Let’s say I offer you the following gamble: You roll a dice, and if you throw a six, I will give you one hundred times your total wealth. Anything else, and you have to give me all that you own, including your retirement savings and your favorite pair of socks. I should point out that I am fantastically rich, and you needn’t worry about my ability to pay up, even in these challenging times. Should you do it? ... The rational answer seems to be “yes”—the expected return on your investment is 1,583 1/3% in the time it takes to throw a dice. But what’s your gut feeling?As he notes, almost no real person would take this bet. You have 5 chances out of 6 of being left destitute, one of being made very much wealthier. Somehow, most of us weight outcomes differently than the simple and supposedly "rational" perspective of maximizing expected return. Why is this? Are we making an error? Or is there some wisdom in this?

Peters' gamble is a variation on the famous St Petersburg "paradox" proposed originally by Nicolas Bernoulli, and later discussed by his brother Daniel. There the question is to determine how much a rational individual should be willing to pay to play a lottery based on a coin flip. In the lottery, if the first flip is heads, you win $1. If the first is tails, you flip again. If the coin now comes up heads, you win $2, otherwise you flip again, and so on. The lottery pays out 2^n (^ meaning exponent) dollars if the head comes up on the nth roll. An easy calculation shows that the expected payout of the lottery is infinite -- given by a sum that does not converge (1*1/2 + 2*(1/2)^2 + 4*(1/2)^3 + ...) = (1/2 + 1/2 + 1/2 + ...). The "paradox" is again why real people do not find this lottery infinitely appealing and generally offer less than $10 or so to play.

This is a paradox, of course, only if you have some reason to think that people should act according to the precepts of maximizing expected return. Are there any such reasons? I don't know enough history of economics and decision theory to say if there are -- perhaps it can be shown that such behavior is rational in some specific sense, i.e. in accordance to some set of axioms? But if so, what the paradox really seems to establish, then, is the limited relevance of such rules to living in a real world (that such rules capture an ineffective version of rationality). Peters' resolution of the paradox shows why (at least for my money!).

His basic idea is that we live in time, and act in time, and have absolutely no choice in the matter. Hence, the most natural way to consider the likely payoff coming from any gamble is to imagine playing the gamble many times in a row (rather than many times simultaneously, as in the ensemble average). Do this indefinitely and you should encounter all the possible outcomes, both good and bad. Mathematically, this way of thinking leads Peters to consider the time average of the growth rate (log return) of the wealth of a player who begins with wealth W and plays the gamble over N periods, in the limit as N goes to infinity. In his paper he goes through a simple calculation and finds the formula for this growth rate:

The third line here is explicitly for the St Petersburg lottery, while the second line holds more generally for any gamble with probability p_i of giving a return r_i (with the sum extending over all possible outcomes).

This immediately gives more sensible guidance on the St Petersburg paradox, as this expected growth rate is positive for cost c sufficiently low, and negative when c becomes too high. Most importantly, how much you ought to be willing to pay depends

**, as this determines how much you can afford to lose before going broke. Notice that this aspect doesn't figure in the ensemble average in any way. It's an initial condition that actually makes the gamble different for players of different wealth. Coincidentally, this result is identical to a solution to the paradox originally proposed by Daniel Bernoulli, who simply postulated a logarithmic utility and supposed that people try to maximize utility, not raw wealth. This idea reflects the fact that further riches tend to matter relatively less to people with more money. In contrast, Peters result emerges without any such arbitrary utility assumptions (plausible though they may be). It is simply the realistic expected growth rate for a person playing this game many times, starting with wealth W. Putting numbers in shows that the payoff becomes positive for a millionaire for a cost c less than around $10. Someone with only $1000 shouldn't be willing to pay more than about $6.**

*on your initial wealth w*It's also useful to go back and work things out for the simpler dice game. One thing to note about the formula is that the average growth rate is NEGATIVE INFINITE for any gamble in which a person stands to lose their entire wealth in one go, no matter how unlikely the outcome. This is true of the dice gamble as laid out before. I was wondering whether this really made any sense, but after some further exploration I now think it does. The secret is to again consider that the person playing has wealth W and that the cost of "losing" isn't the entire wealth, but some cost c. A simple calculation then shows that the time average growth rate for the dice game takes the form shown in the figure below, showing the growth rate versus the c/w, the cost as a fraction of the players' wealth.

Here you see that the payoff is positive, and the gamble worth taking, if the cost is less than about 60% of the player's wealth. If more than that, the time average growth rate is negative. And, if becomes strongly more negative as c/w approaches 1, with the original game recovered for c/w=1. Again, everything makes more sense when a person's initial wealth is taken into account. This initial condition really matters and the question of the likely payoff of a gamble depends strongly on it, as lower wealth means higher chance of going bankrupt quicker and then being out of the game entirely. The possibility of losing all your wealth on one turn, no matter how unlikely, becomes decisive because this becomes certain in the long run.

Again, this way of thinking likely has significance far beyond this paradox. It's really pointing out that ensemble averages are very misleading as guides to decision making, especially when the quantities in question, potential gains and losses, become larger. If they remain small compared to the overall wealth of a person (or a portfolio), then the ensemble and time averages turn out to be the same, giving a formula in which initial wealth doesn't matter. But when potential gains/losses become large, then the initial condition really does matter and the ensemble average is dangerous. These points are made very well in this Towers Watson article I mentioned in an earlier post.

Which brings me to one final point. Ivan in comments suggested that perhaps Peters has changed the initial problem by looking at the time average rather than the ensemble average, and so has not actually resolved the St Petersburg paradox. I'm not yet entirely sure what I think about this. The paradox, if I'm right, is why people don't act in accordance with the precepts of expected return calculated using the ensemble average. To my mind, Peters' perspective resolves this entirely as it shows that this ensemble average simply gives very poor advice on many occasions. In particular, it makes it seem that a person's initial wealth should have no bearing on the question. If you face gambles, and face them repeatedly as we all do throughout life in one form or another, then thinking of facing them sequentially, as we do, makes sense. But that's not, as I say, my final view..... this is one of those things that gets deeper and deeper the more you mull it over....

"Peters' gamble is a variation on the famous St Petersburg "paradox" proposed originally by Nicolas Bernoulli, and later discussed by his brother Daniel."

ReplyDeleteThat it is not even a remote variation of the paradox. Peter's gamble is a game with just one trial and six outcomes. Bernoulli's paradox involves a game with possibly infinite trials and two outcomes for each.

Peter concludes that:

"Thus, the St Petersburg paradox relies for its existence on the assumption that

the expected gain (or growth factor or exponential growth rate) is the relevant

quantity for an individual deciding whether to take part in the lottery."

I think this is a straw man. The paradox poses the question whether the expected gain is a relavant quantity, it does not assume it is. There are many other possibilities that players may consider for relevant quantities.

I have a quick comment.

ReplyDeleteThe quantity of interest is the average player's wealth =. Instead, Peters considers its average exponential growth rate =(1/t) (in the limit t going to infinity).

Now, Jensen's inequality implies that exp(t) \leq = .

Therefore, the average always *underestimates* the gains. For instance, a negative is still compatible with a wealth growing over time.

In my previous comment, most of the equations weren't publish (?). Here's another try:

ReplyDeleteI have a quick comment.

The quantity of interest is the average player's wealth < W(t) >=< exp(gt) >. Instead, Peters considers its average exponential growth rate < g >=(1/t)< ln W(t) > (in the limit t going to infinity).

Now, Jensen's inequality implies that exp(< g >t) \leq < exp(gt) > = < W >.

Therefore, the average < g > always *underestimates* the gains. For instance, a negative < g > is still compatible with a wealth growing over time.

Same anonymous. You wrote:

ReplyDelete"These points are made very well in this Towers Watson article I mentioned in an earlier post."

That article starts with a gross misconception of returns and I am sorry you did not catch that. A 10% return with 0.5 probability and -10% return with 0.5% probability has nothing to do with two consecutive returns of 10% and -10%. That is just a possible sequence like heads first and then tails. This is what they write in their footnote:

"We have been advised that some people, anchored in ensemble averages, will find this result hard to accept and so warrants further explanation.

Consider betting $1 on the coin toss and playing for two rounds. Because it is a fair coin, one round will be heads and the other tails. If we get

heads first, our $1 becomes $1.10 (a 10% gain). In the second round we get tails, a 10% loss. 10% of $1.10 is $0.11, so we end up with $0.99.

Alternatively, if we get tails first, our $1 falls to $0.90. The subsequent heads wins us $0.09 and we, again, end up with $0.99. In the real world we

can get lots of heads in a row, but we can also get lots of tails in a row. If the coin is fair we will see half of each side and an expected loss of 1% per

round. This illustrates the fundamental difference between an additive dynamic and multiplicative dynamic."

Do you agree? I hope not. It seems you are quoting papers you have not even read. The above statements are for laughs (I do not want to be more harsh hear).

It appears that there are two anonymous here. The first and fourth post came from the same anonymous.

ReplyDeleteJesus, the more I look at the Peter and thh other paper you quoted the more I worry about the future of this world. Where has humility and common sense gone? Jesus (or whatever God you want), oh my!

"This is a paradox, of course, only if you have some reason to think that people should act according to the precepts of maximizing expected return. Are there any such reasons?"

ReplyDeleteTo clarify, economic theory says that if people's preferences satisfy certain general properties, people maximize expected utility of payoff, for _some_ utility function. The function will depend on particular preferences, which are taken as given (they are input to the model). So from the point of view of economic theory, there is nothing that requires people to maximize expected payoffs themselves - they would do so only if they had specific (so called risk-neutral) preferences.

What Peters suggests is replacing "arbitrary" preferences by criterion of maximizing long-term geometric average rate of return. So naturally one then asks - why exactly this criterion? Is it supposed to be a descriptive theory about how people actually behave, or normative theory how they should behave? It's not really clear. And besides, this is in fact rather old idea, which was subject of controversies already decades ago. Paul Samuelson had a critical paper on the topic, famously written in monosyllabic words only (nerd humor...): Why we should not make mean log of wealth big though years to act are long, Journal of Banking & Finance, vol. 3 (1979), issue 4 (Dec.), p. 305-307.

Thanks, Ivan, illuminating as always...

DeleteI think Peters is NOT trying to propose a descriptive theory (although in passing the theory does fit better with what people do on these types of gambles). Rather, he's making a normative claim that playing such gambles on the time average strategy will generally lead to higher wealth. The implicit suggestion, on my reading, is that people actually know this instinctively and act on it. [This is true of traders especially, apparently, who tend to have strict rules limiting the amount of their total account they will expose to any one position, < 5% say, based on experience of being blown up in the past; such limits offer the best way to grow a portfolio in the long run. There are some good stories about this in the book Market Wizards by Jack Schwager.]

Samuelson's paper indeed seems to make the point in so many distinct syllables. So is this a widely accepted view? I was under the impression that modern portfolio theory aims to maximize the expected return (over a short interval) for a chosen level of risk, rather than the expected time averaged return. Is this because Samuelson's view didn't win out?

I think most economists would agree with Samuelson. MPT assumes that for given volatility, people prefer high return, which identifies set of efficient portfolios. But which of these portfolios would be chosen depends on how they weigh tradeoff between return and volatility. Somebody who is very risk averse might invest almost all of his wealth in risk-free bonds, which would lead to less than maximal long-term growth rate - but they would prefer this, since their trajectory of wealth would be less volatile than trajectory that would maximize long-term geometric average return.

DeleteI see that you are avoiding dealing with the blunter in the Towers Watson paper. Let me remind you:

ReplyDelete"Because it is a fair coin, one round will be heads and the other tails."

You avoiding dealing with this basic issue makes me wonder.: Do you understand probability or not? Because if you did, then you would have not quoted that paper. Except if you did not read it, win which case that raises issues too. So, as a long time reader, I am asking you kindly to deal with this issue.

thanks for pointing this out... I hadn't even noticed that error and read right past it!

DeleteOf course it is a blunder and incorrect. I assume you are referring to the wording in their footnote number 7. While I read past that, I did wonder while reading about the formula for T(r) on the same page: why does it only include the sequence in which a win and loss alternate, and not those in which you get two wins or two losses in a row? I worried about this and looked into it and eventually satisfied myself that this equation is correct, although they have not given you any details here to see why. For this, go to Peters' PNAS paper (the one I've linked to) and look at his eq. 6.9. This is the general result he derives for the time average and it shows that you take the product of each possible return raised to the power of its likelihood. This gives you the equation quoted for T(r).

So, I think the language in the footnote is sloppy, but not intended to pull any tricks. But you're very right -- it sounds suspicious!!

Dear Mark,

ReplyDeleteThanks for the answer. In the expected value equation for a coin toss for two trials, there is a win amount and a loss amount at each trial and four equal probability outcomes. In the paradox you make an initial bet and then each trial wins W, or nothing and stops. These are two different games. Peters equation applies to the former. The other authors take that formula and compare it to the two trial coin toss expected value. IMO, that is wrong.

"The paradox, if I'm right, is why people don't act in accordance with the precepts of expected return calculated using the ensemble average."

ReplyDeleteYou do not understand the paradox and you making silly statements. Neither Peters does. The paradox is simply about how much you would pay for a game with infinite expected value. It doesn't mean that you would pay an infinite sum to play it. Expected values are realized only at the limit of large numbers.

"Most importantly, how much you ought to be willing to pay depends on your initial wealth w, as this determines how much you can afford to lose before going broke. "

In the St Petersburg "paradox" you never go broke. In the worse case you make $1. Get the facts straight, you and Peters.

It must be very embarrassing for you and also quite revealing about your understanding of the subject that you have linked to the Towers Watson article. Your only choice at this point is to apologize publicly. How in the world it is possible in a game of coin tossing with equal chances of 10% win and 10% loss to end up with a -1% expected per period loss? Are you serious?

ReplyDeleteBecause the calculation is based on the bankroll fluctuating. let's say you have 100 dollars. You win your first you gain ten dollars. But now if your bankroll is 110 dollars, you lose 11. you're down to 99 dollars. That's really not that hard to figure out....if you pull the 10 you win out of the game, only then would you be breaking even.

ReplyDeleteNot to be rude, but I worry about you all, esp the leading Anonymous. 0.9 and 1.1 are not symmetric.

ReplyDeleteTo spell it out, 1-e and 1+e are symmetric and 0.1>>e (e=epsilon)

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