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 ==

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>{{cite journal |last1=Camerer |first1=Colin F. |title=An experimental test of several generalized utility theories |journal=Journal of Risk and Uncertainty |date=April 1989 |volume=2 |issue=1 |pages=61–104 |doi=10.1007/BF00055711 |s2cid=154335530 }}</ref><ref>{{cite journal |last1=Starmer |first1=Chris |last2=Sugden |first2=Robert |title=Probability and juxtaposition effects: An experimental investigation of the common ratio effect |journal=Journal of Risk and Uncertainty |date=June 1989 |volume=2 |issue=2 |pages=159–178 |doi=10.1007/BF00056135 |s2cid=153567599 }}</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>{{cite journal |last1=Wu |first1=George |title=An empirical test of ordinal independence |journal=Journal of Risk and Uncertainty |date=1994 |volume=9 |issue=1 |pages=39–60 |doi=10.1007/BF01073402 |s2cid=153558846 }}</ref><ref>{{cite journal |last1=Ballinger |first1=T. Parker |last2=Wilcox |first2=Nathaniel T. |title=Decisions, Error and Heterogeneity |journal=The Economic Journal |date=July 1997 |volume=107 |issue=443 |pages=1090–1105 |doi=10.1111/j.1468-0297.1997.tb00009.x |s2cid=153823510 }}</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">{{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|date=July 2024}}

[[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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=== 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 }}{{Bare URLpn|date=July PDF2024}}</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|date=July 2024}}</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 |hdl=10197/7121 |hdl-access=free }}</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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* The maximum likelihood estimator - which is the [[Kemeny–Young method]] - is hard to compute (it is <math>\Theta^P_2</math>-complete).<ref>{{cite journal |last1=Hemaspaandra |first1=Edith |last2=Spakowski |first2=Holger |last3=Vogel |first3=Jörg |title=The complexity of Kemeny elections |journal=Theoretical Computer Science |date=December 2005 |volume=349 |issue=3 |pages=382–391 |doi=10.1016/j.tcs.2005.08.031 }}</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 ==

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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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[[Category:Utility]]

[[Category:Economics]]

[[Category:Utility function types]]