Cooperative Game Theory & Governance
Shapley-Shubik Weighted Voting Power Index Formula
Calculate Lloyd Shapley and Martin Shubik’s (1954) exact 6-permutation pivotal voting power index for three shareholder blocks or legislative parties against quota Q.
See why a 2% swing shareholder can wield the exact same voting power as a 49% giant!
Corporate board seats and parliamentary vote weights are famously deceptive! Lloyd Shapley and Martin Shubik (1954) measured true voting power by ordering all n! sequential coalition-formation permutations (for 3 parties, 3! = 6 permutations: 1-2-3, 1-3-2, 2-1-3, 2-3-1, 3-1-2, 3-2-1) and counting the fraction of orderings in which Party i is the PIVOTAL voter whose arrival pushes cumulative votes from < Q to >= Quota Q! Unlike the Banzhaf index (which counts unordered winning coalitions), Shapley-Shubik power shares ALWAYS sum to 1.0000 automatically without ad-hoc renormalization!
Verify on Shareholder_A w_1 = 49 votes (B2), Shareholder_B w_2 = 49 votes (B3), Swing_Founder_C w_3 = 2 votes (B4), and Majority_Quota Q = 51 votes (B5): Although Founder C holds only 2.00% of equity votes, check all 6 arrival orders: (1-2-3 -> B pivotal at 98), (1-3-2 -> C pivotal at 51!), (2-1-3 -> A pivotal at 98), (2-3-1 -> C pivotal at 51!), (3-1-2 -> A pivotal at 51), (3-2-1 -> B pivotal at 51)! Each player is pivotal in 2 of 6 permutations, yielding Shapley-Shubik Power SS_A = 0.3333, SS_B = 0.3333, and SS_C = 0.3333 (33.33% equal power — a +31.33 percentage-point leverage wedge for Founder C)! Now test w_1 = 45, w_2 = 40, w_3 = 15 under a 60-vote supermajority (Q = 60): Party A becomes pivotal in 4/6 = 0.6667 of permutations while B and C split 0.1667 each!
| Shareholder / Bloc [51: 49, 49, 2] | Raw Vote Share | Pivotal Permutations (/ 6) | Shapley-Shubik Power |
|---|---|---|---|
| Shareholder_A (49 votes) | 0.4900 (49.0%) | 2 (2-1-3, 3-1-2) | 0.3333 (33.33%) |
| Shareholder_B (49 votes) | 0.4900 (49.0%) | 2 (1-2-3, 3-2-1) | 0.3333 (33.33%) |
| Swing_Founder_C (2 votes) | 0.0200 (2.0%) | 2 (1-3-2, 2-3-1) | 0.3333 (33.33%) |
| Supermajority [60: 45, 40, 15] | 45% / 40% / 15% | 4 / 1 / 1 | 0.6667 / 0.1667 / 0.1667 |
Exact 6-Permutation Shapley-Shubik Power Index Formula
Player i with weight w_i (facing peers w_j, w_k and quota Q) is pivotal in: 2 permutations as 1st entrant if w_i >= Q; 1 permutation after j if w_j < Q <= w_j + w_i; 1 permutation after k if w_k < Q <= w_k + w_i; and 2 permutations as 3rd entrant if w_j + w_k < Q <= w_1 + w_2 + w_3!
=LET(w_a, B2, w_b, B3, w_c, B4, q_th, B5, w_tt, w_a + w_b + w_c, IF(OR(w_a < 0, w_b < 0, w_c < 0, q_th <= w_tt / 2, q_th > w_tt), "Invalid Proper Voting Game", LET(pv_c, 2 * N(w_c >= q_th) + N(AND(w_a < q_th, w_a + w_c >= q_th)) + N(AND(w_b < q_th, w_b + w_c >= q_th)) + 2 * N(AND(w_a + w_b < q_th, w_tt >= q_th)), ROUND(pv_c / 6, 4))))Power-Over-Vote-Share Leverage Wedge Formula
Enforce proper majority/supermajority game rules: Q > (w_1 + w_2 + w_3) / 2 (so two disjoint coalitions cannot pass contradictory motions simultaneously) and Q <= w_1 + w_2 + w_3!
=IF(ISTEXT(E4), E4, ROUND(E4 - B4 / SUM(B2:B4), 4))Try the example
Use the original values or change them to test your understanding.