Bart Smeulders

Goodness-of-Fit Measures for Revealed Preference Tests: Complexity Results and Algorithms

We provide results on the computational complexity of goodness-of-fit measures (i.e., Afriat’s efficiency index, Varian’s efficiency vector-index, and the Houtman-Maks index) associated with several revealed preference axioms (i.e., WARP, SARP, GARP, and HARP). These results explain the computational difficulties that have been observed in literature when computing these indices. Our NP-hardness results are obtained by reductions from the independent set problem. We also show that this reduction can be used to prove that no approximation algorithm achieving a ratio of O(n1 − Δ), Δ ;> 0 exists for Varian’s index, nor for Houtman-Maks’ index (unless P = NP). Finally, we give an exact polynomial-time algorithm for finding Afriat’s efficiency index.

A Note on Testing Axioms of Revealed Preference

This Note presents an algorithm for testing the generalized axiom of revealed preference that runs in quadratic time. We show that this algorithm can be used to solve a more general problem on graphs. Furthermore, we prove a lower bound on the running time of any algorithm for testing different axioms of revealed preference.

Exact algorithms for the Equitable Traveling Salesman Problem

Given a weighted graph G = (V, E), the Equitable Traveling Salesman Problem (ETSP) asks for two perfect matchings in G such that (1) the two matchings together form a Hamiltonian cycle in G and (2) the absolute difference in costs between the two matchings is minimized. The problem is shown to be NP-Hard, even when the graph G is complete. We present two integer programming models to solve the ETSP problem. One model is solved through branch-and-bound-and-cut, whereas the other model is solved through a branch-and-price-and-cut framework. A simple local search heuristic is also implemented. We conduct computational experiments on different types of instances, often derived from the TSPLib. It turns out that the behavior of the different approaches varies with the type of instances; however, the branch-and-bound-and-cut approach implemented in Cplex seems to work best overall.

Testing a mixture model of single-peaked preferences

Single-Peaked preferences play an important role in the social choice literature. In this paper, we look at necessary and sufficient conditions for aggregated choices to be consistent with a mixture model of single-peaked preferences for a given ordering of the alternatives. These conditions can be tested in time polynomial in the number of choice alternatives. In addition, algorithms are provided which identify the underlying ordering of choice alternatives if the ordering is unknown. These algorithms also run in polynomial time, providing an efficient test for the mixture model of single-peaked preferences.

An O(n2) Algorithm for Testing the Generalized Axiom of Revealed Preference

This note presents an algorithm for testing the generalized axiom of revealed preference in time O(n2), where n is the number of observations in a given data set. Furthermore, we prove a lower bound of (n log n) on the running time of any algorithm for testing different axioms of revealed preference.

Revealed preference theory: an algorithmic outlook

Revealed preference theory is a domain within economics that studies rationalizability of behavior by (certain types of) utility functions. Given observed behavior in the form of choice data, testing whether certain conditions are satisfied gives rise to a variety of computational problems that can be analyzed using operations research techniques. In this survey, we provide an overview of these problems, their theoretical complexity, and available algorithms for tackling them. We focus on consumer choice settings, in particular individual choice, collective choice and stochastic choice settings.