## A Formula for Incorporating Weights into Scoring Rules (1998)

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Venue: | Theoretical Computer Science |

Citations: | 18 - 1 self |

### BibTeX

@ARTICLE{Fagin98aformula,

author = {Ronald Fagin and Edward L. Wimmers},

title = {A Formula for Incorporating Weights into Scoring Rules},

journal = {Theoretical Computer Science},

year = {1998},

volume = {239},

pages = {309--338}

}

### Years of Citing Articles

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### Abstract

A "scoring rule" is an assignment of a value to every tuple (of varying sizes). This paper is concerned with the issue of how to modify a scoring rule to apply to the case where weights are assigned to the importance of each argument. We give an explicit formula for incorporating weights that can be applied no matter what the underlying scoring rule is. The formula is surprisingly simple, in that it involves far fewer terms than one might have guessed. It has three further desirable properties. The first desirable property is that when all of the weights are equal, then the result is obtained by simply using the underlying scoring rule. Intuitively, this says that when all of the weights are equal, then this is the same as considering the unweighted case. The second desirable property is that if a particular argument has zero weight, then that argument can be dropped without affecting the value of the result. The third desirable property is that the value of the result is a continuous ...

### Citations

3651 |
Fuzzy sets
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- 1965
(Show Context)
Citation Context ...s and a single score per attribute takes place in fuzzy logic. In particular, a score must be assigned to a conjunction A 1sA 2 that is a function of the scores of A 1 and A 2 . In his original paper =-=[Za65]-=-, Zadeh defined the score of A 1sA 2 to be the min of the scores of A 1 and A 2 . Similarly, he defined the score of the disjunction A 1 A 2 to be the max of the scores of A 1 and A 2 . Zadeh's choice... |

1457 |
Decisions with Multiple Objectives: Preferences and Value tradeoffs
- Keeney, Raiffa
- 1976
(Show Context)
Citation Context ...ncludes work on computing optimal indifference curves (which depend on user-supplied weightings). That work is only tangentially related to ours, since the focus there is on computing optimality. See =-=[KR76] for a mor-=-e complete discussion. In Example 8 of Section 6, we discussed voting schemes as an example where there is a "competing" approach to ours for incorporating weights. As we noted, future resea... |

554 | The QBIC project: Querying images by content using color, texture and shape - Niblack, Barber, et al. - 1993 |

463 |
Fuzzy sets as a basis for a theory of a possibility, Fuzzy Sets and Systems
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Citation Context ...ations in Fuzzy Set Theory”). Their underlying scenario and goals are actually quite di erent from ours (for example, instead of dealing with probabilities i, they deal with possibility distributions =-=[36]-=-). Nonetheless, it is instructive to compare the explicit formula that they obtain for the “weighted min”, and see how it fares under our criteria. Let X be a tuple over I with range [0; 1], let f(X )... |

358 | Combining Fuzzy Information from Multiple Systems
- Fagin
- 1996
(Show Context)
Citation Context ...ch is being developed at the IBM Almaden Research Center, and which provides access to a variety of data sources, including multimedia. See [CHS+95, CHN+95] for a discussion of the Garlic system, and =-=[Fa96] and [CG96-=-] (along with Section 11 of this paper) for a discussion of algorithms with a low middleware cost for computing distributed scores (where different "black boxes" produce the various scores t... |

340 |
Evaluation techniques for storage hierarchies
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- 1970
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Citation Context ...e). Typically, to make room for this page, another page is removed from the cache according to some page replacement algorithm. For example, the LRU (“least recently used”) page replacement algorithm =-=[21]-=- removes the page that has been least recently used. There are some situations where certain pages are more expensive to retrieve than others. For example, if a page is retrieved from a remote site, w... |

278 |
Manipulation of Voting Schemes: a General Result
- Gibbard
- 1973
(Show Context)
Citation Context ...he best alternative under that criterion, the secondbest, and so on. There is then an aggregation method that decides on an overall ranking of the alternatives, based on the criteria. A voting scheme =-=[2, 16]-=-, such as that for judging gure skating, ts this paradigm, where we think of the contestants as the alternatives, the judges (or voters) as the criteria, and the rule that tells the order of nish of t... |

207 | Manipulation of voting schemes - Gibbard - 1973 |

156 |
Integral representation without additivity
- Schmeidler
- 1986
(Show Context)
Citation Context ...ta (X) is a continuous function of \Theta, for each fixed X. 4 In earlier versions of this paper, we referred to comonotonicity as order-equivalence. We changed the name to coincide with Schmeidler's =-=[Sc86]-=- term he uses in the context of Choquet integrals (see Section 10.2). Proof: The proof is given in Section 7. Since we allow the possibility that the unweighted rule not be a continuous function of X ... |

142 | Competitive Algorithms for On-line Problems - Manasse, McGeoch, et al. - 1988 |

135 | Fuzzy integral in multicriteria decision making
- Grabisch
- 1995
(Show Context)
Citation Context ... unweighted rule, no matter what the unweighted rule is. 10.2 Choquet integral The Choquet integral [Ch55, Sc86] of a function generates a weighted version of the function. A nice overview appears in =-=[Gr96]. Le-=-t g be a function with finite domain X and range [0; 1]. Let �� be a set function (that is, a function with domain the power set of X) and range [0; 1]. It is common to make various assumptions ab... |

107 | Towards heterogeneous multimedia information systems
- CAREY, HAAS, et al.
- 1995
(Show Context)
Citation Context ...e score. Such an approach is taken by the Garlic system, which is being developed at the IBM Almaden Research Center, and which provides access to a variety of data sources, including multimedia. See =-=[6, 10]-=- for a discussion of the Garlic system, and [14] and [7] (along with Section 11 of this paper) for a discussion of algorithms with a low middleware cost for computing distributed scores (where di eren... |

88 | Memory versus randomization in online algorithms - Raghavan, Snir - 1987 |

83 | Optimizing Queries over Multimedia Repositories
- Chaudhuri, Gravano
- 1996
(Show Context)
Citation Context ... developed at the IBM Almaden Research Center, and which provides access to a variety of data sources, including multimedia. See [CHS+95, CHN+95] for a discussion of the Garlic system, and [Fa96] and =-=[CG96] (along wi-=-th Section 11 of this paper) for a discussion of algorithms with a low middleware cost for computing distributed scores (where different "black boxes" produce the various scores that must be... |

74 | New results on server problems
- Chrobak, Karloff, et al.
- 1990
(Show Context)
Citation Context ...=d · g(X ); (12) where d is the degree of di culty of the dive. 6 This example is joint with Alain Azagury. 7 We note that there are several papers on weighted versions of page replacement algorithms =-=[9, 20, 25, 29, 33, 34]-=-.s322 R. Fagin, E.L. Wimmers / Theoretical Computer Science 239 (2000) 309–338 Let us say that the set I of judges is of size seven (this is the standard number of judges in international competition)... |

73 | Online file caching - Young - 1998 |

70 |
Social Choice and Individual Values 2nd Edition
- Arrow
- 1963
(Show Context)
Citation Context ...he best alternative under that criterion, the secondbest, and so on. There is then an aggregation method that decides on an overall ranking of the alternatives, based on the criteria. A voting scheme =-=[2, 16]-=-, such as that for judging gure skating, ts this paradigm, where we think of the contestants as the alternatives, the judges (or voters) as the criteria, and the rule that tells the order of nish of t... |

70 |
Theory of capacities, Ann
- Choquet
- 1953
(Show Context)
Citation Context ...hodology has the advantage that it always gives us a (simple) way to obtain a weighted rule from an unweighted rule, no matter what the unweighted rule is. 10.2. Choquet integral The Choquet integral =-=[8, 27]-=- of a function generates a weighted version of the function. A nice overview appears in [17]. Let g be a function with nite domain X and range [0; 1]. Let be a set function (that is, a function with d... |

51 |
Social choice and multicriterion decision-making
- KJ, Raynaud
- 1986
(Show Context)
Citation Context ...to the jth component of the rule f that we described above. Scoring in figure skating fits the paradigm of multicriterion decision-making, which has a large literature in economics (see, for example, =-=[AR86]-=-). Here, there are a set of alternatives, and a set of relevant criteria. For each criterion, there is a permutation of the set of alternatives, which tells the best alternative under that criterion, ... |

47 | The QUBIC Project: Querying Images by Content Using - Niblack - 1993 |

46 | On-line caching as cache size varies - Young - 1991 |

44 |
Automatic Information Retrieval
- Salton
- 1980
(Show Context)
Citation Context ...e weighted rule (22) for min over tuples with a di erent range than [0; 1], and in particular with range (−∞; +∞). Method 3. The third method is from a paper by Salton et al. on information retrieval =-=[26]-=-, and deals with the case where a version of the Euclidean distance is used in the unweighted case. Let X be a tuple over I, let �� i∈I f(X )= x2 i (23) card(I)s334 R. Fagin, E.L. Wimmers / Theoretica... |

42 |
On the analytic formalism of the theory of fuzzy sets
- Bellman, Giertz
- 1973
(Show Context)
Citation Context ...the scores of A1 and A2. Similarly, he de ned the score of the disjunction A1∨A2 to be the max of the scores of A1 and A2. Zadeh’s choices were later justi ed by a famous result of Bellman and Giertz =-=[5]-=-, which was extended and simpli ed by Yager [32], Voxman and Goetschel [30], Dubois and Prade [12], and Wimmers [31]. They showed that min and max are the unique choices that should be assigned to the... |

35 |
Criteria Aggregation and Ranking of Alternatives in the Framework of Fuzzy Set Theory
- Dubois, Prade
- 1984
(Show Context)
Citation Context ...of A 1 and A 2 . Zadeh's choices were later justified by a famous result of Bellman and Giertz [BG73], which was extended and simplified by Yager [Ya82], Voxman and Goetschel [VG83], Dubois and Prade =-=[DP84]-=-, and Wimmers [Wi98]. They showed that min and max are the unique choices that should be assigned to the conjunction and disjunction, respectively, that fulfill certain natural conditions. There is a ... |

28 | Querying Multimedia Data from Multiple Repositories by Content: the Garlic Project
- Cody, al
- 1995
(Show Context)
Citation Context ...e score. Such an approach is taken by the Garlic system, which is being developed at the IBM Almaden Research Center, and which provides access to a variety of data sources, including multimedia. See =-=[6, 10]-=- for a discussion of the Garlic system, and [14] and [7] (along with Section 11 of this paper) for a discussion of algorithms with a low middleware cost for computing distributed scores (where di eren... |

27 |
Weighted minimum and maximum operations in fuzzy set theory
- Dubois, Prade
- 1986
(Show Context)
Citation Context ...methods deals only with one or a few particular unweighted rules, rather than, as in our approach, with arbitrary unweighted rules. Method 1. The rst method is inspired by a paper by Dubois and Prade =-=[13]-=-. It deals with the case where the min function is used in the unweighted case (in fact, the title of Dubois and Prade’s paper is “Weighted Minimum and Maximum Operations in Fuzzy Set Theory”). Their ... |

15 |
Fuzzy Sets as a Basis for Theory of
- Zadeh
- 1978
(Show Context)
Citation Context ...ns in Fuzzy Set Theory"). Their underlying scenario and goals are actually quite different from ours (for example, instead of dealing with probabilities ` i , they deal with possibility distribut=-=ions [Za78]). Nonethe-=-less, it is instructive to compare the explicit formula that they obtain for the "weighted min", and see how it fares under our criteria. Let X be a tuple over I with range [0; 1], let f(X) ... |

12 |
Quelques commentaires sur les prolongements de capacit\’es, Se’minaire de
- Dellacherie
- 1971
(Show Context)
Citation Context ...0 (x) ? g 0 (y). Since in our case we have g(x i ) = ` i , it is clear that Schmeidler's notion of comonotonicity gives exactly our notion of comonotonicity (this is why we use the term). Dellacherie =-=[De70] sho-=-ws that the Choquet integral is additive for comonotonic functions. It follows easily that the Choquet integral of \Theta (with respect to an arbitrary ��) is locally linear. Marinacci (personal c... |

12 | A charcterization of Min as a procedure for exploiting valued preference relations and related results - Pirlot - 1995 |

8 | Social choice and individual values”, Yales - Arrow - 1963 |

6 |
Rating Skating
- Bassett, Persky
- 1994
(Show Context)
Citation Context ...ther than our method) to find a weighted version 8 For example, take `1 ; a1 ; a2 ; b1 ; b2 respectively to be :93; :1; 0; 0; 1. 9 For simplicity, we are ignoring ties throughout this example. 10 See =-=[BP94]-=- for a discussion of the rules of figure skating. of a voting scheme (at least in the case where the weights are rational). We simply allow multiple copies of voters, where the number of multiple copi... |

6 |
Some Procedures for Selecting Fuzzy Set-Theoretic Operations
- Yager
- 1982
(Show Context)
Citation Context ...f the disjunction A 1 A 2 to be the max of the scores of A 1 and A 2 . Zadeh's choices were later justified by a famous result of Bellman and Giertz [BG73], which was extended and simplified by Yager =-=[Ya82]-=-, Voxman and Goetschel [VG83], Dubois and Prade [DP84], and Wimmers [Wi98]. They showed that min and max are the unique choices that should be assigned to the conjunction and disjunction, respectively... |

4 | Practical and Theoretical Issues in Prefetching and Caching - Tomkins - 1997 |

3 |
Weighted Minimum and Maximum
- Dubois, Prade
- 1986
(Show Context)
Citation Context ...thods deals only with one or a few particular unweighted rules, rather than, as in our approach, with arbitrary unweighted rules. Method 1: The first method is inspired by a paper by Dubois and Prade =-=[DP86]. It deals-=- with the case where the min function is used in the unweighted case (in fact, the title of Dubois and Prade's paper is "Weighted Minimum and Maximum Operations in Fuzzy Set Theory"). Their ... |

3 | Allowing users to weight search terms - Fagin, Maarek - 2000 |

3 |
Minimal Bellman-Giertz Theorems
- Wimmers
(Show Context)
Citation Context ...eh's choices were later justified by a famous result of Bellman and Giertz [BG73], which was extended and simplified by Yager [Ya82], Voxman and Goetschel [VG83], Dubois and Prade [DP84], and Wimmers =-=[Wi98]-=-. They showed that min and max are the unique choices that should be assigned to the conjunction and disjunction, respectively, that fulfill certain natural conditions. There is a large literature on ... |

3 |
A note on the characterization of the max and min operators
- Voxman, Goetschel
- 1983
(Show Context)
Citation Context ... A1∨A2 to be the max of the scores of A1 and A2. Zadeh’s choices were later justi ed by a famous result of Bellman and Giertz [5], which was extended and simpli ed by Yager [32], Voxman and Goetschel =-=[30]-=-, Dubois and Prade [12], and Wimmers [31]. They showed that min and max are the unique choices that should be assigned to the conjunction and disjunction, respectively, that ful ll certain natural con... |

3 |
On-line le caching
- Young
- 2002
(Show Context)
Citation Context ...=d · g(X ); (12) where d is the degree of di culty of the dive. 6 This example is joint with Alain Azagury. 7 We note that there are several papers on weighted versions of page replacement algorithms =-=[9, 20, 25, 29, 33, 34]-=-.s322 R. Fagin, E.L. Wimmers / Theoretical Computer Science 239 (2000) 309–338 Let us say that the set I of judges is of size seven (this is the standard number of judges in international competition)... |

3 |
Fuzzy Set Theory. 3rd ed
- Zimmermann
- 1996
(Show Context)
Citation Context ...respectively, that ful ll certain natural conditions. There is a large literature on other possible choices for scoring rules in fuzzy logic: see, for example, the discussion in Zimmermann’s textbook =-=[37]-=-. Another important case where there are multiple attributes and a single score per attribute arises in queries in multimedia database systems, which we now discuss. A database system faces the task o... |

2 |
On the Analytic Formalism of the Theory
- Bellman, Giertz
- 1973
(Show Context)
Citation Context ... of A 1 and A 2 . Similarly, he defined the score of the disjunction A 1 A 2 to be the max of the scores of A 1 and A 2 . Zadeh's choices were later justified by a famous result of Bellman and Giertz =-=[BG73]-=-, which was extended and simplified by Yager [Ya82], Voxman and Goetschel [VG83], Dubois and Prade [DP84], and Wimmers [Wi98]. They showed that min and max are the unique choices that should be assign... |

2 |
A Linear Transform Scheme for Combining Weights into Scores
- Sung
- 1998
(Show Context)
Citation Context ...ons, where, say, computational simplicity is more important than strict monotonicity and translation invariance, when (21) would be preferable to use rather than the weighting formula. Method 2: Sung =-=[Su98]-=- recently wrote a follow-on to our paper, in which he considers other possible desiderata. In doing so, he develops techniques for obtaining a weighted rule from an unweighted rule that applies in cer... |

2 |
A Note on the Characterization
- Voxman, Goetschel
- 1983
(Show Context)
Citation Context ...be the max of the scores of A 1 and A 2 . Zadeh's choices were later justified by a famous result of Bellman and Giertz [BG73], which was extended and simplified by Yager [Ya82], Voxman and Goetschel =-=[VG83]-=-, Dubois and Prade [DP84], and Wimmers [Wi98]. They showed that min and max are the unique choices that should be assigned to the conjunction and disjunction, respectively, that fulfill certain natura... |

1 | Additive Homogeneity of Logical Connectives for Membership Functions, in Analysis of Fuzzy - Alsina, Trillas - 1987 |

1 | Theory of Capacities, Ann. de l'Institut Fourier 5 - Choquet - 1955 |

1 |
Fuzzy Set Theory, Third Edition
- Zimmermann
- 1996
(Show Context)
Citation Context ...espectively, that fulfill certain natural conditions. There is a large literature on other possible choices for scoring rules in fuzzy logic: see, for example, the discussion in Zimmermann's textbook =-=[Zi96]-=-. Another important case where there are multiple attributes and a single score per attribute arises in queries in multimedia database systems, which we now discuss. A database system faces the task o... |

1 | Allowing users to weight search terms in information retrieval - Fagin, Maarek - 1998 |