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A necessary 4-cycle condition for dice representability of reciprocal relations

Karel De Loof (UGent) , Bernard De Baets (UGent) and Hans De Meyer (UGent)
(2013) 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH. 11(2). p.151-170
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Abstract
The comparison of independent random variables can be modeled by a set of dice and a reciprocal relation expressing the winning probability of one dice over another. It is well known that dice transitivity is a necessary 3-cycle condition for a reciprocal relation to be dice representable, i.e. to be the winning probability relation of a set of dice. Although this 3-cycle condition is sufficient for a rational-valued reciprocal relation on a set of three elements to be dice representable, it has been shown that this is no longer the case for sets consisting of four or more elements. In this contribution, we provide a necessary 4-cycle condition for dice representability of reciprocal relations. Moreover, we show that our condition is sufficient in the sense that a given rational-weighted 4-cycle and reciprocally weighted inverse cycle, both fulfilling the 4-cycle condition, can be extended to a winning probability graph representing a dice-representable reciprocal relation on four elements.
Keywords
DECISION-MAKING, CHOICE, CYCLE-TRANSITIVITY, RANDOM-VARIABLES, Winning probability relation, Reciprocal relation, Independent random variables, Dice representability, Dice transitivity

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MLA
De Loof, Karel, et al. “A Necessary 4-Cycle Condition for Dice Representability of Reciprocal Relations.” 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH, vol. 11, no. 2, 2013, pp. 151–70, doi:10.1007/s10288-012-0214-z.
APA
De Loof, K., De Baets, B., & De Meyer, H. (2013). A necessary 4-cycle condition for dice representability of reciprocal relations. 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH, 11(2), 151–170. https://doi.org/10.1007/s10288-012-0214-z
Chicago author-date
De Loof, Karel, Bernard De Baets, and Hans De Meyer. 2013. “A Necessary 4-Cycle Condition for Dice Representability of Reciprocal Relations.” 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH 11 (2): 151–70. https://doi.org/10.1007/s10288-012-0214-z.
Chicago author-date (all authors)
De Loof, Karel, Bernard De Baets, and Hans De Meyer. 2013. “A Necessary 4-Cycle Condition for Dice Representability of Reciprocal Relations.” 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH 11 (2): 151–170. doi:10.1007/s10288-012-0214-z.
Vancouver
1.
De Loof K, De Baets B, De Meyer H. A necessary 4-cycle condition for dice representability of reciprocal relations. 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH. 2013;11(2):151–70.
IEEE
[1]
K. De Loof, B. De Baets, and H. De Meyer, “A necessary 4-cycle condition for dice representability of reciprocal relations,” 4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH, vol. 11, no. 2, pp. 151–170, 2013.
@article{4167395,
  abstract     = {{The comparison of independent random variables can be modeled by a set of dice and a reciprocal relation expressing the winning probability of one dice over another. It is well known that dice transitivity is a necessary 3-cycle condition for a reciprocal relation to be dice representable, i.e. to be the winning probability relation of a set of dice. Although this 3-cycle condition is sufficient for a rational-valued reciprocal relation on a set of three elements to be dice representable, it has been shown that this is no longer the case for sets consisting of four or more elements. In this contribution, we provide a necessary 4-cycle condition for dice representability of reciprocal relations. Moreover, we show that our condition is sufficient in the sense that a given rational-weighted 4-cycle and reciprocally weighted inverse cycle, both fulfilling the 4-cycle condition, can be extended to a winning probability graph representing a dice-representable reciprocal relation on four elements.}},
  author       = {{De Loof, Karel and De Baets, Bernard and De Meyer, Hans}},
  issn         = {{1619-4500}},
  journal      = {{4OR-A QUARTERLY JOURNAL OF OPERATIONS RESEARCH}},
  keywords     = {{DECISION-MAKING,CHOICE,CYCLE-TRANSITIVITY,RANDOM-VARIABLES,Winning probability relation,Reciprocal relation,Independent random variables,Dice representability,Dice transitivity}},
  language     = {{eng}},
  number       = {{2}},
  pages        = {{151--170}},
  title        = {{A necessary 4-cycle condition for dice representability of reciprocal relations}},
  url          = {{http://doi.org/10.1007/s10288-012-0214-z}},
  volume       = {{11}},
  year         = {{2013}},
}
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