01:640:103 Chap.4 Shapley-Shubik Power Distribution
Shapley-Shubik Power Distribution
Power as the deciding vote
The Shapley-Shubik approach imagines the players joining a coalition one at a time. A sequential coalition lists every player in the order they decide to vote yes. Adding weights from the left, the pivotal player is the one whose arrival first brings the total up to the quota.
Every ordering has exactly one pivotal player, and a player's Shapley-Shubik power index is the fraction of all orderings in which it is pivotal. Lloyd Shapley, one of the two economists behind the index, received the Nobel Prize in Economics in 2012.
Two measures, two answers
Banzhaf and Shapley-Shubik ask different questions, so they can disagree.
In the guide's funding board with quota 10 and weights 6, 4, 3 and 2, Banzhaf gives the largest member half the power while Shapley-Shubik gives it 14 of 24 orderings, about 58%.
Both agree that a dummy has no power and a dictator has all of it, and both can be computed from a complete list of winning coalitions without weights.
How fast the work grows
With N players there are N factorial orderings: 6 for three players, 24 for four, 120 for five and more than three and a half million for ten.
A computer listing a million orderings a second would need about fifteen days for fifteen players and tens of thousands of years for twenty. The course uses this to make a wider point: some problems are exact in principle but impractical to solve by listing every case.
Reading an ordering quickly
You do not need to add every weight in every ordering.
Keep a running total from the left and stop the moment it reaches the quota; the player just added is pivotal and the rest of the ordering is irrelevant. Grouping the orderings by first player, as the text does, prevents omissions: with four players there are four blocks of six.
When a player is a dummy it can be skipped entirely, since it is never pivotal.
A useful check: the pivot counts for all players must add up to N factorial exactly, because each ordering contributes one pivot and only one.
What this chapter covers
- 01
Sequential coalitions
- 02
Pivotal players
- 03
The Shapley-Shubik power index
- 04
Comparing Shapley-Shubik with Banzhaf
- 05
Counting orderings with factorials
Worked example · free
Both power measures for one system
- 3There are 4! = 24 orderings. A is pivotal whenever it arrives after at least 2 votes are already in, which happens in 14 of them.
- 2B is pivotal in 6 orderings and C and D in 2 each; 14 + 6 + 2 + 2 = 24.
- 2Banzhaf: A is critical 5 times, B 3, C and D once each, so T = 10.
Key terms
- Sequential coalition
- A sequential coalition is an ordering of all the players, read as the order in which they join a yes vote.
- Pivotal player
- The pivotal player is the one whose arrival first brings a sequential coalition's running total up to the quota.
- Shapley-Shubik index
- The Shapley-Shubik index is the fraction of all orderings in which a player is pivotal.
- Factorial
- A factorial N! is the product N × (N − 1) × … × 2 × 1, the number of orderings of N players.
- Grand coalition
- The grand coalition is the coalition containing every player in the voting system.
Shapley-Shubik Power Distribution FAQ
When do Banzhaf and Shapley-Shubik give the same distribution?
Sometimes they coincide, as in several three-player systems, but nothing guarantees it. They measure different things, critical membership versus completing a coalition, so always state which index an answer uses.
How many sequential coalitions are there with six players?
Six factorial, which is 6 × 5 × 4 × 3 × 2 × 1 = 720. Each ordering has exactly one pivotal player, so the pivot counts for all six players must add to 720, which gives a quick way to catch an ordering you missed or counted twice.
Is a player with veto power always pivotal?
No. A veto player must be present before the quota is reached, but in some orderings another player arrives last and completes the total. The veto player is pivotal only in the orderings where its own arrival crosses the quota.
Exam move
List orderings in blocks by first player so none is missed, and check that the pivot counts add to N factorial before converting them to percentages.
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