30. Lose to Win:

Parrondo’s Paradox

The Question:

You enter a casino which offers two games of chance, both of which guarantee that if you play them often enough, you will lose all your money. No surprise there.

Are there any guaranteed-to-lose-games which, if you play them both in a certain order, guarantee unlimited winnings? And, if yes, could you monetize this strategy?

The Paradox:

Yes…and no!

The first thing you do upon entering the casino is to buy some chips. Then you read the rules of the two games that the casino offers.

For example, Game P says that if the number of chips that you hold is even, you gain a chip, if it is odd you lose three chips. Game Q says the opposite: if the number of chips is odd, you win a chip, if it is even, you lose three.

Playing either of the games over and over again is sure to bankrupt you. Whenever you win a chip, you will lose three at the next go. In the end, you’ll have nothing.

Now what’s the winning strategy? Easy: buy an even number of chips and play P, then Q, then P again, then Q again, and so on. Playing P at first, you gain a chip; now you hold an odd number of chips and you play Q: you gain another chip. Now you have an even number again and you play P. So you gain another chip. Think about it…you win one chip at every go! You’ll win and win and win. After a while, you’ll be sick of winning, as a certain American (ex-)President is fond of saying.

Background:

The paradoxical combination of two games[1] was devised by the Spanish physicist Juan Parrondo as a pedagogical illustration of a thought experiment: the so-called Brownian Ratchet. If it existed it would be a molecular device that would – if it worked – violate the Second Law of Thermodynamics. The Brownian Ratchet consists of two chambers, with a paddle in one, that is connected by a rod to a ratchet in another. The ratchet can only rotate in one direction because a safety catch prevents it from going backwards. Whenever the paddle in its chamber is hit by a forward going molecule it turns –and, via the rod, moves the ratchet forward by a notch. If a backward going molecule hits the paddle, nothing happens since the safety catch prevents the ratchet from going backwards.

If the ratchet and the paddle were not connected by the rod, the paddle would move randomly forward and backward, while the ratchet would remain stationary. Nothing interesting would happen. If it worked as designed, however, the Brownian ratchet would be a perpetuum mobile: connected to the paddle and able to move in one direction only, the ratchet could perpetually lift weights without any energy input.

***

The link between the Brownian Ratchet and the combination of the two casino games may seem a wee bit tenuous. What the physics professor wanted to illustrate was how dependence between two systems (the ratchet and the paddle, or Games P and Q) can lead to entirely different probabilistic outcomes than if the two systems were operating independently of each other.

Dénouement:

In reality, the Brownian Ratchet won’t work because the safety catch, residing in the same temperature as the paddle, is also subject to the molecules’ Brownian motion. Thus, it bounces up and down and the ratchet is not prevented from going backwards. The only way the Brownian Ratchet can work is if the temparature is higher in the paddle chamber than in the ratchet chamber. But it would only work for a while, until the heat in both chambers has become equal.

***

By sequencing the games judiciously, the winnings are accumulated in a similar manner as the weight is lifted when the ratchet is turned forwards. By switching to the other game, the accumulated chips are trapped in your pocket in a similar manner as the lifted weight is trapped above ground.)

So, can you make money based on the paradox? Yes, in principle you could. Let’s say, you have total wealth of 5000$. You keep 75% in cash and invest 25% in a stock that every week randomly either doubles in value or falls to a third of its former value. If you hold on to that portfolio, the cash will remain at its initial level but the value of your stock will, on average, rise half the time, and fall half the time. Since losses (minus two thirds) outweigh the gains (plus 100%), the total value will drift inevitably downwards.

But now re-balance your portfolio every week: if the stock has risen so that it represents more than a fourth of your portfolio, reduce your investment to 25% and add the remainder to your cash. If the value of the stock is less than one fourth of the portfolio, use cash to stock up on the stock so it is 25% again. It turns out that the strategy of re-balancing this losing stock will, in the long run, add about 2% to the portfolio’s value each week. (See the technical supplement.)

How come? Well, by re-balancing the portfolio, you stop the compounding of the average loss. Instead, by converting the gains when the stock is doing well, profits are parked in cash, in a similar manner as the lifted weight is trapped above ground in the Brownian Ratchet.

But stocks do not usually double in value or fall by two thirds every week. And for reasonable volatilities, transaction costs would eat up all of the minuscule profits. So, unfortunately, there’s no money to be made.

Technical supplement:

Let’s say, the investment horizon is T weeks. Since the stock doubles in value half the time, and falls to a third half the time, the average value of the stock after T periods is 2T/2/3T/2 of its initial value. After T periods, the stock will have fallen to 2/3T/2, i.e., to 0.8T, which tends towards zero.

However, by re-balancing the portfolio, the value of the stock grows on average to 117% every two weeks (because half the time it rises to 200%, and half the time it falls to 33%). With one quarter of the portfolio invested in the stock, the weighted average of the portfolio’s growth is (1/4)117% + (3/4)100% = 104% per two weeks, or about 2% per week.

***

Applications of Parrondo’s Paradox are active area of research in disciplines such as engineering, electronics, biology, economics, ecology. To give just one example, in agriculture both sparrows and insects can eat all the crops. However, by having a combination of sparrows and insects, a healthy crop can be harvested.

[1] Parrondo actually devised more sophisticated, but also more complicated, games.

Corrections, comments, observations: