| dc.description.abstract | In many scenarios, players exhibit inherent limitations in various aspects of their capability to generate maximally rational play in strategic games. Modeling such capability limitations and elucidating their implications will advance our understanding of the strategic interactions among players. In this thesis, I study two novel settings where players have limited capabilities. I formalize a hierarchy of capabilities and study related equilibrium concepts, computational complexity, solution algorithms, and the impact of varying capabilities on game outcomes.
The first limited-capability setting is limited-perception games. I focus on a class of oneshot limited-perception games. Such games extend simultaneous-move normal-form games by presenting each player with an individualized perception of the true game. Players’ payoffs are determined by the true game hidden from players. The accuracy of a player’s perception is determined by the player’s capability level, with a higher level corresponding to a more accurate perception. I study both capability-oblivious and capability-aware players. A capability-oblivious player does not know they have limited perception and therefore plays the optimal strategy of their perceived game. I present payoff bounds and other predictable behavior of capability-oblivious players in a special class of limited-perception games. A capability-aware player reasons with the set of possible true payoff functions and other players’ perceptions and incentives to maximize their own objective (e.g., the worst-case payoff) based on their limited perception. I present novel formalizations of simultaneousmove equilibria and show the hardness of equilibrium solving. I further present positive results that (i) an approximate equilibrium has a compact, tractable representation; and (ii) a few classes of zero-sum games can be efficiently solved.
The aforementioned efficiently solvable zero-sum games are reduced to solving nonsmooth convex programs. To this end, I present the Trust Region Adversarial Functional Subdifferential (TRAFS) algorithm for constrained optimization of unstructured nonsmooth convex Lipschitz functions. Unlike previous methods that assume a subgradient oracle model, I propose the functional subdifferential, defined as a set of subgradients that simultaneously captures sufficient local information for effective minimization while being easy to compute for a wide range of functions. Intriguingly, the TRAFS design also incorporates game-theoretical thinking. In each iteration, TRAFS solves a zero-sum game between the optimizer and a local approximation of the objective function to guarantee progress. The optimizer has access to step vectors in a local ℓ2 -bounded trust region; the local approximation uses the functional subdifferential. TRAFS finds an approximate solution with an absolute error up to ϵ in O(1/ϵ) or O(\sqrt{1/ϵ}) 1/ϵ iterations depending on whether the objective function is strongly convex, improving the previously best-known bounds of O((1/ϵ)^2) and O(1/ϵ) in these settings. TRAFS makes faster progress if the functional subdifferential satisfies a locally quadratic property; as a corollary, TRAFS achieves linear convergence (i.e., O(log 1/ϵ)) for strongly convex smooth functions. In the numerical experiments, TRAFS solves twice as many problems compared to the second-best method and is on average 39.1x faster on problems solved by both methods.
The second limited-capability setting is limited-strategy games where a player’s capability limits the strategies available to them. I work with a formalization where a player’s strategy space is defined as programs in a Domain-Specific Language (DSL). A player’s capability limits the size of programs available to that player. I focus on characterizing the impact of player capability on game outcomes. I study a new game model called McDncDa derived from network congestion games. I show that it is computationally hard to determine whether an McDncDa instance is capability-positive (i.e., whether increasing a player’s capability level leads to a better payoff). I then study a parameterized special class of McDncDa called MGMG. I show that MGMG is always capability-positive, and it is socially capabilitypositive (i.e., the sum of all players’ payoffs always gets better if every player’s capability level is increased by one) if some resources in the game have increasing returns to scale despite the existence of multiple equilibria. | |
