Economics & Market Microstructure
Banzhaf Voting Power Index Formula: Weighted Quota, Critical Swings & Power vs. Seat Share
Expose why nominal shareholdings mislead in weighted board voting [Quota q; w_1, w_2, w_3] by counting critical swing coalitions and computing the Normalized Banzhaf Power Index.
Why a 48% shareholder and a 4% shareholder can hold the EXACT same voting power!
Introduced by Lionel Penrose (1946) and John F. Banzhaf III (1965), the Banzhaf Power Index proves that voting power in corporate boards, IMF/EU councils, and venture governance equals how often a voter’s defection flips a winning coalition (>= Quota q) into a losing coalition (< q)—NOT their raw share percentage!
Consider weighted board [Quota = 51; Voter A = 48, Voter B = 48, Voter C = 4]: no single member reaches 51 alone, ANY two members exceed 51 (48+48=96, 48+4=52), and in the grand coalition A+B+C=100 nobody is critical because 100 - 48 = 52 >= 51! Thus A, B, and C each cast 2 critical swings out of 6 total swings—giving the 4% shareholder C an identical 33.33% Banzhaf voting power as each 48% partner! Conversely, in [Quota = 51; A = 52, B = 35, C = 13], Voter A has 100% dictator power while B and C are 0% dummies!
| Weighted Board Game [Quota q; w_A, w_B, w_C] | Critical Swings [eta_A, eta_B, eta_C] | Normalized Banzhaf Power [beta_A, beta_B, beta_C] |
|---|---|---|
| [51; 48, 48, 4] (4% Tie-Breaker) | [2, 2, 2] (Total = 6) | [33.33%, 33.33%, 33.33%] (Equal Power!) |
| [60; 45, 35, 20] (Supermajority) | [3, 1, 1] (Total = 5) | [60.00%, 20.00%, 20.00%] |
| [51; 52, 35, 13] (Majority Controlling) | [4, 0, 0] (Total = 4) | [100.00%, 0.00%, 0.00%] (A = Dictator) |
Critical Swing Count eta_A Across All 4 Coalitions Containing Voter A (3-Voter Board)
Counts how many of the 4 coalitions containing A ({A}, {A,B}, {A,C}, {A,B,C}) meet quota B2 WITH A (w >= q) but fail WITHOUT A (w - w_A < q).
=IF(OR(B2 <= (B3 + B4 + B5) / 2, B2 > B3 + B4 + B5), "Quota Must Exceed 50% and <= Total Votes", IF(B3 >= B2, 1, 0) + IF(AND(B3 + B4 >= B2, B4 < B2), 1, 0) + IF(AND(B3 + B5 >= B2, B5 < B2), 1, 0) + IF(AND(B3 + B4 + B5 >= B2, B4 + B5 < B2), 1, 0))Voter C Normalized Banzhaf Power Index (%) = 100 * eta_C / (eta_A + eta_B + eta_C)
Divides Voter C’s critical swings by the sum of all three voters’ critical swings.
=IF(OR(B2 <= (B3 + B4 + B5) / 2, B2 > B3 + B4 + B5), "Quota Must Exceed 50% and <= Total Votes", ROUND(100 * (IF(B5 >= B2, 1, 0) + IF(AND(B3 + B5 >= B2, B3 < B2), 1, 0) + IF(AND(B4 + B5 >= B2, B4 < B2), 1, 0) + IF(AND(B3 + B4 + B5 >= B2, B3 + B4 < B2), 1, 0)) / ((IF(B3 >= B2, 1, 0) + IF(AND(B3 + B4 >= B2, B4 < B2), 1, 0) + IF(AND(B3 + B5 >= B2, B5 < B2), 1, 0) + IF(AND(B3 + B4 + B5 >= B2, B4 + B5 < B2), 1, 0)) + (IF(B4 >= B2, 1, 0) + IF(AND(B3 + B4 >= B2, B3 < B2), 1, 0) + IF(AND(B4 + B5 >= B2, B5 < B2), 1, 0) + IF(AND(B3 + B4 + B5 >= B2, B3 + B5 < B2), 1, 0)) + (IF(B5 >= B2, 1, 0) + IF(AND(B3 + B5 >= B2, B3 < B2), 1, 0) + IF(AND(B4 + B5 >= B2, B4 < B2), 1, 0) + IF(AND(B3 + B4 + B5 >= B2, B3 + B4 < B2), 1, 0))), 2))Try the example
Use the original values or change them to test your understanding.