Incentive Design for Congestion Games with Unincentivizable Users

Incentives are an effective tool to alter user preferences to promote more efficient group behavior. It is often assumed that these incentives can be levied on any and every user in the system; in many settings, this is not the case. Accordingly, how should a system operator design incentives that only affect a fraction of the users? The network routing literature contains many results showing the effectiveness of monetary taxes to influence self-interested users’ behavior and improve system efficiency. These results typically assume that all users in the network are influenced by incentives; however, this need not be the case if incentives are opted into or if some users do not experience or are unaffected by monetary fees. In this work, we address the problem of designing incentives for populations of users where a fraction of the population is not influenced by incentives in their decision-making process. By focusing on the setting of parallel-network selfish routing problems, surprisingly we find that the tolls that are optimal when the full population is incentivizable remain optimal when only a fraction of the population is incentivizable, though at reduced effectiveness. To measure the impact that the unincentivizable users have on the efficacy of the optimal tolling scheme, we derive worst-case performance bounds in a simple class of networks when only a fraction of the users can be incentivized.