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Identification in Stochastic Choice

Peter Caradonna, Christopher Turansick

Abstract

We characterize the identified sets of a wide range of stochastic choice models, including random utility, various models of boundedly-rational behavior, and dynamic discrete choice. In each of these settings, we show two distributions over choice rules are observationally equivalent if and only if they can be obtained from one another via a finite sequence of simple swapping transforms. We leverage this to obtain complete descriptions of both the defining inequalities and extreme points of these identified sets. In cases where choice frequencies vary smoothly with some parameters, we provide a novel global-inverse result for practically testing identification.

Identification in Stochastic Choice

Abstract

We characterize the identified sets of a wide range of stochastic choice models, including random utility, various models of boundedly-rational behavior, and dynamic discrete choice. In each of these settings, we show two distributions over choice rules are observationally equivalent if and only if they can be obtained from one another via a finite sequence of simple swapping transforms. We leverage this to obtain complete descriptions of both the defining inequalities and extreme points of these identified sets. In cases where choice frequencies vary smoothly with some parameters, we provide a novel global-inverse result for practically testing identification.
Paper Structure (30 sections, 27 theorems, 72 equations, 1 figure, 3 tables)

This paper contains 30 sections, 27 theorems, 72 equations, 1 figure, 3 tables.

Table of Contents

  1. Introduction
  2. The Random Utility Model
  3. The Geometry of Identification
  4. Compatible Pairs and the Ryser Subspace
  5. Support Restrictions and Extreme Points
  6. Order-Based Restrictions
  7. Parametric Restrictions
  8. Beyond Random Utility
  9. Related Literature
  10. Conclusion
  11. Graphical Results
  12. Definitions & Preliminaries
  13. Graphical Results and Proofs
  14. Graphical Support Restrictions
  15. Graphical Ordered Results
  16. ...and 15 more sections

Key Result

Proposition 1

Let $\succ_1, \succ_2 \in \mathcal{L}$ be $k$-compatible, with $k$-conjugates $\succ_1'$ and $\succ_2'$. Then the uniform distributions on $\{\succ_1, \succ_2\}$ and $\{\succ_1', \succ_2'\}$ are observationally equivalent.

Figures (1)

  • Figure 1: By \ref{['conjugatelemma']}, $\mu_{12}$ and $\mu_{34}$ are observationally equivalent. By the linearity of \ref{['measuretochoiceprobmap']}, both are also equivalent to any distribution in their convex hull, and any pair of distributions belonging to some common translation of this set (e.g. the dashed segments) are observationally equivalent.

Theorems & Definitions (51)

  • Example 1
  • Proposition 1
  • Example 2
  • Theorem 1
  • Example 3
  • Example 4
  • Example 5
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • ...and 41 more