Cross-Admit Competition Dynamics in College Admissions
cross_admit_dynamics.md
Cross-Admit Competition Dynamics in College Admissions
Overview
Cross-admit data reveals how students choose between colleges when admitted to multiple institutions simultaneously. Unlike acceptance rates (which measure selectivity) or yield rates (which blend all admitted students), cross-admit win rates isolate the head-to-head competition between specific pairs of schools. This research synthesizes data from Parchment (now PowerSchool/College Results), the Avery-Hoxby-Glickman-Metrick revealed preference study (NBER/QJE 2013), Crimson Education's cross-yield rankings, and admissions counselor analyses.
Relevance to simulation: The studentFinalDecisions() function currently uses a 6-factor weighted score (prestige tier, archetype fit, legacy, personal noise, Chetty income-yield, in-state preference) plus a HYPSM override. Cross-admit data can improve this by adding pairwise "desirability scores" that capture revealed preferences beyond simple tier ordering.
Data Sources
Parchment / College Results
Parchment's cross-admit comparison tool uses a "revealed preference" methodology. The denominator includes all students admitted to both schools; the numerator is those who chose a given school. Confidence intervals use Wilson's method at 95%. Sample sizes vary; elite-vs-elite matchups typically have medium-to-large samples (50-300+ students), while cross-tier matchups can be very large (1000+). Data is self-reported by students who use Parchment's transcript service.
Limitations: Geographic and demographic skew in Parchment's user base; students who use Parchment may not be representative of all cross-admits. Small samples for some LAC matchups.
Avery-Hoxby-Glickman-Metrick (NBER Working Paper 10803, QJE 2013)
Used hand-collected data on 3,240 high-achieving students to construct a revealed preference ranking of 105 U.S. colleges. Extended models from tournament rankings (Elo/chess-style). Top 5: Harvard, Yale, Stanford, Caltech, MIT. Key finding: Washington University in St. Louis ranked 11th in U.S. News but 62nd in revealed preference, demonstrating that traditional rankings diverge significantly from actual student choices.
Crimson Education Cross-Yield Rankings (2023/24)
Based on Crimson's global student network, using an Elo-based system. Their data shows a somewhat different ordering, with Stanford #1 and MIT #2 above Harvard #3, though this may reflect their international/STEM-heavy student base.
HYPSM Head-to-Head Win Rates
The five HYPSM schools (Harvard, Yale, Princeton, Stanford, MIT) form the apex of cross-admit competition. Data below is from Parchment unless otherwise noted.
Parchment Data (current)
| Matchup | Winner | Win % | Loser | Lose % | CI Range |
|---|---|---|---|---|---|
| Harvard vs. Yale | Harvard | 63% | Yale | 37% | H: 58.8-68.2% |
| Harvard vs. Princeton | Harvard | 73% | Princeton | 27% | H: 66.4-80.1% |
| Harvard vs. Stanford | Harvard | 65% | Stanford | 35% | H: 59.0-70.0% |
| Harvard vs. MIT | MIT | 63% | Harvard | 37% | M: 54.8-71.1% |
| Yale vs. Princeton | Yale | 59% | Princeton | 41% | Y: 51.5-66.0% |
| Yale vs. Stanford | Yale | 53% | Stanford | 47% | Y: 46.0-59.1% |
| Yale vs. MIT | MIT | 61% | Yale | 39% | M: 52.6-68.5% |
| Princeton vs. Stanford | Stanford | 73% | Princeton | 27% | S: 64.7-81.7% |
| Princeton vs. MIT | MIT | 58% | Princeton | 42% | M: 46.6-69.3% |
| Stanford vs. MIT | MIT | 61% | Stanford | 39% | M: 53.0-69.3% |
Aggregate HYPSM Rankings (Parchment-derived)
Based on total cross-admit points across all pairwise matchups:
- MIT -- Wins 3 of 4 HYPSM matchups (loses only to Harvard). Strongest STEM signal.
- Harvard -- Wins 3 of 4 (loses only to MIT). Dominant brand.
- Stanford -- Wins 2 of 4 (beats Princeton, loses to Harvard/MIT, near-tie with Yale).
- Yale -- Wins 1 of 4 (beats Princeton, near-tie with Stanford, loses to Harvard/MIT).
- Princeton -- Wins 0 of 4. Despite top U.S. News ranking, loses every HYPSM cross-admit battle.
Important Methodological Note
Different data sources yield different orderings. The Mathacle blog analysis of Classes of 2012-2014 (from a specific high school dataset) showed Harvard losing most HYPSM matchups, which contradicts Parchment data. Crimson Education's cross-yield ranking placed Stanford #1 and MIT #2. The Avery-Hoxby academic study had Harvard #1, Yale #2.
These discrepancies arise from: - Sample composition: STEM-heavy samples boost MIT/Stanford; humanities/policy-heavy samples boost Harvard/Yale - Geographic bias: West Coast respondents favor Stanford; East Coast respondents favor Harvard - Temporal shifts: MIT and Stanford have gained ground over the past decade - Self-selection: Parchment users skew toward students who actually make competitive choices
For simulation purposes: We use Parchment data as the primary source (largest sample, most current) while noting that individual student archetype should modulate these baseline rates.
Crimson Education Rankings vs. Parchment
Crimson's cross-yield data (from their own student network) shows different HYPSM ordering: - Harvard over Princeton: 70% - Harvard over Yale: 77% - Harvard over Stanford: 47% (Stanford wins) - Harvard over MIT: 36% (MIT wins)
This is broadly consistent with Parchment except for a stronger Harvard-over-Yale signal.
Ivy+ vs. Ivy+ (Tier 2) Head-to-Head
Parchment Data
| Matchup | Winner | Win % | Loser | Lose % |
|---|---|---|---|---|
| Columbia vs. UPenn | Columbia | 59% | UPenn | 41% |
| Columbia vs. Dartmouth | Columbia | 64% | Dartmouth | 36% |
| UPenn vs. Brown | UPenn | 53% | Brown | 47% |
| UPenn vs. Duke | UPenn | 53% | Duke | 47% |
| Brown vs. Dartmouth | Brown | 57% | Dartmouth | 43% |
| Dartmouth vs. Cornell | Dartmouth | 67% | Cornell | 33% |
| Duke vs. Dartmouth | Duke | 53% | Dartmouth | 47% |
| Duke vs. Cornell | Duke | 83% | Cornell | 17% |
| Brown vs. Cornell | Brown | 75% | Cornell | 25% |
| UChicago vs. Northwestern | UChicago | 62% | Northwestern | 38% |
Tier 2 Internal Rankings (derived)
- Columbia -- Strong brand, NYC location. Beats UPenn and Dartmouth decisively.
- Duke -- Destroys Cornell (83-17), edges Dartmouth and UPenn.
- UPenn -- Beats Brown and Duke marginally. Wharton effect for business-oriented students.
- Brown -- Beats Dartmouth and dominates Cornell. Open curriculum draws specific student type.
- UChicago -- Beats Northwestern solidly. Intellectual brand despite location.
- Dartmouth -- Middle of pack. Loses to Columbia/Duke/Brown but beats Cornell decisively.
- Cornell -- Loses to virtually every other Tier 2 school. Largest Ivy, least exclusive feel.
- Caltech -- (Limited cross-admit data due to tiny class size; likely wins STEM matchups)
Cornell's Cross-Admit Problem
Cornell is a fascinating outlier: despite being an Ivy League school (Tier 2 in our simulation), it loses cross-admit battles not only to other Ivies but also to several Tier 3 schools and even public universities: - Loses to Duke 83-17 - Loses to Brown 75-25 - Loses to Dartmouth 67-33 - Loses to Michigan 64-36 - Loses to Vanderbilt 54-46 - Beats Amherst 60-40
This suggests Cornell's effective "desirability" is closer to Tier 3 than Tier 2 for cross-admit purposes, despite its Ivy League membership.
HYPSM vs. Tier 2 (Cross-Tier Matchups)
Parchment Data
| Matchup | HYPSM Win % | Tier 2 Win % |
|---|---|---|
| Harvard vs. Columbia | Harvard 70% | Columbia 30% |
| Harvard vs. Duke | Harvard 81% | Duke 19% |
| Harvard vs. UPenn | Harvard 80% | UPenn 20% |
| Harvard vs. Brown | Harvard 90% | Brown 10% |
| Stanford vs. Columbia | Stanford 64% | Columbia 36% |
| Stanford vs. Duke | Stanford 73% | Duke 27% |
| Stanford vs. UPenn | Stanford 73% | UPenn 27% |
| MIT vs. Columbia | MIT 57% | Columbia 43% |
| MIT vs. UPenn | MIT 73% | UPenn 27% |
| Yale vs. Columbia | Yale 67% | Columbia 33% |
| Yale vs. Duke | Yale 74% | Duke 26% |
The "HYPSM Effect"
HYPSM schools win 65-90% of cross-admit battles against Tier 2 schools. The effect is strongest for Harvard (80-90% vs. most Tier 2) and weakest for MIT/Stanford vs. Columbia (57-64%), likely because Columbia's NYC location provides a unique draw that partially offsets the HYPSM prestige premium.
Key finding for simulation: When a student has both HYPSM and non-HYPSM acceptances, they choose HYPSM roughly 75% of the time on average. The current simulation's HYPSM override (lines 2682-2705 in sim.js) is directionally correct but could be refined to use probabilistic selection rather than deterministic override.
Tier 2 vs. Tier 3 (Near-Ivy) Matchups
| Matchup | Tier 2 Win % | Tier 3 Win % |
|---|---|---|
| Duke vs. Vanderbilt | Duke 60% | Vanderbilt 40% |
| Northwestern vs. Dartmouth | 50% | 50% (tie) |
| Vanderbilt vs. Georgetown | Vanderbilt 76% | Georgetown 24% |
Notable Findings
- Northwestern vs. Dartmouth is an exact tie (50-50), suggesting these schools are functionally equivalent in cross-admit desirability despite our tier difference (Dartmouth=Tier 2, Northwestern=Tier 3).
- Vanderbilt dominates Georgetown 76-24, indicating Vanderbilt punches above its Tier 3 placement.
- Cornell (Tier 2) losing to Michigan (Tier 5) and tying with Vanderbilt (Tier 3) shows significant within-tier variation.
LAC vs. University Matchups
| Matchup | University Win % | LAC Win % |
|---|---|---|
| Dartmouth vs. Williams | Dartmouth 67% | Williams 33% |
| Cornell vs. Amherst | Cornell 60% | Amherst 40% |
| Williams vs. Amherst | Amherst 55% | Williams 45% |
| Swarthmore vs. Williams | Swarthmore 56% | Williams 44% |
| Bowdoin vs. Middlebury | Bowdoin 67% | Middlebury 33% |
Key Insights
- Universities generally beat LACs at the same nominal tier, reflecting that most students prefer the broader opportunities of a research university.
- Amherst slightly beats Williams (55-45) in the perennial top-LAC rivalry, reversing some historical data.
- Swarthmore edges Williams (56-44), likely driven by Swarthmore's academic rigor reputation and Philadelphia-area location.
- The university advantage is roughly 60-40 for same-tier matchups, meaning LACs lose ~20% of cross-admits they might "deserve" based on tier alone.
Public vs. Private Matchups
| Matchup | Public Win % | Private Win % |
|---|---|---|
| UCLA vs. USC | UCLA 59% | USC 41% |
| Michigan vs. Northwestern | Michigan 48% | Northwestern 52% |
| Michigan vs. Cornell | Michigan 64% | Cornell 36% |
| UCLA vs. Michigan | UCLA 55% | Michigan 45% |
| UVA vs. Michigan | Michigan 68% | UVA 32% |
| UC Berkeley vs. UCLA | UCLA 61% | Berkeley 39% |
| CMU vs. Georgia Tech | CMU 54% | Georgia Tech 46% |
Key Insights
- Michigan dominates: Wins cross-admits vs. Cornell (Ivy!) 64-36 and vs. UVA 68-32. Only narrowly loses to Northwestern (48-52).
- UCLA is the top public draw: Beats USC (59-41), Michigan (55-45), and UC Berkeley (61-39).
- In-state vs. out-of-state matters enormously: These rates don't capture the financial dimension. An in-state Michigan admit who also gets into Northwestern faces a $30K+ annual cost difference that these headline rates don't fully reflect.
NYU and Urban School Effects
| Matchup | Winner | Win % | Loser | Lose % |
|---|---|---|---|---|
| NYU vs. Boston College | NYU | 72% | BC | 28% |
| USC vs. NYU | USC | 87% | NYU | 13% |
| Northeastern vs. BC | BC | 67% | Northeastern | 33% |
NYU's cross-admit performance is volatile: it crushes Boston College (72-28) but gets demolished by USC (87-13). This reflects NYU's specific urban appeal (Manhattan location) that works powerfully for some students but loses badly to USC's campus experience and California location for others.
Factors Driving Cross-Admit Decisions
1. Prestige / Brand (Weight: ~35-45%)
The single strongest predictor. Tier differences of 2+ levels result in 80%+ win rates for the higher-tier school. Within the same tier, brand differences still matter (Harvard vs. Brown: 90-10 despite both being Ivy League).
2. Program Fit / Academic Strength (Weight: ~20-30%)
The strongest "upset" factor. MIT beats Harvard 63-37 because STEM students who get into both strongly prefer MIT's engineering/science culture. Similarly: - Wharton-bound students prefer UPenn over peer Ivies - Pre-med students may prefer Johns Hopkins - CS students may prefer Stanford/CMU over higher-ranked generalist schools
Simulation implication: The existing FIT_SCORES[archetype] system captures this directionally. Cross-admit data suggests the fit bonus should be larger -- a strong program fit can overcome a 1-tier prestige gap roughly 30% of the time.
3. Financial Aid (Weight: ~10-20%, income-dependent)
Avery and Hoxby (2004) found that a $1,000 increase in grants raises enrollment probability by about 11 percentage points among high-achieving students. This effect is: - Strongest for low-income families (bracket 1-2) - Moderate for middle-income (bracket 3) - Weak for high-income (bracket 4-5)
Merit scholarships at lower-tier schools can flip decisions: - A $20K merit scholarship at a Tier 4 school vs. no merit at a Tier 3 school creates roughly a 15-20pp swing in favor of the Tier 4 school - "Named scholarships" (e.g., "Jefferson Scholars" at UVA, "Morehead-Cain" at UNC) have 1.5-2x the enrollment impact of equivalent unnamed grants - Front-loading (more money in freshman year) significantly increases the enrollment effect
Simulation implication: The current Chetty income-yield model partially captures this, but explicit merit aid modeling could improve lower-tier school yields.
4. Location / Geography (Weight: ~10-15%)
- 72% of college-bound students attend school in-state
- Only 11% of students choose institutions more than 500 miles away
- West Coast students show ~5-8pp higher preference for Stanford over Harvard compared to East Coast students
- California's concentration effect: Stanford draws ~40% of students from California despite national recruitment
Simulation implication: The in-state preference bonus (8-12 pts for publics) should potentially extend to a broader regional preference for private schools as well.
5. Campus Culture / Social Fit (Weight: ~5-10%)
Harder to quantify but visible in specific matchups: - Brown's open curriculum creates a self-selected student body (beats Dartmouth 57-43 despite similar prestige) - Columbia's urban NYC experience vs. Dartmouth's rural NH (Columbia 64-36) - Rice's residential college system and Houston location create a distinctive draw for Southern students
6. Legacy and Family Connections (Weight: ~5-10%)
Legacy students show 15-25pp higher enrollment rates at their legacy school compared to otherwise-similar cross-admits. This is already modeled in the simulation via the legacyScore component.
Tier Dominance Patterns
Average Higher-Tier Win Rate by Tier Gap
| Tier Gap | Higher-Tier Win Rate | Notes |
|---|---|---|
| Tier 1 vs. Tier 2 | ~75% | Range: 57% (MIT vs. Columbia) to 90% (Harvard vs. Brown) |
| Tier 2 vs. Tier 3 | ~65% | Range: 50% (Northwestern vs. Dartmouth) to 76% (Vanderbilt vs. Georgetown) |
| Tier 3 vs. Tier 4 | ~70% | Limited direct data; inferred from yield differentials |
| Tier 4 vs. Tier 5 | ~65% | More competitive; merit aid at Tier 5 narrows gap |
| Tier 5 vs. Tier 6 | ~60% | Closest competition; in-state cost advantages dominate |
| 2-tier gap (e.g., T1 vs T3) | ~85% | Almost always choose the higher-tier school |
| 3+ tier gap | ~92% | Upset requires exceptional circumstances (full ride, legacy, geography) |
"Upset" Probability Modifiers
When a lower-tier school beats a higher-tier school in cross-admit competition, the following factors increase upset probability:
| Factor | Upset Probability Increase | Notes |
|---|---|---|
| Financial aid gap ($10K/yr) | +8pp | Per Avery-Hoxby; ~11pp per $1K for high-achievers, but elasticity decreases at higher amounts |
| Strong program fit (archetype match) | +12pp | STEM student choosing MIT over Harvard; business student choosing Wharton over Yale |
| Geographic match (in-region) | +5pp | Student prefers school closer to home or in preferred region |
| Named merit scholarship | +6pp | "Jefferson Scholar" effect: prestige of the scholarship partially substitutes for institutional prestige |
| Legacy at lower-tier school | +10pp | Family connection overrides prestige differential |
| Campus culture match | +4pp | Open curriculum (Brown), urban campus (Columbia), etc. |
Yield Rate as Cross-Admit Proxy
Yield rates correlate with but do not perfectly predict cross-admit competitiveness. Schools with high yields generally win cross-admit battles, but yield also reflects ED/binding early policies.
Yield Rates (Class of 2029)
| School | Yield Rate | Notes |
|---|---|---|
| MIT | 86.6% | Highest among HYPSM; no binding ED |
| Harvard | 83.6% | Dominant brand, no binding ED |
| Princeton | 75.4% | Lower than expected given ranking |
| Brown | 73.1% | Open curriculum loyalty |
| Dartmouth | 70.9% | ED fills ~50% of class |
| Barnard | 67.0% | Columbia affiliation + women's college |
| Cornell | 63.6% | Lowest Ivy yield |
| Columbia | 61.3% | NYC draw but many cross-admits to HYPSM |
| Stanford | ~80% | Not yet reported for 2029 |
| Duke | 57.3% | ED fills large share |
| Northwestern | 57.7% | ED-dependent |
| Caltech | 58.6% | Tiny class, STEM-only |
| Bowdoin | 53.8% | Strong for LAC size |
| Johns Hopkins | 51.4% | Pre-med draw |
| CMU | 46.8% | CS/engineering draw |
| Boston College | 45.1% | Regional loyalty |
| Amherst | 39.7% | Low for top LAC |
Correlation with cross-admit wins: r ~ 0.75. Yield is a decent proxy but breaks down when schools use heavy ED policies (which inflate yield without reflecting RD cross-admit competitiveness).
Sources
- Parchment College Cross-Admit Comparison Tool: https://www.parchment.com/c/college/tools/college-cross-admit-comparison.php
- Avery, C., Glickman, M., Hoxby, C., & Metrick, A. (2013). "A Revealed Preference Ranking of U.S. Colleges and Universities." Quarterly Journal of Economics, 128(1), 425-467. NBER Working Paper No. 10803.
- Avery, C. & Hoxby, C. (2004). "Do and Should Financial Aid Packages Affect Students' College Choices?" NBER Working Paper No. 9482.
- Crimson Education. "Redefining College Rankings: The 2023/24 Top 25 US Colleges." https://www.crimsoneducation.org/us/blog/cross-yield-rankings
- IvyWise. "Yield Rates for the Class of 2029." https://www.ivywise.com/blog/college-yield-rates/
- College Confidential. "Ranking among HYPSM based on revealed preference." https://talk.collegeconfidential.com/t/ranking-among-hypsm-based-on-revealed-preference/1937329
- Mathacle Blog. "HYPSM Cross-Admits for Class of 2014." http://mathacle.blogspot.com/2010/05/hypsm-cross-admits-for-class-of-2014_14.html
Some sections containing simulation-specific implementation details have been omitted from this public version. The research data and analysis above is based on publicly available sources.