Stochastic games with heterogeneous players

In game theory, a fundamental research subject is the analysis of games with homogeneous players, which are identical in terms of their strategy sets, payoffs, and decision-making processes. These models provide valuable insights into collective behaviour and equilibrium dynamics. A prominent example are mean-field games (MFGs), which are particularly useful when analysing systems with a large number of players, such as populations or economic markets. In MFGs, each player interacts with the collective, but their individual influence on the overall system is negligible. By focusing on a representative player, one can often obtain a tractable description of the equilibrium and system behaviour as the number of players approaches infinity.

However, this homogeneity assumption is often unrealistic in many practical scenarios. Real-world players are rarely identical; they may have different preferences, resources, and patterns of interaction. As we move towards models that capture heterogeneity among the players, the complexity of the system increases, but often so does the accuracy of the model.

In stochastic games, where the environment evolves over time and players make decisions based on both current and future states, heterogeneity becomes particularly significant. Unlike homogeneous players, heterogeneous players must consider not only the collective state but also how their individual characteristics, as well as their interactions with others, influence the system's dynamics subject to uncertainty.

A natural way to model interactions between players is through graphs, where each node represents a player, and each edge represents an interaction. This framework gives rise to stochastic games on graphs, where players' decisions are influenced by their neighbours. When players are densely connected, meaning that all players interact with many others, the system begins to resemble the mean-field setting. This dense connectivity allows for the emergence of graphon games, which extend mean-field ideas to settings where players interact through a graphon, i.e. a limiting object of a dense graph sequence.

While the dense regime and graphon games offer a tractable framework for studying large-scale interactions, the sparse regime remains far more challenging and less understood. In the sparse regime, players are connected by relatively few edges, meaning that localized interactions play a much more significant role. Here, the interaction structure is irregular, and individual players can exert significant influence over their neighbours. Unlike the dense regime, where mean-field approximations hold, in the sparse regime, the local network structure becomes critical, and the impact of each player's position in the graph becomes more pronounced.

The aim of this research project is the derivation of explicit equilibria in a class of stochastic games on graphs and graphons, as well as of the convergence of the first to the latter as the number of players approaches infinity. This is done by applying methods from stochastic analysis, variational calculus and operator theory. Further research projects may focus on the less tractable sparse regime.