Revenue Sharing in Hierarchical Organizations: A New Interpretation of the Generalized Banzhaf Value

Affiliation(s)

Department of Economics, University of Ottawa, Ottawa, Canada.

école Normale Supérieure, Yaoundé, Cameroon.

Department of Economics, University of Ottawa, Ottawa, Canada.

école Normale Supérieure, Yaoundé, Cameroon.

ABSTRACT

This paper examines the distribution of earnings in a new model of hierarchical multi-task organizations. Each such organization is defined by a finite set of workers and tasks together with a production function that maps each allocation of workers to the tasks into an aggregate output. Tasks are ordered in degree of importance so that aggregate output increases when a worker climbs up the organization ladder. We show that the generalized Banzhaf value proposed by Freixas [1] can be used as a theory of revenue sharing in such organizations, and provide a new interpretation and formulation of this sharing rule, proving that a worker's pay is proportional to the difference between his marginal productivity at the top level and at the bottom level of the hierarchy summed over all the possible configurations of the organization. This new formulation also facilitates computation.

This paper examines the distribution of earnings in a new model of hierarchical multi-task organizations. Each such organization is defined by a finite set of workers and tasks together with a production function that maps each allocation of workers to the tasks into an aggregate output. Tasks are ordered in degree of importance so that aggregate output increases when a worker climbs up the organization ladder. We show that the generalized Banzhaf value proposed by Freixas [1] can be used as a theory of revenue sharing in such organizations, and provide a new interpretation and formulation of this sharing rule, proving that a worker's pay is proportional to the difference between his marginal productivity at the top level and at the bottom level of the hierarchy summed over all the possible configurations of the organization. This new formulation also facilitates computation.

Cite this paper

R. Pongou, B. Tchantcho and N. Tedjeugang, "Revenue Sharing in Hierarchical Organizations: A New Interpretation of the Generalized Banzhaf Value,"*Theoretical Economics Letters*, Vol. 2 No. 4, 2012, pp. 369-372. doi: 10.4236/tel.2012.24068.

R. Pongou, B. Tchantcho and N. Tedjeugang, "Revenue Sharing in Hierarchical Organizations: A New Interpretation of the Generalized Banzhaf Value,"

References

[1] J. Freixas, “The Banzhaf Measures for Games with Several Levels of Approval in the Input and Output,” Annals of Operations Research, Vol. 137, No. 1, 2005, pp. 45-66. doi:10.1007/s10479-005-2244-9

[2] R. Pongou, B. Tchantcho and N. Tedjeugang, “On-the-Job Trial Tournament: Productivity Rank and Earnings,” Mimeo, 2012.

[3] J. Banzhaf, “Weighted Voting Doesn’t Work: A Mathematical Analysis,” Rutgers Law Review, Vol. 19, No.2, 1965, pp. 317-343.

[4] J. Coleman, “Control of Collectivities and the Power of a Collectivity to Act,” In: B. Lieberman, Ed., Social Choice, Gordon and Breach, New York, 1971, pp. 269-300.

[5] P. Dubey and L. Shapley, “Mathematical Properties of the Banzhaf Power Index,” Mathematics of Operations Research, Vol. 4, No. 2, 1979, pp. 99-131. doi:10.1287/moor.4.2.99

[6] A. Laruelle and F. Valenciano, “Shapley-Shubik and Banzhaf Indices Revisited,” Mathematics of Operations Research, Vol. 26, No. 1, 2001, pp. 89-104. doi:10.1287/moor.26.1.89.10589

[7] D. Leech, “The Relationship between Shareholding Concentration and Shareholder Voting Power in British Companies: A Study of the Application of Power Indices for Simple Games,” Management Science, Vol. 34, No. 4, 1988, pp. 509-529. doi:10.1287/mnsc.34.4.509

[8] D. Leech, “Computing Power Indices for Large Voting Games,” Management Science, Vol. 49, No. 6, 2003, pp. 831-838. doi:10.1287/mnsc.49.6.831.16024

[9] V. Feltkamp, “Alternative Axiomatic Characterizations of the Shapley and Banzhaf Values,” International Journal of Game Theory, Vol. 24, No. 2, 1995, pp. 179-186. doi:10.1007/BF01240041

[10] J. Freixas and W. S. Zwicker, “Weighted Voting, Abstention, and Multiple Levels of Approval,” Social Choice and Welfare, Vol. 21, No. 3, 2003, pp. 399-431. doi:10.1007/s00355-003-0212-3

[11] L. Diffo Lambo and J. Moulen, “Ordinal Equivalence of Power Notions in Voting Games,” Theory and Decision, Vol. 53, No. 4, 2002, pp. 313-325. doi:10.1023/A:1024158301610

[12] J. Freixas, D. Marciniako and M. Pons, “On the Ordinal Equivalence of the Johnston, Banzhaf and Shapley Power Indices,” European Journal of Operation Research, Vol. 216, No. 2, 2011, pp. 367-375. doi:10.1016/j.ejor.2011.07.028

[13] B. Tchantcho, L. Diffo Lambo, R. Pongou and B. Mbama Engoulou, “Voters’ Power in Voting Games with Abstention: Influence Relation and Ordinal Equivalence of Power Theories,” Games and Economic Behavior, Vol. 64, No. 1, 2008, pp. 335-350. doi:10.1016/j.geb.2007.10.014

[14] R. Pongou, B. Tchantcho and L. Diffo Lambo, “Political Influence in Multi-Choice Institutions: Cyclicity, Anonymity, and Transitivity,” Theory and Decision, Vol. 70, No. 2, 2011, pp. 157-178. doi:10.1007/s11238-010-9211-x

[1] J. Freixas, “The Banzhaf Measures for Games with Several Levels of Approval in the Input and Output,” Annals of Operations Research, Vol. 137, No. 1, 2005, pp. 45-66. doi:10.1007/s10479-005-2244-9

[2] R. Pongou, B. Tchantcho and N. Tedjeugang, “On-the-Job Trial Tournament: Productivity Rank and Earnings,” Mimeo, 2012.

[3] J. Banzhaf, “Weighted Voting Doesn’t Work: A Mathematical Analysis,” Rutgers Law Review, Vol. 19, No.2, 1965, pp. 317-343.

[4] J. Coleman, “Control of Collectivities and the Power of a Collectivity to Act,” In: B. Lieberman, Ed., Social Choice, Gordon and Breach, New York, 1971, pp. 269-300.

[5] P. Dubey and L. Shapley, “Mathematical Properties of the Banzhaf Power Index,” Mathematics of Operations Research, Vol. 4, No. 2, 1979, pp. 99-131. doi:10.1287/moor.4.2.99

[6] A. Laruelle and F. Valenciano, “Shapley-Shubik and Banzhaf Indices Revisited,” Mathematics of Operations Research, Vol. 26, No. 1, 2001, pp. 89-104. doi:10.1287/moor.26.1.89.10589

[7] D. Leech, “The Relationship between Shareholding Concentration and Shareholder Voting Power in British Companies: A Study of the Application of Power Indices for Simple Games,” Management Science, Vol. 34, No. 4, 1988, pp. 509-529. doi:10.1287/mnsc.34.4.509

[8] D. Leech, “Computing Power Indices for Large Voting Games,” Management Science, Vol. 49, No. 6, 2003, pp. 831-838. doi:10.1287/mnsc.49.6.831.16024

[9] V. Feltkamp, “Alternative Axiomatic Characterizations of the Shapley and Banzhaf Values,” International Journal of Game Theory, Vol. 24, No. 2, 1995, pp. 179-186. doi:10.1007/BF01240041

[10] J. Freixas and W. S. Zwicker, “Weighted Voting, Abstention, and Multiple Levels of Approval,” Social Choice and Welfare, Vol. 21, No. 3, 2003, pp. 399-431. doi:10.1007/s00355-003-0212-3

[11] L. Diffo Lambo and J. Moulen, “Ordinal Equivalence of Power Notions in Voting Games,” Theory and Decision, Vol. 53, No. 4, 2002, pp. 313-325. doi:10.1023/A:1024158301610

[12] J. Freixas, D. Marciniako and M. Pons, “On the Ordinal Equivalence of the Johnston, Banzhaf and Shapley Power Indices,” European Journal of Operation Research, Vol. 216, No. 2, 2011, pp. 367-375. doi:10.1016/j.ejor.2011.07.028

[13] B. Tchantcho, L. Diffo Lambo, R. Pongou and B. Mbama Engoulou, “Voters’ Power in Voting Games with Abstention: Influence Relation and Ordinal Equivalence of Power Theories,” Games and Economic Behavior, Vol. 64, No. 1, 2008, pp. 335-350. doi:10.1016/j.geb.2007.10.014

[14] R. Pongou, B. Tchantcho and L. Diffo Lambo, “Political Influence in Multi-Choice Institutions: Cyclicity, Anonymity, and Transitivity,” Theory and Decision, Vol. 70, No. 2, 2011, pp. 157-178. doi:10.1007/s11238-010-9211-x