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Who Bears an Employee’s Special Annual Payment?

  • Tobias Hiller EMAIL logo
Published/Copyright: February 25, 2020
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Review of Law & Economics
From the journal Review of Law & Economics Volume 17 Issue 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.

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Published Online: 2020-02-25

© 2020 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  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
Keywords for this article
Shapley value; Banzhaf value; Owen value; special annual payment; TV-L

Articles in the same Issue

  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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