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 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:
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.
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
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.