The evil of weighted voting

In some weighted voting systems, one person is allowed to cast more votes than another.

Yet person who casts two votes does not simply have more power than any of the others, but more power than all the others combined. Voting power in a weighted-voting scheme is not proportional to the actual vote cast, but is measured in terms of the probability that one will cast a deciding vote, being a critical voter in the coalition, a concept embraced by federal courts in cases like Franklin v. Krause, 32 N.Y.2d 234, 344 N.Y.S.2d 885, 298 N.E.2d 68 (1973), appeal dismissed, 415 U.S. 904, 94 S.Ct. 1397, 39 L.Ed.2d 461 (1974).

Consider a committee of five in which one individual casts two votes, with four others casting one each; six votes are cast in total. If the threshold is 60% of six votes, then a tally of 6-0, 5-1, or 4-2 results in approval, but a tally of 3-3, 2-4, 1-5, or 0-6 results in defeat. (See spreadsheet and text file.)

In this situation, an individual with one vote casts a deciding vote by voting "aye" to reach 4-2 or "nay" to reach 3-3. An individual with two votes casts a deciding vote by voting "aye" to reach 5-1 or 4-2, or by voting "nay" to reach 3-3 or 2-4.

If a yes-no decision is made by five people, then there are 2^5=32 possible distribuitions of votes: one case of 6-0, four of 5-1, seven of 4-2, eight of 3-3, seven of 2-4, four of 1-5, and one of 0-6.

If all distributions are considered equally likely (the conventional random assumption in probability modeling), then, over a large number of votes, one single-voter can be expected to cast a deciding vote 8/32 or 25% of the time. Of any two single-voters, the probability that at least one will cast a deciding vote rises to 12/32 or 38%. Of any three single-voters, the probability that one or more will cast a deciding vote rises to 14/32 or 44%. Of any four single-voters, the probability that one or more will cast a deciding vote rises to 15/32 or 47%.

Yet the double-voter will cast a deciding vote 20/32 or 63% of the time.

That means that a lone individual with two votes has more power than all four of the others combined.

To quantify power in other weighted-voting or bloc-voting schemes, use a Banzhaf calculator.

Links:

• Weighted voting at Wikipedia
• Voting power
• Game theory at Britannica
• Banzhaf calculator
• John F. Banzhaf III

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