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Implementing efficient graphs in connection networks

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Abstract

We consider the problem of sharing the cost of a network that meets the connection demands of a set of agents. The agents simultaneously choose paths in the network connecting their demand nodes. A mechanism splits the total cost of the network formed among the participants. We introduce two new properties of implementation. The first property, Pareto Nash implementation (PNI), requires that the efficient outcome always be implemented in a Nash equilibrium and that the efficient outcome Pareto dominates any other Nash equilibrium. The average cost mechanism and other asymmetric variations are the only mechanisms that meet PNI. These mechanisms are also characterized under strong Nash implementation. The second property, weakly Pareto Nash implementation (WPNI), requires that the least inefficient equilibrium Pareto dominates any other equilibrium. The egalitarian mechanism (EG) and other asymmetric variations are the only mechanisms that meet WPNI and individual rationality. EG minimizes the price of stability across all individually rational mechanisms.

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Notes

  1. This framework can be considered as a benchmark to more general problems where agents get different utility from the path choice, for instance, links facing congestion and idiosyncratic intensities of use of a link/path by agents, among others.

  2. Here efficient outcome is the outcome that maximizes the sum of utilities of the agents. In our framework, this is equivalent to the outcome that minimizes the sum of costs, that is, the cost of the network fulfilling all the connection demands. This notion of efficiency is different from other optimality notions studied in the literature, such as Myerson (1981), Ulku (2012).

  3. See for instance Moulin (2008) for a comparison of three cost-sharing mechanisms using the price of anarchy. See Juarez (2008) for a comparison of two mechanisms in the problem of commons using the worst-absolute surplus loss.

  4. Notice that the problem of implementation we consider here differs from the one considered in the traditional literature since there is no private information on the part of agents. However, it is the same problem in the sense that the planner has an objective function (here efficiency) and the cost-sharing mechanism induces a game whose equilibrium is the outcome obtained.

  5. Apart from the property of being immune to unilateral deviation, the Pareto optimal NE is also immune to deviation by the grand coalition. Also, pre-play communication leads to the payoff-dominant Pareto optimal NE in many games. See for instance Calcagno and Lovo (2010), Cooper et al. (1992), Kim (1996).

  6. Note that contrary to PNI, WPNI may not implement the best possible outcome. The definition of WPNI does not rule out the existence of another mechanism whose equilibria (possibly not Pareto ranked) are less inefficient than the equilibria of a WPNI mechanism. However, in our model, as we will see in the results, the mechanisms characterized under WPNI are less inefficient than any possible mechanism that satisfies individual rationality.

  7. In this example, the cost of a link in the network is proportional to the distance between the nodes that this link connects.

  8. The choice of path is not a strategy for the telephone user, and thus, the setting is not exactly the same. But, the cost-sharing mechanism has a similar motivation; namely, it is simpler than charging every caller differently based on the path used.

  9. The Shapley mechanism even though it looks like a natural mechanism in this setting, fails basic tests such as efficiency, symmetry at equilibrium and continuity. It also does not satisfy minimal information, since the cost-share of an agent depends on the number of users of his demanded links.

  10. We use the word strategy to be consistent with the game that will be defined in Sect. 2.1.

  11. Continuous with the Euclidean distance as a function of the costs in the network.

  12. It is particularly crucial in the proofs of the key Separability Lemma (Lemma 4) and the proofs of Theorems 1 and 2.

  13. The reader should not confuse this definition with other definitions of efficiency irrespective of incentive-compatibility considerations.

  14. Note that this property and some others discussed in this section are true not just for the network cost-sharing problems but also for other cost-sharing problems. However, as we will see soon, in the network cost-sharing framework, this property together with symmetry characterizes the AC mechanism. Indeed, all our major characterization results use networks in an essential sense.

  15. See Monderer and Shapley (1996a, b) for convergence of fictitious play and best response (br) dynamics. See Sandholm (2001) for more general dynamics.

  16. SNI is more demanding than the original strong NE (Aumann 1959) that requires strict improvements by all agents involved in the deviation.

  17. Notice that there could be more than one solution \(\lambda ^*\) to the system; however, they will give the same cost-shares to the agents.

  18. See Young (1987) for a characterization of the symmetric parametric rules; see Moulin (2002) for a more detailed description of the rules.

  19. Surprisingly, in both cases, the traditional br tatonnement, where at every step an agent picks the path that minimizes his cost-share, starting from some profiles will converge to a NE.

  20. The case of weak inequality follows by continuity of the mechanism.

  21. Notice that we can do that since all expressions are positive.

  22. It is important to note that just one such configuration is enough, since PoS is a measure of the performance of the best NE in the worst case example.

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Authors and Affiliations

  1. Department of Economics, University of Hawaii, 2424 Maile Way, Saunders Hall 542, Honolulu, HI, 96822, USA

    Ruben Juarez

  2. School of Computing and Information Sciences, Florida International University, Miami, FL, 33199, USA

    Rajnish Kumar

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  1. Ruben Juarez
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  2. Rajnish Kumar
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Correspondence to Ruben Juarez.

Additional information

We would like to thank our advisor, Hervé Moulin, for his invaluable insights at the various stages of this paper. This paper significantly benefited from the detailed comments of two referees and the editor. Critical comments from Gaurab Aryal, Gustavo Bergantinos, Anna Bogomolnaia, Francis Bloch, Anirban Kar, Tim Roughgarden, Sudipta Sarangi, Arunava Sen, and Katya Sherstyuk have been helpful. Ruben Juarez acknowledges the financial support from the Air Force Office of Scientific Research, Young Investigator Program, under grant FA9550-11-1-0173.

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Juarez, R., Kumar, R. Implementing efficient graphs in connection networks. Econ Theory 54, 359–403 (2013). https://doi.org/10.1007/s00199-012-0720-4

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