The purpose of this study is to present a systematic analysis of the long-term behavior of the agents of an artificial society under varying payoff functions in finite neighborhood binary games. By assuming the linearity of the payoffs of both cooperating and defecting agents, the type of the game is determined by four fundamental parameters. By fixing the values of three of them and systematically varying the fourth one we can observe a transition from Prisoner\'s Dilemma to Leader Game through Chicken and Benevolent Chicken Games. By using agent-based simulation we are able to observe the long-term behavior of the artificial society with different and gradually changing payoff structure. The difference between different games is explored and the effect of the transition from one game to the other on the society is investigated. The results depend on the personality types of the agents. In this study greedy and Pavlovian agents are considered. In the first case, we observe the most significant change in trajectory structure between Prisoner\'s Dilemma and Chicken Games showing significant difference in the behavioral patterns of the agents. Almost no changes can be observed between Benevolent Chicken and Leader Games, and only small change between Chicken and Benevolent Chicken. The trajectories change from always converging to regularly oscillating patterns with systematically altering amplitude and central values. The results are very similar whether the agents consider themselves as members of their neighborhoods or not. With Pavlovian agents no significant difference can be observed between the four games, the trajectories always converge and the limits smoothly and monotonically depend on the value of the varying parameter.
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