Modeling the Non-Expected Choice: A Weighted Utility Logit∗ Pia Koskenoja Tampere University of Technology Institute of Transportation Engineering Room FA 208, P O Box 541 (Korkeakoulunkatu 8), FI-33101 Tampere, Finland pia.koskenoja@tut.fi Abstract This work derives and simulates two choice models 2 Utility Functionals of the Logit Models applying the weighted utility theory, a generaliza- Following the tradition of logit models I formulate a utility tion of the expected utility theory. It shows one set function that is separable in attributes. The simplest utility of assumptions, which justify the practice of in- function has one sure attribute and one risky attribute. In a cluding the mean and the variance of a risky alter- transportation context these can be monetary cost of travel native into a linear utility function of the choice and travel time, respectively. In the case of discrete distribu- model. A Monte Carlo simulation provides empiri- tion of the risky alternative, the utility functional is: cal evidence on the robustness of the models. p (t i ) w (t i ) U (t i ) V (⋅ ) = b0 − b1 ∑ − b2 c 1 Introduction i ∑ p (t m ) w (t m ) m Allais paradox shows that our choices commonly violate the axioms of von Neumann-Morgenstein expected utility the- where p(ti) denotes the probability of possible travel time ory. But we still commonly apply the expected utility theory outcome ti, w(ti) the weight the decision maker places on the when we model our choices. One possible remedy to this outcome ti, U(ti) the utility of the outcome ti, and c the sure discrepancy is to build a choice model that uses one gener- monetary cost. alization of the expected utility theory, the weighted utility An exponential works well as the weight function. theory. This paper presents two binomial logit models, which w (t i ) = exp( α t i ) assume that the decision maker has weighed utility prefer- ences. The models have been written into a context of a If α = 0, the weight function gets a value one throughout transportation problem, but naturally they can be applied to the domain and reduces the weighted utility expression to an any choice between two risky alternatives. expected utility. If α > 0, the traveler emphasizes the poten- Axiomatically weighted utility differs from expected util- tial of longer travel times. Correspondingly, if α < 0, the ity by a weaker version of the independence axiom. traveler behaves as if he would consider the shorter travel Weighted utility was first axiomatized by Chew and Mac- times as "more weighty" than what expected utility would Crimmon, [1979]. Chew [1982] proved that weighted utility warrant. behavior cannot be derived from expected utility by trans- For the model with continuous distribution of the risky forming the risky variables. Further axiomatic work has attribute the assumptions are: t~N(µ,σ2), U = -b1t, and w(t) = been continued by Chew [1983], Fishburn [1981, 1983] and exp(αt). With these assumptions the utility functional is: Nakamura [1984, 1985]. Fishburn [1988] contains an in- formative presentation of the weighted utility theory. 1 ( − t −µ2  ) The descriptive strength of weighted utility has been 2πσ ∫ t ⋅ exp (αt ) ⋅ exp   dt 2σ 2  tested in empirical laboratory experiments [Chew and V (⋅) = b0 − b1 − b 2 c. Waller, 1986; Camerer 1989; Conlisk 1989]. I do not know 1 ( ) − t−µ2  ( ) of any choice models where weighted utility is applied. 2πσ ∫ exp α t ⋅ exp   2σ 2  dt  This form has the welcome property that it simplifies to ( V (⋅ ) = b0 − b1 µ + ασ 2 − b2 c . ) * The support of Yrjö Jahnson Foundation and NSF grants DMS 9313013 and DMS 9208758 are gratefully acknowledged. This is a welcome find because it justifies the commonly is assumed to not contain a risk premium for the unreliable practiced ad hoc inclusion of the risky attribute’s variance as attribute, like in the value-of-time estimation. If the true a fully separate explanatory variable in addition to the mean preferences driving the choices comply with weighted util- in the utility expression of an estimated choice model. On ity, the parameters estimated from a mean value utility the other hand, it demonstrates that this common practice is model will produce estimates that include a risk premium. not compatible with the expected utility theory. A demon- stration of this property in a 3-outcome space is available 4 Conclusions from the author by request. The model simulations demonstrated that the weighted util- 2.1 Parameter restrictions ity logit models give reliable estimates in a wide range of true weighted utility risk preferences. Especially the discrete It is customary to require that a utility function exhibits risk version of the model poses possibilities for situations where aversion and monotonicity. the decision maker tends to succumb to Allais paradox and Risk aversion is defined to mean that the utility of the bases his decisions on a small number of perceived possible expected outcome is preferred to the utility of a gamble. Assuming two arbitrary outcomes, the requirement of risk realizations of the risky alternative. aversion simplifies to a requirement that the ratio of weight functions of the outcomes cannot equal to one, that is, α References should not equal zero. This requirement reflects the fact that [Camerer, 1989] C.F. Camerer. An Experimental Test of this particular formulation of weighted utility reduces to Several Generalized Utility Theories. Journal of Risk expected utility only in the case of risk neutrality. and Uncertainty. 2:61-104, 1989. Monotonicity of utility function in outcomes generalizes into a requirement that the utility functional exhibits first [Chew, 1982] S.H.Chew. A Mixture set axiomatization of order stochastic dominance (FSD). For the discrete model it weighted utility theory. Discussion Paper 82-4, College is possible to arbitrarily define the range of outcomes as [L1, of Business and Public Administration, University of L2] and thus the range for V[{p(ti)}] as [-b1L2,-b1L1]. The Arizona, Tuscon, 1982. definitions lead to two conditions for FSD: α < 1/(L2-L1) [Chew, 1983] S.H. Chew. A generalization of the quasilin- and α > 0. If the risky attribute has an infinite range of out- ear mean with applications to the measurement of in- comes, the decision maker violates monotonicity if she is come inequality and decision theory resolving the Allais risk averse, that is, if her α ≠ 0. paradox. Econometrica, 51:1065-1092, 1983. [Chew and MacCrimmon 1979] S.H. Chew and K.R. Mac- 3 Monte Carlo Simulations Crimmon. Alpha-nu choice theory: a generalization of The Monte Carlo simulations consisted of rounds of first expected choice theory. Working paper 669, Faculty of creating the true choices according to three models: a con- Commerce and Business Administration, University of tinuous risky attribute, a discrete risky attribute, and a sure British Columbia, Vancouver, Canada, 1979. attribute, and later taking the created choice data as given [Chew and Waller 1986] S.H. Chew and W.S. Waller. Em- and estimating the three models on each data set. pirical Tests of Weighted Utility Theory. Journal of The weighted utility formulations worked well. In all the Mathematical Psychology,30:55-72, 1986. simulation runs the continuous model specification gave [Conlisk 1989] J. Conlisk. Three variants on the Allais Ex- more consistent results than the discrete one, which should ample. American Economic Review,79:392-407, 1989. be expected due to the simpler functional form. The true b- parameters were more consistently retrieved in both specifi- [Fishburn 1981] Fishburn. An axiomatic characterization of cations than α. When true value of α was set to strongly skew-symmetric bilinear functionals, with applications violate FSD, only the continuous model specification was to utility theory. Economic Letters,8:311-313, 1981. able to converge reliably and retrieve the correct values. But [Fishburn 1983] Fishburn. Transitive measurable utility. when true α was set to 0.15, which still moderately violated Journal of Economic Theory 31:293-317, 1983. FSD, the discrete model formulation converged each time [Fishburn 1988] Fishburn. Nonlinear Preference and Utility and the mean of the 50 parameter estimates (0.1864) was Theory. Johns Hopkins University Press, Baltimore, within two standard deviations of the true value of 0.15. 1988. When the true α-value was set to not violate FSD, the weighted utility models retrieved the true parameters very [Nakamura 1984] Y. Nakamura. Nonlinear measurable util- well. The same held when the true behavior was created by ity analysis. Ph.D. dissertation, University of California, mean value utility, that is to say the true α had a value zero. Davis, 1984. When the true behavior was generated by weighted pref- [Nakamura 1985] Y. Nakamura. Weighted linear utility. erences, but was estimated by mean value utility model, the Preprint. Department of Precision Engineering. Osaka estimated parameters were consistently downward biased University, Osaka, Japan, 1985. towards a point where their proportions stay true. This dem- onstration is something that should be taken into account in the interpretations of models where the ratio of parameters