Proof systems and transformation games

Yoram Bachrach*, Michael Zuckerman, Michael Wooldridge, Jeffrey S. Rosenschein

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Scopus citations


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.

Original languageAmerican English
Pages (from-to)1-30
Number of pages30
JournalAnnals of Mathematics and Artificial Intelligence
Issue number1
StatePublished - Jan 2013

Bibliographical note

Funding Information:
Acknowledgements This work was partially supported by Israel Science Foundation grant #898/05, by Israel Ministry of Science and Technology grant #3-6797, and by the European Research Council under Advanced Grant 291528 (“RACE”).


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


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