The classification of critical points of charged topological black holes (TBHs) in anti-de Sitter spacetime (AdS) under the Power Maxwell Invariant (PMI)-massive gravity is accomplished within the framework of black hole chemistry (BHC). Considering the grand canonical ensemble (GCE), we show that $$d=4$$d = 4 black hole have only one topological class, whereas $$d\ge 5$$d ≥ 5 black holes belong to two different topology classes. Furthermore, the conventional critical point characterized by negative topological charge coincides with the maximum extreme point of temperature; and the novel critical point featuring opposite topological charge corresponds to the minimum extreme point of temperature. With increasing pressure, new phases emerge at the novel critical point while disappear from the conventional one. Moreover, a atypical van der Waals (vdW) behavior is found in $$d\ge 6$$d ≥ 6 dimensions, and the anomaly disappears at the traditional critical point. In the limit of nonlinearity parameter $$s\rightarrow 1$$s → 1 , different topology classes are only obtained in the GCE and they may not exist within the canonical ensemble. With the absence of electric potential $$\Phi $$Φ , the neutral TBHs share the same topological classification results as the charged TBHs in the GCE of Maxwell-massive gravity.

To address issues like the tragedy of the commons, reputation mechanisms prove effective in promoting the emergence of cooperative behavior in social dilemmas. Previous research assumed that the increment of reputation is a fixed constant, however, the historical behavior of individuals with time preferences affects their reputation fluctuations on different scales. Inspired by this, we propose a framework for spatial public goods game that incorporates a reputation discount accumulation model with time preferences. In this model, players are classified as either long-sighted players or short-sighted players based on the significance of their historical behavior. Compared with short-sighted players, long-sighted players are more concerned with the impact of historical behaviors on the next game. Simulation results show that long-sighted cooperative players can resist the invasion of short-sighted defectors, and short-sighted defection strategies are eventually replaced by long-sighted cooperative strategies. This indicates that a higher discount factor facilitates the generation and maintenance of cooperation.

Based on a tripartite game model among suppliers of public goods, consumers, and the government, a tripartite repeated game model is constructed to analyze the evolution mechanism of which suppliers supply at low prices, consumers purchase, and the government provides incentives, and to establish the dynamics system of a repeated game. The equilibrium points of the evolutionary game are solved, and among them, the equilibrium points are found to satisfy the parameter conditions of ESS. The numerical simulation is employed to verify the impact of penalty coefficients and discount factors on the stability of strategies, which are adopted by the three players in a tripartite repeated game on public goods, and scenario analyses are conducted. The research results of this paper could provide a reference for the government, suppliers, and consumers to make rapid decisions, who are in the supply chain of public goods, especially quasi-public goods, such as coal, water, electricity, and gas, and help them to obtain stable incomes and then ensure the stable operation of the market.