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A PARAMETRIC, RESOURCE-BOUNDED GENERALIZATION OF LÖB’S THEOREM, AND A ROBUST COOPERATION CRITERION FOR OPEN-SOURCE GAME THEORY

Published online by Cambridge University Press:  02 April 2019

ANDREW CRITCH
ANDREW CRITCH*
Affiliation:
MACHINE INTELLIGENCE RESEARCH INSTITUTE 2030 ADDISON STREET, BERKELEY, CA94704, USA E-mail: [email protected]: http://intelligence.org/
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Abstract

This article presents two theorems: (1) a generalization of Löb’s Theorem that applies to formal proof systems operating with bounded computational resources, such as formal verification software or theorem provers, and (2) a theorem on the robust cooperation of agents that employ proofs about one another’s source code as unexploitable criteria for cooperation. The latter illustrates a capacity for outperforming classical Nash equilibria and correlated equilibria, attaining mutually cooperative program equilibrium in the Prisoner’s Dilemma while remaining unexploitable, i.e., sometimes achieving the outcome (Cooperate, Cooperate), and never receiving the outcome (Cooperate, Defect) as player 1.

Keywords

03B7003B4503F0368T2791A3562C9991A0591A0691A10Löbian cooperationbounded rationalityprogram equilibriumproof lengthPrisoner’s Dilemma
Type
Articles
Information
The Journal of Symbolic Logic , Volume 84 , Issue 4 , December 2019 , pp. 1368 - 1381
Copyright
Copyright © The Association for Symbolic Logic 2019 

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References

REFERENCES

Bárász, M., Christiano, P., Fallenstein, B., Herreshoff, M., LaVictoire, P., and Yudkowsky, E., Robust cooperation in the prisoner’s dilemma, arXiv preprint, 2014, arXiv:1401.5577.Google Scholar
Boolos, G., The Logic of Provability, Cambridge University Press, New York, 1993.Google Scholar
Cori, R. and Lascar, D., Mathematical Logic: A Course with Exercises, Part II, Oxford University Press, New York, 2001.Google Scholar
Fortnow, L., Program equilibria and discounted computation time, TARK ’09: 12th Conference on Theoretical Aspects of Rationality and Knowledge, ACM Press, 2009, pp. 128–133.CrossRefGoogle Scholar
LaVictoire, P., Fallenstein, B., Yudkowsky, E., Barasz, M., Christiano, P., and Herreshoff, M., Program equilibrium in the prisoner’s dilemma via Löb’s theorem, Multiagent Interaction without Prior Coordination: Papers from the AAAI-14 Workshop, AAAI Publications, 2014.Google Scholar
Megill, N., Metamath: A Computer Language for Pure Mathematics, Lulu Press, Morrisville, NC, 2007.Google Scholar
Pudlák, P., On the lengths of proofs of finistic consistency statements in first order theories, Logic Colloquium ’84 (Paris, J. B., Wilkie, A. J., and Wilmers, G. M., editors), North-Holland, Amsterdam, 1986, pp. 165–196.CrossRefGoogle Scholar
Pudlák, P., The lengths of proofs, Handbook of Proof Theory (Buss, S. R., editor), Studies in Logic and the Foundations of Mathematics, vol. 137, Elsevier, Amsterdam, 1998, pp. 547–637.CrossRefGoogle Scholar
Tarau, P., Bijective size-proportionate Gödel numberings for term algebras, unpublished manuscript, 2013.Google Scholar
Tennenholtz, M., Program equilibrium. Games and Economic Behavior, vol. 49 (2004), no. 2, pp. 363–373.CrossRefGoogle Scholar
Tsai, S.-C., Chang, J.-C., and Chen, R.-J., A space-efficient Gödel numbering with Chinese remainder theorem, Proceedings of the 19th Workshop on Combinatorial Mathematics and Computation Theory, 2002, pp. 192–195.Google Scholar