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A boundedly rational user equilibrium (BRUE) is achieved in a transportation system when all users are satisfied with their current travel choices. The theoretical and behavioral background for such a state is given in this paper. The properties of a BRUE in an idealized commuting system with a single bottleneck are investigated, and conditions for the existence of a BRUE are given. More general situations with multiple bottlenecks are also addressed. In general, BRUE flows are not unique, raising methodological and practical issues in flow prediction.