Margin of victory for tournament solutions

Tournament solutions are frequently used to select winners from a set of alternatives based on pairwise comparisons between them. Prior work has shown that several common tournament solutions tend to select large winner sets and therefore have low discriminative power. In this paper, we propose a general framework for refining tournament solutions. In order to distinguish between winning alternatives, and also between non-winning ones, we introduce the notion of margin of victory (MoV) for tournament solutions. MoV is a robustness measure for individual alternatives: For winners, the MoV captures the distance from dropping out of the winner set, and for non-winners, the distance from entering the set. In each case, distance is measured in terms of which pairwise comparisons would have to be reversed in order to achieve the desired outcome. For common tournament solutions, including the top cycle, the uncovered set, and the Banks set, we determine the complexity of computing the MoV and provide bounds on the MoV for both winners and non-winners. We then reveal a number of structural insights on the MoV by investigating fundamental properties such as monotonicity and consistency with respect to the covering relation. Furthermore, we provide experimental evidence on the extent to which the MoV notion refines winner sets in tournaments generated according to various stochastic models. Our results can also be viewed from the perspective of bribery and manipulation.

[1]  P. Faliszewski,et al.  Control and Bribery in Voting , 2016, Handbook of Computational Social Choice.

[2]  Mihalis Yannakakis,et al.  On Limited Nondeterminism and the Complexity of the V-C Dimension , 1996, J. Comput. Syst. Sci..

[3]  David Cary,et al.  Estimating the Margin of Victory for Instant-Runoff Voting , 2011, EVT/WOTE.

[4]  Uzi Vishkin,et al.  On Finding a Minimum Dominating Set in a Tournament , 1988, Theor. Comput. Sci..

[5]  Gerhard J. Woeginger,et al.  Banks winners in tournaments are difficult to recognize , 2003, Soc. Choice Welf..

[6]  Matthias Mnich,et al.  When Does Schwartz Conjecture Hold? , 2015, IJCAI.

[7]  Stéphane Airiau,et al.  Refinements and Randomised Versions of Some Tournament Solutions , 2017, AAMAS.

[8]  Toby Walsh,et al.  PrefLib: A Library for Preferences http://www.preflib.org , 2013, ADT.

[9]  Felix Brandt,et al.  Bounds on the disparity and separation of tournament solutions , 2015, Discret. Appl. Math..

[10]  Stephanie Kovalchik,et al.  Extension of the Elo rating system to margin of victory , 2020 .

[11]  Felix Brandt,et al.  Tournament Solutions , 2016, Handbook of Computational Social Choice.

[12]  Alex D. Scott,et al.  The minimal covering set in large tournaments , 2012, Soc. Choice Welf..

[13]  Jens Vygen,et al.  The Book Review Column1 , 2020, SIGACT News.

[14]  Arun Rajkumar,et al.  Dueling Bandits: Beyond Condorcet Winners to General Tournament Solutions , 2016, NIPS.

[15]  Piotr Faliszewski,et al.  Llull and Copeland Voting Computationally Resist Bribery and Constructive Control , 2009, J. Artif. Intell. Res..

[16]  Yongjie Yang,et al.  Possible winner problems on partial tournaments: a parameterized study , 2017, J. Comb. Optim..

[17]  Weibin Han,et al.  A refinement of the uncovered set in tournaments , 2018, Theory and Decision.

[18]  Ronald L. Rivest,et al.  Computing the Margin of Victory in IRV Elections , 2011, EVT/WOTE.

[19]  Andrew V. Goldberg,et al.  Finding minimum-cost circulations by canceling negative cycles , 1989, JACM.

[20]  Ulrike Schmidt-Kraepelin,et al.  Refining Tournament Solutions via Margin of Victory , 2019, AAAI.

[21]  R. Downey,et al.  Parameterized Computational Feasibility , 1995 .

[22]  Ulrike Schmidt-Kraepelin,et al.  Margin of Victory in Tournaments: Structural and Experimental Results , 2020, ArXiv.

[23]  Felix Brandt,et al.  A note on the McKelvey uncovered set and Pareto optimality , 2016, Soc. Choice Welf..

[24]  Felix A. Fischer,et al.  PageRank as a Weak Tournament Solution , 2007, WINE.

[25]  M. Klein.  A Primal Method for Minimal Cost Flows with Applications to the Assignment and Transportation Problems , 1966 .

[26]  Yongjie Yang,et al.  Approval Voting with Intransitive Preferences , 2017, AAMAS.

[27]  Jennifer Ryan,et al.  Tournament games and positive tournaments , 1995, J. Graph Theory.

[28]  P. Erdös.  On a Problem in Graph Theory , 1963 .

[29]  C. L. Mallows.  NON-NULL RANKING MODELS. I , 1957 .

[30]  Warut Suksompong,et al.  Tournaments in Computational Social Choice: Recent Developments , 2021, IJCAI.

[31]  Piotr Faliszewski,et al.  On the Robustness of Winners: Counting Briberies in Elections , 2020, ArXiv.

[32]  Mark Fey,et al.  Choosing from a large tournament , 2008, Soc. Choice Welf..

[33]  R. Graham,et al.  A Constructive Solution to a Tournament Problem , 1971, Canadian Mathematical Bulletin.

[34]  Saket Saurabh,et al.  On Succinct Encodings for the Tournament Fixing Problem , 2019, IJCAI.

[35]  Edith Elkind,et al.  On elections with robust winners , 2013, AAMAS.

[36]  Felix Brandt,et al.  On the structure of stable tournament solutions , 2016, Economic Theory.

[37]  D. R. Fulkerson,et al.  Maximal Flow Through a Network , 1956 .

[38]  Felix Brandt,et al.  On the Discriminative Power of Tournament Solutions , 2014, OR.

[39]  Abelard Podgorski.  Tournament Decision Theory , 2020 .

[40]  Felix Brandt,et al.  On the Fixed-Parameter Tractability of Composition-Consistent Tournament Solutions , 2011, IJCAI.

[41]  Jean-François Laslier,et al.  Tournament Solutions And Majority Voting , 1997 .

[42]  Palash Dey.  Query Complexity of Tournament Solutions , 2017, AAAI.

[43]  Olivier Hudry,et al.  A survey on the complexity of tournament solutions , 2009, Math. Soc. Sci..

[44]  Felix Brandt,et al.  Extending tournament solutions , 2014, Social Choice and Welfare.

[45]  Warut Suksompong,et al.  Robust bounds on choosing from large tournaments , 2020, Social choice and welfare.

[46]  Lirong Xia,et al.  Computing the margin of victory for various voting rules , 2012, EC '12.

[47]  Toby Walsh,et al.  Manipulating Tournaments in Cup and Round Robin Competitions , 2009, ADT.

[48]  Toby Walsh,et al.  Fixing balanced knockout and double elimination tournaments , 2018, Artif. Intell..

[49]  Virginia Vassilevska Williams.  Fixing a Tournament , 2010, AAAI.

[50]  Dorothea Baumeister,et al.  Complexity of Election Evaluation and Probabilistic Robustness , 2020 .

[51]  Krishna P. Gummadi,et al.  Minimizing Margin of Victory for Fair Political and Educational Districting , 2019, AAMAS.

[52]  Virginia Vassilevska Williams,et al.  Bribery in Balanced Knockout Tournaments , 2019, AAMAS.

[53]  Ali Ridha Mahjoub,et al.  Max Flow and Min Cut with bounded-length paths: complexity, algorithms, and approximation , 2010, Math. Program..

[54]  Krishnendu Chatterjee,et al.  Robust Draws in Balanced Knockout Tournaments , 2016, IJCAI.

[55]  Thomas Erlebach,et al.  Length-bounded cuts and flows , 2006, TALG.

[56]  Virginia Vassilevska Williams,et al.  Fixing Tournaments for Kings, Chokers, and More , 2015, IJCAI.

[57]  Felix A. Fischer,et al.  Possible and necessary winners of partial tournaments , 2012, AAMAS.

[58]  Yoav Shoham,et al.  On the complexity of schedule control problems for knockout tournaments , 2009, AAMAS.

[59]  S. Berg.  Paradox of voting under an urn model: The effect of homogeneity , 1985 .

[60]  Warut Suksompong,et al.  Who Can Win a Single-Elimination Tournament? , 2015, AAAI.

[61]  Andrew Klapper,et al.  On the complexity of bribery and manipulation in tournaments with uncertain information , 2012, J. Appl. Log..

[62]  Y. Narahari,et al.  Estimating the Margin of Victory of an Election Using Sampling , 2015, IJCAI.

[63]  Rolf Niedermeier,et al.  A Refined Complexity Analysis of Fair Districting over Graphs , 2021, ArXiv.

[64]  Sarit Kraus,et al.  On the evaluation of election outcomes under uncertainty , 2008, Artif. Intell..

[65]  Stefan Szeider,et al.  Rigging Nearly Acyclic Tournaments Is Fixed-Parameter Tractable , 2017, AAAI.