Content deleted Content added
No edit summary |
Fix cite date error |
||
Line 18:
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" />
== 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
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
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.
Line 49:
== References ==
{{Reflist}}
<references group="" responsive="1"></references>
|