# Inverse Stackelberg Solutions for Games with Many Followers

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Static Games

**Definition**

**1.**

**(1)**- ${u}^{\ast}\in \mathcal{E}\left(\alpha \right).$
**(2)**- ${J}_{0}[{\alpha}^{\ast},{u}^{\ast}]={max}_{\alpha}{max}_{u\in \mathcal{E}(\alpha )}{J}_{0}[\alpha ,u]$.

**Lemma**

**1.**

**(1)**- If ${u}^{\u266e}\in \mathcal{E}\left(\alpha \right)$, then $(\alpha [{u}^{\u266e}],{u}^{\u266e})\in \mathcal{B}$;
**(2)**- If the strategy of the leader (${u}_{0}^{\u266e}$), and the profile of the followers’ strategies (${u}^{\u266e}$) are $({u}^{\u266e},{u}^{\u266e})\in \mathcal{B}$, then an incentive strategy of the leader α exists such that ${u}^{\u266e}\in \mathcal{E}\left(\alpha \right)$.

**Proof.**

**Theorem**

**1.**

**Proof.**

**Example**

**1.**

a, b | 0, 0 |

0, 0 | b, a |

0, 0 | a, b |

b, a | 0, 0 |

## 3. Inverse Stackelberg Solution for Differential Games

**Definition**

**2.**

**(1)**- ${u}^{\ast}\in {\mathcal{E}}_{d}({\alpha}^{\ast})$
**(2)**- ${J}_{0}[{\alpha}^{\ast},{u}^{\ast}]={max}_{\alpha}{max}_{u\in {\mathcal{E}}_{d}\left(\alpha \right)}{J}_{0}[\alpha ,u].$

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**(1)**- ${i}_{1},\dots ,{i}_{n}$ is a permutation of $1,\dots ,n$;
**(2)**- ${\tau}_{{i}_{1}}\le {\tau}_{{i}_{2}},\dots ,{\tau}_{{i}_{n}}$;
**(3)**- for each k, ${t}_{{i}_{k}}$ is the greatest time such that ${u}_{{i}_{k}}={u}_{{i}_{k}}^{\u266e}$ on $[0,{\tau}_{{i}_{k}}]$.

**Theorem**

**2.**

## 4. Existence of the Inverse Stackelberg Solution for Differential Game

**Theorem**

**3.**

**(1)**- $x\mapsto {\sigma}_{i}\left(x\right)$ is concave;
**(2)**- ${g}_{i}(t,x,{u}_{0},u)={g}_{i}^{0}(t,x,{u}_{-i})+{g}_{i}^{1}(t,{u}_{0},{u}_{-i})+{g}_{i}^{2}(t,u)$ and the function $x\mapsto {g}_{i}^{0}(t,x,{u}_{-i})$ is concave.

**Proof.**

## Funding

## Conflicts of Interest

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Averboukh, Y. Inverse Stackelberg Solutions for Games with Many Followers. *Mathematics* **2018**, *6*, 151.
https://doi.org/10.3390/math6090151

**AMA Style**

Averboukh Y. Inverse Stackelberg Solutions for Games with Many Followers. *Mathematics*. 2018; 6(9):151.
https://doi.org/10.3390/math6090151

**Chicago/Turabian Style**

Averboukh, Yurii. 2018. "Inverse Stackelberg Solutions for Games with Many Followers" *Mathematics* 6, no. 9: 151.
https://doi.org/10.3390/math6090151