Compliance-and-defiance dilemma game best strategy for three and four agents

Abstract

Dilemma noncooperative games involving three or four agents are studied, where strategic interaction is based on selecting between the compliance strategy and the defiance strategy. When the agent applies the compliance strategy, it increases the agent’s loss amount; contrariwise, when the agent applies the defiance strategy, it does not increase the agent’s loss amount. It is assumed that if only one agent defies, it does not much affect the system, and there are no fines. When two or three agents defy, every agent is fined by the same amount. The objective is to determine and analyze the agent’s best strategy in the dilemma games with three and four agents dealing with the full-defiance situation cost as a variable, where partial-defiance situation cost is deduced from this variable. The agent’s best strategy is determined depending on the full-defiance situation cost. The best strategy for every agent is to defy (by applying the defiance pure strategy) with a probability that ensures the global minimum to the agent’s loss on the probabilistic interval [0; 1] by a given value of the full-defiance situation cost. In the three-agent dilemma game, the best strategy is to fully defy if this cost does not exceed 2/3, whereupon the agent’s loss is equal to the cost. As the cost is increased off value 2/3, the best strategy probability exponentially-like decreases, while the agent’s minimized loss increases in the same manner. Nevertheless, the best strategy ensures the agent’s minimized loss does not exceed 1 (a conditional unit), which is the cost of full compliance. In the four-agent dilemma game, the best strategy is to fully defy if the full-defiance situation cost does not exceed 8/9, whereupon the agent’s loss is equal to the cost. As the cost is increased off value 8/9, the best strategy probability exponentially-like decreases dropping down off value 1/4, which is also the best strategy (being far more favorable for the system) by when the full-defiance situation cost is 8/9. Similarly to the agent’s minimized loss in the three-agent game, the agent’s minimized loss in the four-agent game increases in the same manner never exceeding 1.