@article{van1998axiomatizations,
  title={Axiomatizations of the normalized Banzhaf value and the Shapley value},
  author={Van den Brink, Rene and Van der Laan, Gerard},
  journal={Social Choice and Welfare},
  volume={15},
  number={4},
  pages={567--582},
  year={1998},
  publisher={Springer}
}

@book{brandt2016handbook,
  title={Handbook of computational social choice},
  author={Brandt, Felix and Conitzer, Vincent and Endriss, Ulle and Lang, J{\'e}r{\^o}me and Procaccia, Ariel D},
  year={2016},
  publisher={Cambridge University Press}
}

@misc{friedman2001elements,
  title={The elements of statistical learning, volume 1 Springer series in statistics Springer},
  author={Friedman, Jerome and Hastie, Trevor and Tibshirani, Robert},
  year={2001},
  publisher={Berlin}
}

@misc{kaggle_2017,
title={2017 Kaggle Machine Learning \& Data Science Survey}, url={https://www.kaggle.com/kaggle/kaggle-survey-2017},
journal={Kaggle},
author={Kaggle},
year={2017}
} 

 


@article{michalak2013efficient,
  title={Efficient computation of the Shapley value for game-theoretic network centrality},
  author={Michalak, Tomasz P and Aadithya, Karthik V and Szczepanski, Piotr L and Ravindran, Balaraman and Jennings, Nicholas R},
  journal={Journal of Artificial Intelligence Research},
  volume={46},
  pages={607--650},
  year={2013}
}

@article{maafa2018algorithms,
  title={Algorithms for computing the Shapley value of cooperative games on lattices},
  author={Maafa, Khaled and Nourine, Lhouari and Radjef, Mohammed Said},
  journal={Discrete Applied Mathematics},
  volume={249},
  pages={91--105},
  year={2018},
  publisher={Elsevier}
}

@Article{s20216075,
AUTHOR = {Wang, Bin and Fan, Shi-dong and Jiang, Pan and Zhu, Han-hua and Xiong, Ting and Wei, Wei and Fang, Zhen-long},
TITLE = {A Novel Method with Stacking Learning of Data-Driven Soft Sensors for Mud Concentration in a Cutter Suction Dredger},
JOURNAL = {Sensors},
VOLUME = {20},
YEAR = {2020},
NUMBER = {21},
ARTICLE-NUMBER = {6075},
ISSN = {1424-8220},
DOI = {10.3390/s20216075}
}

@book{higham,
  author    = {Nicholas J. Higham},
  title     = {Accuracy and stability of numerical algorithms, Second Edition},
  publisher = {{SIAM}},
  year      = {2002},
  doi       = {10.1137/1.9780898718027},
  isbn      = {978-0-89871-521-7},
  timestamp = {Mon, 16 Sep 2019 14:43:12 +0200},
  biburl    = {https://dblp.org/rec/books/daglib/0007208.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@misc{patel2020high,
      title={High Dimensional Model Explanations: an Axiomatic Approach},
      author={Neel Patel and Martin Strobel and Yair Zick},
      year={2020},
      eprint={2006.08969},
      archivePrefix={arXiv},
      primaryClass={cs.LG}
}

@article{Sliwinski_Strobel_Zick_2019, title={Axiomatic Characterization of Data-Driven Influence Measures for Classification}, volume={33}, number={01}, journal={Proceedings of the AAAI Conference on Artificial Intelligence}, author={Sliwinski, Jakub and Strobel, Martin and Zick, Yair}, year={2019}, month={Jul.}, pages={718-725} }

@inproceedings{datta2,
author = {Datta, Amit and Datta, Anupam and Procaccia, Ariel D. and Zick, Yair},
title = {Influence in Classification via Cooperative Game Theory},
year = {2015},
isbn = {9781577357384},
publisher = {AAAI Press},
booktitle = {Proceedings of the 24th International Conference on Artificial Intelligence},
pages = {511–517},
numpages = {7},
location = {Buenos Aires, Argentina},
series = {IJCAI'15}
}

@TECHREPORT{RePEc:ivi:wpasad:1999-10,
title = {On the Choice of a Power Index},
author = {Laruelle, Annick},
year = {1999},
institution = {Instituto Valenciano de Investigaciones Económicas, S.A. (Ivie)},
type = {Working Papers. {Serie AD}},
abstract = {In this paper, we review and compare the main power indices to be found in the literature, that is to say, the Shapley-Shubik index, the Banzhaf index, the Johnston index, the Deegan-Packel index, and the Holler-Packel index. Three diferent approaches are used. First, the power indices are compared on the basis of a priori properties that they should satisfy. Second, a players power indices are compared on the basis of a priori properties that they shoud satisfy. Second, a players power is interpreted as the probability of her or his vote being necessary to pass a proposal. We present two models consistent with this approach: a model in which this probability arises from the assumptions made on the coalition formation and model, in which it arises from the assumptions made on the players voting behavior. Third, a players power is dened as her or his probability of playing a crucial role. Some power indices are derived from diferent assessments concerning the probability of playing such a role. Illustrations of the choice of an index are given for actual decision-making processes.},
keywords = {Collective Decision Making; Power Indices}
}

@article{10.2307/3689345,
 ISSN = {0364765X, 15265471},
 abstract = {The Banzhaf index of power in a voting situation depends on the number of ways in which each voter can effect a "swing" in the outcome. It is comparable-but not actually equivalent-to the better-known Shapley-Shubik index, which depends on the number of alignments or "orders of support" in which each voter is pivotal. This paper investigates some properties of the Banzhaf index, the main topics being its derivation from axioms and its behavior in weighted-voting models when the number of small voters tends to infinity. These matters have previously been studied from the Shapley-Shubik viewpoint, but the present work reveals some striking differences between the two indices. The paper also attempts to promote better communication and less duplication of mathematical effort by identifying and describing several other theories, formally equivalent to Banzhaf's, that are found in fields ranging from sociology to electrical engineering. An extensive bibliography is provided.},
 author = {Pradeep Dubey and Lloyd S. Shapley},
 journal = {Mathematics of Operations Research},
 number = {2},
 pages = {99--131},
 publisher = {INFORMS},
 title = {Mathematical Properties of the Banzhaf Power Index},
 volume = {4},
 year = {1979}
}




@Article{Lehrer1988,
author={Lehrer, E.},
title={An axiomatization of the Banzhaf value},
journal={International Journal of Game Theory},
year={1988},
month={Jun},
day={01},
volume={17},
number={2},
pages={89-99},
abstract={An axiomatization of the Banzhaf value is given. It is based on a version of three axioms, which are common to all the semi-values, and on an additional reduction axiom.},
issn={1432-1270},
doi={10.1007/BF01254541},
}




@Article{RePEc:inm:ormoor:v:26:y:2001:i:1:p:89-104,
  author={Annick Laruelle and Federico Valenciano},
  title={{Shapley-Shubik and Banzhaf Indices Revisited}},
  journal={Mathematics of Operations Research},
  year=2001,
  volume={26},
  number={1},
  pages={89-104},
  month={February},
  keywords={Power indices; voting power; collective decision-making; simple games; axiomatization},
  doi={10.1287/moor.26.1.89.1058},
  abstract={We provide a new axiomatization of the Shapley-Shubik and the Banzhaf power indices in the domain of simple superadditive games by means of transparent axioms. Only anonymity is shared with the former characterizations in the literature. The rest of the axioms are substituted by more transparent ones in terms of power in collective decision-making procedures. In particular, a clear restatement and a weaker alternative for the transfer axiom are proposed. Only one axiom differentiates the characterization of either index, and these differentiating axioms provide a new point of comparison. In a first step both indices are characterized up to a zero and a unit of scale. Then both indices are singled out by simple normalizing axioms.},
}

@article{scikit-learn,
 title={Scikit-learn: Machine Learning in {P}ython},
 author={Pedregosa, F. and Varoquaux, G. and Gramfort, A. and Michel, V.
         and Thirion, B. and Grisel, O. and Blondel, M. and Prettenhofer, P.
         and Weiss, R. and Dubourg, V. and Vanderplas, J. and Passos, A. and
         Cournapeau, D. and Brucher, M. and Perrot, M. and Duchesnay, E.},
 journal={Journal of Machine Learning Research},
 volume={12},
 pages={2825--2830},
 year={2011}
}

@Article{MomoKenfack2019,
author={Momo Kenfack, Joseph Armel
and Tchantcho, Bertrand
and Tsague, Bill Proces},
title={On the ordinal equivalence of the Jonhston, Banzhaf and Shapley--Shubik power indices for voting games with abstention},
journal={International Journal of Game Theory},
year={2019},
month={Jun},
day={01},
volume={48},
number={2},
pages={647-671},
abstract={The aim of this paper is twofold. We extend the well known Johnston power index usually defined for simple voting games, to voting games with abstention and we provide a full characterization of this extension. On the other hand, we conduct an ordinal comparison of three power indices: the Shapley--Shubik, Banzhaf and newly defined Johnston power indices. We provide a huge class of voting games with abstention in which these three power indices are ordinally equivalent. This is clearly a generalization of the work by Freixas et al. (Eur J Oper Res 216:367--375, 2012) and a twofold extension of Parker (Games Econ Behav 75:867--881, 2012) in the sense that, the ordinal equivalence emerges for three power indices (not just for the Shapley--Shubik and the Banzhaf indices), and it holds for a class of games strictly larger than the class of I-complete (3,2) games namely semi I-complete (3,2) games.},
issn={1432-1270},
doi={10.1007/s00182-018-0650-x},
}



@TechReport{RePEc:ema:worpap:2007-25,
  author={Fabrice Barthélémy and Mathieu Martin and Vincent Merlin},
  title={{On the performance of the Shapley Shubik and Banzhaf power indices for the allocations of mandates}},
  institution={THEMA},
  type={THEMA Working Papers},
  number={2007-25},
  abstract={A classical problem in the power index literature is to design a voting mechanism such as the distribution of power of the different players is equal (or closer) to a pre established target. This tradition is especially popular when considering two tiers voting mechanisms: each player votes in his own jurisdiction to designate a delegate for the upper tier; and the question is to assign a certain number of mandates for each delegate according the population of the jurisdiction he or she represents. Unfortunately, there exist several measures of power, which in turn imply different distributions of the mandates for the same pre established target. The purposes of this paper are twofold: first, we calculate the probability that the two most important power indices, the Banzhaf index and the Shapley-Shubik index, lead to the same voting rule when the target is the same. Secondly, we determine which index on average comes closer to the pre established target.},
  keywords={Banzhaf; Shapley-Shubik; power indices},
}


@article{power,
author = {Dowding, Keith},
year = {2003},
month = {10},
pages = {},
title = {Power: A Philosophical Analysis, 2nd edition},
volume = {2},
journal = {Contemporary Political Theory},
doi = {10.1057/palgrave.cpt.9300080}
}

@Inbook{Koczy2009,
author="K{\'o}czy, L{\'a}szl{\'o} {\'A}.",
title="Measuring Voting Power: The Paradox of New Members vs. the Null Player Axiom",
bookTitle="Towards Intelligent Engineering and Information Technology",
year="2009",
publisher="Springer Berlin Heidelberg",
pages="67--78",
abstract="Qualified majority voting is used when decisions are made by voters of different sizes. In such situations the voters' influence on decision making is far from obvious; power measures are used for an indication of the decision making ability. Several power measures have been proposed and characterised by simple axioms to help the choice between them. Unfortunately the power measures also feature a number of so-called paradoxes of voting power. In this paper we show that the Paradox of New Members follows from the Null Player Axiom. As a corollary of this result it follows that there does not exist a power measure that satisfies the axiom, while not exhibiting the Paradox.",
isbn="978-3-642-03737-5"
}




@article{doi:10.1111/1467-9248.00356,
author = {Dennis Leech},
title ={An Empirical Comparison of the Performance of Classical Power Indices},
journal = {Political Studies},
volume = {50},
number = {1},
pages = {1-22},
year = {2002}
}

@inbook{kurz,
 author ={Kurz, Sascha},
 title ="Which Criteria Qualify Power Indices for Applications? A Comment to 'The Story of the Poor Public Good Index'",
 date= "December 14, 2018",
 publisher =  "SSRN"
}


@Inbook{Hart1989,
author="Hart, Sergiu",
title="Shapley Value",
bookTitle="Game Theory",
year="1989",
publisher="Palgrave Macmillan UK",
address="London",
pages="210--216",
}



@inproceedings{Lundberg2017,
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@inproceedings{GrecoLS15,
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@article{Lundberg2020,
author={Lundberg, Scott M.
and Erion, Gabriel
and Chen, Hugh
and DeGrave, Alex
and Prutkin, Jordan M.
and Nair, Bala
and Katz, Ronit
and Himmelfarb, Jonathan
and Bansal, Nisha
and Lee, Su-In},
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@inproceedings{SundararajanN20,
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@ARTICLE{Kislaya90,
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@article{Matsui2001,
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  volume={263},
  pages={305-310}
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@Article{Banzhaf65,
  author = {Banzhaf, J.F.},
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}

@book{Breiman,
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@inproceedings{ChenG16,
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               Raia Hadsell and
               Maria{-}Florina Balcan and
               Hsuan{-}Tien Lin},
  title     = {Understanding Global Feature Contributions With Additive Importance
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  year      = {2020},
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@article{Breiman2004,
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@article{AasJL21,
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               Anders L{\o}land},
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@article{brophy2020trex,
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               Alina Beygelzimer and
               Florence d'Alch{\'{e}}{-}Buc and
               Emily B. Fox and
               Roman Garnett},
  title     = {MonoForest framework for tree ensemble analysis},
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@inproceedings{ribeiro2018anchors,
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@article{Ribeiro2016WhySI,
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@InProceedings{shrikumar17a,
  title = 	 {Learning Important Features Through Propagating Activation Differences},
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@article{Bach2015OnPE,
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@article{Roth,
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@article{Shapley53,
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@article{Chebrolu2005,
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@article{HuynhThu2010InferringRN,
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@inproceedings{Kindermans2018LearningHT,
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@misc{shappackage,
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title={{SHAP} Python package},
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year={2022}
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@misc{bostondataset,
key={BS},
title={{BOSTON} dataset},
url={https://www.cs.toronto.edu/~delve/data/boston/bostonDetail.html},
year={2022}

}

@misc{vehicledataset,
key={VI},
title={{VEHICLE INSURANCE} dataset},
url={https://www.kaggle.com/anmolkumar/health-insurance-cross-sell-prediction?select=train.csv},
year={2022}
}

@misc{flightsdataset,
key={FL},
title={{FLIGHTS} dataset},
url={https://www.kaggle.com/abdurrehmankhalid/delayedflights},
year={2022}
}

@misc{Saabas,
  title={Treeinterpreter Python package},
  author={A. Saabas},
  url={https://github.com/andosa/treeinterpreter},
  year={2022}
}

@article{Baehrens10a,
  author  = {David Baehrens and Timon Schroeter and Stefan Harmeling and Motoaki Kawanabe and Katja Hansen and Klaus-Robert M{{\"u}}ller},
  title   = {How to Explain Individual Classification Decisions},
  journal = {Journal of Machine Learning Research},
  year    = {2010},
  volume  = {11},
  number  = {61},
  pages   = {1803-1831}
}


@inproceedings{Plumb18,
 author = {Plumb, Gregory and Molitor, Denali and Talwalkar, Ameet S},
 booktitle = {Advances in Neural Information Processing Systems},
 pages = {2515--2524},
 publisher = {Curran Associates, Inc.},
 title = {Model Agnostic Supervised Local Explanations},
 volume = {31},
 year = {2018}
}
@article {Bordag2021.04.18.21254782,
	author = {Bordag, Natalie and Z{\"u}gner, Elmar and L{\'o}pez-Garc{\'\i}a, Pablo and Kofler, Selina and Tomberger, Martina and Al-Baghdadi, Abdullah and Schweiger, Jessica and Erdem, Yasemin and Magnes, Christoph and Hidekazu, Saiki and Wadsak, Wolfgang and Erxleben, Bj{\"o}rn-Thoralf and Prietl, Barbara},
	title = {Towards fast, routine blood sample quality evaluation by Probe Electrospray Ionization (PESI) metabolomics},
	elocation-id = {2021.04.18.21254782},
	year = {2021},
	doi = {10.1101/2021.04.18.21254782},
	publisher = {Cold Spring Harbor Laboratory Press},
	eprint = {https://www.medrxiv.org/content/early/2021/04/23/2021.04.18.21254782.full.pdf},
	journal = {medRxiv}
}
@article{Pham2019MultimodalDO,
  title={Multimodal Detection of Parkinson Disease based on Vocal and Improved Spiral Test},
  author={Hung N. Pham and T. T. Do and Kelvin Yi Jie Chan and Gopa Sen and Andy Han and Pier Lim and Teresa Siew Loon Cheng and Quang H. Nguyen and Binh P. Nguyen and Matthew C. H. Chua},
  journal={2019 International Conference on System Science and Engineering (ICSSE)},
  year={2019},
  pages={279-284}
}

@article{abs-1905-04610,
  author    = {Scott M. Lundberg and
               Gabriel G. Erion and
               Hugh Chen and
               Alex J. DeGrave and
               Jordan M. Prutkin and
               Bala Nair and
               Ronit Katz and
               Jonathan Himmelfarb and
               Nisha Bansal and
               Su{-}In Lee},
  title     = {Explainable {AI} for Trees: From Local Explanations to Global Understanding},
  journal   = {CoRR},
  volume    = {abs/1905.04610},
  year      = {2019},
  url       = {http://arxiv.org/abs/1905.04610},
  eprinttype = {arXiv},
  eprint    = {1905.04610},
  timestamp = {Fri, 26 Nov 2021 16:33:35 +0100},
  biburl    = {https://dblp.org/rec/journals/corr/abs-1905-04610.bib},
  bibsource = {dblp computer science bibliography, https://dblp.org}
}

@inproceedings{Van_den_Broeck_Lykov_Schleich_Suciu_2021,
  author    = {Guy Van den Broeck and
               Anton Lykov and
               Maximilian Schleich and
               Dan Suciu},
  title     = {On the Tractability of {SHAP} Explanations},
  booktitle = {Thirty-Fifth {AAAI} Conference on Artificial Intelligence, {AAAI}
               2021, Thirty-Third Conference on Innovative Applications of Artificial
               Intelligence, {IAAI} 2021, The Eleventh Symposium on Educational Advances
               in Artificial Intelligence, {EAAI} 2021, Virtual Event, February 2-9,
               2021},
  pages     = {6505--6513},
  publisher = {{AAAI} Press},
  year      = {2021},
}

@article{Arenas_Barcelo_Bertossi_Monet_2021,
title={The Tractability of SHAP-Score-Based Explanations for Classification over Deterministic and Decomposable Boolean Circuits},
volume={35},
number={8},
journal={Proceedings of the AAAI Conference on Artificial Intelligence},
author={Arenas, Marcelo and Barceló, Pablo and Bertossi, Leopoldo and Monet, Mikaël},
year={2021},
month={May},
pages={6670-6678}
}

@inproceedings{Patel2021GametheoreticVS,
  title={Game-theoretic Vocabulary Selection via the Shapley Value and Banzhaf Index},
  author={Roma Patel and Marta Garnelo and Ian M. Gemp and Chris Dyer and Yoram Bachrach},
  booktitle={NAACL},
  year={2021}
}