Random utility model: Difference between revisions

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== 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 |journal=Econometrica |volume=62 |issue=6 |pages=1291–1326 |doi=10.2307/2951750 |jstor=2951750 |s2cid=120069179 }}</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.

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There are various RUMs, which differ in the assumptions on the probability distributions of the agent's utility, A popular RUM 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=193–202 |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> 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 book |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 ==