Effect of emotional migration on Cooperation for public goods games on continuous two-dimensional space

Abstract

Emotion helps people hide from traitors and follow partners. It greatly affects the actual migration in a public goods game (PGG). How this factor affect the migration and the corresponding cooperation level has still not been investigated yet. An emotional migration in a PGG on continuous two-dimensional space was investigated. An emotional index was proposed to describe the different attitude of a player to his different neighbors. An emotional migration model was proposed by defining a weight function that a cooperator goes with a neighbor who makes him happier and gets away from a neighbor who makes him angry. Results show that the emotional migration mechanism acts as a double-edged sword, with its effects intricately intertwined with migratory velocity and interaction radius. When both migratory velocity and interaction radius are low, stable emotional communication gradually solidifies the foundation of trust. This process motivates individuals to actively engage in public goods provision, fostering a virtuous cycle of win-win cooperation. Conversely, when migratory velocity surges and the interaction scope expands exponentially, excessive emotional migration disrupts the trust equilibrium, undermines resource allocation order, and renders cooperative strategies unsustainable. Ultimately, this leads to the collapse of the entire cooperative system.

Introduction

Public goods games are key models for studying human cooperative behavior. They abstract the predicament of collective actions in reality and provide an analytical framework that integrates theory with practice for social interaction research and institutional design. Public goods games can not only reveal the mechanisms of cooperation evolution but also provide innovative policy insights for global issues such as climate change governance and resource allocation^{1, 2, 3, 4–5}. Previous research on public goods games largely relied on the assumption of static networks, in which group relationship networks were assumed to be fixed and unchanging^{6, 7, 8, 9, 10–11}. For example, Wang^{12, 13–14} formulated incentive mechanisms and rules regulating strategy updating for the Prisoner’s Dilemma game, investigated optimal protocols across diverse network structures, and centered on optimization objectives such as cost minimization. Li^{15, 16, 17–18} analyzed the effects of varying conditions on cooperation levels by developing a computational model that integrates factors including the regulation of interaction strength, individual satisfaction, reputation assessment, and the co-evolution of strategies and network structures. However, in the Internet era, individual connections are dynamic and open: participants’ strategies and interaction patterns change in real time in response to environmental changes, others’ behaviors, and individual mobility. Compared with static models, game processes in dynamic network environments can more accurately reflect the evolution of real-world cooperation, offering a more powerful analytical perspective for studying behaviors in complex social systems.

People may migrate for many reasons, such as for better living environment, educational resources or job opportunities, et atc. Many people investigated the effect of migration on cooperative level on different network structures, such as on a square lattice, a complex network, a continuous two-dimensional space, et atc. In terms of a square lattice, Zhang¹⁹ investigated an evolutionary game of players migrating on a square lattice in search of better environments. And he found that the fast migration of defectors significantly strengthens cooperator clusters. Lütz²⁰ investigated individuals migrating on a square lattice and found that a few immigrants can trigger cooperation across the native population. Ren²¹ introduced a neighbor-considered migration strategy on a square lattice and found that it helps cooperative clusters evade the invasion of defectors. Yang²² proposed an aspiration-induced migration and claimed that this migration can greatly expand the cooperator cluster. For the complex structure, Wang²³ explained the effect of node degrees on the evolution of fairness and found that individual migration leads to a rise of acceptance level of populations. Yang²⁴ found that the cooperator cluster is hard to be formed when individuals migrating on a network with a high degree. Dhakal²⁵ investigated the migration of players in a scale-free network and found that the migration induced by climate change improves the cooperation level. In terms of the continuous two-dimensional space, Xiao²⁶ studied the cooperation level of mobile individuals in a two-dimensional public goods game, and claimed that the escape of cooperators away from defectors can greatly improve the cooperation level. Chen²⁷ proposed a payoff-driven migration on continuous space and found that this kind of migration can promote the cooperation. Compared with these three network structures, the migration of players on continuous two-dimensional space is more acceptable and more in line with that in actual situations.

Except for the network structure, the migratory direction is also a hot topic. At the early stage, many previous works investigated the migration with a random direction^{28, 29–30}. This is not consisted with that in reality, because individuals usually migrate with some regular directions. Many people thinks that they migrates in a direction with some rational consideration, such as, environment factors, payoffs, etc. Li³¹ investigatd the success-driven migration mixed with the mean-payoff-driven migration, and found that the mixed migration can effectively eliminate the betrayal clusters. Zhang³² proposed a cooperator-following migration model and found that the appropriate migration speeds promote cooperation. Xiao²⁷ proposed an environment-driven migration according to the differences between the local and global cooperative environments, and found that this migration effectively improves cooperation. Zhang³³ proposed a historical payoff migration model by supposing that individuals decides whether to migrate by comparing the current payoffs with the historical ones obtained in last round. He found that the historical-payoff-dependent migration can greatly improve the cooperation. Funk³⁴ proposed a directed migration model for individuals moving towards higher cooperator densities, and found that population densities rise, and the risk of extinction is reduced in this migration.

The above migrations were conducted under a hypothesis of rational analysis. In fact, emotion can be generated in the game and it also greatly affects migration in reality. Many researchers have proved that emotion affects the decision making in updating their strategies and in contributing their investment. Wang³⁵ proposed an strategy model based on the emotion and found that the emotional decision improve the cooperation level in a PGG. Chen³⁶ proposed an emotional probability for players imitating their neighbors’ behavior and found that a large memory length decreases the cooperation level. Long^37,38 introduced heterogeneous investments induced by emotion and found that emotion can improve cooperation. Except that, emotion also affects the decision making in where to move. As we know, one individual usually likes to follow those who pleased him and deviates from those who disgusted him. Until now, there is few people explaining about this problem. Considering about this, we proposed an emotional migration model and investigated its cooperation level on continuous two-dimensional space.

This paper is organized as follows. The modeling section introduces the emotional migration model for a PGG. The results and discussions section presents simulation results and discussions. The last section draws conclusions.

Model

A two-dimensional continuous planar network is employed. The connectivity of nodes is dynamically determined by their spatial distance, with connection relationships being entirely dependent on the actual spatial positions of the nodes. Within the same network, the connected counterparts of different nodes may vary significantly. Individuals can move continuously across the plane, rendering the topological structures dynamic and non-deterministic, which evolve with node positions or density. The core feature of structured networks resides in “rule-driven static determinism”-the distribution of nodes and their connections are governed by preset topological rules, forming a fixed structure that allows for precise prediction. For individual mobility scenarios, the core requirements are “dynamic adaptation to positional changes” and “compliance with physical proximity constraints.” Notably, the dynamic topology, nearest-neighbor connection rules, and self-organization capabilities of two-dimensional continuous planar networks precisely align with these requirements. In contrast, the fixed topology and preset rules of structured networks struggle to accommodate the uncertainties introduced by mobility. Therefore, two-dimensional continuous planar networks constitute a better choice for individual mobility scenarios. A continuous two-dimensional plane with side length L and a periodic boundary is introduced in this paper. N players are randomly distributed in this plane. For a player i, its position vector is defined as , and its neighbor set within a distance of R could also be defined as,

1

where, is the neighbor set, the position vector, R the interaction radius, t the time. The subscripts (i, j) represent different players.

People produce some different emotional attitudes when playing with different neighbors. As shown in Fig. 1, a cooperator usually shows a pleased mood when playing with a cooperative neighbor, but feels regretful to a defective one. While a defector remains rational to every neighbour because he contributes nothing in this game. In our previous research^30,31, an emotion function was proposed to describe the emotion of a player to his neighbors, as follows,

2

Where, is the emotional index of player i to player j, the emotional increment.

Fig. 1 [Images not available. See PDF.]

Emotion of a player when facing different neighbors.

This generated emotion affects the migratory velocity in reality. As we know, a player is easy to follow in the footstep of a neighbor who pleased him and runs counter to a neighbor who makes him disgust. Here, we define a migratory velocity to describe the location change. A player’s location dynamically updates according to this migratory velocity as,

3

where, is the migratory velocity vector. It includes a velocity magnitude and a direction. We defined that everyone migrates with a same velocity magnitude (v) and a dynamically changing direction (θ).

In reality, a player usually determines his migratory direction based on his neighbors’ directions. Thus, the new migratory direction is a comprehensive superposition of the existing directions of all his neighbors,

4

Where, is the weight of a player i following the migratory direction of player j. This weight is greatly affected by the emotion. Usually, a player is more willing to go with a neighbor who makes him happier. Also, a player goes in the opposite direction of a neighbor who makes him angry. Thus, a weight function could be constructed as follows,

5

If the sum of absolute values of emotional indexes is 0, it means that a defector remains rational and will not go with anyone. That’s to say, a defector keeps still under a rational condition.

At the first time, players are evenly distributed in the plane and play games with their neighbors. Every cooperator contributes a homogeneous investment to every neighbor including himself,

6

Where, is the investment of player i to player j, t the time, n the number in the neighbor set.

Thus, the payoff of player i obtained from player j could be calculated as,

7

Where, u is the payoff, the multiplication factor, s the strategy value which is equal to 1 for a cooperator and 0 for a defector.

Players update their strategies by means of richest-following rule^37,38. That’s to say, people imitate the behavior of a neighbor who obtains the largest payoff. If multiple neighbors get the same largest payoff, he will randomly choose one neighbor.

In this simulation, the side length of a square plane is 10 and the number of players is 1000. Initially, these players keep a rational attitude and take a cooperative behavior or defective behavior with a same probability. Before starting this game, players are randomly distributed in this plane and have a randomly migratory direction, . Firstly, every player tries to look for his neighbors in a radius of R. Also, every player contributes to his neighbors according to his degree in this network. And the corresponding payoff could be calculated according to Eq. (7). Secondly, emotion is generated according to the behavior of these players. And players update their strategies synchronously according to the richest-following rule. Finally, all players migrate according to the migration rules in Eqs. (3)-(5). Up to this point, all players have finished the first-round game, which also means they have updated their locations and strategies. Subsequently, they will play the next game according to their new locations and strategies. To obtain the stable cooperation level, each Monte Carlo simulation is conducted in a sufficiently large steps (1 × 10⁴). The calculated result in last 1000 steps is averaged to be regarded as a stable cooperation level. Moreover, this cooperation level is recalculated and re-averaged over 20 independent trials with random initial conditions.

Results and discussions

Table 1 describes cooperative fractions for different multiplication factors. A small multiplication factor results in no cooperator in PGG because no one is willing to do this little return business. As shown in Fig. 2, the cooperative fraction remains 0 until the multiplication factor exceeds a critical value for 3 at v = 0, and the cooperative fraction slowly increases when the multiplication factor is larger than this value. It could be found that the critical multiplication factor under a migration mode is smaller than that at a stationary state. That’s to say, an emotional migration could break a deadlock that no one wants to invest because players could search their satisfied partners once players can move.

Table 1. Cooperative fractions under different multiplication factors.

Multiplication factor (r0)	Cooperative fraction (fc)
	R = 0.5,v = 0	R = 0.5,v = 0.1	R = 0.5,v = 0.15
1	0	0	0.001
1.5	0	0.008	0.021
2	0	0.02098	0.0213
2.5	0	0.102	0.20844
3	0.008	0.28709	0.3303
3.5	0.67806	0.85624	0.87464
4	0.72627	0.89661	0.92175
4.5	0.81201	0.95449	0.95445
5	0.88	0.98	0.98851
5.5	0.9046	0.98302	0.998
6	0.90453	0.991	0.99748
6.5	0.91	0.997	0.997
7	0.916	0.98398	0.997
7.5	0.913	0.993	0.997
8	0.918	0.98857	0.992

Migratory velocity also affects players in seeing their new partners. Table 2 describes cooperative fractions for different migratory velocities. It could be found that the migratory velocity improves the cooperation fraction firstly and then reduces it slowly as shown in Fig. 3. As we know, a large velocity gives cooperators some power to gather together and keeps away from defectors. A cooperator may quickly find his pleased partners by dynamically changing their location when he has a large velocity. That’s is a good news for players to improve the cooperative level. However, this migratory velocity is not suggested to be too large. A large velocity means an oversensitive reaction to surrounding neighbors. Especially at the early stage when cooperators show a random migratory direction, a larger velocity results in a more chaos migration, which is a bad news for improving the cooperative level. In reality, the investment of PGGs is usually conducted on a favorable geographical location and a stable investment environment. A small migration is benefit for people to pursue a favorable geographical location for building a cooperative team. In other words, a migration can weed out the defectors who only want to reap without sowing in cooperative clusters and can create more shareholder value for cooperators. On the other hand, a large migratory velocity is not allowed because cooperative clusters become more and more unstable. Especially, under a condition of an invasion of defectors, people flee in all directions, resulting in a disintegration of cooperative clusters. That’s to say, only a small migration is suggested to be adopt to get rid of the traitors and to improve the stability of the investment environment.

Fig. 2 [Images not available. See PDF.]

Cooperative fraction versus multiplication factor.

Table 2. Cooperative fractions under different migratory velocities.

Migratory velocity (v)	Cooperative fraction (fc)
	R = 0.3, r0 = 5	R = 0.35, r0 = 5
0	0.75	0.71039
0.05	0.881	0.81802
0.1	0.878	0.84048
0.15	0.887	0.8595
0.2	0.91	0.864
0.25	0.945	0.88151
0.3	0.9467	0.69416
0.35	0.71061	0.519
0.4	0.54351	0.12811
0.45	0.22505	0
0.5	0	0
0.55	0	0
0.6	0	0

Fig. 3 [Images not available. See PDF.]

Cooperative fraction versus migratory velocity.

The interaction radius directly affect the migratory direction. Table 3 describes cooperative fractions under different interaction radius. It could be found that the cooperative fraction firstly increases and then decreases with the interaction radius. As we know, an appropriate increase of interaction radius sharply increases the number of players, which results in a climb of individual income. This is helpful for expanding cooperative clusters. But an oversize radius reduces the cooperative level as shown in Fig. 4. When the radius is too large, too many players participant in the game. It is hard to form a consistent direction for so many players, because players also have many choices to dynamically update their behaviors according to their payoffs. Moreover, some opportunists usually change to be defectors with a minor frequency for a maximum personal interest. These treacheries further exacerbate the inconsistency in the migratory direction. That’s to say, the interaction radius should not to be too large in the actual investment. Don’t put all your eggs in one basket. As we know, a small increase of an interaction radius can diversify players’ investments and can reduce project risks. But people usually has no enough energy to manage the diversified investments when the interaction radius is too large.

Emotion helps people hide from traitors and follow partners. It has distinct characteristics of love and hate in real life, which is just like a double-edged sword. It improves the cooperative level when the migratory velocity and the interaction radius are small but induces the collapse of the cooperative system when the migratory velocity and the interaction radius become large. That’s to say, both the migratory velocity and the interaction radius should not be too large. Otherwise, emotion may play negative effects on the cooperative level in PGGs.

Table 3. Cooperative fractions under different interaction radius.

Interaction radius (R)	Cooperative fraction (fc)
	v = 0.2	v = 0.25
0.1	0.154	0.106
0.2	0.487	0.3383
0.3	0.801	0.709
0.4	0.85105	0.745
0.5	0.915	0.758
0.6	0.29446	0.203
0.7	0.118	0.0205
0.8	0	0
0.9	0	0

Fig. 4 [Images not available. See PDF.]

Cooperative fraction versus interaction radius.

Conclusions

An emotional migration model for public goods games on continuous two-dimensional space was proposed and the emotional effect on the cooperation level of public goods games was investigated. An emotional index was proposed to describe different attitudes of a player to his different neighbors. A weight function was defined by employing this emotional index that a cooperator goes with his neighbor who makes him happy and goes away from a neighbor who makes him angry. The new migratory direction for every player is calculated by using this weight function and the current directions of all his neighbors. Thus, the locations of everyone are dynamically updated by using this new direction and a migratory velocity. It could be found that emotion is just like a double-edged sword in PGGs. It improves the cooperative level when the migratory velocity and the interaction radius are small but induces a collapse of the cooperative system when the migratory velocity and the interaction radius become large. A proper increase of the migratory velocity can quickly gather cooperators together and form a big cooperative cluster with a consistent direction. But an excessive velocity results in an unstable and disordered cluster because cooperators are easy to jump into other clusters, especially when facing with some defective neighbors. A proper increase of the interaction radius can enhance the profit and the cooperative level by increasing the number of players. But this radius should not to be too large because cooperators can hardly form a consistent direction under a huge number of players.

Author contributions

Hui Long was the principal contributor to the study. She made significant contributions to data collection and provided valuable suggestions for model improvement. Youxing Ji conducts formula derivation and theoretical analysis, and he writes the first draft of the article. Min Tan does programming and conducts data analysis.

Data availability

All the data were obtained through computer simulation of the theoretical model. All data generated or analyzed during this study are included in the published article.

Declarations

Competing interests

The authors declare no competing interests.

Publisher’s note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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