Dice thoughts

A few weeks ago, while I was going for a walk in the woods, I came up with a thought experiment. Later, I tweeted it out, and later still, I asked the listeners of Slate Money to write in with their answers.

The question:

I will roll a die as many times as you like. Every time it comes up 1/2/3/4/5, I double your money. If it’s a 6 you lose everything. You need to specify ex ante how much your initial stake is, and how many times I roll. What do you choose?

The obvious conditions apply: I am a gazillionaire with no counterparty risk, and you can only play the game once, on your own, with no partners.

This question is interesting on a bunch of different levels.

First, by forcing you to state up front how many times I am going to roll, this question is mathematically equivalent to me just rolling that number of dice at the same time — with you getting zero if at least one of those dice comes up 6. Psychologically, however, there’s a big difference.

Let’s say you choose 5 rolls on a stake of $100. You immediately think of a scenario where your money doubles four times in a row — you’re up to $1,600 — and then a 6 comes on the fifth roll. It feels as though you’ve lost $1,600 rather than just $100, and that outcome is psychologically worse than if a 6 came on the first roll.

Second, the question forces people to think hard about what it means to “expect” a certain outcome. Take that choice of 5 rolls on a stake of $100 again. The median outcome is that you end up with $0: there’s roughly a 60% chance of that happening. On the other hand, you have a 40% chance of winning $3,200, which means the “expected value” of the bet is $1,286.

What’s more, the more rolls you specify, the higher the expected value becomes. If you roll 20 times, your 2.6% chance of winning $105 million works out to an “average” payout (whatever that means) of $2.7 million. But few of us would consider it rational to expect to win $2.7 million if we specified $100 and 20 rolls. In fact, one thing we know for certain is that we won’t win $2.7 million! It’s either $105 million or nothing.

Third is the utility of money. For most of us, the utility of having $40 million is not so far removed from the utility of having $20 million. So it doesn’t make a lot of sense to take an 86% chance of doubling $20 million into $40 million. Better to just stop at $20 million.

Indeed, for many people the utility of money turns negative after a certain point. Just ask most lottery winners, or Johnny Depp. Getting enough money to buy a house and thereby have shelter for the rest of your life — that’s great. But getting enough money to buy 10 houses — that can create more misery than happiness, and even surprisingly rapid bankruptcy.

Almost no one has the self-awareness to know their own utility-of-money curve. But it’s not hard to think of people, from gambling addicts to Warren Buffett, who would not be better off if they won this bet. Such people might well do best by not playing at all.

Fourth is the question of whether this is an investment or a gamble. Most people’s answers fall into one of two broad buckets: There are the people who will stake roughly as much as they might risk at a poker game, and then there are the people who will stake roughly as much as they might invest in the stock market.

For the people who choose a poker-sized bet, the obvious question is whether and why they would ever invest in, say, stocks. The expected return characteristics of this bet are vastly superior to anything you can get in the stock market, so if you’re OK with stocks, shouldn’t you be OK with this? Is the difference that stocks go to zero slowly, if they go to zero? But that still implies that most people have a point at which they will sell a stock that is slowly declining. Which in turn violates the tenets of buy-and-hold investing.

For people making an investment-sized bet, the obvious question is how much to bet. That’s where the Kelly criterion comes in. Ian Chan on Twitter helpfully did a bunch of the mathematics so I don’t have to, but if you have a set amount of money, then according to the Kelly criterion the amount you should bet starts at 67% of that amount for 1 roll, and then declines to 59% for 2 rolls, 52% for 3 rolls, and so on down to 2.6% for 20 rolls. (From about 6 rolls onwards, the Kelly criterion percentage is very close to the probability of winning.)

Ian works backwards from that number, and basically asks: How much are you willing to tolerate losing, expressed as a percentage of the amount of money you have? If it’s, say, 9% of your wealth, then you should work backwards from there and decide to go for 13 rolls.

I take a different tack, which is to ask: How much money would you like to win? This is, after all, by far the best and most generous opportunity you will ever receive in your life. If you could sell this opportunity on the open market, there’s any number of individuals who would pay you hundreds of millions or even billions of dollars for it. So it seems a bit of a waste to just try to win $100 or so.

So I start by asking: What is an amount of money that would significantly improve my life and livelihood? To bet for any less than that seems silly, given the incredible opportunity in front of me. But to bet for more than that also seems like a bit of a waste: I’m risking losing more money, or a higher probability of ending up with zero, for a relatively marginal improvement in utility.

Let’s say the amount I alight upon is the aforementioned $20 million. And let’s say I have $250,000 in liquid assets. (Should people include non-liquid assets, or expected future income, in this calculation? Let’s say no for the time being, although that’s a question that can be debated.)

In order to get to $20 million in 7 rolls I would need to bet $156,000, which is 62% of my wealth. But according to the Kelly criterion at 7 rolls I shouldn’t bet more than 27% of my wealth, so that’s a non-starter. So increase the rolls: To get there in 10 rolls I’d need to bet $19,531, which is 7.8% of my wealth. That’s well below the 16% Kelly criterion, so that’s one option.

On the other hand, at 10 rolls I only have a 16% chance of winning. I can improve that to 19% by betting $39,000 on 9 rolls. That’s 15% of my wealth, which is still within the Kelly criterion of 19%.

So in this case I would probably bet either $19,500 on 10 rolls, or $39,000 on 9 rolls. In both cases I’ll expect to lose a sum that I could easily suffer in the stock market. This bet, however, comes with much higher expected returns. It’s a gain that will really transform my life if I hit, so the bargain seems like a good one.

All of which raises the toughest question of all: How on earth am I going to persuade my wife that I should spend $39,000 on a roll of the dice?

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2 Responses to Dice thoughts

  1. Steve Strickland says:

    Similar to the St. Petersburg Paradox

  2. Robert Rosenkranz says:

    On your analysis your penultimate bet is 10 million of your then 10.25 million net worth. Obviously not a bet you would make. So how can it be rational to commit yourself to doing it? I think you are falling into a “house money” fallacy. Each extra roll of the dice you commit to is a marginal bet which should, at the margin, be rational.

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