The basic social dilemma is frequently captured by a public goods game where participants decide simultaneously whether to support a common pool or not and after the enhanced contributions are distributed uniformly among all competitors. What if the result of common efforts is not distributed immediately, but it is reinvested and added to the pool for a next round? This extension may not only result in an enhanced benefit for group members but also opens new strategies for involved players because they may act in distinct rounds differently. In this work we focus on the simplest case when two rounds are considered, but the applied multiplication factors dedicated to a certain round can be different. We show that in structured populations the winning strategy may depend sensitively on the ratio of these factors and the last round has a special importance to reach a fully cooperative state. We also observe that it may pay for defectors to support the first round and after enjoy the extra benefit of accumulated contributions. Full cooperator strategy is only viable if the second round ensures a premium benefit of investments. 

Tactical cooperation of defectors in a multi-stage public goods game

Involution game with spatio-temporal heterogeneity of social resources

The conflict between individual and collective interests is in the heart of every social dilemmas established by evolutionary game theory. We cannot avoid these conflicts but sometimes we may choose which interaction framework to use as a battlefield. For instance some people like to be part of a larger group while other persons prefer to interact in a more personalized, individual way. Both attitudes can be formulated via appropriately chosen traditional games. In particular, the prisoner's dilemma game is based on pair interaction while the public goods game represents multi-point interactions of group members. To reveal the possible advantage of a certain attitude we extend these models by allowing players not simply to change their strategies but also let them to vary their attitudes for a higher individual income. We show that both attitudes could be the winner at a specific parameter value. Interestingly, however, the subtle interplay between different states may result in a counterintuitive evolutionary outcome where the increase of the multiplication factor of public goods game drives the population to a fully defector state. We point out that the accompanying pattern formation can only be understood via the multipoint or multi-player interactions of different microscopic states where the vicinity of a particular state may influence the relation of two other competitors.

/pdf/cooperation-and-competition-between-pair-and-multi-player-3v6rxs6c75.pdf

Cooperation and competition between pair and multi-player social games in spatial populations

The basic social dilemma is frequently captured by a public goods game where participants decide simultaneously whether to support a common pool or not and after the enhanced contributions are distributed uniformly among all competitors. What if the result of common efforts is {\it not} distributed immediately, but it is reinvested and added to the pool for a next round? This extension may not only result in an enhanced benefit for group members but also opens new strategies for involved players because they may act in distinct rounds differently. In this work we focus on the simplest case when two rounds are considered, but the applied multiplication factors dedicated to a certain round can be different. We show that in structured populations the winning strategy may depend sensitively on the ratio of these factors and the last round has a special importance to reach a fully cooperative state. We also observe that it may pay for defectors to support the first round and after enjoy the extra benefit of accumulated contributions. Full cooperator strategy is only viable if the second round ensures a premium benefit of investments.

Tactical cooperation of defectors in a multi-stage public goods game.

Taxes are an essential and uniformly applied institution for maintaining modern societies. However, the levels of taxation remain an intensive debate topic among citizens. If each citizen contributes to common goals, a minimal tax would be sufficient to cover common expenses. However, this is only achievable at high cooperation level; hence, a larger tax bracket is required. A recent study demonstrated that if an appropriate tax partially covers the punishment of defectors, cooperation can be maintained above a critical level of the multiplication factor, characterizing the synergistic effect of common ventures. Motivated by real-life experiences, we revisited this model by assuming an interactive structure among competitors. All other model elements, including the key parameters characterizing the cost of punishment, fines, and tax level, remain unchanged. The aim was to determine how the spatiality of a population influences the competition of strategies when punishment is partly based on a uniform tax paid by all participants. This extension results in a more subtle system behavior in which different ways of coexistence can be observed, including dynamic pattern formation owing to cyclic dominance among competing strategies. 

Supporting punishment via taxation in a structured population

We study the evolution of cooperation on a two-dimension square lattice where local common resources are considered changeable depending on the actions of individuals. We build a simple modified model by introducing an exogenous variable on the basis of the traditional public goods game with two classical pure strategies of cooperation (C) and defection (D). In this approach, cooperation can be significantly promoted than that in the traditional model. We make a statistical analysis and discuss the conditions for the emergence and development of cooperation on lattices. We intuitively show in detail the evolutionary process of cooperation in infinite populations. However, it becomes stricter for cooperation to emerge and maintain than in well-mixed populations. Furthermore, we find that defectors still survive even though cooperators can possibly obtain the highest payoffs in a single game. The modified model can be applied in some scenarios with environmental feedback which expands the application of the traditional public goods game.

Environmental feedback in spatial public goods game

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Environmental feedback in spatial public goods game