ABSTRACT
In Section 5.6.1, the Plackett-Luce model is given as a multistage model, which we extend in this chapter. Section 5.13.1 notes that it is also a Thurstonian model. The basic model is based on the positive constants Vi, i = 1, ... , m, where Vi is associated with Object i. The larger the parameter, the more preferred is the object. The Vi's are proportional to the probability that Object i is ranked first:
(8.1)
where W E Tm is a random ordering. The key assumption for the first-order model is that the conditional probability that Object j is ranked k given the Objects ranked 1 through k - 1 is also proportional to the same parameters:
The probability of any ordering is then easily obtained in stages, the same stages as in the ¢ component model of Section 5.6.2, but with different probabilities. Thus, as in (5.13),
(8.3)
Note that the parameters are identifiable only up to a multiplicative constant. This model assumes no interaction between the
objects, as formalized by the Luce choice axiom in Section 5.13.1. In the Voting data, for example, this property is likely not to hold, since people who rank Candidate A first are more prone to rank Candidate C high than are those who rank Candidates D or E first. See Example 2.15.
