A Rational Case for Playing with House Money
Playing with house money or the house money effect is a well documented and talked about increase in risk appetite after prior gains. Traditionally, this is seen as irrational behavior. But maybe it’s more rational than you’d think. Let’s have a look!
Behavioral economist Richard Thaler studies human decision making and all of our cognitive biases.
A cognitive bias is a systematic error in thinking, in the sense that a judgment deviates from what would be considered desirable from the perspective of accepted norms or correct in terms of formal logic.
At least that’s how Dan Ariely defines it in his book Predictably Irrational. So a bias is seen as irrational behavior, a flaw that we’d rather not have.
One of those is called mental accounting. In this phenomenon, people regard money differently based on the source of income. For example, a person would treat $10,000 they worked hard for in another way than a sudden inheritance of $10,000.
A specific type of mental accounting is what is dubbed the house money effect. Imagine someone in a casino, with a budget of $100. They get lucky and quickly double up to $200. This person would likely be more aggressive with the newly won $100, the money they won “from the house”. Hence, house money effect.
Thaler describes it as follows:
After a gain, subsequent losses that are smaller than the original gain can be integrated with the prior gain, mitigating the influence of loss aversion and facilitating risk-seeking.
(To study risk-seeking behavior, Thaler compared bets with the same expected value or EV. The difference between the options was that the “riskier” ones had a wider range of outcomes. For example, not risky would be to win $5 for sure, risky would be to have a 50% chance to win $10 and 50% chance to win $0. Both options have the same EV of $5.)
The essence of the idea is that until the winnings are completely depleted, losses are coded as reduction in a gain, as if losing some of “their” money doesn’t hurt as much as losing one’s own cash.
The house money effect is deemed one of those silly biases.
Statistician-writer-flâneur-deadlifter Nassim Taleb doesn’t like Thaler. And that’s an understatement. They even fight it out on the Twitter streets. For him, Thaler is one of them Intellectual Yet Idiots (or IYIs). And if you know Taleb, you also know he considers it his duty to aggressively call those people out.
Tagging the house money effect as irrational, makes Nassim angry, because the way he looks at it, it’s perfectly rational.
(Even Wikipedia seems to know about it:
“There are also controversies over some of these biases as to whether they count as useless or irrational, or whether they result in useful attitudes or behavior.”)
To understand his point of view, we need to talk about Kelly.
First, you must survive
Imagine a rigged coin that lands on heads 60% of the time and you get to bet on it at 1-to-1 odds. So for every $1 you wager, you have a 60% chance of winning $1 and a 40% chance of losing $1. This sweet game has an expected value (or EV) of 0.6*$1–0.4*$1 = $0.2 per $1 wagered. As far as gambling goes, edges this big rarely appear, so you are pretty stoked to play this game.
Say you have a bankroll of $1,000 to gamble with. And you can keep on playing this coin flipping game as long as you want. How would you approach this? What would your bet size be?
If you bet all of the $1,000 at once and you lose the flip, it’s game over. No more future bets. It’s pretty intuitive that this bet is simply too big. We want to make sure we don’t go broke, so we can keep playing this highly profitable game. When you’re out of money, you essentially lose out on all of the EV of future bets that you can’t make anymore.
So you become very conservative and start making bets of $1. You likely won’t go bust and you can keep playing basically forever. But this feels too conservative. We’d like to maximize our winnings over this series of bets and using $1 increments is too slow. Maybe there’s a bigger sizing that balances making money and not going broke?
The Kelly criterion has the answer. It’s a formula for bet sizing maximizing the expected geometric growth rate of your bankroll for a series of bets. This is what matters for an individual gambler. Basically, it makes sure you can keep betting and get the most out of it, without going to $0.
For this game, with these odds, a Kelly bet would be 20% of your bankroll. If you’re interested how to calculate this, check out Wikipedia.
So in our example, our first bet will be $200. If we lose, the next bet will be smaller: 20% of our remaining bankroll of $800 = $160. But if we win, we increase our bet size: it becomes 20% of $1200 = $240.
The EV of a $240 bet is the same whether you have $800 or $1200. But as you start winning, it makes sense to increase your bet size.
Semantics?
Taleb likes to bring up John L. Kelly (and practitioners using the Kelly criterion) as one of the few people who get it, unlike almost everybody else involved in decision theory. Kelly showed that you should bet bigger after prior gains. Thus, concludes Taleb, the house money effect is rational.
But the Kelly criterion is more specific than just increasing your risk as you win. It doesn’t care whether the money comes from the casino or from your initial bankroll. Plus the risk in Thaler’s experiments and risk in increasing your bet size in the Kelly example is not an apples to apples comparison.
So I’m not sure this is a logical conclusion.
You can hear Taleb talk about it for a couple of minutes around the 14:55 mark.
Remember that both Thaler and Taleb accept that mental accounting and playing with house money exist. Taleb isn’t arguing it doesn’t. He just gets angry when people call it a bias and say it’s irrational.
Taleb:
Behavioral economists have something called mental accounting, which states … that treating money according to the source is irrational because these are one-period models. That’s how they view the world, as a one-shot experiment. They don’t view the world as repetition. A repetition of bets. So, if you look at the world as a repetition of bets, under condition of survival, then mental accounting is not only not irrational but is necessary. Any other strategy would be effectively irrational.
For Taleb, what is rational is what helps you survive. Rationality is risk management. So these “silly” biases could help us with these tail risks that would otherwise ruin us.
“If a cognitive “bias” is helpful, it is not a bias.”
I started researching and writing this post 100% on team Taleb. But as I looked more into the details, it was harder to be only on his side. At some point, I was simply wondering: “Can’t we all just get along?”
What is considered irrational in one framework, can be rational in another one. What is irrational in an isolated one-shot experiment, can be rational when you zoom out. That’s why I don’t even think the rational versus irrational discussion matters too much. In my opinion, it’s mostly semantics based on context.
So instead of telling you whether it’s rational or irrational, I’ll let you draw your own conclusions.
House Money in Practice
So we just saw how the house money effect is a natural form of risk management and in that way can be seen as rational.
The problem with it in practice is that the individual is not all of a sudden going to make the perfect Kelly bet size. And maybe even worse, it can lead to risk-seeking behavior that gives up expected value.
Poker
We’ve all seen that guy at a poker table that divided his chip stack in two: his buy-in, and his profit for the session. He’s ready to gamble it up with his profits but does not want to turn his winning session into a losing session.
In the rigged coin flip example, a player can increase his risk by raising his bet size. But in poker, risk-seeking behavior usually occurs by making sub-optimal plays. Playing more hands, chasing draws… It’s not just a change in “risk”, it’s a change in expected value.
Simply put: the player starts playing worse.
As a professional poker player, you should practice proper bankroll management. The amount you sit at the table with, the size of the bets you make and pots you play, should be such a small part of your bankroll that you don’t have to change your behavior based on how well the session is going. This makes sure you can focus solely on expected value.
For example, a recreational player might come to the casino with a “bankroll” of $400, and sit at the table with $200. You, as a professional, would want a bankroll of let’s say $10,000 to sit at that very same table with $200. Winning or losing $200 doesn’t really affect your bankroll.
While the house money effect might be rational from a survival perspective, when it happens at the casino, you want to be the one that capitalizes on others “falling victim to it”. If you’re prepared, a poker session is the isolated experiment where you can just play for max EV. You manage your risk of ruin before and after the session and have such a big cushion you don’t need to worry about it during.
Trump
Casinos aren’t the only places where the house money effect can be observed.
So yes, biases can be a naive form of risk management, but no, I don’t think the president should use it as an excuse to gamble it up.
Whether you call them biases or not, irrational or rational, the tendencies are real. While maybe helpful for survival, they do cause “unwanted” behavior in certain situations.
But luckily, you can choose to play in specific small subgames of life where those biases can be exploited.
Where you get rewarded for resisting your own biases.
And where you can collect the EV others transfer to you because of their biases.
Poker is one of those. And if you look for others, you will find them.
Originally published at https://mydomainiskarl.com on June 17, 2019.
