Generalized Random Utility Models with Multiple Types
We propose a model for demand estimation in multi-agent, differentiated product settings and present an estimation algorithm that uses reversible jump MCMC techniques to classify agents' types. Our model extends the popular setup in Berry, Levinsohn and Pakes (1995) to allow for the data-driven classification of agents' types using agent-level data. We focus on applications involving data on agents' ranking over alternatives, and present theoretical conditions that establish the identifiability of the model and uni-modality of the likelihood/posterior. Results on both real and simulated data provide support for the scalability of our approach.
This paper addresses the problem of identifying the 
      type of each agents from his/her partial preference data, in order to use 
      this information to better estimate the underlying preferences for each 
      type. The authors propose a Generalized RUM to model the behavior of such 
      clustered agents. A reversible jump MCMC technique is used to estimate the 
      latent variables, including the types of the agents. A theoretical 
      analysis of the identifiability of the model and uni-modality of the 
      likelihood posterior are presented. Quality There are 
      three contributions of this paper. The new GRUM model, theoretical 
      analysis, and inference algorithm. The model is a generalization of RUM 
      model to multiple agents with types, which is new. Theoretical guarantees 
      are interesting, but have limitations discussed below in the significance 
      section. The inference algorithm is quite standard and the numerical 
      analysis is not impressive, either on the simulated data or the real world 
      datasets. Only the performance of estimating the number of clusters is 
      addressed, while the main problem is in clustering agents. A much more 
      relevant numerical simulation would be simulating how the number of 
      misclustering depends on problem parameters such as number of clusters, 
      how much the matrix W differ between clusters, missing data, etc. 
      Clarity Some claims could be better explained. On 
      page 2, it is not clear what the authors mean by the first paragraph of 
      section 1.1. Which aspects of the model eliminates unrealistic 
      substitution patterns? and avoid the situation where removing the top 
      choices result in the same alternative choice? On page 2, it is 
      not clear from the numerical results that "the clustering of types 
      provides a better fit to real world data". On page 5, in the 
      definition of `nice' pdfs, $\phi^{(n)}(x)$ is used without proper 
      definition, which makes the conditions difficult to understand. For 
      instance, given \phi is a pdf, g_n should be non-negative. But 
      g_n(x1)/g_n(x_2) converges to -1. The definition of `nice' cdf's 
      is not intuitive and no explanation is given as to why the model might not 
      be identifiable if noise cdf is not `nice'. On page 7, it is 
      claimed that "It can be seen that GRUM with 3 types has significantly 
      better performance than...". However, from the table, it seems like the 
      gain is only marginal. How significant is the gain of 2~3 % in the log 
      posterior? Originality This paper extends the definition 
      of RUM model to the setting where there are multiple alternatives and 
      multiple agents. The correlation between the agents are modelled via types 
      that an agent belongs to. RJ-MCMC approach seems to be quite standard. 
      Significance The main results on the theoretical 
      guarantees are interesting, but the application is limited. For theorem 1, 
      unimodality is only established when the types are known (as clearly 
      explained in the paper). This limits the convergence of MCMC approach, and 
      it is not clear how long one should run the MCMC in practice. This paper 
      does not explain why the proposed problem is difficult. Why has this 
      problem not been addresses so far, as the authors claim? Further, because 
      there is no comparisons either in theoretical results or numerical 
      results, it is difficult to judge how good the proposed algorithm is.