Selection Procedures in Competitive Admission

1. Problem Statement

The paper investigates how organizations (firms or universities) design their selection procedures (tests and acceptance rules) in a competitive environment where they vie for candidates of unknown productivity. The core tension lies in the interaction between two factors:

1. Statistical Properties: The accuracy and difficulty of the test used to screen candidates.
2. Strategic Selection: How candidates, who are privately informed about their own productivity, choose which firm to apply to based on the firm's selection procedure.

The author asks: How does competition between firms shape the statistical properties (accuracy and difficulty) of the tests they employ, and how does this differ from a monopolistic or non-competitive setting?

2. Methodology and Model Setup

The Game Structure

• Agents: Two firms and a continuum of candidates with mass 1. Candidates have private types $\theta \in [\underline{\theta}, \bar{\theta}]$ representing productivity, distributed according to CDF $F$.
• Firms' Strategy: Firms simultaneously post a selection procedure $s = (t, \alpha)$, consisting of:
  • A test $t$ (a Blackwell experiment with binary signals $h$ and $l$).
  • An acceptance rule $\alpha: \{h, l\} \to [0, 1]$ (probability of acceptance given the signal).
• Candidates' Strategy: Candidates observe the procedures and choose to apply to exactly one firm (due to application costs or capacity constraints like internships). They maximize their probability of acceptance.
• Payoffs:
  • Candidates: $u = 1$ if accepted, $0$ otherwise.
  • Firms: Expected productivity of accepted candidates. The baseline model assumes no capacity constraints and fixed wages (or no wage offers), meaning firms want to accept any candidate with positive expected productivity.

Key Concepts: Accuracy and Difficulty

The author utilizes two partial orders to characterize tests, moving beyond standard Blackwell informativeness:

1. Accuracy ($\succeq_a$): Based on Lehmann (1988). A test is more accurate if it provides more precise information about the type, satisfying the Monotone Likelihood Ratio Property (MLRP). Higher accuracy means the likelihood ratio of signals is more sensitive to the type.
2. Difficulty ($\succeq_d$): Based on Hancart (2026). A test is more difficult if a high signal shifts posterior beliefs more strongly toward high types, while a low signal shifts them more strongly toward low types. Intuitively, a difficult test is harder to pass for everyone, but especially for low types.

Equilibrium Concept

The paper focuses on Symmetric Subgame Perfect Equilibria (SSPE).

3. Key Results

A. The Baseline Model (No Capacity Constraints, No Wage Offers)

In the standard competitive setting where firms can accept unlimited candidates and cannot offer differentiated wages:

• Theorem 1: There exists a unique symmetric equilibrium where firms choose a test that is maximally accurate but minimally difficult (among those providing payoff-relevant information).
  • Mechanism: Competition drives profits to zero (Bertrand-style logic).
  • Positive Selection into Accuracy: Higher types benefit more from a more accurate test.
  • Negative Selection into Difficulty: If a firm makes a test more difficult, it must relax its acceptance rule (accept more low types) to attract candidates. Low types benefit disproportionately from this leniency. Thus, a more difficult test attracts a "worse" pool of candidates (negative selection).
  • Result: Firms end up with "misguided learning." They possess precise information (high accuracy) but use it poorly. They accept candidates even after a low signal (if the test is easy enough) or set the difficulty so low that the high signal conveys little new information. The posterior expectation after a high signal is barely positive (often zero).

B. Comparative Statics

• Stronger Applicant Pool: If the distribution of candidates shifts to the right (in the likelihood ratio order), the equilibrium test becomes less difficult, and acceptance probabilities increase for all candidates.
• Improved Technology: If the set of feasible tests expands (better technology), average acceptance probabilities increase, though some low types may be worse off.

C. Extensions: Capacity Constraints and Wage Competition

The paper contrasts the baseline with two modifications that alter the selection incentives:

1. Capacity Constraints (Proposition 4):

   • Setup: Firms have a fixed capacity $k$.
   • Result: Firms use the most difficult test possible.
   • Mechanism: Once at capacity, firms cannot "undercut" by accepting more candidates. They only care about the quality of the marginal candidate. A difficult test ensures that the high signal is highly informative (high posterior mean). Since they only accept high signals, selection into the difficult test becomes positive (high types prefer the difficult test because it signals their quality more effectively). This leads to "cherry-picking."

2. Wage Competition (Proposition 6):

   • Setup: Firms can offer wages conditional on the signal.
   • Result: Firms use the most difficult test.
   • Mechanism: Competition shifts from acceptance probability to wages. Firms compete by offering higher wages to high-signal candidates. High types are willing to take the risk of a difficult test in exchange for a higher wage if they pass. This creates positive selection into difficult tests.

D. Asymmetric Equilibria (Proposition 5)

• Without Capacity Constraints: No asymmetric equilibria exist; firms always converge to the same procedure.
• With Capacity Constraints: A two-tier equilibrium can emerge. One firm becomes "selective" (attracting only high types with a difficult/accurate test), and the other becomes "safe" (attracting lower types). This endogenously differentiates ex-ante identical firms.

4. Contributions

1. Endogenizing Test Design: Unlike previous literature that treats testing technology as fixed, this paper models the choice of both accuracy and difficulty as strategic variables.
2. The "Misguided Learning" Paradox: The paper demonstrates that competition can lead to an equilibrium where firms acquire maximal information (accuracy) but fail to utilize it optimally (minimal difficulty), resulting in suboptimal selection compared to a monopolist.
3. New Orders for Test Comparison: It formally applies and extends the concepts of Accuracy (Lehmann) and Difficulty (Hancart) to competitive screening games, showing how these dimensions drive candidate self-selection.
4. Contextualizing Selectivity: The model offers a theoretical explanation for the observed decline in selectivity in US college admissions (Hoxby, 2009). It suggests that as the applicant pool improves or testing technology advances, institutions may rationally lower their difficulty thresholds to maximize applicant volume, even if they maintain high accuracy.

5. Significance and Implications

• Policy Relevance: The findings suggest that simply increasing the "rigor" of admissions (difficulty) without considering capacity constraints or wage structures may not lead to better selection. In unconstrained markets, firms may actually lower difficulty to attract a broader pool, leading to "lemon-dropping" (accepting low-quality candidates).
• Institutional Design: For universities or firms facing capacity constraints, the model predicts a shift toward "cherry-picking" (high difficulty), whereas those in oversupply markets will shift toward "lemon-dropping" (low difficulty, high accuracy).
• Literature Connection: It bridges the gap between information design (certification) and competitive search markets, showing how the strategic interaction between the designer (firm) and the agent (candidate) fundamentally alters the statistical properties of the information revealed.

In summary, Hancart provides a rigorous game-theoretic framework showing that the nature of competition (constrained vs. unconstrained, wage vs. non-wage) dictates whether organizations use tests to filter out the worst (easy tests) or identify the best (difficult tests).