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Extra info for Active Networks: IFIP-TC6 4th International Working Conference, IWAN 2002 Zurich, Switzerland, December 4–6, 2002 Proceedings
Based on these observations, one can devise an eﬃcient algorithm for identifying the most preferred arm when the underlying distribution is Mallows. The algorithm proposed in  for the MPI problem, called MallowsMPI, is similar to the one used for ﬁnding the largest element in an array. However, since a stochastic environment is assumed in which the outcomes of pairwise comparisons are random variables, a single comparison of two arms ai and aj is not enough; instead, they are compared until 1/2 ∈ / qi,j − ci,j , qi,j + ci,j .
PAC Rank Elicitation. In a subsequent work by Busa-Fekete et al. , an extended version of the top-k selection problem is considered. In the PAC rank elicitation problem, the goal is to ﬁnd a ranking that is “close” to the ranking produced by the ranking procedure with high probability. To make this problem feasible, more practical ranking procedures are considered. In fact, the problem of ranking procedures like Copeland is that a minimal change of a value qi,j ≈ 12 may strongly inﬂuence the induced order relation CO .
Busa-Fekete and E. H¨ ullermeier with h(k, φ) = k/(1 − φk ). Based on this result, one can show that the “margin” min |1/2 − qi,j | i=j φ 1 , 1+φ ). around 1/2 is relatively wide; more speciﬁcally, there is no qi,j ∈ ( 1+φ Moreover, the result also implies that qi,j − qi,k = O( φ ) for arms ai , aj , ak satisfying ri = rj − = rk − − 1 with 1 < , and qi,k − qi,j = O( φ ) for arms ai , aj , ak satisfying ri = rj + = rk + + 1 with 1 < . Therefore, deciding whether an arm aj has higher or lower rank than ai (with respect to r) is easier than selecting the preferred option from two candidates aj and ak for which j, k = i.