Abstract
A decision maker chooses in a probabilistic manner if she does not necessarily prefer the same choice alternative when repeatedly presented with the same choice set. Probabilistic choice may occur for a variety of reasons such as unobserved attributes of choice alternatives, imprecision of preferences, or random errors/noise in decisions. The Luce choice model (also known as strict utility or multinomial logit) is derived from the choice axiom (also known as the independence from irrelevant alternatives). This axiom postulates that the relative likelihood of choosing one choice alternative A over another choice alternative B is not affected by the presence or absence of other choice alternatives in the choice set. This paper presents a dual choice axiom: the relative probability of NOT choosing A over the probability of NOT choosing B is independent from irrelevant alternatives. A new model of probabilistic choice is derived from this dual axiom. This model coincides with Luce’s choice model only in the case of a binary choice. The new model has similar properties as the Luce choice model: the higher is the utility of a choice alternative, the higher is the probability that a decision maker chooses this alternative and the lower is the probability that he or she chooses any other alternative. The new model differs from the Luce choice model in two aspects: utility of choice alternatives is bounded (from above and below) and choice probabilities are more sensitive to differences in utility of choice alternatives.
Similar content being viewed by others
Notes
Machina (1985) and Chew et al. (1991) develop models of probabilistic choice under risk as a result of deliberate randomization by decision makers with (deterministic) quasi-concave preferences. Hey and Carbone (1995) find that conscious randomization cannot rationalize their experimental data but Agranov and Ortoleva (2017) reach the opposite conclusion.
The Fechner model is an econometric model of discrete choice with random errors (noise, attention slips, carelessness) additive on the (latent) utility scale (cf. Becker et al. 1963, pp. 44–45; Hey and Orme 1994, p. 1301). Binary Luce (1959) model is a special case of binary Fechner model when such errors are drawn from the logistic distribution (cf. Yellott 1977).
They also consider a related model where a decision maker selects the best and the worst alternative from the choice set.
Luce (1959, Axiom 1, part (i)) formulated his choice axiom in a different form: the probability that a decision maker chooses an alternative from set T that lies in the nonempty set R ⊂ T is equal to the probability that this decision maker chooses an alternative from set T that lies in the nonempty set S ⊂ T multiplied by the probability that the decision maker chooses an alternative from set S that lies in set R ⊂ S. Yet, it can be easily shown that this formulation implies the independence from irrelevant alternatives property (2), cf. Luce (1959, Lemma 3).
These sets are defined by equation \( \frac{U(A)}{U(B)}+\frac{U(A)}{U(C)}=\frac{1+p}{1-p} \) subject to bounds (11) and (12).
In Luce (1959) choice model PLuce(A|{A,B,C}) = p is generated by utility ratios \( \frac{U(B)}{U(A)}+\frac{U(C)}{U(A)}=\frac{1-p}{p} \).
The first-order stochastic dominance in choice under risk, state-wise dominance in choice under uncertainty/ambiguity or the first-order temporal dominance in choice over time.
Blavatskyy (2018) recently proposed its generalization to choice among n > 2 alternatives.
This function must satisfy restriction F(v1, v2, …, vn − 1) + F(−v1, v2 − v1, …, vn − 1 − v1) + F(−v2, v1 − v2, v3 − v2, …, vn − 1 − v2) + … + F(−vn − 1, v1 − vn − 1, …, vn − 2 − vn − 1) = 1 for all v1, v2, …, vn − 1 ∈ ℝ, which implies inter alia F(0, …, 0) = 1/n.
References
Agranov, M., & Ortoleva, P. (2017). Stochastic choice and preferences for randomization. Journal of Political Economy, 125(1), 40–68.
Andersen, S., Harrison, G. W., Lau, M. I., & Rutström, E. E. (2008). Eliciting risk and time preferences. Econometrica, 76(3), 583–618.
Ballinger, P., & Wilcox, N. (1997). Decisions, error and heterogeneity. Economic Journal, 107(443), 1090–1105.
Becker, G. M., DeGroot, M. H., & Marschak, J. (1963). Stochastic models of choice behavior. Behavioral Science, 8, 41–55.
Birnbaum, M. (2005). Three new tests of independence that differentiate models of risky decision making. Management Science, 51(9), 1346–1358.
Birnbaum, M., & Navarrete, J. (1998). Testing descriptive utility theories: Violations of stochastic dominance and cumulative independence. Journal of Risk Uncertainty, 17(1), 49–78.
Blavatskyy, P. (2006). Violations of betweenness or random errors? Economics Letters, 91(1), 34–38.
Blavatskyy, P. R. (2009). How to extend a model of probabilistic choice from binary choices to choices among more than two alternatives. Economics Letters, 105(3), 330–332.
Blavatskyy, P. R. (2012). Probabilistic choice and stochastic dominance. Economic Theory, 50(1), 59–83.
Blavatskyy, P. (2014). Stronger utility. Theory and Decision, 76(2), 265–286.
Blavatskyy, P. (2017). Probabilistic intertemporal choice. Journal of Mathematical Economics, 73, 142–148.
Blavatskyy, P. (2018). Fechner’s strong utility model for choice among n>2 alternatives: Risky lotteries, Savage acts, and intertemporal payoffs. Journal of Mathematical Economics, 79, 75–82.
Blavatskyy, P., & Maafi, H. (2018). Estimating representations of time preferences and models of probabilistic intertemporal choice on experimental data. Journal of Risk and Uncertainty, 56(3), 259–287.
Butler, D. J., & Loomes, G. C. (2007). Imprecision as an account of the preference reversal phenomenon. American Economic Review, 97(1), 277–297.
Butler, D. J., & Loomes, G. C. (2011). Imprecision as an account of violations of independence and betweenness. Journal of Economic Behavior and Organization, 80(3), 511–522.
Camerer, C. F. (1989). An experimental test of several generalized utility theories. Journal of Risk and Uncertainty, 2(1), 61–61.
Camerer, C., & Ho, T. (1994). Violations of the betweenness axiom and nonlinearity in probability. Journal of Risk and Uncertainty, 8(2), 167–196.
Carbone, E. (1997). Investigation of stochastic preference theory using experimental data. Economics Letters, 57(3), 305–311.
Carbone, E., & Hey, J. (1995). A comparison of the estimates of EU and non-EU preference functionals using data from pairwise choice and complete ranking experiments. Geneva Papers on Risk and Insurance Theory, 20(1), 111–133.
Chew, S., Epstein, L., & Segal, U. (1991). Mixture symmetry and quadratic utility. Econometrica, 59(1), 139–163.
Coller, M., & Williams, M. (1999). Eliciting individual discount rates. Experimental Economics, 2(2), 107–127.
de Condorcet, M. (1785). Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Paris: L’Imprimerie Royale.
Debreu, G. (1960). Individual choice behavior: A theoretical analysis by R. Duncan Luce. American Economic Review, 50, 186–188.
Estes, W. K. (1960). A random-walk model for choice behavior. In K. J. Arrow, S. Karlin, & P. Suppes (Eds.), Mathematical methods in the social sciences (pp. 265–276). Stanford: Stanford University Press.
Falmagne, J.-C. (1985). Elements of psychophysical theory. New York: Oxford University Press.
Fechner, G. (1860). Elements of psychophysics. New York: Holt, Rinehart and Winston.
Harless, D., & Camerer, C. (1994). The predictive utility of generalized expected utility theories. Econometrica, 62(6), 1251–1289.
Harrison, G., Lau, M., & Williams, M. (2002). Estimating individual discount rates in Denmark: A field experiment. American Economic Review, 92(5), 1606–1617.
Hey, J. D. (2001). Does repetition improve consistency? Experimental Economics, 4(1), 5–54.
Hey, J., & Carbone, E. (1995). Stochastic choice with deterministic preferences: An experimental investigation. Economics Letters, 47, 161–167.
Hey, J. D., & Orme, C. (1994). Investigating generalisations of expected utility theory using experimental data. Econometrica, 62(6), 1291–1326.
Hey, J. D., Lotito, G., & Maffioletti, A. (2010). The descriptive and predictive accuracy of theories of decision making under uncertainty/ambiguity. Journal of Risk and Uncertainy, 41(2), 81–111.
Holt, C. A., & Laury, S. K. (2002). Risk aversion and incentive effects. American Economic Review, 92(5), 1644–1655.
Loomes, G. (2005). Modelling the stochastic component of behaviour in experiments: Some issues for the interpretation of data. Experimental Economics, 8(4), 301–323.
Loomes, G., & Sugden, R. (1995). Incorporating a stochastic element into decision theories. European Economic Review, 39(3-4), 641–648.
Loomes, G., & Sugden, R. (1998). Testing different stochastic specifications of risky choice. Economica, 65(260), 581–598.
Loomes, G., Moffatt, P., & Sugden, R. (2002). A microeconomic test of alternative stochastic theories of risky choice. Journal of Risk and Uncertainty, 24(2), 103–130.
Luce, R. D. (1959). Individual choice behavior. New York: Wiley.
Luce, R. D., & Suppes, P. (1965). Preference, utility, and subjective probability. In Handbook of mathematical psychology (Vol. III, pp. 249–410). New York: Wiley.
Machina, M. (1985). Stochastic choice functions generated from deterministic preferences over lotteries. Economic Journal, 95(379), 575–594.
Marley, A. A. J. (1965). The relation between the discard and regularity conditions for choice probabilities. Journal of Mathematical Psychology, 2(2), 242–253.
Marley, A. A. J., & Louviere, J. J. (2005). Some probabilistic models of best, worst, and best-worst choices. Journal of Mathematical Psychology, 49(6), 464–480.
McFadden, D. (1976). Quantal choice analysis: A survey. Annals of Economic and Social Measurement, 5, 363–390.
McKelvey, R., & Palfrey, T. (1995). Quantal response equilibria for normal form games. Games and Economic Behavior, 10(1), 6–38.
Meier, S., & Sprenger, C. (2015). Temporal stability of time preferences. Review of Economics and Statistics, 97(2), 273–286.
Rieskamp, J., Busemeyer, J., & Mellers, B. (2006). Extending the bounds of rationality. Journal of Economic Literature, 44(3), 631–661.
Samuelson, P. (1937). A note on measurement of utility. The Review of Economic Studies, 4(2), 155–161.
Savage, L. J. (1954). The foundations of statistics. New York: Wiley.
Starmer, C., & Sugden, R. (1989). Probability and juxtaposition effects: An experimental investigation of the common ratio effect. Journal of Risk and Uncertainty, 2(2), 159–178.
von Neumann, J., & Morgenstern, O. (1947). Theory of games and economic behavior (Second ed.). Princeton: Princeton University Press.
Warner, J., & Pleeter, S. (2001). The personal discount rate: Evidence from military downsizing programs. American Economic Review, 91(1), 33–53.
Wilcox, N. (2008). Stochastic models for binary discrete choice under risk: A critical primer and econometric comparison. In J. C. Cox & G. W. Harrison (Eds.), Research in experimental economics (Vol. 12, pp. 197–292). Bingley: Emerald.
Wilcox, N. (2011). Stochastically more risk averse: A contextual theory of stochastic discrete choice under risk. Journal of Econometrics, 162(1), 89–104.
Wu, G., & Gonzalez, R. (1996). Curvature of the probability weighting function. Management Science, 42(12), 1676–1690.
Yellott Jr., J. I. (1977). The relationship between Luce’s choice axiom, Thurstone’s theory of comparative judgement, and the double exponential distribution. Journal of Mathematical Psychology, 15(2), 109–144.
Funding
Pavlo Blavatskyy is a member of the Entrepreneurship and Innovation Chair, which is part of LabEx Entrepreneurship (University of Montpellier, France) and funded by the French government (Labex Entreprendre, ANR-10-Labex-11-01).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
1.1 Proof of Proposition 1
For any two alternatives A,B∈S, dual choice axiom implies Eq. (3). Similarly, for any two alternatives B,C∈S, dual choice axiom implies Eq. (14).
Multiplying Eq. (3) by Eq. (14) yields Eq. (15).
According to dual choice axiom the left-hand side of Eq. (15) is equal to P(C, A)/P(A, C). Using this result we can rewrite Eq. (15) as Eq. (16).
Equation (16) is known as the product rule (e.g., Estes 1960, p. 272; Luce and Suppes 1965, definition 25, p. 341). According to Theorem 48 in Luce and Suppes (1965, p. 350), a binary choice probability function P : S × S → ℝ satisfies the product rule (16) if and only if it is a binary strict utility (4). Q.E.D.
1.2 Proof of Proposition 2
Using the result of proposition 1, for any two alternatives A,B∈S, we can rewrite dual choice axiom (3) as Eq. (17).
Equation (17) can be rearranged as Eq. (18).
Since Eq. (18) holds for any alternative B∈S, we can sum it over all B∈S to obtain Eq. (19).
The left-hand side of Eq. (19) is equal to one due to (1). Thus, we can rewrite Eq. (19) as Eq. (20).
A simple rearranging of Eq. (20) then yields model of probabilistic choice (5). Q.E.D.
Rights and permissions
About this article
Cite this article
Blavatskyy, P.R. Dual choice axiom and probabilistic choice. J Risk Uncertain 61, 25–41 (2020). https://doi.org/10.1007/s11166-020-09332-7
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11166-020-09332-7
Keywords
- Decision making
- Probabilistic choice
- Choice axiom
- Independence from irrelevant alternatives
- Luce choice model
- Strict utility