Evolutionary Equilibria in Mean Field Games: Theory and Applications to Token Economies
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Modern societies increasingly rely on shared infrastructures such as transportation networks, energy systems, cloud platforms, communication channels, and other resources whose value depends on how many users access them at the same time. When users act selfishly, the resulting allocation is often inefficient from a societal perspective. Classical mechanism-design solutions address this problem through monetary prices or tolls. Yet money can compromise fairness, because access then depends on wealth. Fairness is especially delicate when users belong to different classes with different needs or constraints, since a desirable allocation may intentionally favor those who are worse off. An appealing alternative is to use non-tradable tokens that users earn and spend when accessing resources. Such token economies can promote turn-taking. Users alternate between more and less favorable outcomes, allowing efficiency and fairness to be reconciled over time. This thesis develops the theoretical and design foundations for such token-based mechanisms in dynamic resource allocation problems. The choices to be priced are not necessarily isolated resources; they are combinations of resources that many agents use repeatedly over time. For example, a traveler may choose a path consisting of several road links, and the travel time of that path depends on the congestion of each link. The design problem is therefore to determine token tolls for combinations of resources so that the collective behavior induced by individual choices leads to fair and efficient long-run outcomes. To address this problem, this thesis models token economies as continuous-time stochastic dynamic games with finitely many boundedly rational agents. Each agent has an individual state, namely the number of tokens in their wallet, and repeatedly chooses actions from the set of resource combinations available to satisfy their needs. These decisions are made according to policies, which map individual states to actions. They affect the agents' future states and shape the aggregate congestion experienced by the population. Because real users cannot be expected to know the full game, compute equilibria, or behave as perfectly rational optimizers, the thesis adopts an evolutionary-game-theoretic perspective. Evolutionary models replace strong rationality assumptions by simple myopic revision protocols: rules by which agents occasionally revise their policies based on currently perceived payoffs. This is a natural level of behavioral detail for large populations. Crucially, the relevant question is not only which outcomes are equilibria, but also whether such outcomes can emerge and persist under plausible behavioral dynamics. The first part of the thesis develops a new evolutionary theory for continuous-time finite-state stochastic dynamic games of many players. A mean-field approximation is introduced and shown to approximate the finite-population game with strong guarantees as the population grows. This approximation makes it possible to study the joint state-policy distribution of the population through a deterministic ordinary differential equation induced by simple evolutionary revision rules. The thesis shows that standard equilibrium concepts for this class of games lack an evolutionary interpretation, because they do not allow individual heterogeneity in the policies used by the players. To overcome this limitation, a new solution concept is introduced: the mixed stationary Nash equilibrium. It admits an evolutionary interpretation and exists under mild conditions. Moreover, for broad classes of meaningful revision protocols, mixed stationary Nash equilibria coincide with the rest points of the proposed mean-field evolutionary dynamics. The thesis then studies their evolutionary stability, establishing conditions under which population trajectories approach (or remain close to) the set of mixed stationary Nash equilibria. This gives equilibrium a design-relevant meaning: if a game is constructed so that a desired population state is a mixed stationary Nash equilibrium, the theory provides tools to show that this state can emerge and persist against strategic deviations. The second part of the thesis applies this theory to the design of token economies for fair and efficient dynamic resource allocation in congestion games. Three performance objectives are considered. Intra-class fairness requires users with the same needs to experience the same long-run average reward, independently of individual factors such as wealth. Inter-class fairness concerns how rewards are distributed across different user classes, for instance to favor classes that are worse off. Efficiency aims to maximize the societal utility of the resources, such as the average reward of the whole population or environmental performance. The design problem is then formulated as the choice of token tolls that guarantee intra-class fairness while optimizing a prescribed trade-off between inter-class fairness and efficiency. The thesis shows that intra-class fairness holds for any choice of tolls, establishes existence and essential uniqueness of the mean-field equilibrium resource flows, and shows convergence of the token economy to a neighborhood of equilibrium from any initial condition. Finally, it derives closed-form integer tolls that induce an equilibrium at the desired optimal trade-off between fairness and efficiency. Altogether, this thesis bridges mechanism design, evolutionary game theory, and control engineering to provide a principled framework for dynamic resource allocation without money. Its main contribution is twofold: it develops a new evolutionary equilibrium theory for continuous-time finite-state dynamic games, and it uses this theory to design token economies whose long-run behavior is fair, efficient, and can emerge robustly under boundedly rational behavior.
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