Theorems in Discrete Mathematics: Akra Bazzi Method, Analyst's Traveling Salesman Theorem, Arrow's Impossibility Theorem, Cameron Erd S Conjecture, Ch

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 34. Chapters: Akra Bazzi method, Analyst's traveling salesman theorem, Arrow's impossibility theorem, Cameron Erd s conjecture, Chomsky Schutzenberger theorem, Curtis Hedlund Lyndon theorem, Dinitz conjecture, Erd s Anning theorem, Erd s Ko Rado theorem, Erd s Szekeres theorem, Erd s Szemeredi theorem, Folkman's theorem, Hales Jewett theorem, Holland's schema theorem, Kemnitz's conjecture, Kruskal's tree theorem, May's theorem, Milliken's tree theorem, Milliken Taylor theorem, Minimax, Myhill Nerode theorem, Noisy-channel coding theorem, Parthasarathy's theorem, Pigeonhole principle, Rado's theorem (Ramsey theory), Schur's theorem, Sprague Grundy theorem, Stanley Wilf conjecture, Star height problem, Star of David theorem, Szemeredi's theorem, Theorem on friends and strangers, Tutte Berge formula, Universal approximation theorem, Van der Waerden's theorem. Excerpt: In social choice theory, Arrow s impossibility theorem, the General Possibility Theorem, or Arrow s paradox, states that, when voters have three or more distinct alternatives (options), no rank order voting system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specific set of criteria. These criteria are called unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of election theory as it is further interpreted by the Gibbard Satterthwaite theorem. The theorem is named after economist Kenneth Arrow, who demonstrated the theorem in his Ph.D. thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare." In short, the theorem states that no rank-order voting system can be designed that satisfies these three "fairness" criteria: Voting systems that use cardinal utility (which conveys more information than rank orders; see the subsection discussing the cardinal utility approach to overcoming the negative conclusion) are not covered by the theorem. The theorem can also be sidestepped by weakening the notion of independence. Arrow, like many economists, rejected cardinal utility as a meaningful tool for expressing social welfare, and so focused his theorem on preference rankings. The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem. The need to aggregate preferences occurs in many different disciplines: in welfare economics, where one attempts to find an economic outcome which would be acceptable