The Team Orienteering Problem (TOP) generalizes many real-world multi-agent scheduling and routing tasks that occur in autonomous mobility, aerial logistics, and surveillance applications. While many flavors of the TOP exist for planning in multi-agent systems, they assume that all the agents cooperate toward a single objective; therefore, they do not extend to settings when they compete in reward-scarce environments. We propose Stochastic Prize-Collecting Orienteering Games (SPCOG) as an extension of the TOP to plan in the presence of self-interested agents operating on a graph, under energy constraints and stochastic transitions. A theoretical discussion on complete and star graphs establishes that there is a unique pure Nash equilibrium in SPCOGs that coincides with the optimal routing solution of an equivalent TOP under rank-based conflict resolution. We propose the concept of Ordinal Rank (OR) as a concise representation of an agents' global rank and its location within a topological, well-defined neighborhood. Empirical evaluations conducted on real-world, road-network graphs under both dynamic and stationary prize distributions show that in parameter-sharing settings, the policies that leverage local information can outperform those policies leverage global information when the former is conditioned on the OR rather than the global rank, indicating that the OR acts as a strong inductive bias in multi-agent games on graphs. The OR-conditioned policies also generalize much better to games with large number of agents compared to global-rank conditioned policies. Finally, we also propose we propose Fictitious Ordinal Response Learning (FORL) as an entropy-regulated algorithm to obtain convergent policies in independent-learning settings in prize-collecting games on graphs.