Shareholder Voting Power

It is a general rule that voting rights in a corporation are allocated to its shareholders in a way that is proportional with their equity shares. This is intended to enshrine a natural principle that the distribution of decision-making power should mirror the distribution of capital invested. However, this ideal is often not achieved. Studies using voting power analysis and empirical evidence from capital markets both have shown that many companies are effectively controlled by a large minority shareholding if the distribution of the [Page 607]remaining share ownership is widely enough dispersed, as is typical in the United Kingdom and the United States. A single 30% shareholding has invariably working control, and a 20% holding is enough for control in many cases. On the other hand, where ownership is very concentrated and there are few shareholders, as in many continental European companies, shareholder voting power often bears little relation to ownership, as is typical in small weighting voting systems.

Two extreme examples illustrate the point. The first example involves a company with three shareholders, two of whom own 49% of the shares each, with the third owning 2%. All three members have the same power (with a normalized Banzhaf index of 33.33% each), because any decision requires any two members to pass. The second example involves a company with one large shareholder with a 30% holding and 70 small shareholders with 1% each. Here the large shareholder is almost totally dominant with almost total power: almost all decisions taken by simple majority require his or her support and the normalized Banzhaf index is 99.05%, while that for each 1% shareholder is 0.0136%. In both these cases, voting power is distributed very differently from share ownership.

Shareholder voting was one of the earliest applications of voting power indices discussed in the literature; it was one of the examples given in Lloyd Shapley and Martin Shubik's first paper. Shapley later developed the theory of oceanic games specifically to deal with the particular characteristics of shareholder voting games, where typically there are many small players with infinitesimal weights—a model that is a reasonable limiting approximation to many U.S. corporations.

Shareholder voting is a good context in which to compare the performance of different power indices because we have some empirical evidence—from market experience—about company control. Dennis Leech has compared the results of the Shapley-Shubik and Banzhaf power indices for shareholders in a large sample of corporations and found that the Shapley-Shubik indices calculated were implausible. For example, a large shareholder owning 40% of a company in which all the other shares are widely held in holdings of infinitesimal size, has virtual control and its power index should be very close to 100%. But its Shapley—Shubik Index is only 66.67%, whereas its Banzhaf index is almost 100%. This leads to the conclusion that the Shapley—Shubik Index is deficient and that in applied work one should use the Banzhaf index.