Suspicious Discontinuities: How Arbitrary Thresholds Drive Human Behavior

The Impact of Arbitrary Thresholds on Data and Behavior

Sharp discontinuities in policy or system design—often called "cliffs"—create incentives for individuals to manipulate their behavior to stay just below a specific threshold. This results in "suspicious discontinuities" in data, where distributions that should be smooth instead show unnatural spikes or drops at specific values. These patterns often reveal how people game systems to maximize their utility, sometimes even by intentionally losing money.

Economic and Tax Policy Discontinuities

U.S. tax and subsidy policies frequently utilize hard income cut-offs that disincentivize earning more and can actually incentivize decreasing income.

The ACA Subsidy Cliff

In the U.S., health insurance subsidies under the Affordable Care Act (ACA) have historically featured hard income cut-offs. For example, an individual earning slightly above the subsidy ceiling (e.g., $55,000) might face a health insurance cost increase of roughly $7,200 per year compared to someone just below the ceiling ($48,560). In such cases, an individual is financially better off reducing their income by $6,440 to qualify for the subsidy than they are earning the higher salary.

Other Welfare Thresholds

Similar discontinuities exist in other U.S. programs, including:

• TANF (Temporary Assistance for Needy Families) income limits.
• Medicaid income limits.
• CHIP (Children's Health Insurance Program) limits for both free and reduced-cost coverage.

To mitigate these effects, economists suggest using slow phase-outs (gradients) rather than sharp thresholds, which reduces the incentive for individuals to intentionally lose money or avoid higher earnings.

Academic and Professional Thresholds

When a specific number is required for success or certification, the data often clusters just above that number, indicating a tendency to "nudge" results.

P-Values in Psychology

In scientific publishing, particularly in psychology, there is a strong incentive to produce p-values below the 0.05 threshold for statistical significance. Analysis of published papers shows a curiously high number of p-values sitting just below 0.05, suggesting that authors may be fudging results, journals may be more likely to accept papers just under the threshold, or authors avoid submitting results that fall slightly above it (e.g., 0.055).

High School Exit Exams

In Polish language exit exams, data shows a spike of scores at exactly 30% (the failing threshold) and a dearth of scores between 23% and 29%. This is attributed to the fact that language grading is subjective; teachers often find a "missing" point to prevent a student from failing, as failing is considered a tragedy for the student and the school. This phenomenon is not seen in math exams, where objective grading makes such "nudges" nearly impossible.

College Admissions and Pell Grants

Universities using Pell Grant eligibility as a proxy for low-income student admission have created a new discontinuity. Students just below the Pell threshold have a significantly higher probability of admission than those just above it. This has led some parents to intentionally reduce their taxable income—through traditional IRAs or losing money on options—to bring their children under the Pell Grant threshold to increase admission chances.

Systemic and Behavioral Discontinuities

Youth Sports and the Relative Age Effect

Youth sports often bucket players by birth year. Because children born early in the year are physically more developed than those born later, they are more likely to be selected for elite teams (like the UEFA Youth League). This creates a sawtooth pattern in participation based on birth month, though the actual on-field value of the player (measured by playing time) is only weakly correlated with birth month.

Marathon Finishing Times

Histograms of millions of marathon finishes show spikes at every half hour and at "round" times (e.g., :10, :15, :20). This suggests that runners who are close to a round time speed up toward the end of the race to achieve a more satisfying numerical milestone.

Restaurant Inspections

NYC restaurant grading shows sharp discontinuities between scores of 13 and 14 (the boundary between an A and B grade) and between 27 and 28 (B and C grade). This indicates that inspectors may use their discretion to nudge a restaurant's score up to the next higher grade.

Detecting Fraud and Collusion

Election Statistics

Russian election data has shown curious spikes in turnout and results at "round" numbers (e.g., 95%), which often indicates fabricated results where the fraudster did not bother to create a smooth distribution.

Procurement Auctions

In Japanese government procurement, analysis of second-round bidding revealed that the second-lowest bidder almost never lowers their bid by more than the first-lowest bidder did. This sharp discontinuity at zero suggests collusion among bidders to ensure a specific firm wins without significantly lowering the price.

Technical Applications: Smoothing the Cliff

In software and hardware engineering, naive queues create a discontinuity: if the queue is full, packets are dropped; if not, they are not. This can be unfair to bursty workloads.

Random Early Detection (RED) is a common solution. By giving incoming items a probability of being dropped based on queue fullness, the system smooths out the discontinuity and mitigates the issues caused by a binary drop/no-drop threshold.