# Computational Behavioral Models for Public Goods Games on Social Networks

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## Abstract

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## 1. Introduction

_{2}emissions. In game theory, this kind of collective-action situation is modeled through public goods games (PGG) [1,2,3]. PGGs are multi-player games in which N agents have the choice to voluntarily contribute to a common pool that will be enjoyed by all and will have an added value for each agent through a multiplication factor $1<r<N$, i.e., before sharing the total contribution among the players, the quantity contributed in the common pool is multiplied by r. This represents the fact that the realization of the public good has an added value for the whole community. In laboratory work, the surplus is provided by the experimenter. The temptation for an agent is to free-ride on the contribution of others and give little or nothing to the common pool. Of course, if enough agents think in this way the public good will not be realized. Indeed, according to game theory based on rational optimizing agents, the dominant individual strategy in not to contribute and the unique Nash equilibrium corresponds to zero contribution from each agent implying no public good provision at all.

## 2. Payoff Satisfaction Model

## 3. Results

## 4. Group Average-Based Model

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**In the example, the node $\alpha $ has four direct neighbors ($\beta ,\gamma ,\delta ,\u03f5$) and it thus participates in five PGGs, i.e. the circle centered in $\alpha $ involving $\{\alpha ,\beta ,\gamma ,\delta ,\u03f5\}$, and four other PGGs represented by circles centered in its neighbors: $\{\alpha ,\beta \},\{\alpha ,\gamma ,\delta \},\{\alpha ,\gamma ,\delta ,\u03f5\},\{\alpha ,\delta ,\u03f5,\zeta ,\eta \}$. Note that a node may indirectly interact with other nodes that are not first neighbors.

**Figure 2.**Average individual contribution as a function of round number for several values of the enhancement factor r as shown in the inset. The average is taken over 40 independent repetitions of 40 rounds each on a network of size 500 and $\alpha =0.3$.

**Figure 3.**Results of the experimental study by Burton-Chellew and West; we refer the reader to [6] for more detailed information.

**Figure 4.**Average contribution from simulation of a small network of 20 agents as a function of the round number for the r values shown in the inset. Averages are taken over 20 independent repetitions of the simulation.

**Figure 5.**Average contribution as a function of round number for several values of the enhancement factor r as shown in the inset: (

**a**) $\alpha =0$; and (

**b**) $\alpha =1$. The average is over 40 independent repetitions of 40 rounds each on a network of size 500.

**Figure 6.**Group size distribution for an instance of a network of 500 nodes having an average degree equals to 6.

**Figure 7.**Average contribution as a function of round number for several values of the enhancement factor r as shown in the inset: (

**a**) mean degree four; and (

**b**) mean degree twelve. The average is over 40 independent repetitions of 40 rounds each on a network of size 500.

**Figure 8.**Average contribution during time in five independent simulations in which agents are trend-following according to the rules of the group average-based model (see Table 2).

**Figure 9.**Average contribution as a function of round number for several values of the enhancement factor r as shown in the inset for the mixed model. The average is over 40 independent repetitions of 40 rounds each on a network of size 500.

**Table 1.**In the payoff satisfaction model, the agent decreases its contribution fraction to PGG when the received payoff ${\pi}_{i}$ is less than the contributed amount ${c}_{i}$. Otherwise, it increases its contribution or keeps the previous one with uniform probability. Here, R represents a uniform randomly generated number in [0,1].

Payoff Satisfaction Model | ||||||
---|---|---|---|---|---|---|

Previous contribution fraction (${c}_{i}$) | 0 | 0.25 | 0.5 | 0.75 | 1.0 | |

$\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{\pi}_{i}\ge {c}_{i}$ | $R<0.5$ | = | = | = | = | = |

$R\ge 0.5$ | + | + | + | + | = | |

${\pi}_{i}<{c}_{i}$ | = | − | − | − | − |

**Table 2.**In the payoff group average model, the agent decreases (increases) its contribution when the latter is larger (smaller) than the average contribution of the group. It remains the same when these two quantities are equal. The amount of decrease (increase) is a random value chosen in $\{0.1,0.2,0.3\}$.

Group Average-Based Model | ||||||
---|---|---|---|---|---|---|

Previous contribution fraction (${c}_{i}$) | 0 | 0.25 | 0.5 | 0.75 | 1.0 | |

${c}_{i}>\langle {c}_{N}\rangle $ | = | − | − | − | − | |

${c}_{i}<\langle {c}_{N}\rangle $ | + | + | + | + | = | |

${c}_{i}=\langle {c}_{N}\rangle $ | = | = | = | = | = |

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Tomassini, M.; Antonioni, A. Computational Behavioral Models for Public Goods Games on Social Networks. *Games* **2019**, *10*, 35.
https://doi.org/10.3390/g10030035

**AMA Style**

Tomassini M, Antonioni A. Computational Behavioral Models for Public Goods Games on Social Networks. *Games*. 2019; 10(3):35.
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**Chicago/Turabian Style**

Tomassini, Marco, and Alberto Antonioni. 2019. "Computational Behavioral Models for Public Goods Games on Social Networks" *Games* 10, no. 3: 35.
https://doi.org/10.3390/g10030035