When prospect theory and poker bubbles collide
What would Danny Kahneman think of my recent tournament play?
Late last Tuesday, as the clock was nearing midnight, I found myself facing a decision that would determine the course of a poker tournament. It was the €2700 Second Chance event, for players (like me) who had busted out of the EPT Barcelona Main and wanted to keep playing. And we were three players away from the stone bubble—that crucial inflection point in every tournament that marks the difference between going home with nothing and making money. With just over 20 big blinds and plenty of players with fewer than 10 big blinds in the field, I was sitting pretty. All I had to do, basically, was fold, and I was guaranteed to profit on this particular event.1
It was a few hands later that I was dealt a lovely ace and jack of clubs in the big blind. The first player to act, who had slightly more chips than I did, raised, everyone folded, and I made the call. So far, so obvious. Then came the flop: an ace of diamonds, an 8 of clubs, and a 4 of hearts. I checked and the raiser made a tiny bet, just over a big blind. That’s when I veered away from the all you have to do is play passively and you will cash script. I decided to raise. After some thought, he called. The turn card: the deuce of clubs. Beautiful. With just about a pot-sized bet left, I decided to go all in. My opponent thought. And thought. And thought some more, shaking his head. Several times, he made as if to fold. And then, at long last, he called. He had an off-suit ace and king. The clubs did not come in on the river. And I went back to my room with nothing.
Let’s just say this up front: I absolutely did not have to bubble. I had plenty of chips. There were plenty of short stacks. I could have quite easily called the first bet instead of raising it. I absolutely could have checked the turn, keeping the pot small and, in the end, losing to a superior ace. Instead, I chose violence.
Some would say this was a clear lapse in judgment, likely the result of fatigue. It was late. I had been playing all day—and all day the day before. I had suffered a cooler earlier in the afternoon to see my EPT Main Event dreams sail away. I was tired and tilted and over it.
I, however, would disagree. I’d be the first to admit that I have moments of poor decision making—and that those moments are likely to multiply when I’m exhausted. But this wasn’t one of them. I was clear, I was thoughtful, and I knew exactly what I was doing. I weighed the options, considered the range of outcomes, and made my choice.
***
The late psychologist Daniel Kahneman won the Nobel Prize for his work on human decision making and what’s known as bounded rationality – that is, the biases and heuristics that make us something other than ideally rational beings. One of the central ideas of his work is a concept known as prospect theory, which essentially says that when we are evaluating gambles (or prospects) under risky conditions, we don’t evaluate gains in the same way we do losses. We tend to be more risk-averse when we can be assured a gain and risk-seeking to avoid a certain loss. And we are far more sensitive to losses than to gains.
Imagine you’re facing the following choice:
a) A sure gain of $240
b) 25% to gain $1000, and 75% chance to gain nothing
Which do you choose? Think about it for a moment before moving on.
Now, imagine you’re choosing between these options:
c) A sure loss of $750
d) 75% chance to lose $1000, and 25% to lose nothing
Which do you choose? Again, take a moment to consider.
When Kahneman and his co-author, Amos Tversky, posed these questions to 150 participants, 84% chose option A over B and 87% chose option D over option C. That, in a nutshell, is prospect theory: risk aversion in the first case, with a preference for the sure gain, and risk seeking in the second, with a preference for the gamble over the sure loss.2 The pattern is not rational—the EV of B is higher than that of A—but it is pervasive, emerging time and time again whenever these kinds of choices are on offer.
Back to my decision on the bubble. You can actually look at it as pretty closely resembling options A and B in the classic prospect theory illustration. With a comfortable stack and plenty of shorties, I can basically guarantee a min cash—a certain gain that will assure me a small profit in this particular tournament. But if I take the riskier approach, I stand to gain many times more than the minimum payout, given that I would then be one of the chip leaders of the tournament. Yes, I may go broke, as the opposing player has slightly more chips than I do, but I also put myself in a position to make substantial money. The appeal of the certain gain is understandable, but as in the classic demonstration of prospect theory, the EV of the gamble is higher—in this case, substantially higher, given how top-heavy poker tournament payouts are.
If we assume that most poker players understand risk-reward fairly well, it would seem like they would all agree and go with the higher variance play.3 But that’s not what happens. Over and over, players overvalue the cash.4 They overvalue their tournament life. And they play it safe, just to make sure they are still standing on the other side. Indeed, one of the calculations I made when I decided to push my position in this particular hand was that I was likely to have higher fold equity—that is, my opponent would throw away his cards more often—than I would at a different stage of the tournament, thus making my aggression even higher in expected value.
A guaranteed gain has intuitive appeal. Of course it does. But what happens if we always decide to play it safe, always live life afraid of busting, always decide to go for the certain outcome and never take the riskier path—even though that path may yield the much higher reward? That certainly is not the kind of life I want.
Even though this time, it didn’t work out, I did go on to place fifth in the €3,000 Mystery Bounty, for a six-figure score. (Listen to me recount part of that experience on the most recent episode of Risky Business.) And yes, I also played that bubble quite aggressively, pushing my edge every time I could. So, on balance, I would call that +EV.
Next week, I’ll talk about when prospect theory is not a fallacy and it may make sense to play it safe. As with everything, context is key.
I only entered one time, so I would indeed be profiting. The math changes if you fire multiple bullets.
The probability percentages are also weighted incorrectly by the human brain, another element of Kahneman and Tversky’s theory that I won’t go into here.
I ran the sim with GTOWizard, specifying that we were on the bubble. My exact hand likes to raise a chunk of the time.
I used to be one of them. I detail Erik Seidel’s reaction to my high min-cash rate in The Biggest Bluff.
Thank you for this great post.
Three polite thoughts for your consideration:
First, there is a wonderful place for a poker and psychology treatise akin to Harrington on Hold’ Em or Super System. Many poker coaches / books give strategies with a discussion of the underlying why. Poker players are more likely to hear about John Nash because of equilibrium in solvers but Kahneman et al are equally important.
I absolutely agree prospect theory is a great framework for your article but I think you could easily analyze play around a bubble in context of Mischel’s marshmallow study as well. That was Seidel’s comment to you in a nutshell —- do you want a min cash now or the larger (but not guaranteed) victory later.
I know there are questions about delaying gratification can be linked to how are brains work. I also wonder how much is associated with access to resources as children and in now. A min cash is much important when it is your full bankroll…
2. I question how much those of us who think in probability and ev are in someways are lying to ourselves. You correctly point out we should pick the higher EV spot. But statistics in life are rarely correct or repeatable. Life choices don’t come with statistics and equally no guarantee of repeating. Perhaps Kahneman is underestimating the what you might call present value in finance. Money today is by definition worth more than the same amount tomorrow.
3. I think another fundamental part of all of this is wagering, risk, gambling etc has a fundamental unfavorable connotation in certain parts of society. I think others would say all of like is risk, chance, and probability. But we often teach these concepts through “gambling” like poker leading people to improperly believe they should abstain. In short, probability needs a better PR team.
This is a great post. It perfectly shows how playing poker is good practice for making a lot of other decisions in life.
These have little to do with prospect theory, but there are some other factors in why the play you made was the right one. I wasn't there, but from your description it sounded like your opponent almost folded. The odds of hitting the flush on the river were not super great so limping in would have meant you were likely to lose the whole pot. Betting gave you an additional chance to win in potentially having your opponent fold.
Also, I think making plays like you did increased the odds of doing as well in the tournament as you did. Succumbing to lose aversion would have prevented you from getting near the bubble.