Parrondo's Paradox

What’s going on

Each round of these games you either win one coin or lose one. Your capital is how many coins you are up (or down, if it is negative).

• Game A: toss a coin that lands your way 49.5% of the time.
• Game B: look at your capital first. If it is a multiple of 3 (…, −3, 0, 3, 6, …), toss a bad coin that wins only 9.5% of the time. Otherwise toss a good coin that wins 74.5% of the time.

Game A clearly loses: on average you lose one coin every hundred rounds. Game B is less obvious, because the good coin looks very good. But the scene above starts with a thousand players playing each game alone for 200 rounds, and both lines go down.

Now press Random A/B: before each round, every player tosses an ordinary coin to decide whether to play A or B this time. The line goes up. Two losing games, mixed together, win. Try AABB… as well: it wins too.

Why B loses on its own

If the capital were a multiple of 3 exactly one time in three, B would use the bad coin a third of the time and win about 0.33 × 9.5% + 0.67 × 74.5% ≈ 53% of its rounds. But B does not leave the capital alone. Picture the capital sitting just above a multiple of 3, say at 4:

• the good coin usually takes it up to 5, and from 5 usually up to 6, a multiple of 3;
• at 6 the bad coin usually knocks it back down to 5.

So the capital keeps falling into the trap at multiples of 3. When you play B alone, about 38.4% of its rounds are played with the bad coin, and B then wins only about 49.6% of the time. That is a loss of almost one coin every hundred rounds.

Why mixing helps

Game A does not look at the capital at all. It just moves it one step up or down, almost at random. Slip A rounds in between and the capital gets shaken out of B’s favourite pattern. With random mixing, B meets a multiple of 3 in only about 34.5% of its rounds, close to one in three, and wins about 52.1% of them.

That gain in B is bigger than the small loss in A. Overall the random mixture gains about 1.6 coins every hundred rounds. The trick is that B’s coin depends on the capital, and A changes the capital. Neither game is any better on its own; together they change when B is played.

Not every mixture works. With plain alternation, ABAB…, B still meets a multiple of 3 almost as often as when played alone, and the players still lose. Press Expected curves to see the exact average for each way of playing.

Try it yourself

1. Remove the handicap

Take away the half-percent handicap: A’s coin now wins exactly 50%, and B’s coins win 10% and 75%. Is A fair? Is B fair? What happens when you mix them at random?

Answer

A is now exactly fair. B is fair too: played alone, it meets a multiple of 3 exactly 5 times in 13, and 5/13 × 10% + 8/13 × 75% = 50%. The random mixture now wins about 2.5 coins every hundred rounds. The half-percent handicap was only there to make the separate games lose, not just break even.

2. Alternation fails

ABAB… loses but AABB… wins. What is different for game B?

Answer

In ABAB… each B round comes right after exactly one A round. In the long run those B rounds meet a multiple of 3 about 38% of the time, nearly as often as B alone, so B still loses. In AABB… two A steps in a row spread the capital out better: B meets a multiple of 3 only about 34.7% of the time and wins.

3. Beat the casino?

Could you use this idea to win in a casino by switching between two losing games, say roulette and dice?

Answer

No. In a casino, every bet loses on average no matter how much money you have. Then every round of any mixture loses on average, so the mixture loses. Parrondo's games need a game, like B, whose odds depend on your capital, plus another game that changes the capital.

Go deeper

• The games were invented by the Spanish physicist Juan Parrondo, and became widely known in 1999, when Gregory Harmer and Derek Abbott described them in Nature.
• Parrondo was thinking about the “flashing ratchet”: tiny particles that drift downhill can be pushed uphill by switching a lopsided force field on and off. Game B plays the part of the lopsided field, and game A the part of switching it off.