Monotonicity Analysis of the Rank-Shapley Value for Super-Additive Games
Chinonso Michael Eze *
University of Nigeria, Nsukka, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
Changes in a cooperative game data can lead to unparalleled changes in the solution part of the game. Similarly, an alteration in sharing weight can cause some influences especially in a sharing scheme that shares the dividend of a coalition based on a specific weight system. This concept is explored under monotonic solutions of cooperative games. However, there is no established definite pattern of effect on solution as a result of various changes in any definite weight function. Hence, understanding the monotonicity of value functions is crucial in ensuring fair and efficient resource allocation in cooperative systems. This work specifically, explores some patterns of changes in rank as a weight function, and their corresponding effect on Rank-Shapley value of cooperative games. The study of the monotonic property of the value presents a basis to support its applicability. The key result of this work is the solution to the difference equation which provides a positive integer that can ensure a desired change in the payoff of players in a cooperative game. Particularly, the result demonstrates that equal rank adjustments lead to zero-sum payoff changes, preserving Pareto-optimality and efficiency. This result presents significant implications in voting systems, decision-making, and economic resource allocations.
Keywords: Monotonicity, rank, sensitivity, Rank-Shapley value