High-Dimensional Linear Preference Model

Multiple-criteria decision analysis (MCDA) is a branch of operations research concerned with choices based on several quantifiable factors. Here, we focus on high-dimensional markets, in which each available product is described by a large number of features. Taking a probabilistic approach, we define a market as a collection of alternatives in a decision-making scenario governed by a linear utility function. Analytic approximations for the market share and its moments are derived in the limit of a large population and a large number of measured features. We identify a single parameter, termed the degree of subjectivity, that places markets on a continuous spectrum ranging from fully objective to fully subjective. At an intermediate value, the market is competitive in the sense that it maximizes the entropy of the market-share distribution. Empirical analysis of several real markets indicates that they can indeed be classified by this parameter, yielding predictable decision patterns and a unified, relative measure of competitiveness across markets. Simulations involving non-linear utility functions and a trained machine-learning classifier provide preliminary evidence that similar behavior may also arise beyond the linear model, suggesting that the degree of subjectivity may be useful as a diagnostic in some broader multi-feature decision problems.