Abstract
Support vector machines (SVMs) have been successfully used to identify individuals’ preferences in conjoint analysis. One of the challenges of using SVMs in this context is to properly control for preference heterogeneity among individuals to construct robust partworths. In this work, we present a new technique that obtains all individual utility functions simultaneously in a single optimization problem based on three objectives: complexity reduction, model fit, and heterogeneity control. While complexity reduction and model fit are dealt using SVMs, heterogeneity is controlled by shrinking the individual-level partworths toward a population mean. The proposed approach is further extended to kernel-based machines, conferring flexibility to the model by allowing nonlinear utility functions. Experiments on simulated and real-world datasets show that the proposed approach in its linear form outperforms existing methods for choice-based conjoint analysis.
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Acknowledgments
The authors thank Olivier Toubia and Bryan Orme for providing the data for the two empirical applications. The first author was funded by FONDECYT project 1160894 and by CONICYT Anillo ACT1106. The second author was supported by FONDECYT projects 1140831 and 1160738. The third author was supported by FONDECYT project 1151395 and FONDEF Project IT13I20031. This research was partially funded by the Complex Engineering Systems Institute, ISCI (ICM-FIC: P05-004-F, CONICYT: FB0816).
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Appendices
Appendix A
Strictly convexity of problem (6)
In order to prove that our Formulation (6) is strictly convex, we first rewrite it in a compact form. For this, we follow the derivation of Yajima (2005) for multicategory SVM. Let us denote by
and
where \(I_J\) denotes the identity matrix of size J. Then, the quadratic term in (6) can be expressed as
Proposition 1
For any \(\theta >0\), the matrix \({\mathcal{Q}}(\theta )\) is symmetric definite positive. Moreover,
Proof
It is clear that the matrix \({\mathcal{Q}}(\theta )\) is symmetric definite positive (cf. (12)). Now, we denote by \(F_i\in {\mathfrak {R}}^{J\times J(N+1)}\) and \(C_i\in {\mathfrak {R}}^{J(N+1)\times J}\) the i-th block (in row) and the i-th block (in column) of \({\mathcal{Q}}(\theta )\) and \({\mathcal{Q}}(\theta )^{-1}\), respectively. Then,
Thus, the result follows. \(\square\)
Appendix B
Dual formulation of problem (6)
Let us denote by \({\varvec{\xi }}_t^k=(\xi _{1t}^k,\ldots ,\xi _{Nt}^k)\in {\mathfrak {R}}^N\), and by \(\mathbf{X}^k_{t}=\left[ \begin{array}{l} 0\\ X^k_{t} \end{array} \right] \in {\mathfrak {R}}^{(N+1)J\times N}\) with
Then, the constraints of the problem (6) can be expressed as follows
With this notation, the Lagrangian function associated to formulation (6) is given by
Then, Problem (6) can be written equivalently as
Hence, the dual formulation (see eg, Bertsekas, 1982) of (6) is given by
The above expression enables us to compute the dual problem based only on the Lagrange multipliers \(\varvec{\alpha }\). The first-order conditions of the inner minimization problem yields to
Since \(\mathbf{s}_t^k\ge 0\), from (16) it follows that \({\varvec{\alpha }}_t^k\le C\mathbf{e}\) for \(t=1,\ldots ,T\), and \(k=2,\ldots ,K\).
Remark 1
Note that using (1), (15), and the notation of \(\mathbf{X}_t^k\), we have that
On the other hand, by using (15) and (16) in (14), we obtain that
Since \({\mathcal{Q}}(\theta )\) is nonsingular, it follows from (15) that the above expression can be written as
The following result allows us to rewrite the above equality.
Proposition 2
For \(\theta >0\), let \(\widetilde{{\mathcal{Q}}}(\theta )= NI_{JN}+\theta \mathcal{J}\in {\mathfrak {R}}^{JN\times JN}\), where \(I_{JN}\) denotes the identity matrix of size JN and
Then, there exists a symmetric matrix \(\widetilde{{\mathcal{Q}}}(\theta )^{1/2}\) satisfying \((\widetilde{{\mathcal{Q}}}(\theta )^{1/2})^2=\widetilde{{\mathcal{Q}}}(\theta )\).
Proof
Let
Note that \(\mathcal{J}^2=NJ\). Then,
\(\square\)
By using the relation (13), Proposition 2, and the definition of \(\mathbf{X}_t^k\), the expression (17) reduces to
Hence, the dual formulation is given by
Remark 2
From (15) and (13), it follows that
and
for \(i=1,\ldots ,N\).
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López, J., Maldonado, S. & Montoya, R. Simultaneous preference estimation and heterogeneity control for choice-based conjoint via support vector machines. J Oper Res Soc 68, 1323–1334 (2017). https://doi.org/10.1057/s41274-016-0013-6
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DOI: https://doi.org/10.1057/s41274-016-0013-6