# On the Likelihood of Cyclic Comparisons

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• Mathematics
• Psychology

## Abstract

C:\ariel\work-new\uzi\transitiv On the Likelihood of Cyclic Comparisons Ariel Rubinstein University of Tel Aviv Cafes University of Tel Aviv and New York University and Uzi Segal Boston College April 2011 Abstract We investigate the procedure of "random sampling" where the alternatives are random variables. When comparing any two alternatives, the decision maker samples each of the alternatives once and ranks them according to the comparison between the two realizations. Our main result is that when applied to three alternatives, the procedure yields a cycle with a probability bounded above by 827 . Bounds are also obtained for other related procedures. Keywords: Transitivity, preference formation, the paradox of nontransitive dice Acknowledgment: We thank Noga Alon and Jason Dana for their comments. Page 1 1. Introduction An experimenter would like to prove that people hold transitive preferences. He asks a psychologist, who thinks otherwise, to suggest 10 triples of lotteries that in his view are likely to lead to cycles. He requires that no two lotteries in the same triple have a common outcome and for simplicity he also requires that each lottery has three outcomes at most. The psychologist provides the experimenter with ten triple of lotteries Ai,Bi,Cii1,…,10. Each of the subjects is asked to make the thirty binary choices, three for each triple. A person is said to reveal a cycle in triple i if his choices from Ai,Bi, Bi,Ci, Ai,Ci are either Ai, Bi, and Ci, or Bi, Ci and Ai. For each subject, the experimenter counts the number of cycles (out of a possible ten), and reports the following results: # of cycles 0 1 2 3 4 5 6 7 8 9 10 % of subjects 73 23 3 1 0 0 0 0 0 0 0 The experimenter claims that the data can be nicely explained by a theory according to which the decision maker activates a transitive preference relation and there is a 3% chance that he makes a mistake when making a choice. We show that results which can be explained in this way are al

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