Set-Valued Solution Concepts in Social Choice and Game Theory: Axiomatic and Computational Aspects

This thesis studies axiomatic and computational aspects of set-valued solution concepts in social choice and game theory. It is divided into two parts. The first part focusses on solution concepts for normal-form games that are based on varying notions of dominance. These concepts are intuitively appealing and admit unique minimal solutions in important subclasses of games. We propose generic algorithms for computing solutions, and study for which classes of games and which properties of the underlying dominance notion the algorithms are sound and efficient. The second part is concerned with social choice functions (SCFs), an important subclass of which is formed by tournament solutions. The complexity of the winner determination problem is determined for a number of SCFs, and a new attractive tournament solution is proposed. Furthermore, the strategyproofness of irresolute SCFs is considered.

[1]  J. Weibull,et al.  Strategy subsets closed under rational behavior , 1991 .

[2]  Henk Norde,et al.  Bimatrix games have quasi-strict equilibria , 1999, Math. Program..

[3]  Jean-François Laslier,et al.  Composition-consistent tournament solutions and social choice functions , 1996 .

[4]  Felix Brandt,et al.  Set-rationalizable choice and self-stability , 2009, J. Econ. Theory.

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

[6]  Bhaskar Dutta.  Covering sets and a new condorcet choice correspondence , 1988 .

[7]  Peter C. Fishburn,et al.  Non-cooperative stochastic dominance games , 1978 .

[8]  Felix Brandt,et al.  Minimal stable sets in tournaments , 2008, J. Econ. Theory.

[9]  David Pearce.  Rationalizable Strategic Behavior and the Problem of Perfection , 1984 .

[10]  Andreu Mas-Colell,et al.  General Possibility Theorems for Group Decisions , 1972 .

[11]  Toby Walsh,et al.  Uncertainty in Preference Elicitation and Aggregation , 2007, AAAI.

[12]  The Strong No Show Paradoxes are a common flaw in Condorcet voting correspondences , 2001, Soc. Choice Welf..

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

[14]  J. Banks,et al.  Positive Political Theory I: Collective Preference , 1998 .

[15]  Jörg Rothe,et al.  Taking the final step to a full dichotomy of the possible winner problem in pure scoring rules , 2010, Inf. Process. Lett..

[16]  Robert Wilson.  The finer structure of revealed preference , 1970 .

[17]  Daniël Paulusma,et al.  The computational complexity of the elimination problem in generalized sports competitions , 2004, Discret. Optim..

[18]  Jean-François Laslier,et al.  More on the tournament equilibrium set , 1993 .

[19]  J. Harsanyi.  Oddness of the number of equilibrium points: A new proof , 1973 .

[20]  Olivier Hudry.  A note on “Banks winners in tournaments are difficult to recognize” by G. J. Woeginger , 2004, Soc. Choice Welf..

[21]  Klaus W. Wagner.  More Complicated Questions About Maxima and Minima, and Some Closures of NP , 1987, Theor. Comput. Sci..

[22]  Phan Minh Dung,et al.  On the Acceptability of Arguments and its Fundamental Role in Nonmonotonic Reasoning, Logic Programming and n-Person Games , 1995, Artif. Intell..

[23]  Salvador Barberà.  Manipulation of social decision functions , 1977 .

[24]  Alexander Schrijver,et al.  Combinatorial optimization. Polyhedra and efficiency. , 2003 .

[25]  K. Binmore.  Interpersonal Comparison of Utility , 2009 .

[26]  Peter C. Fishburn,et al.  Even-chance lotteries in social choice theory , 1972 .

[27]  E. Rowland.  Theory of Games and Economic Behavior , 1946, Nature.

[28]  Ariel D. Procaccia,et al.  Complexity of unweighted coalitional manipulation under some common voting rules , 2009, IJCAI 2009.

[29]  P. Fishburn.  Condorcet Social Choice Functions , 1977 .

[30]  Richard D. McKelvey,et al.  Covering, Dominance, and Institution Free Properties of Social Choice , 1986 .

[31]  E. Maskin.  Nash Equilibrium and Welfare Optimality , 1999 .

[32]  Nadja Betzler,et al.  Towards a dichotomy for the Possible Winner problem in elections based on scoring rules , 2009, J. Comput. Syst. Sci..

[33]  M. Satterthwaite.  Strategy-proofness and Arrow's conditions: Existence and correspondence theorems for voting procedures and social welfare functions , 1975 .

[34]  Salvador Barberà,et al.  THE MANIPULATION OF SOCIAL CHOICE MECHANISMS THAT DO NOT LEAVE "TOO MUCH" TO CHANCE' , 1977 .

[35]  Felix A. Fischer,et al.  Computing the minimal covering set , 2008, Math. Soc. Sci..

[36]  W. Gaertner.  A Primer in Social Choice Theory , 2006 .

[37]  P. Fishburn.  Probabilistic Social Choice Based on Simple Voting Comparisons , 1984 .

[38]  Allan M. Feldman.  Manipulation and the Pareto rule , 1979 .

[39]  H. P. Young,et al.  An axiomatization of Borda's rule , 1974 .

[40]  Krzysztof R. Apt,et al.  Uniform Proofs of Order Independence for Various Strategy Elimination Procedures , 2004, ArXiv.

[41]  Thomas Schwartz.  Cyclic tournaments and cooperative majority voting: A solution , 1990 .

[42]  H. Moulin.  Axioms of Cooperative Decision Making , 1988 .

[43]  Federico Poloni.  Of Note , 2009 .

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

[45]  Felix A. Fischer,et al.  On the Hardness and Existence of Quasi-Strict Equilibria , 2008, SAGT.

[46]  Yann Chevaleyre,et al.  Possible Winners when New Candidates Are Added: The Case of Scoring Rules , 2010, AAAI.

[47]  Henk Norde,et al.  The consistency principle for set-valued solutions and a new direction for normative game theory , 2001, Math. Methods Oper. Res..

[48]  Paul E. Dunne,et al.  Computational properties of argument systems satisfying graph-theoretic constraints , 2007, Artif. Intell..

[49]  David Gale,et al.  Review: R. Duncan Luce and Howard Raiffa, Games and decisions: Introduction and critical survey , 1958 .

[50]  Paul D. Seymour,et al.  A counterexample to a conjecture of Schwartz , 2013, Soc. Choice Welf..

[51]  William S. Zwicker,et al.  Monotonicity properties and their adaptation to irresolute social choice rules , 2012, Soc. Choice Welf..

[52]  J. Richelson.  Conditions on social choice functions , 1977 .

[53]  Felix Brandt,et al.  Characterization of dominance relations in finite coalitional games , 2010 .

[54]  P. Fishburn.  The Theory Of Social Choice , 1973 .

[55]  Jörg Rothe,et al.  Exact Complexity of the Winner Problem for Young Elections , 2001, Theory of Computing Systems.

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

[57]  B. Debord.  Caractérisation des matrices des préférences nettes et méthodes d'agrégation associées , 1987 .

[58]  Hervé Moulin,et al.  Fair division and collective welfare , 2003 .

[59]  Toby Walsh,et al.  Possible and necessary winners in voting trees: majority graphs vs. profiles , 2011, AAMAS.

[60]  B. D. Finetti.  La prévision : ses lois logiques, ses sources subjectives , 1937 .

[61]  Vincent Conitzer,et al.  Complexity of (iterated) dominance , 2005, EC '05.

[62]  M. Breton,et al.  The Bipartisan Set of a Tournament Game , 1993 .

[63]  Richard J. Zeckhauser,et al.  Voting Systems, Honest Preferences and Pareto Optimality , 1973, American Political Science Review.

[64]  Krzysztof R. Apt,et al.  Direct Proofs of Order Independence , 2010, ArXiv.

[65]  Meir Kalech,et al.  Practical voting rules with partial information , 2010, Autonomous Agents and Multi-Agent Systems.

[66]  Jacobus Petrus Maria Hurkens.  Games, rules, and solutions , 1995 .

[67]  P. Gärdenfors.  Manipulation of social choice functions , 1976 .

[68]  M. R. Rao,et al.  Combinatorial Optimization , 1992, NATO ASI Series.

[69]  S. Vajda.  Some topics in two-person games , 1971 .

[70]  Charles R. Plott,et al.  Axiomatic Social Choice Theory: An Overview and Interpretation , 1976 .

[71]  J. H. Smith.  AGGREGATION OF PREFERENCES WITH VARIABLE ELECTORATE , 1973 .

[72]  W. Riker.  The art of political manipulation , 1987 .

[73]  Eric van Damme,et al.  Non-Cooperative Games , 2000 .

[74]  Felix A. Fischer,et al.  A computational analysis of the tournament equilibrium set , 2008, Soc. Choice Welf..

[75]  Klaus Ritzberger,et al.  Strategic games beyond expected utility , 2011 .

[76]  John C. Harsanyi,et al.  Общая теория выбора равновесия в играх / A General Theory of Equilibrium Selection in Games , 1989 .

[77]  J. Edmonds.  Paths, Trees, and Flowers , 1965, Canadian Journal of Mathematics.

[78]  H. Moulin.  Condorcet's principle implies the no show paradox , 1988 .

[79]  Peter C. Ordeshook,et al.  Symmetric Spatial Games Without Majority Rule Equilibria , 1976, American Political Science Review.

[80]  Tuomas Sandholm,et al.  Algorithms for Closed Under Rational Behavior (CURB) Sets , 2010, J. Artif. Intell. Res..

[81]  Leslie M. Marx,et al.  Order Independence for Iterated Weak Dominance , 2000, Games Econ. Behav..

[82]  Eitan Zemel,et al.  The Complexity of Eliminating Dominated Strategies , 1993, Math. Oper. Res..

[83]  John Duggan,et al.  Dominance-based Solutions for Strategic Form Games , 1998 .

[84]  E. Damme.  Refinements of the Nash Equilibrium Concept , 1983 .

[85]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[86]  Andrés Perea,et al.  A one-person doxastic characterization of Nash strategies , 2007, Synthese.

[87]  John Duggan,et al.  Mixed refinements of Shapley's saddles and weak tournaments , 2001, Soc. Choice Welf..

[88]  H. Raiffa,et al.  GAMES AND DECISIONS; INTRODUCTION AND CRITICAL SURVEY. , 1958 .

[89]  T. Tideman,et al.  Independence of clones as a criterion for voting rules , 1987 .

[90]  Joaquín Pérez Ortega,et al.  An extension of the Moulin No Show Paradox for voting correspondences , 2009, Soc. Choice Welf..

[91]  J. Banks.  Sophisticated voting outcomes and agenda control , 1984 .

[92]  Toby Walsh,et al.  Dealing with Incomplete Agents' Preferences and an Uncertain Agenda in Group Decision Making via Sequential Majority Voting , 2008, KR.

[93]  Felix A. Fischer,et al.  The Computational Complexity of Choice Sets , 2007, TARK '07.

[94]  L. A. Goodman,et al.  Social Choice and Individual Values , 1951 .

[95]  Jérôme Lang,et al.  Voting procedures with incomplete preferences , 2005 .

[96]  Michel Le Breton,et al.  On the Uniqueness of Equilibrium in Symmetric Two-Player Zero-Sum Games with Integer Payoffs , 2007 .

[97]  Thomas Schwartz,et al.  On the possibility of rational policy evaluation , 1970 .

[98]  Jerry S. Kelly,et al.  STRATEGY-PROOFNESS AND SOCIAL CHOICE FUNCTIONS WITHOUT SINGLEVALUEDNESS , 1977 .

[99]  Toby Walsh,et al.  Winner determination in voting trees with incomplete preferences and weighted votes , 2011, Autonomous Agents and Multi-Agent Systems.

[100]  M. Dufwenberg,et al.  Existence and Uniqueness of Maximal Reductions Under Iterated Strict Dominance , 2002 .

[101]  Eitan Zemel,et al.  On the order of eliminating dominated strategies , 1990 .

[102]  Nicolas Houy,et al.  Still more on the Tournament Equilibrium Set , 2009, Soc. Choice Welf..

[103]  Roger B. Myerson,et al.  Game theory - Analysis of Conflict , 1991 .

[104]  Jérôme Monnot,et al.  Possible winners when new alternatives join: new results coming up! , 2011, AAMAS.

[105]  Larry Samuelson,et al.  Dominated strategies and common knowledge , 1992 .

[106]  Vincent Conitzer,et al.  Determining Possible and Necessary Winners under Common Voting Rules Given Partial Orders , 2008, AAAI.

[107]  T. Tideman,et al.  Complete independence of clones in the ranked pairs rule , 1989 .

[108]  Kevin Roberts,et al.  Interpersonal Comparability and Social Choice Theory , 1980 .

[109]  Svetlana Obraztsova,et al.  On the Complexity of Voting Manipulation under Randomized Tie-Breaking , 2011, IJCAI.

[110]  S. Barberà.  Strategy-proof social choice , 2010 .

[111]  Lin Zhou,et al.  Multi-valued strategy-proof social choice rules , 2002, Soc. Choice Welf..

[112]  John Duggan,et al.  Dutta's Minimal Covering Set and Shapley's Saddles , 1996 .

[113]  Alan D. Taylor,et al.  Social choice and the mathematics of manipulation , 2005 .

[114]  I. Good.  A note on condorcet sets , 1971 .

[115]  William S. Zwicker.  The voters' paradox, spin, and the Borda count , 1991 .

[116]  Felix Brandt,et al.  Group-Strategyproof Irresolute Social Choice Functions , 2010, IJCAI.

[117]  Bhaskar Dutta,et al.  On the tournament equilibrium set , 1990 .

[118]  Ariel Rubinstein,et al.  A Course in Game Theory , 1995 .

[119]  Thierry Marchant,et al.  AND DECISION MODELS : Stepping stones for the analyst , 2003 .

[120]  Begoña Subiza Martínez,et al.  Condorcet choice correspondences for weak tournaments , 1997 .

[121]  Bhaskar Dutta,et al.  Comparison functions and choice correspondences , 1999 .

[122]  W. Bossert,et al.  Interpersonal Comparisons of Well‐Being , 2004 .

[123]  Rolf Niedermeier,et al.  A logic for causal reasoning , 2003, IJCAI 2003.

[124]  Jörg Rothe,et al.  Exact analysis of Dodgson elections: Lewis Carroll's 1876 voting system is complete for parallel access to NP , 1997, JACM.

[125]  David C. Parkes,et al.  A Complexity-of-Strategic-Behavior Comparison between Schulze's Rule and Ranked Pairs , 2012, AAAI.