Parrondo's Paradox

[pdonald]

Have you heard of Parrondo's Paradox? (This has nothing to do with martingale/progressive betting.)

Imagine there are two casino games: Game A and Game B. Each game is random, and each of them has odds stacked against the player (they have a negative expected outcome). Each time either game is played, if you win, you win $1, if you lose, you lose $1. If you play Game A continuously you’ll gradually lose money. The longer you play, the more money you will lose; not good for your retirement plans! Similarly, if you play Game B continuously you’ll gradually lose money.

However, what if I told you that if you alternate playing these two games using the pattern: A, B, B, A, B, B, A, B, B, … that the odds will switch to being in your favor?!?!

Huh? Yes, that’s right! By alternating play you can turn these two loss making games into a money making exercise with a positive expected outcome. This is Parrondo’s Paradox.

Before you get all excited about making a killing at the casino, let me point out that the two games are not independent but all casino games are.

But I have a feeling most of us here have played games where we make a decision based on the current value of our cash balance. Trading several soccer markets or playing some poker, perhaps?

:Kerching

[doubleback]

A, B, B, A

[Cran]

However, what if I told you that if you alternate playing these two games using the pattern: A, B, B, A, B, B, A, B, B, … that the odds will switch to being in your favor?!?!

:Boom

[Nikol]

But I have a feeling most of us here have played games where we make a decision based on the current value of our cash balance. Trading several soccer markets or playing some poker, perhaps?

:Kerching

Usually feelings are bad in our job, especially if the odds are bad.
What if I tell you that this is simple probability theory and statistics and if the odds in the 1st game are against you and the odds in the second game are against you, no matter how you switch the games, patterns or money managment you will always lose ?!

[pdonald]

I would think you didn't read the article.

[Morgans Choice]

Have you heard of Parrondo's Paradox? (This has nothing to do with martingale/progressive betting.)



Before you get all excited about making a killing at the casino, let me point out that the two games are not independent but all casino games are.

But I have a feeling most of us here have played games where we make a decision based on the current value of our cash balance. Trading several soccer markets or playing some poker, perhaps?

:Kerching

Hi P,

my memory is a bit hazy, but I think that I have read about what appears to be a variant of this method. It involved "splitting" the runners in a horse race into two groups (halves) and backing one of those groups. That gives the "two losing games" and the interdependance. It's not an idea which appealed immediately, in spite of the impressive mathematics,

cheers, MC

[flok]

very interesting article and concepts, but i think they're real on in theory.
the game A of the example can be reproduced in reality, betting @2 and win or lose the same stake.
but the game B, where you have 2 possible bets, and one of them allows you to win 75% of the times... how do you reproduce it in reality?
you can't, because if you want a chance of 75% you must bet 3 to win 1, you cant bet 1 to win 1 or you'd reduce your chance to 50% and the whole theory of the article would collapse

[Tytteboevs]

Why would this be a paradox?

If it was able to work, it would not be a paradox but pure math.

Its "articles" like this, that keeps money in the business coming from the less work willing.

So, if that would work, I reckon this also works:

1: I have one bill i need to pay
2: I have another bill I need to pay
3: I pay none of them, and suddenly i got money!
Best regards
T

[pdonald]

Why would this be a paradox?

I believe it is called a paradox because it is not clear right away how that is possible, it requires further analysis to understand.

You say it doesn't work i.e. it is wrong. Is there a problem with the proof in the article? Can you please tell us what it is?

[pdonald]

my memory is a bit hazy, but I think that I have read about what appears to be a variant of this method. It involved "splitting" the runners in a horse race into two groups (halves) and backing one of those groups. That gives the "two losing games" and the interdependance. It's not an idea which appealed immediately, in spite of the impressive mathematics

Thanks for contributing the first real answer in the thread.

While reading the Wikipedia page about this paradox, I came across an article by Michael Stutzer in Mathematical Scientist about this.

If you have a betting strategy with a 51% probability of winning, the expected value of your bank after 1000 bets is naturally more i.e. 2.717 * bank. But it turns out this value is too positively skewed and the lower median is actually 0.778 * bank which means you're more likely to lose money than win it. However, if you diversify your funds, it will produce a Parrondo Paradox and the lower median value then becomes 1.2 * bank!

If you're interested in reading more about it, here's a link to the article: A Simple Parrondo Paradox (PDF)

[Nikol]

Ok I've read it now. Its specificly said - the games have not to be indipendent. This means they have to be related. And if they are related he is talking about arbitrage! THe only think I couldn't understand is what is so paradoxal about an arbitrage ?

[WhyAlwaysMe]

If you ever gamble with 50/50 coin flips you will get some insight as to why that median is so low. Do 1000 and your result will be in a narrow band. But along the way you can get on some crazy sequences, with 11 or 12 consecutive wins/losses and there will be times when you won't believe what you see (always loss ). A constantly adjusting balance is introducing just enough variance to give a mildly skewed result. However I did think a 51/49 edge would raise the median to close to 1 because 1000 trials is quite a lot of +EV.

It's interesting stuff. As an offshoot it does throw light on a common cycle with tipping services. The service starts and goes on a really good run. Having shared in the success the clients raise their stakes. Tipping service starts to revert to mean and now clients have lost money even though the service may be showing a profit to a £1 stake.

[rphi6876]

I believe it is called a paradox because it is not clear right away how that is possible, it requires further analysis to understand.

You say it doesn't work i.e. it is wrong. Is there a problem with the proof in the article? Can you please tell us what it is?

I wouldn't worry too much about people not believing this. I found many here think "opinion" counts for much more than maths, empirical results or academic journals. Unfortunately I have learnt this the hard way.

I love Tytteboevs response. "Why would this be a paradox? If it was able to work, it would not be a paradox but pure math and then suggest it couldn't work as this is so counter intuitive"

Definition of Paradox from oxford dictionary: 1. a statement or proposition which, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems logically unacceptable or self-contradictory.

2.A seemingly absurd or self-contradictory statement or proposition that when investigated or explained may prove to be well founded or true


Sure sounds like a paradox to me. But this surely cant be true. So I have chosen to not read the proof - dismiss any academic research and rename this as Parrondo's Witchcraft and Sorcery- as surely my gut instinct is infinitely smarter than all of those professional mathematicians.:Stupid

[rphi6876]

Ok I've read it now. Its specificly said - the games have not to be indipendent. This means they have to be related. And if they are related he is talking about arbitrage! THe only think I couldn't understand is what is so paradoxal about an arbitrage ?

Arbitrage in sports markets suggests you manage to get (at least) one bet on with positive expected value. Ie. in soccer lay the win at 2 on betfair and back with a bookie at 2.4. Fair value could be in three possible locations.

1) less than 2- implies that your lay has negative expected value but your back has large positive expected value.
2) between 2 and 2.4 - implies both your lay and back have positive expected value.
3) greater than 2.4- implies your lay has large positive expected value but your back has negative expected value.

In all three cases at least one bet has a positive expected value. This paradox suggests both bets on their own have negative expected value. A feature which arbitrage does not have. So......

P.S. I have written "less than" instead of using the symbol "<" as those mysterious witchcraft symbols can be a bit confusing.

[pdonald]

very interesting article and concepts, but i think they're real on in theory.

We can easily generalize Parrondo's Paradox. Perhaps I should have linked to the Wikipedia page to begin with.

A combination of losing strategies becomes a winning strategy.

or

There exist pairs of games, each with a higher probability of losing than winning, for which it is possible to construct a winning strategy by playing the games alternately.

The article I stumbled upon a while ago and wanted to share because it explained it well in an easy to understand language with lots of visualizations just happens to illustrate the paradox with a betting example involving biased coins, a real world example with nuts and a couple of other examples.

It doesn't mean those are the only examples. Here's a simple one that I thought of: if we drink just water we will die of starvation, if eat just dry food we will die of thirst, but if we drink some water and then eat some food we will live (but that is not always the case: if you drink water but eat too little food you may still die). Here are a few more about poker.

Human minds, apart from Tytteboevs's, have evolved to be capable of abstract thought and I had this in mind when I posted about it, perhaps together we can come up with interesting cases producing a Parrando Paradox?

We can easily simulate a coin flip on the sports betting exchanges, but in the other game you don't necessarily need to win +1 and lose -1 with those probabilities, there could be other variants as well. You could even try to think outside the box. While Game A could be tossing the coin, Game B could playing the slots 10% of the time and playing poker with your 7 year old niece the remainder of the time, etc.

I'll post a comment from a leading Parrondo's Paradox researcher about whether this is a paradox or not and hopefully we can finally stop arguing about it:

Is Parrondo's paradox really a "paradox"? This question is sometimes asked by mathematicians, whereas physicists usually don't worry about such things. The first thing to point out is that "Parrondo's paradox" is just a name, just like the "Braess paradox" or "Simpson's paradox." Secondly, as is the case with most of these named paradoxes they are all really apparent paradoxes. People drop the word "apparent" in these cases as it is a mouthful, and it is obvious anyway. So no one claims these are paradoxes in the strict sense. In the wide sense, a paradox is simply something that is counterintuitive. Parrondo's games certainly are counterintuitive—at least until you have intensively studied them for a few months. The truth is we still keep finding new surprising things to delight us, as we research these games. I have had one mathematician complain that the games always were obvious to him and hence we should not use the word "paradox." He is either a genius or never really understood it in the first place. In either case, it is not worth arguing with people like that.