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Weighted Voting and Power Indices

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Now · Weighted Voting Games

Weighted Voting Games

Each player

i

has weight

w_i

, quota

Q

. Coalition

S

wins if

\sum_{i \in S} w_i \geq Q

Index	Formula	Interpretation
Shapley-Shubik	$\phi_i = \frac{\text{\# permutations where } i \text{ is pivotal}}{n!}$	Probability of being pivotal given random order
Banzhaf	$\beta_i = \frac{\text{\# winning coalitions where } i \text{ is critical}}{2^{n-1}}$	Number of times $i$ can change outcome

Example: UN Security Council (5 permanent, 10 rotating, 9 votes needed).

Permanent members: Shapley-Shubik index $\approx 0.196$ each
Rotating members: $\approx 0.00186$ each
Permanent members have $\approx 100\times$ the power Join Discord PreviousCooperative GamesNextBargaining