In science, paradoxes often appear when familiar rules are pushed into unfamiliar territory. One of them is Parrondo’s paradox, a curious mathematical result showing that when two losing strategies are combined, they can produce a winning outcome. This might sound like trickery but the paradox has deep connections to how randomness and asymmetry interact in the physical world. In fact its roots can be traced back to a famous thought experiment explored by the US physicist Richard Feynman, who analysed whether one could extract useful work from random thermal motion. The link between Feynman’s thought experiment and Parrondo’s paradox demonstrates how chance can be turned into order when the conditions are right.

Imagine two games. Each game, when played on its own, is stacked against you. In one, the odds are slightly less than fair, e.g. you win 49% of the time and lose 51%. In another, the rules are even more complex, with the chances of winning and losing depending on your current position or capital. If you keep playing either game alone, the statistics say you will eventually go broke.

But then there’s a twist. If you alternate the games — sometimes playing one, sometimes the other — your fortune can actually grow. This is Parrondo’s paradox, proposed in 1996 by the Spanish physicist Juan Parrondo.

The answer to how combining losing games can result in a winning streak lies in how randomness interacts with structure. In Parrondo’s games, the rules are not simply fair or unfair in isolation; they have hidden patterns. When the games are alternated, these patterns line up in such a way that random losses become rectified into net gains.

Say there’s a perfectly flat surface in front of you. You place a small bead on it and then you constantly jiggle the surface. The bead jitters back and forth. Because the noise you’re applying to the bead’s position is unbiased, the bead simply wanders around in different directions on the surface. Now, say you introduce a switch that alternates the surface between two states. When the switch is ON, an ice-tray shape appears on the surface. When the switch is OFF, it becomes flat again. This ice-tray shape is special: the cups are slightly lopsided because there’s a gentle downward slope from left to right in each cup. At the right end, there’s a steep wall. If you’re jiggling the surface when the switch is OFF, the bead diffuses a little towards the left, a little towards the right, and so on. When you throw the switch to ON, the bead falls into the nearest cup. Because each cup is slightly tilted towards the right, the bead eventually settles near the steep wall there. Then you move the switch to OFF again.

As you repeat these steps with more and more beads over time, you’ll see they end up a little to the right of where they started. This is Parrando’s paradox. The jittering motion you applied to the surface caused each bead to move randomly. The switch you used to alter the shape of the surface allowed you to expend some energy in order to rectify the beads’ randomness.

The reason why Parrondo’s paradox isn’t just a mathematical trick lies in physics. At the microscopic scale, particles of matter are in constant, jittery motion because of heat. This restless behaviour is known as Brownian motion, named after the botanist Robert Brown, who observed pollen grains dancing erratically in water under a microscope in 1827. At this scale, randomness is unavoidable: molecules collide, rebound, and scatter endlessly.

Scientists have long wondered whether such random motion could be tapped to extract useful work, perhaps to drive a microscopic machine. This was Feynman’s thought experiment as well, involving a device called the Brownian ratchet, a.k.a. the Feynman-Smoluchowski ratchet. The Polish physicist Marian Smoluchowski dreamt up the idea in 1912 and which Feynman popularised in a lecture 50 years later, in 1962.

Picture a set of paddles immersed in a fluid, constantly jolted by Brownian motion. A ratchet and pawl mechanism is attached to the paddles (see video below). The ratchet allows the paddles to rotate in one direction but not the other. It seems plausible that the random kicks from molecules would turn the paddles, which the ratchet would then lock into forward motion. Over time, this could spin a wheel or lift a weight.

In one of his physics famous lectures in 1962, Feynman analysed the ratchet. He showed that the pawl itself would also be subject to Brownian motion. It would jiggle, slip, and release under the same thermal agitation as the paddles. When everything is at the same temperature, the forward and backward slips would cancel out and no net motion would occur.

This insight was crucial: it preserved the rule that free energy can’t be extracted from randomness at equilibrium. If motion is to be biased in only one direction, there needs to be a temperature difference between different parts of the ratchet. In other words, random noise alone isn’t enough: you also need an asymmetry, or what physicists call nonequilibrium conditions, to turn randomness into work.

Let’s return to Parrondo’s paradox now. The paradoxical games are essentially a discrete-time abstraction of Feynman’s ratchet. The losing games are like unbiased random motion: fluctuations that on their own can’t produce net gain because the gains become cancelled out. But when they’re alternated cleverly, they mimic the effect of adding asymmetry. The combination rectifies the randomness, just as a physical ratchet can rectify the molecular jostling when a gradient is present.

This is why Parrondo explicitly acknowledged his inspiration from Feynman’s analysis of the Brownian ratchet. Where Feynman used a wheel and pawl to show how equilibrium noise can’t be exploited without a bias, Parrondo created games whose hidden rules provided the bias when they were combined. Both cases highlight a universal theme: randomness can be guided to produce order.

The implications of these ideas extend well beyond thought experiments. Inside living cells, molecular motors like kinesin and myosin actually function like Brownian ratchets. These proteins move along cellular tracks by drawing energy from random thermal kicks with the aid of a chemical energy gradient. They demonstrate that life itself has evolved ways to turn thermal noise into directed motion by operating out of equilibrium.

Parrondo’s paradox also has applications in economics, evolutionary biology, and computer algorithms. For example, alternating between two investment strategies, each of which is poor on its own, may yield better long-term outcomes if the fluctuations in markets interact in the right way. Similarly, in genetics, when harmful mutations alternate in certain conditions, they can produce beneficial effects for populations. The paradox provides a framework to describe how losing at one level can add up to winning at another.

Feynman’s role in this story is historical as well as philosophical. By dissecting the Brownian ratchet, he demonstrated how deeply the laws of thermodynamics constrain what’s possible. His analysis reminded physicists that intuition about randomness can be misleading and that only careful reasoning could reveal the real rules.

In 2021, a group of scientists from Australia, Canada, France, and Germany wrote in Cancers that the mathematics of Parrondo’s paradox could also illuminate the biology of cancerous tumours. Their starting point was the observation that cancer cells behave in ways that often seem self-defeating: they accumulate genetic and epigenetic instability, devolve into abnormal states, sometimes stop dividing altogether, and often migrate away from their original location and perish. Each of these traits looks like a “losing strategy” — yet cancers that use these ‘strategies’ together are often persistent.

The group suggested that the paradox arises because cancers grow in unstable, hostile environments. Tumour cells deal with low oxygen, intermittent blood supply, attacks by the immune system, and toxic drugs. In these circumstances, no single survival strategy is reliable. A population of only stable tumour cells would be wiped out when the conditions change. Likewise a population of only unstable cells would collapse under its own chaos. But by maintaining a mix, the group contended, cancers achieve resilience. Stable, specialised cells can exploit resources efficiently while unstable cells with high plasticity constantly generate new variations, some of which could respond better to future challenges. Together, the team continued, the cancer can alternate between the two sets of cells so that it can win.

The scientists also interpreted dormancy and metastasis of cancers through this lens. Dormant cells are inactive and can lie hidden for years, escaping chemotherapy drugs that are aimed at cells that divide. Once the drugs have faded, they restart growth. While a migrating cancer cell has a high chance of dying off, even one success can seed a tumor in a new tissue.

On the flip side, the scientists argued that cancer therapy can also be improved by embracing Parrondo’s paradox. In conventional chemotherapy, doctors repeatedly administer strong drugs, creating a strategy that often backfires: the therapy kills off the weak, leaving the strong behind — but in this case the strong are the very cells you least want to survive. By contrast, adaptive approaches that alternate periods of treatment with rest or that mix real drugs with harmless lookalikes could harness evolutionary trade-offs inside the tumor and keep it in check. Just as cancer may use Parrondo’s paradox to outwit the body, doctors may one day use the same paradox to outwit cancer.

On August 6, physicists from Lanzhou University in China published a paper in Physical Review E discussing just such a possibility. They focused on chemotherapy, which is usually delivered in one of two main ways. The first, called the maximum tolerated dose (MTD), uses strong doses given at intervals. The second, called low-dose metronomic (LDM), uses weaker doses applied continuously over time. Each method has been widely tested in clinics and each one has drawbacks.

MTD often succeeds at first by rapidly killing off drug-sensitive cancer cells. In the process, however, it also paves the way for the most resistant cancer cells to expand, leading to relapse. LDM on the other hand keeps steady pressure on a tumor but can end up either failing to control sensitive cells if the dose is too low or clearing them so thoroughly that resistant cells again dominate if the dose is too strong. In other words, both strategies can be losing games in the long run.

The question the study’s authors asked was whether combining these two flawed strategies in a specific sequence could achieve better results than deploying either strategy on its own. This is the sort of situation Parrondo’s paradox describes, even if not exactly. While the paradox is concerned with combining outright losing strategies, the study has discussed combining two ineffective strategies.

To investigate, the researchers used mathematical models that treated tumors as ecosystems containing three interacting populations: healthy cells, drug-sensitive cancer cells, and drug-resistant cancer cells. They applied equations from evolutionary game theory that tracked how the fractions of these groups shifted in different conditions.

The models showed that in a purely MTD strategy, the resistant cells soon took over, and in a purely LDM strategy, the outcomes depended strongly on drug strength but still ended badly. But when the two schedules were alternated, the tumor behaved differently. The more sensitive cells were suppressed but not eliminated while their persistence prevented the resistant cells from proliferating quickly. The team also found that the healthy cells survived longer.

Of course, tumours are not well-mixed soups of cells; in reality they have spatial structure. To account for this, the team put together computer simulations where individual cells occupied positions on a grid; grew, divided or died according to fixed rules; and interacted with their neighbours. This agent-based approach allowed the team to examine how pockets of sensitive and resistant cells might compete in more realistic tissue settings.

Their simulations only confirmed the previous set of results. A therapeutic strategy that alternated between MTD and LDM schedules extended the amount of time before the resistant cells took over and while the healthy cells dominated. When the model started with the LDM phase in particular, the sensitive cancer cells were found to compete with the resistant cancer cells and the arrival of the MTD phase next applied even more pressure on the latter.

This is an interesting finding because it suggests that the goal of therapy may not always be to eliminate every sensitive cancer cell as quickly as possible but, paradoxically, that sometimes it may be wiser to preserve some sensitive cells so that they can compete directly with resistant cells and prevent them from monopolising the tumor. In clinical terms, alternating between high- and low-dose regimens may delay resistance and keep tumours tractable for longer periods.

Then again this is cancer — the “emperor of all maladies” — and in silico evidence from a physics-based model is only the start. Researchers will have to test it in real, live tissue in animal models (or organoids) and subsequently in human trials. They will also have to assess whether certain cancers, followed by a specific combination of drugs for those cancers, will benefit more (or less) from taking the Parrando’s paradox way.

As Physics reported on August 6:

[University of London mathematical oncologist Robert] Noble … says that the method outlined in the new study may not be ripe for a real-world clinical setting. “The alternating strategy fails much faster, and the tumor bounces back, if you slightly change the initial conditions,” adds Noble. Liu and colleagues, however, plan to conduct in vitro experiments to test their mathematical model and to select regimen parameters that would make their strategy more robust in a realistic setting.

That Vulture piece. If you haven’t already, read it but be warned: it’s just as disturbing as everyone is saying it is.

One paragraph in particular I found more unsettling than the rest — not because it presents one more awful detail but because I just didn’t know, while I was reading the piece, what I was expected to make of it.

In various interviews over the years, Gaiman has called The Ocean at the End of the Lane his most personal book. While much of it is fantastical, Gaiman has said “that kid is me.” The book is set in Sussex, where Gaiman grew up. In the story, the narrator survives otherworldly evil with the help of a family of magical women. As a child, Gaiman had no such friends to call on. “I was going back to the 7-year-old me and giving myself a peculiar kind of love that I didn’t have,” he told an interviewer in 2017. “I never feel the past is dead or young Neil isn’t around anymore. He’s still there, hiding in a library somewhere, looking for a doorway that will lead him to somewhere safe where everything works.”

It paints a fuller picture of Neil Gaiman but the article as a whole says nothing about why that’s necessary. The piece is overall well-composed; the writer and editors have obviously gone to great lengths to produce it. As a result, most parts of the piece justify themselves. But I couldn’t say why they saw fit to include parts like this paragraph, which (re-)humanise Gaiman unto some unclear end. Ultimately, there’s nothing in these narratives to suggest the perpetrators of sexual violence ever lost track of the difference between right and wrong.

Reality is strange and truths pointing at contradicting implications can coexist, but this coexistence defies logic and I find that quite disturbing. This isn’t that abstract, disconnected “cold logic” so much as that it’s not just about the coexistence of truths, that somewhere between the past and the future the fellow had a chance to weigh right and wrong and picked the wrong thing to do. The buck stops there and whatever happened to him in his childhood — as long as it didn’t leave him with schizophrenia, psychosis or any other condition that took away his ability to tell if what he was doing was wrong — ceases to matter.

I realise I might be thinking about this in an incomplete way, but since that big #MeToo wave, I’ve always had in mind those nonsensical statements by some men that they were depressed or anxious and that they’d go to therapy and ‘fix’ themselves. None of these conditions could have interfered with these men’s ability to tell right from wrong (which in some legal systems would have been required to mount an insanity defence), but by blaming them the men stigmatised them in a horrible way. Since then, bad men pointing to bad childhoods, as if the past offers some kind of mitigating background, has only been confusing.

In fact, I wonder if the two truths really do coexist. Maybe they don’t because somewhere in between, these men decided the past doesn’t matter any more.

To me the line that Gaiman hadn’t ever been to therapy was a red flag. When there’s emotional trauma, therapy is useful — as many of us know — to reprogramme emotional pathways that have become hijacked and distorted to respond in harmful ways to even benign stimuli. But there was still an opportunity in front of these men to do the right thing. Self-discipline was still possible and they possessed the agency for it.

Humans are both logical and emotional beings. At times like this, however, in a rush to remind ourselves of how a life of emotion can lead to discomfiting truths — like how a childhood of suffering trauma needn’t preclude an adulthood of inflicting trauma or that antithetical narratives of reality can be equally and simultaneously legitimate — we seem to forget humans are still capable of logic, and thus of restraint and forethought. And well-exercised restraint and forethought often lead to compassionate actions towards others. This logic only comes to the fore when we choose to do the right thing.

Gaiman had this choice, the doorway in the library. Now, he doesn’t get to pretend he didn’t do to his past what he did to his work and what he did to all those women, turning them into his fantasy.

Saying “choosing to do the right thing is easier than done” risks trivialising the difficulty, but again, the right thing here is to look for help before rather than succumb, more so because a man of Gaiman’s tremendous wealth and privileges is also bound to find it.

Self-discipline in this context often reminds me of a piece from a decade ago about how tough the road can be for people with illegal sexual preferences, so to speak, without also being a cul-de-sac. The piece is a long-form deep-dive (trigger warnings: child sexual abuse and discussions of suicide) into the inner lives of people who identify as paedophiles but who also recognise their urges are wrong and refuse to act on them. The narrative is careful enough to be fair to all the dramatis personae (fair of course doesn’t mean nice). Where you draw parallels with the Gaiman story, if it all, is up to you; I found the following passages particularly enlightening:

Dr. Klaus Beier doesn’t believe in sexual reconditioning. He leads the team behind Prevention Project Dunkelfeld, a therapeutic program based in Germany that targets potential offenders. He believes that minor attraction is a fixed part of someone’s makeup, that it’s “fate and not choice.” His program is considered the global gold standard of preventive treatment, and its practitioners help adults manage their attraction to children rather than try to change it. “In my view, it’s not the inclination that’s a problem,” he said. “And I wouldn’t condemn the inclination, I’d condemn the behavior.”

Later:

When [Mike] started his teaching placement, he created a strict set of rules: staying away from the bathroom area where possible and avoiding any physical contact with the children. He said he would tense up when the more playful kids approached him for a hug. This wasn’t so much to keep the children safe, he told me, as to ensure people wouldn’t become wary of him. Which was something his dad warned him about when he first started. “He’s like, ‘People are going to be suspicious of you simply because you’re a guy. Don’t do anything stupid,’” he said. “And, honestly, I wonder if that had anything to do with it, because that kind of freaked me out.”

...

Like Adam, Mike grew increasingly depressed while grappling with his desires. He never made active plans to commit suicide, but told me that he thought about it and knew what to do if the time came. “If I had a sawn-off shotgun, that would be it,” he said. “I don’t want to take pills because I could come back from that.” Ultimately, he felt he couldn’t kill himself and leave his family with no context for what he had done, and instead hoped that God would take care of matters for him. “But at the same time, I was still… It would be nice if I got hit by a car or I got really sick,” he said.

More than anyone else I spoke with, Mike seems like he could benefit from having a professional to talk to, and not just because of his proximity to children. I was taken by his urgent need to disclose information others might have a hard time expressing. Late one afternoon we were sitting in his car in the parking lot of a different shopping mall. After hours of conversation, I suggested that we wrap for the day and he flat-out refused, telling me in an uncharacteristically abrupt tone that he had to get it out. We continued speaking until the encroaching shadows finally met and turned to darkness, stopping only when the center’s cleaning staff began arriving for their night’s work.

Later:

Now the largest pedophile support group in the U.S., [Virtuous Paedophiles’] 318 active members are clear in their belief that sex with children is wrong. The founders, Ethan Edwards and Nick Devin (also pseudonyms), both family men with children, enact this policy with tight moderation. If someone is seen to be voicing the opinion that minor sex is acceptable, he gets a warning. Repeat offenders are ousted from the group. The membership list is also restricted to those aged 18 and over, lest they be accused of wrongdoing.

While Adam contributes to discussions there from time to time, his focus remains on the young men who come to his own group for help. James, for one, speaks with a clear reverence for Adam. Though his status as a sex offender means he must attend court-mandated therapy, it is Adam and the others that he credits with helping keep him on the right path. It’s also not lost on him that, for everyone else, it is the only lifeline they have. “If they want help, if they want to be better, to try and fix their behavior and be a better person, he’s never given up on them,” he said. “He didn’t give up on me, he didn’t give up on Mike, he never gave up on any of us.”

You’ve got to look.