’t actually make sense.

Imagine you’re on the second story of a burning house. You can either jump out the window or stay in the house, there’s no other escape route. In the short term, jumping out the window feels risky, and it is: you might get hurt or even die from the fall. But if you stay in the house, you’ll certainly be engulfed in the fire, so jumping is clearly the best option even though it feels risky in the moment.

Where this metaphor breaks down is, in the burning house, eventually the fire will spread to your room, so you’ll viscerally realize there is no safety in staying put. In contrast, an NFL coach can punt on fourth down for their whole career without ever feeling the win probability they’re giving away. The downside of their decision never rises to the level of a crisis, but that doesn’t mean it’s not real.

And what about situations where there is no safe choice? That is unfortunately the situation we find ourselves in during a pandemic that has already claimed hundreds of thousands of lives in the United States, with vaccines that are proven to be effective but also have risks associated with them.

When several people who had received the J&J vaccine died of blood clots, the CDC paused distribution of the vaccine out of an “abundance of caution.” But is it really more cautious to pause a vaccine when thousands of people are still getting sick? On the face of it, the numbers seemed to overwhelmingly point towards continuing to administer the J&J vaccine: the lives saved would greatly outnumber the lives lost.

Some have argued that the CDC’s advisors behaved like NFL coaches, favoring a decision that felt safe but, according to the numbers, was a big mistake. In another similarity to NFL coaches, they may have been optimizing for their own job security, consciously or not: it’s hard to blame someone for vaccines not given, but they could certainly be blamed for deaths resulting from side effects of the J&J vaccine if they resumed after knowing the risks.

It’s true that there are emotions that spur us to take risks, but there are also emotions - perhaps more prevalent, if not more powerful - that pull us towards playing us safe. This gets us into trouble when the safe choice is not really safe, it just kicks consequences down the road to a distance where they don’t feel urgent. In fields that are tractable by mathematical analysis, the optimal strategy, stripped of emotion, often turns out to be bolder than anyone imagined.

Thanks to James Somers for feedback on earlier drafts of this piece.