Every spring a small number of high-school seniors face a problem most families would envy: they get into both Harvard and Stanford. Or Princeton and MIT. What they decide in those weeks tells us something acceptance rates and rankings cannot — which schools students actually prefer when the choice is real.
Harvard, Yale, Princeton, Stanford, MIT — HYPSM, in admissions shorthand — sit at the apex of the cross-admit pyramid. Their applicants frequently overlap, which means we can watch them compete for the same students. Parchment, which collects self-reported decisions, has done the bookkeeping for ten pairwise matchups.
The headline result: Princeton, ranked #1 by U.S. News, loses every single head-to-head matchup against the other four. Stanford takes 73 of every 100 Princeton cross-admits. MIT takes 58. Yale takes 59. Even Harvard, which beats all four other Ivies, beats Princeton by the widest margin: 73-27.
The other surprise lives at MIT. Despite being ranked below Harvard on most lists, MIT wins three of its four HYPSM matchups. The lone loss is to Harvard — and even that is closer than the rankings suggest.
Pairwise wins are noisy. Aggregate the records and you get a pecking order.
In 2013, Christopher Avery, Mark Glickman, Caroline Hoxby, and Andrew Metrick published a study in the Quarterly Journal of Economics that treated college choice as a tournament. Each cross-admit decision is a "match." Apply Elo — the rating system used in chess — and you get a ranking based on what students do, not what they say.
Their top five from 3,240 high-achieving students: Harvard, Yale, Stanford, Caltech, MIT. Crimson Education's 2024 cross-yield ranking, drawn from a STEM-heavy international pool, instead crowns Stanford and ranks MIT second. The ordering depends on whose sample you're looking at — but the cluster at the top is remarkably stable.
The chart at right is our calibrated Elo curve for the top 30 colleges, anchored to the Parchment win rates. The gap from Harvard (2100) to Princeton (1950) implies Harvard wins ~70% of the time. Observed: 73%.
One school's place on this curve is jarring. It is also a clue.
Cornell is an Ivy. By every status convention — Ivy+ in our tiering, Ancient Eight by tradition — it belongs at or near the top. In cross-admit data it does not.
Cornell loses to Duke 83-17. To Brown 75-25. To Dartmouth 67-33. To Michigan, a public flagship, 64-36. It ties Vanderbilt. Across nearly every matchup Parchment can sample, Cornell is the school students walk away from when something else is on the table.
The likely culprits: Cornell admits more students per class than any other Ivy (~3,200 vs. ~1,650 at Princeton), giving it a less exclusive feel. The Ithaca location reads as remote to many applicants. And Cornell's seven undergraduate colleges fragment its brand — the College of Agriculture is admissions-easier than Engineering, which dilutes the perception of a single Ivy bar.
Cornell is a special case. Pull back, and tier still rules.
Stack all the matchups by how many tiers separate the contestants and a clean gradient emerges. A one-tier gap is a coin-flip with a thumb on the scale — roughly 65-75% in favor of the higher tier. A two-tier gap collapses to about 85-15. Three tiers or more and the upper school wins virtually every time.
The gradient is steepest at the top. HYPSM beats Ivy+ 75% on average, with Harvard running the table at 80-90% against most of Tier 2. The gradient is shallowest at the bottom: Top LAC / Public only beats Selective Public 60-40, because in-state cost advantages start eating into prestige.
Tier difference, in other words, is the single best predictor of who wins a cross-admit. Everything else — archetype fit, financial aid, geography, legacy — mostly modulates how big the favorite's edge is.
If tier predicts cross-admit, what about yield rate? Does it map cleanly?
Yield — the share of admitted students who actually enroll — is the metric deans worry about. It is also published. So it gets used as a stand-in for desirability everywhere from U.S. News to high-school newsletters.
For the Class of 2029, MIT yields 86.6% — the highest figure ever reported by an HYPSM school. Harvard sits at 83.6%. Stanford ~80%. The cross-admit rankings line up.
Then yield breaks down. Dartmouth's 70.9% yield is inflated by Early Decision, which fills roughly half its class with binding commitments. Duke (57.3%) and Northwestern (57.7%) lean even harder on ED. Compare these to Caltech (58.6%), which uses no binding ED at all — that 58.6% is unalloyed choice.
Yield is a number. Cross-admit data is a number. They disagree about who's winning. Whose numbers do we trust?
Three serious sources have tried to measure cross-admit competitiveness in the last fifteen years. They disagree.
Parchment works from self-reported decisions on its transcript platform. Sample is huge but skews toward students who comparison-shop — and toward East Coast and humanities majors. Crimson Education's 2024 ranking uses its own client base, which is global and STEM-heavy. Avery-Hoxby hand-collected decisions from 3,240 high-achievers in the early 2000s, treating each cross-admit as a tournament match.
The chart at right shows the top-10 ordering side by side. Harvard tops two of the three. Stanford tops Crimson. Yale rises in Avery-Hoxby (a humanities-heavy sample, a more academic era) and falls in Crimson. WashU does not appear in anyone's top 10 — but lands at #11 in U.S. News. Avery-Hoxby has WashU at #62 by revealed preference.
Three honest counts of the same thing, and the schools are not in the same order. The lesson is not that any source is wrong; it is that "best college" is partly about who is doing the choosing.
Tier explains most of the variance. The remaining 25% — the upsets — are where the interesting causality lives.
When a lower-tier school wins a cross-admit it shouldn't, six things tend to be in the room. The chart at right ranks them by the percentage-point bump they add to the underdog's win probability.
Program fit is the largest single factor (+12 pp). A STEM student admitted to MIT and Harvard chooses MIT. A future banker admitted to Wharton and Yale takes Wharton. The 12-point bump is enough to flip a 65-35 baseline into a coin flip.
Financial aid comes next (+8 pp per $10K/year of difference). Avery and Hoxby found that an extra thousand dollars of grant aid raises enrollment probability by about 11 percentage points among high achievers. The effect is biggest at low incomes (income bracket 1: 1.5x multiplier) and nearly disappears for full-pay families.
Legacy adds about 10 points, named merit scholarships 6, geographic match 5, and campus-culture match 4. Stack two of these on a one-tier favorite and the underdog wins.
Six factors, one chart. The simulation reads them all.
Strip away the rankings and the marketing brochures and look at what 100,000 cross-admits actually did. Harvard wins more than it loses. MIT and Stanford crowd it from below. Princeton, despite its #1 perch, is the school students leave when something else is offered. WashU and the elite LACs trail their rankings.
These differences matter for a personalized admissions simulator. A naive model that picks the higher-tier school every time is right about 75% of the time. The remaining 25% — archetype, geography, money, legacy — is where any honest forecast lives or dies. Our simulation uses Elo ratings, archetype-specific multipliers, and an Avery-Hoxby-calibrated aid elasticity to capture that last quarter.
Rankings are easy to publish and easy to game. Cross-admit data is messier but harder to fake. When the choice is real, students tell us where the prestige is.
For the small group of students with multiple acceptances at the top of the system, college choice is a revealed preference: a private decision that aggregates into a public signal. Parchment reads that signal one way, Crimson another, the economists yet another. None of the three honestly look like the U.S. News list.
What does that mean for an applicant deciding where to apply? Tier still matters most — the gap between HYPSM and the rest of the Ivy+ pool is wider than rankings suggest. But within a tier, the cross-admit data exposes which schools are punching above weight (Duke, Brown, Vanderbilt) and which are punching below (Princeton, Cornell, WashU).
For our simulation, we use Parchment win rates as the primary calibration target, fold Avery-Hoxby aid elasticity into the financial-fit term, and let archetype-program matches modulate the base Elo. The 25% of cross-admits that don't follow tier are the most interesting 25%.