# Team Incentives under Moral and Altruistic Preferences: Which Team to Choose?

## Abstract

**:**

## 1. Introduction

## 2. The Model

**Assumption**

**1.**

## 3. Studying the Benchmark: The Contractible Effort Case

**Proposition**

**1.**

**Proposition**

**2.**

## 4. Moving to the Second Best: Non-Contractible Efforts

**Proposition**

**3.**

**Corollary**

**1.**

**Proposition**

**4.**

**Corollary**

**2.**

**Lemma**

**1.**

**Theorem**

**1.**

- 1.
- increasing returns to efforts (${p}_{2}-{p}_{1}\ge {p}_{1}-{p}_{0}$), the principal is better off hiring a team of altruistic agents over a team of moral agents,
- 2.
- decreasing returns to efforts (${p}_{2}-{p}_{1}<{p}_{1}-{p}_{0}$) and
- the outside option is zero ($\overline{u}=0$), the principal prefers a team of moral agents; or
- the outside option is positive, and the degree of morality is sufficiently low ($\overline{u}>0$, $\kappa \to 0$); the principal prefers a team of moral agents only if ${p}_{2}>{\overline{p}}_{2}\in (0,1)$.

## 5. Concluding Remarks

## Acknowledgments

## Conflicts of Interest

## Appendix A. Multiplicity of Equilibria

${\mathbf{e}}_{\mathbf{B}}=\mathbf{1}$ | ${\mathbf{e}}_{\mathbf{B}}=\mathbf{0}$ | |

${e}_{A}=1$ | $U(1,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{A},\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{B};\phantom{\rule{0.166667em}{0ex}}\theta )$ | $U(1,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{A},\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{B};\phantom{\rule{0.166667em}{0ex}}\theta )$ |

${e}_{A}=0$ | $U(0,\phantom{\rule{0.166667em}{0ex}}1,\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{A},\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{B};\phantom{\rule{0.166667em}{0ex}}\theta )$ | $U(0,\phantom{\rule{0.166667em}{0ex}}0,\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{A},\phantom{\rule{0.166667em}{0ex}}{\mathbf{w}}_{B};\phantom{\rule{0.166667em}{0ex}}\theta )$ |

**Lemma**

**A1.**

## Appendix B. Obtaining the Borch Rule for Asymmetric Efforts under Homo Moralis Preferences

**Proposition**

**A1.**

## Appendix C. Proofs

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**3.**

**Proof**

**of**

**Corollary**

**1.**

**Proof**

**of**

**Proposition**

**4.**

**Proof**

**of**

**Corollary**

**2.**

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Theorem**

**1.**

**Proof**

**of**

**Proposition**

**5.**

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1 | See [1]. |

2 | See the Society for Human Resource Management survey results at https://www.shrm.org/hr-today/trends-and-forecasting/research-and-surveys/Pages/criminalbackgroundcheck.aspx. |

3 | See the Society for Human Resource Management survey results at https://www.shrm.org/hr-today/trends-and-forecasting/research-and-surveys/Pages/creditbackgroundchecks.aspx. |

4 | See, for instance, [2]. |

5 | For a broader debate on the definition of altruism from an evolutionary biology view, see [19]. |

6 | |

7 | Reference [24] study optimal incentives for teams in a repeated setting. |

8 | in reality, agents may differ in their respective cost of effort, but this is not pursued in this paper because it does not qualitatively change the results, and at the same time, it adds a more cumbersome notation. |

9 | The assumption that $u(\xb7)$ is strictly concave can be relaxed, and the same model below can be solved in a setting with risk-neutral agents and limited liability constraints, where the qualitative results are not changed from the analysis below. |

10 | The normal form game here is comprised of the set of players $\{A,B\}$, the common set of pure strategies $S=\{0,1\}$ and payoff function $U({e}_{i},{e}_{j},{\mathbf{w}}_{i},{\mathbf{w}}_{j},\theta )$. |

11 | This is the reason why contracts are indexed by i. |

12 | Since $1\ge {p}_{2}>{p}_{1}>{p}_{0}\ge 0$ and ${u}^{\prime}>0$ by assumption, the first-order conditions imply that ${\lambda}_{i}+{\lambda}_{j}\theta >0$ for $i,j\in \{A,B\}$, $i\ne j$. |

13 | In Appendix B, I show that relaxing both these assumptions leads to a Borch rule for moral agents that demands nonconstant wages when efforts are observable, in stark contracts to the literature with selfish and altruistic agents. |

14 | In Appendix A, I show that the optimal contracts for both moral and altruistic agents may lead to a multiplicity of equilibria in their effort choices, as in [17]. |

15 | See Appendix C for the proof. |

16 | For symmetric equilibrium choices of effort. |

17 | If ${p}_{2}-{p}_{1}={p}_{1}-{p}_{0}$, both lines coincide with the $\left(IC\right)$ for altruistic agents. |

18 | This intuition in indeed right and integrates the proof of Theorem 1 in the Appendix. |

19 | Given the symmetry of the problem, I focus attention on agent A and drop the subscripts. The same results would hold for agent B by simply reversing the effort choices ${e}_{A}$ and ${e}_{B}$. |

20 | As shown by the Borch rules $\left(B{R}_{S}\right)$, $\left(B{R}_{Alt}\right)$. |

**Figure 2.**(

**a**) Comparing principal’s profits for ${p}_{2}-{p}_{1}\ge {p}_{1}-{p}_{0}$; (

**b**) comparing principal’s profits for ${p}_{2}-{p}_{1}<{p}_{1}-{p}_{0}$ and $\overline{u}=0$; (

**c**) comparing the ratio of principal’s profits for ${p}_{2}-{p}_{1}<{p}_{1}-{p}_{0}$ and $\overline{u}>0$.

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Sarkisian, R.
Team Incentives under Moral and Altruistic Preferences: Which Team to Choose? *Games* **2017**, *8*, 37.
https://doi.org/10.3390/g8030037

**AMA Style**

Sarkisian R.
Team Incentives under Moral and Altruistic Preferences: Which Team to Choose? *Games*. 2017; 8(3):37.
https://doi.org/10.3390/g8030037

**Chicago/Turabian Style**

Sarkisian, Roberto.
2017. "Team Incentives under Moral and Altruistic Preferences: Which Team to Choose?" *Games* 8, no. 3: 37.
https://doi.org/10.3390/g8030037