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Don’t Be a Dummy – Measuring Shareholder Voting Power

John F. Banzhaf III is a law professor who in the 1960s became interested in voting and voting power.  See Weighted Voting Doesn’t Work: A Mathematical Analysis, 19 Rutgers Law Review 317 (1965).  In ensuing years, Professor Banzhaf has attracted controversy for his anti-smoking and anti-obesity activism.

Determining voting power is key to knowing where the balance of power lies.  Our political system is primarily, but not exclusively based on the principle of one person one vote.   The California General Corporation Law rejects per capita voting.  Instead, each shareholder’s voting power is based on the number of shares held of record by that shareholder with each outstanding share having one vote on each matter submitted to the shareholders.  Cal. Corp. Code § 700(a)  There are two exceptions to this rule.  First, cumulative voting gives each shareholder a number of votes equal to the number of votes that the shareholder is “normally entitled” multiplied by the number of directors to be elected (the idea cumulative voting in the political arena was the idea that caused so much trouble for Professor Lani Guinier).  Cal. Corp. Code § 708.  Second, the articles of incorporation may vary this rule.

Section 700 prescribes the voting rule but it doesn’t tell you how to determine voting power. This is where the Banzhaf Power Index offers some insight.  Basically, the Banzhaf Power Index is calculated based on the relative number of situations in which a shareholder has a “swing” vote (i.e., the power to change a winning coalition into a losing coalition or vice versa).  This is illustrated in the following example.

A corporation has four stockholders who own the following number of shares with each share entitled to one vote:

Stockholder Number of Shares
A 51
B 20
C 19
D 10

Under a voting rule that requires the affirmative vote of a majority of the outstanding shares, Stockholder A has complete control and the remaining stockholders have no power to change the result.  If the voting rule is changed to require the affirmative vote of at least 75% of the shares, Stockholder A retains substantial power but is no longer a autocrat.  Now, there are a total of 4 winning coalitions: (A+B+C+D), (A+B+C), (A+B+D), and (A+C+D) and Shareholder A can change the result in all four of these coalitions.  Shareholders B, C & D can now change the result (by switching votes) in each of two coalitions.  Thus, the relative voting powers of Shareholders A, B, C & D expressed as percentages are 40%, 20%, 20%, and 20%.  If a shareholder has no power to change the result, that shareholder is referred to as a “dummy”.

Note that Shareholder D’s voting power is equivalent to that of Shareholder B even though Stockholder D holds half as many shares.  This insight is as important as it is occult.  Thus, knowing how to determine a shareholder’s Banzhaf Power Index can help a shareholder assess its voting power and avoid being a dummy.

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