the second hundred · metaphor 137
A negotiation can only reach the compromises that lie between the positions people have actually stated. Blend them however you like — but you cannot mix your way to an outcome no one ever put on the table.
Every deal is built from what was said out loud. Each party stakes out a position — a price, a deadline, a set of terms — and the space of possible settlements is exactly the region you can reach by averaging those stated positions: sixty percent of hers, forty percent of his; a little from each corner of the room. Anything in that region is a deal you can defend, because it's a genuine blend of things people asked for.
And anything outside it is not a compromise at all — it's a new proposal. If the settlement everyone secretly wants sits beyond the reachable region, no amount of splitting the difference will get you there. Someone has to say something new: put a fresh position on the table and stretch the region to include it. Below is the room, and the region, drawn live.
The reachable region
Put each stated position as a point. A compromise that gives weight to several of them — say, thirty percent to one, forty-five to another, twenty-five to a third — lands at the weighted average of those points. Require only that the weights are non-negative and add to one (you can't give a party negative presence in the deal, and the weights are shares of a whole), and the set of every reachable weighted average has an exact shape: the convex hull of the stated points.
That's the shaded region. Its corners are the positions people actually staked out; its interior is every blend of them. A remarkable fact — Carathéodory's theorem — says that in the plane, any reachable compromise is a mix of at most three of the stated positions. So the panel can hand you a concrete recipe: this exact settlement equals so-much of Alice, so-much of Bob, so-much of Dana. Not a vague "somewhere in the middle" — named shares that sum to a hundred percent.
A position that sits inside the region contributed by others is redundant: it is itself already a blend of the others, so it adds nothing new the deal couldn't already reach. Only the positions on the frontier — the hull's corners — actually enlarge what's possible. And a target outside the region is unreachable by any blend: the deal everyone wants requires a position no one has yet spoken.
What to try
Move the black target ring around. Inside the region it turns green and the panel prints the exact blend that reaches it — watch the shares shift as you slide it toward one corner and that party's percentage climbs. Push it across the boundary and it turns red: the readout reports how far outside reach it now is, and the nearest deal you could actually strike, drawn as a dashed line to the region's edge.
Now drag a stated position outward: the frontier stretches to follow, and a target that was impossible becomes reachable — that is what it looks like to put something new on the table. Drag a corner inward until it slips inside the others and watch it drop from the frontier count to the redundant count: a position that no longer shapes any deal. Tap empty space to add a voice; the region grows only if the new position reaches past the old frontier.
The mapping
The hull is the honest boundary of a negotiation. The corners are leverage: only the parties whose positions sit on the frontier shape what's possible; a party whose demand is already a blend of others' has, mathematically, no new leverage — their ask changes nothing the deal couldn't already offer. The interior is the zone of settlement: every point in it is a defensible compromise, backed by a specific recipe of who gave what.
And the boundary is the hard news. A mediator who keeps proposing points outside the hull is not compromising — they're inventing, and pretending it's a split. The only way to reach an outside target is to change the inputs: get someone to state a new position, or move an old one. This is why breakthroughs so often arrive as a sentence no one had said yet. It didn't split a difference; it added a corner and grew the reachable world.
Read as life lessons
The reachable set is spanned by positions actually voiced. Unspoken hopes contribute nothing to the hull. If you want an option on the table, someone has to put it there — silence has zero weight.
A position inside the others is redundant: it's expressible as a mix of them, so it adds no reach. Restating a moderate view others already bracket gives you no leverage — only a frontier position moves the boundary.
An outside target can't be split into existence. Reaching it requires a new vertex — a genuinely new proposal that stretches the region. Creativity in a deal is geometric: it enlarges what "compromise" can even mean.
In the wild
Any blend of available assets or foods lands in the convex hull of the ingredients. You can't achieve a nutrient mix, or a risk-return point, that lies outside the hull of what's actually on the shelf.
A screen's gamut is the hull of its primaries; a colour outside it simply can't be shown. Alloys, blends, and mixtures live inside the hull of their components — the recipe is a set of convex weights.
The set of outcomes a coalition can guarantee is a convex region spanned by its members' positions. A platform outside it needs a new member or a new plank — it can't be assembled from the current ones.
The mapping, exactly
| Mathematics | Life |
|---|---|
| a stated point | A position someone actually put on the table — a price, a term, a demand voiced out loud. |
| convex weights wᵢ≥0, Σwᵢ=1 | The shares of a compromise — how much of each party's ask the deal honours, adding to one whole. |
| the convex hull | The reachable region: every settlement you can build by blending the stated positions, and no others. |
| a hull vertex | A position with real leverage — one whose absence would shrink what's possible. |
| an interior point | A redundant demand — already a blend of others, adding no new reach to the deal. |
| a point outside the hull | An outcome no blend can reach — attainable only if someone tables a genuinely new position. |
The honest model
The region is a true convex hull, recomputed on every drag by Andrew's monotone-chain algorithm — sort the points, sweep once for the lower boundary and once for the upper. The "on the frontier" count is the number of hull vertices; "redundant" is every stated point strictly inside, each one genuinely expressible as a blend of the others.
The target's status is decided by testing which side of every hull edge it falls on — inside the convex region if and only if it lies on the interior side of them all. When it's inside, the panel fan-triangulates the hull, finds the triangle containing the target, and solves for that triangle's barycentric weights — three non-negative numbers summing to one. Those are the printed shares, and multiplying the three corner positions by them lands exactly back on the target. When it's outside, the panel projects the target onto every edge to find the nearest reachable point, and reports the gap. Every number is measured from the geometry you built; nothing is scripted.
Where the metaphor tears
The hull assumes a blend of two workable positions is itself workable. Real deals aren't convex: half a bridge is not half as good as a bridge, it's useless; a compromise between two coherent plans can be a plan nobody can execute. Some interior points are worse than either extreme they average. The region tells you what's reachable by mixing, not what's viable.
To draw a hull you must place each demand as coordinates and treat compromise as arithmetic on them. But loyalty, dignity, and identity don't average: you can't give someone thirty percent of an apology or split a matter of principle down the middle. When the axes are incommensurable, the geometry is a picture, not a calculator.
A point can sit deep in the hull and still be politically dead, because someone would rather walk away than accept it. The hull ignores intensity, vetoes, and the outside option. A reachable compromise is only a candidate; whether anyone will actually sign is a question the geometry never asks.