Shareholder Voting Power and Ownership Control of Companies

Abstract

The pattern of ownership and control of British industry is unusual compared with most other countries in that ownership is relatively dispersed. Typically the largest shareholder in any large listed company is likely to own a voting minority of the shares. Majority ownership by a single shareholder is unusual. It is not uncommon for the largest shareholding to be under 20 % and in many cases it is much less than that. A broadly similar pattern is observed in the USA. Two inferences about corporate governance are conventionally drawn from this, following the early work of Berle and Means: (1) All but the very largest shareholders are typically too small to have any real incentive to participate in decision making; (2) All but the very largest shareholdings are too small to have any real voting power. The question of voting power is the focus of this chapter.

An earlier version of this chapter has been published in Power Measures, Vol II (Homo Oeconomicus 19(3)), edited by Manfred J. Holler and Guillermo Owen, Munich: Accedo Verlag, 2002: 345–371. The ideas and results in this chapter have been presented in different forms at the conferences of the European Association for Research in Industrial Economics in Turin, September 1999, “Ownership and Economic Performance”, Oslo May 2000 and the Game Theory Society, Games 2000, Bilbao, July 2000, and to staff seminars at City, Oxford, Oslo, Warwick and Cambridge Universities.

Appendix: 1 Proof that the power curve is concave

To show this, consider a block consisting of the largest k shareholders with combined shareholding ${\text{s}}_{\text{k}} = \sum\limits_{i = 1}^{k} {{\text{w}}_{\text{i}} }$. The power index for the block is PI_k defined as the swing probability for the coalition k in the voting model in which the votes of shareholders k + 1, k + 2,…,n are treated randomly as defined. Let x_i be the number of votes cast by shareholder i. Then x_i has the probability distribution, Pr(x_i = w_i) = Pr [x_i = 0] = 1/2, independently for all i. Define the random variable ${\text{Y }} = \sum\limits_{{{\text{i}} = {\text{k}} + 2}}^{\text{n}} {{\text{x}}_{\text{i}} }$. The swing probability PI_k can be written:

$$ {\text{PI}}_{\text{k}} = \, 0. 5 {\text{ P}}_{\text{r}} [0. 5 { } - {\text{ s}}_{\text{k}} < {\text{Y}} < 0. 5\left] { \, + \, 0. 5 {\text{ P}}_{\text{r}} } \right[0. 5 { } - {\text{ s}}_{\text{k}} < {\text{Y }} + {\text{ w}}_{{{\text{k}} + 1}} < 0. 5] \, . $$

Denoting the cumulative probability distribution function for Y by the function P(Y), this can be written as,

$$ {\text{PI}}_{\text{k}} = \, \left[ {{\text{P}}\left( {0. 5} \right) \, - {\text{ P}}\left( {0. 5 { } - {\text{ s}}_{\text{k}} } \right) \, + {\text{ P}}\left( {0. 5 { } - {\text{ w}}_{{{\text{k}} + 1}} } \right) \, - {\text{ P}}\left( {0. 5 { } - {\text{ s}}_{{{\text{k}} + 1}} } \right)} \right]/ 2. $$

Now consider the index for coalition k + 1 of size s_k+1: PI_k+1 = P(0.5) – P(0.5 –s_k+1) Therefore the change in the index is:

$$ {\text{PI}}_{{{\text{k}} + 1}} - {\text{ PI}}_{\text{k}} = \, \left[ {{\text{P}}\left( {0. 5} \right) \, - {\text{ P}}\left( {0. 5 { } - {\text{ w}}_{{{\text{k}} + 1}} } \right) \, + {\text{ P}}\left( {0. 5 { } - {\text{ s}}_{\text{k}} } \right) \, - {\text{ P}}\left( {0. 5 { } - {\text{ s}}_{{{\text{k}} + 1}} } \right)} \right]/ 2 { }. $$

This expression is always non-negative if w_k+1 ≥ 0. It is decreasing as w_k+1 → 0, since P(0.5) → P(0.5 – w_k+1) and P(0.5 – s_k) → P(0.5 – s_k+1). Therefore the power curve is concave increasing as drawn in Fig. 1.