Life is a game, and everyone (or rather everything) is a player—even bacteria. Players do not even have to be consciously aware that they are playing. This great game might even be woven into the very DNA of all living things.
On November 24, 2021, Magnus Carlsen and Ian Nepomniachtchi sat opposite each other in a showdown for the world chess championship, blatantly disregarding a certain fundamental flaw of the game: the fact that it is either futile or unfair. In other words, we know that it is true that either each player has a strategy available to them that will avoid defeat under any circumstances, or there exists a guaranteed winning method available to only one side—a strategy so potent it will clinch a victory even if the opponent were God himself (assuming he doesn’t use any supernatural tricks). But does this knowledge really detract from the game? Consider soccer: matches also end either in victory for one side or in a tie, yet we don’t label the sport itself as “unfair” or “futile.” The real charm of any game lies in the fact that the outcome depends on the skills and talent of the players; sometimes it’s also luck. The electrifying combination of these three factors is what captivates spectators. This holds true for football, tennis, and bridge. But not for chess.
Before We Play
If chess is unfair, this would mean that, theoretically at least, one could formulate a foolproof method for winning any game—a method that could even be followed by a child, as long as they can read chess notation. Then, even if a Carlsen or Nepomniachtchi were sitting opposite, no matter how valiant a struggle they fought, they would inevitably lose. The child would additionally need to have the ability to choose white or black at the beginning, because such a foolproof method would work only for one particular color. It is suspected that, if such a method existed, it would be for white. Conversely, if chess is a futile game, that means one can devise a strategy that will always avoid defeat for both white and black.
A World Championship match consists of fourteen rounds, in which each player gets seven turns playing white and seven playing black. Imagine if both competitors knew a surefire victory strategy. If chess is a futile game, this knowledge would result in every round ending in a draw. Or, if it’s an unfair game, the player playing the advantageous color—the one for which the winning recipe works—would always win each round. The overall match would also therefore end in a draw, 7 to 7. How is it possible, then, that Magnus nevertheless defeated his opponent 7.5 to 3.5 after drawing the first five games? Could it be that a grandmaster like Nepomniachtchi did not know that there was a method for winning? Or that he simply forgot it?
As the saying goes: in theory, there’s no difference between theory and practice. But in practice, there is. The fact that chess is certainly either a futile or an unfair game is a theorem proven back in 1912 by the German mathematician Ernst Zermelo. His work contributed to the emergence of the relatively new (now just over a century old) branch of mathematics known as game theory.
Mathematical theorems are the bedrock of science. They rest upon mathematical proofs, which consist of sequential logical statements, each following from the ones coming before. This contrasts with fields like biology, sociology, and even physics, where theories frequently get revised or replaced in response to new discoveries. In mathematics, knowledge is definitive and timeless: a theorem is either eternally true or it is simply not true at all.
So, if a foolproof method exists that ensures that every world championship chess match will end in a draw, why did Carlsen nevertheless emerge victorious? The catch is that, while such a method does theoretically exist (of that we are certain), the actual method itself remains unknown. Its discovery would essentially render the game of chess meaningless. Yet, this method might be so complex that the entire universe might not contain enough atoms to construct a device capable of storing it.
Fortunately, not all games are as intricate as chess. Who would not like to earn a few pennies defeating curious passersby at an outdoor fair by challenging them to a very simple game? I am talking about the old Chinese game of nim, which is known to belong to the family of unfair games. To play, you set up, say, four stacks containing varying amounts of tokens in front of your willing (in other words: unsuspecting) volunteer. The players then take turns, removing as many tokens as they wish from any single stack during their turn. The game continues alternately until no tokens are left on the table. The player that finds themselves unable to move (there being no tokens remaining) loses. The number of stacks is arbitrary and can be agreed upon by the players—so you can even give your opponent the illusion of control by letting them decide the stack sizes and who moves first.
In the case of nim, not only do we know that there definitely exists a method whereby one of the players is able to win no matter how well the opponent plays—we also actually know what it is. This strategy was discovered by the mathematician Charles Bouton in 1902. It is simple enough for someone relatively good at doing calculations in their head to apply it in practice but also complicated enough for a layperson not to be able to see through it easily.
Which of the players can use it depends on the distribution of tokens. If we write the number of tokens in each stack as a sum of different powers of two (for example, twelve is eight plus four, or two cubed plus two squared), then we need to check whether any of those powers appears an odd number of times in the set of stacks. If so (e.g., if eight—two cubed—appears in the sum for three different stacks), we can pop open the champagne because it means that we’ve got ourselves a lucky distribution. Bouton’s method states that to win, in the next move we should take away the right number of tokens from one of the stacks so that, when we are finished, each of the powers of two now appears an even number of times. The reliability of this method is confirmed by mathematical theorems. It turns out that a move leading to an even configuration of powers of two is always possible, whereas for the opponent the opposite is true—every move they could make always leads back to a distribution in which one of the powers of two appears an odd number of times. As a result, the opponent will never be able to make the last, winning move, because—if we keep following the method—they will never reach a state where each of the powers of two appears an even number of times (as is the case with zero), so they have no chance of collecting all the tokens. And so, they will always eventually lose.
A potential problem arises when we allow the opponent to choose the initial configuration of tokens and to play first—because they may find themselves in a situation where one of the powers of two is odd, opening up for them the strategy ensuring victory. However, since the opponent most likely does not realize this, they are highly likely to make a mistake and bring about a distribution that instead guarantees us the win. So, if we know the mathematical method but the opponent does not, that leverage makes our probability of winning very high.
The Prisoner’s Dilemma
Game theory does not, however, deal exclusively with the theoretical underpinnings of recreational games such as these. On the contrary: its development has been propelled by important applications in economics. Significantly, one of the most famous precursors of game theory, John Nash, received a Nobel Prize precisely in the field of economics—rather than the Fields Medal, which is considered the equivalent of the Nobel Prize in mathematics.
Naturally, not all human decisions are susceptible to quantitative analysis allowing us to predict the future. But it is nevertheless astounding how many of our decisions can indeed be modeled using relatively simple types of games, in which appropriate actions yield specific numerically determined rewards or punishments. In the simplest type of such games, the sum of rewards obtained by all players always equals zero—as in the case of poker (the winner’s gain equals the sum of the opponents’ losses).
It turns out that when individual participants can choose from a variety of strategies, each of which is associated with specific rewards or punishments, there always exists a certain configuration of those strategies such that it does not pay for any of the players to change tactics. This is known as an equilibrium, and it does not have to be beneficial for all the participants—in fact, in the case of a zero-sum game, it certainly will not be. Interestingly, however, in a non-zero-sum game, an equilibrium point can actually be unfavorable for all players at once, even when another possible arrangement exists that is more beneficial for each of the players.
At first glance, it seems difficult to imagine a situation getting stuck at a point where everyone incurs a loss (or fails to gain maximal benefit), and yet any attempt to break free from this folly results in even greater losses. However, such a configuration is possible. The most famous example of such a situation is known as the prisoner’s dilemma.
Let’s say that a certain pair of criminals awaiting trial were proven to have broken into a store, for which they face a two-year prison sentence. However, they were not proven to have stolen the safe, which disappeared from the back room. If one of the accused would just reveal where it is hidden, a large sum of money could be recovered.
The prosecutor decides to interrogate the detainees separately and offers each of them a deal: if he decides to cooperate and tell them where the safe is, his sentence will be reduced to just one year. But in such a case, if his accomplice does not cooperate, the latter will get a sentence of five years in prison, as there will be grounds to convict him of a more serious crime.
But what if both of them own up to having stolen the safe? The prosecutor cannot reduce the sentence for both, as that would lead to the absurd situation where the criminals receive lower sentences for confessing to a greater crime than they were originally proven to have committed. Nor can he sentence both to five years in prison, as that would take away their motivation to confess. Ultimately, the prosecutor decides that if both suspects confess, they will each serve four years. So what will the detainees decide to do?
It would be most beneficial, overall, for both to keep their mouths shut. But by choosing not to cooperate with the prosecutor, each of runs the risk that the other detainee will betray them. Then, they will end up with not two years in prison, but five. This is where game theory comes in, which in this case suggests that the situation in which both of them confess is the equilibrium—even though that entails an additional two years behind bars for each of them. Empirical research shows that in similar situations, the “players” of such a game most often choose to admit their guilt, even though such a solution is by no means optimal overall. It would be best if both kept quiet—they would each be sentenced to two years. The lack of certainty that the accomplice will act the same way, however, makes silence too risky an option.
But what if these same criminals, after serving their time, one day found themselves in such a situation again? After the first sentencing, they would already know whether their colleague is prone to betrayal or tends to remain loyal. Perhaps they would be wiser the second time around? And what if they found themselves in such a situation ten times, and both always kept silent, but then the eleventh time one of them snitched? Is there a chance that on the twelfth they would go back to sticking together? And what if these are not two detainees at a police station but rather two strains of bacteria living in a symbiosis based solely on a certain kind of “trust”? Could bacteria be aware of being involved in such a game, and on what basis would they “decide” upon a strategy? Of course, bacteria do not actually know they are playing anything. But it doesn’t matter—to play, one doesn’t even have to be consciously aware of it.
The Foundation of the Universe
One of the most fascinating successes of game theory is that it proves to be applicable well beyond the realm of human behaviors or corporate strategies; it also applies to non-human communities. It turns out that game theory can describe, with surprising accuracy, the behaviors of animals and even single-cell organisms. And so, yes, everyone and everything is playing a game! The simple “prisoner’s dilemma” scheme sometimes accurately explains the behavior of entire ecosystems. Bacteria of one species either remain loyal, continuing to hold up their end of the symbiotic “agreement,” or they don’t, reaping benefits from the contribution of a partner species while remaining passive themselves. In such a situation, an equilibrium would be reached if both parties stopped contributing—but that would not be beneficial for either of them. Bacterial strains naturally inclined to cooperate have had an edge in survival, and so cooperation has become quite common in nature.
Loyalty can sometimes lead to extraordinary effects. It is hypothesized that eukaryotic cells (those with a chromosome-containing nucleus)—which are the basic building blocks of all complex living organisms, including humans—originated from a symbiotic relationship between aerobic and anaerobic bacteria. But, at some point, the former engulfed the latter. This theory, known as endosymbiosis, suggests that one of the most pivotal evolutionary episodes in Earth’s history, occurring during the Mesoproterozoic era about 1.3 billion years ago, may have been a result of these ancient symbiotic “games.”
From two children playing tic-tac-toe, to Magnus Carlsen’s masterful chess gambits, to the strategic maneuvers of corporations and the blind competition among bacterial strains—it’s evident that game-playing pervades the living world, with stakes ranging from simple pleasure to survival. Game-playing seems to be a thoroughly human pastime, but the fact that it underpins so many process that have little to do with either entertainment or humans prompts a fascinating question: Does its ubiquity perhaps stem from some fundamental biological principles, or perhaps even from deeper, more elemental laws of physics? Alternatively, could it be that our scientific understanding is flawed, suggesting that it’s not really the laws of physics that underpin the workings of the universe but something else entirely?
“This is a sore point, an Achilles heel in the Physics of the present Universum. The microworld currently is the main arena of the Players’ construction activity. […] They are making revisions, they are putting laws now moribund back into service. This is the reason they maintain their silence, which is a ‘strategic quiet’. They inform none of the ‘outsiders’ of what they are doing, or even of the very fact of the Game. A knowledge of the existence of the Game, after all, places all of Physics in an altogether different light.” This is an excerpt from a speech by Professor Alfred Testa, a Nobel Prize laureate, in which he lays out the mechanisms governing the universe. His perspective may be considered unconventional, radical, even iconoclastic within the realm of contemporary science. Testa suggests that game theory’s influence is not just widespread but all-pervasive, to the extent that its architects, including Testa himself, seem to essentially grant it a monopoly on depicting reality—encompassing not just living entities but also the inanimate world. In his view, physics is merely a secondary structure, built upon a more fundamental framework of game theory.
However, if a curious reader ventures to verify exactly when and where this speech was given and to find Professor Testa listed among the Nobel laureates, they’ll discover no record of him. The reason is simple: Professor Testa is a figment of the imagination, a character from Polish science-fiction writer Stanisław Lem’s short story “The New Cosmogony.” Lem was known for such unexpected maneuvers—just like Magnus Carlsen. In 2023, in a move that caught many off-guard, Carlsen withdrew from the world championships, a tournament he had dominated since 2013. The chess community was abuzz with anticipation for his rematch with Ian Nepomniachtchi, making his withdrawal all the more disappointing. Carlsen’s decision, shrouded in mystery, serves as a reminder that sometimes the act of withdrawing from playing—without revealing one’s motives—can itself be a strategic part of the game.
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