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Optimal Nash Equilibria for Bandwidth Allocation

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Web and Internet Economics (WINE 2020)

Part of the book series: Lecture Notes in Computer Science ((LNISA,volume 12495))

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Abstract

In bandwidth allocation, competing agents wish to transmit data along paths of links in a network, and each agent’s utility is equal to the minimum bandwidth she receives among all links in her desired path. Recent market mechanisms for this problem have either focused on only Nash welfare  [9], or ignored strategic behavior  [21]. We propose a nonlinear variant of the classic trading post mechanism, and show that for almost the entire family of CES welfare functions (which includes maxmin welfare, Nash welfare, and utilitarian welfare), every Nash equilibrium of our mechanism is optimal. We also prove that fully strategyproof mechanisms for this problem are impossible in general, with the exception of maxmin welfare. More broadly, our work shows that even small modifications (such as allowing nonlinear constraints) can dramatically increase the power of market mechanisms like trading post.

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Notes

  1. 1.

    The price of anarchy  [25] concept applies only when \(\varPsi \) can be written as the maximization of some cardinal function. This is true when \(\varPsi \) denotes Nash welfare maximization, but is not true in general.

  2. 2.

    This happens when there is a good that has large enough supply that is not the “rate limiting factor” for any agent; see Sect. 3 and the full version of the paper  [35] for additional discussion.

  3. 3.

    They study Leontief utilities, which is a generalization of bandwidth allocation to the setting where agents may desire goods in different proportions.

  4. 4.

    A slightly weaker version of continuity is often used: given an allocation \(\mathbf {x}\), the sets \(\{\mathbf {y}: \varPhi (\mathbf {x}) \ge \varPhi (\mathbf {y})\}\) and \(\{\mathbf {y}: \varPhi (\mathbf {x}) \le \varPhi (\mathbf {y})\}\) should be closed. This weaker version only requires a welfare ordering and does not require that this ordering be expressed by a function. However, any such ordering which also satisfies the rest of our axioms is indeed representable by a welfare function  [30], and so both sets of axioms end up specifying the same set of welfare functions/orderings.

  5. 5.

    Without the Pigou-Dalton principle, \(\rho > 1\) is also allowed. This can result in unnatural cases where it is optimal to give one agent everything and the rest none, even when this does not maximize the sum of utilities.

  6. 6.

    This actually does not include maxmin welfare, which obeys weak monotonicity but not strict monotonicity.

  7. 7.

    The reader may notice that for \(\rho = 0\) – which corresponds to Nash welfare – this constraint reduces to the standard linear constraint of \(\sum _j b_{ij} \le 1\), which is what we should expect: we know from  [9] that trading post with the linear constraint leads to good Nash welfare.

  8. 8.

    It is worth noting that the result of [9] holds for Leontief utilities, a generalization of bandwidth allocation utilities.

  9. 9.

    The mechanism for this result is unrelated to trading post: our trading post approach breaks down for both maxmin welfare and utilitarian welfare. This is because \(g_j(x) = q_j x^{1-\rho }\) is not a valid price curve when \(\rho \rightarrow -\infty \) or when \(\rho = 1\).

  10. 10.

    A plethora of other names have been applied to this mechanism as well, including the proportional share mechanism  [18], the Chinese auction  [28], and the Tullock contest in rent seeking  [10].

  11. 11.

    Recently, this topic has garnered significant attention in the computer science community as well (see  [42] for an algorithmic exposition).

  12. 12.

    There is some work treating price-taking market models as strategic games; see e.g., [1, 8, 9].

  13. 13.

    Specifically, an equilibrium is guaranteed to exist as long agent utilities are continuous, quasi-concave, and non-satiated. The full Arrow-Debreu model also allows for agents to enter to market with goods themselves and not only money; the necessary conditions on utilities are slightly more complex in that setting.

  14. 14.

    The conditions for the correspondence between Fisher market equilibria and Nash welfare are slightly stricter than those for market equilibrium existence, but are still quite general. Sufficient criteria were given in [16] and generalized slightly by [22].

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Acknowledgements

This work would not have been possible without my advisor Ashish Goel, who I would like to thank for continued guidance and feedback. This research was supported by NSF Graduate Research Fellowship under grant DGE-1656518.

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Authors and Affiliations

  1. Stanford University, Stanford, USA

    Benjamin Plaut

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  1. Benjamin Plaut
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  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Xujin Chen

  2. Shanghai University of Finance and Economics, Shanghai, China

    Nikolai Gravin

  3. Goethe University Frankfurt, Frankfurt am Main, Germany

    Martin Hoefer

  4. University of Illinois at Urbana-Champaign, Urbana, IL, USA

    Ruta Mehta

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Plaut, B. (2020). Optimal Nash Equilibria for Bandwidth Allocation. In: Chen, X., Gravin, N., Hoefer, M., Mehta, R. (eds) Web and Internet Economics. WINE 2020. Lecture Notes in Computer Science(), vol 12495. Springer, Cham. https://doi.org/10.1007/978-3-030-64946-3_6

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