Tag: game theory

• A little ignorance can be a good thing

  Picture a city where most drivers use the same navigation app. At 9 am, the app says one side street is the quickest shortcut to get from area A to area B. Thousands of commuters accept this option and drive through that street, and soon it becomes jammed with traffic. As a result, the route that was until a few minutes ago a shortcut on the app becomes undesirable, even as nearby streets remain underused.

  Now imagine a small change. Say some drivers don’t fully trust the label “fast”, or they see slightly different estimates, so they spread out across several streets. As a result no single street bears the entire traffic load, and in turn the average travel time (across all commuters) falls even though no road has been widened and no signal has been reconfigured.

  According to a preprint paper uploaded to arXiv in March, this limited ignorance on the drivers’ part effectively reduced congestion created by completely informed yet selfish choices.

  This view of the traffic should be reminiscent of any public space you’ve navigated in India, where seemingly small selfish choices add up to inconveniences that affect many people. Motorists spill into junctions when the signal is on red, with motorcyclists in particular clogging the sidewalk and even blocking the path of oncoming traffic. Commuters rush at bus and train doors rather than queuing in, thus slowing the boarding process as well as rendering existing queues useless. Families often reserve extra seats or rush baggage carousels and stall others. Each act yields a momentary private gain but together they congest and frustrate.

  Curiously, the new study posits that if people who route themselves through a crowded network know a little less, the whole system might work a little better.

  Prior work has already shown that when every user pursues the quickest route for themselves, the system settles into a state that isn’t globally optimal. The study began by quantifying this shortfall with a “price of anarchy”. The authors — all from Ohio State University — also defined a parallel idea that they called the “price of ignorance”, which tracked how outcomes change when users are uncertain about the network’s links.

  Next, the model mixed two kinds of links. “Slow” links had a fixed travel time that didn’t change with traffic. “Fast” links were faster when they were underused but slowed down in a linear fashion as more users pile on. The study’s model then assumed that users didn’t know for sure which kind of link they were facing. Instead, the model continued, they planned their route using a perceived cost that blended the two possibilities. The authors ‘measured’ the users’ ignorance using a single parameter denoted ɑ. If ɑ = 0, the users had no ignorance and complete knowledge; if ɑ = 1, the users were completely ignorant. The more the users were ignorant, the more they’d think all routes are equally suitable.

  The network took the form of a directed square lattice; ‘directed’ means users could only move through it in a fixed way, from left to right. Each link between two points in the lattice was said to be “fast” with probability p and “slow” with probability 1 – p. Users chose those routes that they believed would minimise travel time. The authors evaluated the true average travel time and compared it to the ɑ = 0, and defined the “price of ignorance” as the ratio of the average travel time with and without ignorance.

  The main finding was stark: the authors found that a small amount of ignorance always helped. For all compositions of the network, increasing ignorance from 0 reduced the average travel time up to a particular threshold. The paper proved analytically that for every probability of a lane being “fast” (i.e. for all values of p), any ɑ ≤ 2/3 guaranteed the price of ignorance was PI ≤ 1. That is to say, a limited ‘amount’ of ignorance softened the traffic by redirecting it into more tempting “fast” links. In this scenario, because some users overestimated the “slow” links and “underestimated” the fast ones, the traffic spread more evenly across the network. This reduced congestion on the fast links — which is good.

  The team also found a special sweet spot, so to speak. Say the network has a tipping point between “not enough fast links” and “plenty of fast links”. Near this tipping point, if users’ ignorance is around ⅔, their self-chosen routes spread out in just the right way. The result matches the best possible routing that a planner might pick. Put another way: imagine some links are marked “fast” and others are marked “slow”. If everyone fully trusted the labels, most people would chase the same link and eventually clog it. But if people trusted the labels only partly (as implied by ɑ = ⅔), some would choose alternative links nearby. This small hesitation thus spreads traffic across several routes.

  Alas, ignorance beyond the helpful range eventually starts to hurt. When people have no idea which links are fast or slow, they spread out almost evenly. That sounds fair but it also ‘wastes’ good options. Even in cities with many quicker routes, some travellers drift to slower side streets or paths that fizzle out, so the average travel time rises. This waste grows further in larger networks. Thus there is a sort of separatrix between “helpful doubt” and “harmful cluelessness”. If fast links are scarce, a planner can tolerate more doubt before the network’s performance drops. If fast routes are plentiful, on the other hand, only near-total cluelessness can cause harm. That is, in very large and well-served networks, things go bad only when people are almost completely in the dark.

  So to be clear, ignorance in the study didn’t mean carelessness or lack of effort. Users still knew the map, could see congestion, and chose the route that looked best to them. What they didn’t know for sure was which roads were actually quicker and which were actually slower. And this specific kind of ignorance had two virtues: first, it prevented users from overreacting to “fast” labels and kept a small subset of links from being overloaded; second, users’ ignorance caused different users to make different routing decisions even when some links looked attractive, thus keeping them from clogging these links in a coordinated way.

  These virtues might sound familiar if you’ve been to other parts of physics. In statistical physics, for example, adding noise to a weak signal can help it cross a valuable threshold, a phenomenon called stochastic resonance. If the amount of this noise is just right, it can improve the system’s response. A familiar example is in hearing assistance. Some hearing aids add a soft, random ‘hiss’ under speech. On its own, this hiss is too weak to notice — but for listeners with mild hearing loss, it helps small sound cues, like faint consonants in Hindi, cross the ear’s detection threshold more reliably. Thus speech becomes somewhat more clear at lower volumes or in quiet rooms.

  In ecology and evolution, some seeds have been known to germinate over multiple seasons because they’re not sure which ones are going to be bad. Similarly, with algorithms and machine learning, a bit of uncertainty can make models work better. During training, a program can turn off some parts of the network at random, so a model doesn’t simply memorise patterns from the data. Small, carefully added noise in the training labels can have a similar effect. In reinforcement learning, letting the program try some actions at random can help keep it from getting stuck on a strategy that looks good early but isn’t actually the best.

  And in behavioural game theory, people don’t always pick the mathematically ‘best’ move. They just pick a pretty good move most of the time. This can help alleviate crowding because as a result not everyone chases the same option at once. A similar idea can help in clinics as well: if a sign or app always says “counter 3 is fastest”, for example, everyone might rush there and block it. If instead the app randomly assigned people to different counters, everyone isn’t steered to the same counter.

  The overall lesson isn’t that ignorance is good in itself but that perfect certainty can produce brittle, crowded choices in systems with congestion or competition. A carefully controlled amount of uncertainty can instead spread the load and pull the system as a whole from a state governed by selfish dynamics to one by social optima.

  2025.11.09

• Lighting the way with Parrondo’s paradox

  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.

  2025.09.07

• Solve all our problems

  

  This is xkcd #1232. When it came out I remember it was to rebut a particular line of argument against NASA’s lunar and interplanetary missions — that the agency was spending large sums of money that would be better spent on “solving problems on Earth”. Considering Earth would always have problems, xkcd and others contended, we’d never be able to go to space if we had to spend all our time, money, and labours fixing them. The snark implied in #1232 was warranted.

  But recently, I saw this comic used in a different context: during a conversation (in a private group) about Elon Musk’s aggression with SpaceX and his plans to colonise the moon and visit Mars in his lifetime. Insofar as #1232 pushed back against space exploration that couldn’t by any measure subtract from public spending on socio-economic welfare and justice, it was clever and good. But in the conversation in the group, #1232 donned a new implication: of reducing any other (even minimally) legitimate criticism of the world’s plans to land probes on the moon, establish lunar bases, and start the human campaign to permanently settle the moon and of Elon Musk’s and SpaceX’s plans to being an argument about spending on space exploration subtracting from more immediately measurable pursuits.

  Two arguments come to mind that are poorly served by such flattening. First: the pace at which SpaceX has been manufacturing satellites, launching rockets, and expanding its satellite constellations is at odds with its, and our, ability to deal with the environmental footprint of these activities. Neither SpaceX nor Musk have made any provisions for the activities to be sustainable and they should asap. Doing so might slow the company down, and the company needs to stop considering this retardation to be undesirable. Yet SpaceX’s supporters have often construed any criticism of the company’s pace to be criticism of the company altogether and as the argument that its money would be better spent doing other things.

  Second: I was recently asked a curious question during a formal engagement at work. Is it ethical for India to spend so much on Gaganyaan considering we live in a world with war, violence, and poverty? Gaganyaan has so far cost the Indian government more than Rs 11,000 crore. But there are a couple underlying assumptions here, leading up to questions of the ethicality of human spaceflight, that are flawed.

  (i) The allocation of resources for various activities isn’t a zero-sum game in India. The national budget is voluminous enough for the government to fund both human spaceflight and poverty alleviation programmes. Also unlike in game theory, fractional outcomes are possible and possibly more desirable. For example, India can make great strides in its poverty alleviation programme if it diverts only 0.1% of its defence spending (Rs 6.2 lakh crore in 2024-2025) that way.

  (ii) Many of us like to believe if we don’t spend money on X, it will be available for Y. (Here, X could be ’spaceflight’ and Y could be ‘alleviating poverty’.) We don’t stop to ask whether the state will divert it to Z instead (say, ‘missiles’). If we’d like to guarantee X → Y, we need to persuade the state to rejig its existing priorities and prevent X → Z. Expecting ISRO to not pursue Gaganyaan with funds provided by the state isn’t reasonable.

  In sum, it seems like the “let’s first fix all problems on Earth” argument has become both straw man and red herring in conversations about off-world human activities whose benefits aren’t entirely clear at the moment. The real problem is of course that the benefits aren’t clear, not that the activities are happening at all, plus the belief that money spared by not performing one activity will automatically become available for the precise alternative activity we’re rooting for.

  2024.12.12

• Remembering John Nash, mathematician who unlocked game theory for economics

  The Wire
  May 25, 2015

  The economist and Nobel Laureate Robert Solow once said, “It wasn’t until Nash that game theory came alive for economists.” He was speaking of the work of John Forbes Nash, Jr., a mathematician whose 27-page PhD thesis from 1949 transformed a chapter in mathematics from a novel idea to a powerful tool in economics, business and political science.

  At the time, Nash was only 21, his age a telltale mark of genius that had accompanied and would accompany him for the rest of his life.

  That life was brought to a tragic close on May 23 when his wife Alicia Nash and he were killed in a car-accident at the New Jersey Turnpike. He was 86 and she was 82; they are survived by two children.

  Alicia (née Larde) met Nash when she took an advanced calculus class from him at the Massachusetts Institute of Technology in the mid-1950s. He had received his PhD in 1950 from Princeton University, spent some time as an instructor there and as a consultant at the Rand Corporation, and had moved to MIT in 1951 determined to take on the biggest problems in mathematics.

  Between then and 1959, Nash made a name for himself as possibly one of the greatest mathematicians since Carl Friedrich Gauss. He solved what was until then believed to be an unsolvable problem in geometry dating from the 19th century. He worked on a cryptography machine he’d invented while at Rand and tried to get the NSA to use it. He worked with the Canadian-American mathematician Louis Nirenberg to develop non-linear partial differential equations (in recognition, the duo was awarded the coveted Abel Prize in 2015).

  He made significant advances in the field of number theory and analysis that – in the eyes of other mathematicians – easily overshadowed his work from the previous decade. After Nash was awarded the Nobel Prize for economics in 1994 for transforming the field of game theory, the joke was that he’d won the prize for his most trivial work.

  In 1957, Nash took a break from the Institute for Advanced Studies in Princeton, during which he married Alicia. In 1958, she became pregnant with John Charles Martin Nash. Then, in 1959, misfortune struck when Nash was diagnosed with paranoid schizophrenia. The illness would transform him, his work and the community of his peers in the next 20 years far beyond putting a dent in his professional career – even as it exposed the superhuman commitments of those who stood by him.

  This group included his family, his friends at Princeton and MIT, and the Princeton community at large, even as Nash was as good as dead for the world outside.

  His colleagues were no longer able to understand his work. He stopped publishing papers after 1958. He was committed to psychiatric hospitals many times but treatment didn’t help. Psychoanalysis was still in vogue in the 1950s and 1960s – while it’s been discredited now, its unsurprising inability to get through to Nash ground at people’s hopes. In these trying times, Alicia Nash became a great source of support.

  Although the couple had divorced in 1963, he continued to write her strange letters – while roaming around Europe, while absconding from Princeton to Roanoke (West Virginia), while convinced that the American government was spying on him.

  She later let him live in her house along with their son, paying the bills by working as a computer programmer. Many believe that his eventual remission – in the 1980s – had been the work of Alicia. She had firmly believed that he would feel better if he could live in a quiet, friendly environment, occasionally bumping into old friends, walking familiar walkways in peace. Princeton afforded him just these things.

  The remission was considered miraculous because it was wholly unexpected. The intensity of Nash’s affliction was exacerbated by the genius tag, by how much of Nash’s brilliance the world was being deprived of. And the deprivation in turn served to intensify the sensation of loss, drawing out each day that he was unable to make sense when he spoke, when he worked. John Moore, a mathematician and friend of the Nashes, thought they could have been his most productive years.

  After journalist Sylvia Nasar’s book A Beautiful Mind, and then an Academy-Award-winning movie based on it, his story became a part of popular culture – but the man himself withdrew from society. Ron Howard, who directed the movie, mentions in a 2002 interview that Nash couldn’t remember large chunks of his life from the 1970s.

  While mood disorders like depression strike far more people – and are these days almost commonplace – schizophrenia is more ruthless and debilitating. Even as scientists think it has a firm neurological basis, a perfect cure is yet to be invented because schizophrenia damages a victim’s mind as much as her/his ability to process social stimuli.

  In Nash’s case, his family and friends among the professors of Princeton and MIT protected him from succumbing to his own demons – the voices in his head, the ebb of reason, the tendency to isolate himself, that are altogether often the first step toward suicide in people less cared for. Moreover, Nash’s own work played a role in his illness. He was convinced for a time that a new global government was on the horizon, a probable outcome in game theory that his work had made possible, and tried to give up his American citizenship. As a result, his re-emergence from the two decades of mental torture were as much about escaping the vile grip of irrationality and paranoia as much as regaining a sense of certainty in the face of his mathematics’ enchanting possibilities.

  A Beautiful Mind closes with Nash’s peers at Princeton learning of his being awarded the Nobel in 1994, and walking up to his table to congratulate him. On screen, Russell Crowe smiles the smile of a simple man, a certain man, revealing nothing of the once-brazen virtuosity that had him dashing into classrooms at Princeton just to scribble equations on the boards, dismissing his colleagues’ work, rearing to have a go at the next big thing in science. By then, that brilliance lay firmly trapped within John Nash’s beautiful but unsettled mind. With his death, and that of Alicia, that mind will now always be known and remembered by the brilliant body of work it produced.

  2015.05.25