Quiz 2

Weighted Voting and Power Indices

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# Weighted Voting and Power Indices ## Weighted Voting Games Each player $i$ has weight $w_i$, quota $Q$. Coalition $S$ wins if $\sum_{i \in S} w_i \geq Q$.

Weighted Voting and Power Indices

Weighted Voting Games

Each player ii has weight wiw_i, quota QQ. Coalition SS wins if ∑i∈Swi≥Q\sum_{i \in S} w_i \geq Q.

Shapley-Shubik vs Banzhaf

IndexFormulaInterpretation
Shapley-Shubikϕi=# permutations where i is pivotaln!\phi_i = \frac{\text{\# permutations where } i \text{ is pivotal}}{n!}Probability of being pivotal given random order
Banzhafβi=# winning coalitions where i is critical2n−1\beta_i = \frac{\text{\# winning coalitions where } i \text{ is critical}}{2^{n-1}}Number of times ii can change outcome
Example: UN Security Council (5 permanent, 10 rotating, 9 votes needed).
  • Permanent members: Shapley-Shubik index ≈0.196\approx 0.196 each
  • Rotating members: ≈0.00186\approx 0.00186 each
  • Permanent members have ≈100×\approx 100\times the power Join Discord PreviousCooperative GamesNextBargaining
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