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Marginal contributions and derivatives for set functions in cooperative games

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Abstract

A cooperative game (Nv) is said to be monotone if \(v(S)\ge v(T)\) for all \(T\subseteq S\subseteq N\), and k-monotone for \(k\ge 2\) if \(v(\cup _{i=1}^k S_i)\ge \sum _{I:\,\emptyset \ne I\subseteq \{1,\ldots , k\}} (-1)^{|I|-1} v(\cap _{i\in I} S_i)\) for all k subsets \(S_1,\ldots ,S_k\) of N. Call a set function v totally monotone if it is monotone and k-monotone for all \(k\ge 2\). To generalize both of marginal contribution and Harsanyi dividend, we define derivatives of v as \(v^{(0)}=v\) and for pairwise disjoint subsets \(R_1,\dots ,R_k\) of N, \(v'_{R_1}(S)=v(S\cup R_1)-v(S)\) for \(S\subseteq N\setminus R_1\), and \(v^{(k)}_{R1,\dots ,R_k}(S)=(v^{(k-1)}_{R_1,\dots ,R_{k-1}})'_{R_k}(S)\) for \(S\subseteq N\setminus \cup _{i=1}^k R_i\). We generalize the equivalence between convexity and monotonicity of marginal contribution of v to total monotonicity and higher derivatives of v from several aspects. We also give the Taylor expansion of any game (set function) v.

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Correspondence to Erfang Shan.

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This was partially supported by the National Nature Science Foundation of China (No. 11971298).

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Li, D.L., Shan, E. Marginal contributions and derivatives for set functions in cooperative games. J Comb Optim (2020) doi:10.1007/s10878-020-00526-y

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Keywords

  • TU-game
  • Total monotonicity
  • Hansaryi dividend
  • Marginal contribution
  • Higher derivative

Mathematics Subject Classification

  • 91A12

JEL Classification

  • C71