*Satish K. Jain*

- Published in print:
- 2008
- Published Online:
- May 2009
- ISBN:
- 9780199239115
- eISBN:
- 9780191716935
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199239115.003.0011
- Subject:
- Economics and Finance, Development, Growth, and Environmental

This chapter extends the earlier work on domain conditions for social rationality under the method of majority decision (MMD) by deriving Inada-type necessary and sufficient conditions for ...
More

This chapter extends the earlier work on domain conditions for social rationality under the method of majority decision (MMD) by deriving Inada-type necessary and sufficient conditions for transitivity and quasi-transitivity for cases which were not covered by the earlier results. It presents a unified approach to the problem of obtaining domain conditions by formulating all conditions in terms of Latin Squares. Also, each characterization is obtained in terms of a single condition. These two together result in considerable simplification of proofs. The chapter also provides new proofs for the earlier results. Regarding acyclicity, it is shown that for any non-trivial set of binary relations containing intransitive binary relations, no condition defined only over triples can be an Inada-type necessary and sufficient condition for acyclicity under the MMD.Less

This chapter extends the earlier work on domain conditions for social rationality under the method of majority decision (MMD) by deriving Inada-type necessary and sufficient conditions for transitivity and quasi-transitivity for cases which were not covered by the earlier results. It presents a unified approach to the problem of obtaining domain conditions by formulating all conditions in terms of Latin Squares. Also, each characterization is obtained in terms of a single condition. These two together result in considerable simplification of proofs. The chapter also provides new proofs for the earlier results. Regarding acyclicity, it is shown that for any non-trivial set of binary relations containing intransitive binary relations, no condition defined only over triples can be an Inada-type necessary and sufficient condition for acyclicity under the MMD.

*Maureen T. Carroll and Steven T. Dougherty*

- Published in print:
- 2015
- Published Online:
- October 2017
- ISBN:
- 9780691164038
- eISBN:
- 9781400881338
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691164038.003.0013
- Subject:
- Mathematics, History of Mathematics

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the ...
More

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.Less

This chapter introduces a new game of tic-tac-toe that fits squarely within the body of work inspired by mathematician Leonhard Euler's findings on the so-called “Graeco-Latin squares” and the surprisingly interesting problem of arranging thirty-six officers of six different ranks and regiments. In his 1782 paper on the subject, Euler begins with the thirty-six-officer problem and ends with a conjecture about the possible sizes of Graeco-Latin squares. The chapter first explains the rules for a game based on Euler's work, and then analyzes it from a game-theoretic perspective to determine winning and drawing strategies. Along the way, the chapter explains Euler's connection to the story.

*Kurt Smith*

- Published in print:
- 2010
- Published Online:
- September 2010
- ISBN:
- 9780199583652
- eISBN:
- 9780191723155
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199583652.003.0013
- Subject:
- Philosophy, History of Philosophy, Metaphysics/Epistemology

This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter ...
More

This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter explains how synthesis, or a synthetic system, is isomorphic to a permutation group. What is more, it is shown how analysis, or the system of concepts resulting from analysis, is isomorphic to a synthetic system. This, it is argued, establishes a sense in which analysis and synthesis are ‘flip sides’ of the same conceptual coin. Since an analytic‐synthetic system is a group, and a group is a genuine mathematical system, it is seen the important role that Descartes's enumeration played in establishing a ‘mathematized’ physics.Less

This chapter shows how the combinatorial nature of bodies expresses the permutation group concept, the latter expressing the conditions underwriting a genuine mathematical system. The chapter explains how synthesis, or a synthetic system, is isomorphic to a permutation group. What is more, it is shown how analysis, or the system of concepts resulting from analysis, is isomorphic to a synthetic system. This, it is argued, establishes a sense in which analysis and synthesis are ‘flip sides’ of the same conceptual coin. Since an analytic‐synthetic system is a group, and a group is a genuine mathematical system, it is seen the important role that Descartes's enumeration played in establishing a ‘mathematized’ physics.