Have you ever seen Ocean’s Eleven? Spoiler alert if you haven’t: in it, we see a ragtag group of lovable crooks and shysters attempt to take down a Las Vegas casino – essentially an impenetrable fortress of anti-thievery measures which, we are repeatedly informed throughout, just about nobody has ever successfully cracked.
At almost every turn, things go sideways. Personalities clash; covers are blown; at one point there’s an honest-to-goodness medical emergency. Yet, at the end of the movie, we see our heroes sauntering out of the joint carrying a cool $160 million in booty while the casino owner is left with nothing. Somehow, the gang pulled through.
We admit, such a denouement owes more than a little to the movie’s storytelling and direction. But it’s also a pretty good demonstration of Parrondo’s Paradox: a strange little concept from the mathematical field of game theory, in which we see a combination of losing strategies ultimately becoming a winner.
The origins of Parrondo’s Paradox: an impossible machine
Initially formulated in 1996, Parrondo’s Paradox comes from a thought experiment known as the Brownian ratchet. In short, this is a tiny device that can convert heat into mechanical work without ever losing thermal energy.
How can it be possible? In theory, the ratchet, or gear, is rotated by the movement of a paddle wheel, which in turn gets its energy from being pelted by randomly moving molecules within some fluid – a word which, in math and physics, can refer to gases as well as liquids, so we can just assume it’s surrounded by air.
Now, let’s say you want to harness the rotations of this little machine to fuel some kind of process. Of course, being hit by molecules moving at random means that the paddle, and therefore the ratchet, is also going to be moving in more than one direction, and that’s no good for generating power – so we’re going to be clever about it. We’ll add a pawl – that is, a spring-loaded claw that allows the ratchet to move in one direction, but not the other.
And voila: a machine that uses only energy from the random movement of air particles around it to spin a ratchet in one direction. It’s a miraculous device: we could, say, hook it up to a little electrical generator, perhaps, or use the gear to lift a small mass, and as long as the Sun continues to heat the Earth and the surrounding molecules carry on with their random movement, the machine’s motion will continue perpetually.
Okay, so that last sentence may have clued you into why the Brownian ratchet is a thought experiment rather than a real-life machine: according to all known laws of physics, it cannot exist.
“It’s not immediately obvious that such a machine should be impossible,” explained Brian Skinner, now an assistant professor of physics at the Ohio State University, in a blog post from 2010. “It certainly doesn’t violate energy conservation, nor does it rely on any ‘zero friction’ assumptions.”
“But, by decree of thermodynamics, Feynman’s ratchet cannot work as a heat engine,” he continued. “It plainly violates the Second Law, which says that useful work can only be obtained by the flow of energy from high to low temperature. This device purports to get energy from a single temperature reservoir: that of the air around it.”
Now, since violating the second law of thermodynamics is not on the cards, what would happen if you actually created a Brownian ratchet? For the answer to that, we turn to legendary physicist Richard Feynman: “When the vanes get kicked, sometimes the pawl lifts up and goes over the end,” he explained in his Feynman Lectures on Physics.
“But sometimes, when it tries to turn the other way, the pawl has already lifted due to the fluctuations of the motions on the wheel side, and the wheel goes back the other way!” he pointed out. “The net result is nothing.”
Finding direction in disorder
So, the Brownian ratchet is a bust – but what if you were to add in a second mechanism to help things along? Like the ratchet, this would also rely on random movement: it would be one-dimensional Brownian motion, and the new ratchet setup would alternate stochastically between the two regimes.
This is the so-called flashing Brownian ratchet – and it’s very interesting to both physicists and other scientists alike. That’s because it has a seemingly impossible property: it’s a device powered by two random processes which nevertheless result in a kind of order.
“This is the inspiration for Parrondo’s Paradox,” explained Gregory Harmer and Derek Abbott in a 1999 paper on the concept. “The individual states are said to be like ‘losing’ games and when they are alternated we get […] a ‘winning’ expectation.”
Parrondo’s Paradox in play
We’ve been talking an awful lot of physics for a concept supposedly from game theory – but luckily, one of the more accessible thought experiments explaining the paradox comes straight from the casino.
For example, suppose you are playing two games, A and B, with the following rules: in game A, you lose $1 every time you play; in game B, you win $3 if you have an even number of dollars left, and you lose $5 if it’s an odd number.
Clearly, both of these games are doomed to leave you impoverished – start with $100, say, and both games will empty your coffers in exactly 100 rounds. But start with game B, and alternate between the two, and you’ve got yourself a winning strategy: for every two rounds, you will net a profit of $2.
Alternatively, consider the two-envelope problem: a game in which you must choose between two envelopes, one of which contains twice as much money as the other. Once a decision is made, the player may open the envelope, and then decide whether or not to switch to the other – swapping their prize for one worth either half or double what they’ve already obtained.
Like the Monty Hall Problem, the solution is annoying and confusing. While common sense tells us that there’s always a 50-50 chance of choosing either envelope – and therefore an equal chance of gaining or losing money by switching – probability theory seems to show that swapping is the better strategy, yielding an expected value for the second envelope which is always 5/4 higher than the one you initially chose.
It was a problem that had confused mathematicians since the 30s. “The apparent paradox arose before because it didn't seem to make sense that opening an envelope and seeing $10 actually tells you anything, and therefore it seemed strange that your expected value of winning is $12.50 by switching,” Abbott noted in 2009.
But thanks to his experience with Brownian ratchets and Parrondo games, he was able to crack the problem: “we resolve this by explaining it in terms of symmetry breaking,” he explained. “Before the envelopes are opened, the situation is symmetrical, so it doesn't matter if you switch envelopes or not. However, once you open an envelope and use [the] strategy [detailed in his recent paper], you break that symmetry, and then switching envelopes helps you in the long run.”
“This solution to the two-envelope problem is a breakthrough in the field of Parrondo's paradox,” he said.
Parrondo’s Paradox in the real world
So far, so abstract – but as counterintuitive as it may remain, there are actually quite a few examples of Parrondo’s paradox being used successfully in the real world. From investment advice to quantum computing and evolution, the seemingly nonsensical concept has turned out to be responsible for some of the most fundamental scientific results in recent years – including some of the strategies utilized in the fight against the COVID-19 pandemic.
In fact, this confusing little paradox might just hold the key to life itself.
“Developments in Parrondo's paradox to date have revealed a potential unifying fundamental characteristic of life itself, more valuable to our understanding of nature than its individual components,” said physicist Jin Ming Koh, who in 2019 co-authored a study applying the paradox to concepts across biology including ecology and evolution, genetics, social and behavioral systems, cellular processes, and disease.
“Every cell, organism and species, and species assemblage and ecosystem, is necessarily mortal,” agreed co-author and Assistant Professor at Singapore University of Technology and Design Kang Hao Cheong, “yet the biosphere persists.”
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