How much do models impact the Pareto Frontier?

I created a metric to measure that.

This is currently based off of Artificial Analysis data for their intelligence index versus cost per task chart.

I was just thinking: maybe we can measure how much a model advances the Pareto Frontier by the angle composed by the triangle of three points: the model itself, and the models immediately to its left and right.

One way we can measure this is by the angle of the triangle created by the points on the Pareto Frontier to the left and right of your main point.

A very "pointy" model would represent a major advancement. The most acute (or narrow) this angle could be is 90 degrees, which is the limit, and the worst it could be is 270 degrees. We can come up with a score normalized between those two extremes.

Of course, this angle will change as you scale the chart. So here, I use a fixed log scale to calculate the scores.

How I calculated it

I sorted the Pareto Frontier by cost. For every model except the two endpoints, I took the direction of the line coming in from the model to its left and the line going out to the model on its right. Then I calculated the oriented angle between them.

I had to use an oriented angle because the ordinary interior angle of a triangle stops at 180 degrees. That would treat an outward point and an inward dent as the same shape. The oriented angle lets the range continue to 270 degrees.

The fixed coordinates are:

x = log10(cost)
y = intelligence / 34.35

The 34.35 is not a claim about intelligence. It just locks the geometry to the version of the Artificial Analysis chart I used, so the same three points keep the same angle every time the calculation is run.

Then I calculate the score like this:

A = 180° + θright − θleft
score = 100 × (270° − A) / 180°

A 90-degree point scores 100. A straight 180-degree section of the frontier scores 50. A 270-degree inward point scores 0. The two endpoints do not get scores because they only have one neighbor.

The bar chart I posted on X subtracts 50 from the normalized score. That makes a straight frontier zero, an outward bend positive, and an inward bend negative. It is the same metric, just centered around zero so the direction is easier to see.

The results

On the August 26, 2026 snapshot, GLM-5.3 max got the highest normalized score at 82.8, or +32.8 on the centered version. Grok 4.6 high scored 69.8, MiMo-V2.5 scored 69.7, and GLM-5.3-Flash scored 65.5.

A model like K3 that creates more of a concave dent in the frontier will get a negative score on this metric.

Kimi K3 max scored 31.9 on the 0-to-100 version, which is -18.1 when centered around zero. GPT-5.6 Sol xhigh scored 16.8, or -33.2 centered.

This is not an overall model ranking. It only measures the shape each model creates between its two immediate neighbors on this particular cost and intelligence frontier.

The overlap problem

One thing this metric fails to account for, however, is when a model barely sneaks onto the Pareto Frontier by being ever so slightly cheaper than another, more intelligent model. For example, if you zoom in on the graph and look at GPT 5.6 Sol xhigh, it sits lower than GLM 5.3 Max while being almost the exact same price. Because it is slightly cheaper, it technically remains on the Pareto Frontier, even though it isn't noticeably cheaper (the circles literally overlap on the graph). This overlap can artificially make a model's angle look much better or worse than it actually is.

In this snapshot, GPT-5.6 Sol xhigh cost $0.6711 per task and scored 59.009 on the Intelligence Index. GLM-5.3 max cost $0.6829 and scored 59.513. GLM was 1.77% more expensive for about half an Intelligence Index point more.

I don't think this is too much of a problem, though, since it will ultimately just increase the score of a model that represents a genuine step up. If we measure these angles across the Pareto Frontier, we can establish a useful index.

I also ran a version with a 2% cost tolerance to see how sensitive the result was. Under that rule, a more intelligent model can knock out a cheaper model if it costs no more than 2% extra. GPT-5.6 Sol xhigh drops off the frontier, and GLM-5.3 max falls from 82.8 to about 50.2.

I would report both versions. The strict score uses the literal Pareto Frontier. The tolerant score shows when a sharp angle mostly comes from two models having almost the same price.

The score is local, too. If a new model becomes one of a model's immediate neighbors, its angle changes. That is expected because the shape of the frontier changed.

The code

I open-sourced the calculation, tests, dated data snapshot, example output, and charts here: github.com/adamholter/pareto-frontier-advancement.

It has no runtime dependencies and runs on Node.js 20 or newer:

npm test
npm run example

The implementation first removes dominated points, sorts the remaining frontier by cost, transforms the coordinates, and calculates the angle at every interior point. The optional cost tolerance reruns the same calculation with a slightly less literal definition of "cheaper."

Not sure if this is a super useful metric, but I've never seen it reported before or really anything like it, so I thought I'd share.

Data source: Artificial Analysis Intelligence Index. Artificial Analysis defines cost per task as the weighted average token cost required to complete one Intelligence Index task. The numbers here are from a dated August 26, 2026 snapshot, so they will not always match the live chart.