K-NCC: Stability Against Group Deviations in Non-cooperative Computation

A function is non-cooperative computable [NCC] if honest agents can compute it by reporting truthfully their private inputs, while unilateral deviations by the players are not beneficial: if a deviation from truth revelation can mislead other agents, then the deviator might end up with a wrong result. Previous work provided full characterization of the boolean functions which are non-cooperatively computable. Later work have extended that study in various directions. This paper extends the study of NCC functions to the context of group deviations. A function is K-NCC if deviations by a group of at most K agents is not beneficial: in order to mislead other agents, at least one group member might compute the wrong outcome. A function which is K-NCC for every K is termed strong-NCC. In this paper we provide a full characterization of the K-NCC functions, for every K, and of strong-NCC functions in particular. We show that the hierarchy ofK-NCC functions is strict. Surprisingly, we also show that an anonymous function is NCC iff it is strong-NCC; that is, an anonymous function which is non-cooperatively computable is stable against deviations by any coalition of the agents. In addition, we show that group deviations are stable: if there exists a deviating coalition of minimal size K, then there is no sub-coalition of it which will benefit by further deviation from the original deviating strategy.