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Who Bears an Employee’s Special Annual Payment?

  • Tobias Hiller EMAIL logo
Veröffentlicht/Copyright: 25. Februar 2020
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Review of Law & Economics
Aus der Zeitschrift Review of Law & Economics Band 17 Heft 1

Abstract

In this note, we analyze the question of who bears an employee’s special annual payment if different external funders pay an employee’s wages over the course of a year. To answer this question, we provide a legal argument and use cooperative game theory.

Keywords: Shapley value; Banzhaf value; Owen value; special annual payment; TV-L
JEL Classification: C71; K31

Acknowledgements

We thank Melanie Welz, Silvia Böhm and Susanne Zerjatke for helpful discussions. In addition, we are grateful to an anonymous referee for comments on this paper.

Shapley payoffs

We exemplify the determination of the Shapley payoffs in Section 3 by the payoff of month September. In general, 12 positions are possible in the rank orders of the months. All positions are equally probable. We calculate the marginal contribution for September at each position. For example, we look at the marginal contribution at Position 4:

1182113⋅13x⋅412+112181113⋅23x⋅412−13x⋅312+1122113⋅x⋅412−23x⋅312.

With probability 1182113 December and two months from Z are in front of September. Their worth is zero. Together with September, they obtain 13x⋅412. The second marginal contribution has the probability 112181113. Now, December, one month from Z and one month from I are in front of September. They obtain a worth of 13x⋅312. Together with September, they obtain a worth of 23x⋅412. In the last constellation for the fourth position, July, August, and December are in front of September (probability 1122113). The marginal contribution is x⋅412−23x⋅312. For the other positions, the marginal contributions (at position 1, the marginal contribution is zero) are as follows:

Position 2

11111⋅13x⋅212

Position 3

1181112⋅13x⋅312+112111223x⋅312−13x⋅212

Position 5

1183114⋅13x⋅512+112182114⋅23x⋅512−13x⋅412+112281114⋅x⋅512−23x⋅412

Position 6

1184115⋅13x⋅612+112183115⋅23x⋅612−13x⋅512+112282115⋅x⋅612−23x⋅512

Position 7

1185116⋅13x⋅712+112184116⋅23x⋅712−13x⋅612+112283116⋅x⋅712−23x⋅612

Position 8

1186117⋅13x⋅812+112185117⋅23x⋅812−13x⋅712+112284117⋅x⋅812−23x⋅712

Position 9

1187118⋅13x⋅912+112186118⋅23x⋅912−13x⋅812+112285118⋅x⋅912−23x⋅812

Position 10

1188119⋅13x⋅1012+112187119⋅23x⋅1012−13x⋅912+112286119⋅x⋅1012−23x⋅912

Position 11

1121881110⋅23x⋅1112−13x⋅1012+1122871110⋅x⋅1112−23x⋅1012

Position 12

1122881111⋅x⋅1212−23x⋅1112

Summing up the marginal contributions and weighting them by 112 gives the Shapley payoff.

Banzhaf payoffs

To calculate the Banzhaf payoffs, we use the Shapley considerations. Again, we consider each position of a player and calculate the marginal contributions to all coalitions. For September, at position 4, we obtain:

1182⋅13x⋅412+112181⋅23x⋅412−13x⋅312+1122⋅x⋅412−23x⋅312=112x.

There are 1182=28 possible coalitions with December together with one month from Z, for example. Joining these coalitions, the marginal contribution is 13x⋅412. Summing up the marginal contributions and multiplying the sums by 1212−1=12048 yields:

BziN,v=148⋅x,i∈ZBzjN,v=19⋅x,j∈IBz12N,v=724⋅x.

Owen payoffs

We exemplify the calculations for PA and September. At position 4, we obtain:

12⋅15⋅112243⋅13x⋅412+11112143⋅23x⋅412−13x⋅312=7720x

With probability 12⋅15, September is at position 4. With probability 112243, December, October and November are in front of September. The coalition with September is worth 13x⋅412. With probability 11112143, December, August and October or November are in front of September. The marginal contribution is 23x⋅412−13x⋅312. For the other positions, we have:

Position 2

12⋅15⋅1141⋅13x⋅212=1720x

Position 3

12⋅15⋅112142⋅13x⋅312+111142⋅23x⋅312−13x⋅212=1216x

Position 5

12⋅15⋅11112244⋅23x⋅512−13x⋅412=160x

Positions 6 and 7 not possible

Position 8

0

Position 9

12⋅15⋅1141⋅23x⋅912−13x⋅812=1144x

Position 10

12⋅15⋅112142⋅23x⋅1012−13x⋅912+111142⋅x⋅1012−23x⋅912=171080x

Position 11

12⋅15⋅112243⋅23x⋅1112−13x⋅1012+11112143⋅x⋅1112−23x⋅1012=19720x

Position 12

12⋅15⋅11112244⋅x⋅1212−23x⋅1112=7180x

Summing up gives the Owen payoff.

References

Auer, B.R., and T. Hiller. 2015. “On the Evaluation of Soccer Players: A Comparison of a New Game-Theoretical Approach to Classic Performance Measures,” 22 (14)Applied Economics Letters 1100–1107.10.1080/13504851.2015.1005809Suche in Google Scholar

Aumann, R.J., and J.H. Drèze. 1974. “Cooperative Games with Coalition Structures,” 3 (4)International Journal of Game Theory 217–237.10.1007/BF01766876Suche in Google Scholar

Banzhaf, J.F. 1965. “Weighted Voting Doesn’t Work: A Mathematical Analysis,” 19 (2)Rutgers Law Review 317–343.Suche in Google Scholar

Casajus, A. 2009. “Outside Options, Component Efficiency, and Stability,” 65 (1)Games and Economic Behavior 49–61.10.1016/j.geb.2007.04.003Suche in Google Scholar

Casajus, A., T. Hiller, and H. Wiese. 2009. “Hierarchie und Entlohnung,” 79 (7/8)Zeitschrift für Betriebswirtschaft 929–954.10.1007/s11573-009-0271-8Suche in Google Scholar

Deutscher Beamtenbund. 2016. Zahlen, Daten, Fakten, Berlin.Suche in Google Scholar

Dubey, P. 1982. “The Shapley Value as Aircraft Landing Fees – Revisited,” 28 (8)Management Science 869–874.10.1287/mnsc.28.8.869Suche in Google Scholar

Dubey, P., and L.S. Shapley. 1979. “Mathematical Properties of the Banzhaf Power Index,” 4 (2)Mathematics of Operations Research 99–131.10.1287/moor.4.2.99Suche in Google Scholar

Federal Labor Court. 2012. 10 AZR 922/11. Decision from 12.12.2012.10.1016/S0958-2118(12)70213-6Suche in Google Scholar

Felsenthal, D.S., and M. Machover. 1998. The Measurement of Voting Power: Theory and Practice, Problems and Paradoxes, Edward Elgar, Cheltenham et al.10.4337/9781840647761Suche in Google Scholar

Felsenthal, D.S., and M. Machover. 2005. “Voting Power Measurement: A Story of Misreinvention,” 25 (2)Social Choice and Welfare 485–506.10.1007/s00355-005-0015-9Suche in Google Scholar

Ginsburgh, V., and I. Zang. 2003. “The Museum Pass Game and Its Value,” 43 (2)Games and Economic Behavior 322–325.10.1016/S0899-8256(03)00013-7Suche in Google Scholar

Hart, S. 1997. Classical Cooperative Theory II: Value-Like Concepts, Cooperation: Game-Theoretic Approaches. Berlin: Springer, pp. 43-49.10.1007/978-3-642-60454-6_6Suche in Google Scholar

Hernández-Lamoneda, L., and F. Sánchez-Sánchez. 2010. “Rankings and Values for Team Games,” 39 (3)International Journal of Game Theory 319–350.10.1007/s00182-009-0178-1Suche in Google Scholar

Hiller, T. 2015. “The Importance of Players in Teams of the German Bundesliga in the Season 2012/2013 - A Cooperative Game Theory Approach,” 22 (4)Applied Economics Letters 324–329.10.1080/13504851.2014.941527Suche in Google Scholar

Hiller, T. 2019. “Structure of Teams - A Cooperative Game Theory Approach,” 40 (5)Managerial and Decision Economics 520–525.10.1002/mde.3021Suche in Google Scholar

Holler, M.J. 1982. “Forming Coalitions and Measuring Voting Power,” 30 (2)Political Studies 262–271.10.1111/j.1467-9248.1982.tb00537.xSuche in Google Scholar

Holler, M.J., and E.W. Packel. 1983. “Power, Luck and the Right Index,” 43 (1)Journal of Economics 21–29.10.1007/BF01283881Suche in Google Scholar

Ju, Y., P. Borm, and P. Ruys. 2007. “The Consensus Value: A New Solution Concept for Cooperative Games,” 28 (3)Social Choice and Welfare 437–452.10.1007/s00355-006-0192-1Suche in Google Scholar

Littlechild, S.C., and G. Owen. 1973. “A Simple Expression for the Shapley Value in a Special Case,” 20(3)Management Science 370–372.10.1287/mnsc.20.3.370Suche in Google Scholar

Littlechild, S.C., and G.F. Thompson. 1977. “Aircraft Landing Fees: A Game Theory Approach,” 8 (1)Bell Journal of Economics 186–204.10.2307/3003493Suche in Google Scholar

Nowak, A.S., and T. Radzik. 1994. “A Solidarity Value for n-Person Transferable Utility Games,” 23 (1)International Journal of Game Theory 43–48.10.1007/BF01242845Suche in Google Scholar

Owen, G. 1977. “Values of Games with a Priori Unions,” in R. Henn and O. Moeschlin, eds. Essays in Mathematical Economics & Game Theory. Springer Verlag, Berlin et al., pp. 76–88.10.1007/978-3-642-45494-3_7Suche in Google Scholar

Penrose, L.S. 1946. “The Elementary Statistics of Majority Voting,” 109 (1)Journal of the Royal Statistical Society 53–57.10.2307/2981392Suche in Google Scholar

Penrose, L.S. 1952. “On the Objective Study of Crowd Behaviour. London: H. K. Lewis & Co.Suche in Google Scholar

Saari, D.G., and K. Sieberg. 2001. “Some surprising properties of power indices,” 36 (2)Games and Economic Behavior 241–263.10.1006/game.2000.0819Suche in Google Scholar

Shapley, L.S. 1953. “A Value for n-Person Games,” in H.W. Kuhn and A.W. Tucker, eds. Contributions to the Theory of Games, Vol. 2. Princeton: Princeton University Press, pp. 307–317.10.1515/9781400881970-018Suche in Google Scholar

Tarifvertrag für den öffentlichen Dienst der Länder, in the version dated 9 March 2013 (2013).Suche in Google Scholar

Tijs, S.H. 1987. “An Axiomatization of the t-Value,” 13 (2)Mathematical Social Sciences 177–181.10.1016/0165-4896(87)90054-0Suche in Google Scholar

van den Brink, R. 2008. “Vertical Wage Differences in Hierarchically Structured Firms,” 30(2)Social Choice and Welfare 225–243.10.1007/s00355-007-0230-7Suche in Google Scholar

van den Brink, R., and van der Laan, G. 1998. “Axiomatizations of the Normalized Banzhaf value and the Shapley value,” 15 (4)Social Choice and Welfare 567–582.10.1007/s003550050125Suche in Google Scholar

Wiese, H. 2007. “Measuring the Power of Parties Within Government Coalitions,” 9 (2)International Game Theory Review 307–322.10.1142/S0219198907001412Suche in Google Scholar

Young, H. 1994. “Cost Allocation,” in R.J. Aumann and S. Hart, eds. Handbook of Game Theory, Vol. 2. Amsterdam: Elsevier, pp. 1193–1235.10.1016/S1574-0005(05)80066-9Suche in Google Scholar

Published Online: 2020-02-25

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Artikel in diesem Heft

  1. Frontmatter
  2. Articles
  3. Equilibria Under Negligence Liability
  4. Does an Inclusive Citizenship Law Promote Economic Development?
  5. The Effects of Enforcement on Corporate Environmental Performance: The Role of Perceived Fairness
  6. Economic Analysis of Accident Law: A New Liability Rule that Induces Socially Optimal Behaviour in Case of Limited Information
  7. To Settle or to Fight to the End? Case-level Determinants of Early Settlement of Investor-State Disputes
  8. The (Mixed) Effects of Minimum Asset Requirements When There is a Possibility of Technological Change
  9. The Effect of Constitutional Commitment to Social Security on Social Expenditure Schemes
  10. Case Note
  11. Who Bears an Employee’s Special Annual Payment?
  12. A Note on the Article by Mark Ramseyer, published in RLE 2020
https://doi.org/10.1515/rle-2019-0022
Schlagwörter für diesen Artikel
Shapley value; Banzhaf value; Owen value; special annual payment; TV-L

Artikel in diesem Heft

  1. Frontmatter
  2. Articles
  3. Equilibria Under Negligence Liability
  4. Does an Inclusive Citizenship Law Promote Economic Development?
  5. The Effects of Enforcement on Corporate Environmental Performance: The Role of Perceived Fairness
  6. Economic Analysis of Accident Law: A New Liability Rule that Induces Socially Optimal Behaviour in Case of Limited Information
  7. To Settle or to Fight to the End? Case-level Determinants of Early Settlement of Investor-State Disputes
  8. The (Mixed) Effects of Minimum Asset Requirements When There is a Possibility of Technological Change
  9. The Effect of Constitutional Commitment to Social Security on Social Expenditure Schemes
  10. Case Note
  11. Who Bears an Employee’s Special Annual Payment?
  12. A Note on the Article by Mark Ramseyer, published in RLE 2020
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