THE FOLD / CO-OP / THE PUSH / THE SHAPLEY
THE SHAPLEY
the unique fair split — average marginal contribution
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
The Shapley value answers: when a group cooperates to make something worth v, how much does each member fairly deserve? Lloyd Shapley’s answer (1953): average each player’s marginal contribution over every order of joining. Imagine the players arriving one at a time in a random sequence; each is credited with how much they add to the coalition already present; average over all n! orders.
This single formula is the only one obeying four fairness axioms at once: efficiency (shares sum to the whole), symmetry (interchangeable players get equal shares), dummy (a player who adds nothing to any coalition gets nothing), and additivity. A jolt follows in voting: with weights 50/30/20 and a 51% quota, the biggest party holds two-thirds of the real power despite only half the seats, while the 30 and 20 parties are exactly equal. It underlies cost-sharing, credit attribution, and — as SHAP — explaining machine-learning predictions.
LIT verified live: for [51; 50,30,20] the Shapley values are 2/3, 1/6, 1/6 — they sum to 1 (efficiency), the two smaller are equal (symmetry) — and a weight-0 player scores 0 (dummy) (window.__shapley). FIG no framing; the axioms and the power-≠-weight result are exact.
This single formula is the only one obeying four fairness axioms at once: efficiency (shares sum to the whole), symmetry (interchangeable players get equal shares), dummy (a player who adds nothing to any coalition gets nothing), and additivity. A jolt follows in voting: with weights 50/30/20 and a 51% quota, the biggest party holds two-thirds of the real power despite only half the seats, while the 30 and 20 parties are exactly equal. It underlies cost-sharing, credit attribution, and — as SHAP — explaining machine-learning predictions.
LIT verified live: for [51; 50,30,20] the Shapley values are 2/3, 1/6, 1/6 — they sum to 1 (efficiency), the two smaller are equal (symmetry) — and a weight-0 player scores 0 (dummy) (window.__shapley). FIG no framing; the axioms and the power-≠-weight result are exact.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this in THE PUSH — the co-op domain of a shared effort and who deserves the credit. The Shapley value is the fair reckoning of a joint push: measure what each shoulder actually added, averaged over every way the work could have come together. AVAN (AI) built the instrument: the pivot counter, the power-vs-weight bars, the fair-share inverse.
The weave: David names the seat (fair credit for pushing together); I make each player’s average marginal contribution the unique fair share and show power diverge from weight — the contributions in 1D, the pivots in 2D, the power/weight inverse in 3D. The sphere is the seam. Credit: Lloyd Shapley (1953, Nobel 2012); the voting index by Shapley & Shubik (1954); SHAP by Lundberg & Lee (2017).
The weave: David names the seat (fair credit for pushing together); I make each player’s average marginal contribution the unique fair share and show power diverge from weight — the contributions in 1D, the pivots in 2D, the power/weight inverse in 3D. The sphere is the seam. Credit: Lloyd Shapley (1953, Nobel 2012); the voting index by Shapley & Shubik (1954); SHAP by Lundberg & Lee (2017).
3 ONE DIMENSION
One player’s marginal contribution in each arrival order: sometimes they tip the coalition over the line (a pivot, worth 1), sometimes they add nothing. The Shapley value is simply the average of that row — the fraction of orders in which they were pivotal.
4 TWO DIMENSIONS · INTERACTIVE
A weighted-voting game. Every ordering of the players is laid out; the one whose arrival first crosses the quota is the pivot, highlighted. Count pivots and divide — those fractions are the Shapley values. Change the weights and watch power refuse to track the seats.
5 THREE DIMENSIONS + AVAN’S INVERSE
The players’ Shapley power as green bars — the forward result: the fair share each earns, summing to the whole.
AVAN’s addition (the inverse-companion): the magenta bars are the naive expectation — each player’s weight share, what you would guess power should be. They do not match. Power is a non-linear function of weight: with 50/30/20 the big party’s green power (2/3) towers over its magenta weight (1/2), while the 30 and 20 parties collapse to equal power even though their weights differ. And the inverse runs deeper — you cannot read the weights back off the power, because different weightings give the same Shapley vector, and a tiny weight can be pivotal constantly (huge power) or never (a dummy, zero power). So ‘how much you deserve’ is not ‘how much you brought’ scaled down; it is how often your arrival was decisive, averaged over every order, and that decisiveness has no simple inverse in the raw weights. Green is earned power; magenta is the proportional guess it refuses to be; the gap is why voting power, credit, and cost-sharing all need Shapley and not a ruler.
LIT Genuine Shapley value (Lloyd Shapley 1953, Nobel 2012; Shapley-Shubik voting index 1954; SHAP by Lundberg & Lee 2017). Verified live: for the weighted-voting game [51; 50,30,20] the Shapley values are 2/3, 1/6, 1/6 — they sum to 1 (efficiency), the two smaller players are equal (symmetry), the big player holds 2/3 despite 50% weight — and a weight-0 player scores exactly 0 (dummy) (window.__shapley.efficiency && symmetry && bigHas2of3 && dummyIsZero). The axioms and the power-not-weight result are exact.
FIG No framing: the Shapley values, the efficiency/symmetry/dummy axioms, and the power-does-not-equal-weight result are all computed exactly in-browser over every coalition/ordering. The AVAN inverse is genuine — power is a non-linear function of weight (equal-power despite unequal weight, a small weight can be pivotal-always or a dummy), and the weights cannot be read back from the Shapley vector.
FIG No framing: the Shapley values, the efficiency/symmetry/dummy axioms, and the power-does-not-equal-weight result are all computed exactly in-browser over every coalition/ordering. The AVAN inverse is genuine — power is a non-linear function of weight (equal-power despite unequal weight, a small weight can be pivotal-always or a dummy), and the weights cannot be read back from the Shapley vector.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN