Content deleted Content added
No edit summary |
m Dating maintenance tags: {{Self-published inline}} {{Pn}} {{Full}} |
||
Line 16:
=== 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>https://scholar.harvard.edu/files/tomasz/files/lisbon32-post.pdf{{full|date=November 2023}}{{self-published inline|date=November 2023}}</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 }}</ref> presents two characterizations for the existence of a unique random utility representation.
== Models ==
There are various random utility models, which differ in the assumptions on the probability distributions of the agent's utility, A popular random utility model 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=November 2023}}</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 |doi=10.2307/2346567 }}</ref> They assume that the random utility terms are generated according to [[Gumbel distribution]]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 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 |title=Automobile Prices in Market Equilibrium |journal=Econometrica |date=1995 |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 }}</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 }}</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 journal |doi=10.1145/1553374.1553423 }}</ref>
|