Urban Driving Games With Lexicographic Preferences and Socially Efficient Nash Equilibria

Abstract

We describe Urban Driving Games (UDGs) as a particular class of differential games that model the interactions and incentives of the urban driving task. The drivers possess a "communal" interest, such as not colliding with each other, but are also self-interested in fulfilling traffic rules and personal objectives. Subject to their physical dynamics, the preference of the agents is expressed via a lexicographic relation that puts as first priority the shared objective of not colliding. Under mild assumptions, we show that communal UDGs have the structure of a lexicographic ordinal potential game which allows us to prove several interesting properties. Namely, socially efficient equilibria can be found by solving a single (lexicographic) optimal control problem and iterated best response schemes have desirable convergence guarantees.

Posetal Games: Efficiency, Existence, and Refinement of Equilibria in Games with Prioritized Metrics

“…Additionally, motivated by the complexity of traffic rules and the need for a transparent and interpretable system, another body of literature looked at techniques to specify objectives in a prioritized fashion [2], [15]- [17], leading to practical results [18].…”

“…a) Jointly Communal Objectives: We consider the idea of players not having contrasting objectives by extending the idea of communal metrics [17,Def. 4] to arbitrary pairs of metrics.…”

“…Note that Eq. ( 4) is not restrictive for many common cost functions used in mobile robotics, including, among others, collision costs, clearance objectives (i.e., keeping a minimum safety distance from obstacles) [17]. Moreover, any pair of personal metrics (i.e., metrics whose outcome depends on the single player strategy only) is trivially jointly communal.…”

“…To ease the reading, in the following we denote with m ∈ M the joint metrics, opposed to the personal ones, as the metrics that depend on the actions and states of many players [17]. For instance, the time taken by an agent to cross an intersection can be computed as a function of its own trajectory only.…”

“…The same reasoning applies for the strict counterpart assuming Pot(γ ) is a minimum element. Note that the existence of such potential allows one to additionally prove the convergence of widely used algorithms, such as iterated strictly-better response schemes as in [17].…”