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We study a new variation of the Traveling Salesman Problem (TSP) called the Budget-Constrained Traveling Salesman Problem (BC-TSP). BC-TSP is inspired by a few emerging network applications, such as robotic sensor networks. We design a prize-driven multi-agent reinforcement learning (MARL) framework to solve the BC-TSP. The main novelty of the framework, named P-MARL, is that it makes a connection between the prize maximization in BC-TSP and the cumulative reward maximization in reinforcement learning (RL) to design a more efficient MARL algorithm. In particular, P-MARL integrates the prizes available at nodes into the reward model of the MARL to guide the cooperative effort of multiple learning agents. Via extensive simulations using synthetic data of state capital cities of the U.S., we show that a) the P-MARL outperforms the existing prize-oblivious MARL work by collecting 28.8 % of more prizes under the same budget constraints, b) it takes two orders of magnitudes of shorter training time than the state-of-the-art deep reinforcement learning-based approach while collecting 45.3 % more prizes under the same budgets, and c) P-MARL collects prizes at least 91.9% of optimal obtained by the Integer Linear Programming (ILP) under different network parameters.