Editing Mechanism design (section)

=== Shapley value ===
{{main|Shapley value}}

Phillips and Marden (2018) proved that for cost-sharing games with concave cost functions, the optimal cost-sharing rule that firstly optimizes the worst-case inefficiencies in a game (the [[price of anarchy]]), and then secondly optimizes the best-case outcomes (the [[price of stability]]), is precisely the Shapley value cost-sharing rule.<ref>{{Cite journal|last1=Phillips|first1=Matthew|last2=Marden|first2=Jason R.|date=July 2018|title=Design Tradeoffs in Concave Cost-Sharing Games|journal=IEEE Transactions on Automatic Control|language=en-US|volume=63|issue=7|pages=2242–2247|doi=10.1109/tac.2017.2765299|issn=0018-9286|s2cid=45923961}}</ref> A symmetrical statement is similarly valid for utility-sharing games with convex utility functions.