# Training, Abilities and the Structure of Teams

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Example**

**1.**

**Example**

**2.**

## 3. Results

#### 3.1. The Firm’s Perspective

**Theorem**

**1.**

- $\left|{G}_{1}\right|<t$ and
- $\left|{G}_{h}\right|=t\forall $${G}_{h}\in \left\{{G}_{2},\dots ,{G}_{m}\right\}$

- ${\widehat{a}}_{i}\le {\widehat{a}}_{j}\forall $$i\in {G}_{1},$$j\in \left\{{G}_{2},\dots ,{G}_{m}\right\}$ and
- ${\widehat{a}}_{l},$$l\in \left\{{G}_{2},\dots ,{G}_{m}\right\},$ with the possibility of a rank order $\rho \in R{O}^{\mathcal{P}}\left(\left\{{G}_{2},\dots ,{G}_{m}\right\}\right)$ with ${\widehat{a}}_{\rho \left(i\right)}\le {\widehat{a}}_{\rho \left(i+1\right)},$$i\in \left\{1,\dots ,\left|N\right|-\left|{G}_{1}\right|-1\right\}$does not affect the stability of $\mathcal{P}.$

**Corollary**

**1.**

- $\left|{G}_{1}\right|<t$ and
- $\left|{G}_{h}\right|=b\xb7t,$$b\in \mathbb{N},$ with $b>1\forall $ ${G}_{h}\in \left\{{G}_{2},\dots ,{G}_{m}\right\}$

- ${\widehat{a}}_{i}\le {\widehat{a}}_{j}\forall $$i\in {G}_{1},$$j\in \left\{{G}_{2},\dots ,{G}_{m}\right\}$ and
- ${\widehat{a}}_{l}={\widehat{a}}_{k},\forall $$l,k\in N\backslash {G}_{1}$ with $\mathcal{P}\left(l\right)=\mathcal{P}\left(k\right)$does not affect the stability of $\mathcal{P}.$

**Corollary**

**2.**

#### 3.2. The Employee’s Perspective

**Corollary**

**3.**

- $\left|\bigcup {G}_{1},\dots ,{G}_{s}\right|<t$
- $\left|{G}_{s+1}\right|,\dots ,\left|{G}_{m}\right|\in \left\{\left.l\right|l=b\xb7t\right\}$ with $b\in {\mathbb{N}}^{+}$
- ${\widehat{a}}_{i}={\widehat{a}}_{j}>0,$$i\ne j,$$i,j\in \left\{{G}_{s+1},\dots ,{G}_{m}\right\}$ and ${\widehat{a}}_{l}=0\forall $$l\in N\backslash \left\{i,j\right\}$

- ${\chi}_{i}(N,\widehat{p},\mathcal{P})\ge {\chi}_{j}(N,\widehat{p},\mathcal{P})$ iff $\left|\mathcal{P}\left(i\right)\right|\le \left|\mathcal{P}\left(j\right)\right|.$

#### 3.3. Summary and Outlook

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Theorem**

**A1.**

- ${\sum}_{i\in \left\{{G}_{1},\dots ,{G}_{q}\right\}}{a}_{i}<z,$${a}_{i}<{a}_{j}\forall $$i\in \left\{{G}_{1},\dots ,{G}_{q}\right\}$ and $j\in \left\{{G}_{q+1},\dots ,{G}_{m}\right\}$
- $\left|{G}_{q+1}\right|,\dots ,\left|{G}_{s}\right|=b\xb7t,$ with $b\in {\mathbb{N}}^{+},$${\sum}_{i\in {G}_{h}}{a}_{i}>z,\forall $ ${\sum}_{i\in {G}_{h}}{a}_{i}-{a}_{j}<z\forall $ $j\in {G}_{h},$${G}_{h}\in \left\{{G}_{q+1},\dots ,{G}_{s}\right\}$ and ${a}_{i}>{a}_{j}$ for $i\in {G}_{l}$, $j\in {G}_{l-1},$$l=m,\dots ,q+1$.

**Corollary**

**A1.**

## Notes

1 | Extensive empirical literature on returns from human capital investments were inspired by [24]. |

2 | Casajus [25] outlined the relation of hedonic games to the TU games, which our article belongs to. |

3 | |

4 | |

5 | The notation is partly based on Hiller [33]. |

6 | In the following, only the application to games $\left(N,p\right)$ is relevant. |

7 | Abe [47] studied the relationship between an FS value and stable coalition structures. |

8 | As noted in the previous section, one could assume that instead of $v\left(\mathcal{P}\left(i\right)\right)$ only a fraction $c,$$0<c<1$ is distributed among the team’s members by $\chi .$ |

## References

- Becker, G.S. Investment in Human Capital: A Theoretical Analysis. J. Political Econ.
**1962**, 70, 9–49. [Google Scholar] [CrossRef] - Hashimoto, M. Firm-Specific Human Capital as a Shared Investment. Am. Econ. Rev.
**1981**, 71, 475–482. [Google Scholar] [CrossRef] - Carmichael, L. Firm-Specific Human Capital and Promotion Ladders. Bell J. Econ.
**1983**, 14, 251–258. [Google Scholar] [CrossRef] - Kahn, C.; Huberman, G. Two-Sided Uncertainty and “Up-or-Out” Contracts. J. Labor Econ.
**1988**, 6, 423–444. [Google Scholar] [CrossRef] - Prendergast, C. The Role of Promotion in Inducing Specific Human Capital Acquisition. Q. J. Econ.
**1993**, 108, 523–534. [Google Scholar] [CrossRef] - Scoones, D.; Bernhardt, D. Promotion, Turnover, and Discretionary Human Capital Acquisition. J. Labor Econ.
**1998**, 16, 122–141. [Google Scholar] [CrossRef] - Lazear, E.P. Firm-Specific Human Capital: A Skill-Weights Approach. J. Political Econ.
**2009**, 117, 914–940. [Google Scholar] [CrossRef] - Coff, R.; Raffiee, J. Toward a Theory of Perceived Firm-Specific Human Capital. Acad. Manag. Perspect.
**2015**, 29, 326–341. [Google Scholar] [CrossRef] - Waldman, M.; Zax, O. Promotion Signaling and Human Capital Investments. Am. Econ. J. Microecon.
**2020**, 12, 125–155. [Google Scholar] [CrossRef] - Katz, E.; Ziderman, A. Investment in General Training: The Role of Information and Labour Mobility. Econ. J.
**1990**, 100, 1147–1158. [Google Scholar] [CrossRef] - Chiang, S.H.; Chiang, S.C. General Human Capital as a Shared Investment under Asymmetric Information. Can. J. Econ.
**1990**, 23, 175–188. [Google Scholar] [CrossRef] - Chang, C.; Wang, Y. Human Capital Investment under Asymmetric Information: The Pigovian Conjecture Revisited. J. Labour Econ.
**1996**, 14, 505–519. [Google Scholar] [CrossRef] - Stevens, M. A Theoretical Model of on-the-Job Training with Imperfect Competition. Oxf. Econ. Pap.
**1994**, 46, 537–562. [Google Scholar] [CrossRef] - Acemoglu, D. Training and Innovation in an Imperfect Labour Market. Rev. Econ. Stud.
**1997**, 64, 445–464. [Google Scholar] [CrossRef] - Loewenstein, M.A.; Spletzer, J.K. Dividing the Costs and Returns to General Training. J. Labor Econ.
**1998**, 16, 142–171. [Google Scholar] [CrossRef] - Acemoglu, D.; Pischke, J.S. The Structure of Wages and Investment in General Training. J. Political Econ.
**1999**, 107, 539–572. [Google Scholar] [CrossRef] - Fella, G. Termination Restrictions and Investment in General Training. Eur. Econ. Rev.
**2005**, 49, 1479–1499. [Google Scholar] [CrossRef] - Molloy, J.C.; Barney, J.B. Who Captures the Value Created with Human Capital? A Market-Based View. Acad. Manag. Perspect.
**2015**, 29, 309–325. [Google Scholar] [CrossRef] - Alewell, D. Die Finanzierung Betrieblicher Weiterbildungsinvestitionen; Gabler Verlag: Wiesbaden, Germany, 1997. [Google Scholar]
- Leuven, E. The Economics of Private Sector Training: A Survey of the Literature. J. Econ. Surv.
**2005**, 19, 91–111. [Google Scholar] [CrossRef] - Lazear, E.P.; Shaw, K.L. Personnel Economics: The Economist’s View of Human Resources. J. Econ. Perspect.
**2007**, 21, 91–114. [Google Scholar] [CrossRef] - De Grip, A.; Sauermann, J. The effect of training on productivity: The transfer of on-the-job training from the perspective of economics. Educ. Res. Rev.
**2013**, 8, 28–36. [Google Scholar] [CrossRef] - Deming, D.J. Four Facts about Human Capital. J. Econ. Perspect.
**2022**, 36, 75–102. [Google Scholar] [CrossRef] - Mincer, J. On-the-Job Training: Costs, Returns, and some Implications. J. Political Econ.
**1962**, 70, 50–79. [Google Scholar] [CrossRef] - Casajus, A. On the stability of coalition structures. Econ. Lett.
**2008**, 100, 271–274. [Google Scholar] [CrossRef] - Banerjee, S.; Konishi, H.; Sonmez, T. Core in a simple coalition formation game. Soc. Choice Welf.
**2001**, 18, 135–153. [Google Scholar] [CrossRef] - Bogomolnaia, A.; Jackson, M.O. The Stability of Hedonic Coalition Structures. Games Econ. Behav.
**2002**, 38, 201–230. [Google Scholar] [CrossRef] - Barber, S.; Bevi, C.; Ponsat, C. Meritocracy, egalitarianism and the stability of majoritarian organizations. Games Econ. Behav.
**2015**, 91, 237–257. [Google Scholar] [CrossRef] - Piccione, M.; Razin, R. Coalition formation under power relations. Theor. Econ.
**2009**, 4, 1–15. [Google Scholar] - Damiano, E.; Li, H.; Suen, W. First in village or second in Rome. Int. Econ. Rev.
**2010**, 51, 263–288. [Google Scholar] [CrossRef] - Watts, A. Formation of segregated and integrated groups. Int. J. Game Theory
**2007**, 35, 505–519. [Google Scholar] [CrossRef] - Herings, P.J.J.; Saulle, R.D.; Seel, C. The Last Will be First, and the First Last: Segregation in Societies with Relative Pay-off Concerns. Econ. J.
**2021**, 131, 2119–2143. [Google Scholar] [CrossRef] - Hiller, T. Abilities and the structure of the firm. Int. Rev. Econ.
**2022**, 69, 339–349. [Google Scholar] [CrossRef] - Hernández-Lamoneda, L.; Sánchez-Sánchez, F. Rankings and values for team games. Int. J. Game Theory
**2010**, 39, 319–350. [Google Scholar] [CrossRef] - Morelli, M.; Park, I.U. Internal hierarchy and stable coalition structures. Games Econ. Behav.
**2016**, 96, 90–96. [Google Scholar] [CrossRef] - Hiller, T. Structure of teams—A cooperative game theory approach. Manag. Decis. Econ.
**2019**, 40, 520–525. [Google Scholar] [CrossRef] - Casajus, A. Outside Options, Component Efficiency, and Stability. Games Econ. Behav.
**2009**, 65, 49–61. [Google Scholar] [CrossRef] - Aumann, R.J.; Drèze, J.H. Cooperative Games with Coalition Structures. Int. J. Game Theory
**1974**, 3, 217–237. [Google Scholar] [CrossRef] - Owen, G. Values of Games with a Priori Unions. In Essays in Mathematical Economics & Game Theory; Henn, R., Moeschlin, O., Eds.; Springer: Berlin/Heidelberg, Germany, 1977; pp. 76–88. [Google Scholar]
- Wiese, H. Measuring the Power of Parties Within Government Coalitions. Int. Game Theory Rev.
**2007**, 9, 307–322. [Google Scholar] [CrossRef] - Alonso-Meijide, J.M.; Carreras, F.; Costa, J.; Garcia-Jurado, I. The proportional partitional Shapley value. Discret. Appl. Math.
**2015**, 187, 1–11. [Google Scholar] [CrossRef] - Kamijo, Y. A Two-Step Shapley Value for Cooperative Games with Coalition Structures. Int. Game Theory Rev.
**2009**, 11, 207–214. [Google Scholar] [CrossRef] - Hiller, T. Quantitative overeducation and cooperative game theory. Econ. Lett.
**2017**, 152, 36–40. [Google Scholar] [CrossRef] - Shapley, L.S. A Value for N-Person Games. In Contributions to the Theory of Games; Kuhn, H.W., Tucker, A.W., Eds.; Princeton University Press: Princeton, NJ, USA, 1953; Volume 2, pp. 307–317. [Google Scholar]
- Wiese, H. Kooperative Spieltheorie; Oldenbourg Verlag: München, Germany, 2005. [Google Scholar]
- Hart, S.; Kurz, M. Endogenous Formation of Coalitions. Econometrica
**1983**, 51, 1047–1064. [Google Scholar] [CrossRef] - Abe, T. Stability and values for games with coalition structures. Econ. Lett.
**2021**, 200, 109750. [Google Scholar] [CrossRef] - Newton, J. Evolutionary game theory: A renaissance. Games
**2018**, 9, 31. [Google Scholar] [CrossRef] - Casajus, A.; Kramm, M.; Wiese, H. Asymptotic stability in the Lovász-Shapley replicator dynamic for cooperative games. J. Econ. Theory
**2020**, 186, 104993. [Google Scholar] [CrossRef]

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Training, Abilities and the Structure of Teams. *Games* **2023**, *14*, 44.
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Hiller T.
Training, Abilities and the Structure of Teams. *Games*. 2023; 14(3):44.
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2023. "Training, Abilities and the Structure of Teams" *Games* 14, no. 3: 44.
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