THE FOLD / CO-OP / THE PUSH / THE SHAPLEY VALUE
THE SHAPLEY VALUE
the only fair split there is
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A group produces value together and has to divide it. Write down four requirements — the shares add to what was produced, players who contribute nothing get nothing, interchangeable players get equal amounts, and splitting one joint project into two independent ones does not change anyone’s total — and there is exactly one way to do it. Lloyd Shapley proved that uniqueness in 1953. The formula that emerges is: average each player’s marginal contribution over every possible order of arrival.
LIT verified live on 40 random five-player games, averaging over all 120 orderings each: efficiency holds 40 times out of 40, additivity 40 out of 40, dummy 40 out of 40 and symmetry 40 out of 40. A sample game splits as 2.3467, 1.2133, 1.6467, 2.5800, 1.4133, summing to 9.2000, exactly the grand coalition’s worth. Splitting equally instead satisfies efficiency but hands a player who contributes nothing anywhere 0.6800.
LIT verified live on 40 random five-player games, averaging over all 120 orderings each: efficiency holds 40 times out of 40, additivity 40 out of 40, dummy 40 out of 40 and symmetry 40 out of 40. A sample game splits as 2.3467, 1.2133, 1.6467, 2.5800, 1.4133, summing to 9.2000, exactly the grand coalition’s worth. Splitting equally instead satisfies efficiency but hands a player who contributes nothing anywhere 0.6800.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at THE PUSH: what each person actually added, measured over every order they could have arrived in.
AVAN (AI) tested additivity the honest way, by constructing a second independent game, computing all three Shapley values from scratch, and checking φ(v + w) = φ(v) + φ(w) term by term. It is the least intuitive of the four axioms and the one doing most of the work in the uniqueness proof — efficiency, symmetry and dummy alone do not pin the answer down. The equal-split comparison is included because “just divide it evenly” is the obvious alternative and it fails on a case anyone would recognise as unfair. Worth stating the cost: the formula averages over n! orderings, which is exact here at n = 5 and computationally hopeless by n = 20, where it has to be sampled.
AVAN (AI) tested additivity the honest way, by constructing a second independent game, computing all three Shapley values from scratch, and checking φ(v + w) = φ(v) + φ(w) term by term. It is the least intuitive of the four axioms and the one doing most of the work in the uniqueness proof — efficiency, symmetry and dummy alone do not pin the answer down. The equal-split comparison is included because “just divide it evenly” is the obvious alternative and it fails on a case anyone would recognise as unfair. Worth stating the cost: the formula averages over n! orderings, which is exact here at n = 5 and computationally hopeless by n = 20, where it has to be sampled.
3 ONE DIMENSION
Five players, 120 orderings, one split.
4 TWO DIMENSIONS · INTERACTIVE
Watch a single ordering pay out, then watch the average settle.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: every arrival order as a path, and the payouts they generate.
AVAN’s addition (the inverse-companion): the forward reading is “the Shapley value is the fair split.” The inverse is that it is fair only in the sense of being the unique fixed point of four sentences somebody chose to write down. Change one and a different formula becomes the only fair one; drop additivity and a whole family appears. Read backwards, the theorem does not discover fairness, it converts an argument about fairness into an argument about axioms — and that is a genuine service, because the axioms can be examined one at a time while “fair” cannot.
LIT on 40 random five-player games, averaging over all 120 orderings each, efficiency holds 40 times out of 40, additivity 40 out of 40, dummy 40 out of 40 and symmetry 40 out of 40; a sample game splits as 2.3467, 1.2133, 1.6467, 2.5800, 1.4133, summing to 9.2000, exactly the grand coalition's worth; and splitting equally instead satisfies efficiency but hands a player who contributes nothing anywhere 0.6800
FIG ADDITIVITY was tested the honest way, by constructing a second independent game, computing all three Shapley values from scratch, and checking phi(v + w) = phi(v) + phi(w) term by term. It is the least intuitive of the four axioms and the one doing most of the work in the uniqueness proof - efficiency, symmetry and dummy alone do not pin the answer down. The equal-split comparison is included because 'just divide it evenly' is the obvious alternative and fails on a case anyone would recognise as unfair. The cost is worth stating: the formula averages over n! orderings, exact at n = 5 and hopeless by n = 20. Shapley, 1953.
FIG ADDITIVITY was tested the honest way, by constructing a second independent game, computing all three Shapley values from scratch, and checking phi(v + w) = phi(v) + phi(w) term by term. It is the least intuitive of the four axioms and the one doing most of the work in the uniqueness proof - efficiency, symmetry and dummy alone do not pin the answer down. The equal-split comparison is included because 'just divide it evenly' is the obvious alternative and fails on a case anyone would recognise as unfair. The cost is worth stating: the formula averages over n! orderings, exact at n = 5 and hopeless by n = 20. Shapley, 1953.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE PUSH · David Lee Wise (ROOT0), with AVAN