Solution
A Seat Worth a Twelfth
Show the problem again
A company has four shareholders with 40, 30, 20 and 10 votes (100 in total). A motion passes if it gets at least 51 votes. In the Shapley-Shubik index, voters join a coalition in a uniformly random order, and the *pivot* is the voter whose arrival first lifts the running total to at least 51. A voter's power is the probability of being the pivot. What is the Shapley-Shubik power of the 10-vote shareholder? Give a reduced fraction.
Worked solution
There are 4! = 24 arrival orders. The 10-vote holder is pivotal exactly when the voters ahead of them total between 41 and 50 votes, so that adding 10 reaches 51 for the first time. Among the other holders (40, 30, 20) the only subset with a total in that range is {30, 20}, which sums to 50. The orders are therefore 30, 20, 10, 40 and 20, 30, 10, 40: 2 orders in all, so the index is 2/24 = **1/12**. For comparison the full index is (5/12, 1/4, 1/4, 1/12): the 30-vote and 20-vote holders have identical power, and weight is not proportional to power.
Source: Shapley-Shubik power index (Shapley and Shubik, 1954); the weighted voting system is invented for AxiomIQ. Statement written for AxiomIQ.