# Moving Information Horizon Approach for Dynamic Game Models

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## Abstract

**:**

## 1. Introduction

## 2. Looking Forward Approach

#### 2.1. Truncated Subgame

**Definition**

**1.**

#### 2.2. Noncooperative Outcomes in the Game Model with a Moving Information Horizon

**Theorem**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.3. Cooperative Game Model with a Moving Information Horizon

**Theorem**

**2.**

**Definition**

**4.**

**Definition**

**5.**

#### 2.4. Resulting Cooperative Solution and Theoremerties

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**Definition**

**9.**

**Definition**

**10.**

**Theorem**

**3.**

#### 2.5. Construction of the Characteristic Function in the Game Model with Dynamic Updating

**Definition**

**11.**

**Theorem**

**4.**

#### 2.6. Relationship of the Solutions in Truncated Subgames and Resulting Solutions

**Theorem**

**5.**

**Theorem**

**6.**

#### 2.7. Random Information Horizon

**Theorem**

**7.**

**Corollary**

**1.**

## 3. Dynamic Oligopoly Marketing Model of Advertising

#### 3.1. Initial Game Model

#### 3.2. Noncooperative Outcomes in a Truncated Subgame

**Theorem**

**8.**

**Theorem**

**9.**

- 1.
- $${\widehat{V}}_{i}^{k}(x,\tau )=\sum _{i\in I}{A}_{i}^{k,\tau}{x}_{i}^{\tau}+{B}^{k,\tau}\left(i\right),\tau \in \{{\overline{T}}_{k-1}+1,\dots ,N-1\},$$$$\begin{array}{c}\phantom{\rule{4.pt}{0ex}}\mathit{where}\phantom{\rule{4.pt}{0ex}}{A}_{i}^{k,\tau}={m}_{i}-{\left(\frac{N-\tau}{N-\tau +1}{\overline{Y}}_{i}^{k,\tau +1}{Z}_{i}\right)}^{2}-\frac{N-\tau}{N-\tau +1}{A}_{i}^{k,\tau +1}(\delta h-1),\hfill \\ {A}_{j}^{k,\tau}=-2{\left(\frac{N-\tau}{N-\tau +1}{\overline{Y}}_{j}^{k,\tau +1}{Z}_{j}\right)}^{2}-\frac{N-\tau}{N-\tau +1}{A}_{j}^{k,\tau +1}(\delta h-1),\phantom{\rule{4pt}{0ex}}j\in I\backslash i,\hfill \\ {B}^{k,\tau}\left(i\right)=\sum _{i\in I}[2{\left(\frac{N-\tau}{N-\tau +1}{\overline{Y}}_{i}^{k,\tau +1}{Z}_{i}\right)}^{2}+\frac{N-\tau}{N-\tau +1}\frac{{A}_{i}^{k,\tau +1}h\delta}{n}]\hfill \\ +\frac{N-\tau}{N-\tau +1}{B}^{k,\tau +1}\left(i\right)-{\left(\frac{N-\tau}{N-\tau +1}{\overline{Y}}_{i}^{k,\tau +1}{Z}_{i}\right)}^{2}.\hfill \end{array}$$
- 2.
- $${V}_{i}^{k}(x,f)=\sum _{i\in I}{A}_{i}^{k,f}{x}_{i}^{f}+{B}^{k,f}\left(i\right),f\in \{k,\dots ,{\overline{T}}_{k-1}\},$$$$\begin{array}{c}\phantom{\rule{4.pt}{0ex}}\mathit{where}\phantom{\rule{4.pt}{0ex}}{A}_{i}^{k,f}={m}_{i}-{\left({\overline{Y}}_{i}^{k+1}{Z}_{i}\right)}^{2}-{A}_{i}^{k,f+1}(\delta h-1),\hfill \\ {A}_{j}^{k,f}=-2{\left({\overline{Y}}_{j}^{k,f+1}{Z}_{j}\right)}^{2}-{A}_{j}^{k,f+1}(\delta h-1),\phantom{\rule{4pt}{0ex}}j\in I\backslash i,\hfill \\ {B}^{k,f}\left(i\right)=\sum _{i\in I}[2{\left({\overline{Y}}_{i}^{k,f+1}{Z}_{i}\right)}^{2}+\frac{{A}_{i}^{k,f+1}h\delta}{n}]+{B}^{k,f+1}\left(i\right)-{\left({\overline{Y}}_{i}^{k,f+1}{Z}_{i}\right)}^{2}\hfill \end{array}$$

#### 3.3. Cooperative Outcomes in a Truncated Subgame

**Theorem**

**10.**

**Theorem**

**11.**

- 1.
- $${\widehat{W}}^{k}(x,\tau )=\sum _{i\in I}{\tilde{C}}_{i}^{k,\tau}{x}_{i}^{\tau}+{\tilde{D}}_{i}^{k,\tau},\tau \in \{{\overline{T}}_{k-1}+1,\dots ,N-1\},$$$$\begin{array}{c}\phantom{\rule{4.pt}{0ex}}\mathit{where}\phantom{\rule{4.pt}{0ex}}{\tilde{C}}_{i}^{k,\tau}={m}_{i}-{\left(\frac{N-\tau}{N-\tau +1}{\widehat{\overline{G}}}_{i}^{k,\tau +1}{Z}_{i}\right)}^{2}-\frac{N-\tau}{N-\tau +1}{\tilde{C}}_{i}^{k,\tau +1}(\delta h-1),\hfill \\ {\tilde{D}}_{i}^{k,\tau}={\left(\frac{N-\tau}{N-\tau +1}{\widehat{\overline{G}}}_{i}^{k,\tau +1}{Z}_{i}\right)}^{2}+\frac{N-\tau}{N-\tau +1}\frac{{\tilde{C}}_{i}^{k,\tau +1}h\delta}{n}+\frac{N-\tau}{N-\tau +1}{\tilde{D}}_{i}^{k,\tau +1};\hfill \end{array}$$
- 2.
- $${W}_{i}^{k}(x,f)=\sum _{i\in I}{\tilde{C}}_{i}^{k,f}{x}_{i}^{f}+{\tilde{D}}_{i}^{k,f},f\in \{k,\dots ,{\overline{T}}_{k-1}\},$$$$\begin{array}{c}\phantom{\rule{4.pt}{0ex}}\mathit{where}\phantom{\rule{4.pt}{0ex}}{\tilde{C}}_{i}^{k,l}={m}_{i}-{\left({\widehat{\overline{G}}}_{i}^{k,l+1}{Z}_{i}\right)}^{2}-{\tilde{C}}_{i}^{k,l+1}(\delta h-1),\hfill \\ {\tilde{D}}_{i}^{k,l}={\left({\widehat{\overline{G}}}_{i}^{k,l+1}{Z}_{i}\right)}^{2}+\frac{{\tilde{C}}_{i}^{k,l+1}\delta h}{n}+{D}_{i}^{k,l+1},\hfill \end{array}$$

#### 3.4. Characteristic Function in a Truncated Subgame

## 4. Numerical Simulation for a Dynamic Advertising Game Model with Updating

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proof of Theorem 4

#### Appendix A.2. Proof of Theorem 5

#### Appendix A.3. Proof of Theorem 6

- 1.
- If players in every truncated subgame choose the imputation ${\xi}^{k}({x}_{k}^{*},k,k+\overline{T})\in C({x}_{k}^{*},k,k+\overline{T})$ calculated using $V(S;{x}^{*},k,k+\overline{T})$ for $k\in \left\{1,\dots ,N-\overline{T}\right\},$ then the resulting imputation $\widehat{\xi}({\tilde{x}}_{k}^{*},N-k)$ belongs to core $\widehat{C}({\tilde{x}}_{k}^{*},N-k)$, calculated using the resulting characteristic function $\overline{V}(S,{\tilde{x}}_{k}^{*},N-k)$.
- 2.
- Core $\widehat{C}({\tilde{x}}_{k}^{*},N-k)$ should not contain imputation $\widehat{\xi}({\tilde{x}}_{k}^{*},N-k)$, for which it is impossible to find the set of imputations in truncated subgame ${\xi}^{k}({x}_{k}^{*},k,k+\overline{T})\in C({x}_{k}^{*},k,k+\overline{T})$.

#### Appendix A.4. Proof of Theorem 8

#### Appendix A.5. Proof of Theorem 9

#### Appendix A.6. Proof of Theorem 10

#### Appendix A.7. Proof of Theorem 11

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**Figure 1.**The behavior of players in the game with truncated information can be modeled using a series of truncated subgames ${\mathsf{\Gamma}}_{k}({x}_{k,0},k,k+\overline{T}),k=1,\dots ,N-\overline{T}.$

**Figure 3.**The feedback Nash equilibrium of the noncooperative case in the initial game (solid line) and the feedback Nash equilibrium of noncooperative case in the game model with dynamic updating (dashed line).

**Figure 4.**Optimal cooperative strategies in the cooperative case of the initial game (solid line) and optimal cooperative strategies in the cooperative game model with dynamic updating (dashed line).

**Figure 5.**Noncooperative trajectory in the initial game (solid line) and resulting noncooperative trajectory (dashed line) in the game model with dynamic updating.

**Figure 6.**Cooperative trajectory in the initial game (solid line) and resulting cooperative trajectory (dashed line) in the game model with dynamic updating.

**Figure 7.**Noncooperative outcomes in the initial game (solid line) and resulting noncooperative outcomes (dashed line) in the game model with dynamic updating.

**Figure 8.**Characteristic function in the initial game model (solid line) and resulting characteristic function (dashed line) in the game model with dynamic updating.

**Figure 9.**Shapley value in the initial game model (solid line) and the resulting Shapley value (dashed line) in the game model with dynamic updating.

**Figure 10.**The resulting noncooperative trajectory with a fixed information horizon (dashed line) and the resulting noncooperative trajectory with a random information horizon (solid line).

**Figure 11.**The resulting noncooperative outcomes with a fixed information horizon (dashed line) and the resulting noncooperative outcomes with a random information horizon (solid line).

**Figure 12.**The resulting cooperative trajectory with a fixed information horizon (dashed line) and the resulting cooperative trajectory with a random information horizon (solid line).

Notation | Explanation |
---|---|

${x}_{i}^{k}\le 1$ | Market share of firm $i\in I\equiv \left\{1,\dots ,n\right\}$ at stage k. |

${u}_{i}^{k}\ge 0$ | Advertising effort rate of firm $i\in I$ at stage k. |

${\rho}_{i}>0$ | Advertising effectiveness parameter of firm $i\in I$. |

$\delta >0$ | Churn parameter. |

${m}_{i}>0$ | Industry sales multiplied by the per unit profit margin of firm $i\in I$. |

$C\left({u}_{i}^{k}\right)$ | Cost of advertising of firm $i\in I$ at stage k, parameterized by ${\left({u}_{i}^{k}\right)}^{2}$. |

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Petrosian, O.; Shi, L.; Li, Y.; Gao, H. Moving Information Horizon Approach for Dynamic Game Models. *Mathematics* **2019**, *7*, 1239.
https://doi.org/10.3390/math7121239

**AMA Style**

Petrosian O, Shi L, Li Y, Gao H. Moving Information Horizon Approach for Dynamic Game Models. *Mathematics*. 2019; 7(12):1239.
https://doi.org/10.3390/math7121239

**Chicago/Turabian Style**

Petrosian, Ovanes, Lihong Shi, Yin Li, and Hongwei Gao. 2019. "Moving Information Horizon Approach for Dynamic Game Models" *Mathematics* 7, no. 12: 1239.
https://doi.org/10.3390/math7121239