1262:(PAPRIKA) method. The method involves the decision-maker repeatedly pairwise comparing and ranking alternatives defined on two criteria or attributes at a time and involving a trade-off, and then, if the decision-maker chooses to continue, pairwise comparisons of alternatives defined on successively more criteria. From the pairwise rankings, the relative importance of the criteria to the decision-maker, represented as weights, is determined.
566:
than A, but it is not discernible without an extremely sensitive scale. Further suppose C is larger than B, but this also is not discernible without an extremely sensitive scale. However, the difference in sizes between apples A and C is large enough that you can discern that C is larger than A without a sensitive scale. In psychophysical terms, the size difference between A and C is above the
554:, all of which can be verified to satisfy (non-stochastic) transitivity within the bounds of errors of estimates of scale locations of entities. Thus, decisions need not be deterministically transitive in order to apply probabilistic models. However, transitivity will generally hold for a large number of comparisons if models such as the BTL can be effectively applied.
565:
Some contend that indifference is not transitive. Consider the following example. Suppose you like apples and you prefer apples that are larger. Now suppose there exists an apple A, an apple B, and an apple C which have identical intrinsic characteristics except for the following. Suppose B is larger
573:
You are confronted with the three apples in pairs without the benefit of a sensitive scale. Therefore, when presented A and B alone, you are indifferent between apple A and apple B; and you are indifferent between apple B and apple C when presented B and C alone. However, when the pair A and C are
517:
For a given decision agent, if the information, objective, and alternatives used by the agent remain constant, then it is generally assumed that pairwise comparisons over those alternatives by the decision agent are transitive. Most agree upon what transitivity is, though there is debate about the
412:
508:
Thurstone used the method of pairwise comparisons as an approach to measuring perceived intensity of physical stimuli, attitudes, preferences, choices, and values. He also studied implications of the theory he developed for opinion polls and political voting (Thurstone, 1959).
251:
1255:, a structured technique for helping people deal with complex decisions. It uses pairwise comparisons of tangible and intangible factors to construct ratio scales that are useful in making important decisions.
1405:"Relative Measurement and its Generalization in Decision Making: Why Pairwise Comparisons are Central in Mathematics for the Measurement of Intangible Factors – The Analytic Hierarchy/Network Process"
1518:
Y. Chevaleyre, P.E. Dunne, U. Endriss, J. Lang, M. LemaĂ®tre, N. Maudet, J. Padget, S. Phelps, J.A. RodrĂguez-Aguilar, and P. Sousa. Issues in
Multiagent Resource Allocation. Informatica, 30:3–31,
1219:
133:
If an individual or organization expresses a preference between two mutually distinct alternatives, this preference can be expressed as a pairwise comparison. If the two alternatives are
1282:
1259:
898:
865:
832:
799:
766:
733:
442:
1096:
1063:
1030:
997:
964:
931:
484:
1129:
463:
230:
is used. Thurstone used the normal distribution in applications of the model. The simple logistic function varies by less than 0.01 from the cumulative normal
557:
Using a transitivity test one can investigate whether a data set of pairwise comparisons contains a higher degree of transitivity than expected by chance.
106:. Thurstone demonstrated that the method can be used to order items along a dimension such as preference or importance using an interval-type scale.
1557:. In T.V. Smith and W.K. Wright (Eds.), Essays in Philosophy by Seventeen Doctors of Philosophy of the University of Chicago. Chicago: Open Court.
1455:
Hansen, Paul; Ombler, Franz (2008). "A new method for scoring additive multi-attribute value models using pairwise rankings of alternatives".
407:{\displaystyle \Pr\{X_{ji}=1\}={\frac {e^{{\delta _{j}}-{\delta _{i}}}}{1+e^{{\delta _{j}}-{\delta _{i}}}}}=\sigma (\delta _{j}-\delta _{i}),}
582:
If pairwise comparisons are in fact transitive in respect to the four mentioned rules, then pairwise comparisons for a list of alternatives (
1510:
1491:
1134:
If the number of alternatives is n, and indifference is not allowed, then the number of possible preference orders for any given
226:
is often applied to pairwise comparison data to scale preferences. The BTL model is identical to
Thurstone's model if the simple
1384:
494:). For example, the scale location might represent the perceived quality of a product, or the perceived weight of an object.
17:
94:
first introduced a scientific approach to using pairwise comparisons for measurement in 1927, which he referred to as the
1143:
52:, or whether or not the two entities are identical. The method of pairwise comparison is used in the scientific study of
1363:
1240:
1529:
Bradley, R.A. and Terry, M.E. (1952). Rank analysis of incomplete block designs, I. the method of paired comparisons.
1412:
117:
in incomplete tournaments, which serves as the basis (even though not credited for a while) for methods such as the
1152:
30:
This article is about pairwise comparisons in psychology. For statistical analysis of paired comparisons, see
1358:
Nikolić D (2012) Non-parametric detection of temporal order across pairwise measurements of time delays.
1277:
95:
1324:
Oliveira, I.F.D.; Zehavi, S.; Davidov, O. (August 2018). "Stochastic transitivity: Axioms and models".
1271:
1252:
567:
1426:
1413:
Review of the Royal
Academy of Exact, Physical and Natural Sciences, Series A: Mathematics (RACSAM)
518:
transitivity of indifference. The rules of transitivity are as follows for a given decision agent.
69:
1299:
871:
838:
805:
772:
739:
706:
551:
502:
420:
223:
219:
215:
122:
1586:
1421:
1294:
31:
1069:
1036:
1003:
970:
937:
904:
1569:
469:
49:
1570:
Die
Berechnung der Turnier-Ergebnisse als ein Maximumproblem der Wahrscheinlichkeitsrechnung
1377:
Decision Making for
Leaders: The Analytic Hierarchy Process for Decisions in a Complex World
1102:
544:
99:
73:
8:
114:
44:
generally is any process of comparing entities in pairs to judge which of each entity is
1536:
David, H.A. (1988). The Method of Paired
Comparisons. New York: Oxford University Press.
1505:"Sequence A000670 (Number of preferential arrangements of n labeled elements)"
448:
231:
35:
1380:
1364:
http://www.danko-nikolic.com/wp-content/uploads/2011/09/Nikolic-Transitivity-2007.pdf
1341:
540:
487:
227:
118:
1142:!. If indifference is allowed, then the number of possible preference orders is the
1464:
1431:
1333:
1304:
570:('jnd') while the size differences between A and B and B and C are below the jnd.
1400:
211:
91:
88:
536:
103:
57:
1337:
1580:
1345:
110:
65:
61:
1404:
210:
In terms of modern psychometric theory probabilistic models, which include
1500:
1481:
498:
501:
for measurement are all closely related and belong to the same class of
1435:
98:. Thurstone linked this approach to psychophysical theory developed by
77:
53:
45:
1468:
1288:
1251:
One important application of pairwise comparisons is the widely used
1283:
Potentially all pairwise rankings of all possible alternatives
1260:
Potentially All
Pairwise RanKings of all possible Alternatives
560:
491:
113:(1929) first described a model for pairwise comparisons for
1504:
1485:
222:
models, are more aptly regarded as measurement models. The
673:
659:
655:
644:
633:
27:
Process of comparing two entities to determine preference
1546:
Thurstone, L.L. (1927). A law of comparative judgement.
141:, the following are the possible pairwise comparisons:
1499:
1480:
1323:
1155:
1105:
1072:
1039:
1006:
973:
940:
907:
874:
841:
808:
775:
742:
709:
472:
451:
423:
254:
190:
The agent is indifferent between both alternatives: "
497:
241:is judged to have more of an attribute than object
234:across the range, given an arbitrary scale factor.
214:(also called the law of comparative judgment), the
1213:
1123:
1090:
1057:
1024:
991:
958:
925:
892:
859:
826:
793:
760:
727:
478:
457:
436:
406:
1578:
1573:, Mathematische Zeitschrift 29, 1929, S. 436–460
1486:"Sequence A000142 (Factorial numbers)"
255:
34:. For the application to electoral systems, see
1379:. Pittsburgh, Pennsylvania: RWS Publications.
237:In the BTL model, the probability that object
1543:: A Theoretical Analysis. New York: J. Wiley.
688:For example, if there are three alternatives
1454:
1214:{\displaystyle \sum _{k=1}^{n}k!S_{2}(n,k),}
280:
258:
1564:. Chicago: The University of Chicago Press.
1457:Journal of Multi-Criteria Decision Analysis
1448:
700:, then the possible preference orders are:
561:Argument for intransitivity of indifference
1146:. It can be expressed as a function of n:
224:Bradley–Terry–Luce (BTL) model
216:Bradley–Terry–Luce (BTL) model
1511:On-Line Encyclopedia of Integer Sequences
1492:On-Line Encyclopedia of Integer Sequences
1425:
535:This corresponds to (xPy or xIy) being a
550:Probabilistic models also give rise to
205:
80:literature, it is often referred to as
14:
1579:
1555:The Measurement of Psychological Value
1352:
1399:
1393:
1374:
1360:Journal of Computational Neuroscience
1258:Another important application is the
577:
1368:
552:stochastic variants of transitivity
24:
1523:
1326:Journal of Mathematical Psychology
1241:Stirling number of the second kind
48:, or has a greater amount of some
25:
1598:
543:, and I being the corresponding
444:is the scale location of object
1375:Saaty, Thomas L. (1999-05-01).
1246:
512:
1317:
1205:
1193:
398:
372:
13:
1:
1310:
1541:Individual Choice Behaviours
574:shown, you prefer C over A.
539:, P being the corresponding
7:
1278:Law of comparative judgment
1265:
893:{\displaystyle c>b>a}
860:{\displaystyle c>a>b}
827:{\displaystyle b>c>a}
794:{\displaystyle b>a>c}
761:{\displaystyle a>c>b}
728:{\displaystyle a>b>c}
437:{\displaystyle \delta _{i}}
128:
125:that was proposed in 1952.
96:law of comparative judgment
10:
1603:
1501:Sloane, N. J. A.
1482:Sloane, N. J. A.
1291:pairwise comparison method
1272:Analytic Hierarchy Process
1253:Analytic Hierarchy Process
568:just noticeable difference
29:
1562:The Measurement of Values
1338:10.1016/j.jmp.2018.06.002
1144:number of total preorders
121:and is equivalent to the
1560:Thurstone, L.L. (1959).
1553:Thurstone, L.L. (1929).
1091:{\displaystyle a=b>c}
1058:{\displaystyle c>a=b}
1025:{\displaystyle a=c>b}
992:{\displaystyle b>a=c}
959:{\displaystyle b=c>a}
926:{\displaystyle a>b=c}
531:If xIy and yIz, then xIz
528:If xIy and yPz, then xPz
525:If xPy and yIz, then xPz
522:If xPy and yPz, then xPz
70:requirements engineering
1300:Stochastic Transitivity
503:stochastic transitivity
479:{\displaystyle \sigma }
220:stochastic transitivity
1295:Preference (economics)
1215:
1176:
1125:
1092:
1059:
1026:
993:
960:
927:
894:
861:
828:
795:
762:
729:
658:=) ... (>
480:
459:
438:
408:
32:paired difference test
1216:
1156:
1126:
1124:{\displaystyle a=b=c}
1093:
1060:
1027:
994:
961:
928:
895:
862:
829:
796:
763:
730:
622:) can take the form:
481:
460:
439:
409:
74:multiagent AI systems
50:quantitative property
1567:Zermelo, E. (1928).
1550:, 34, 278–286.
1548:Psychological Review
1533:, 39, 324–345.
1362:, 22(1)" pp. 5–19.
1153:
1103:
1070:
1037:
1004:
971:
938:
905:
872:
839:
806:
773:
740:
707:
545:equivalence relation
490:(the inverse of the
470:
449:
421:
252:
212:Thurstone's approach
206:Probabilistic models
100:Ernst Heinrich Weber
18:Pairwise comparisons
1539:Luce, R.D. (1959).
123:Bradley–Terry model
42:Pairwise comparison
1514:. OEIS Foundation.
1495:. OEIS Foundation.
1436:10.1007/bf03191825
1211:
1121:
1088:
1055:
1022:
989:
956:
923:
890:
857:
824:
791:
758:
725:
476:
455:
434:
404:
167:The agent prefers
144:The agent prefers
36:round-robin voting
1386:978-0-9620317-8-6
603:, ...,
578:Preference orders
541:strict weak order
488:logistic function
458:{\displaystyle i}
364:
228:logistic function
119:Elo rating system
82:paired comparison
16:(Redirected from
1594:
1515:
1496:
1473:
1472:
1469:10.1002/mcda.428
1452:
1446:
1445:
1443:
1442:
1429:
1409:
1401:Saaty, Thomas L.
1397:
1391:
1390:
1372:
1366:
1356:
1350:
1349:
1321:
1305:Condorcet method
1285:(PAPRIKA) method
1220:
1218:
1217:
1212:
1192:
1191:
1175:
1170:
1130:
1128:
1127:
1122:
1097:
1095:
1094:
1089:
1064:
1062:
1061:
1056:
1031:
1029:
1028:
1023:
998:
996:
995:
990:
965:
963:
962:
957:
932:
930:
929:
924:
899:
897:
896:
891:
866:
864:
863:
858:
833:
831:
830:
825:
800:
798:
797:
792:
767:
765:
764:
759:
734:
732:
731:
726:
485:
483:
482:
477:
464:
462:
461:
456:
443:
441:
440:
435:
433:
432:
413:
411:
410:
405:
397:
396:
384:
383:
365:
363:
362:
361:
360:
359:
358:
345:
344:
343:
321:
320:
319:
318:
317:
304:
303:
302:
287:
273:
272:
179: >
156: >
21:
1602:
1601:
1597:
1596:
1595:
1593:
1592:
1591:
1577:
1576:
1526:
1524:Further reading
1477:
1476:
1463:(3–4): 87–107.
1453:
1449:
1440:
1438:
1427:10.1.1.455.3274
1407:
1398:
1394:
1387:
1373:
1369:
1357:
1353:
1322:
1318:
1313:
1268:
1249:
1230:
1187:
1183:
1171:
1160:
1154:
1151:
1150:
1138:-value is
1104:
1101:
1100:
1071:
1068:
1067:
1038:
1035:
1034:
1005:
1002:
1001:
972:
969:
968:
939:
936:
935:
906:
903:
902:
873:
870:
869:
840:
837:
836:
807:
804:
803:
774:
771:
770:
741:
738:
737:
708:
705:
704:
684:
671:
653:
642:
631:
621:
612:
602:
595:
588:
580:
563:
515:
471:
468:
467:
450:
447:
446:
428:
424:
422:
419:
418:
392:
388:
379:
375:
354:
350:
349:
339:
335:
334:
333:
329:
322:
313:
309:
308:
298:
294:
293:
292:
288:
286:
265:
261:
253:
250:
249:
208:
131:
92:L. L. Thurstone
89:psychometrician
39:
28:
23:
22:
15:
12:
11:
5:
1600:
1590:
1589:
1575:
1574:
1565:
1558:
1551:
1544:
1537:
1534:
1525:
1522:
1521:
1520:
1516:
1497:
1475:
1474:
1447:
1420:(2): 251–318.
1392:
1385:
1367:
1351:
1315:
1314:
1312:
1309:
1308:
1307:
1302:
1297:
1292:
1286:
1280:
1275:
1267:
1264:
1248:
1245:
1228:
1222:
1221:
1210:
1207:
1204:
1201:
1198:
1195:
1190:
1186:
1182:
1179:
1174:
1169:
1166:
1163:
1159:
1132:
1131:
1120:
1117:
1114:
1111:
1108:
1098:
1087:
1084:
1081:
1078:
1075:
1065:
1054:
1051:
1048:
1045:
1042:
1032:
1021:
1018:
1015:
1012:
1009:
999:
988:
985:
982:
979:
976:
966:
955:
952:
949:
946:
943:
933:
922:
919:
916:
913:
910:
900:
889:
886:
883:
880:
877:
867:
856:
853:
850:
847:
844:
834:
823:
820:
817:
814:
811:
801:
790:
787:
784:
781:
778:
768:
757:
754:
751:
748:
745:
735:
724:
721:
718:
715:
712:
686:
685:
680:
666:
651:
640:
629:
617:
607:
600:
593:
586:
579:
576:
562:
559:
537:total preorder
533:
532:
529:
526:
523:
514:
511:
475:
454:
431:
427:
415:
414:
403:
400:
395:
391:
387:
382:
378:
374:
371:
368:
357:
353:
348:
342:
338:
332:
328:
325:
316:
312:
307:
301:
297:
291:
285:
282:
279:
276:
271:
268:
264:
260:
257:
218:, and general
207:
204:
130:
127:
109:Mathematician
104:Gustav Fechner
58:voting systems
26:
9:
6:
4:
3:
2:
1599:
1588:
1587:Psychometrics
1585:
1584:
1582:
1572:
1571:
1566:
1563:
1559:
1556:
1552:
1549:
1545:
1542:
1538:
1535:
1532:
1528:
1527:
1517:
1513:
1512:
1506:
1502:
1498:
1494:
1493:
1487:
1483:
1479:
1478:
1470:
1466:
1462:
1458:
1451:
1437:
1433:
1428:
1423:
1419:
1415:
1414:
1406:
1403:(June 2008).
1402:
1396:
1388:
1382:
1378:
1371:
1365:
1361:
1355:
1347:
1343:
1339:
1335:
1331:
1327:
1320:
1316:
1306:
1303:
1301:
1298:
1296:
1293:
1290:
1287:
1284:
1281:
1279:
1276:
1273:
1270:
1269:
1263:
1261:
1256:
1254:
1244:
1242:
1238:
1234:
1227:
1208:
1202:
1199:
1196:
1188:
1184:
1180:
1177:
1172:
1167:
1164:
1161:
1157:
1149:
1148:
1147:
1145:
1141:
1137:
1118:
1115:
1112:
1109:
1106:
1099:
1085:
1082:
1079:
1076:
1073:
1066:
1052:
1049:
1046:
1043:
1040:
1033:
1019:
1016:
1013:
1010:
1007:
1000:
986:
983:
980:
977:
974:
967:
953:
950:
947:
944:
941:
934:
920:
917:
914:
911:
908:
901:
887:
884:
881:
878:
875:
868:
854:
851:
848:
845:
842:
835:
821:
818:
815:
812:
809:
802:
788:
785:
782:
779:
776:
769:
755:
752:
749:
746:
743:
736:
722:
719:
716:
713:
710:
703:
702:
701:
699:
695:
691:
683:
679:
675:
669:
665:
661:
657:
650:
646:
639:
635:
628:
625:
624:
623:
620:
616:
610:
606:
599:
592:
585:
575:
571:
569:
558:
555:
553:
548:
546:
542:
538:
530:
527:
524:
521:
520:
519:
510:
506:
504:
500:
495:
493:
489:
473:
465:
452:
429:
425:
401:
393:
389:
385:
380:
376:
369:
366:
355:
351:
346:
340:
336:
330:
326:
323:
314:
310:
305:
299:
295:
289:
283:
277:
274:
269:
266:
262:
248:
247:
246:
244:
240:
235:
233:
229:
225:
221:
217:
213:
203:
201:
197:
194: =
193:
188:
186:
182:
178:
174:
170:
165:
163:
159:
155:
151:
147:
142:
140:
136:
126:
124:
120:
116:
115:chess ranking
112:
111:Ernst Zermelo
107:
105:
101:
97:
93:
90:
85:
83:
79:
75:
71:
67:
66:public choice
63:
62:social choice
59:
56:, attitudes,
55:
51:
47:
43:
37:
33:
19:
1568:
1561:
1554:
1547:
1540:
1530:
1508:
1489:
1460:
1456:
1450:
1439:. Retrieved
1417:
1411:
1395:
1376:
1370:
1359:
1354:
1329:
1325:
1319:
1257:
1250:
1247:Applications
1236:
1232:
1225:
1223:
1139:
1135:
1133:
697:
693:
689:
687:
681:
677:
667:
663:
648:
637:
626:
618:
614:
608:
604:
597:
590:
583:
581:
572:
564:
556:
549:
534:
516:
513:Transitivity
507:
496:
445:
416:
242:
238:
236:
209:
199:
195:
191:
189:
184:
180:
176:
172:
168:
166:
161:
157:
153:
149:
145:
143:
138:
134:
132:
108:
86:
81:
41:
40:
499:Rasch model
54:preferences
1531:Biometrika
1441:2008-12-22
1311:References
87:Prominent
78:psychology
1422:CiteSeerX
1346:0022-2496
1332:: 25–35.
1289:PROMETHEE
1239:) is the
1158:∑
474:σ
426:δ
390:δ
386:−
377:δ
370:σ
352:δ
347:−
337:δ
311:δ
306:−
296:δ
46:preferred
1581:Category
1266:See also
670:−1
611:−1
129:Overview
1503:(ed.).
1484:(ed.).
1235:,
596:,
589:,
486:is the
1424:
1383:
1344:
1224:where
696:, and
613:, and
417:where
198:" or "
183:" or "
160:" or "
1519:2006.
1408:(PDF)
1274:(AHP)
672:(>
654:(>
643:(>
632:(>
492:logit
232:ogive
171:over
148:over
76:. In
1509:The
1490:The
1381:ISBN
1342:ISSN
1083:>
1044:>
1017:>
978:>
951:>
912:>
885:>
879:>
852:>
846:>
819:>
813:>
786:>
780:>
753:>
747:>
720:>
714:>
245:is:
137:and
102:and
72:and
1465:doi
1432:doi
1418:102
1334:doi
674:XOR
660:XOR
656:XOR
645:XOR
634:XOR
200:xIy
185:yPx
175:: "
162:xPy
152:: "
1583::
1507:.
1488:.
1461:15
1459:.
1430:.
1416:.
1410:.
1340:.
1330:85
1328:.
1243:.
692:,
676:=)
662:=)
647:=)
636:=)
547:.
505:.
466:;
256:Pr
202:"
187:"
164:"
84:.
68:,
64:,
60:,
1471:.
1467::
1444:.
1434::
1389:.
1348:.
1336::
1237:k
1233:n
1231:(
1229:2
1226:S
1209:,
1206:)
1203:k
1200:,
1197:n
1194:(
1189:2
1185:S
1181:!
1178:k
1173:n
1168:1
1165:=
1162:k
1140:n
1136:n
1119:c
1116:=
1113:b
1110:=
1107:a
1086:c
1080:b
1077:=
1074:a
1053:b
1050:=
1047:a
1041:c
1020:b
1014:c
1011:=
1008:a
987:c
984:=
981:a
975:b
954:a
948:c
945:=
942:b
921:c
918:=
915:b
909:a
888:a
882:b
876:c
855:b
849:a
843:c
822:a
816:c
810:b
789:c
783:a
777:b
756:b
750:c
744:a
723:c
717:b
711:a
698:c
694:b
690:a
682:n
678:A
668:n
664:A
652:3
649:A
641:2
638:A
630:1
627:A
619:n
615:A
609:n
605:A
601:3
598:A
594:2
591:A
587:1
584:A
453:i
430:i
402:,
399:)
394:i
381:j
373:(
367:=
356:i
341:j
331:e
327:+
324:1
315:i
300:j
290:e
284:=
281:}
278:1
275:=
270:i
267:j
263:X
259:{
243:i
239:j
196:y
192:x
181:x
177:y
173:x
169:y
158:y
154:x
150:y
146:x
139:y
135:x
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.