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Proof systems and transformation games

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Abstract

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.

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Authors and Affiliations

  1. Microsoft Research, Cambridge, UK

    Yoram Bachrach

  2. Hebrew University, Jerusalem, Israel

    Michael Zuckerman & Jeffrey S. Rosenschein

  3. University of Oxford, Oxford, UK

    Michael Wooldridge

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  1. Yoram Bachrach
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  2. Michael Zuckerman
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  3. Michael Wooldridge
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  4. Jeffrey S. Rosenschein
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Correspondence to Yoram Bachrach.

Additional information

This is an extended version of a conference paper with the same title [21], which was presented at the 35th International Symposium on Mathematical Foundations of Computer Science (MFCS 2010), in Brno, Czech Republic. This new version contains additional results regarding computing the Shapley value in Transformation Games as well as additional results on Transformation Games with a bounded number of resources and transformations. We have also expanded our examination of how the Transformation Game model can capture multiagent interactions in both supply chains and proof systems, and included revised and extended proofs.

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Bachrach, Y., Zuckerman, M., Wooldridge, M. et al. Proof systems and transformation games. Ann Math Artif Intell 67, 1–30 (2013). https://doi.org/10.1007/s10472-012-9323-9

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