{"type":"ScholarlyArticle","title":"On the Ordinal Invariance of Power Indices on Coalitional Games","authors":[{"name":"Jean-Paul Doignon","slug":"jean-paul-doignon","given":"Jean-Paul","family":"Doignon","affiliation":"Département de Mathématique ,"},{"name":"Stefano Moretti","slug":"stefano-moretti","given":"Stefano","family":"Moretti","affiliation":"Université Paris Dauphine , PSL Research University , CNRS , Lamsade , 75016 Paris , France; c .p. 216 , Université Libre de Bruxelles , Bruxelles , Belgium"},{"name":"Meltem Ozturk","slug":"meltem-ozturk","given":"Meltem","family":"Öztürk","affiliation":"Université Paris Dauphine , PSL Research University , CNRS , Lamsade , 75016 Paris , France"}],"abstract":"In a coalitional game, the coalitions are weakly ordered according to their worths in the game.  When moreover a power index is given, the players are ranked according to the real numbers they are assigned by the power index.  If any game inducing the same ordering of the coalitions generates the same ranking of the players then, by definition, the game is (ordinally) stable for the power index, which in turn is ordinally invariant for the game.  If one is interested in ranking players of a game which is stable, re-computing the power indices when the coalitional worths slightly fluctuate or are uncertain becomes useless.  Bivalued games are easy examples of games stable for any power index which is linear.  Among general games, we characterize those that are stable for a given linear index.  Note that the Shapley and Banzhaf scores, frequently used in AI, are particular semivalues, and all semivalues are linear indices.  To check whether a game is stable for a specific semivalue, it suffices to inspect the ordering of the coalitions and to perform some direct computation based on the semivalue parameters.","doi":"10.24963/ijcai.2022/37","url":"https://ijcai.org/proceedings/2022/37","pdf":"https://ijcai.org/proceedings/2022/0037.pdf","markdown":"https://ijcai.org/proceedings/2022/37.md","bibtex":"https://ijcai.org/proceedings/2022/bibtex/37","volume":{"slug":"2022","title":"Proceedings of the Thirty-First International Joint Conference on Artificial Intelligence","short_title":"IJCAI 2022","url":"https://ijcai.org/proceedings/2022/","year":2022,"location":"Vienna","dates":"23-29 July 2022","editors":"Luc De Raedt","isbn":"978-1-956792-00-3","publisher":"International Joint Conferences on Artificial Intelligence Organization","doi_prefix":"10.24963/ijcai.2022"},"section":"Agent-based and Multi-agent Systems","track":"Main Track","number":37,"pages":{"first":258,"last":264},"keywords":["Agent-based and Multi-agent Systems: Cooperative Games","Agent-based and Multi-agent Systems: Computational Social Choice"],"videos":[{"label":"Video #1","url":"https://slideslive.com/38986704","length_sec":84,"provider":"slideslive"},{"label":"Video #2","url":"https://slideslive.com/38984847","length_sec":377,"provider":"slideslive"}],"references":[{"position":1,"raw":"[Algaba et al., 2003] Encarnación Algaba, Jesús Mario Bil- bao, Fernández Garcıa, and J.J. López. Computing power indices in weighted multiple majority games. Mathemati- cal Social Sciences, 46(1):63-80, 2003.","title":"Computing power indices in weighted multiple majority games","authors":"Algaba","year":"2003","venue":"Mathematical Social Sciences","doi":null,"url":null,"internal_url":null},{"position":2,"raw":"[Amgoud et al., 2017] Leila Amgoud, Jonathan Ben-Naim, and Srdjan Vesic. Measuring the intensity of attacks in ar- gumentation graphs with Shapley value. In Proceedings of the Twenty-Sixth International Joint Conference on Artifi- cial Intelligence, IJCAI 2017, pages 63-69, 2017.","title":"Measuring the intensity of attacks in argumentation graphs with Shapley value","authors":"Leila Amgoud, Jonathan Ben-Naim, Srdjan Vesic","year":"2017","venue":"Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence, IJCAI 2017","doi":null,"url":null,"internal_url":"/proceedings/2017/10"},{"position":3,"raw":"[Banzhaf III, 1964] John F. Banzhaf III. Weighted voting doesn't work: A mathematical analysis. Rutgers Univer- sity Law Review, 19:317-343, 1964.","title":"Weighted voting doesn't work: A mathematical analysis","authors":"John F Banzhaf, Iii","year":"1964","venue":"Rutgers University Law Review","doi":null,"url":null,"internal_url":null},{"position":4,"raw":"[Bernardi et al., 2019] Giulia Bernardi, Roberto Lucchetti, and Stefano Moretti. Ranking objects from a preference relation over their subsets. Social Choice and Welfare, 52(4):589-606, 2019.","title":"Ranking objects from a preference relation over their subsets","authors":"Giulia Bernardi, Roberto Lucchetti, Stefano Moretti","year":"2019","venue":"Social Choice and Welfare","doi":null,"url":null,"internal_url":null},{"position":5,"raw":"[Carreras and Freixas, 2008] Francesc Carreras and Josep Freixas. On ordinal equivalence of power measures given by regular semivalues. Mathematical Social Sciences, 55(2):221-234, 2008.","title":"Francesc Carreras and Josep Freixas. On ordinal equivalence of power measures given by regular semivalues","authors":"Freixas Carreras","year":"2008","venue":"Mathematical Social Sciences","doi":null,"url":null,"internal_url":null},{"position":6,"raw":"[Deegan and Packel, 1978] John Deegan and Edward W. Packel. A new index of power for simple n-person games. International Journal of Game Theory, 7(2):113- 123, 1978.","title":"A new index of power for simple n-person games","authors":"John Deegan, Edward W Packel","year":"1978","venue":"International Journal of Game Theory","doi":null,"url":null,"internal_url":null},{"position":7,"raw":"[Dubey et al., 1981] Pradeep Dubey, Abraham Neyman, and Robert James Weber. Value theory without effi- ciency. Mathematics of Operations Research, 6(1):122- 128, 1981.","title":"Value theory without efficiency","authors":"Pradeep Dubey, Abraham Neyman, Robert James Weber","year":"1981","venue":"Mathematics of Operations Research","doi":null,"url":null,"internal_url":null},{"position":8,"raw":"[Grabisch and Roubens, 2000] Michel Grabisch and Marc Roubens. Application of the Choquet integral in multi- criteria decision making. Fuzzy Measures and Integrals- Theory and Applications, pages 348-374, 2000.","title":"Application of the Choquet integral in multicriteria decision making","authors":"Roubens Grabisch, Michel Grabisch, Marc Roubens","year":"2000","venue":"Fuzzy Measures and Integrals-Theory and Applications","doi":null,"url":null,"internal_url":null},{"position":9,"raw":"[Haret et al., 2018] Adrian Haret, Hossein Khani, Stefano Moretti, and Meltem Öztürk. Ceteris paribus majority for social ranking. In Proceedings of the Twenty-Seventh In- ternational Joint Conference on Artificial Intelligence, IJ- CAI 2018, pages 303-309, 2018.","title":"Ceteris paribus majority for social ranking","authors":"Adrian Haret, Hossein Khani, Stefano Moretti, Meltem Öztürk","year":"2018","venue":"Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence","doi":null,"url":null,"internal_url":"/proceedings/2018/42"},{"position":10,"raw":"[Holler and Nurmi, 2013] Manfred J. Holler and Hannu Nurmi. Power, Voting, and Voting Power: 30 years after. Springer, 2013.","title":"Power, Voting, and Voting Power: 30 years after","authors":"Manfred J Holler, Hannu Nurmi","year":"2013","venue":null,"doi":null,"url":null,"internal_url":null},{"position":11,"raw":"[Holler, 1982] Manfred J. Holler. Forming coalitions and measuring voting power. Political Studies, 30(2):262-271, 1982.","title":"Forming coalitions and measuring voting power","authors":"Manfred J Holler","year":"1982","venue":"Political Studies","doi":null,"url":null,"internal_url":null},{"position":12,"raw":"[Hunter and Konieczny, 2006] Anthony Hunter and Sébastien Konieczny. Shapley inconsistency values. In Proceedings of Knowledge Representation and Reasoning, KR 2006, volume 6, pages 249-259. 2006.","title":"Shapley inconsistency values","authors":"Anthony Hunter, Sébastien Konieczny","year":"2006","venue":"Proceedings of Knowledge Representation and Reasoning","doi":null,"url":null,"internal_url":null},{"position":13,"raw":"[Lambo and Moulen, 2002] Lawrence Diffo Lambo and Joël Moulen. Ordinal equivalence of power notions in voting games. Theory and Decision, 53(4):313-325, 2002.","title":"Lawrence Diffo Lambo and Joël Moulen. Ordinal equivalence of power notions in voting games","authors":"Moulen Lambo","year":"2002","venue":"Theory and Decision","doi":null,"url":null,"internal_url":null},{"position":14,"raw":"[Lucchetti et al., 2015] Roberto Lucchetti, Stefano Moretti, and Fioravante Patrone. Ranking sets of interacting objects via semivalues. TOP, 23(2):567-590, 2015.","title":"Ranking sets of interacting objects via semivalues","authors":"Roberto Lucchetti, Stefano Moretti, Fioravante Patrone","year":"2015","venue":"TOP","doi":null,"url":null,"internal_url":null},{"position":15,"raw":"[Lundberg and Lee, 2017] Scott M. Lundberg and Su-In Lee. A unified approach to interpreting model predictions. In 31st International Conference on Neural Information Processing Systems, pages 4768-4777, 2017.","title":"A unified approach to interpreting model predictions","authors":"M Scott, Su-In Lundberg, Lee","year":"2017","venue":"31st International Conference on Neural Information Processing Systems","doi":null,"url":null,"internal_url":null},{"position":16,"raw":"[Michalak et al., 2013] Tomasz P. Michalak, Karthik V. Aa- dithya, Piotr L. Szczepanski, Balaraman Ravindran, and Nicholas R. Jennings. Efficient computation of the Shap- ley value for game-theoretic network centrality. Journal of Artificial Intelligence Research, 46:607-650, 2013.","title":"Efficient computation of the Shapley value for game-theoretic network centrality","authors":"P Tomasz, Michalak, V Karthik, Piotr L Aadithya, Balaraman Szczepanski, Nicholas R Ravindran, Jennings","year":"2013","venue":"Journal of Artificial Intelligence Research","doi":null,"url":null,"internal_url":null},{"position":17,"raw":"[Myerson, 1977] Roger Myerson. Graphs and cooperation in games. Mathematics of Operations Research, 2(3):225- 229, 1977.","title":"Roger Myerson. Graphs and cooperation in games","authors":"Myerson","year":"1977","venue":"Mathematics of Operations Research","doi":null,"url":null,"internal_url":null},{"position":18,"raw":"[Roth, 1985] Alvin E. Roth. The college admissions prob- lem is not equivalent to the marriage problem. Journal of Economic Theory, 36(2):277 -288, 1985.","title":"The college admissions problem is not equivalent to the marriage problem","authors":"Alvin E Roth","year":"1985","venue":"Journal of Economic Theory","doi":null,"url":null,"internal_url":null},{"position":19,"raw":"[Schmeidler, 1969] David Schmeidler. The nucleolus of a characteristic function game. SIAM Journal on Applied Mathematics, 17(6):1163-1170, 1969.","title":"The nucleolus of a characteristic function game","authors":"David Schmeidler","year":"1969","venue":"SIAM Journal on Applied Mathematics","doi":null,"url":null,"internal_url":null},{"position":20,"raw":"[Shapley, 1953] Lloyd S. Shapley. A value for n-person games. Contributions to the Theory of Games, 2(28):307- 317, 1953.","title":"A value for n-person games","authors":"Lloyd S Shapley","year":"1953","venue":"Contributions to the Theory of Games","doi":null,"url":null,"internal_url":null}],"license_note":"Copyright International Joint Conferences on Artificial Intelligence."}