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Multi-dimensional Test Design

Xiaoyun Qiu, Liren Shan

TL;DR

The paper analyzes jointly designing tests and testing procedures when a principal screens a two-dimensional agent type under two technologies: manipulation (misrepresentation) and investment (genuine improvement). It shows a sharp dichotomy: manipulation favors a fixed-order sequential mechanism with stringent tests, while investment favors a simultaneous (or random-order) mechanism with less stringent tests, with conditions where random-order sequential can match first-best outcomes. The results are developed via geometric and constructive arguments about agent best responses under different procedures, establishing dominance relationships between mechanism classes and extending to perfect tests, cheap-talk, and robustness to cost functions. The framework connects to real-world regulatory and organizational settings, explaining why European and US banking regulations or merger approvals may be shaped by whether agents tamper with signals or invest in genuine improvements. Overall, the work highlights how the dimensionality of types and the agent’s technology determine the optimal institutional design for multi-criterion screening.

Abstract

How should one jointly design tests and the arrangement of agencies to administer these tests (testing procedure)? To answer this question, we analyze a model where a principal must use multiple tests to screen an agent with a multi-dimensional type, knowing that the agent can change his type at a cost. We identify a new tradeoff between setting difficult tests and using a difficult testing procedure. We compare two settings: (1) the agent only misrepresents his type (manipulation) and (2) the agent improves his actual type (investment). Examples include interviews, regulations, and data classification. We show that in the manipulation setting, stringent tests combined with an easy procedure, i.e., offering tests sequentially in a fixed order, is optimal. In contrast, in the investment setting, non-stringent tests with a difficult procedure, i.e., offering tests simultaneously, is optimal; however, under mild conditions offering them sequentially in a random order may be as good. Our results suggest that whether the agent manipulates or invests in his type determines which arrangement of agencies is optimal.

Multi-dimensional Test Design

TL;DR

The paper analyzes jointly designing tests and testing procedures when a principal screens a two-dimensional agent type under two technologies: manipulation (misrepresentation) and investment (genuine improvement). It shows a sharp dichotomy: manipulation favors a fixed-order sequential mechanism with stringent tests, while investment favors a simultaneous (or random-order) mechanism with less stringent tests, with conditions where random-order sequential can match first-best outcomes. The results are developed via geometric and constructive arguments about agent best responses under different procedures, establishing dominance relationships between mechanism classes and extending to perfect tests, cheap-talk, and robustness to cost functions. The framework connects to real-world regulatory and organizational settings, explaining why European and US banking regulations or merger approvals may be shaped by whether agents tamper with signals or invest in genuine improvements. Overall, the work highlights how the dimensionality of types and the agent’s technology determine the optimal institutional design for multi-criterion screening.

Abstract

How should one jointly design tests and the arrangement of agencies to administer these tests (testing procedure)? To answer this question, we analyze a model where a principal must use multiple tests to screen an agent with a multi-dimensional type, knowing that the agent can change his type at a cost. We identify a new tradeoff between setting difficult tests and using a difficult testing procedure. We compare two settings: (1) the agent only misrepresents his type (manipulation) and (2) the agent improves his actual type (investment). Examples include interviews, regulations, and data classification. We show that in the manipulation setting, stringent tests combined with an easy procedure, i.e., offering tests sequentially in a fixed order, is optimal. In contrast, in the investment setting, non-stringent tests with a difficult procedure, i.e., offering tests simultaneously, is optimal; however, under mild conditions offering them sequentially in a random order may be as good. Our results suggest that whether the agent manipulates or invests in his type determines which arrangement of agencies is optimal.

Paper Structure

This paper contains 75 sections, 33 theorems, 21 equations, 25 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Simultaneous mechanisms
  4. Sequential mechanisms
  5. An example and preliminary analysis
  6. Preliminary analysis
  7. Manipulation
  8. Optimal sequential mechanism
  9. Cheap-talk
  10. Comparison to simultaneous mechanisms
  11. Investment
  12. Optimal sequential mechanism
  13. Comparison to simultaneous mechanisms
  14. Robustness
  15. Robustness to tests and qualified region
  16. ...and 60 more sections

Key Result

Theorem 1

Suppose $M=\varnothing$. For any distribution $\mathcal{F}$, the optimal sequential mechanism uses

Figures (25)

  • Figure 1: Hiring requirements for a senior scientist position
  • Figure 2: Two-step strategy: $\boldsymbol{x}^0\rightarrow \boldsymbol{x}^1\rightarrow \boldsymbol{x}^2$
  • Figure 3: Stringency of procedures: random-order without disclosure vs fixed-order
  • Figure 4: Lack of nested structure: random-order procedure with disclosure vs fixed-order procedures
  • Figure 5: An oversimplified intuition
  • ...and 20 more figures

Theorems & Definitions (75)

  • Remark
  • Theorem 1
  • Lemma 1
  • Lemma 2
  • Theorem 2
  • Proposition 1
  • Theorem 3
  • Lemma 3
  • proof : Proof of \ref{['thm: true effort sequential']}
  • Proposition 2
  • ...and 65 more