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In a peer-to-peer network, nodes are typically required to route packets for each other. This leads to a problem of "free-loaders", nodes that use the network but refuse to route other nodes' packets. In this paper we study ways of designing incentives to discourage free-loading. We model the interactions between nodes as a "random matching game", and describe a simple reputation system that provides incentives for good behavior. Under certain assumptions, we obtain a stable subgame-perfect equilibrium. We use simulations to investigate the robustness of this scheme in the presence of noise and malicious nodes, and we examine some of the design trade-offs. We also evaluate some possible adversarial strategies, and discuss how our results might apply to real peer-to-peer systems.