Random utility model: Difference between revisions

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In [[economics]], a '''random utility model''' ('''RUM'''),<ref>{{cite journal |id={{ProQuest|1303217712}} |last1=Manski |first1=Charles F |title=The Structure of Random Utility Models |journal=Theory and Decision |volume=8 |issue=3 |date=July 1977 |pages=229229–254 |doi=10.1007/BF00133443 }}</ref><ref>{{cite journalbook |doi=10.1007/978-0-387-75857-2_3 |chapter=Random Utility Theory |title=Transportation Systems Analysis |series=Springer Optimization and Its Applications |date=2009 |last1=Cascetta |first1=Ennio |volume=29 |pages=89–167 |isbn=978-0-387-75856-5 }}</ref> also called '''stochastic utility model''',<ref>{{cite journal |doi=10.1016/0304-4076(75)90032-9 |title=Maximum score estimation of the stochastic utility model of choice |date=1975 |last1=Manski |first1=Charles F. |journal=Journal of Econometrics |volume=3 |issue=3 |pages=205–228 }}</ref> is a mathematical description of the preferences of a person, whose choices are not deterministic, but depend on a random state variable.

== Background ==

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== The representation problem ==

Block and [[Jacob Marschak|Marschak]]<ref name=":1">{{cite journalbook |doi=10.1007/978-94-010-9276-0_8 |chapter=Random Orderings and Stochastic Theories of Responses (1960) |title=Economic Information, Decision, and Prediction |date=1974 |last1=Block |first1=H. D. |pages=172–217 |isbn=978-90-277-1195-3 }}</ref> presented the following problem. Suppose we are given as input, a set of ''choice probabilities'' ''P_a,B'', describing the probability that an agent chooses alternative ''a'' from the set ''B''. We want to ''rationalize'' the agent's behavior by a probability distribution over preference relations. That is: we want to find a distribution such that, for all pairs ''a,B'' given in the input, ''P_a,B'' = Prob[a is weakly preferred to all alternatives in B]. What conditions on the set of probabilities ''P_a,B'' guarantee the existence of such a distribution?{{fact}}

[[Jean-Claude Falmagne|Falmagne]]<ref name=":2">{{cite journal |last1=Falmagne |first1=J.C. |title=A representation theorem for finite random scale systems |journal=Journal of Mathematical Psychology |date=August 1978 |volume=18 |issue=1 |pages=52–72 |doi=10.1016/0022-2496(78)90048-2 }}</ref> solved this problem for the case in which the set of alternatives is finite: he proved that a probability distribution exists iff a set of polynomials derived from the choice-probabilities, denoted ''Block-Marschak polynomials,'' are nonnegative. His solution is constructive, and provides an algorithm for computing the distribution.

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Block and [[Jacob Marschak|Marschak]]<ref name=":1" /> proved that, when there are at most 3 alternatives, the random utility model is unique ("identified"); however, when there are 4 or more alternatives, the model may be non-unique.<ref name=":3" /> For example,<ref>https://scholar.harvard.edu/files/tomasz/files/lisbon32-post.pdf{{Bare URL PDF}}</ref> we can compute the probability that the agent prefers w to x (w>x), and the probability that y>z, but may not be able to know the probability that both w>x and y>z. There are even distributions with disjoint supports, which induce the same set of choice probabilities.

Some conditions for uniqueness were given by [[Jean-Claude Falmagne|Falmagne]].<ref name=":2" /> Turansick<ref name=":0">{{cite journal |last1=Turansick |first1=Christopher |title=Identification in the random utility model |journal=Journal of Economic Theory |date=July 2022 |volume=203 |pages=105489 |doi=10.1016/j.jet.2022.105489 |arxiv=2102.05570 }}</ref> presents two characterizations for the existence of a unique random utility representation.

== Models ==

There are various RUMs, which differ in the assumptions on the probability distributions of the agent's utility, A popular RUM is was developed by Luce<ref>{{cite book |last1=Luce |first1=R. Duncan |title=Individual Choice Behavior: A Theoretical Analysis |date=2012 |publisher=Courier Corporation |isbn=978-0-486-15339-1 }}{{pn}}</ref> and Plackett.<ref>{{cite journal |last1=Plackett |first1=R. L. |title=The Analysis of Permutations |journal=Applied Statistics |date=1975 |volume=24 |issue=2 |pages=193193–202 |doi=10.2307/2346567 |jstor=2346567 }}</ref>

The Plackett-Luce model was applied in [[econometrics]],<ref name="McFadden Conditional Logit Analysis">{{cite book |last1=McFadden |first1=Daniel |chapter=Conditional Logit Analysis of Qualitative Choice Behavior |pages=105–142 |editor1-last=Zarembka |editor1-first=Paul |title=Frontiers in Econometrics |date=1974 |publisher=Academic Press |isbn=978-0-12-776150-3 }}</ref> for example, to analyze automobile prices in [[market equilibrium]].<ref>{{Cite journal |last1=Berry |first1=Steven |last2=Levinsohn |first2=James |last3=Pakes |first3=Ariel |date=1995 |title=Automobile Prices in Market Equilibrium |journal=Econometrica |volume=63 |issue=4 |pages=841–890 |doi=10.2307/2171802 |jstor=2171802 }}</ref> It was also applied in [[Machine learning in earth sciences|machine learning]] and [[information retrieval]].<ref>{{cite journal |last1=Liu |first1=Tie-Yan |title=Learning to Rank for Information Retrieval |journal=Foundations and Trends® in Information Retrieval |date=2007 |volume=3 |issue=3 |pages=225–331 |doi=10.1561/1500000016}}</ref><ref>{{cite journal |last1=Liu |first1=Tie-Yan |title=Learning to Rank for Information Retrieval |journal=Foundations and Trends® in Information Retrieval |date=2007 |volume=3 |issue=3 |pages=225–331 |doi=10.1561/1500000016 }}</ref> It was also applied in [[Social choice theory|social choice]], to analyze an opinion poll conducted during the [[1997 Irish presidential election|Irish presidential election]].<ref>{{cite journal |last1=Gormley |first1=Isobel Claire |last2=Murphy |first2=Thomas Brendan |title=A grade of membership model for rank data |journal=Bayesian Analysis |date=June 2009 |volume=4 |issue=2 |doi=10.1214/09-BA410 }}</ref> Efficient methods for [[expectation-maximization]] and [[Expectation propagation]] exist for the Plackett-Luce model.<ref>{{cite journal |last1=Caron |first1=François |last2=Doucet |first2=Arnaud |title=Efficient Bayesian Inference for Generalized Bradley–Terry Models |journal=Journal of Computational and Graphical Statistics |date=January 2012 |volume=21 |issue=1 |pages=174–196 |doi=10.1080/10618600.2012.638220 |arxiv=1011.1761 }}</ref><ref>{{cite journal |last1=Hunter |first1=David R. |title=MM algorithms for generalized Bradley-Terry models |journal=The Annals of Statistics |date=February 2004 |volume=32 |issue=1 |doi=10.1214/aos/1079120141 }}</ref><ref>{{cite journalbook |doi=10.1145/1553374.1553423 |chapter=Bayesian inference for Plackett-Luce ranking models |title=Proceedings of the 26th Annual International Conference on Machine Learning |date=2009 |last1=Guiver |first1=John |last2=Snelson |first2=Edward |pages=377–384 |isbn=978-1-60558-516-1 }}</ref>

== Application to social choice ==

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* ''Combining Revealed Preferences and Stated Preferences:'' to combine advantages of these two data types.

Blavatzkyy<ref>{{cite journal |last1=Blavatskyy |first1=Pavlo R. |title=Stochastic utility theorem |journal=Journal of Mathematical Economics |date=December 2008 |volume=44 |issue=11 |pages=1049–1056 |doi=10.1016/j.jmateco.2007.12.005 |url=http://www.econ.uzh.ch/static/wp_iew/iewwp311.pdf }}</ref> studies stochastic utility theory based on choices between lotteries. The input is a set of ''choice probabilities'', which indicate the likelihood that the agent choose one lottery over the other.

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