## Abstract

A cooperative game (*N*, *v*) 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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This was partially supported by the National Nature Science Foundation of China (No. 11971298).

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### Cite this article

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