The Shapley value in the non differentiable case (Q1115363)

The paper extends the so-called diagonal formula for the Shapley value on spaces of non atomic games like pNA and bv'NA to a diagonal formula for a value on a much wider class of games, including majority games and n- handed glove markets. For instance, if the worth v(S) of coalition S is expressed as a function of finitely many non atomic probabilities \(\mu_ 1,\mu_ 2,...,\mu_ n\) by \(v(S)=f(\mu_ 1(S),...,\mu_ n(S))\), \(f\in C^ 1\), \(f(0)=0\), then the diagonal formula for the Shapley value \(\Phi\) takes the form \[ [\Phi (v)](S)=\Sigma \mu_ i(S)\int^{1}_{0}\frac{\partial f}{\partial x_ i}(t,t,...,t)dt. \] The diagonal formula considered depends on the differentiability of the function f along the diagonal. In the paper the diagonal formula is extended by taking the derivative not on the diagonal, but at some small perturbation of it and by averaging the result for some probability distribution over perturbations. A weak form of uniqueness is also proved, in the sense that there is only one such probability distribution over perturbations that would yield a value. An explicit formula for the value of majority games and n-handed glove markets is given.