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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</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 |last=Turansick |first=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 |volume=203 |pages=105489 |doi=10.1016/j.jet.2022.105489 |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>{{Cite book |last=Luce |first=R. Duncan |url=https://books.google.co.il/books?hl=iw&lr=&id=ERQsKkPiKkkC&oi=fnd&pg=PP1&dq=R.+Duncan+Luce.+Individual+Choice+Behavior:+A+Theoretical+Analysis.+Wiley,+1959.&ots=2jvv-vZggj&sig=0WtE8Ggx-CQamUsPjRarmOO5YUI&redir_esc=y#v=onepage&q=R.%20Duncan%20Luce.%20Individual%20Choice%20Behavior:%20A%20Theoretical%20Analysis.%20Wiley,%201959.&f=false |title=Individual Choice Behavior: A Theoretical Analysis |date=2012-06-22 |publisher=Courier Corporation |isbn=978-0-486-15339-1 |language=en}}</ref> and Plackett.<ref>{{Cite web |url=https://academic.oup.com/crawlprevention/governor?content=%2fjrsssc%2farticle-abstract%2f24%2f2%2f193%2f6953554 |access-date=2023-11-07 |website=academic.oup.com}}</ref> They assume that the random utility terms are generated according to [[Gumbel distribution|Gumbel distributions]]s with fixed shape parameter. In the Plackett-Luce model, the likelihood function has a simple analytical solution, so [[maximum likelihood estimation]] can be done in polynomial time.
 
The Plackett-Luce model was applied in [[econometrics]],<ref>{{Cite journal |last=D |first=Mcfadden |date=1974 |title=Conditional Logit Analysis of Qualitative Choice Behavior |url=https://cir.nii.ac.jp/crid/1572824500127838080 |journal=Frontiers in Econometrics}}</ref> for example, to analyze automobile prices in [[market equilibrium]].<ref>{{Cite journal |last=Berry |first=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 |issn=0012-9682}}</ref> It was also applied in [[Machine learning in earth sciences|machine learning]] and [[information retrieval]].<ref>{{Cite journal |last=Liu |first=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 |language=English |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>{{Cite journal |last=Gormley |first=Isobel Claire |last2=Murphy |first2=Thomas Brendan |date=June 2009-06 |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 |volume=4 |issue=2 |pages=265–295 |doi=10.1214/09-BA410 |issn=1936-0975}}</ref> Efficient methods for [[expectation-maximization]] and [[Expectation propagation]] exist for the Plackett-Luce model.<ref>{{Cite journal |last=Caron |first=François |last2=Doucet |first2=Arnaud |date=January 2012-01 |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 |language=en |volume=21 |issue=1 |pages=174–196 |doi=10.1080/10618600.2012.638220 |issn=1061-8600}}</ref><ref>{{Cite journal |last=Hunter |first=David R. |date=February 2004-02 |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 |volume=32 |issue=1 |pages=384–406 |doi=10.1214/aos/1079120141 |issn=0090-5364}}</ref><ref>{{Cite journal |last=Guiver |first=John |last2=Snelson |first2=Edward |date=2009-06-14 |title=Bayesian inference for Plackett-Luce ranking models |url=https://doi.org/10.1145/1553374.1553423 |journal=Proceedings of the 26th Annual International Conference on Machine Learning |series=ICML '09 |___location=New York, NY, USA |publisher=Association for Computing Machinery |pages=377–384 |doi=10.1145/1553374.1553423 |isbn=978-1-60558-516-1}}</ref>
 
Azari, [[David C. Parkes|Parkes]] and Xia<ref name=":4">{{Cite journal |last=Azari |first=Hossein |last2=Parks |first2=David |last3=Xia |first3=Lirong |date=2012 |title=Random Utility Theory for Social Choice |url=https://proceedings.neurips.cc/paper/2012/hash/a512294422de868f8474d22344636f16-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=25}}</ref> extend the Plackett-Luce model: they consider RUM in which the random utilities can be drawn from any distribution in the [[Exponential family]]. They prove conditions under which the log-likelihood function is concave, and the set of global maxima solutions is bounded for a family of RUMs where the shape of each distribution is fixed and the only latent variables are the means.
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== References ==
{{Reflist}}
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