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Title: Optimal discrete choice designs via hypergraphs Authors:  Frank Roettger - Université de Genève (Switzerland) [presenting]
Rainer Schwabe - Otto-von-Guericke University Magdeburg (Germany)
Thomas Kahle - Otto-von-Guericke University Magdeburg (Germany)
Abstract: Multinomial discrete choice models are studied to compare $m$ unstructured alternatives in choice sets of constant size $k<m$. To describe the choice sets, we use a representation by uniform hypergraphs, where each choice set forms a hyperedge on $k$ vertices in the complete hypergraph. In this model, finding $D$-optimal designs corresponds to a parameterized convex optimization problem. This relates by the Kiefer--Wolfowitz equivalence theorem to directional derivatives being non-positive in general and zero for choice sets in the support of the design. We show that via the linear Farris transform of the inverse information matrix, the system of directional derivatives is determined from a hyperedge-edge incidence matrix. For designs supported on $\binom{m}{2}$ choice sets the number of equations in the system of directional derivatives equals the number of unknowns, such that the system has a unique solution when the corresponding submatrix of the hyperedge-edge incidence matrix is invertible. In this case, the design weights are easily obtained from the inverse information matrix. In the Bradley--Terry paired comparison model ($k=2$), the edge-edge incidence matrix is diagonal, and therefore always invertible. This implies that for designs supported on all $\binom{m}{2}$ edges, we always obtain the optimal design weights via a simple matrix inversion. Furthermore, this allows us to derive optimal weights for any design that is supported on a decomposable graph.