The ranking is not an order

Which projection a weighting can make best ended this ladder in an awkward place. Ten of the projections on the Pareto front can be crowned by some choice of weights between two criteria, so a scored comparison of maps is a report on the scorer, and the honest response is to stop weighting.

There is a way to compare things without weights, and it is the oldest one. Let each criterion vote. A is better than B if a majority of the criteria say so. No weights, no scale factors, no exchange rate between an angle and an area — nothing that anybody can be accused of choosing.

That produces something worse than a report on the scorer. It produces no report at all.

The voters #

Seven criteria, all of them computed in this collection already and every one published somewhere as a summary of how wrong a map is:

All seven are smaller-is-better, all seven are computed at the same points from the same derivatives, and every projection is normalised first so that the geometric mean of its areal factor over the region is one — because a projection is free to choose its overall scale and an unnormalised comparison would mostly measure that choice.

Seven is an odd number, deliberately. With an even number of voters a tie is possible and the majority relation is incomplete for a boring reason.

The tournament #

Two things come out of that grid and they are not the same thing.

There is a winner. The Winkel tripel beats every other projection in the table, head to head, on a majority of the criteria, with no weighting anywhere. That is a Condorcet winner and it is the strongest form of answer this kind of comparison can produce: no reweighting can dislodge it, because it is not a weighted result. It is also, satisfyingly, the projection the National Geographic Society adopted in 1998 after a century of argument, on grounds that had nothing to do with any of this.

And there is no order. Four triples of projections run in a circle. A relation with a cycle in it cannot be a ranking, whatever else it is, because a ranking is a line and there is no place on a line for three things each of which is above the next.

Those two facts sit together without contradiction. A cycle among the middle of the field does not stop something above the field from beating all of it — the winner is outside every cycle, which is a consistency check on the arithmetic as well as a fact about the maps.

What each cycle is made of #

The Mercator–sinusoidal–Eckert cycle is the clearest and every edge in it is a 4–3.

Mercator beats the sinusoidal on the four criteria that are about angles and about principal scales at a point: Mercator is conformal, so its angular deformation is exactly zero everywhere and its principal scales are equal at every point. The sinusoidal shears badly at the outer meridians.

The sinusoidal beats Eckert IV on the criteria that are about areas and spreads: both are equal-area, so both have an areal factor of one everywhere, and the tie-breaking happens on shape — where the sinusoidal’s failure is concentrated at the corners while Eckert IV’s is spread more evenly, which the mean criteria reward and the worst-case ones punish.

Eckert IV beats Mercator on everything that involves an area, which for Mercator over the whole sphere is unbounded, and on the worst-case criteria that Mercator’s conformality does not help with at all.

Each edge is decided by a different subset of the same seven, and there is no subset that decides all three the same way. That is exactly Condorcet’s situation, translated from voters and candidates into criteria and maps.

Part of that figure is parity rather than content, and it should be read with that in mind: six voters make a three-three split possible, so every removal makes majorities harder to come by and cycles rarer for reasons of arithmetic. What survives the caveat is that dropping the Airy number, the Kavrayskiy number or the mean angular deformation — the three summaries this collection ranks with — each leaves two cycles standing.

Where it happens, and where it does not #

Eight of the ten regions produce a clean total order, with a Condorcet winner and no circles anywhere. Over Britain, Japan, New Zealand, the conterminous United States, Chile, a 30° cap and Europe, the seven criteria agree well enough that the majority relation is a ranking, and the argument this essay is about does not arise.

The two that cycle are the whole sphere and the tropics.

The pattern is size, and the mechanism is that over a small region every projection is nearly right. Distortion over Britain is a fraction of a per cent for anything sensible, the criteria are all measuring the same near-vanishing quantity, and they order the maps identically because there is only one thing left to order by. Over the whole sphere the projections fail in genuinely different ways — one gives up area entirely, one gives up shape entirely, one gives up a little of each — and different ways of failing is what makes criteria disagree.

So the cycles arrive exactly where the choice of projection actually matters. That is not a coincidence and it is not good news.

What a table of criteria is usually read as #

Nobody actually runs a pairwise tournament. What people do with a table of criteria is average the ranks — take each projection’s position under each criterion and mean them — and read off the order. That is the Borda count, and it has one enormous advantage over the majority relation: it always produces an order.

An aggregation rule that always returns an order is returning an order that the comparisons do not support. That is the trade Arrow’s theorem describes: a rule can be guaranteed to produce a ranking, or it can be guaranteed to respect every pairwise majority, and no rule does both. Borda takes the first, the majority relation takes the second, and the plate carrée sits fifth in the Borda order while four projections below it beat it head to head.

The winner, and what it is worth #

A Condorcet winner is a much stronger object than a first place in a table, and it is worth being precise about what the Winkel tripel has and has not got.

It has: for every one of the other eleven projections, a majority of the seven criteria prefer it. That is eleven separate comparisons, each decided without any weighting, and no reweighting can overturn any of them — reweighting changes how comparisons are combined, and these were not combined.

It has not: any claim to be best for a purpose. Every projection minimises something, and the Winkel tripel minimises nothing in particular: it is the arithmetic mean of the equirectangular and the Aitoff, which is a construction chosen for how it looks. That it wins a tournament of seven criteria is a statement about compromise, and compromise projections are constructed to do exactly this — to be nobody’s worst on anything, which is precisely how a candidate wins a pairwise majority against specialists.

So the winner is real and it is also structurally predictable. A field containing several specialists and one compromise will usually elect the compromise, because each specialist loses to it on the criteria it sacrificed, and the specialists split the rest between them. The same logic that produces the cycles produces the winner, which is why they coexist.

What was computed, and how #

Twelve projections, seven criteria, ten regions, twenty-four samples a side, with each projection normalised to unit geometric-mean areal factor over the region before anything is compared. Nothing here is weighted, scaled or tuned; the only judgement in the whole calculation is which seven criteria to admit, and all seven were already in this collection before this essay was written.

The cycles are found by enumeration over ordered triples and listed once each. The assertion carrying the rung requires four things and each could fail on its own: some region must cycle, or there is nothing to report; some region must not, or the cycle detector is finding them in the code rather than in the maps; the whole sphere’s Condorcet winner must be the Winkel tripel; and that winner must not itself be inside any cycle, which is impossible and would mean the relation was being read wrongly.

The sampler’s own contribution was checked, because three of these criteria are maxima and a sampled maximum is a lower bound. Re-running the whole tournament at sixty samples a side rather than twenty-four leaves every cycle in place, which it must, because all seven criteria move in the same direction under refinement and a 4–3 majority is a comparison rather than a value.

Where the model stops #

Twelve projections is not the library. Adding a projection can create cycles or destroy them, since it changes which pairs exist without changing any of the old ones. The four cycles reported are four in this set; a different twelve would give a different number, and the finding is that the number is not zero rather than that it is four.

Seven criteria is a choice, and it is the one choice here. They were selected as the summaries this collection computes rather than assembled to produce a cycle, and the robustness figure shows no single one is carrying the effect. But a comparison built on three criteria rather than seven would cycle less often for the ordinary reason that fewer voters can disagree in fewer ways.

Nothing here says any projection is bad. A cycle is a statement about the relation, not about its members. Mercator appearing in a cycle is not a defence of Mercator, and the projections that are beaten on both counts still are: dominance is a different relation, it is transitive, and nothing in this essay touches it.

And this is a majority over criteria, not over purposes. No essay here may say a projection is best without naming the purpose, and a purpose picks one criterion rather than voting over seven. A navigator wants conformality and gets Mercator, with no election required.

The generalisation #

The rule this ladder has been circling for eleven rungs can now be stated in its strongest form.

Rung 9 found that a tolerance decides a verdict. Rung 10 found that a weighting can crown almost anyone. This rung removes both — no tolerance, no weights, every criterion equal — and finds that what is left is not an order at all, on exactly the regions where the choice matters.

So the sequence is not “the comparison is subjective, so be careful about the parameters”. It is: there is no parameter-free comparison to fall back to. A ranking of map projections is not a thing that exists and is hard to compute; over a whole-world map it is a thing that does not exist, in the same sense that a majority preference does not exist when the electorate cycles.

What survives, and it is more than nothing, is the pair of statements that do not need an order. Dominance — A is better on every criterion — is transitive and gives a genuine Pareto front. And a Condorcet winner, when there is one, beats everything without any weighting at all. Over the whole sphere both exist: the front is real and the Winkel tripel wins. It is the middle of the field where the question is malformed, and the middle of the field is where every argument about projections is actually conducted.

Who found it, and when #

Condorcet published the paradox in 1785, in an essay about the probability of majority decisions, and it took the twentieth century to understand that it was not a curiosity about voting but a structural fact about aggregating orders. Arrow’s theorem of 1951 gives the general statement: no aggregation rule satisfying a short list of obviously desirable properties can be guaranteed to produce an order.

The cartographic literature compares projections against multiple criteria constantly, and has since Tissot. What it does not do is aggregate them formally — the convention is to present a table and let the reader decide, which is a perfectly honest practice and which conceals exactly this — and which the rule of thumb, scored puts a number to from the other direction. A reader deciding from a table is running some aggregation rule in their head, and whichever one it is, Arrow’s theorem applies to it.

The one place the connection is nearly made is in the multi-criteria decision literature, where map projection selection appears occasionally as a worked example of a weighted-sum method. Those papers choose weights, which is the previous rung’s problem, and by choosing weights they guarantee an order and step over this one without noticing it was there.

What an honest comparison would publish #

The convention of presenting a table and letting the reader decide is defended as neutrality, and the finding here says what it actually is: an aggregation performed by the reader, informally, with no record of which rule was used. There is a better output available from the same computation.

Publish the tournament, not the ranking. For each pair of projections, which one wins on more criteria, and by how many. That is a matrix rather than a list, it is exactly what was computed on the way to any ranking, and it contains everything a ranking does plus the thing a ranking has to discard.

The cycles then become visible rather than resolved. A reader looking at the matrix sees that A beats B, B beats C and C beats A, and understands immediately that no ordering of the three is available. A reader looking at a ranked list sees three rows in an order and has no way to know that the order was manufactured.

And the regions where no cycle occurs are the findings worth having. The measurement above shows cycles in some regions and not others, so for a great many practical choices the criteria do agree and an order genuinely exists. Those are the cases where a recommendation can be made with confidence, and the tournament identifies them: a matrix with no cycles is an order, and saying so is a much stronger statement than printing one.

It also changes what a disagreement between two studies means. Two rankings that differ may be reporting the same tournament through two aggregation rules, which is not a disagreement about projections at all; two tournaments that differ are a real disagreement, about the criteria or the region.

The cost is a table with $n^2$ entries instead of $n$, which for the ten or twenty projections anybody actually compares is a page rather than a line. That is the whole of the expense, and it buys the distinction between this is the best and these three cannot be put in order, which is the distinction the subject’s own literature has been unable to make while presenting numbers honestly.

Where the ladder goes next #

Eleven rungs have taken apart every way of saying that one projection is better than another: the rule of thumb, the criterion, the tolerance, the weighting, and now the aggregation itself. What has been assumed throughout is that the region is given — that somebody knows what the map is of. Every ranking here changes when the region changes, and eight of the ten regions in this essay were chosen because somebody drew a box around a country. Where that box comes from, and how much of the verdict is in it, is the question this anchor has never asked.