Optimal designs for discrete choice models via graph Laplacians

In discrete choice experiments, the information matrix depends on the model parameters. Therefore, $D$-optimal designs are only locally optimal in the parameter space. This dependence renders the optimization problem very difficult, as standard theory encodes $D$-optimality in systems of highly nonlinear equations and inequalities. In this work, we connect design theory for discrete choice experiments with Laplacian matrices of connected graphs. We rewrite the $D$-optimality criterion in terms of Laplacians via Kirchhoff's matrix tree theorem, and show that its dual has a simple description via the Cayley--Menger determinant of the Farris transform of the Laplacian matrix. This results in a drastic reduction of complexity and allows us to implement a gradient descent algorithm to find locally $D$-optimal designs. For the subclass of Bradley--Terry paired comparison models, we find a direct link to maximum likelihood estimation for Laplacian-constrained Gaussian graphical models. This implies that every locally $D$-optimal design is a rational function in the parameter when the design is supported on a chordal graph. Finally, we study the performance of our algorithm and demonstrate its application on real and simulated data.