  Within mobility systems, the presence of self-interested users can lead to
aggregate routing patterns that are far from the societal optimum which could
be achieved by centrally controlling the users' choices. In this paper, we
design a fair incentive mechanism to steer the selfish behavior of the users to
align with the societally optimal aggregate routing. The proposed mechanism is
based on an artificial currency that cannot be traded or bought, but only spent
or received when traveling. Specifically, we consider a parallel-arc network
with a single origin and destination node within a repeated game setting
whereby each user chooses from one of the available arcs to reach their
destination on a daily basis. In this framework, taking faster routes comes at
a cost, whereas taking slower routes is incentivized by a reward. The users are
thus playing against their future selves when choosing their present actions.
To capture this complex behavior, we assume the users to be rational and to
minimize an urgency-weighted combination of their immediate and future
discomfort. To design the optimal pricing, we first derive a closed-form
expression for the best individual response strategy. Second, we formulate the
pricing design problem for each arc to achieve the societally optimal aggregate
flows, and reformulate it so that it can be solved with gradient-free
optimization methods. Our numerical simulations show that it is possible to
achieve a near-optimal routing whilst significantly reducing the users'
perceived discomfort when compared to a centralized optimal but urgency-unaware
policy.
