Matpic
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Ladder

The core — the ladder

2 distinct arguments against one idea, from the one that introduces it to the one that assumes the rest.
  1. The splits no group can beat, for three partners. The triangle of ways to split a fixed total between three players, with each coalition's demand drawn as a straight cut across it, and the region surviving every cut shaded.

    A split nobody can walk away from

    Every way of dividing what a group earns is a point of a triangle, and every coalition's threat to leave cuts a straight line across it. What survives all the cuts is the set of stable divisions — and for one three-player game there is nothing left.

    rung 1 · applied
  2. The quietest loudest complaint in any two of three decide. The triangle of all splits of what a three-player group is worth, with the split minimising the largest excess marked, the average split beside it, and the loudest complaint named.

    The objection nobody can make louder

    When no split of the winnings survives every group's objection, the core is empty and the question changes — which split makes the loudest objection as quiet as it can be? Sorting the complaints and minimising them in dictionary order picks exactly one split, always, whether or not the core exists.

    rung 2 · applied

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