Takeover Algorithms

Gianfranco Gambarelli
Department of Mathematics, University of Bergamo

Abstract

A power index provides a reliable forecast of the expected division of power in a certain committee, where the ``weights" of the participants are known. Among the numerous indices published, the most well known and applied are those of Shapley-Shubik and Banzhaf-Coleman. Interesting applicationa of power indices are found when it is necessary to study how variations in the index of a certain player would be affected by changes in his weight. For example, in finance, a shareholder can evaluate the outcome of a potential take-over bid, by knowing a priori how the configuration of a major shareholding would change. He could also estimate how his controlling position would change, if he were to buy a block of shares from another major shareholder.

This paper extends the technique proposed in a previous work by Arcaini and Gambarelli for the computation of the variations of the Shapley-Shubik index in the above quoted cases, to the Banzhaf-Coleman index, and consequently provides a rapid method for the calculation of the latter.

Keywords: Power index, Banzhaf-Coleman, Shaplex-Shubik, takeover, OPA, shares, seats, government.

References

1

Arcaini, G. and G. Gambarelli, 1986. `Algorithm for Automatic Computation of the Power Variations in Share Trading', Calcolo 23(1), 13-19.

2

Banzhaf, J.F., 1965. `Weighted Voting Doesn't Work: A Mathematical Analysis', Rugers Law Review 19, 317-343.

3

Coleman, J.S., 1964. Introduction to Mathematical Sociology. Free Press of Glencoe, New York.

4

Gambarelli, G., 1983. `Common Behaviour of Power Indices', Internat. Journal of Game Theory 12(4), 237-244.

5

Gambarelli, G., 1994. `Power Indices for Political and Financial Decision Making', Annals of Operation Research 51,165-173.

6

Gambarelli, G. and G. Owen, 19994. `Indirect Control of Corporations', Internat. Journal of Game Theory 23(4), 287-302.

7

Shapley, L.S. and M. Shubik, 1954. `A Method for Evaluating the Distributions of Poer in a Committee System', American Political Science Review 48, 787-792.