## Abstract

In the traditional maximum-flow problem, the goal is to transfer maximum flow in a network by directing, in each vertex in the network, incoming flow to outgoing edges. The problem corresponds to settings in which a central authority has control on all vertices of the network. Today’s computing environment, however, involves systems with no central authority. In particular, in many applications of flow networks, the vertices correspond to decision-points controlled by different and selfish entities. For example, in communication networks, routers may belong to different companies, with different destination objectives. This suggests that the maximum-flow problem should be revisited, and examined from a game-theoretic perspective. We introduce and study *multi-player flow games* (MFGs, for short). Essentially, the vertices of an MFG are partitioned among the players, and a player that owns a vertex directs the flow that reaches it. Each player has a different target vertex, and the objective of each player is to maximize the flow that reaches her target vertex. We study the stability of MFGs and show that, unfortunately, an MFG need not have a Nash Equilibrium. Moreover, the price of anarchy and even the price of stability of MFGs are unbounded. That is, the reduction in the flow due to selfish behavior is unbounded. We study the problem of deciding whether a given MFG has a Nash Equilibrium and show that it is \(\Sigma _2^P\)-complete, as well as the problem of finding optimal strategies for the players (that is, best-response moves), which we show to be NP-complete. We continue with some good news and consider a variant of MFGs in which flow may be swallowed. For example, when routers in a communication network may drop messages. We show that, surprisingly, while this model seems to incentivize selfish behavior, a Nash Equilibrium that achieves the maximum flow always exists, and can be found in polynomial time. Finally, we consider MFGs in which the strategies of the players may use non-integral flows, which we show to be stronger.

## Keywords

Game theory for practical applications Noncooperative games: computation Methodologies for agent-based systems Noncooperative games: theory and analysis## Notes

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