The Banzhaf index in complete and incomplete shareholding structures: A new algorithm

Abstract

In this global world many firms present a complex shareholding structure with indirect participation, such that it may become difficult to assess a firm's controllers. Furthermore, if there are numerous dominant shareholders, the control can be shared between them. Determining who has the most influence often is a difficult task. To measure this influence, game theory allows modeling voting game and computing the Banzhaf index. This paper first offers a new algorithm to compute this index in all structures and suggests some modelisations of the floating shareholder. Then, our model is applied to a real case study: The French Group Lafarge. This exemplary case demonstrates how the float's structure and hidden coalition can impact the power relationship between dominant shareholders. Abstract:In this global world many firms present a complex shareholding structure with indirect participation, such that it may become difficult to assess a firm's controllers. Furthermore, if there are numerous dominant shareholders, the control can be shared between them. Determining who has the most influence often is a difficult task. To measure this influence, game theory allows modeling voting game and computing the Banzhaf index. This paper first offers a new algorithm to compute this index in all structures and suggests some modelisations of the floating shareholder. Then, our model is applied to a real case study: The French Group Lafarge. This exemplary case demonstrates how the float's structure and hidden coalition can impact the power relationship between dominant shareholders. JEL Classifications: C63, G32, L22

Corporate Control with Cross-Ownership

“…This section extends Levy's (2011) algorithm to corporate structures with cross-ownership. Levy's (2011) algorithm computes the classical Banzhaf index thanks to the coincidence matrix.…”

“…In particular, the current definition of the Banzhaf (1965) index does not accommodate multiple equilibria. Moreover, multiple equilibria are detrimental to the convergence of existing algorithms, which compute the ultimate shareholders' control stakes in a set of interlocked firms (Crama and Leruth, 2007;Levy, 2011).…”

“…Under the first, control stakes are consolidated through input-output matrix algebra (Brioschi et al, 1989;Claessens et al, 2000;Chapelle and. Secondly, the Banzhaf (1965) power index from game theory gives the probability that an ultimate shareholder decisively influences the decisions made in a subordinated firm (Crama and Leruth, 2007;Levy, 2009 and2011). These two approaches work well as long as the ownership structure is a "directed 4 acyclic graph", which de facto excludes cross-ownership.…”

“…To address this issue, Crama and Leruth (2007) use Monte-Carlo simulations. Alternatively, Levy's (2011) (12) is taken over all possible voting states of the sources, and then divided by the number of such possibilities.…”

Power Indices and the Measurement of Control in Corporate Structures

“…The previous approaches provide different ways of modeling the float in weighted majority games; it would be interesting to validate them in a variety of applications (see Leech [2002] for investigations along these lines; see also , Levy [2011]). More importantly, perhaps, the impact of the float in general games associated with multilevel pyramidal structures appears to have been disregarded so far, and needs to be appraised both from the theoretical and from the computational points of view.…”