# Countervailing Conflicts of Interest in Delegation Games

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model

- Step 1. Nature selects the state $\theta \in \{1,2\}$.
- Step 2. The agent privately observes a signal $\sigma \in \{1,2\}$.
- Step 3. Both players know the cost $c\in \left[0,C\right]$.
- Step 4. The agent decides whether to pursue a project, $P\in \{1,2\}$, or opt for no project, $P=\u2300$.
- Step 5. Payoffs are realized for both players, marking the end of the game.

- Step 2${}^{\prime}$. After observing $\sigma \in \left\{1,2\right\}$, the agent sends a message $m\in \left\{1,2\right\}$ to the principal.3 The principal observes m without noise.
- Step 4${}^{\prime}$. The principal decides whether to pursue the recommended project, $P=m$, or opt for no project, $P=\u2300$.

- Step 4″. The principal decides whether to pursue a project, $P\in \{1,2\}$, or opt for no project, $P=\u2300$.

**Definition**

**1.**

- In a truthful equilibrium (T), ${\beta}^{d}\left(1\right)={\beta}^{d}\left(2\right)=1$.
- In a pandering-toward-1 equilibrium (P1), ${\beta}^{d}\left(1\right)=1$ and ${\beta}^{d}\left(2\right)\in \left(0,1\right)$.
- In a pandering-toward-2 equilibrium (P2), ${\beta}^{d}\left(1\right)\in \left(0,1\right)$ and ${\beta}^{d}\left(2\right)=1$.
- In a zero-1 equilibrium (Z1), ${\beta}^{d}\left(1\right)=1$ and ${\beta}^{d}\left(2\right)=0$.
- In a zero-2 equilibrium (Z2), ${\beta}^{d}\left(1\right)=0$ and ${\beta}^{d}\left(2\right)=1$.

## 3. Results—Comparison among Three Mechanisms

#### 3.1. Example 1

**Proposition**

**1.**

**Proof.**

#### 3.2. Example 2

**Proposition**

**2.**

**Proof.**

## 4. Future Research

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Full Delegation

- $\gamma $ is sufficiently large so that the agent’s cost can be larger than the expected payoff from a project (i.e., the outside option can be chosen), including $\gamma =1$.
- $\gamma $ is sufficiently small so that the agent’s cost is always smaller than the expected payoff from a project (i.e., the outside option is never chosen), including $\gamma =0$.

**Claim**

**1.**

- (a)
- If $t\in \left[\frac{1-\alpha}{\alpha},\frac{\alpha}{1-\alpha}\right]$, ${P}^{Fd}\left(c,\sigma \right)\in \left\{\sigma ,\u2300\right\}$ for $\forall \sigma $ such that:$${P}^{Fd}\left(c,\sigma \right)=\left\{\begin{array}{cc}1\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\sigma =1\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\gamma \xb7c\le \alpha \hfill \\ 2\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}\sigma =2\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\gamma \xb7c\le \alpha \xb7t\hfill \\ \u2300\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$
- (b)
- If $t<\frac{1-\alpha}{\alpha}$, ${P}^{Fd}\left(c,\sigma \right)\in \left\{1,\u2300\right\}$ for $\forall \sigma $ such that:$${P}^{Fd}\left(c,\sigma \right)=\left\{\begin{array}{cc}1\hfill & \mathit{if}[\sigma =1\gamma \xb7c\le \alpha ]\phantom{\rule{4.pt}{0ex}}\mathit{or}\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}[\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\gamma \xb7c\le 1-\alpha ]\hfill \\ \u2300\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$
- (c)
- If $t>\frac{\alpha}{1-\alpha}$, ${P}^{Fd}\left(c,\sigma \right)\in \left\{2,\u2300\right\}$ for $\forall \sigma $ such that:$${P}^{Fd}\left(c,\sigma \right)=\left\{\begin{array}{cc}2\hfill & \mathit{if}\phantom{\rule{4.pt}{0ex}}[\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\gamma \xb7c\le \left(1-\alpha \right)\xb7t]\phantom{\rule{4.pt}{0ex}}\mathit{or}\phantom{\rule{4.pt}{0ex}}\mathit{if}\phantom{\rule{4.pt}{0ex}}[\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\gamma \xb7c\le \alpha \xb7t]\hfill \\ \u2300\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

- (1)
- For sufficiently large γ:
- (a)
- If $t\in \left[\frac{1-\alpha}{\alpha},\frac{\alpha}{1-\alpha}\right]$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{\sigma ,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{2\left(2\gamma +2tx\gamma -{t}^{2}-1\right){\alpha}^{2}}{4{\gamma}^{2}C}.$$
- (b)
- If $t<\frac{1-\alpha}{\alpha}$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{1,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{\left(2\gamma -1\right)\left(2{\alpha}^{2}-2\alpha +1\right)}{4{\gamma}^{2}C}.$$
- (c)
- If $t>\frac{\alpha}{1-\alpha}$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{2,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{2\left(2x\gamma -t\right)t\left(2{\alpha}^{2}-2\alpha +1\right)}{4{\gamma}^{2}C}.$$

- (2)
- For sufficiently small γ:
- (a)
- If $t\in \left[\frac{1-\alpha}{\alpha},\frac{\alpha}{1-\alpha}\right]$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{\sigma ,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{x\alpha +\alpha -C}{2}.$$
- (b)
- If $t<\frac{1-\alpha}{\alpha}$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{1,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{1-C}{2}.$$
- (c)
- If $t>\frac{\alpha}{1-\alpha}$, i.e., ${P}^{Fd}\left(c,\sigma \right)\in \left\{2,\u2300\right\}$ for $\forall \sigma $:$${U}^{Fd}=\frac{x-C}{2}.$$

**Proof.**

- (1)
- Suppose that $\gamma $ is sufficiently large $\left(\mathrm{e}.\mathrm{g}.,\phantom{\rule{4.pt}{0ex}}\gamma =1\right)$.
- (a)
- If $t\in \left[\frac{1-\alpha}{\alpha},\frac{\alpha}{1-\alpha}\right]$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1}{\underbrace{{\int}_{0}^{\frac{1\xb7\alpha}{\gamma}}\left(1\xb7\alpha -c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{\frac{t\xb7\alpha}{\gamma}}\left(x\xb7\alpha -c\right)dF\left(c\right)}}.\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2C}\left(\frac{\left(2\gamma -1\right){\alpha}^{2}}{2{\gamma}^{2}}+\frac{\left(2x\gamma -t\right)t{\alpha}^{2}}{2{\gamma}^{2}}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{\left(2\gamma +2tx\gamma -{t}^{2}-1\right){\alpha}^{2}}{4{\gamma}^{2}C}\hfill \end{array}$$
- (b)
- If $t<\frac{1-\alpha}{\alpha}$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}}{\underbrace{{\int}_{0}^{\frac{1\xb7\alpha}{\gamma}}\left(1\xb7\alpha -c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{\frac{1\xb7\left(1-\alpha \right)}{\gamma}}\left(1\xb7\left(1-\alpha \right)-c\right)dF\left(c\right)}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2C}\left(\frac{\left(2\gamma -1\right){\alpha}^{2}}{2{\gamma}^{2}}+\frac{\left(2\gamma -1\right){\left(1-\alpha \right)}^{2}}{2{\gamma}^{2}}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{\left(2\gamma -1\right)\left(2{\alpha}^{2}-2\alpha +1\right)}{4{\gamma}^{2}C}\hfill \end{array}$$
- (c)
- If $t>\frac{\alpha}{1-\alpha}$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}}{\underbrace{{\int}_{0}^{\frac{t\xb7\left(1-\alpha \right)}{\gamma}}\left(x\xb7\left(1-\alpha \right)-c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{\frac{t\xb7\alpha}{\gamma}}\left(x\xb7\alpha -c\right)dF\left(c\right)}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2C}\left(\frac{\left(2x\gamma -t\right){\left(\alpha -1\right)}^{2}t}{2{\gamma}^{2}}+\frac{\left(2x\gamma -t\right)t{\alpha}^{2}}{2{\gamma}^{2}}\right)\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{\left(2x\gamma -t\right)t\left(2{\alpha}^{2}-2\alpha +1\right)}{4{\gamma}^{2}C}\hfill \end{array}$$

- (2)
- Suppose that $\gamma $ is sufficiently small $\left(\mathrm{e}.\mathrm{g}.,\phantom{\rule{4.pt}{0ex}}\gamma =0\right)$.
- (a)
- If $t\in \left[\frac{1-\alpha}{\alpha},\frac{\alpha}{1-\alpha}\right]$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1}{\underbrace{{\int}_{0}^{C}\left(1\xb7\alpha -c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{C}\left(x\xb7\alpha -c\right)dF\left(c\right)}}.\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2C}\left(\frac{1}{2}C\left(2x\alpha -C\right)+\frac{1}{2}C\left(2\alpha -C\right)\right)\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2}\left(x\alpha +\alpha -C\right)\hfill \end{array}$$
- (b)
- If $t<\frac{1-\alpha}{\alpha}$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}}{\underbrace{{\int}_{0}^{C}\left(1\xb7\alpha -c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{C}\left(1\xb7\left(1-\alpha \right)-c\right)dF\left(c\right)}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1}{2C}\left(\frac{1}{2}C\left(2\alpha -C\right)+\frac{1}{2}C\left(2-C-2\alpha \right)\right)\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{1-C}{2}\hfill \end{array}$$
- (c)
- If $t>\frac{\alpha}{1-\alpha}$:$$\begin{array}{ccc}{U}^{Fd}\hfill & =\hfill & \frac{1}{2}\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}1\phantom{\rule{4.pt}{0ex}}}{\underbrace{{\int}_{0}^{C}\left(x\xb7\left(1-\alpha \right)-c\right)dF\left(c\right)}}+\underset{\mathrm{Payoffs}\phantom{\rule{4.pt}{0ex}}\mathrm{given}\phantom{\rule{4.pt}{0ex}}\sigma \phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}2\phantom{\rule{4.pt}{0ex}}}{\frac{1}{2}\underbrace{{\int}_{0}^{C}\left(x\xb7\alpha -c\right)dF\left(c\right)}}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{C\left(x-C\right)}{2C}\mathrm{given}\phantom{\rule{4.pt}{0ex}}F\left(c\right)=\frac{c}{C}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & =\hfill & \frac{x-C}{2}\hfill \end{array}$$

#### Appendix A.2. Veto-Based Delegation

**Claim**

**2.**

- (1)
- T exists iff:$$\begin{array}{cc}t\in & \left[max\left\{\frac{1-\alpha}{\alpha x}-\frac{\gamma \left(1-{x}^{2}\right)}{2x},0\right\},\frac{\alpha}{x\left(1-\alpha \right)}\xb7\frac{2-\gamma \left(1-{x}^{2}\right)}{2}\right].\end{array}$$
- (2)
- P1 exists iff:$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in \hfill & \left(max\left\{0,{\displaystyle \underset{\mu \in \left(1/2,\alpha \right)}{min}}\left(-\frac{\gamma}{2{\alpha}^{2}x}{\left(\mu -\frac{1-\alpha}{\gamma}\right)}^{2}+\frac{x\gamma}{2}+\frac{{\left(1-\alpha \right)}^{2}}{2{\alpha}^{2}x\gamma}\right)\right\},\right.\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \left.max\left\{\frac{1-\alpha}{\alpha x}-\frac{\gamma \left(1-{x}^{2}\right)}{2x},0\right\}\right)\hfill \end{array}$$
- (3)
- P2 exists iff:$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in & \left(\frac{\alpha}{x\xb7\left(1-\alpha \right)}\xb7\frac{2-\gamma \left(1-{x}^{2}\right)}{2},{\displaystyle \underset{\mu \in \left[1/2,\alpha \right]}{max}}\frac{{\alpha}^{2}\left(2-\gamma \right)+\gamma {\mu}^{2}{x}^{2}}{2\mu \left(1-\alpha \right)x}\right).\end{array}$$
- (4)
- Only Z1 exists if (https://www.overleaf.com/project/654eff35961581028f1bf052):$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in & \left(0,{\displaystyle \underset{\mu \in \left(1/2,\alpha \right)}{min}}\left(-\frac{\gamma}{2{\alpha}^{2}x}{\left(\mu -\frac{1-\alpha}{\gamma}\right)}^{2}+\frac{x\gamma}{2}+\frac{{\left(1-\alpha \right)}^{2}}{2{\alpha}^{2}x\gamma}\right)\right).\end{array}$$
- (5)
- Only Z2 exists if:$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in & \left({\displaystyle \underset{\mu \in \left[1/2,\alpha \right]}{max}}\frac{{\alpha}^{2}\left(2-\gamma \right)+\gamma {\mu}^{2}{x}^{2}}{2\mu \left(1-\alpha \right)x},\infty \right).\end{array}$$

**Proof.**

#### Appendix A.3. Communication (No Delegation)

**Claim**

**3.**

- (1)
- T exists if:$$\begin{array}{cc}t\in \left[max\left\{\frac{1-\alpha}{\alpha x}-\frac{\gamma \left(1-{x}^{2}\right)}{2x},0\right\},\frac{\alpha}{x\xb7\left(1-\alpha \right)}\xb7\frac{2-\gamma \left(1-{x}^{2}\right)}{2}\right]& \mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge \frac{1-\alpha}{\alpha}.\end{array}$$
- (2)
- P1 exists if:$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in \hfill & \left(max\left\{0,{\displaystyle \underset{\mu \in \left(1/2,\alpha \right)}{min}}\left(-\frac{\gamma}{2{\alpha}^{2}x}{\left(\mu -\frac{1-\alpha}{\gamma}\right)}^{2}+\frac{x\gamma}{2}+\frac{{\left(1-\alpha \right)}^{2}}{2{\alpha}^{2}x\gamma}\right)\right\},\right.\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \left.max\left\{\frac{1-\alpha}{\alpha x}-\frac{\gamma \left(1-{x}^{2}\right)}{2x},0\right\}\right)\phantom{\rule{4.pt}{0ex}}\mathit{and}\phantom{\rule{4.pt}{0ex}}\hfill \\ x\ge \hfill & \frac{1-\alpha}{\alpha}.\phantom{\rule{4.pt}{0ex}}\hfill \end{array}$$
- (3)
- P2 exists if:$$\begin{array}{cc}\phantom{\rule{4pt}{0ex}}t\in \left(\frac{\alpha}{x\xb7\left(1-\alpha \right)}\xb7\frac{2-\gamma \left(1-{x}^{2}\right)}{2},{\displaystyle \underset{\mu \in \left[\frac{1}{1+x},\alpha \right]}{max}}\frac{{\alpha}^{2}\left(2-\gamma \right)+\gamma {\mu}^{2}{x}^{2}}{2\mu \left(1-\alpha \right)x}\right)& \mathit{and}\phantom{\rule{4.pt}{0ex}}x\ge \frac{1-\alpha}{\alpha}.\end{array}$$

**Proof.**

#### Appendix A.4. Comparison among Three Mechanisms

## Notes

1 | In 2009, Brickley, Smith, and Zimmerman [17] discussed a common design, particularly prevalent among large firms, which involves the distribution of top executive responsibilities among a CEO and other senior executives. They also presented case studies highlighting the inclusion of outside directors, as exemplified by Sony Corporation. |

2 | A positive probability exists that the outside option will be selected, regardless of which player makes the decision. |

3 | If the message set contains at least two elements (equal to the number of states), the outcome remains the same. |

## References

- Crawford, V.; Sobel, J. Strategic Information Transmission. Econometrica
**1982**, 50, 1431–1451. [Google Scholar] [CrossRef] - Che, Y.-K.; Dessein, W.; Kartik, N. Pandering to Persuade. Am. Econ. Rev.
**2013**, 103, 47–79. [Google Scholar] [CrossRef] - Chiba, S.; Leong, K. An Example of Conflicts of Interest as Pandering Disincentives. Econ. Lett.
**2015**, 131, 20–23. [Google Scholar] [CrossRef] - Dessein, W. Authority and Communication in Organizations. Rev. Econ. Stud.
**2002**, 69, 811–838. [Google Scholar] [CrossRef] - Milgrom, P.; Roberts, J. Economics, Organization and Management; Prentice Hall: Upper Saddle River, NJ, USA, 1992. [Google Scholar]
- Krishna, V.; Morgan, J. Contracting for Information under Imperfect Commitment. RAND J. Econ.
**2008**, 39, 905–925. [Google Scholar] [CrossRef] - Holmström, B. On Incentives and Control in Organizations. Ph.D. Dissertation, Stanford University, Stanford, CA, USA, 1977. [Google Scholar]
- Alonso, R.; Matouschek, N. Optimal Delegation. Rev. Econ. Stud.
**2008**, 75, 259–293. [Google Scholar] [CrossRef] - Goltsman, M.; Hörner, J.; Pavlov, G.; Squitani, F. Mediation, arbitration and negotiation. J. Econ. Theory
**2009**, 144, 1397–1420. [Google Scholar] [CrossRef] - Amador, M.; Bagwell, K. Money burning in the theory of delegation. Games Econ. Behav.
**2020**, 121, 382–412. [Google Scholar] [CrossRef] - Kartik, N.; Kleiner, A.; Van Weelden, R. Delegation in Veto Bargaining. Am. Econ. Rev.
**2021**, 111, 4046–4087. [Google Scholar] [CrossRef] - Gilligan, T.W.; Krehbiel, K. Collective Decision-Making and Standing Committees: An Information Rationale for Restrictive Amendment Procedures. J. Law Econ. Organ.
**1987**, 3, 287–335. [Google Scholar] - Krishna, V.; John Morgan, J. Asymmetric Information and Legislative Rules: Some Amendments. Am. Political Sci. Rev.
**2001**, 95, 435–452. [Google Scholar] [CrossRef] - Martin, E.M. An Informational Theory of the Legislative Veto. J. Law Econ. Organ.
**1997**, 13, 319–343. [Google Scholar] [CrossRef] - Volden, C. Delegating Power to Bureaucracies: Evidence from the States. J. Law Econ. Organ.
**2002**, 18, 187–220. [Google Scholar] [CrossRef] - Gilardi, F. The Formal Independence of Regulators: A Comparison of 17 Countries and 7 Sectors. Swiss Political Sci. Rev.
**2005**, 11, 139–167. [Google Scholar] - Brickley, J.; Smith, C.; Zimmerman, J. Managerial Economics and Organizational Architecture; McGraw-Hill: New York, NJ, USA, 2009. [Google Scholar]
- Christie, A.; Joye, M.; Watts, R. Decentralization of the Firm: Theory and Evidence. J. Corp. Financ.
**2003**, 9, 3–36. [Google Scholar] - Colombo, M.; Delmastro, M. Delegation of Authority in Business Organizations: An Empirical Test. J. Ind. Econ.
**2004**, 52, 53–80. [Google Scholar] [CrossRef] - Marino, A.M. Delegation Versus Veto in Organizational Games of Strategic Communication. J. Public Econ. Theory
**2007**, 9, 979–992. [Google Scholar] [CrossRef] - Mylovanov, T. Veto-Based Delegation. J. Econ. Theory
**2008**, 138, 297–307. [Google Scholar] [CrossRef] - Chen, S.-S.; Ko, P.-S.; Tsai, C.-S.; Lee, J.-Y. Managerial Delegation and Conflicting Interest in Unionized Duopoly with Firm Heterogeneity. Mathematics
**2022**, 10, 4201. [Google Scholar] [CrossRef]

**Figure 3.**Principal’s welfare (given $\gamma $ = 0 and $x<$ (1 $-\alpha $)/$\alpha $). The straight line is for $Vd$. The dashed line is for $Nd$.

**Figure 5.**Principal’s welfare (given $\gamma $ = 1 and $x>$ (1 $-\alpha $)/$\alpha $). The dashed line is for $Fd$. The straight line is for $Vd$.

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Chiba, S.; Leong, K.
Countervailing Conflicts of Interest in Delegation Games. *Games* **2023**, *14*, 71.
https://doi.org/10.3390/g14060071

**AMA Style**

Chiba S, Leong K.
Countervailing Conflicts of Interest in Delegation Games. *Games*. 2023; 14(6):71.
https://doi.org/10.3390/g14060071

**Chicago/Turabian Style**

Chiba, Saori, and Kaiwen Leong.
2023. "Countervailing Conflicts of Interest in Delegation Games" *Games* 14, no. 6: 71.
https://doi.org/10.3390/g14060071