Annals of Mathematics and Artificial Intelligence

, Volume 67, Issue 1, pp 1-30

First online:

Proof systems and transformation games

  • Yoram BachrachAffiliated withMicrosoft Research Email author 
  • , Michael ZuckermanAffiliated withHebrew University
  • , Michael WooldridgeAffiliated withUniversity of Oxford
  • , Jeffrey S. RosenscheinAffiliated withHebrew University

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We introduce Transformation Games (TGs), a form of coalitional game in which players are endowed with sets of initial resources, and have capabilities allowing them to derive certain output resources, given certain input resources. The aim of a TG is to generate a particular target resource; players achieve this by forming a coalition capable of performing a sequence of transformations from a combined set of initial resources to the target resource. TGs can model a number of natural settings, such as cooperative proof systems, where a collection of agents having different expertise work together to derive a proof for a target theorem, or supply chains, where agents cooperate to create a target product from base resources. After presenting the TG model, and discussing its interpretation, we consider possible restrictions on the transformation chain, resulting in different coalitional games. Following the basic model, we consider the computational complexity of several problems in TGs, such as testing whether a coalition wins, checking if a player is a dummy or a veto player, computing the core of the game, computing power indices, and checking the effects of possible restrictions on the coalition. Finally, we consider extensions to the model in which transformations have associated costs.


Cooperative game theory Supply chains The core The Shapley value Power indices

Mathematics Subject Classifications (2010)

91A12 68Q25