Applicability of the Analytical Solution to N-Person Social Dilemma Games - Merlone, Ugo; Sandbank, Daren R.; Szidarovszky, Ferenc (Frontiers in Applied Mathematics and Statistics, 2018)
The purpose of this study is to present an analysis of the applicability of an analytical solution to the $N-$person social dilemma game. Such solution has been earlier developed for Pavlovian agents in a cellular automaton environment with linear payoff functions and also been verified using agent based simulation. However no discussion has been offered for the applicability of this result in all Prisoners 'Dilemma game scenarios or in other $N-$person social dilemma games such as Chicken or Stag Hunt. In this paper it is shown that the analytical solution works in all social games where the linear payoff functions are such that each agent's cooperating probability fluctuates around the analytical solution without cooperating or defecting with certainty. The social game regions where this determination holds are explored by varying payoff function parameters. It is found by both simulation and a special method that the analytical solution applies best in Chicken when the payoff parameter $S$ is slightly negative and then the analytical solution slowly degrades as $S$ becomes more negative. It turns out that the analytical solution is only a good estimate for Prisoners' Dilemma games and again becomes worse as $S$ becomes more negative. A sensitivity analysis is performed to determine the impact of different initial cooperating probabilities, learning factors, and neighborhood size.