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Selfish Routing and Proportional Resource Allocation

A Joint Bound on the Loss of Optimality

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Abstract

We consider two optimization problems, multicommodity flow and resource allocation, from a game-theoretic point of view. We show that two known bounds on the respective losses of optimality can be derived from the same geometric quantity, which yields a simple proof of either result.

Keywords

  • Nash Equilibrium
  • Equilibrium Allocation
  • Equilibrium Flow
  • Feasible Allocation
  • Large Rectangle

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Fig. 1
Fig. 2

Notes

  1. 1.

    Note that in (1) and (2), the cost of each path and arc, respectively, is fixed in advance according to its load experienced in some equilibrium. Such equilibrium costs are, in fact, unique, even if instances admit more than one equilibrium flow.

  2. 2.

    Note that this allocation cannot arise from an equilibrium; only the bid of Player 1 would be positive, who could bid less and still receive the entire resource.

  3. 3.

    Similarly to the case of uncoordinated routing in Sect. 2, the equilibrium allocation is known to be unique.

  4. 4.

    For (7), we also use that \(x_i \le 1\) and that \(U_i\) is non-negative for all i, which implies that \(\sum _{i=1}^n U_i(x_i) x_i \le \sum _{i=1}^n U_i(x_i)\).

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Schulz, A.S. (2015). Selfish Routing and Proportional Resource Allocation. In: Schulz, A., Skutella, M., Stiller, S., Wagner, D. (eds) Gems of Combinatorial Optimization and Graph Algorithms . Springer, Cham. https://doi.org/10.1007/978-3-319-24971-1_9

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