Economics & Market Microstructure
Shapley Value Formula (3-Player Cooperative Game): Exact Coalition Marginal Allocation
Compute the fair Shapley Value payoff allocation for a 3-player cooperative game (marketing attribution, joint logistics, or R&D consortium) across all 3! = 6 arrival permutations and verify grand-coalition efficiency.
Why Lloyd Shapley’s 1953 axiomatic value is the unique fair payoff division of synergy!
Formulated by Nobel laureate Lloyd S. Shapley (1953), the Shapley Value is the unique cooperative game-theory allocation satisfying Efficiency (individual payoffs sum exactly to grand coalition worth v({1,2,3})), Symmetry, Dummy/Null-Player, and Additivity! Today it powers multi-touch marketing attribution, joint-venture cost sharing, and SHAP feature attribution.
For n = 3 players with standalone values v({1}), v({2}), v({3}), pairwise values v({1,2}), v({1,3}), v({2,3}), and grand coalition v({1,2,3}) (with v(empty) = 0), Player 1 enters first in 2 of 6 permutations (weight 1/3), second after {2} or {3} in 1 permutation each (weight 1/6), and last after {2,3} in 2 permutations (weight 1/3): phi_1 = v({1})/3 + (v({1,2}) - v({2}))/6 + (v({1,3}) - v({3}))/6 + (v({1,2,3}) - v({2,3}))/3!
| Player in Joint Logistics Game (v1=10, v2=20, v3=30, v12=42, v13=54, v23=62, v123=90) | Exact 6-Permutation Shapley Formula | Shapley Payoff |
|---|---|---|
| Player 1 (phi_1) | 10/3 + (42-20)/6 + (54-30)/6 + (90-62)/3 | 20.3333 |
| Player 2 (phi_2) | 20/3 + (42-10)/6 + (62-30)/6 + (90-54)/3 | 29.3333 |
| Player 3 (phi_3) | 30/3 + (54-10)/6 + (62-20)/6 + (90-42)/3 | 40.3333 (Sum = 90.0000!) |
Player 1 Exact Shapley Value phi_1 = v1/3 + (v12 - v2)/6 + (v13 - v3)/6 + (v123 - v23)/3
Computes Player 1’s exact expected marginal contribution across all 6 coalition arrival orders.
=IF(B8 < MAX(B2, B3, B4, B5, B6, B7), "Non-Monotonic Coalition Values", ROUND(B2 / 3 + (B5 - B3) / 6 + (B6 - B4) / 6 + (B8 - B7) / 3, 4))Grand-Coalition Efficiency Check (phi_1 + phi_2 + phi_3 = v123)
Verifies that the sum of all three players’ Shapley allocations equals grand coalition value B8.
=ROUND((B2 / 3 + (B5 - B3) / 6 + (B6 - B4) / 6 + (B8 - B7) / 3) + (B3 / 3 + (B5 - B2) / 6 + (B7 - B4) / 6 + (B8 - B6) / 3) + (B4 / 3 + (B6 - B2) / 6 + (B7 - B3) / 6 + (B8 - B5) / 3), 4)Try the example
Use the original values or change them to test your understanding.