Marginal contributions and derivatives for set functions in cooperative games

A cooperative game (N, v) is said to be monotone if $$v(S)\ge v(T)$$v(S)≥v(T) for all $$T\subseteq S\subseteq N$$T⊆S⊆N, and k-monotone for $$k\ge 2$$k≥2 if $$v(\cup _{i=1}^k S_i)\ge \sum _{I:\,\emptyset \ne I\subseteq \{1,\ldots , k\}} (-1)^{|I|-1} v(\cap _{i\in I} S_i)$$v(∪i=1kSi)≥∑I:∅≠I⊆{1,…,k}(-1)|I|-1v(∩i∈ISi) for all k subsets $$S_1,\ldots ,S_k$$S1,…,Sk of N. Call a set function v totally monotone if it is monotone and k-monotone for all $$k\ge 2$$k≥2. To generalize both of marginal contribution and Harsanyi dividend, we define derivatives of v as $$v^{(0)}=v$$v(0)=v and for pairwise disjoint subsets $$R_1,\dots ,R_k$$R1,⋯,Rk of N, $$v'_{R_1}(S)=v(S\cup R_1)-v(S)$$vR1′(S)=v(S∪R1)-v(S) for $$S\subseteq N\setminus R_1$$S⊆N\R1, and $$v^{(k)}_{R1,\dots ,R_k}(S)=(v^{(k-1)}_{R_1,\dots ,R_{k-1}})'_{R_k}(S)$$vR1,⋯,Rk(k)(S)=(vR1,⋯,Rk-1(k-1))Rk′(S) for $$S\subseteq N\setminus \cup _{i=1}^k R_i$$S⊆N\∪i=1kRi. 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.