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The infamous coin toss

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In 2011 I gave a 15-minute talk to a lay audience in London. The topic I had chosen was ergodicity breaking, and the challenge was clear: how do you get this across? I invented a coin-toss gamble, which has since become a go-to illustration of ergodicity breaking and a very intuitive way of explaining how ergodicity economics differs from other approaches to economics, and how its concepts may apply to problems unrelated to economics.

The gamble I’m about to describe, now sometimes called the Peters coin toss,’ is discussed in detail in my 2016 paper with Murray Gell-Mann, there’s a youTube video about it on the ergodicity.tv channel, and my public talk from 2011 is also available online. In this post, I will present the basic setup, hint at its significance, and then mention a few generalizations.

Imagine I offer you the following gamble. I toss a fair coin, and if it comes up heads I’ll add 50% to your current wealth; if it comes up tails I will take away 40% of your current wealth. A fun thing to do in a lecture on the topic is to pause at this point and ask the audience if they’d like to take the gamble. Some will say yes, other no, and usually an interesting discussion of people’s motivations emerges. Often, the question comes up whether we’re allowed to repeat the gamble, and we will see that this leads naturally to the ergodicity problem.

The ergodicity problem, at least the part of it that is important to us, boils down to asking whether we get the same number when we average a fluctuating quantity over many different systems and when we average it over time. If we try this for the fluctuating wealth in the Peters coin toss the answer is no, and this has far-reaching consequences for economic theory.

Let’s start with averaging wealth, x_i(t) over an ensemble of many different systems. In our case this corresponds to N different players, each starting with x_i=\$100, say, and each tossing a coin independently. After the coins have been tossed, about half of the people will have thrown heads, and the other half tails. As the number of players goes to infinity, N\to\infty, the proportions of heads and tails will approach 1/2 exactly, and half the players will have$150, the other half $60. In this limit, we know what the ensemble average will be, namely \langle x(1)\rangle = 1/2\left(\$150 +\$60\right) = \$105. For historical reasons, this average is also called the expected value, and for the Peters coin toss, it grows by 5% in every round of the gamble so that

\langle x(t)\rangle = \$100\times 1.05^t . We can recover this average numerically by setting up a Monte Carlo simulation: let a large number, N, of agents play the game for T rounds, then average over N. You can see what happens for T=10 as you increase N in the app below. Choose the number of players and hit “Simulate.” The red straight line shows the expected value for the first 10 rounds (starting at 1). The light grey lines are the wealth trajectories of individual players, and the solid blue line is the average over the ensemble. To see that the gamble is not ergodic, let’s now find the average value of wealth in a single trajectory in the long-time limit (not in the large-ensemble limit). Here, as T grows, again the proportions of heads and tails converge to 1/2. But, crucially, a head and a tail experienced sequentially is different from two different agents experiencing them. Starting at x_1(0)=\$100, heads takes us to x_1(1)=\$150, and following this sequentially with tails, a 40% loss, takes us down to x_1(2)=\$90 — we have lost 10% over two rounds, or approximately 5% per round. Since we lose 5% per round, averaged over time, individual wealth is guaranteed to approach zero (or negative infinity on logarithmic scales) in the long-time limit T\to\infty. You can try this out too, in the app below, by simulating the game for different numbers of repetitions.

Choose the number of rounds to play and hit “Simulate.” The red straight line shows the expected value, the green straight line decays at the time-average growth rate. The light grey lines are the wealth trajectories of individual players, and the solid blue line is the wealth of the one individual we’re simulating (all starting at 1 as before).

We have thus arrived at the intriguing result that wealth averaged over many systems grows at 5% per round, but wealth averaged in one system over a long time shrinks at about 5% per round. Plotted on logarithmic vertical scales, this gives one of the iconic images of ergodicity economics, the featured image of this post (taken from my 2019 Nature Physics paper).

The significance of this ergodicity breaking cannot be overstated. First, all living processes, including economic growth processes are similar to the coin toss in the sense that they rely on self-reproduction. The number of rabbits, or viruses, or the dollars in your trading account, grow in a self-reproducing noisy multiplicative way (until some carrying capacity is reached), just like wealth in the Peters coin toss. Second, most mainstream economic decision theories are based on the concept of expected value, and all of that has to be questioned in the presence of ergodicity breaking. Third, one core problem of economics and politics is to address conflicts between an individual, for example a citizen, and a collective, for example a state. This is the question of societal organization, institutions, Rousseau’s social contract and so on. This problem can seem puzzling, and it often attracts naive answers, because the collective consists of individuals. How, then, can the interests of the individual be misaligned with those of the collective? One important answer is ergodicity breaking.

The coin toss is often the starting point for more detailed investigations. We may allow players to withhold some of their wealth and only subject a fraction of it to the coin toss dynamics. This gives us Kelly betting and optimal leverage. We can allow players to pool their resources, which leads to the ergodicity solution of the cooperation puzzle, and to the emergence of complexity. Or players may pool a proportion of their wealth, leading to reallocating geometric Brownian motion and an intriguing perspective on the dynamics of wealth inequality.

I’ll end here and encourage you to gamble away and meditate on the many curious stories this simple little coin toss has to tell us.

40 responses to “The infamous coin toss”

1. 悪名高いコイントス – 世界の話題を日本語でザックリ素早く確認！

[…] この記事はHackerNewsに掲載された下記の記事およびそれに対するHackerNews上のコメントを元に作成されています。The infamous coin toss […]

1. Sandy

I wonder if this implies value investing is better at the individual level then growth investing although you will get individual examples of incredible growth investor success.

If a value investor buy stocks where when he is wrong it is a ‘value trap; which usually means they tread water and go nowhere but importantly dont lose much then the ergoditicy equation will be much more flattering for the individual than a growth investor where if he is wrong it is huge drawdown.

2. Tom

I submit that this has nothing to do with ‘ergodicity’, and everything to do with an incorrectly set up simulation.

The construction of this problem, with the high volatility in each period forces many accounts to ruin in any simulation. These depleted accounts return 0%, and skew the returns. This is the source of the contradictory results.

If instead, at the end of each period, you keep a separate tally of profits, and take them when positive, or replace them with new funds when negative, then the math works out as expected. Each account over time will achieve a 5% return.

Math is math, and if the pool expects a return of 5% per period, and the simulation is not delivering it, then the simulation must be incorrect.

I am happy to provide 150 lines of Java code that proves this out.

1. Andy

Where do the new funds come from to replace the loss?

1. MDM

From the US Treasury, of course.

2. Tom

Beg, borrow, steal, work, earn, sell…etc, etc.

Don’t invest your life savings in one basket just because you are GUARANTEED a 5% expected return when it is THIS volatile.

3. Tom

The math that follows “To see that the gamble is not ergodic, let’s now find the average value of wealth in a single trajectory in the long-time limit..” would get a failing grade in a freshman statistics course.

It fails to consider all possible sequences of heads and tails, which are (TT, TH,HT and HH).

The expected value of the balance for an individual over a 2-period sequence would be the average of the 4 balances given these sequences

TT:36, TH:90, HT:90, HH:225 -> (36 + 90 + 90 + 225) / 4 = 110.25, not 90 as is suggested by the “naive answer” given above!!

Not surprisingly using B * (1+R) ^2 = 110.25, where B = 100, we see that indeed R = 5%.

1. HR

Yes, but the point is that three of the four sequences result in a loss. This ratio will worsen as the sequences get longer. In the long run almost everyone will loose money.

1. Tom

one word: wrong.

Can we set this up where YOU are the payer, and I am the client…please please?

1. HR

No, it’s not wrong. Think of this in the extreme version: tails you lose all of your 100 bucks and you’re out, heads you win 210$and you put all that on the table in the next round. In the long run everyone goes bust except for the one lucky bastard who gets extremely wealthy. This single lucky bastard ensures that the average return per period is still 5%. 1. Tom I’m glad I have to compete in this capitalist society that don’t understand math and probability. 2. Joshua Brooks I’m not sure why he didn’t show his simulation’s code, but here is a go program for it: “ package main import ( “fmt” “math/rand” ) func main() { wealth := 1000.0 for i := 0; i = 2, the majority of players will likely come out behind) b) if you win, you’ll win a lot 1. Joshua Brooks Well apparently this blog doesn’t like code blocks 4. jcb Isn’t this exercise related to Shannon’s demon? 5. David Above is a great means of demonstrating the difference between average and median. An interesting corollary for investment sponsors on the link between investment debt sizing & duration of the game. To follow Tom’s simulation above, with replacement & profit taking what amount of borrowing credit as a percentage of starting wealth is needed for your almost worst case (5%-ile)? 1. Tom I ran a simulation with 50 investors over 10,000 periods. The biggest draw down was 1,000, given a working balance of 100. Only half or so had any drawdown to speak of, and most of those were 200-400. All participants made about 50k over the 10k periods. 1. David Your policy of replacement is also an option for investment sizing. I don’t think the above was meant to swear us off opportunities with positive expected value, just how to approach the consequences of volatility on a single path. If the drawdown is 10x the working balance, would you agree that we’d need some risk management? Is the Kelly fraction the optimal bet sizing compared to constant bet replacement? 1. Tom I 100% agree. But also, all of the accounts would RECOVER from percentage losses, IF the data types used to simulate didn’t round down to ZERO at some point. The math as stated is so flawed…example: “Since we lose 5% per round, averaged over time, individual wealth is guaranteed to approach zero”, this is just plainly incorrect on its face. 1. Gottfried Can you give an example of how such a recovery would look in practice? For example, assume that at some point in the game you have left an amount of$0.0001. What sequence of heads/tails would be required to recover to the starting amount of \$1 and how likely is it that this sequence occurs in practice?

2. GET_A_LIFE_TOM

Hey Tom,

Do you know what’s funnier than someone with an overflow in their code?

An arrogant jerk who think they are smarter than anyone else, with an overflow in their code.

1. Ole Peters

Dear GET_A_LIFE_TOM

I don’t think you should lash out at Tom like this. He’s puzzled by the mathematical properties of the coin toss, and he’s not alone in this. His mistake is his failure to question his beliefs about himself and his mathematical expertise, and those are hard things to question. You can help him by explaining how the mathematics works, and you will be more likely to succeed if you do it using kinder language.

1. Joshua Brooks

I do think this language is not perfect:

we have lost 10% over two rounds, or approximately 5% per round. Since we lose 5% per round, averaged over time, individual wealth is guaranteed to approach zero

Perhaps I am missing something, but I don’t think this all follows. The fact that we lose 10% on a HT sequence doesn’t prove that we lose 5% per round generally, and I’m not sure saying we lose 5% per round is accurate, since the expected value of the game is good. Rather it would be better to show that out of the four permutations, 3 out of 4 of them end up in the red, and that this discrepancy gets wider with greater numbers of rounds.

6. Rupert Nagler

Dear Ole,
Great work, hopefully helps some more people to understand the pitfalls of traditional economic theories!
I for myself tried a small simulation to help people understand the implications of non-ergodic growth and the need for cooperation through redistribution, see:
https://rnagler.shinyapps.io/prosperityshiny/

Following your article “The ergodicity solution of the cooperation puzzle” what we need to be prosperous are well-designed wealth- and heritage taxes.
Best regards from Austria.

1. Ole Peters

Many thanks. Glad you enjoyed this post. Another blog post about cooperation is already in the works… watch this space.

7. Tom

The point of my posts is being missed.

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*** The MATH behind this ENTIRE post is FLAWED ***
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8. Michael Edesess

See my article here (https://www.advisorperspectives.com/articles/2021/03/01/understanding-fat-tail-returns), from which I believe Peters took his example, though he didn’t reference my article. (I had not seen him use this specific example before I published it and sent him my paper.) Consider a much simpler example of the same phenomenon. Suppose your fire insurance company does their actuarial math poorly and offers you a policy that will lose for them in aggregate. Even so, you will almost certainly lose too because you pay premiums but never collect, unless you happen to be one of the few “lucky” policy-owners who actually has a fire and gets a big payoff. The insuror loses, while on average — but only on average — the premium-payers gain. This has nothing to do with “ergodicity economics,” and certainly does not have “far-reaching consequences for economic theory.” The math is fun and cute but doesn’t add any practical or theoretical value here and shouldn’t pretend to.

1. Tom

Do you have an opinion on the correctness of the math in this discussion following “But, crucially, a head and a tail experienced sequentially is different from two different agents experiencing them…”.

Here’s mine.

It’s wrong, and the entire piece is based on this failed math.

No failure in making a mistake, I’ve made plenty. Not being able to see it, or refusing to correct it, now that gets uncomfortable.

1. Curzio Malaparte

The entire piece is based on presenting things in a confusing way.

“let’s now find the average value of wealth in a single trajectory in the long-time limit”

That “time-average” depends on the trajectory and the limit doesn’t converge to a single number independent of the trajectory.

“Since we lose 5% per round, averaged over time, individual wealth is guaranteed to approach zero”

It’s not “guaranteed” unless we fail to make the (important) distinction between “with certainty” and “almost surely”.

The simpler example of tripling your money or losing everything on a coin flip can be clearer.

After T rounds you have 3^T your initial amount or zero. The probability of the former is 1/2^T, the probability of the latter is 1-1/2^T. The expected value is (3/2)^T.

If we were to calculate “the average value of wealth in a single trajectory in the long-time limit” we would see that it depends on the trajectory. It will be close to zero for most (but not all) trajectories.

The average over trajectories of “the average value of wealth in a single trajectory in the long-time limit” is like the average in the long-time limit of the average over trajectories of wealth at each point and it diverges.

1. Gottfried

“After T rounds you have 3^T your initial amount or zero. The probability of the former is 1/2^T, the probability of the latter is 1-1/2^T.”

I think you are on to something interesting!

Let T go to infinity, what is the probability of having 3^T your intial amount and what is the probability of having zero?
Note that your statement about the probabilities of these outcomes does not “depend on the trajectory”. It plainly and correctly states the general result: After T rounds you have either this or that, and the probabilities for each of these are p and 1-p.

“If we were to calculate “the average value of wealth in a single trajectory in the long-time limit” we would see that it depends on the trajectory. It will be close to zero for most (but not all) trajectories.”

Could you give an example of a realistic (probability of occuring is non-zero) long-term trajectory that results in a wealth other than zero? What sequence of heads/tails would that be?

1. Curzio Malaparte

> Could you give an example of a realistic (probability of occuring is non-zero) long-term trajectory that results in a wealth other than zero?

Could you give an example of a realistic (probability of occuring is non-zero) long-term trajectory that results in zero wealth?

I think we agree that the probability of any particular trajectory of length T is 1/2^T and the probability of any particular infinite trajectory is zero.

Hopefully we can also agree that the distinction between “with certainty” and “almost surely” is important, that despite having said “let’s now find the average value of wealth in a single trajectory in the long-time limit” such thing was not actually found, and that it’s at best very misleading to say that it’s “guaranteed” that you will get heads if you play long enough.

I think that all that ambiguity doesn’t help readers to understand whether the blog post is onto something interesting or not.

2. Curzio Malaparte

I just said:

> I think we agree that the probability of any particular trajectory of length T is 1/2^T and the probability of any particular infinite trajectory is zero.

I’m thinking now that in my variant we could also say that trajectories stop when they get to zero so indeed all the losing trajectories would have non-zero probability because they would be finite. (That’s not true in the original problem though – all the trajectories are infinite and never reach zero.)

The rest of my points stand, of course. There is no reason to be sloppy with the terminology and the exposition if the objective is to produce enlightenment rather than astonishment.

3. Curzio Malaparte

(When I said “we could also say that trajectories stop when they get to zero” I was still thinking of “coin-flip outcomes” trajectories. But the original question was implicitly about “wealth” trajectories which can extend to infinity after they get to zero and have different probabilities depending on when that happened.)

4. Gottfried

“Could you give an example of a realistic (probability of occuring is non-zero) long-term trajectory that results in zero wealth?”

Literally every possible sequence of length T except the special sequence “T heads” results in zero wealth.

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15. Alex

Interesting posting!

I wrote a quick Python (v2.7 / v3.x) program to look in to the 50%/40% example. In case anyone is interested: https://pastes.io/hu482qcjjf

It was fun to play with the dance between how many C notes you have in your wallet and how many times you flip a coin on each C note. And how you can otherwise dink with gambler’s ruin, gambler’s eureka, and at least one ruin-mitigation strategy.

I was surprised by the dramatic effects of a fuzzy run of heads-you-win.

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