Amplitude of weighted representation of voting games with several levels of approval

Abstract

In this paper, we evaluate in the context of weighted (j, k)-simple games, the maximal degree of perturbations which may be allowed, in voters weights and/or in the quotas, without changing the structure of the game. For this purpose, we extend on (j, k)-simple games the notion of amplitude well known for ordinary simple games. Recall that, (j, k)-simple games provide a model of decision making in which each voter has j levels of approval (inputs), while k levels of approval are permitted as collective decision (outputs). Here, the j inputs are qualitatively ordered, same are the k outputs. Ordinary simple games correspond to the particular case $j=k=2$. Our results generalize those obtained by Freixas and Puente (Qüestiió 23(1):43–60, 1999) on ordinary simple games. We illustrate by computing the amplitude of some real world examples like the United Nations Security Council which is a (3, 2)-simple game.