# IDP-Core: Novel Cooperative Solution for Differential Games

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Problem Statement and Preliminary Information

#### 2.1. Differential Game Model

#### 2.2. Cooperative Differential Game Model

- (1)
- Determination of a strategy set for players which maximizes the sum of their payoffs or determination of strategies corresponding to cooperative behavior. These strategies ${u}^{*}=({u}_{1}^{*},\dots ,{u}_{n}^{*})$ are called optimal, the corresponding trajectory is called the cooperative trajectory and denoted by ${x}^{*}\left(t\right)$.
- (2)
- Determination of the allocation rule for the maximum joint payoff of players corresponding to the optimal strategies ${u}^{*}\left(t\right)$ and determination of optimal trajectory ${x}^{*}\left(t\right)$. Namely, the determination of a cooperative solution as a subset of the imputation set.

#### 2.3. Core

**Definition**

**1.**

**Theorem**

**1.**

#### 2.4. Non-Emptiness of Core in Static Games

**Theorem**

**2.**

#### 2.5. Time-Consistency of Cooperative Solution and Imputation Distribution Procedure

**Definition**

**2.**

**Definition**

**3.**

## 3. IDP-Core and Dominance of Imputation Distribution Procedures

#### 3.1. Dominance of Imputation Distribution Procedures

**Definition**

**4.**

**Definition**

**5.**

#### 3.2. IDP-Core

**Definition**

**6.**

**Theorem**

**3.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**1.**

**Proposition**

**3.**

**Proof.**

## 4. Application of Linear Programming Methods for Nonemptiness Properties

**Theorem**

**4.**

**Proof.**

## 5. Differential Game Model of Resource Extraction

#### 5.1. Cooperative Strategies and Cooperative Trajectory

#### 5.2. Characteristic Function

#### 5.3. IDP-Core

#### 5.4. Non-Emptiness of IDP-Core

#### 5.5. Core and IDP-Core

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Axes: ${\beta}_{1}$, ${\beta}_{3}$, t. ${\beta}_{2}$ can be found using the equality in (25).

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Petrosian, O.; Zakharov, V. IDP-Core: Novel Cooperative Solution for Differential Games. *Mathematics* **2020**, *8*, 721.
https://doi.org/10.3390/math8050721

**AMA Style**

Petrosian O, Zakharov V. IDP-Core: Novel Cooperative Solution for Differential Games. *Mathematics*. 2020; 8(5):721.
https://doi.org/10.3390/math8050721

**Chicago/Turabian Style**

Petrosian, Ovanes, and Victor Zakharov. 2020. "IDP-Core: Novel Cooperative Solution for Differential Games" *Mathematics* 8, no. 5: 721.
https://doi.org/10.3390/math8050721