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In [[economics]], a '''random utility model''' ('''RUM'''),<ref>{{Citecite webjournal |id={{ProQuest|1303217712}} |last1=Manski |first1=Charles F |title=The Structure of Random Utility Models -|journal=Theory ProQuestand Decision |urlvolume=https://www.proquest.com/openview/7acf07ef00e4d7b837b4de87994aed40/1?cbl8 |issue=1818302&pq-origsite=gscholar3 |access-date=2023-11-07July 1977 |websitepages=www.proquest.com229–254 |languagedoi=en10.1007/BF00133443 }}</ref><ref>{{Citationcite |last=Cascettabook |firstdoi=Ennio |title=Random Utility Theory |date=2009 |url=https://doi.org/10.1007/978-0-387-75857-2_3 |workchapter=TransportationRandom SystemsUtility Analysis: Models and ApplicationsTheory |pagestitle=89–167Transportation |editor-last=CascettaSystems |editor-first=Ennio |access-date=2023-11-07Analysis |series=Springer Optimization and Its Applications |volumedate=292009 |placelast1=Boston, MACascetta |publisherfirst1=Springer USEnnio |languagevolume=en29 |doipages=10.1007/978-0-387-75857-2_389–167 |isbn=978-0-387-7585775856-25 }}</ref>, also called '''stochastic utility model''',<ref>{{Citecite journal |lastdoi=Manski |first=Charles F10. |date=19751016/0304-084076(75)90032-019 |title=Maximum score estimation of the stochastic utility model of choice |urldate=https://dx1975 |last1=Manski |first1=Charles F.doi.org/10.1016/0304-4076%2875%2990032-9 |journal=Journal of Econometrics |volume=3 |issue=3 |pages=205–228 |doi=10.1016/0304-4076(75)90032-9 |issn=0304-4076}}</ref> is a mathematical description of the preferences of a person, whose choices are not deterministic, but depend on a random state variable.
== Background ==
A basic assumption in classic economics is that the choices of a rational person choices are guided by a [[Preference (economics)|preference relation]], which can usually be described by a [[utility function]]. When faced with several alternatives, the rational person will choose the alternative with the highest utility. The utility function is not visible; however, by observing the choices made by the person, we can "reverse-engineer" his utility function. This is the goal of [[revealed preference]] theory.{{fact|date=July 2024}}
In practice, however, people are not rational. Ample empirical evidence shows that, when faced with the same set of alternatives, people may make different choices.<ref>{{Citecite journal |lastlast1=Camerer |firstfirst1=Colin F. |date=1989-04-01 |title=An experimental test of several generalized utility theories |url=https://doi.org/10.1007/BF00055711 |journal=Journal of Risk and Uncertainty |languagedate=enApril 1989 |volume=2 |issue=1 |pages=61–104 |doi=10.1007/BF00055711 |s2cid=154335530 |issn=1573-0476}}</ref><ref>{{Citecite journal |last1=Starmer |first1=Chris |last2=Sugden |first2=Robert |date=1989-06-01 |title=Probability and juxtaposition effects: An experimental investigation of the common ratio effect |url=https://doi.org/10.1007/BF00056135 |journal=Journal of Risk and Uncertainty |languagedate=enJune 1989 |volume=2 |issue=2 |pages=159–178 |doi=10.1007/BF00056135 |s2cid=153567599 |issn=1573-0476}}</ref><ref>{{Cite journal |last1=Hey |first1=John D. |last2=Orme |first2=Chris |date=1994 |title=Investigating Generalizations of Expected Utility Theory Using Experimental Data |url=https://www.jstor.org/stable/2951750 |journal=Econometrica |volume=62 |issue=6 |pages=1291–1326 |doi=10.2307/2951750 |jstor=2951750 |s2cid=120069179 |issn=0012-9682}}</ref><ref>{{Citecite journal |lastlast1=Wu |firstfirst1=George |date=1994-02-01 |title=An empirical test of ordinal independence |url=https://doi.org/10.1007/BF01073402 |journal=Journal of Risk and Uncertainty |languagedate=en1994 |volume=9 |issue=1 |pages=39–60 |doi=10.1007/BF01073402 |s2cid=153558846 |issn=1573-0476}}</ref><ref>{{Citecite journal |last1=Ballinger |first1=T. Parker |last2=Wilcox |first2=Nathaniel T. |date=1997-07-01 |title=Decisions, Error and Heterogeneity |url=https://academic.oup.com/ej/article/107/443/1090-1105/5065251 |journal=The Economic Journal |languagedate=enJuly 1997 |volume=107 |issue=443 |pages=1090–1105 |doi=10.1111/j.1468-0297.1997.tb00009.x |s2cid=153823510 |issn=0013-0133}}</ref> To an outside observer, their choices may appear random.
One way to model this behavior is called '''stochastic rationality'''. It is assumed that each agent has an unobserved ''state'', which can be considered a random variable. Given that state, the agent behaves rationally. In other words: each agent has, not a single preference-relation, but a [[Probability distribution|''distribution'']] over preference-relations (or utility functions).{{fact|date=July 2024}}
== The representation problem ==
Block and [[Jacob Marschak|Marschak]]<ref name=":1">{{Citationcite |last=Blockbook |firstdoi=H. D10.1007/978-94-010-9276-0_8 |titlechapter=Random Orderings and Stochastic Theories of Responses (1960) |date=1974 |url=https://doi.org/10.1007/978-94-010-9276-0_8 |worktitle=Economic Information, Decision, and Prediction: Selected Essays: Volume I Part I Economics of Decision |pages=172–217 |editor-last=Marschak |editor-first=Jacob |access-date=2023-11-071974 |serieslast1=Theory and Decision LibraryBlock |placefirst1=DordrechtH. |publisher=Springer NetherlandsD. |languagepages=en172–217 |doiisbn=10.1007/978-9490-010277-92761195-0_83 |isbn=978-94-010-9276-0}}</ref> presented the following problem. Suppose we are given as input, a set of ''choice probabilities'' ''P<sub>a,B</sub>'', 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<sub>a,B</sub>'' = Prob[a is weakly preferred to all alternatives in B]. What conditions on the set of probabilities ''P<sub>a,B</sub>'' guarantee the existence of such a distribution?{{fact|date=July 2024}}
[[Jean-Claude Falmagne|Falmagne]]<ref name=":2">{{Citecite journal |lastlast1=Falmagne |firstfirst1=J. C. |date=1978-08-01 |title=A representation theorem for finite random scale systems |url=https://dx.doi.org/10.1016/0022-2496%2878%2990048-2 |journal=Journal of Mathematical Psychology |date=August 1978 |volume=18 |issue=1 |pages=52–72 |doi=10.1016/0022-2496(78)90048-2 |issn=0022-2496}}</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.
Barbera and Pattanaik<ref name=":3">{{Cite journal |last1=Barberá |first1=Salvador |last2=Pattanaik |first2=Prasanta K. |date=1986 |title=Falmagne and the Rationalizability of Stochastic Choices in Terms of Random Orderings |url=https://www.jstor.org/stable/1911317 |journal=Econometrica |volume=54 |issue=3 |pages=707–715 |doi=10.2307/1911317 |jstor=1911317 |issn=0012-9682}}</ref> extend this result to settings in which the agent may choose sets of alternatives, rather than just singletons.
=== Uniqueness ===
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>{{cite conference |title=Stochastic Choice |first1=Tomasz |last1=Strzalecki
|conference=Hotelling Lectures in Economic Theory, Econometric Society European Meeting |___location=Lisbon |date=25 August 2017 |url=https://scholar.harvard.edu/files/tomasz/files/lisbon32-post.pdf }}{{pn|date=July 2024}}</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">{{Citecite journal |lastlast1=Turansick |firstfirst1=Christopher |date=2022-07-01 |title=Identification in the random utility model |url=https://www.sciencedirect.com/science/article/pii/S0022053122000795 |journal=Journal of Economic Theory |date=July 2022 |volume=203 |pages=105489 |doi=10.1016/j.jet.2022.105489 |arxiv=2102.05570 |s2cid=231861383 |issn=0022-0531}}</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>{{Citecite book |lastlast1=Luce |firstfirst1=R. Duncan |url=https://books.google.com/books?id=ERQsKkPiKkkC&dq=R.+Duncan+Luce.+Individual+Choice+Behavior%3A+A+Theoretical+Analysis.+Wiley%2C+1959.&pg=PP1 |title=Individual Choice Behavior: A Theoretical Analysis |date=2012-06-22 |publisher=Courier Corporation |isbn=978-0-486-15339-1 }}{{pn|languagedate=enJuly 2024}}</ref> and Plackett.<ref>{{Citecite webjournal |urllast1=https://academicPlackett |first1=R.oup L.com/jrsssc/article-abstract/ |title=The Analysis of Permutations |journal=Applied Statistics |date=1975 |volume=24/ |issue=2 |pages=193–202 |doi=10.2307/193/69535542346567 |access-datejstor=2023-11-072346567 }}</ref>
The [[Plackett-Luce model]] was applied in [[econometrics]],<ref name="McFadden Conditional Logit Analysis">{{Citecite journalbook |lastlast1=DMcFadden |firstfirst1=McfaddenDaniel |date=1974 |titlechapter=Conditional Logit Analysis of Qualitative Choice Behavior |urlpages=https://cir.nii.ac.jp/crid/1572824500127838080105–142 |journaleditor1-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 |url=https://www.jstor.org/stable/2171802 |journal=Econometrica |volume=63 |issue=4 |pages=841–890 |doi=10.2307/2171802 |jstor=2171802 |issn=0012-9682}}</ref> It was also applied in [[Machine learning in earth sciences|machine learning]] and [[information retrieval]].<ref>{{Citecite journal |lastlast1=Liu |firstfirst1=Tie-Yan |date=2009-06-26 |title=Learning to Rank for Information Retrieval |url=https://www.nowpublishers.com/article/Details/INR-016 |journal=Foundations and Trends® in Information Retrieval |languagedate=English2007 |volume=3 |issue=3 |pages=225–331 |doi=10.1561/1500000016 |issn=1554-0669}}</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>{{Citecite journal |last1=Gormley |first1=Isobel Claire |last2=Murphy |first2=Thomas Brendan |date=June 2009 |title=A grade of membership model for rank data |url=https://projecteuclid.org/journals/bayesian-analysis/volume-4/issue-2/A-grade-of-membership-model-for-rank-data/10.1214/09-BA410.full |journal=Bayesian Analysis |date=June 2009 |volume=4 |issue=2 |pages=265–295 |doi=10.1214/09-BA410 |s2cidhdl=5355945210197/7121 |issn=1936hdl-0975access=free }}</ref> Efficient methods for [[expectation-maximization]] and [[Expectation propagation]] exist for the Plackett-Luce model.<ref>{{Citecite journal |last1=Caron |first1=François |last2=Doucet |first2=Arnaud |date=January 2012 |title=Efficient Bayesian Inference for Generalized Bradley–Terry Models |url=http://www.tandfonline.com/doi/abs/10.1080/10618600.2012.638220 |journal=Journal of Computational and Graphical Statistics |languagedate=enJanuary 2012 |volume=21 |issue=1 |pages=174–196 |doi=10.1080/10618600.2012.638220 |arxiv=1011.1761 |s2cid=42955305 |issn=1061-8600}}</ref><ref>{{Citecite journal |lastlast1=Hunter |firstfirst1=David R. |date=February 2004 |title=MM algorithms for generalized Bradley-Terry models |url=https://projecteuclid.org/journals/annals-of-statistics/volume-32/issue-1/MM-algorithms-for-generalized-Bradley-Terry-models/10.1214/aos/1079120141.full |journal=The Annals of Statistics |date=February 2004 |volume=32 |issue=1 |pages=384–406 |doi=10.1214/aos/1079120141 |issn=0090-5364}}</ref><ref>{{Citecite book |last1doi=Guiver10.1145/1553374.1553423 |first1chapter=JohnBayesian |last2=Snelsoninference |first2=Edwardfor Plackett-Luce ranking models |title=Proceedings of the 26th Annual International Conference on Machine Learning |chapter=Bayesian inference for Plackett-Luce ranking models |date=2009-06-14 |chapter-urllast1=https://doi.org/10.1145/1553374.1553423Guiver |seriesfirst1=ICML '09John |___locationlast2=New York, NY, USASnelson |publisherfirst2=Association for Computing MachineryEdward |pages=377–384 |doi=10.1145/1553374.1553423 |isbn=978-1-60558-516-1|s2cid=16965626 }}</ref>
== Application to social choice ==
* It ignores the strength of agents' expressed preferences. An agent who prefers a "much more than" b and an agent who prefers a "a little more than b" are treated the same.
* It allows for cyclic preferences. There is a positive probability that an agent will prefer a to b, b to c, and c to a.
* The maximum likelihood estimator - which is the [[Kemeny–Young method]] - is hard to compute (it is <math>\Theta^P_2</math>-complete).<ref>{{Citecite journal |last1=Hemaspaandra |first1=Edith |last2=Spakowski |first2=Holger |last3=Vogel |first3=Jörg |date=2005-12-16 |title=The complexity of Kemeny elections |url=https://www.sciencedirect.com/science/article/pii/S0304397505005785 |journal=Theoretical Computer Science |date=December 2005 |volume=349 |issue=3 |pages=382–391 |doi=10.1016/j.tcs.2005.08.031 |issn=0304-3975}}</ref>
RUM provides an alternative model: there is a ground-truth vector of utilities; each agent draws a utility for each alternative, based on a probability distribution whose mean value is the ground-truth. This model captures the strength of preferences, and rules out cyclic preferences. Moreover, for some common probability distributions (particularly, the Plackett-Luce model), the maximum likelihood estimators can be computed efficiently.{{fact|date=July 2024}}
== Generalizations ==
Walker and Ben-Akiva<ref>{{Citecite journal |last1=Walker |first1=Joan |last2=Ben-Akiva |first2=Moshe |date=2002-07-01 |title=Generalized random utility model |url=https://www.sciencedirect.com/science/article/pii/S0165489602000239 |journal=Mathematical Social Sciences |seriesdate=RandomJuly Utility Theory and Probabilistic measurement theory2002 |volume=43 |issue=3 |pages=303–343 |doi=10.1016/S0165-4896(02)00023-9 |issn=0165-4896}}</ref> generalize the classic RUM in several ways, aiming to improve the accuracy of forecasts:
* ''Flexible Disturbances'': allowing a richer [[Covariance structure modeling|covariance structure]], estimating unobserved heterogeneity, and random parameters;
* ''Combining Revealed Preferences and Stated Preferences:'' to combine advantages of these two data types.
Blavatzkyy<ref>{{Citecite journal |lastlast1=Blavatskyy |firstfirst1=Pavlo R. |date=2008-12-01 |title=Stochastic utility theorem |url=https://www.sciencedirect.com/science/article/pii/S030440680700136X |journal=Journal of Mathematical Economics |date=December 2008 |volume=44 |issue=11 |pages=1049–1056 |doi=10.1016/j.jmateco.2007.12.005 |issnurl=0304-4068http://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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There are different types of RUMs, depending on the distributional assumptions and functional forms of the utility functions. Some common examples are:
* [[Logit model]]: The utility function is linear in the observed variables, and the unobserved component follows a [[Gumbel distribution]]. This model has a closed-form expression for the choice probabilities, and satisfies the [[independence of irrelevant alternatives]] (IIA) property, which means that the relative odds of choosing any two alternatives are unaffected by the availability of other alternatives.<ref> name="McFadden, D. (1974). Conditional logitLogit analysis of qualitative choice behavior. In P. Zarembka (Ed.), ''Frontiers in econometrics'' (pp. 105-142). New York: Academic Press.<Analysis"/ref>.
* [[Multivariate probit model]]: The utility function is linear in the observed variables, and the unobserved component follows a [[normal distribution]]. This model does not have a closed-form expression for the choice probabilities, and does not satisfy the IIA property. It is more flexible than the logit model, but also more computationally demanding<ref>Train, K. (2009). ''Discrete choice methods with simulation'' (2nd ed.). Cambridge: Cambridge University Press.{{pn}}{{isbn missing}}</ref>
* Nested logit model: The utility function is linear in the observed variables, and the unobserved component follows a Gumbel distribution. This model relaxes the IIA property by allowing the alternatives to be grouped into subsets, or nests, such that the IIA property holds within each nest, but not across nests. This model can capture the correlation among alternatives that share some common characteristics<ref>Ben-Akiva, M., & Lerman, S. (1985). ''Discrete choice analysis: Theory and application to travel demand''. Cambridge, MA: MIT Press.{{pn}}{{isbn missing}}</ref>.
* [[Mixed logit model]]: The utility function is linear in the observed variables, and the unobserved component follows a general distribution that can vary across individuals. This model allows for heterogeneity in preferences and random taste variation among individuals. It can also accommodate flexible substitution patterns among alternatives<ref>Train, K. (2003). ''Discrete choice methods with simulation''. Cambridge: Cambridge University Press.{{pn}}{{isbn missing}}</ref>
The RUMs can be estimated using various methods, such as [[maximum likelihood]], [[method of moments]], or [[Bayesian inference]]. The data used for estimation can be either aggregate or individual level. Aggregate data are data that have been summarized for each unique combination of the independent variables, such as market shares or voting outcomes. Individual level data are data that record the choices of each individual, such as survey responses or purchase histories.{{fact}}
The RUMs have many applications and extensions in various domains. For example, they can be used to model the choice of transportation modes, routes, or destinations; the choice of products, brands, or attributes; the choice of health care providers, treatments, or insurance plans; the choice of education, occupation, or ___location; the choice of political candidates, parties, or policies; and so on. They can also be extended to incorporate dynamic, strategic, or social aspects of choice behavior.{{fact}}
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== References ==
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[[Category:Utility]]
[[Category:Economics]]
[[Category:Utility function types]]
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