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The Expected Value Paradox – in a nutshell.

May 3, 2019

To illustrate the Expected Value Paradox, let us propose a coin-tossing game, in which you gain 50% of what you bet if the coin lands Heads and lose 40% if it lands Tails. What is the expected value of a single play of this game?

The Expected Value can be calculated as the sum of the probabilities of each possible outcome in the game times the return if that outcome occurs.

Say, for example, the unit stake for each play of the game is £10. In this case, the gain if the coin lands Heads is 50% x £10 = £5, and the loss if the coin lands Tails is 40% x £10 = £4.

In this case, the expected value (given a fair coin, with 0.5 chance of Heads and 0.5 chance of Tails) = 0.5 x £5 – 0.5 x £4 = £0.5, or 50 pence.

So the Expected Value of the game is 5%. This is the positive net expectation for each play of the game (toss of the coin).

Let’s see how this plays out in an actual experiment in which 100 people play the game. What do we expect would be the average final balance of the players?

The expected gain from the 50 players tossing Heads = 50 x £5 = £250.

The expected loss from the 50 players tossing Tails = 50 x £4 = £200.

So, the net gain over 100 players = £250 – £200 = £50.

The average net gain of the 100 player = £50/100 = £0.5, or 50 pence.

Expected Value = 0.5 x £1.5 + 0.5 x 60p. = £1.05. As above, this is an expected gain of 5%.

From two coin tosses, our best estimate is 25 Heads-Heads, 25 Tails-Tails, 25 Heads-Tails and 25 Tails-Heads.

The Expected Value over the two coin tosses = 0.25 x (1.5)2 + 0.25 x (0.6)2 + 0.25 (1.5 x 0.6) + 0.25 (0.6 x 1.5) = £1.0575.

However many coin tosses the group throws, the Expected Value is positive.

Take now the case of one person playing the game through time. Say there are four coin tosses, for a stake of £10.

From four coin tosses, our best estimate is 2 Heads and 2 Tails.

Expected value for 2 Heads and 2 Tails = £10 x 1.5 x 1.5 x 0.6 x 0.6.

Expected value goes from £10 to £15 to £22.50 to £13.50 to £8.10. This is a net loss.

To clarify, we bet £10. The coin lands Heads. We now have £15. We bet £15 now on the next coin toss. It lands Heads again. We now have £22.50. We bet £22.50 now on the next coin toss. It lands Tails. Now we are back to £13.50. We bet this £13.50 on the next coin toss. It lands Tails again and we are down to £8.10. This is a net loss on the original stake of £10.

If we throw the same number of Heads and Tails after tossing the coin N times, we would expect more generally to earn the following.

1.5N/2 x 0.6N/2 = (1.5 x 0.6)N/2 = 0.9N/2

Eventually, all the stack used for betting is lost.

Herein lies the paradox. When many people play the game a fixed number of times, the average return is positive, but when a fixed number of people play the game many times, they should expect to lose most of their money.

This is a demonstration of the difference between what is termed ‘time averaging’ and ‘ensemble averaging.’

Thinking of the game as a random process, time averaging is taking the average value as the process continues. Ensemble averaging is taking the average value of many processes running for some fixed amount of time.

Processes where there is a difference between time and ensemble averaging are called ‘ergodic processes.’ In the real world, however, many processes, including notably in finance, are non-ergodic.

Say that in an election two parties, A and B, attract some percentage of voters, x% and y% respectively. This is not the same thing as saying that over the course of their voting lives, each individual votes for party A in x% of elections and for party B in y% of elections. These two concepts are distinct.

Again, if we wish to determine the most visited parts of a city, we could take a snapshot in time of how many people are in neighbourhood A, how many in neighbourhood B, etc. Alternatively, we could follow a particular individual or a few individuals, over a period of time and see how often they visit neighbourhood A, neighbourhood B, etc. The first analysis (the ensemble) may not be representative over a period of time, while the second (time) may not be representative of all the people.

An ergodic process is one which in which the two types of statistic give the same results. In an ergodic system, time is irrelevant and has no direction. Say, for example, that 100 people rolled a die once, and the total of the scores is divided by 100. This finite-time average approaches the ensemble average as more and more people are included in the sample. Now, take the case of a single person rolling a die 100 times, and the total scored is divided by 100. This finite-time average would eventually approach the time average.

An implication of ergodicity is that the result ensemble averaging will be the same as time averaging.

And here is the key point: In the case of ensemble averages, it is the size of the sample that eventually removes the randomness from the sample. In the case of time averages, it is the time devoted to the process that removes randomness.

In the dice rolling example, both methods give the same answer, subject to errors. In this sense, rolling dice is an ergodic system.

However, if we now bet on the results of the dice rolling game, wealth does not follow an ergodic system. If a player goes bankrupt, he stays bankrupt, so the time average of wealth can approach zero over time as time passes, even though the ensemble value of wealth may increase.

As a new example take the case of 100 people visiting a casino, with a certain amount of money. Some may win, some may lose, but we can infer the house edge by counting the average percentage loss of the 100 people. This is the ensemble average. This is different to one person going to the casino 100 days in a row, starting with a set amount. The probabilities of success derived from a collection of people does not apply to one person. The first is the ‘ensemble probability’, the second is the ‘time probability’ (the second is concerned with a single person through time).

Here is the key point: No individual person has sure access to the returns of the market without infinite pockets and an absence of so-called ‘uncle points’ (the point at which he needs, or feels the need, to exit the game). To equate the two is to confuse ensemble averaging with time averaging.

If the player/investor has to reduce exposure because of losses, or maybe retirement or other change of circumstances, his returns will be divorced from those of the market or the game. The essential point is that success first requires survival. This applies to an individual in a different sense to the ensemble.

So where does the money lost by the non-survivors go? It gets transferred to the survivors, some of whom tend to scoop up much or most of the pool, i.e. the money is scoped up by the tail probability of those who keep surviving, which may just be by blind good luck, just as the non-survivors may have been forced out of the game/market by blind bad luck. So the lucky survivors (and in particular the tail-end very lucky survivors) more than compensate for the effect of the unlucky entrants.

The so-called Kelly approach to investment strategy, discussed in a separate chapter, is an investment approach which seeks to respond to the survivor issue.

Say, for example, that the probability of Heads from a coin toss is 0.6, and Heads wins a dollar, but Tails (with a probability of 0.4) loses a dollar. Although the Expected Value of this game is positive, if the response of an investor in the game is to stake all their bankroll on each toss of the coin, the expected time until bankroll bankruptcy is just 1/(1-0.6) = 2.5 tosses of the coin.

The Kelly strategy to optimise the growth rate if the bankroll is to invest a fraction of the bankroll equal to the difference in the likelihood you will win or lose.

In the above example, it means we should in each game bet the fraction of x = 0.6 – 0.4 = 0.2 of the bankroll.

The optimal average growth rate becomes: 0.6 log (1.2) + 0.4 log (0.8) = 0.2.

If we bet all our bankroll on each coin toss, we will most likely lose the bankroll. This is balanced out over all players by those who with low probability win a large bankroll. For the real-life player, however, it is most relevant to look at the time-average of what may be expected to be won.

In trying to maximise Expected Value, the probability of bankroll bankruptcy soon gets close to one. It is better to invest, say, 20% of bankroll in each game, and maximise long-term average bankroll growth.

In the coin-toss example, it is like supposing that various “I”s are tossing a coin, and the losses of the many of them are offset by the huge profit of the relatively small number of “I”s who do win. But this ensemble-average does not work for an individual for whom a time-average better reflects the one timeline in which that individual exists.

Put another way, because the individual cannot go back in time and the bankruptcy option is always actual, it is not possible to realise the small chance of making the tail-end upside of the positive expectation value of a game/investment without taking on the significant risk of non-survival/bankruptcy. In other words, the individual lives in one universe, on one time path, and so is faced with the reality of time-averaging as opposed to an ensemble average in which one can call upon the gains of parallel investors/game players on parallel timelines in essentially parallel worlds.

To summarise, the difference between 100 people going to a casino and one person going to the casino 100 times is the difference between understanding probability in conventional terms and through the lens of path dependency.

 

References and Links

Time for a change: Introducing irreversible time in economics. https://www.gresham.ac.uk/lectures-and-events/time-for-a-change-introducing-irreversible-time-in-economics

What is ergodicity? https://larspsyll.wordpress.com/2016/11/23/what-is-ergodicity-2/

Non-ergodic economics, expected utility and the Kelly criterion. https://larspsyll.wordpress.com/2012/04/21/non-ergodic-economics-expected-utility-and-the-kelly-criterion/

Ergodicity. http://squidarth.com/math/2018/11/27/ergodicity.html

Ergodicity. http://nassimtaleb.org/tag/ergodicity/

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