# Insider Trading with Semi-Informed Traders and Information Sharing: The Stackelberg Game

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Model and Methodology

- (a)
- Profit maximization of insider 1,$$\begin{array}{c}\hfill E[\tilde{z}-p({\tilde{x}}_{1}+{\tilde{x}}_{2}+\tilde{u})){\tilde{x}}_{1}|{\tilde{s}}_{1}]\ge E[\tilde{z}-p{\tilde{x}}_{1}^{\prime}+{\tilde{x}}_{2}+\tilde{u})){\tilde{x}}_{1}^{\prime}\left|{\tilde{s}}_{1}\right]\end{array}$$
- (b)
- Profit maximization of insider 2,$$\begin{array}{c}\hfill E[\tilde{z}-p({\tilde{x}}_{1}+{\tilde{x}}_{2}+\tilde{u})){\tilde{x}}_{1}|{\tilde{x}}_{1},{\tilde{s}}_{2}]\ge E[\tilde{z}-p({\tilde{x}}_{1}+{\tilde{x}}_{2}^{\prime}+\tilde{u})){\tilde{x}}_{2}^{\prime}|{\tilde{x}}_{1},{\tilde{s}}_{2}]\end{array}$$
- (c)
- Semi-strong market efficiency: The pricing rule $p(.)$ satisfies,$$p\left(\tilde{r}\right)=E\left[\tilde{z}\right|\tilde{r}].$$

## 4. Results and Analysis

**Proposition 1.**

**Proof.**

**Lemma 1.**

## 5. Discussion and Conclusions

**Proposition 2.**

**Lemma 2.**

**Proposition 3.**

**Lemma 3.**

**Subgame 1: Neither Investor Shares Information (${A}_{L}={A}_{H}=\varnothing $)**.

**Subgame 2:**L

**Shares Information but**H

**Does Not (${A}_{L}=\left\{\tilde{s}\right\}$ and ${A}_{H}=\varnothing $)**.

**Subgame 3:**H

**Shares Information (${A}_{H}=S$)**.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof of Proposition 1.**

**Theorem**

**A1.**

**Proof.**

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**Figure 1.**Market depth parameter $\lambda $ and the profits of the two insiders in the Stackelberg and Cournot settings, when they observe the same signal.

**Figure 2.**The market depth parameter $\lambda $ and the profits of the two insiders in the Stackelberg and Cournot settings when insider 2 is fully informed and insider 1 is partially informed.

**Figure 3.**$\delta \Pi ={\Pi}_{2}-{\Pi}_{1}$: the difference between the two insiders’ profits in the Stackelberg setting when they observe the same signal.

H | |||
---|---|---|---|

Not Share | Share | ||

L | $Not\phantom{\rule{3.33333pt}{0ex}}share$ | $\frac{h{\sigma}_{u}}{2\sqrt{(1+h)(3h+4)}},\frac{(4+h){\sigma}_{u}}{4\sqrt{(1+h)(3h+4)}}$ | $\frac{{\sigma}_{u}}{2\sqrt{3}},\frac{{\sigma}_{u}}{4\sqrt{3}}$ |

$Share$ | $\frac{h{\sigma}_{u}}{2\sqrt{(1+h)(3h+4)}},\frac{(4+h){\sigma}_{u}}{4\sqrt{(1+h)(3h+4)}}$ | $\frac{{\sigma}_{u}}{2\sqrt{3}},\frac{{\sigma}_{u}}{4\sqrt{3}}$ |

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**MDPI and ACS Style**

Daher, W.; Karam, F.; Ahmed, N.
Insider Trading with Semi-Informed Traders and Information Sharing: The Stackelberg Game. *Mathematics* **2023**, *11*, 4580.
https://doi.org/10.3390/math11224580

**AMA Style**

Daher W, Karam F, Ahmed N.
Insider Trading with Semi-Informed Traders and Information Sharing: The Stackelberg Game. *Mathematics*. 2023; 11(22):4580.
https://doi.org/10.3390/math11224580

**Chicago/Turabian Style**

Daher, Wassim, Fida Karam, and Naveed Ahmed.
2023. "Insider Trading with Semi-Informed Traders and Information Sharing: The Stackelberg Game" *Mathematics* 11, no. 22: 4580.
https://doi.org/10.3390/math11224580