Evolutionary game dynamics driven by setting a ceiling in payoffs of defectors

The collective cooperation dilemma, where players are involved in a public goods game and must decide whether to make an individual contribution or not, has gained increasing attention from game theorists and control engineers. Different from the traditional definition, defectors here will face the payoff restriction after adopting the strategy of free riding others' contributions. Specially, we introduce payoff ceilings applied to the gained benefits of the defectors, which leads to an extensive public goods game model. When the number of cooperators is lower than the threshold value, the current model can be transferred to the setups of the general public goods game. Then further increasing the number of cooperators will bring the restricted payoffs for defectors. Analytical results suggest that the payoff ceilings provide us with more insight into the resolution of cooperative dilemma. Particularly, together with other parameters (e.g., the amplification factor and the size of gaming population) of the introduced model, lower values of the payoff ceilings can better facilitate the domination of cooperation strategy at the steady state.