It turns out electrons and fish follow the exact same mathematics.
Game theory is a branch of mathematics that considers how groups of players choose a strategy to solve complex problems. When the number of players is large, researchers will model the game with a mean-field approach that averages over the behavior of all the players.
Amazingly, it turns out that mean-field game theory can be translated in terms of the Schrödinger equation — the foundation of quantum mechanics, which is a field of physics that studies the smallest particles in the universe.
According to a team of French physicists, it is possible to translate a huge number of problems in game theory into the language of quantum mechanics. In a new paper, published in Physical Review Letters, they explain how electrons and fish follow the exact same mathematics.
Most people have heard of Schrödinger’s cat — the cat considered both dead and alive inside a box until someone opens it — however, Schrödinger is famous to physicists for being the first to write down an equation that fully describes the weird stuff that happens when you try to do experiments on the fundamental particles of matter.
Schrödinger realized that you can’t describe electrons or atoms as billiard balls that will be exactly where you expect them to be exactly when you expect them to be there. Instead, you need to assume that particles have positions that are spread out in space, and that they only have some probability of appearing where you think they are going to be at any point in time.
Working with these types of probabilities — rather than specific positions — researchers could exactly predict the results of experiments that had puzzled physicists at the start of the twentieth century using Schrödinger’s equation. It tells you the relationship between how these probabilities change in time and the way they change in space.
Now, you may be wondering what any of this has to do with mean-field game theory. Igor Swiecicki, from France’s Laboratoire de Physique Théorique Orsay, used the example of a school of fish.
Generally, the fish move as a single group, with individuals moving around randomly within it. However, every once in awhile, a fish may wander away from the others when it spots a piece of food, but once it grabs the food it swims back to its school for safety.
This means that the fish have some distribution — they are concentrated within the group and are rarer to spot the further away you get from it. In other words, if you pick a particular spot in space, there’s some probability that you chose somewhere with a fish and some probability you chose somewhere without a fish. And as the school swims past your spot, the probability of finding a fish there goes up, but after the fish moves beyond that point, the probability goes down, according to ScienceAlert.
To the researchers’ surprise, the probability of finding a fish changes exactly like the probability of finding an electron does — the fish follow Schrödinger’s equation.
So what does this mean for game theory?
It may advance dramatically since several game theory problems can now be translated into the framework of quantum mechanics. Physicists have been using Schrödinger’s equation to solve extremely complicated problems for nearly 100 years, but since mean-field game theory has only been around for about 10 years, there are still numerous unanswered questions waiting to be solved.
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