# Vertical Relationships with Hidden Interactions

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model

#### 3.1. Information

#### 3.2. Organizational Structure

- In the competitive structure $\left(C={C}^{m}\right)$, the agents can only play non-cooperatively, and collusion between them is impossible. Therefore, the agents cannot coordinate on reporting their types to the principal to jointly maximize their information rents.
- In the cooperative structure $\left(C={C}^{c}\right)$, the agents can engage in collusion; side-contracting between the agents is possible, and they can coordinate on reporting their types.2 In the cooperative structure, however, a Good agent can choose to help a Bad agent through knowledge transfer; with a Good agent’s help, a Bad agent becomes Good with probability ${\psi}_{G}$ and remains Bad with probability ${\psi}_{B}=1-{\psi}_{G}$.

#### 3.3. Report and Production

#### 3.4. Payoffs

#### 3.5. Timing of the Game

#### 3.5.1. Competitive Structure ($C={C}^{m}$)

- The principal offers $\{{Q}_{k},$ ${t}_{ij,}$ ${\gamma \}}_{k\in \{G,M,B\}}^{i,j\in \{G,B\}}$ to the agents.
- If the offers are accepted, the agents learn their types.
- Each agent decides whether or not to sabotage the other agent.
- Depending on sabotage, each agent’s type is revised.
- Reports are made to the principal and the contracts are executed.

#### 3.5.2. Cooperative Structure ($C={C}^{c}$)

- The principal offers $\{{Q}_{k},$ ${t}_{ij,}$ ${\gamma \}}_{k\in \{G,M,B\}}^{i,j\in \{G,B\}}$ to the agents.
- If the offers are accepted, the agents learn their types.
- Each agent decides whether or not to sabotage/help the other agent.
- Depending on sabotage/help, each agent’s type is revised.
- The agents decide whether or not to collude.7
- Reports are made to the principal and the contracts are executed.

#### 3.6. The First-Best Outcome

## 4. Without Hidden Interactions

#### 4.1. Optimal Outcome in the Competitive Structure (${C}^{m}$)

**Lemma**

**1.**

- ${Q}_{G}^{m}={Q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{Q}_{M}^{m}={Q}_{M}^{*}\phantom{\rule{0.277778em}{0ex}}$ with $\gamma =1;\phantom{\rule{0.277778em}{0ex}}{Q}_{B}^{m}<{Q}_{B}^{*}$.
- ${t}_{GG}^{m}=(1-{\theta}_{G}){Q}_{G}^{*}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{m}=(1-{\theta}_{G}){Q}_{M}^{*}+\Delta \theta {Q}_{B}^{m}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{m}=0;$ ${t}_{BB}^{m}=(1-{\theta}_{B}){Q}_{B}^{m}/2$.
- ${u}_{GG}^{m}=0;\phantom{\rule{0.277778em}{0ex}}{u}_{GB}^{m}=\Delta \theta {Q}_{B}^{m}/2;\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{m}={u}_{BB}^{m}=0$.

#### 4.2. Optimal Outcome in the Cooperative Structure (${C}^{c}$)

**Lemma**

**2.**

- ${Q}_{G}^{c}={Q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{Q}_{M}^{c}={Q}_{M}^{*}\phantom{\rule{0.277778em}{0ex}}$ with $\phantom{\rule{0.277778em}{0ex}}\gamma =1;\phantom{\rule{0.277778em}{0ex}}{Q}_{B}^{c}<{Q}_{B}^{m}$ $(<{Q}_{B}^{*})$.
- ${t}_{GG}^{c}=(1-{\theta}_{G}){Q}_{G}^{*}/2+\Delta \theta {Q}_{B}^{c}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{c}=(1-{\theta}_{G}){Q}_{M}^{*}+\Delta \theta {Q}_{B}^{c}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{c}=0;$ ${t}_{BB}^{c}=(1-{\theta}_{B}){Q}_{B}^{c}/2$.
- ${u}_{GG}^{c}={u}_{GB}^{c}=\Delta \theta {Q}_{B}^{c}/2;\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{c}={u}_{BB}^{c}=0$.

**Proposition 1.**

## 5. With Hidden Interactions

#### 5.1. Optimal Outcome in the Competitive Structure (${C}^{m}$)

**Claim**

**1.**

**Lemma**

**3.**

- ${Q}_{G}^{\widehat{m}}={Q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{Q}_{M}^{\widehat{m}}={Q}_{M}^{*}\phantom{\rule{0.277778em}{0ex}}$with $\gamma =1;\phantom{\rule{0.277778em}{0ex}}{Q}_{B}^{\widehat{m}}<{Q}_{B}^{*}$.
- ${t}_{GG}^{\widehat{m}}=(1-{\theta}_{G}){Q}_{G}^{*}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{\widehat{m}}=(1-{\theta}_{G}){Q}_{M}^{*}+\Delta \theta {Q}_{B}^{\widehat{m}}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{\widehat{m}}=0;$ ${t}_{BB}^{\widehat{m}}=(1-{\theta}_{B}){Q}_{B}^{\widehat{m}}/2$.
- ${u}_{GG}^{\widehat{m}}=0;\phantom{\rule{0.277778em}{0ex}}{u}_{GB}^{\widehat{m}}=\Delta \theta {Q}_{B}^{\widehat{m}}/2;\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{\widehat{m}}={u}_{BB}^{\widehat{m}}=0$.

**Lemma**

**4.**

- ${Q}_{G}^{\tilde{m}}={Q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{Q}_{M}^{\tilde{m}}={Q}_{M}^{*}\phantom{\rule{0.277778em}{0ex}}$ with $\phantom{\rule{0.277778em}{0ex}}\gamma =1;\phantom{\rule{0.277778em}{0ex}}{Q}_{B}^{\tilde{m}}<{Q}_{B}^{\widehat{m}}$ $(<{Q}_{B}^{*})$.
- ${t}_{GG}^{\tilde{m}}=(1-{\theta}_{G}){Q}_{G}^{*}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{\tilde{m}}=(1-{\theta}_{G}){Q}_{M}^{*}+\Delta \theta {Q}_{B}^{\tilde{m}}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{\tilde{m}}=0;$ ${t}_{BB}^{\tilde{m}}=(1-{\theta}_{B}){Q}_{B}^{\tilde{m}}/2$.
- ${u}_{GG}^{\tilde{m}}={u}_{GB}^{\tilde{m}}=\Delta \theta {Q}_{B}^{\tilde{m}}/2;\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{\tilde{m}}={u}_{BB}^{\tilde{m}}=0$.

**Lemma**

**5.**

#### 5.2. Optimal Outcome in the Cooperative Structure (${C}^{c}$)

**Claim**

**2.**

**Lemma**

**6.**

- ${Q}_{G}^{\tilde{c}}={Q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{Q}_{M}^{\tilde{c}}={Q}_{M}^{*}\phantom{\rule{0.277778em}{0ex}}$ with $\phantom{\rule{0.277778em}{0ex}}\gamma =1;\phantom{\rule{0.277778em}{0ex}}{Q}_{B}^{\tilde{c}}<{Q}_{B}^{\tilde{m}}<{Q}_{B}^{\widehat{m}}$ $(<{Q}_{B}^{*})$.
- ${t}_{GG}^{\tilde{c}}=(1-{\theta}_{G}){Q}_{G}^{*}/2+\Delta \theta {Q}_{B}^{\tilde{c}}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{\tilde{c}}=(1-{\theta}_{G}){Q}_{M}^{*}+\Delta \theta {Q}_{B}^{\tilde{c}}/2;\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{\tilde{c}}=0;$ ${t}_{BB}^{\tilde{c}}=(1-{\theta}_{B}){Q}_{B}^{\tilde{c}}/2$.
- ${u}_{GG}^{\tilde{c}}={u}_{GB}^{\tilde{c}}=\Delta \theta {Q}_{B}^{\tilde{c}}/2;\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{\tilde{c}}={u}_{BB}^{\tilde{c}}=0$.

**Proposition 2.**

- When $\Delta v$ is small, ${C}^{m}$ is optimal.
- When $\Delta v$ is intermediate, $\left\{\begin{array}{cc}{C}^{m}\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}\mathit{optimal}\hfill & if\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}{\varphi}_{B}\phantom{\rule{4.pt}{0ex}}\mathit{and}/\mathit{or}\phantom{\rule{4.pt}{0ex}}{\psi}_{G}\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}\mathit{small}.\hfill \\ {C}^{c}\phantom{\rule{4.pt}{0ex}}\mathit{is}\phantom{\rule{4.pt}{0ex}}\mathit{optimal}\hfill & if\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{4.pt}{0ex}}{\varphi}_{B}\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}{\psi}_{G}\phantom{\rule{4.pt}{0ex}}\mathit{are}\phantom{\rule{4.pt}{0ex}}\mathit{large}.\hfill \end{array}\right.$
- When $\Delta v$ is large, ${C}^{c}$ is optimal.

## 6. Production Technology and Organizational Structure

**Lemma**

**7.**

- ${q}_{G}^{\overline{m}}={q}_{G}^{*};\phantom{\rule{0.277778em}{0ex}}{q}_{M}^{\overline{m}}<{q}_{M}^{*};\phantom{\rule{0.277778em}{0ex}}{q}_{B}^{\overline{m}}<{q}_{B}^{*}$.
- ${t}_{GG}^{\overline{m}}=(1-{\theta}_{G}){q}_{G}^{*}/2+\Delta \theta {q}_{M}^{m};\phantom{\rule{0.277778em}{0ex}}{t}_{GB}^{\overline{m}}=(1-{\theta}_{G}){q}_{M}^{*}+\Delta \theta {q}_{B}^{\overline{m}};\phantom{\rule{0.277778em}{0ex}}{t}_{BG}^{\overline{m}}={t}_{BB}^{\overline{m}}=(1-{\theta}_{B}){q}_{B}^{\overline{m}}/2$.
- ${u}_{GG}^{\overline{m}}=\Delta \theta {q}_{M}^{m};\phantom{\rule{0.277778em}{0ex}}{u}_{GB}^{\overline{m}}=\Delta \theta {q}_{B}^{\overline{m}};\phantom{\rule{0.277778em}{0ex}}{u}_{BG}^{\overline{m}}={u}_{BB}^{\overline{m}}=0$.

**Claim**

**3.**

**Proposition 3.**

## 7. Discussion

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Lemma**

**2.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Claim**

**1.**

**Proof**

**of**

**Lemma**

**3.**

**Proof**

**of**

**Lemma**

**4.**

**Proof**

**of**

**Lemma**

**5.**

**Proof**

**of**

**Claim**

**2**

**and**

**Lemma**

**6.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Lemma**

**7.**

**Proof**

**of**

**Claim**

**3.**

**Proof**

**of**

**Proposition**

**3.**

## Notes

1 | In traditional principal–agent models, the possibility of collusion among agents limits the principal’s welfare (e.g., Tirole 1986 [10]; Kofman and Lawarrée 1993 [11]; Laffont and Martimort 1997 [12], 1998 [13]; Khalil and Lawarrée 2006 [14]). The possibility of collusion in the traditional models corresponds to the integrated organizational structure in our model, in which the agents cannot affect each other’s task environment. |

2 | This is a standard assumption; see Tirole (1992) [15] for a discussion of enforceability of side-contracts. |

3 | In our model, an agent only reports their updated type. If an agent reports both their initial type and the updated type, then there is no hidden interaction. When reporting both the initial and the updated type, it is necessary to impose a random shock in our model (i.e., the agent’s task environment can be changed by chance) to generate the prospect of hidden interactions; such a model adds more cases to our current setting without changing the qualitative result. |

4 | Thus, we assume that an agent has a right to protest the other agent’s report in court if an agent’s type is reported by the other agent, and that resolution is prohibitively costly. Suppose that an agent can acquire hard evidence on the other agent’s type with a strictly positive probability. Then, the principal can achieve the first-best outcome if the penalty applied to a misreporting agent can be unlimited. |

5 | Due to the Inada conditions, ${Q}_{ij}$ is strictly positive at the optimum. |

6 | In the principal’s problem, constraints are identical for agents of the same type, which implies that ${q}_{GG}={Q}_{G}/2$ and ${q}_{BB}={Q}_{B}/2$. |

7 | The result will be the same as long as the agents can collude after learning their types and before reporting them. |

8 | The strictly decreasing schedule, in particular ${Q}_{G}^{*}>{Q}_{M}^{*},$ is due to the fact that ${\Theta}_{k}$ enters the value function (common value). We discuss the robustness related to this issue later. |

9 | Constant marginal costs are adopted for expositional purposes; this allows us to exaggerate the intuition. We discuss this issue at greater length in the concluding section. |

10 | See Winter (2009) [21], who showed that, in a team production context, rewards may affect performance in a nonmonotonic way, i.e., a higher reward in the case of success may reduce agents’ incentives to exert effort. |

11 | See Daft (2009) [22] for an example. |

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Notations | Descriptions |
---|---|

${\theta}_{i},{\theta}_{j}\in \left\{{\theta}_{G},{\theta}_{B}\right\}$ | the agent’s type |

${\mu}_{G},{\mu}_{B}$ | the distribution of each type |

${\varphi}_{B}$ | $\mathrm{The}\phantom{\rule{4.pt}{0ex}}\mathrm{probabilty}\phantom{\rule{4.pt}{0ex}}\mathrm{that}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}Good\phantom{\rule{4.pt}{0ex}}\mathrm{agent}\phantom{\rule{4.pt}{0ex}}\mathrm{becomes}\phantom{\rule{4.pt}{0ex}}Bad\phantom{\rule{4.pt}{0ex}}\mathrm{due}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{sabotage}$ |

${\psi}_{B}$ | $\mathrm{The}\phantom{\rule{4.pt}{0ex}}\mathrm{probabilty}\phantom{\rule{4.pt}{0ex}}\mathrm{that}\phantom{\rule{4.pt}{0ex}}\mathrm{a}\phantom{\rule{4.pt}{0ex}}Bad\phantom{\rule{4.pt}{0ex}}\mathrm{agent}\phantom{\rule{4.pt}{0ex}}\mathrm{becomes}\phantom{\rule{4.pt}{0ex}}Good\phantom{\rule{4.pt}{0ex}}\mathrm{due}\phantom{\rule{4.pt}{0ex}}\mathrm{to}\phantom{\rule{4.pt}{0ex}}\mathrm{help}$ |

${q}_{ij}$ | individual output level |

${Q}_{ij}={q}_{ij}+{q}_{ji}$ | total output level |

${Q}_{G},{Q}_{M},{Q}_{B}$ | ${Q}_{G}\equiv {Q}_{GG},{Q}_{M}\equiv {Q}_{GB},\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}{Q}_{B}\equiv {Q}_{BB}$ |

$v({Q}_{ij},{\Theta}_{k})$ | value function |

${\Theta}_{k}\in \{{\Theta}_{G},{\Theta}_{M},{\Theta}_{B}\}$ | ${\Theta}_{G}\equiv 2{\theta}_{G},\phantom{\rule{1.em}{0ex}}{\Theta}_{M}\equiv {\theta}_{G}+{\theta}_{B},\phantom{\rule{1.em}{0ex}}{\Theta}_{B}\equiv 2{\theta}_{B}$ |

${\pi}_{ij}$ | The principal’s ex post payoff |

${t}_{ij}$ | monetary transfer |

$\gamma $ | the proportion of ${Q}_{M}$ produced by the $Good$ agent |

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**MDPI and ACS Style**

Kim, J.; Shin, D.
Vertical Relationships with Hidden Interactions. *Games* **2023**, *14*, 69.
https://doi.org/10.3390/g14060069

**AMA Style**

Kim J, Shin D.
Vertical Relationships with Hidden Interactions. *Games*. 2023; 14(6):69.
https://doi.org/10.3390/g14060069

**Chicago/Turabian Style**

Kim, Jaesoo, and Dongsoo Shin.
2023. "Vertical Relationships with Hidden Interactions" *Games* 14, no. 6: 69.
https://doi.org/10.3390/g14060069