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Existence in multidimensional screening with general nonlinear preferences

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Abstract

We generalize the approach of Carlier (J Math Econ 35, 129–150, 2001) and provide an existence proof for the multidimensional screening problem with general nonlinear preferences. We first formulate the principal’s problem as a maximization problem with G-convexity constraints and then use G-convex analysis to prove existence.

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Notes

  1. It is worth mentioning that in some literature, the monopolist’s objective is to design a product line \(\tilde{Y}\) (i.e., a subset of \(\mathrm{cl}(Y)\)) and a price menu \(\tilde{p}: \tilde{Y} \rightarrow \mathbf {R}\) that jointly maximize the overall monopolist profit. Then, given \(\tilde{Y}\) and \(\tilde{p}\), an agent of type x chooses the product y(x) that solves

    $$\begin{aligned} \max _{y \in \tilde{Y}} G(x,y, \tilde{p}(y)):= u(x). \end{aligned}$$

    Allowing the price to take value \(\bar{z}\) (which may be \(+\infty \)), and assuming Assumption 1, the effect of designing a product line \(\tilde{Y}\) and price menu \(\tilde{p}: \tilde{Y}\rightarrow \mathbf {R}\) is equivalent to that of designing a price menu \(p : \mathrm{cl}(Y)\rightarrow (-\infty , +\infty ]\), which equals \(\tilde{p}\) on \(\tilde{Y}\) and maps \(\mathrm{cl}(Y) {\setminus } \tilde{Y}\) to \(\bar{z}\), such that no agents choose to purchase any product from \(\mathrm{cl}(Y) {\setminus } \tilde{Y}\), which is less attractive than the outside option \(y_{\emptyset }\) according to Assumption 1. In this paper, we use the latter as the monopolist’s objective.

    For any given price menu \(p: \mathrm{cl}(Y)\rightarrow (-\infty , +\infty ]\), one can construct a mapping \(y: X \rightarrow \mathrm{cl}(Y)\) such that each y(x) solves the maximization problem in (2.1). But such mapping is not necessarily unique, without the single-crossing type assumptions. Therefore, we adopt in (2.2) the total profit as a functional of both price menu p and its corresponding mapping y.

  2. In Trudinger (2014), this point-to-set mapping \(\partial ^G u\) is also called G-normal mapping; see this paper for more properties related to G-convexity.

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  1. Department of Mathematics, University of Toronto, Toronto, ON, M5S 2E4, Canada

    Kelvin Shuangjian Zhang

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This paper is based on Chapter 3 of the author’s thesis Zhang (2018). The author would like to express his deepest gratitude to his Ph.D. advisor Robert J. McCann for leading him to this project, and for his guidance and inspiration throughout. The author is grateful to Xianwen Shi, Alfred Galichon, Guillaume Carlier and Ivar Ekeland for stimulating conversations and encouragement, as well as to Georg Nöldeke and Larry Samuelson for sharing their work in preprint form and vital remarks. This project was initiated during the Fall of 2013 when the author was in residence at the Mathematical Sciences Research Institute in Berkeley CA, under a program supported by National Science Foundation Grant No. 0932078 000, and progressed during the Fall 2014 program of the Fields Institute for the Mathematical Sciences.

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Zhang, K.S. Existence in multidimensional screening with general nonlinear preferences. Econ Theory 67, 463–485 (2019). https://doi.org/10.1007/s00199-018-1170-4

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