Tail probabilities for triangular arrays

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January 2014, 1(1): 45-56. doi: 10.3934/jdg.2014.1.45

Tail probabilities for triangular arrays

Drew Fudenberg ^1, and David K. Levine ^2,

1.	Department of Economics, Harvard University, Littauer Center, 1805 Cambridge Street, Cambridge, MA 02138, United States
2.	Department of Economics, Washington University in St. Louis, 1 Brookings Dr., St. Louis MO 63130-4899, United States

Received June 2012 Revised January 2013 Published June 2013

Abstract
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Different discrete time triangular arrays representing a noisy signal of players' activities can lead to the same limiting diffusion process yet lead to different limit equilibria. Whether the limit equilibria are equilibria of the limiting continuous time game depends on the limit properties of test statistics for whether a player has deviated. We provide an estimate of the tail probabilities along these arrays that allows us to determine the asymptotic behavior of the best test and thus of the best equilibrium.

Keywords: tail probabilities, Triangular array, continuous-time games., limit equilibria.

Mathematics Subject Classification: Primary: 60G50, 91A20; Secondary: 60B10, 91A25, 91B7.

Citation: Drew Fudenberg, David K. Levine. Tail probabilities for triangular arrays. Journal of Dynamics and Games, 2014, 1 (1) : 45-56. doi: 10.3934/jdg.2014.1.45

References:

[1]	W. Feller, "An Introduction to Probability Theory and Its Applications," Volume II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
[2]	D. Fudenberg and D. K. Levine, Continuous time limits of repeated games with imperfect public monitoring, Review of Economic Dynamics, 10 (2007), 173-192.
[3]	D. Fudenberg and D. K. Levine, Repeated games with frequent signals, Quarterly Journal of Economics, 124 (2009), 233-265.
[4]	J. A. Mirlees, Notes on welfare economics, information and uncertainty, in "Essays on Economic Behavior Under Uncertainty" (eds. M. Balch, D. McFadden and S.-Y. Wu), North Holland, Amsterdam, 1974.
[5]	Y. Sannikov, Games with imperfectly observed actions in continuous time, Econometrica, 75 (2007), 1285-1329. doi: 10.1111/j.1468-0262.2007.00795.x.
[6]	Y. Sannikov and A. Skrzypacz, Impossibility of collusion under imperfect monitoring with flexible production, American Economic Review, 97 (2007), 1794-1823.
[7]	T. Sadzik and E. Stacchetti, Agency Models with Frequent Actions: A Quadratic Approximation Method, in "Workshop on Macroeconomic Applications of Dynamic Games and Contracts" (Stonybrook), (2012).

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References:

[1]	W. Feller, "An Introduction to Probability Theory and Its Applications," Volume II, Second edition, John Wiley & Sons, Inc., New York-London-Sydney, 1971.
[2]	D. Fudenberg and D. K. Levine, Continuous time limits of repeated games with imperfect public monitoring, Review of Economic Dynamics, 10 (2007), 173-192.
[3]	D. Fudenberg and D. K. Levine, Repeated games with frequent signals, Quarterly Journal of Economics, 124 (2009), 233-265.
[4]	J. A. Mirlees, Notes on welfare economics, information and uncertainty, in "Essays on Economic Behavior Under Uncertainty" (eds. M. Balch, D. McFadden and S.-Y. Wu), North Holland, Amsterdam, 1974.
[5]	Y. Sannikov, Games with imperfectly observed actions in continuous time, Econometrica, 75 (2007), 1285-1329. doi: 10.1111/j.1468-0262.2007.00795.x.
[6]	Y. Sannikov and A. Skrzypacz, Impossibility of collusion under imperfect monitoring with flexible production, American Economic Review, 97 (2007), 1794-1823.
[7]	T. Sadzik and E. Stacchetti, Agency Models with Frequent Actions: A Quadratic Approximation Method, in "Workshop on Macroeconomic Applications of Dynamic Games and Contracts" (Stonybrook), (2012).

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