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<p>The desirability relation introduced by Isbell (Duke Math J 25(3):423–439, 1958) in simple games compares players according to their degrees of influence in the game. This relation has been shown to be a preorder on the set of players and a subpreorder of the preorders induced by the well-known Shapley–Shubik and Banzhaf–Coleman power indices. Many extensions of this relation to more general classes of voting games are available in the literature (Tchantcho et al. in Games Econ Behav 64(1):335–350, 2008; Pongou et al. in Oper Res Perspect 1(1):42–49, 2014; Freixas et al. in Decis Support Syst 57:172–177, 2014b and others). However, none of these extensions preserve the above properties. Thus, we consistently extend the desirability relation to games with several levels of approval in terms of input and output. Our extension is not only a preorder but also a subpreorder of the preorders induced by the Shapley–Shubik and Banzhaf–Coleman power indices. A complete study of the achievable hierarchies of this new desirability relation is conducted.</p>

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Consistent desirability relation in games with multiple levels of approval and achievable hierarchies

  • 期刊论文

  • 发表日期：

    2025年11月11日

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  • Joseph Siani,
  • Narcisse Tedjeugang,
  • Bertrand Tchantcho

摘要

The desirability relation introduced by Isbell (Duke Math J 25(3):423–439, 1958) in simple games compares players according to their degrees of influence in the game. This relation has been shown to be a preorder on the set of players and a subpreorder of the preorders induced by the well-known Shapley–Shubik and Banzhaf–Coleman power indices. Many extensions of this relation to more general classes of voting games are available in the literature (Tchantcho et al. in Games Econ Behav 64(1):335–350, 2008; Pongou et al. in Oper Res Perspect 1(1):42–49, 2014; Freixas et al. in Decis Support Syst 57:172–177, 2014b and others). However, none of these extensions preserve the above properties. Thus, we consistently extend the desirability relation to games with several levels of approval in terms of input and output. Our extension is not only a preorder but also a subpreorder of the preorders induced by the Shapley–Shubik and Banzhaf–Coleman power indices. A complete study of the achievable hierarchies of this new desirability relation is conducted.