Invariance of the distribution of the maximum

Fosgerau, Mogens and Lindberg, Per Olov and Mattsson, Lars-Göran and Weibull, Jörgen (2015): Invariance of the distribution of the maximum.

Abstract

Many models in economics involve probabilistic choices where each decision-maker selects the best alternative from a finite set. Viewing the value of each alternative as a random variable, the analyst is then interested in the choice probabilities, that is, the probability for an alternative to give the maximum value. Much analytical power can be gained, both for positive and normative analysis, if the maximum value is statistically independent of which alternative obtains the highest value. This note synthesizes and generalizes previous results on this invariance property. We provide characterizations of the property within a wide class of distributions that comprises the McFadden GEV class, show implications in several directions, and establish connections with copulas. We illustrate the usefulness of the invariance property by way of a few examples.

• Item Type: MPRA Paper
• Language: English
• Keywords: Choice; random utility; extreme value; leader-maximum; invariance; independence
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C25 - Discrete Regression and Qualitative Choice Models ; Discrete Regressors ; Proportions ; Probabilities; D - Microeconomics > D0 - General > D01 - Microeconomic Behavior: Underlying Principles

References

Anas, A., and C.M. Feng (1988): "Invariance of expected utility in logit models", Economics Letters 27 (1), 41-45.

Baye, M.R. and H.C. Hoppe (2003): "The strategic equivalence of rent-seeking, innovation, and patent-race games", Games and Economic Behavior 44 (2), 217-226.

de Palma, A., and K. Kilani (2007): "Invariance of conditional maximum utility", Journal of Economic Theory 132 (1), 137-146.

Hale, J.K. (1969): Ordinary Differential Equations. New York: Wiley and Sons.

Joe, H. (1997): Multivariate Models and Dependence Concepts. London: Chapman & Hall.

Karlin, S., and Y. Rinott (1980): "Classes of orderings of measures and related correlation inequalities I: Multivariate totally positive distributions", Journal of Multivariate Analysis 10, 467-498.

Krishna, V. (2002): Auction Theory. Cambridge (USA): Academic Press.

Lindberg, P.O. (2012): Contributions to Probabilistic Discrete Choice. Ph.D. Thesis, Department of Transport Science, KTH Royal Institute of Technology, Stockholm. (http://kth.diva-portal.org/smash/get/diva2:528102/FULLTEXT01.pdf)

Lindberg, P.O., E. A. Eriksson, and L.-G. Mattsson (1995): "Invariance of achieved utility in random utility models", Environment and Planning A 27 (1), 121-142.

Mattsson, L.-G., J. Weibull, and P.O. Lindberg (2014): "Extreme values, invariance and choice probabilities", Transportation Research Part B 59, 81-95.

McFadden, D. (1974): "Conditional logit analysis of qualitative choice behavior", in Zarembka (ed.), Frontiers in Econometrics. New York: Academic Press.

McFadden, D. (1978): "Modelling the choice of residential location", in Karlqvist, Lundqvist, Snickars, Weibull (eds.), Spatial Interaction Theory and Planning Models. Amsterdam: North-Holland.

McFadden, D. (1981): "Econometric models of probabilistic choice", in Manski, McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications. Cambridge (USA): MIT Press.

Milgrom, P., and R. Weber (1980): "A theory of auctions and competitive bidding", Econometrica 50, 1089-1122.

Nelsen, R. B. (2006): An Introduction to Copulas, Vol. 2. Berlin: Springer Series in Statistics.

Perloff, J. M., and S. C. Salop (1985): "Equilibrium with product differentiation", Review of Economic Studies 52(1), 107--120.

Resnick, S. and R. Roy (1990a): "Multivariate extremal processes, leader processes and dynamic choice models", Advances in Applied Probability 22 (2), 309-331.

Resnick, S., and R. Roy (1990b): "Leader and maximum independence for a class of discrete choice models", Economics Letters 33 (3), 259-263.

Robertson, C. A. and D. J. Strauss (1981): "A characterization theorem for random utility variables", Journal of Mathematical Psychology 23 (2): 184--89.

Sattinger, M. (1984): "Value of an additional firm in monopolistic competition", Review of Economic Studies 51(2), 321--332.

Strauss, D. (1979): "Some results on random utility models", Journal of Mathematical Psychology 20 (1): 35--52.

Train, K., and W. W. Wilson (2008): "Estimation on stated-preference experiments constructed from revealed-preference choices", Transportation Research Part B 42 (3), 191-203.