Evaluating Mechanism and Related Axiomatic Results under Multiple Considerations

Under many interactive environments in the real world, there is often a need to evaluate the minimization effects and subsequent allocation outcomes derived from these interactions under multiple considerations. For instance, in the context of product sales, it is necessary to evaluate how to minimize the manufacturing costs of various producing factors, and sometimes, from a holistic perspective, it may even be necessary to evaluate situations with minimal sales benefits. On the other hand, in order to evaluate related effects derived from interactions and subsequent allocation outcomes, many game-theoretical studies are based on interactive models to formulate evaluating mechanisms, and then they apply axiomatic processes to analyze the rationality of these mechanisms. Therefore, this study first proposes a mechanism for evaluating the minimization effects and subsequent allocation outcomes under multiple considerations. Additionally, considering that different environmental impacts result from varying participation factors, this study also presents several weighted derivatives based on participation factors and their behaviors. Concurrently, we utilize axiomatic results to demonstrate the mathematical correctness and practicality for these evaluating mechanisms.