Shareholder voting power
Are raw numbers of shares always the best indicator of the extent to which an individual can influence the outcome of a particular motion? To examine this question, we apply the tools of cooperative game theory that were introduced in Chapter 3, with only a few minor modifications. Suppose, for example, that a particular motion comes before shareholders. Then the motion will either pass or not pass (fail). If motions must pass by a strict majority of voting shares in favour, then (in the absence of abstentions) there are only two relevant kinds of coalitions that can form: those which can force a motion to pass (winning coalitions), and those which cannot (losing coalitions).
Consider the simplest case, where there are N shareholders, each of whom owns one share with voting rights. Without loss of generality, with transferable utility we can set v(S) = 1 if the coalition S is winning, and v(S) = 0 if S is not winning, so that:
The cooperative game described by the characteristic function in equation (10.27) is called a simple majority game.
As a complete model for analysing corporate voting, however, it is inadequate, because it does not allow for the possibility that shareholders may not hold the same number of shares. To allow for this possibility, we consider the notion of a weighted majority game, which consists of a set N of individuals, a collection of weights {w_{t} > 0 : i = 1,..., N} and a quota q. We represent a weighted majority game by:
In this environment, a coalition S is winning if:
Setting q = N +1 and w_{{} = 1 for all i gets us back to the situation in
which each shareholder holds the same number of shares and there is simple majority rule.
If shareholder i owns w_{i} voting shares then we have a weighted majority game, and the characteristic function of the weighted majority game is:
How influential is an individual shareholder? As we saw in Chapter 7 and earlier in this chapter, the concept of 'pivotalness' is critical for examining the extent to which an individual shareholder's decision affects the outcome in any given situation.
This idea of pivotalness can be extended to the analysis of shareholder voting. The concept of a shareholder's voting power captures the likelihood that a shareholder will influence the outcome of a particular motion that will in turn affect the firm's future direction. If a shareholder can purchase shares which have voting rights attached to them and such a purchase will increase the shareholder's influence, then those shares may be more valuable to that particular shareholder. Hence there may be a close link between an individual shareholder's voting power and the value that they attach to shares  which means that voting power may, in some circumstances, influence the company's share price.
The motivation behind the concept of voting power can be illustrated using the following simple example. Suppose that a company has 36 voting shares, and three shareholders: A, B and C. Suppose that the distribution of shares is as in Table 10.5.1.
With 36 shares in total, to pass a motion would require a clear majority or 19 shares. Using our notation, the representation of this game is therefore:
Table 10.5.1 An example of shareholder voting
A 
B 
C 

Shares 
11 
17 
8 
Percentage of Shares 
0.30556 
0.47222 
0.22222 
Winning Coalitions 
{A, B} 
{B, A} 
{C, A} 
{A, C} 
{B, C} 
{C, B} 

Containing Shareholder i 
{A, B, C} 
{A, B, C} 
{A, B, C} 
This situation illustrates the basic point that using raw numbers of shares to determine voting power or influence is clearly inappropriate. If we just looked at the raw numbers of shares, it would seem reasonable to think that shareholder B, who holds 17 shares (or over 47 per cent of voting rights), should be more influential than each of the other two shareholders. However, this reasoning is incorrect: all of the winning coalitions in the above game require at least two of the shareholders. Therefore, no shareholder could be said to have more influence than the other in this situation. In this situation, any reasonable measure of voting power should have each shareholder having equal power or influence. Since raw percentages of shares do not have this property, they are unsuitable as a measure of voting power and shareholder influence.
The concept of voting power simply attempts to formalise this reality that raw numbers of shares are not always a reasonable indicator of how likely certain shareholder will be pivotal (in the sense that they can turn a losing coalition into a winning one). We now turn to three such measures that have been developed and used in the literature.