# Moving Information Horizon Approach for Dynamic Game Models

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

## References

- Owen, G. Game Theory; Emerald Group Publishing Limited: Bingley, UK, 2013. [Google Scholar]
- Nash, J.F. Equilibrium points in n-person games. Proc. Natl. Acad. Sci. USA
**1950**, 36, 48–49. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nash, J.F. Non-cooperative games. Ann. Math.
**1951**, 54, 286–295. [Google Scholar] [CrossRef] - Basar, T.; Olsder, G.J. Dynamic Noncooperative Game Theory; SIAM: Philadelphia, PA, USA, 1999; Volume 23. [Google Scholar]
- Chander, P.; Tulkens, H. A core-theoretic solution for the design of cooperative agreements on transfrontier pollution. In Public Goods, Environmental Externalities and Fiscal Competition; Springer: Berlin/Heidelberg, Germany, 2006; pp. 176–193. [Google Scholar]
- Haurie, A. A note on nonzero-sum differential games with bargaining solution. J. Optim. Theory Appl.
**1976**, 18, 31–39. [Google Scholar] [CrossRef] - Petrosyan, L. Time-consistency of solutions in multi-player differential games. Vestn. Leningr. State Univ.
**1977**, 4, 46–52. [Google Scholar] - Petrosyan, L.; Danilov, N. Stability of solutions in non-zero sum differential games with transferable payoffs. Vestn. Leningr. Univ.
**1979**, 1, 52–59. [Google Scholar] - Petrosyan, L. Strongly time-consistent differential optimality principles. Vestn. St. Petersburg Univ. Math.
**1993**, 26, 40–46. [Google Scholar] - Petrosyan, L.; Yeung, D.W.K. Dynamically stable cooperative solutions in randomly furcating differential games. Proc. Steklov Inst. Math.
**2006**, 253, S208–S220. [Google Scholar] [CrossRef] - Jørgensen, S.; Yeung, D.W. Inter-and intragenerational renewableresource extraction. Ann. Oper. Res.
**1999**, 88, 275–289. [Google Scholar] [CrossRef] - Jørgensen, S.; Martin-Herran, G.; Zaccour, G. Agreeability and time consistency in linear-state differential games. J. Optim. Theory Appl.
**2003**, 119, 49–63. [Google Scholar] [CrossRef] - Petrosian, O. Looking Forward Approach in Cooperative Differential Games. Int. Game Theory Rev.
**2016**, 18, 1640007. [Google Scholar] - Yeung, D.; Petrosian, O. Infinite Horizon Dynamic Games: A New Approach via Information Updating. Int. Game Theory Rev.
**2017**, 19, 1–16. [Google Scholar] [CrossRef] - Gromova, E.; Petrosian, O. Control of information horizon for cooperative differential game of pollution control. In Proceedings of the International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy’s Conference), Moscow, Russia, 1–3 June 2016; pp. 1–4. [Google Scholar]
- Petrosian, O.L. Looking Forward Approach in Cooperative Differential Games with Infinite-Horizon Vestn. St. Petersburg Univ. Math.
**2016**, 10, 18–30. [Google Scholar] - Petrosian, O.; Barabanov, A. Looking Forward Approach in Cooperative Differential Games with Uncertain Stochastic Dynamics. J. Optim. Theory Appl.
**2017**, 172, 328–347. [Google Scholar] [CrossRef] - Petrosian, O.; Nastych, M.; Volf, D. Differential game of oil market with moving informational horizon and non-transferable utility. In Proceedings of the Constructive Nonsmooth Analysis and Related Topics (Dedicated to the Memory of VF Demyanov) (CNSA), St. Petersburg, Russia, 22–27 May 2017; pp. 1–4. [Google Scholar]
- Petrosian, O.; Tur, A. Hamilton-Jacobi-Bellman Equations for Non-cooperative Differential Games with Continuous Updating. Commun. Comput. Inf. Sci.
**2019**. to be published. [Google Scholar] - Kuchkarov, I.; Petrosian, O. On Class of Linear Quadratic Non-cooperative Differential Games with Continuous Updating. Lect. Notes Comput. Sci.
**2019**. to be published. [Google Scholar] - Petrosian, O.; Nastych, M.; Volf, D. Non-cooperative Differential Game Model of Oil Market with Looking Forward Approach. Frontiers of Dynamic Games, Game Theory and Management, St. Petersburg, 2017; Petrosyan, L.A., Mazalov, V.V., Zenkevich, N., Eds.; Birkhäuser: Basel, Switzerland, 2018. [Google Scholar]
- Huang, J.; Leng, M.; Liang, L. Recent developments in dynamic advertising research. Eur. J. Oper. Res.
**2012**, 220, 591–609. [Google Scholar] [CrossRef] - Feichtinger, G.; Hartl, R.F.; Sethi, S.P. Dynamic optimal control models in advertising: Recent developments. Manag. Sci.
**1994**, 40, 195–226. [Google Scholar] [CrossRef] - Jorgensen, S.; Zaccour, G. Differential Games in Marketing. International Series in Quantitative Marketing; Springer: New York, NY, USA, 2004. [Google Scholar]
- He, X.; Prasad, A.; Sethi, S.P.; Gutierrez, G.J. A survey of Stackelberg differential game models in supply and marketing channels. J. Syst. Sci. Syst. Eng.
**2007**, 16, 385–413. [Google Scholar] [CrossRef] - Aust, G.; Buscher, U. Vertical cooperative advertising in a retailer duopoly. Comput. Ind. Eng.
**2014**, 72, 247–254. [Google Scholar] [CrossRef] - Jørgensen, S.; Zaccour, G. A survey of game-theoretic models of cooperative advertising. Eur. J. Oper. Res.
**2014**, 237, 1–14. [Google Scholar] [CrossRef] - Prasad, A.; Sethi, S.P.; Naik, P.A. Optimal Control of an Oligopoly Model of Advertising. IFAC Proc. Vol.
**2009**, 42, 91–96. [Google Scholar] [CrossRef] - Helbing, D.; Brockmann, D.; Chadefaux, T.; Donnay, K.; Blanke, U.; Woolley-Meza, O.; Moussaid, M.; Johansson, A.; Krause, J.; Schutte, S.; et al. Saving Human Lives: What Complexity Science and Information Systems can Contribute. Phys. Soc.
**2015**, 158, 735–781. [Google Scholar] [CrossRef] [PubMed] - Yeung, D.W.; Petrosyan, L.A. Subgame Consistent Economic Optimization: An Advanced Cooperative Dynamic Game Analysis; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2011. [Google Scholar]
- Yeung, D. On differential games with a feedback Nash equilibrium. J. Optim. Theory Appl.
**1994**, 82, 181–188. [Google Scholar] [CrossRef] - Bellman, R. Dynamic Programming; Dover Publications: New York, NY, USA, 2003. [Google Scholar]
- Petrosjan, L.A. Differential Pursuit Games; World Scientific Publishing Co Pte Ltd.: Singapore, 1993. [Google Scholar]
- Yeung, D.W.; Petrosjan, L.A. Cooperative Stochastic Differential Games; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2006. [Google Scholar]
- Yeung, D.W.; Petrosyan, L.A. Subgame consistent cooperative solution of dynamic games with random horizon. J. Optim. Theory Appl.
**2011**, 150, 78–97. [Google Scholar] [CrossRef] - Shi, L.; Petrosian, O. A Dynamic Oligopoly Marketing Model of Advertising. In Proceedings of the 2017 International Conference on Eleventh International Conference “Game Theory and Management”, St. Petersburg, Russia, 28–30 June 2017; pp. 207–223. [Google Scholar]

**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 ${\left({u}_{i}^{k}\right)}^{2}$. |

© 2019 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

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