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Nash-implementation of competitive equilibria via a bounded mechanism

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Abstract

We deal with the problem of providing incentives for the implementation of competitive outcomes in a pure-exchange economy with finitely many households. We construct a feasible price-quantity mechanism, which fully implements Walras equilibria via Nash equilibria in fairly general environments. Traders’ preferences need neither to be ordered nor continuous. In addition, the mechanism is such that no pure strategy is weakly dominated, hence is bounded (in the sense of Jackson 1992). In particular it makes no use of any integer game.

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Notes

  1. For an introduction to strategic market game literature in imperfectly competitive environments, see Giraud (2003). For a survey, see Dubey (1994).

  2. See for instance Stahn (1996, 1998, 1999).

  3. See Simon et al. (1995) for an extension of the previous ideas to games with infinitely many strategies, as well as Weyers (2003) for an use of the iterated elimination of weakly dominated strategies in strategic market games.

  4. See, e.g., Jackson et al. (1994), Ohseto (1994) and Sjöström (1994).

  5. This restriction is standard, see, e.g., Hurwicz (1979), Schmeidler (1980), Postlewaite and Wettstein (1989), and harmless if one thinks of current markets.

  6. A market selling order obtains whenever \(z_{j,\ell }^{i}-z_{i,\ell }^{j}<0\) and \(p_{\ell }^{i}=0\). Since the set of conceivable prices is unbounded, there is no market buying order in our mechanism (the same restriction is encountered in Mertens (2003), but for different reasons). This is harmless, since no rational player would ever be willing to play such a risky strategy, while, in ‘real’ financial markets, practical constraints provide natural upper bounds for the buying prices, so that every buying order is de facto a limit price order.

  7. Similar punishments rules are used in Dubey (1982), Peck et al. (1992) or Weyers (1999) for instance.

  8. In Weyers (1999), a double round of elimination of weakly dominated strategies is needed in order to get rid of the autarkic Nash equilibrium.

  9. This condition is known to be necessary for the Maskin-monotonicity of the Walrasian correspondence (see Maskin (1985)), hence for its Nash-implementability. In the conclusion of that paper, we briefly discuss how to get rid of this condition by replacing the Walrasian correspondence by the correspondence of constrained Walrasian equilibria.

  10. For convenience, we use here the \(\left\Vert\cdot \right\Vert_{\infty }\) to define open balls \(B_{\varepsilon }(x)\) of radius \(\varepsilon \) around \(x\).

  11. Koutsougeras (2003) showed that this is not the case in Shapley-Shubik games with multiple trading-posts: to be more precise, consider a Shapley-Shubik game with several trading-posts for the same commodity (or, equivalently, with one trading-post per commodity but perfectly substitutable commodities). It may happen that the same commodity is priced differently at different trading-posts at some active NE, so that the “law of one price” does not hold. Otherwise stated, the number of trading-posts per commodity influences the NE prices ! In our framework, we assumed that there is one trading-post per individual, but this restriction is immaterial at every active NE, since the same commodity will get the same price at each open trading-post.

  12. Remember that \(N\ge 3\).

  13. \(H_{i_{0}}^{2}\) is the set of commodities for which agent \(i_{0}\) is the only one who quotes the Walrasian price. When \(i_{0}\) is identical to \(i_{\ell }^{+}\) or \(i_{\ell }^{-}\), one observes that \(\ell \notin H_{i_{\ell }^{+}}^{2}\) and \(\ell \notin H_{i_{\ell }^{-}}^{2}\). This means, otherwise stated, that this deviation with respect to commodity \(\ell \) does neither affect \(i_{\ell }^{+}\) nor \(i_{\ell }^{-}\).

  14. For examples of Nash-implementation of this latter correspondence, see Postlewaite and Wettstein (1989) or Wettstein (1995). Observe that every CE is a constrained competitive equilibrium, every interior CCE is a CE, and the First Welfare Theorem applies to CCE as well.

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Acknowledgments

We thanks an anonymous referee and an associated editor for very helpful comments on an early version of this paper. As usually, the reminding errors remain ours.

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Authors and Affiliations

  1. C.E.S., Université de Paris I Panthéon-Sorbonne, Maison des Sciences Economiques, 106-112, Boulevard de l’Hôpital, 75 647, Paris Cedex 13, France

    Gael Giraud

  2. Aix-Marseille University (Aix-Marseille School of Economics), CNRS & EHESS, Château Lafarge, Route des Milles, 13 290, Les Milles, France

    Hubert Stahn

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  1. Gael Giraud
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Correspondence to Hubert Stahn.

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Giraud, G., Stahn, H. Nash-implementation of competitive equilibria via a bounded mechanism. Rev Econ Design 17, 43–62 (2013). https://doi.org/10.1007/s10058-012-0138-2

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