Imagine that a corporation holds a shareholder vote on a project like a merger, and, under the corporation’s bylaws, each shareholder can cast a number of votes equal to the square root of the number of shares that he holds. This might seem like a gimmick, but it actually provides a natural, smooth form of minority shareholder protection without the external intervention of the courts. In fact, under reasonable conditions square root voting (SRV) ensures that the project is approved if and only if it maximizes the value of the firm and thus achieves the efficiency goals of minority shareholder protection without the messy legal procedures that usually accompany them.
To see why, suppose that a corporation proposes a merger. An investor believes that the merger will increase the value of the corporation by $1000 per share, and that a vote in favor of the merger increases the probability of approval by 0.01. Thus, the marginal benefit from buying a share is $10. The investor can buy shares of the corporation at an opportunity cost of $1 (in the sense that she would rather use her money in another way and she incurs brokerage fees, but can otherwise sell the share for net expected profit of $0 if the merger is not approved). Because of the square-root rule, however, she must buy the square of the number of shares for every vote she casts. Thus, if she wants to cast 1 vote, she must buy 1 share at a cost of $1; if she wants to cast 2 votes, she must buy 2 shares at a cost of $4; if she wants to buy 3 shares, she must pay $9; and so on.
In deciding how many shares she buys, the investor will set the marginal benefit equal to the marginal cost—in this case, 10=2v, or v*=5. So she buys 5 shares. An investor who believed that the merger would cause the share price to rise by $500 would by contrast be willing to buy 2.5 shares (rounding up or down)—so half the willingness-to-pay for the merger translates into half the number of votes bought. Because everyone buys in proportion to the intensity of his or her preference, the vote will result in approval of the merger if and only if the merger is efficient (in the sense of reflecting everyone’s beliefs about efficiency).
SRV is derived from Quadratic Vote Buying (QVB), a mechanism developed by Weyl and which we applied to corporate governance. Under QVB, people have the option to buy votes by paying the square of the number of votes they want to cast, and the collected funds are returned to everyone pro rata. QVB can be adapted to shareholder voting, as we argue, but SRV may be simpler and more intuitive as it does not require the collection and redistribution of funds.
If we are right that SRV or QVB can ensure that corporations undergo mergers and other major transactions only when it is efficient, then these voting mechanisms should also give shareholders a stronger incentive to monitor corporations and give corporations a stronger incentive to hold votes than they have under the existing system. This will further reduce managerial agency costs and help generate optimal corporate behavior. Furthermore, because these systems protect minority shareholders directly by making the voting rule resilient against abuse and manipulation, they obviate the need for prohibitions against “empty voting” or intervention by courts through “appraisal remedies” to ensure a firm is not sold for a price below its value. By providing a smooth version of minority shareholder protection that does not require the intervention of inefficient courts, SRV or QVB addresses many of the problems that the corporate governance movement has tried to solve, with limited success, for decades.
The full paper is available for download here.