A Paradox in Measuring Corporate Control

When major investors own shares in multiple firms in the same industry, the cross ownership changes competitive interests and moves the equilibrium closer to the monopoly solution. In the presence of cross ownership, the traditional measure of industry concentration, the Hirschman-Herfindahl Index, fails to accurately reflect the level of competition in the market. We extend the literature on cross ownership and develop a robust measure, a modified Hirschman-Herfindahl index, that captures the competitive effects of cross ownership. The modified Hirschman-Herfindahl index requires a measure of the degree of control exercised by the various shareholders of the firm. One candidate for this measure is the Shapley value—see Owen (1982), Shapley (1953), Shapley and Shubik (1954) and Leech (1988). Some recent empirical studies of the premium paid for voting rights on shares of stock have used the Shapley value as predictor of the premium, and with some success—see Rydqvist (1987) and Zingales (1994, 1995). We show that the Shapley value can generate paradoxical results when ownership interests in corporations can be held indirectly through layered corporate entities and when there exists cross ownership of shares at the different corporate levels. We illustrate this fact with a couple of simple examples based on patterns of corporate ownership in the cable television industry where layered corporate entities and cross ownership is common. The fact that the cross ownership itself may undermine the validity of a commonly proposed measure is important to note. We show that the paradox exposed by the case of the cable television industry is an example of a well known paradox in defining a measure of control in voting games—see Brams and Affuso (1976), Fischer and Schotter (1978).