How to Bet on Anything

(If you want to win)

Introduction

You’re in Las Vegas for a business convention when you’re approached by a funny-looking guy on the street.

It’s me, and I’m holding an ordinary, 6-sided die. I propose a game, as follows:

You roll the die, and either:

Receive the amount shown on the die in dollars, or Roll the die again, and forfeit this roll

You can roll the die at most 3 times, and you only get to keep your last roll. If you re-roll, you forfeit the previous roll.

“I’m not an artist, so bear with me” — Every math teacher, and now me

Here are 2 examples:

You roll a 4, re-roll a 5, and take the $5. You roll a 3, re-roll another 3, then re-roll a 2. You must take the $2 because you used up all 3 rolls.

Now that you understand the rules, here’s the catch: it costs $4.50 to play my game. Do you play? If so, what’s your strategy?

Take a moment to think about your strategy before reading the next section. Better yet, take a moment to ponder whether this game is actually a useful abstraction for countless other real-world scenarios. Oops, I forgot to say *spoiler alert*.

Strategy

The first question to think about is: should you play at all? In order to answer that, we first have to know: what’s the best strategy to play? If the best strategy makes you money on average, you’d probably want to play.

If you’re not sure what the optimal strategy is yet, simplify the game.

Easy mode: Pretend you only get to roll the die once. In this case there is no strategy, because you can’t take any action other than rolling. So what would you be willing to pay to play a 1-roll-game?

Well, the average face value of the die is (1+2+3+4+5+6)/6 = 3.5. So if you could play for anything less than $3.50, you’d expect to make money from this 1-roll-game, on average.

Building up: Okay, so now let’s pretend you get up to 2 rolls. What would you be willing to pay to play a 2-roll-game? There’s not much strategy to this game, because you only get one choice. You can choose to accept your first roll, or you can choose to roll a second time.

A clever gambler should recognize that you’d want to accept a 4, 5, or 6 on the first roll. If you get a 1, 2, or 3 on your first roll, you should roll again. Why? Because you’re expected to make $3.50 from the second roll, on average. If your first roll is higher than 3.5, you’re unlikely to improve on the second roll.

So what would you pay to play this 2-roll-game?

Looking at just the first roll, there’s a 50% chance of rolling a 4, 5, or 6, in which case the average payout is $5. There’s also a 50% chance of rolling a 1, 2, or 3, in which case you’ll re-roll, because that’s the better choice. Then, it’s just a 1-roll-game all over again, which has an average value of 3.5.

The expected value of a 2-roll-game is $5 x 0.5 + $3.5 x 0.5 = $4.25

For any price less than $4.25, you’d expect to make money from this 2-roll-game, on average.

Putting it all together: Now, you get 3 rolls. This might seem more complicated, but it isn’t. After your first roll, you can decide to keep your roll or, essentially, play another 2-roll-game. Knowing that your remaining two rolls are worth $4.25 if used, you’d be well-advised to accept 5 or 6 on your first roll and re-roll for anything less. In other words, we have our full 3-roll-game strategy!

Roll once, if you get a 5 or a 6, keep it. Otherwise, re-roll. If you get a 4, 5, or 6 on the second roll, keep it. Otherwise, roll one last time and accept your fate.

So, what would you be willing to pay to enter this 3-roll-game? There’s a 2/6=1/3 chance of rolling a 5 or 6 on the first roll (with an average of $5.50), otherwise, there’s a 4/6=2/3 chance you’ll use your extra roll(s), which averages you $4.25.

The expected value is $5.5 x 1/3 + $4.25 x 2/3 = $4.666666….

For anything less than $4.67, you’d expect to make money from this game, on average. For $4.50, you’d expect to make nearly 17 cents per game!

Big Picture

This is a silly game. You’re more likely to encounter it in a brainteaser than a casino. But, could you play this game with more rolls? A die with more sides? The numbers get bigger, but that’s what computers are for! In fact, you can use this strategy for any random turn-based game without much extra thought. Here are some examples:

We’re on to your game, Howie.

Deal or No Deal: You’re nearing the end of Howie Mandel’s hit NBC game show, Deal or No Deal, with 6 cases left, each holding a dollar amount from $1 to $1M. Each round, you have the option to take a buyout, or keep playing.

This is essentially the same problem we just solved. With 6 remaining cases, it’s like a 6-sided die where each “face” has the value of your future buyout offer*, if that case were opened. In the next round, the “die” has 5 sides, then 4, etc. In the last round, it’s like a coin flip with the value of the remaining two cases on either side of the coin. The value of that game is just the average of the remaining cases, since there will be no more buyout offers.

Securities Trading: Now that you’ve beaten me at my own game and bankrupted NBC, you decide to take your talents to Wall Street. You’ve discovered a hot new stock called DeepQuantumChain (DQC)— a blockchain-powered quantum computing artificial intelligence company whose stock is listed at $1.00/share.

DeepQuantumChain is competing with Microsoft for a client whose business would surely bring DQC’s stock price to $1.50/share, you reason. If DQC doesn’t land the client, you suspect the stock will fall to $0.50/share. Your research suggests it’s essentially a coin-toss whether they get the contract, but the contract expires every year. Lastly, if you don’t invest in DQC, you’re certain you can make 8%/year by investing in an index fund, and that’s your best alternative.

In this “game”, you can flip the coin as many times as you want** (i.e., invest in DQC and don’t sell your stock until they win the contract — each “coin flip” represents owning the stock for a year). On one side of this coin is $1.50, and $0.50 is on the other. If this were the whole game, you’d just keep playing until you get the $1.50. However, each “flip” takes one full year, because it represents waiting for the contract to expire, which costs you 8% in foregone returns by not investing in index funds. So what’s the stock worth? More than $1? Take a look at the possible outcomes:

(50% chance): DQC gets the contract right away, you get $1.50

(25% chance): DQC gets the contract in 1 year, you get $1.50, which is worth $1.50 * 0.92 = $1.38 today

(12.5% chance): DQC gets the contract in 2 years, you get $1.50, which is worth $1.50 * 0.92* 0.92 ~= $1.27 today

(6.25% chance): DQC gets the contract in 3 years, you get $1.50, which is worth $1.50 * 0.92 * 0.92 * 0.92 ~= $1.17 today

You can take the sum of these computations for many flips (years) and have something very close to the true value after multiplying by the probabilities. That would mean evaluating this formula:

As n gets large. Here, i is the number of years that have gone by. Since we can’t evaluate for n = ∞, using n=100 is just fine. A computer won’t even know the difference. I used python and got around $1.39

If you’re not satisfied with that result and want an exact answer, you might notice that each term in the series above differs from the previous by a constant amount (0.92 * 0.5), so you could obtain the exact resulting value, V, of the stock simply by relating it to the subsequent year.

V = 0.5 * 0.92 * V + 0.5 * 1.5

In which case we solve for V and see that the stock is indeed worth around $1.39/share. You buy all the stock that you can for up to $1.38. You take your profits and buy a yacht with internet, so you can read my blog from anywhere.

Congratulations ~