A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules
Abstract
:1. Introduction
- This paper puts forward a new evolutionary networked game model called the Mixed Rules NEG. This model can well describe the real situation in which players adopt mixed rules to play with its neighbors in reality.
- We give two strategy updating rules and establish their strict algebraic forms by the method STP. Then, we construct the algebraic expression of the evolutionary process under mixed rules.
- We discuss the Nash Equilibrium under the game.
- A class of networked game’s strategy optimization is discussed. We obtain two results on whether and when the game will achieve an optimal strategy.
2. Preliminaries
2.1. Semi-Tensor Product
- (1)
- is the field of real numbers;
- (2)
- is the n dimensional Euclidean space;
- (3)
- is the set of real matrices;
- (4)
- is the i-th column of matrix M; is the set of columns of M;
- (5)
- is the i-th column of matrix M; is the set of columns of M;
- (6)
- denotes the row stacking form of A;
- (7)
- ;
- (8)
- is called a logical matrix if the columns of L are the form of . That is, Col (L). Denote by the set of logical matrices;
- (9)
- Assume , then , and its shorthand form is ;
- (10)
- is a probabilistic vector, if , and . The set of k dimensional probabilistic vectors is denoted by . The set of probabilistic matrices is denoted by .
- (11)
- .
2.2. Networked Evolutionary Game
- 1.
- denotes the players’ set;
- 2.
- denotes the strategy set;
- 3.
- denotes the payoff matrix, element denotes the payoff of when player 1 takes strategy and player 2 takes strategy .
- A topology about the network, denoted by a undirected graph , where denotes the nodes, denotes the vertex. There is a adjacent matrix corresponding to the graph, if there is a edge between i and j, then , otherwise, . Denote by the neighborhood of player i where if and only if ;
- A fundamental network game G, such that if , then player i and j play FNG with strategies and , respectively;
- A strategy updating rule (SUR) , that is, player i takes the rule to determine the next step’s strategy, i.e.,
- (I)
- (Myopic best response adjustment rule) Each player holds the opinion that its neighbors will make the same decisions as in their last step, and the strategy choice at the present time is the best response against its neighbors’ strategies in the last step. It can be expressed asIf the set is not a singleton, then we choose the minimum, i.e., .
- (II)
- (Unconditional Imitation Rule) Each player adopts the strategy of the player with the largest gain in the neighborhood at the initial moment, i.e.,In addition, we adopt the same priority as rule (I).
- (III)
- (Repeating Rule) Each player will choose the same strategy as the last step, i.e.,
3. Main Result
3.1. Problem Formulation
3.2. Mixed Rules NEG Dynamics
- (A)
- If player i chooses rule (I), we can transform the payoff function into algebraic form by the first rule as followsThe r-th block means player i’s payoff when others’ strategy profile is at time t. We deduce the best response strategies of all the other players by lettingmeans the player i’s best response strategy when others strategy profile . Let , the algebraic forms of strategy dynamics of player i can be expressed as:
- (B)
- If player i chooses rule (II), we need an initial strategy profile, and we assume that player i’s neighbor has the maximum payoff at the initial time, then we can imply the dynamics of player i as:We can find that the dynamics of rule (II) needs the initial strategy profile.
- (C)
- If player i chooses rule (III), its strategy dynamics can be expressed as
3.3. Nash Equilibrium Seeking
Algorithm 1: Seeking of Nash Equilibrium. |
|
3.4. Strategy Consensus Analysis
4. Illustrate Example
- Finite networks: is a connected undirected graph, as in Figure 1a.
- A fundamental networked game: every pair of players play the game G and G’s payoff bi-matrix is shown in Table 1.
- The SUR F is consisted of , which are described in (I)–(III), respectively.
(1, 1) | (0, 1) | |
(1, 0) | (2, 2) |
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
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(1, 1) | (0, 2) | |
(2, 0) | (−1, −1) |
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Gui, Y.; Du, C.; Gao, L. A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules. Mathematics 2022, 10, 3612. https://doi.org/10.3390/math10193612
Gui Y, Du C, Gao L. A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules. Mathematics. 2022; 10(19):3612. https://doi.org/10.3390/math10193612
Chicago/Turabian StyleGui, Yalin, Chengyuan Du, and Lixin Gao. 2022. "A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules" Mathematics 10, no. 19: 3612. https://doi.org/10.3390/math10193612
APA StyleGui, Y., Du, C., & Gao, L. (2022). A Matrix Approach to the Modeling and Analysis of Network Evolutionary Games with Mixed Strategy Updating Rules. Mathematics, 10(19), 3612. https://doi.org/10.3390/math10193612