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Genericity with Infinitely Many Parameters

Author

Listed:
  • Anderson Robert M.

    () (UC Berkeley)

  • Zame William R.

    () (UCLA)

Abstract

Genericity analysis is widely used to show that desirable properties that fail in certain "knife-edge" economic situations nonetheless obtain in "typical" situations. For finite-dimensional spaces of parameters, the usual notion of genericity is full Lebesgue measure. For infinite dimensional spaces of parameters (for instance, the space of preferences on a finite-dimensional commodity space, no analogue of Lebesgue measure is available; the lack of such an analogue has prompted the use of less compelling topological notions of genericity. Christensen (1974) and Hunt, Sauer and Yorke (1992) have proposed a measure-theoretic notion of genericity, which Hunt, Sauer and Yorke call prevalence, which coincides with full Lebesgue measure in Euclidean space and which extends to infinite-dimensional vector spaces. This notion is not directly applicable in most economic settings because the relevant parameter sets are small subsets of vector spaces -- especially cones or order intervals -- not vector spaces themselves. We adapt the notion to economically relevant environments by defining two notions of prevalence relative to a convex set in a topological vector space. The first notion is very easy to understand and apply, and has all of the properties one would desire except that it is not closed under countable unions; the second notion contains the first and has all the good properties of the first notion except simplicity; it is closed under countable unions. We provide four economic applications: 1) generic existence of equilibrium in financial models, 2) generic finiteness of the number of pure strategy Nash equilibria and Pareto inefficiency of "non-vertex" Nash equilibria for games with a continuum of actions and smooth payoffs, 3) generic regularity of exchange economies when some agents are constrained to have 0 endowment of some goods, 4) generic single-valuedness of the core of transferable utility games.

Suggested Citation

  • Anderson Robert M. & Zame William R., 2001. "Genericity with Infinitely Many Parameters," The B.E. Journal of Theoretical Economics, De Gruyter, vol. 1(1), pages 1-64, February.
  • Handle: RePEc:bpj:bejtec:v:advances.1:y:2001:i:1:n:1
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    Citations

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    Cited by:

    1. Chris Shannon & William R. Zame, 2002. "Quadratic Concavity and Determinacy of Equilibrium," Econometrica, Econometric Society, pages 631-662.
    2. Attar, Andrea & Piaser, Gwenael & Porteiro, Nicolas, 2007. "A note on Common Agency models of moral hazard," Economics Letters, Elsevier, vol. 95(2), pages 278-284, May.
    3. Aviad Heifetz & Zvika Neeman, 2006. "On the Generic (Im)Possibility of Full Surplus Extraction in Mechanism Design," Econometrica, Econometric Society, vol. 74(1), pages 213-233, January.
    4. Dirk Bergemann & Stephen Morris, 2005. "Robust Mechanism Design," Econometrica, Econometric Society, vol. 73(6), pages 1771-1813, November.
    5. John W. Patty, 2005. "Generic Difference of Expected Vote Share and Probability of Victory Maximization in Simple Plurality Elections with Probabilistic Voters," Public Economics 0502006, EconWPA.
    6. Riedel, Frank, 2005. "Generic determinacy of equilibria with local substitution," Journal of Mathematical Economics, Elsevier, vol. 41(4-5), pages 603-616, August.
    7. Fang, Fang & Stinchcombe, Maxwell B. & Whinston, Andrew B., 2010. "Proper scoring rules with arbitrary value functions," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1200-1210, November.
    8. Heifetz, Aviad & Shannon, Chris & Spiegel, Yossi, 2007. "What to maximize if you must," Journal of Economic Theory, Elsevier, vol. 133(1), pages 31-57, March.
    9. Chris Shannon & William R. Zame, 1999. "Quadratic Concavity and the Determinancy of Equilibrium," UCLA Economics Working Papers 791, UCLA Department of Economics.
    10. Stinchcombe, Maxwell B., 2011. "Balance and discontinuities in infinite games with type-dependent strategies," Journal of Economic Theory, Elsevier, vol. 146(2), pages 656-671, March.
    11. Hans Haller & Roger Lagunoff, 2000. "Genericity and Markovian Behavior in Stochastic Games," Econometrica, Econometric Society, vol. 68(5), pages 1231-1248, September.
    12. repec:eee:econom:v:199:y:2017:i:2:p:213-220 is not listed on IDEAS
    13. Klishchuk, Bogdan, 2015. "New conditions for the existence of Radner equilibrium with infinitely many states," Journal of Mathematical Economics, Elsevier, vol. 61(C), pages 67-73.
    14. Philippe Jehiel & Moritz Meyer-ter-Vehn & Benny Moldovanu & William R. Zame, 2006. "The Limits of ex post Implementation," Econometrica, Econometric Society, vol. 74(3), pages 585-610, May.
    15. Eddie Dekel & Yossi Feinberg, 2006. "Non-Bayesian Testing of a Stochastic Prediction," Review of Economic Studies, Oxford University Press, vol. 73(4), pages 893-906.
    16. Wojciech Olszewski & Alvaro Sandroni, 2006. "Strategic Manipulation of Empirical Tests," Discussion Papers 1425, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
    17. Alexei Zakharov & Constantine Sorokin, 2014. "Policy convergence in a two-candidate probabilistic voting model," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 43(2), pages 429-446, August.
    18. Jäger, Gerhard & Metzger, Lars P. & Riedel, Frank, 2011. "Voronoi languages," Games and Economic Behavior, Elsevier, vol. 73(2), pages 517-537.
    19. Castro, Luciano I. de, 2007. "Affiliation, equilibrium existence and the revenue ranking of auctions," UC3M Working papers. Economics we074622, Universidad Carlos III de Madrid. Departamento de Economía.
    20. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2007. "The Dynamic Evolution of Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), pages 251-286.
    21. Aviad Heifetz & Chris Shannon & Yossi Spiegel, 2007. "The Dynamic Evolution of Preferences," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), pages 251-286.
    22. Beggs, A.W., 2015. "Regularity and robustness in monotone Bayesian games," Journal of Mathematical Economics, Elsevier, vol. 60(C), pages 145-158.

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