This book is a survey on the problem of choosing from a tournament. It brings together under a unified and self-contained presentation results and concepts from Graph Theory, Choice Theory, Decision Science and Social Choice which were discovered in the last ten years. Classical scoring and ranking methods are introduced, including the Slater orderings, as well as new statistical methods for describing a tournament, graph-theoretical methods based on the covering relation and game-theoretical methods. As an illustration, results are applied to the classical problem of Majority Voting: How to deal with the Condorcet Paradox.…mehr
This book is a survey on the problem of choosing from a tournament. It brings together under a unified and self-contained presentation results and concepts from Graph Theory, Choice Theory, Decision Science and Social Choice which were discovered in the last ten years. Classical scoring and ranking methods are introduced, including the Slater orderings, as well as new statistical methods for describing a tournament, graph-theoretical methods based on the covering relation and game-theoretical methods. As an illustration, results are applied to the classical problem of Majority Voting: How to deal with the Condorcet Paradox.
The book presents classical scoring and ranking methods as well as new statistical, graph-theoretical and game-theoretical methods for the description of a tournament and applies them to the problem of Majority Voting.
Inhaltsangabe
Organisation of the Book.- 1 Generalities.- 1.1 Definitions and Notations ..- 1.2 Finite Tournaments.- 1.3 Decomposition.- 1.4 Regularity.- 1.5 Useful Notions about General Binary Relations.- 2 Tournament Solutions.- 2.1 Majority Voting and Tournaments.- 2.2 Solution Concepts.- 2.3 Monotonicity, Strong Superset Property and Independence of Losers.- 2.4 Composition-Consistency and Regularity.- 2.5 Composition-Consistent Hulls.- 3 Scoring and Ranking Methods.- 3.1 Copeland Solution.- 3.2 Iterative Matrix Solutions.- 3.3 Markov Solution.- 3.4 Slater Solution.- 4 Multivariate Descriptions.- 4.1 Complete Euclidean Description.- 4.2 Multidimensional Scaling.- 5 Covering.- 5.1 Covering Relation and Uncovered Set.- 5.2 Iterations of the Uncovered Set.- 5.3 Dutta's Minimal Covering Set.- 5.4 Weak Covering à la Laffond and Lainé.- 5.5 Weak Covering à la Levchenkov.- 6 Tournament Game.- 6.1 Tournament Game in Pure Strategies.- 6.2 Tournament Game in Mixed Strategies.- 6.3 Properties of the Bipartisan Set.- 6.4 Method of the Minimal Gain.- 6.5 Interpretation of Tournament Games.- 7 The Contestation Process.- 7.1 Banks' Solution.- 7.2 The Tournament Equilibrium Set.- 8 Tournament Algebras and Binary Trees.- 8.1 Definition of a Tournament Algebra.- 8.2 Binary Trees.- 8.3 An Algebraic Solution: The Top-Cycle.- 8.4 An Algebraic Solution: The Banks' set.- 8.5 Properties of Algebraic Solutions.- 9 Copeland Value of a Solution.- 9.1 Definition of the Copeland Value.- 9.2 Computation of Some Copeland Values.- 10 From Tournaments to Choice and Voting.- 10.1 Generalized Tournaments.- 10.2 Social Choice.- 10.3 Voting with Mediators.- 10.4 Voting with Agendas.- Annex - Summary Tables.- A.1 Relations between the Main Solutions.- A.2 Properties of the Main Solutions.- A.3 Games andTournaments Concepts.- A.4 An Example.- Index of Main Notations.- References.
Organisation of the Book.- 1 Generalities.- 1.1 Definitions and Notations ..- 1.2 Finite Tournaments.- 1.3 Decomposition.- 1.4 Regularity.- 1.5 Useful Notions about General Binary Relations.- 2 Tournament Solutions.- 2.1 Majority Voting and Tournaments.- 2.2 Solution Concepts.- 2.3 Monotonicity, Strong Superset Property and Independence of Losers.- 2.4 Composition-Consistency and Regularity.- 2.5 Composition-Consistent Hulls.- 3 Scoring and Ranking Methods.- 3.1 Copeland Solution.- 3.2 Iterative Matrix Solutions.- 3.3 Markov Solution.- 3.4 Slater Solution.- 4 Multivariate Descriptions.- 4.1 Complete Euclidean Description.- 4.2 Multidimensional Scaling.- 5 Covering.- 5.1 Covering Relation and Uncovered Set.- 5.2 Iterations of the Uncovered Set.- 5.3 Dutta's Minimal Covering Set.- 5.4 Weak Covering à la Laffond and Lainé.- 5.5 Weak Covering à la Levchenkov.- 6 Tournament Game.- 6.1 Tournament Game in Pure Strategies.- 6.2 Tournament Game in Mixed Strategies.- 6.3 Properties of the Bipartisan Set.- 6.4 Method of the Minimal Gain.- 6.5 Interpretation of Tournament Games.- 7 The Contestation Process.- 7.1 Banks' Solution.- 7.2 The Tournament Equilibrium Set.- 8 Tournament Algebras and Binary Trees.- 8.1 Definition of a Tournament Algebra.- 8.2 Binary Trees.- 8.3 An Algebraic Solution: The Top-Cycle.- 8.4 An Algebraic Solution: The Banks' set.- 8.5 Properties of Algebraic Solutions.- 9 Copeland Value of a Solution.- 9.1 Definition of the Copeland Value.- 9.2 Computation of Some Copeland Values.- 10 From Tournaments to Choice and Voting.- 10.1 Generalized Tournaments.- 10.2 Social Choice.- 10.3 Voting with Mediators.- 10.4 Voting with Agendas.- Annex - Summary Tables.- A.1 Relations between the Main Solutions.- A.2 Properties of the Main Solutions.- A.3 Games andTournaments Concepts.- A.4 An Example.- Index of Main Notations.- References.
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