# The Evolution of Cooperation in Multigames with Uniform Random Hypergraphs

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## Abstract

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## 1. Introduction

## 2. Model

#### 2.1. Hypergraph Structure

**Uniform random hypergraph:**G-order uniform random hypergraphs differ from general hypergraphs in that each hyperedge contains a fixed number of G nodes, G is the order of the URH, and the generation of hyperedges is random, with each hyperedge having an equal probability of being selected. To generate the URH, we utilized Wang’s proposed method [41], which involves constructing a tree structure for selecting nodes and generating a random number to determine a hyperedge. This method is flexible as it considers all node combinations and has high computational efficiency. Figure 1 shows a schematic diagram of a $G=3$ URH, indicating that each hyperlink in the hypergraph contains 3 nodes.

#### 2.2. Evolutionary Multigames

**Payoff matrix:**In a network, players are provided with a set of strategies, denoted by s, that includes two options: cooperation (C) and defection (D). At the start of the evolution, players are equally likely to select any of the options. As the evolution proceeds, these players participate in games based on their chosen strategies. If both players opt for cooperation, they both receive a reward, denoted by R. Conversely, if both players choose to defect, they both receive punishment, represented as P. In the case where a cooperator interacts with a defector, the defector gets a temptation payoff of T, while the cooperator receives a sucker’s payoff of S.

**Payoff function:**To simulate evolutionary dynamics in our model, we use Monte Carlo simulations. During a paired game interaction, each player plays a game with every player in each hyperlink where they exist, based on their perception mechanism. Afterward, the resulting payoffs are aggregated and averaged to obtain the normalized payoff ${P}_{i}$ for each player.

**Strategy update mechanism:**During the evolution process, rational players often adopt the strategy of their higher-earning neighbors, with a certain probability, to increase their own payoffs. If neighbor j has a different strategy from player i, the probability of player i adopting the strategy of neighbor j can be determined using the Fermi function [39]:

Algorithm 1 Monte Carlo simulation |

## 3. Results and Discussion

#### 3.1. The Impact of Multigame Matrix Differences on the Evolution of Cooperation in Hypergraphs

#### 3.2. The Impact of Network Structure on the Evolution of Cooperation in Hypergraphs

#### 3.2.1. The Influence of G and L on the Fraction of Cooperators

#### 3.2.2. Exploring the Impact of Hyperdegree on the Evolution of Cooperation

#### 3.2.3. Microcosmic Analysis of Node Evolution with Varied Hyperdegrees

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**A uniform random hypergraph for $G=3$. In this graph, the colored circles denote hyperedges in the hypergraph, while the dots and numbers represent nodes and their assigned numbers, respectively.

**Figure 2.**How the parameter $\theta $ affects the fraction of cooperators ($fc$). The parameter $\theta $ represents the difference in the payoff matrix of two players with different perceptions. The horizontal axis represents $\theta $, while the vertical axis displays the fraction of cooperators. Three curves are plotted, corresponding to PD players, SD players, and all players. The data is presented for two hypergraphs: $G=3$ and $G=7$, which are shown in (

**a**,

**b**). The parameter G represents the order of the hypergraph.

**Figure 3.**The heatmap illustrates the observed fraction of cooperation in the G-L parameter space of the hypergraph. The color intensity in the graph corresponds to the fraction of cooperators, with darker shades indicating higher values of $fc$. These results are obtained under the conditions of $b=1.2$ and $\theta =0.4$. The parameter b represents the temptation payoff of the defector, and $\theta $ represents the difference in the payoff matrix of two players with different perceptions.

**Figure 4.**The depicted figure illustrates the fraction of cooperators among nodes with distinct hyperdegrees. The x-axis denotes the hyperdegree, while the y-axis indicates the $fc$ for nodes with that hyperdegree. The results of the three hypergraphs ($G=3,5,7$) are illustrated by three distinct curves, which uncover the correlation between node hyperdegree and cooperative behavior. Different values of G represent hypergraphs with different orders.

**Figure 5.**How node influence changes over time for both low-hyperdegree and high-hyperdegree groups, represented by (

**a**,

**b**), respectively. Two curves, each with a different color, show the influence of defectors and cooperators within the group. The horizontal axis represents time t, while the vertical axis represents influence, which indicates the probability of neighboring nodes adopting the group’s strategy.

**Figure 6.**An abstract evolution process, highlighting the co-evolutionary relationship among high-hyperdegree nodes. The colored dots represent nodes within the network, while the black circles signify hyperedges.

**Figure 7.**The abstract evolution of low-hyperdegree nodes in a network, with colored dots representing the nodes and black circles representing hyperedges. The arrows indicate possible pathways of evolution. Fork generation groups face an evolutionary pitfall, as they are all defectors and cannot access cooperative information.

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**MDPI and ACS Style**

Xu, H.; Zhang, Y.; Jin, X.; Wang, J.; Wang, Z.
The Evolution of Cooperation in Multigames with Uniform Random Hypergraphs. *Mathematics* **2023**, *11*, 2409.
https://doi.org/10.3390/math11112409

**AMA Style**

Xu H, Zhang Y, Jin X, Wang J, Wang Z.
The Evolution of Cooperation in Multigames with Uniform Random Hypergraphs. *Mathematics*. 2023; 11(11):2409.
https://doi.org/10.3390/math11112409

**Chicago/Turabian Style**

Xu, Haozheng, Yiwen Zhang, Xing Jin, Jingrui Wang, and Zhen Wang.
2023. "The Evolution of Cooperation in Multigames with Uniform Random Hypergraphs" *Mathematics* 11, no. 11: 2409.
https://doi.org/10.3390/math11112409