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

作者：De Loof K^*; De Baets B; De Meyer H

来源：4OR-A Quarterly Journal of Operations Research, 2013, 11(2): 151-170.

DOI：10.1007/s10288-012-0214-z

• Dice representability
• Dice transitivity
• Independent random variables
• Reciprocal relation
• Winning probability relation

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.