# A Game-Theoretic Approach for Modeling Competitive Diffusion over Social Networks

^{*}

## Abstract

**:**

## 1. Introduction

**active**if it receives a message from one of the agents (players) in the diffusion process. However, it is clear that some of the received messages do not have any effect on a node and cannot convince the inactive node to choose that agent’s option. Therefore, it can be concluded that the total number of infected nodes is not the most suitable criterion for optimizing the diffusion process. Hence, in the current paper, a node is called active if there is a received message that can change its own tendency, and players seek to maximize the sum of individuals’ tendencies toward their options, i.e., the level of society’s desire to choose the options taken by the players is maximized. All of these considerations bring the proposed model closer to reality compared to previous studies.

## 2. The Game

## 3. The Influence Model

**active**and

**inactive**—are allocated to the nodes at each step in order to explain the state of nodes. In this influence model, a node is called active if its tendency has changed by receiving a message and it is inactive if it has not received any message or if the received messages cannot change its initial tendency. At the first step, all of the nodes are inactive. Nodes face three different decision-making situations: first, accepting the incoming message such that the node’s tendency changes; second, forwarding the message; and third, selecting the target nodes to forward the message to. It should be noted that, at the first step (t = 1), nodes receive a message from social change agents (players), but at subsequent steps ($t\ge 2$), they receive messages from their neighbors.

- Node i does not receive any message from players and remains inactive.
- Node i receives (only) a message from the kth social change agent. In this case, the effect of the received message on node i depends on the sender’s social skill and the consistency of i’s tendency toward the content of the received message. From a mathematical point of view, the magnitude of this effect is calculated by $({\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}}))$. If this value is lower than node i’s low threshold ${\theta}_{l}^{i}$, then it does not influence node i and cannot change its tendency. That is:$$If\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})<{\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{i}(2)={\alpha}_{i}(1),$$$$If\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{i}(2)=\frac{{\alpha}_{i}(1)+tex{t}_{{p}_{k}}}{2},$$$${x}_{ik}(t)=\left\{\begin{array}{c}1,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}if\phantom{\rule{3.33333pt}{0ex}}node\phantom{\rule{3.33333pt}{0ex}}i\phantom{\rule{3.33333pt}{0ex}}decides\phantom{\rule{3.33333pt}{0ex}}to\phantom{\rule{3.33333pt}{0ex}}send\phantom{\rule{3.33333pt}{0ex}}message\phantom{\rule{3.33333pt}{0ex}}tex{t}_{{P}_{k}}\phantom{\rule{3.33333pt}{0ex}}in\phantom{\rule{3.33333pt}{0ex}}step\phantom{\rule{3.33333pt}{0ex}}t\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill \\ 0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}o.w\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\hfill \end{array}\right.$$$$if\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{ik}(2)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{i{k}^{\prime}}(2)=0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}$$
- Node i receives both messages $tex{t}_{{p}_{1}}$ and $tex{t}_{{p}_{2}}$ from both players. In this case, node i faces a decision-making situation in which it evaluates the influence of both messages and decides how to act by drawing a comparison between these messages. Realistically, node i selects one of the incoming messages based on the its social skill (node i’s social skill) and the consistency of its tendency toward the content of the received messages. The mathematical representation of this situation is as follows:$$\begin{array}{c}\hfill If\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{1}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{z}_{i1}(2)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}else\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{z}_{i1}(2)=0,\end{array}$$$$\begin{array}{c}\hfill if\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{2}}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{z}_{i2}(2)=1\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}else\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{z}_{i2}(2)=0,\end{array}$$$$\begin{array}{c}\hfill {\alpha}_{i}(2)=\frac{{\alpha}_{i}(1)+{z}_{i1}(2)\xb7tex{t}_{{p}_{1}}+{z}_{i2}(2)\xb7tex{t}_{{p}_{2}}}{1+{z}_{i1}(2)+{z}_{i2}(2)}\end{array}$$$$\begin{array}{c}\hfill If\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{y}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{iy}(2)=1,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{i{y}^{\prime}}(2)=0,\end{array}$$$$\begin{array}{c}\hfill if\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{y}-{\alpha}_{i}{(1)\parallel}_{2}}{2\sqrt{2}}})<{\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{iy}(2)=0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{x}_{i{y}^{\prime}}(2)=0,\end{array}$$

- It does not receive any message and remains inactive.
- It receives (only) one type of message from its neighbors; e.g., message $tex{t}_{{p}_{k}}$. In this case, according to the magnitude of the impact of the received messages (that depends on the senders’ influence on i, social skill of i, and message content), node i decides how to act. That is:$$\begin{array}{c}\hfill If\sum _{j\in {I}_{i}:{y}_{jik}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{i}(t)=\frac{{\alpha}_{i}(t-1)+tex{t}_{{p}_{k}}}{2},\end{array}$$$$\begin{array}{c}\hfill if\phantom{\rule{3.33333pt}{0ex}}\sum _{j\in {I}_{i}:{y}_{jik}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{k}}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{x}_{ik}(t)=1,\end{array}$$
- It receives both messages $tex{t}_{{p}_{1}}$ and $tex{t}_{{p}_{2}}$ from its active neighbors. It will encounter a decision-making situation. The message that fits best with the node’s tendency and has been forwarded by neighbors which have a considerable influence on i will be accepted and changes i’s tendency. That is:$$\begin{array}{c}\hfill If\sum _{j\in {I}_{i}:{y}_{ji1}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{1}}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{z}_{i1}(t)=1\phantom{\rule{3.33333pt}{0ex}}else\phantom{\rule{3.33333pt}{0ex}}{z}_{i1}(t)=0,\end{array}$$$$\begin{array}{c}\hfill if\sum _{j\in {I}_{i}:{y}_{ji2}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{{p}_{2}}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{l}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{z}_{i2}(t)=1\phantom{\rule{3.33333pt}{0ex}}else\phantom{\rule{3.33333pt}{0ex}}{z}_{i2}(t)=0,\end{array}$$$$\begin{array}{c}\hfill {\alpha}_{i}(t)=\frac{{\alpha}_{i}(t-1)+{z}_{i1}(t)\xb7tex{t}_{{p}_{1}}+{z}_{i2}(t)\xb7tex{t}_{{p}_{2}}}{1+{z}_{i1}(t)+{z}_{i2}(t)},\end{array}$$$$\begin{array}{c}\hfill If\sum _{j\in {I}_{i}:{y}_{jiq}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{q}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})\ge {\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{x}_{iq}(t)=1,\phantom{\rule{3.33333pt}{0ex}}{x}_{i{q}^{\prime}}(t)=0\end{array}$$$$\begin{array}{c}\hfill if\sum _{j\in {I}_{i}:{y}_{jiq}(t)=1}{w}_{ji}\xb7{\beta}_{i}\xb7(1-{\displaystyle \frac{\parallel tex{t}_{q}-{\alpha}_{i}{(t-1)\parallel}_{2}}{2\sqrt{2}}})<{\theta}_{h}^{i}\phantom{\rule{3.33333pt}{0ex}}then\phantom{\rule{3.33333pt}{0ex}}{x}_{iq}(t)=0,\phantom{\rule{3.33333pt}{0ex}}{x}_{i{q}^{\prime}}(t)=0,\end{array}$$

## 4. Results

**Price Of Anarchy (POA)**is a concept in economics and game theory that measures how the efficiency of a system degrades due to selfish behavior of its agents. It is a general notion that can be extended to diverse systems and notions of efficiency. So, in the following, to better consider the efficiency of the outcomes, the price of anarchy for the solutions will be calculated first, and then account will be taken of two settings in which players send their message to the network separately as in a one-player game. To calculate the price of anarchy, we define a measure of efficiency of each outcome that is called welfare function; $\mathbf{Welf}:\mathbf{S}\u27f6\mathbf{R}$ where

**S**is the set of strategy profiles and $\mathbf{Welf}(\mathbf{s})={\sum}_{\mathbf{i}\in \mathbf{N}}{\mathbf{u}}_{\mathbf{i}}(\mathbf{s})$. Suppose that

**NE**denotes the set of Nash equilibrium of the game. The Price of Anarchy is then defined as:

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

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${\mathit{\theta}}_{\mathit{h}}^{\mathit{i}}$, ${\mathit{\theta}}_{\mathit{l}}^{\mathit{i}}$ | $\mathit{\delta}$ | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{2}}$’s Payoff | Best Initial Node for ${\mathit{P}}_{\mathbf{1}}$ | Best Initial Node for ${\mathit{P}}_{\mathbf{2}}$ | Best Content for ${\mathit{P}}_{\mathbf{1}}$ | Best Content for ${\mathit{P}}_{\mathbf{2}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{1}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{2}}$ |
---|---|---|---|---|---|---|---|---|---|

0.1 | 8.4581 | 1.3689 | 19 | 2 | $\{1,0\}$ | $\{-1,1\}$ | 17 | 1 | |

8.8581 | 1.3689 | 19 | 2 | $\{1,0\}$ | $\{0,1\}$ | 17 | 1 | ||

$0<{\theta}_{l}^{i}<0.3$ | 0.3 | 5.0740 | 1.8705 | 19 | 12 | $\{1,0\}$ | $\{0,1\}$ | 9 | 4 |

4.4740 | 1.8705 | 19 | 18 | $\{1,0\}$ | $\{0,1\}$ | 8 | 5 | ||

$0.3<{\theta}_{h}^{i}<1$ | 0.5 | 0.9569 | 1.7562 | 2 | 20 | $\{1,0\}$ | $\{-1,1\}$ | 5 | 4 |

0.7 | 0.9524 | 1.5324 | 2 | 20 | $\{1,0\}$ | $\{-0.5,1\}$ | 1 | 4 | |

0.9 | 0.7735 | 0.6785 | 12 | 15 | $\{1,-0.4\}$ | $\{-0.6,1\}$ | 1 | 2 | |

${\theta}_{l}^{i}=0.5$ ${\theta}_{h}^{i}=0.5$ | 0.1 | 8.4581 | 0.8689 | 4 | 2 | $\{1,-0.6\}$ | $\{-0.9,1\}$ | 17 | 1 |

0.3 | 3.8740 | 1.8705 | 19 | 13 | $\{1,-0.1\}$ | $\{0,1\}$ | 8 | 5 | |

0.5 | 0.6271 | 0.3592 | 4 | 9 | $\{1,-0.4\}$ | $\{0,1\}$ | 3 | 4 | |

0.7 | There is not any Nash equilibrium. | ||||||||

0.9 | 0.0113 | 0.6363 | 19 | 2 | $\{1,0\}$ | $\{-1,1\}$ | 1 | 1 |

${\mathit{\theta}}_{\mathit{h}}^{\mathit{i}}$, ${\mathit{\theta}}_{\mathit{l}}^{\mathit{i}}$ | $\mathit{\delta}$ | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{2}}$’s Payoff | Best Initial Node for ${\mathit{P}}_{\mathbf{1}}$ | Best Initial Node for ${\mathit{P}}_{\mathbf{2}}$ | Best Content for ${\mathit{P}}_{\mathbf{1}}$ | Best Content for ${\mathit{P}}_{\mathbf{2}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{1}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{2}}$ |
---|---|---|---|---|---|---|---|---|---|

${\theta}_{l}^{i}=0$ ${\theta}_{h}^{i}=0.4$ | 0.1 | 23.3373 | 59.7413 | 81 | 23 | $\{1,0\}$ | $\{-0.6,1\}$ | 87 | 76 |

0.3 | 26.6065 | 54.7455 | 81 | 17 | $\{1,0\}$ | $\{-0.6,1\}$ | 93 | 65 | |

0.5 | There is not any Nash equilibrium. | ||||||||

0.7 | 3.5958 | 17.1273 | 27 | 113 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 12 | 17 | |

0.9 | 2.7059 | 16.3655 | 125 | 37 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 1 | 2 | |

2.7059 | 16.6155 | 125 | 37 | $\{1,0\}$ | $\{-0.2,1\}$ | 1 | 2 | ||

$0<{\theta}_{l}^{i}<0.4$ $0.4<{\theta}_{h}^{i}<1$ | 0.1 | 14.537 | 32.2413 | 93 | 39 | $\{1,-0.5\}$ | $\{-0.6,1\}$ | 77 | 86 |

31.537 | 32.2413 | 93 | 25 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 76 | 87 | ||

0.3 | 21.0968 | 39.0414 | 89 | 23 | $\{1,-1\}$ | $\{-0.2,1\}$ | 60 | 93 | |

21.0968 | 39.0414 | 89 | 28 | $\{1,-1\}$ | $\{-0.2,1\}$ | 60 | 93 | ||

0.5 | 23.3014 | 33.7797 | 22 | 3 | $\{1,0\}$ | $\{-0.2,1\}$ | 68 | 59 | |

23.8083 | 33.7797 | 22 | 49 | $\{1,0\}$ | $\{-0.2,1\}$ | 68 | 59 | ||

0.7 | 3.2396 | 16.7611 | 35 | 93 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 12 | 13 | |

3.8620 | 11.1159 | 84 | 46 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 12 | 11 | ||

3.2396 | 17.1159 | 84 | 93 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 10 | 13 | ||

0.9 | 3.1313 | 16.7649 | 54 | 144 | $\{1,-0.5\}$ | $\{-0.6,1\}$ | 2 | 2 | |

3.5313 | 16.7649 | 54 | 144 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 2 | 2 | ||

${\theta}_{l}^{i}=0.5$ ${\theta}_{h}^{i}=1$ | 0.1 | 2.7095 | 16.1442 | 1 | 115 | $\{1,0\}$ | $\{-0.2,1\}$ | 1 | 1 |

2.7541 | 16.1442 | 1 | 160 | $\{1,-1\}$ | $\{-0.6,1\}$ | 1 | 1 | ||

0.3 | 2.5541 | 16.1442 | 1 | 160 | $\{1,0\}$ | $\{-1,1\}$ | 1 | 1 | |

0.5 | 2.5541 | 16.1442 | 1 | 160 | $\{1,0\}$ | $\{-1,1\}$ | 1 | 1 | |

0.7 | 2.5541 | 16.1442 | 1 | 160 | $\{1,0\}$ | $\{-1,1\}$ | 1 | 1 | |

0.9 | 2.5541 | 15.8623 | 10 | 160 | $\{1,0\}$ | $\{-1,1\}$ | 1 | 1 | |

${\theta}_{l}^{i}=1$, ${\theta}_{h}^{i}=1$ | The diffusion process does not happen. |

MGDB | MGTB | MGSB | MGEB | MRND | |
---|---|---|---|---|---|

Player 1 | 30 | 30 | 157 | 30 | 25 |

Player 2 | 85 | 85 | 32 | 32 | 88 |

Initial Node for ${\mathit{P}}_{\mathbf{1}}$ Based on Well-Known Strategies | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{2}}$’s Payoff | Best initial Node for ${\mathit{P}}_{\mathbf{2}}$ | Best Content for ${\mathit{P}}_{\mathbf{1}}$ | Best Content for ${\mathit{P}}_{\mathbf{2}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{1}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{2}}$ | |
---|---|---|---|---|---|---|---|---|

MGDB | 30 | 1.8135 | 16.4623 | 46 | $\{1,-1\}$ | $\{-0.2,1\}$ | 0 | 2 |

MGTB | 1.8135 | 16.4623 | 46 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 0 | 2 | |

MGEB | 1.8135 | 16.4623 | 46 | $\{1,0\}$ | $\{-0.2,1\}$ | 0 | 2 | |

MGSB | 157 | 2.3033 | 16.2556 | 46 | $\{1,-1\}$ | $\{-0.2,1\}$ | 1 | 2 |

2.3033 | 16.5056 | 46 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 1 | 2 | ||

2.3033 | 16.7556 | 46 | $\{1,0\}$ | $\{-0.2,1\}$ | 1 | 2 | ||

MRND | 25 | 1.8164 | 16.1319 | 46 | $\{1,-1\}$ | $\{-0.2,1\}$ | 1 | 2 |

1.8164 | 16.1319 | 46 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 1 | 2 | ||

1.8164 | 16.1319 | 46 | $\{1,0\}$ | $\{-0.2,1\}$ | 1 | 2 |

Initial Node for ${\mathit{P}}_{\mathbf{2}}$ Based on Well-Known Strategies | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{2}}$’s Payoff | Best initial Node for ${\mathit{P}}_{\mathbf{1}}$ | Best Content for ${\mathit{P}}_{\mathbf{1}}$ | Best Content for ${\mathit{P}}_{\mathbf{2}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{1}}$ | The Number of Inffected Nodes by ${\mathit{P}}_{\mathbf{2}}$ | |
---|---|---|---|---|---|---|---|---|

MGDB MGTB | 85 | 4.0984 | 15.1517 | 35 | $\{1,-0.5\}$ | $\{-1,1\}$ | 14 | 1 |

4.2984 | 15.1517 | 35 | $\{1,-0.5\}$ | $\{-0.6,1\}$ | 14 | 1 | ||

4.4684 | 15.1517 | 35 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 14 | 1 | ||

MGSB | 32 | 4.1371 | 15.3215 | 35 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 14 | 8 |

MGEB | ||||||||

MRND | 88 | 3.7802 | 14.4450 | 46 | $\{1,-0.5\}$ | $\{-0.2,1\}$ | 14 | 2 |

$\mathit{\delta}$ | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{4}}$’s Payoff | Best Initial Node for P_{1} | Best Initial Node for P_{2} | Best Content for Comprtitor | The Number of Inffected Nodes by P_{1} | The Number of Inffected Nodes by P_{2} | |
---|---|---|---|---|---|---|---|---|

Player 1 selects $\{1,-1\}$ | 0.5 | 4.5542 | 28.3361 | 22 | 48 | $\{-0.2,1\}$ | 31 | 82 |

4.5542 | 29.3747 | 41 | 48 | $\{-0.2,1\}$ | 31 | 84 | ||

0.7 | 2.8340 | 17.4060 | 54 | 46 | $\{-0.2,1\}$ | 2 | 11 | |

Player 2 selects $\{-1,1\}$ | 0.5 | $-1.9469$ | 19.4809 | 135 | 84 | $\{1,-1\}$ | 36 | 47 |

$-1.9469$ | 19.1998 | 158 | 84 | $\{1,-1\}$ | 36 | 47 | ||

$-1.9469$ | 19.1994 | 158 | 84 | $\{1,0\}$ | 36 | 47 | ||

0.7 | 3.7878 | 16.2260 | 84 | 66 | $\{1,0\}$ | 12 | 2 | |

3.2051 | 16.2260 | 84 | 144 | $\{1,0\}$ | 12 | 4 |

$\mathit{\delta}$ | POA | ${\mathit{P}}_{\mathbf{1}}$’s Payoff | ${\mathit{P}}_{\mathbf{2}}$’s Payoff | s${}_{1}$ | s${}_{2}$ |
---|---|---|---|---|---|

0.1 | 1.087 | 82.5373 | 7.7413 | (1, {1, 0}) | (1, {$-1$, 1}) |

0.3 | 1.022 | 67.7967 | 15.3790 | (54, {1, 0}) | (101, {$-0.2$, 1}) |

0.5 | NON | 29.1317 | 25.2867 | (54, {1, 0}) | (76, { $-0.2$, 1}) |

0.7 | 1.118 | 3.99 | 19.1732 | (76, {1, 0}) | (54, {$-0.2$, 01 }) |

0.9 | 1.059 | 3.4349 | 16.7610 | (76, {1, 0}) | (54, {$-0.2$, 01 }) |

$\mathit{\delta}$ | Payoff | Best Initial Node | Best Message Content | The Number of Infected Nodes |
---|---|---|---|---|

0.1 | 82.5373 | 1, 2, 10 | $\{1,-0.5\}$ | 163 |

0.3 | 64.9419 | 125 | $\{1,0\}$ | 144 |

0.5 | 24.2832 | 76 | $\{1,0\}$ | 84 |

0.7 | 4.6257 | 31 | $\{1,-0.5\}$ | 14 |

0.9 | 3.0649 | 125 | $\{1,-0.5\}$ | 1 |

$\mathit{\delta}$ | Its Payoff | Best Initial Node | Best Message Content | The Number of Infected Nodes |
---|---|---|---|---|

0.1 | 89.2413 | 92, 1, 17 | $\{-0.6,1\}$ | 163 |

0.3 | 72.7808 | 75 | $\{-0.2,1\}$ | 145 |

0.5 | 36.1180 | 75 | $\{-0.2,1\}$ | 85 |

0.7 | 18.0687 | 107 | $\{-0.2,1\}$ | 15 |

0.9 | 16.4780 | 37 | $\{-0.2,1\}$ | 2 |

© 2018 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**

Jafari, S.; Navidi, H. A Game-Theoretic Approach for Modeling Competitive Diffusion over Social Networks. *Games* **2018**, *9*, 8.
https://doi.org/10.3390/g9010008

**AMA Style**

Jafari S, Navidi H. A Game-Theoretic Approach for Modeling Competitive Diffusion over Social Networks. *Games*. 2018; 9(1):8.
https://doi.org/10.3390/g9010008

**Chicago/Turabian Style**

Jafari, Shahla, and Hamidreza Navidi. 2018. "A Game-Theoretic Approach for Modeling Competitive Diffusion over Social Networks" *Games* 9, no. 1: 8.
https://doi.org/10.3390/g9010008