# The Cauchy Spinner Game

“It is now the scientific consensus that our risk-avoidance mechanism is not mediated by the cognitive modules of our brain, but rather by the emotional ones. This may have made us fit for the Pleistocene era. Our risk machinery is designed to run away from tigers; it is not designed for the information-laden modern world.”

-Nassim Taleb[1]

Imagine you are working at the local fairground midway and you’ve been assigned to operate an unusual carnival game called the “Cauchy Spinner”. The game is a spin the wheel type game; you spin a wheel with an arrow on it, and you win the amount that the arrow points to.

The unusual thing about this game is that instead of the amount you win being arranged around the perimeter of the wheel, the payouts are shown as a long row of signs on the ground that seem to lead off forever into the distance. When the wheel stops, the amount a player wins depends on the distance to the spot on the ground where the arrow is pointing. The arrow on the wheel points both directions, so that at least one end of the arrow always points at the ground, as in the diagram below.

Since this is just a thought experiment, assume a few ideal conditions apply here: the ground is perfectly flat, the payout signs trail off infinitely far in both directions, and when you spin the wheel, it could stop at any angle with equal probability.

So here’s the question: how much do you have to charge people for each spin to make sure that you will make money in the long term?

One possible way you could try to answer this question is to give the wheel a few spins and see what kind of payouts typically come up. We don’t need to actually build one to do this, we can use some math and some true random numbers to simulate the payouts we would get when we play this game[2]. Lets give our virtual wheel twenty spins and see what we get:

 $0.19$3.32 $2.75$0.03 $1.84$0.27 $0.70$0.40 $0.18$5.11 $0.44$0.54 $0.13$3.00 $1.48$0.26 $0.15$4.20 $0.70$0.64

After twenty spins the biggest win was $5.11 and the smallest one was only 3¢. We’ve paid out a total of$26.33 over twenty spins, so the average payout was a little more than a dollar. Maybe $2 is a fair price to charge for a spin then. Lets give the wheel a few more spins to make sure this was a typical result: $24.12 $5.27$0.10 $0.88$0.35 $1.13$34.73 $2.28$0.43 $2.70$1.13 $1.04$0.84 $0.13$0.16 $0.70$1.06 $0.96$3.04 $1.47 Most of the results are similar to before, with two big exceptions. One player won$24.12, which is several times larger than the next largest win so far, and then another player won $34.73. Because of these big wins the average payout has gone up to$2.72. Maybe we’re going to have to charge $5 dollars for a spin. Lets try another twenty spins: $1.11 $0.09$2.57 $1.56$0.16 $0.47$4.98 $1.44$6.83 $0.29$0.33 $1.30$0.33 $0.32$0.02 $0.54$0.88 $1.11$1,280.68 $3.08 This time a single win has cost you vastly more money than all the other payouts so far combined! This one win has pushed the average payout up to over$20.

In trying to answering the question of what a typical result is like, we have now encountered a new question: after how many trial spins will we have a good idea of what a typical outcome is? And the answer to this question is surprising…you never will. In this game, if you keep spinning the wheel eventually one huge payout will always come along that will be vastly larger than all the other wins so far combined. For this particular game the average[3] payout is infinite; eventually someone would come along and win enough money to negate all your profits and this is true no matter how much you charge for a spin.

Here’s an example of an even simpler game where something similar happens. I put a penny on the table and you flip a fair coin. If it lands heads, you win the penny. If it lands tails, I double the amount of money on the table and you flip the coin again. The pot keeps doubling with every tails until the first heads comes up, and then you win the pot.

We can ask the same question as before, how much would you have to charge people to play the game to make money in the long run? The surprising answer is the same: an infinite amount.

Here’s the proof. To find out how much money someone wins on average when they play a game, you add up all the possible payouts multiplied by the probability of that payout occurring. In half of the games the player wins the 1¢ on the first round, and the other half the time the pot doubles to 2¢. In the games where the pot doubles, half of the time the player wins the 2¢ and and half the time it doubles again to 4¢. By now you can probably see the pattern:

 Payout Probability 1¢ 1/2 2¢ 1/4 4¢ 1/8 8¢ 1/16 … …

In half of the games the player wins 1¢. Winning 1¢ every other game is the same over the long run as winning ½¢ in every game. In a quarter of the games the player wins 2¢. Winning 2¢ every fourth game is also the same as winning ½¢ every game. The pattern continues over all possible payouts, of which there are an infinite number each of which contribute ½¢ on average to the winnings of each game. The average payout over the long run is therefore:

$\frac{1\cent}{2}+\frac{2\cent}{4}+\frac{4\cent}{8}+\frac{8\cent}{16}+\frac{16\cent}{32}+\cdots=\tfrac{1}{2}\cent+\tfrac{1}{2}\cent+\tfrac{1}{2}\cent+\tfrac{1}{2}\cent+\tfrac{1}{2}\cent+\cdots=\infty\cent$

Adding together an infinite number of ha’pennies equals infinite money. So just like the Cauchy Spinner game, there’s no amount you can charge people to play where you expect to make money over the long term.

If you play the same game but instead of doubling the pot when tails is showing you merely add a penny to the pot each time, this effect disappears, and the game does have a finite average payout (2¢ per game in this case). It’s not that big wins never happen, it’s just that they happen so rarely that most of the payout money still comes from the many small wins, and not from the rare big wins.

Many of the phenomena in the world, like the penny-adding game, are in some sense well-behaved. To use a term coined by Nassim Taleb, they come from mediocristan[4]. For example, most of the molecules in a gas are moving at some average speed, with a few travelling faster or slower, but you won’t find a single gas molecule that is going so fast that its speed is vastly greater than all the other molecules combined. Similarly, you can go around measuring people’s heights and, even though some people are taller than others, you will never encounter a single person so incredibly tall that they’re taller than everyone else you measured put together. In mediocristan, the typical cases are good indicators of what to expect.

But some phenomena are from extremeistan, like the Cauchy Spinner game, or the penny doubling game. In this world rare black swan events crop up often enough that they end up being more important than the common, typical cases. Often there is no typical case, and in some ways these phenomena are fundamentally unpredictable.

Humans are poorly equipped to reason about these phenomena. We like to infer future events by looking at examples from the past, but for some events the past is not necessarily a good indicator of the future. The advent of computers and the Internet is an example of such a black swan, an event that had a massive impact and changed the world, but that nobody could have predicted very far in advance. Similarly, if catastrophic climate change were to occur, we couldn’t learn what to expect it to be like by looking at historical climate data. There is no typical case.

Note: For those interested in reading more about the impact of rare events, the best-selling book The Black Swan by Nassim Taleb is the seminal tome. Interestingly, note also that the massive popularity of this book was not something that could have been predicted in advance.

If you’d like you can give the Cauchy Spinner twenty simulated spins yourself using Wolfram|Alpha. You can also compare the results to “well behaved” randomness. Notice that a single huge number will often dominate all the other results with the Cauchy Spinner, but this does not happen in the well behaved case.

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