The Shapley value in the non differentiate case

@article{Mertens1988TheSV,
  title={The Shapley value in the non differentiate case},
  author={Jean-François Mertens},
  journal={International Journal of Game Theory},
  year={1988},
  volume={17},
  pages={1-65}
}
  • J. Mertens
  • Published 1988
  • Economics
  • International Journal of Game Theory
The Shapley value is shown to exist even when there are essential non differentiabilities on the diagonal. 

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We prove the existence of a (unique) Aumann-Shapley value on the space on non-atomic gamesQn generated byn-handed glove games. (These are the minima ofn non-atomic mutually singular probability

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The idea of “marginal contribution” is best captured in the game theoretic concept of value. The relation between it and the usual economic equilibrium can be stated as the following Value Principle:

Values of Non-Atomic Games

The "Shapley value" of a finite multi- person game associates to each player the amount he should be willing to pay to participate. This book extends the value concept to certain classes of

Values and Derivatives

TLDR
An averaging process is used to reinterpret and then prove the diagonal formula for much larger spaces of games, including spaces in which the games cannot be considered differentiable and may even have jumps (e.g., voting games).