The Original Banzhaf Power Index Problem

Nassau County is located on the western part of Long Island, New York, and is governed by a Board of Supervisors. The county is divided into 6 districts and, based on 1964 population figures, a total of 115 votes were assigned to the Board with a majority of 58 votes needed to carry a motion. The allocation of the 115 votes is given in the following table:

Hempstead #1 31

Hempstead #2 31

Oyster Bay 28

North Hempstead 21

Long Beach 2

Glen Cove 2

In 1965, lawyer John Banzhaf argued that the 3 largest districts held all the power on the County Board. Show that Banzhaf was right.

Solution: This is the weighted voting system [58: 31, 31, 28, 21, 2, 2] with 58 votes also needed to block.

COALITION WEIGHT TYPE CRITICAL VOTERS

{ } 0 losing

{A} 31 losing

{B} 31 losing

{C} 28 losing

{D} 21 losing

{E} 2 losing

{F} 2 losing

{A, B} 62 winning, blocking A, B, A, B

{A, C} 59 winning, blocking A, C, A, C

{A, D} 52 losing

{A, E} 33 losing

{A, F} 33 losing

{B, C} 59 winning, blocking B, C, B, C

{B, D} 52 losing

{B, E} 33 losing

{B, F} 33 losing

{C, D} 49 losing

{C, E} 30 losing

{C, F} 30 losing

{D, E} 23 losing

{D, F} 23 losing

{E, F} 4 losing

{A, B, C} 90 winning, blocking

{A, B, D} 83 winning, blocking A, B, A, B

{A, B, E} 64 winning, blocking A, B, A, B

{A, B, F} 64 winning, blocking A, B, A, B

{A, C, D} 80 winning, blocking A, C, A, C

{A, C, E} 61 winning, blocking A, C, A, C

{A, C, F} 61 winning, blocking A, C, A, C
{A, D, E} 54 losing

{A, D, F} 54 losing

{A, E, F} 35 losing

{B, C, D} 80 winning, blocking B, C, B, C
{B, C, E} 61 winning, blocking B, C, B, C

{B, C, F} 61 winning, blocking B, C, B, C

{B, D, E} 54 losing

{B, D, F} 54 losing

{B, E, F} 35 losing

{C, D, E} 51 losing

{C, D, F} 51 losing

{C, E, F} 32 losing

{D, E, F} 25 losing
{A, B, C, D} 111 winning, blocking

{A, B, C, E} 92 winning, blocking
{A, B, C, F} 92 winning, blocking

{A, B, D, E} 85 winning, blocking A, B, A, B
{A, B, D, F} 85 winning, blocking A, B, A, B

{A, B, E, F} 66 winning, blocking A, B, A, B

{A, C, D, E} 82 winning, blocking A, C, A, C

{A, C, D, F} 82 winning, blocking A, C, A, C

{A, C, E, F} 63 winning, blocking A, C, A, C
{A, D, E, F} 56 losing
{B, C, D, E} 82 winning, blocking B, C, B, C

{B, C, D, F} 82 winning, blocking B, C, B, C
{B, C, E, F} 63 winning, blocking B, C, B, C

{B, D, E, F} 56 losing

{C, D, E, F} 53 losing

{A, B, C, D, E} 113 winning, blocking

{A, B, C, D, F} 113 winning, blocking
{A, B, C, E, F} 94 winning, blocking

{A, B, D, E, F} 87 winning, blocking A, B, A, B

{A, C, D, E, F} 84 winning, blocking A, C, A, C

{B, C, D, E, F} 84 winning, blocking B, C, B, C

{A, B, C, D, E, F} 115 winning, blocking

So the Banzhaf power index is (32, 32, 32, 0, 0, 0) or, as percentages, (33.3%, 33.3%, 33.3%, 0%, 0%, 0%).

Based on Banzhaf’s work, the weights assigned to each district were changed and have been revised several more times since 1965.