# Research

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### Selected papers

• Toward negotiable reinforcement learning: shifting priorities in Pareto optimal sequential decision-making
(2017, under review)
• Description
• Exhibits a recursive formula for characterizing how a Pareto-optimal policy, as judged by players with common knowledge of differing beliefs and values, must provably obey over time.
• BibTeX
@misc{Cri17,
author="Andrew Critch",
title = "{Toward negotiable reinforcement learning: shifting priorities in Pareto optimal sequential decision-making}",
year = {2017},
howpublished = "{arXiv:1701.01302}",
}

• latest pdf
• arxiv
• Logical Inductors
(2016, under review)
• Description
• Presents an algorithm which assigns probabilities to mathematical statements using Brouwer’s fixed point theorem, in a well-calibrated, self-consistent manner, long before proofs verifying or refuting the statements can be found. The algorithm predicts the outputs of computations in a well-calibrated way, including its own outputs. It is also has reasonable beliefs about itself (as a mathematical object), including accurate beliefs about its own approximate consistency, and beliefs that its future beliefs are more accurate than its current beliefs. All these properties follow from a single, elegant criterion, which makes them highly natural as criteria for “good reasoning” for a bounded reasoner thinking about mathematics, and in particular, computations.
• BibTeX
@misc{Gar16,
author="Scott Garrabrant, Tsvi Benson Tilsen, Andrew Critch, Nate Soares, Jessica Taylor",
title = "{Logical Induction}",
year = {2016},
howpublished = "{arXiv:1609.03543}",
}

• arXiv
• pdf
• abridged pdf
• latest beamer slides
• latest powerpoint slides
• Parametric Bounded Löb’s Theorem and Robust Cooperation of Bounded Agents
(2016, under review)
• Description
• Establishes a bounded version of Löb’s theorem applicable to (computable) algorithms. The result is then used to demonstrate the existence of various algorithmic agents which robustly cooperate with one another on a singe-instance Prisoner’s Dilemma when they “know who heir opponent is”, with no incentive for either agent to defect. This result departs from the classical Nash equilibrium solution (D,D) for classically rational agents, illustrating that the framework of classical game theory is not adequate for agents that reason about one another’s source codes.
• BibTeX
@misc{Cri16,
author="Andrew J. Critch",
title = "{Parametric Bounded Lob's Theorem and Robust Cooperation of Bounded Agents}",
year = {2016},
howpublished = "{arXiv:1602.04184}",
}

• arXiv
• pdf
• Algebraic Geometry of Matrix Product States
with Jason Morton (2012), in SIGMA 10 (2014), 095, 10 pages
• Description
We exhibit polynomial constraints on classes of representations of entangled qubits as matrix product states.

• BibTeX
@article{critch2014algebraic,
title={Algebraic geometry of matrix product states},
author={Critch, Andrew and Morton, Jason},
journal={SIGMA},
volume={10},
number={095},
pages={095},
year={2014}
}

• arXiv
• pdf
• SIGMA
• Binary hidden Markov models and varieties
(2012), in the Journal of Algebraic Statistics, Vol . 4, No. 1, 2013.
• BibTeX
@misc{Cri12,
author="{Andrew J. Critch}",
title = "{Binary hidden Markov models and varieties}",
year = {2012},
howpublished = "{arXiv:1206.0500}",
}

• arXiv
• pdf
• Jalgstat
• A note on the proportionality between some consistency indices in the AHP
• @misc{BCF10,
author="Matteo Brunelli and Andrew J. Critch and Michele Fedrizzi",
title = "{A note on the proportionality between some consistency indices in the AHP}",
year = {2010},
howpublished = "{arXiv:1203.6431v1}",
}

• Pushing the limit (generalized limits and limit extrema in topology)
in the Atlantic Electronic Journal of Mathematics, vol. 1, iss. 1 (2006), pp. 47-55.
• @article{Cri06a,
author="{Andrew J. Critch}",
title="{Generalized limits and limit extrema}",
journal="Atlantic Electronic Journal of Mathematics",
year={2006},
volume={1},
number={1},
pages={47 - 55},
}

• Resolving the Banach-Tarski paradox: inseparability of rigid bodies
B.Sc. Honors thesis, Memorial University of Newfoundland (2006).
• @misc{Cri06b,
author="{Andrew J. Critch}",
title = "{Resolving the Banach-Tarski paradox: inseparability of rigid bodies}",
year = {2006},
howpublished = "{B.Sc. Honors thesis, Memorial University of Newfoundland}",
}


### Software

• Robust Rental Harmony
with Jacob Tsimerman and Chelsea Voss (2015) A novel algorithm and accompanying web-app interface for splitting rent in a robustly envy-free way between housemates.
• Tensors.m2
with Claudiu Raicu (2012), began at the WFU2012 Macaulay2 developers workshop. A Macaulay2 package for studying varieties of tensors and tensor networks.
• Credence Calibration Game
with Alexei Andreev and Zachary Alethia (2012). A game using information-theoretic scoring to calibrate players’ subjective belief strengths to more accurately match their success rates.