An Axiomatic Theory of Tournament Aggregation

An axiomatic theory for aggregation of individual preferences is developed. Many authors, since K. J. Arrow, have studied the case where every individual preference is an ordering. We study here the case where every individual preference is a tournament for instance, in “paired comparisons”. The original results obtained can be compared to those of the classic theory. For example, we prove, in this context, a generalisation of Arrow's theorem and we emphasize duality between Arrow's results and Black-Inada-Sen's results technically by means of a Galois connection between two lattices. We used social functions defined by means of “families of majorities” simple games,

[1]  B. G. Mirkin,et al.  15 – On the Problem of Reconciling Partitions* , 1975 .

[2]  Peter C. Fishburn,et al.  Collective rationality versus distribution of power for binary social choice functions , 1977 .

[3]  L. Shapley.  Simple games: an outline of the descriptive theory. , 2007, Behavioral science.

[4]  Donald J. Brown.  Aggregation of Preferences , 1975 .

[5]  Robert B. Wilson,et al.  Social choice theory without the Pareto Principle , 1972 .

[6]  R. P. Dilworth.  Review: G. Birkhoff, Lattice theory , 1950 .

[7]  Amartya Sen,et al.  A Possibility Theorem on Majority Decisions , 1966 .

[8]  B. Monjardet,et al.  Une autre preuve du théorème d'Arrow , 1978 .

[9]  Dieter Sondermann,et al.  Arrow's theorem, many agents, and invisible dictators☆ , 1972 .

[10]  J. Moon.  Topics on tournaments , 1968 .

[11]  Yasusuke Murakami.  Formal Structure of Majority Decision , 1966 .

[12]  Kit Fine.  Some Necessary and Sufficient Conditions for Representative Decision on Two Alternatives , 1972 .

[13]  Robert B. Wilson.  On the theory of aggregation , 1975 .

[14]  Bernard Monjardet,et al.  Tournois et ordres n~dians pour une opinion , 1973 .

[15]  K. May.  Intransitivity, Utility, and the Aggregation of Preference Patterns , 1954 .

[16]  C. Berge.  Graphes et hypergraphes , 1970 .

[17]  D. Black.  The theory of committees and elections , 1959 .

[18]  A. Sen,et al.  Collective Choice and Social Welfare , 2017 .

[19]  Y. Murakami.  Logic and Social Choice , 1968 .

[20]  H. Young,et al.  A Consistent Extension of Condorcet’s Election Principle , 1978 .

[21]  M. Kendall,et al.  ON THE METHOD OF PAIRED COMPARISONS , 1940 .

[22]  J. Blin.  Preference aggregation and statistical estimation , 1973 .

[23]  Peter C. Fishburn.  The Theory of Representative Majority Decision , 1971 .

[24]  Ken-ichi Inada,et al.  A Note on the Simple Majority Decision Rule , 1964 .

[25]  A. Sen,et al.  Social Choice Theory: A Re-Examination , 1977 .

[26]  J. Laffont.  Aggregation and revelation of preferences , 1979 .

[27]  J. H. Blau,et al.  A Direct Proof of Arrow's Theorem , 1972 .

[28]  P. Fishburn.  Transitive Binary Social Choices and Intraprofile Conditions , 1973 .

[29]  M. Kendall.  Rank Correlation Methods , 1949 .

[30]  A. Astié,et al.  Comparaisons par paires et problèmes de classement. Estimation et tests statistiques , 1970 .

[31]  Bernard Monjardet.  Éléments ipsoduaux du treillis distributif libre et familles de Sperner ipsotransversales , 1975, J. Comb. Theory, Ser. A.

[32]  J. H. Blau.  THE EXISTENCE OF SOCIAL WELFARE FUNCTIONS , 1957 .

[33]  Robert A. Wilson,et al.  The postulates of game theory , 1972 .

[34]  K. Arrow.  Social Choice and Individual Values , 1951 .

[35]  Kenneth O. May,et al.  A Set of Independent Necessary and Sufficient Conditions for Simple Majority Decision , 1952 .

[36]  Robert B. Wilson,et al.  The game-theoretic structure of Arrow's general possibility theorem , 1972 .

[37]  A. Tversky.  Intransitivity of preferences. , 1969 .

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

[39]  J. Neumann,et al.  Theory of games and economic behavior , 1945, 100 Years of Math Milestones.