# Surprising applications of the theory of games?

I am currently studying the applications of games in quantum information theory and related fields and I am aware of its uses in places like model theory and set theory. So I was curious, what are some other somewhat surprising places(somewhat because I am aware of the subjectivity of the term) where games were/are useful tools?

• I presume you want to limit this to applications in mathematics? (Otherwise, there are many applications in politics, economics, finance, ...). Mar 21, 2022 at 13:38
• This related question might be relevant: mathoverflow.net/questions/267051/… Mar 21, 2022 at 14:55
• The existence of a Nash equilibrium in two player games implies Brouwer’s fixed point theorem. Mar 21, 2022 at 17:13
• That MO question and answers are pretty much what I had in mind when I asked, thank you. Also thank you for the references @DanRomik, the second one especially seems interesting Mar 21, 2022 at 18:30
• Schmidt $(\alpha,\beta)$ game and its variants have some fruitful applications to number theory, more specifically in the metric theory of Diophantine approximation Mar 21, 2022 at 23:54

I think evolutionary biology is a major application, if you're accepting answers outside of math. The central notion of an evolutionarily stable strategy (ESS) is a Nash equilibrium.

For applications of game theory to elections, this is a classic reference:

A game-theoretic model of party affiliation of candidates and office holders

We develop a formal model of ambition theory, extending it to account for the choice of party affiliation. We begin by translating the expected utility, “calculus of candidacy” to the choice party affiliation. The model is then used to develop two game-theoretic models of affiliation. The first game models the affiliation decisions of an incumbent and a challenger within a single constituency. Our analysis shows these decisions to be fundamentally interdependent. Switches in affiliation can occur because of shifts in the electoral support for the parties, but also because politicians want to avoid contested primaries. Moving beyond one district, we show how the affiliation decisions of candidates running for different offices or in different districts are also interdependent. The analysis indicates that when electoral strength depends on who runs, politicians affiliated with a decaying political party are involved in a collective-action game.

There is a very popular application of game theory in eXplainable Artificial Intelligence (XAI). The concept of Shapley values is used to attribute a score to components of the input (features) of machine learning models (so-called black boxes).

Perhaps one of the most cited work on this idea is the following: A Unified Approach to Interpreting Model Predictions

As far as I know, the main idea originated from: An Efficient Explanation of Individual Classifications using Game Theory

Generative Adversarial Networks (GAN). are a a huge area in artificial intelligence right now. They are the state of the art in the field of generative modeling- which is the task of generating new samples from some (possibly very complex!) distribution from which you have samples (training data).

For example, this person does not exist is a website containing pictures of (fake!) human faces generated by a GAN. The GAN is generating samples from the distribution over pixel space of human faces. There is a whole class at Stanford on them right now, if you want to learn more.

At a high level, GANs are essentially two neural networks, a generator, and a discriminator, playing a game with each other. The generator is trying to generate samples from an unknown distribution, while the discriminator network is trying to determine whether the generate sample is real training data, or generated by the first model. You can model this as a zero-sum game between the two networks, where one network's gain is the other's loss.

• I think this is very much in line with Surb's response above right? Mar 25, 2022 at 13:39
• Same broad field, but different applications. Mine is a type of generative model- theirs is explaining how models that we already use work (peering inside of the black box). Mar 25, 2022 at 20:56

Games are used to solve the problem of "full abstraction for the lambda-calculus", which should count as fairly surprising.

Spelling it out a bit: a game can define what a computer program means, satisfying some stringent requirements. The problem is stated such that we need a "meaning" that is not syntactic, i.e. it is quite unlike the source program, and is also expected to be unlike the state of a concrete machine that implements it. And furthermore, it is expected that the distinct objects in the semantics (the "meanings") should exactly match the behaviorally distinct programs. It has turned out that over time, games have been the only kind of solution to that puzzle (I believe). Wikipedia has a brief overview: https://en.wikipedia.org/wiki/Game_semantics

• The theory of surreal numbers also has some connection to games: en.wikipedia.org/wiki/Surreal_number#Games Mar 25, 2022 at 6:46
• I haven't heard about the full abstract model problem until now, thank you Mar 25, 2022 at 12:14

To me, the most surprising has been it's application to numerical solutions of PDEs, see here: Owhadi, Houman, and Clint Scovel. Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design. Vol. 35. Cambridge University Press, 2019.

• This sounds very interesting indeed. Mar 25, 2022 at 12:15