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Values and Derivatives


  • Jean-François Mertens

    (Core, 34 Voie Du Roman Pays, 1348 Louvain-La-Neuve, Belgium)


The diagonal formula in the theory of nonatomic games expresses the idea that the Shapley value of each infinitesimal player is his marginal contribution to the worth of a “perfect sample” of the population of all players, when averaged over all possible sample sizes. The concept of marginal contribution is most easily expressed in terms of derivatives; as a result, the diagonal formula has heretofore only been established for spaces of games that are in an appropriate sense differentiable (such as p NA or p NAD). In this paper we use an averaging process to reinterpret and then prove the diagonal formula for much larger spaces of games, including spaces (like bv 'NA) in which the games cannot be considered differentiable and may even have jumps (e.g., voting games). The new diagonal formula is then used to establish the existence of values on even larger spaces of games, on which it had not previously been known that there exists any operator satisfying the axioms for the Shapley value.

Suggested Citation

  • Jean-François Mertens, 1980. "Values and Derivatives," Mathematics of Operations Research, INFORMS, vol. 5(4), pages 523-552, November.
  • Handle: RePEc:inm:ormoor:v:5:y:1980:i:4:p:523-552

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    References listed on IDEAS

    1. Forges, Francoise, 1992. "Repeated games of incomplete information: Non-zero-sum," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 6, pages 155-177 Elsevier.
    2. Kalai, Ehud & Lehrer, Ehud, 1993. "Rational Learning Leads to Nash Equilibrium," Econometrica, Econometric Society, vol. 61(5), pages 1019-1045, September.
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    6. Bergin, James, 1989. "A characterization of sequential equilibrium strategies in infinitely repeated incomplete information games," Journal of Economic Theory, Elsevier, vol. 47(1), pages 51-65, February.
    7. Shalev Jonathan, 1994. "Nonzero-Sum Two-Person Repeated Games with Incomplete Information and Known-Own Payoffs," Games and Economic Behavior, Elsevier, vol. 7(2), pages 246-259, September.
    8. Sorin, Sylvain, 1999. "Merging, Reputation, and Repeated Games with Incomplete Information," Games and Economic Behavior, Elsevier, vol. 29(1-2), pages 274-308, October.
    9. Zamir, Shmuel, 1992. "Repeated games of incomplete information: Zero-sum," Handbook of Game Theory with Economic Applications,in: R.J. Aumann & S. Hart (ed.), Handbook of Game Theory with Economic Applications, edition 1, volume 1, chapter 5, pages 109-154 Elsevier.
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