750:
794:, a preference relation with a loop is not transitive. For if it is, each option in the loop is preferred to each option, including itself. This can be illustrated for this example of a loop among A, B, and C. Assume the relation is transitive. Then, since A is preferred to B and B is preferred to C, also A is preferred to C. But then, since C is preferred to A, also A is preferred to A.
885:
The first argument of the relation is a row and the second one is a column. Ones indicate the relation holds, zero indicates that it does not hold. Now, notice that the following statement is true for any pair of elements x and y drawn (with replacement) from the set {rock, scissors, paper}: If x
822:
is an example. The relation over rock, paper, and scissors is "defeats", and the standard rules of the game are such that rock defeats scissors, scissors defeats paper, and paper defeats rock. Furthermore, it is also true that scissors does not defeat rock, paper does not defeat scissors, and rock
814:
possesses cycles but is transitive. Now, consider the relation "is an enemy of" and suppose that the relation is symmetric and satisfies the condition that for any country, any enemy of an enemy of the country is not itself an enemy of the country. This is an example of an antitransitive relation
979:
tends to eliminate "intransitive loops" when large numbers of voters participate because the overall assessment criteria for voters balances out. For instance, voters may prefer candidates on several different units of measure such as by order of social consciousness or by order of most fiscally
669:
513:
173:
437:
258:
753:
Sometimes, when people are asked their preferences through a series of binary questions, they will give logically impossible responses: 1 is better than 2, and 2 is better than 3, but 3 is better than
518:
787:
are examples. Real combative relations of competing species, strategies of individual animals, and fights of remote-controlled vehicles in BattleBots shows ("robot
Darwinism") can be cyclic as well.
324:: in some instances lodge A recognizes lodge B, and lodge B recognizes lodge C, but lodge A does not recognize lodge C. Thus the recognition relation among Masonic lodges is intransitive.
762:
is often used when speaking of scenarios in which a relation describes the relative preferences between pairs of options, and weighing several options produces a "loop" of preference:
347:
relation is not transitive, but it still contains some transitivity: for instance, humans feed on rabbits, rabbits feed on carrots, and humans also feed on carrots.
89:
84:
62:
A relation is transitive if, whenever it relates some A to some B, and that B to some C, it also relates that A to that C. Some authors call a relation
664:{\displaystyle {\begin{aligned}&\forall a,b,c:aRb\land aRc\implies \lnot (bRc)\\&\forall a,b,c:aRc\land bRc\implies \lnot (aRb)\end{aligned}}}
1081:
966:. Economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).
357:
983:
In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units of measure in assessing candidates.
1343:
180:
1056:
1039:
1243:
1218:
499:. If player A defeated player B and player B defeated player C, A can have never played C, and therefore, A has not defeated C.
823:
does not defeat paper. Finally, it is also true that no option defeats itself. This information can be depicted in a table:
17:
810:
Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, an
290:
3 (6 is a multiple of 3), but 2 is neither a multiple nor a divisor of 3. This does not imply that the relation is
1005:
815:
that does not have any cycles. In particular, by virtue of being antitransitive the relation is not transitive.
910:
method in which ranking several candidates can produce a loop of preference when the weights are compared (see
1085:
955:
819:
776:
698:
503:
1207:
1254:
1312:
1166:
811:
799:
694:
680:
496:
337:
47:
43:
1000:
While each voter may not assess the units of measure identically, the trend then becomes a single
1302:
1277:
1190:
1001:
959:
1182:
1035:
944:
917:
889:
Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.
780:
1208:
Leutwyler, K. (2000). Mating
Lizards Play a Game of Rock-Paper-Scissors. Scientific American.
886:
defeats y, and y defeats z, then x does not defeat z. Hence the relation is antitransitive.
1320:
1269:
1174:
1157:(2002). "Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors".
1154:
976:
907:
784:
1244:
Bar-Hillel, M., & Margalit, A. (1988). How vicious are cycles of intransitive choice?
1060:
749:
687:
313:, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the
39:
1316:
1170:
911:
69:
1337:
899:
1281:
1194:
936:
996:
20% favor a 40/60 weighting between social consciousness and fiscal conservatism
963:
903:
791:
321:
27:
1273:
993:
50% favor 50/50 weighting between social consciousness and fiscal conservatism
990:
30% favor 60/40 weighting between social consciousness and fiscal conservatism
940:
932:
310:
701:) unique relation is always anti-transitive. An example of the former is the
951:
1186:
168:{\displaystyle \lnot \left(\forall a,b,c:aRb\land bRc\implies aRc\right).}
16:
This article is about intransitivity in mathematics. For other uses, see
1178:
66:
if it is not transitive, that is, (if the relation in question is named
1325:
1290:
506:, each of the following formulas is equivalent to antitransitivity of
1219:
Atherton, K. D. (2013). A brief history of the demise of battle bots.
1307:
748:
320:
Another example that does not involve preference loops arises in
1032:
Wild Health: Lessons in
Natural Wellness from the Animal Kingdom
790:
Assuming no option is preferred to itself i.e. the relation is
432:{\displaystyle \forall a,b,c:aRb\land bRc\implies \lnot (aRc).}
46:. This may include any relation that is not transitive, or the
690:. On a 3-element set, the depicted cycle has both properties.
1255:"Complexity and intransitivity in technological development"
686:
An antitransitive relation on a set of ≥4 elements is never
253:{\displaystyle \exists a,b,c:aRb\land bRc\land \lnot (aRc).}
317:
relation among life forms is intransitive, in this sense.
1153:
Kerr, Benjamin; Riley, Margaret A.; Feldman, Marcus W.;
54:, which describes a relation that is never transitive.
282:. This relation is intransitive since, for example, 2
1034:(paperback ed.). Houghton Mifflin. p. 141.
1008:
agrees is a preferred balance of candidate criteria.
516:
360:
183:
92:
72:
491:
A second example of an antitransitive relation: the
1262:Journal of Systems Science and Systems Engineering
663:
431:
252:
167:
78:
947:), potentially leading to unresolvable conflicts.
1291:"Intransitivity in Theory and in the Real World"
928:more than half the time" need not be transitive.
8:
935:, intransitivity often occurs in a person's
638:
634:
568:
564:
407:
403:
147:
143:
1324:
1306:
1236:Foundations of Rational Choice Under Risk
954:intransitivity can occur in a consumer's
517:
515:
359:
182:
91:
71:
825:
736:is antitransitive, so is each subset of
484:is even, or vice-versa. In either case,
1017:
797:Therefore such a preference loop (or
679:An antitransitive relation is always
7:
920:demonstrate that the relation "die
262:For example, consider the relation
1238:. Oxford: Oxford University Press.
1139:by left uniqueness, contradicting
639:
592:
569:
522:
408:
361:
354:if this never occurs at all, i.e.
229:
184:
101:
93:
23:Property of mathematical relations
14:
962:that does not conform to perfect
177:This statement is equivalent to
1253:Klimenko, Alexander Y. (2014).
924:rolls a higher number than die
902:, in probabilistic outcomes of
898:Intransitivity can occur under
1344:Properties of binary relations
1057:"Guide to Logic, Relations II"
654:
642:
635:
584:
572:
565:
423:
411:
404:
286:6 (2 is a divisor of 6) and 6
244:
232:
144:
1:
18:Intransitive (disambiguation)
1289:Klimenko, Alexander (2015).
464:is odd, is intransitive. If
294:(see below); for example, 2
975:It has been suggested that
456:on the integers, such that
309:As another example, in the
1360:
893:Occurrences in preferences
452:For example, the relation
441:Many authors use the term
343:In the example above, the
266:on the integers such that
15:
1274:10.1007/s11518-014-5245-x
1028:in fact eat grass – see
725:cannot be the mother of
336:is used to refer to the
1246:Theory and Decision, 24
1155:Bohannan, Brendan J. M.
1082:"IntransitiveRelation"
755:
665:
433:
254:
169:
80:
1030:Engel, Cindy (2003).
820:rock, paper, scissors
777:Rock, paper, scissors
752:
666:
434:
340:of antitransitivity.
255:
170:
81:
1117:would hold for some
964:economic rationality
812:equivalence relation
514:
497:knockout tournaments
358:
181:
90:
70:
44:transitive relations
1317:2015Entrp..17.4364K
1179:10.1038/nature00823
1171:2002Natur.418..171K
958:. This may lead to
772:C is preferred to A
769:B is preferred to C
766:A is preferred to B
693:An irreflexive and
38:) is a property of
960:consumer behaviour
756:
661:
659:
429:
250:
165:
76:
34:(sometimes called
1326:10.3390/e17064364
1301:(12): 4364–4412.
1234:Anand, P (1993).
1165:(6894): 171–174.
1143:by irreflexivity.
918:Intransitive dice
883:
882:
803:) is known as an
781:intransitive dice
709:is the mother of
480:are both odd and
338:stronger property
274:is a multiple of
79:{\displaystyle R}
48:stronger property
1351:
1330:
1328:
1310:
1285:
1259:
1239:
1221:
1216:
1210:
1205:
1199:
1198:
1150:
1144:
1138:
1103:
1097:
1096:
1094:
1093:
1084:. Archived from
1078:
1072:
1071:
1069:
1068:
1059:. Archived from
1053:
1047:
1045:
1022:
977:Condorcet voting
950:Analogously, in
937:system of values
908:Condorcet voting
826:
670:
668:
667:
662:
660:
590:
520:
447:antitransitivity
438:
436:
435:
430:
328:Antitransitivity
278:or a divisor of
259:
257:
256:
251:
174:
172:
171:
166:
161:
157:
85:
83:
82:
77:
52:antitransitivity
40:binary relations
1359:
1358:
1354:
1353:
1352:
1350:
1349:
1348:
1334:
1333:
1288:
1257:
1252:
1233:
1230:
1228:Further reading
1225:
1224:
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1213:
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1023:
1019:
1014:
973:
895:
747:
676:
658:
657:
588:
587:
512:
511:
460:if and only if
356:
355:
332:Often the term
330:
270:if and only if
179:
178:
100:
96:
88:
87:
68:
67:
60:
36:nontransitivity
24:
21:
12:
11:
5:
1357:
1355:
1347:
1346:
1336:
1335:
1332:
1331:
1286:
1268:(2): 128–152.
1250:
1241:
1229:
1226:
1223:
1222:
1211:
1200:
1145:
1098:
1073:
1048:
1040:
1016:
1015:
1013:
1010:
998:
997:
994:
991:
980:conservative.
972:
969:
968:
967:
948:
929:
915:
912:voting paradox
894:
891:
881:
880:
877:
874:
871:
867:
866:
863:
860:
857:
853:
852:
849:
846:
843:
839:
838:
835:
832:
829:
806:
805:intransitivity
802:
774:
773:
770:
767:
761:
760:intransitivity
746:
743:
742:
741:
732:If a relation
730:
717:the mother of
691:
684:
675:
672:
656:
653:
650:
647:
644:
641:
637:
633:
630:
627:
624:
621:
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583:
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563:
560:
557:
554:
551:
548:
545:
542:
539:
536:
533:
530:
527:
524:
521:
519:
472:, then either
448:
444:
443:intransitivity
428:
425:
422:
419:
416:
413:
410:
406:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
353:
352:antitransitive
350:A relation is
346:
335:
329:
326:
316:
293:
292:antitransitive
249:
246:
243:
240:
237:
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103:
99:
95:
75:
65:
59:
58:Intransitivity
56:
32:intransitivity
22:
13:
10:
9:
6:
4:
3:
2:
1356:
1345:
1342:
1341:
1339:
1327:
1322:
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1314:
1309:
1304:
1300:
1296:
1292:
1287:
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1275:
1271:
1267:
1263:
1256:
1251:
1249:
1248:(2), 119-145.
1247:
1242:
1237:
1232:
1231:
1227:
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1215:
1212:
1209:
1204:
1201:
1196:
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1172:
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1164:
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1149:
1146:
1142:
1137:
1133:
1128:
1124:
1120:
1116:
1112:
1108:
1102:
1099:
1088:on 2016-03-03
1087:
1083:
1077:
1074:
1063:on 2008-09-16
1062:
1058:
1052:
1049:
1043:
1041:0-618-34068-8
1037:
1033:
1027:
1021:
1018:
1011:
1009:
1007:
1004:on which the
1003:
995:
992:
989:
988:
987:
984:
981:
978:
970:
965:
961:
957:
953:
949:
946:
942:
938:
934:
930:
927:
923:
919:
916:
913:
909:
906:, and in the
905:
901:
900:majority rule
897:
896:
892:
890:
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869:
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827:
824:
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804:
801:
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795:
793:
788:
786:
785:Penney's game
782:
778:
771:
768:
765:
764:
763:
759:
751:
744:
739:
735:
731:
728:
724:
720:
716:
712:
708:
705:relation. If
704:
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671:
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631:
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619:
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531:
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504:transposition
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104:
97:
73:
63:
57:
55:
53:
49:
45:
42:that are not
41:
37:
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1294:
1265:
1261:
1245:
1235:
1214:
1203:
1162:
1158:
1148:
1140:
1135:
1131:
1126:
1122:
1118:
1114:
1110:
1106:
1101:
1090:. Retrieved
1086:the original
1076:
1065:. Retrieved
1061:the original
1051:
1031:
1025:
1020:
999:
985:
982:
974:
925:
921:
888:
884:
818:The game of
817:
809:
796:
789:
775:
757:
737:
733:
726:
722:
718:
714:
710:
706:
702:
507:
501:
495:relation in
492:
490:
485:
481:
477:
473:
469:
465:
461:
457:
453:
451:
440:
349:
342:
334:intransitive
331:
319:
308:
306:12 as well.
303:
299:
295:
287:
283:
279:
275:
271:
267:
263:
261:
176:
64:intransitive
61:
51:
35:
31:
25:
956:preferences
941:preferences
904:game theory
792:irreflexive
681:irreflexive
322:freemasonry
28:mathematics
1308:1507.03169
1092:2006-07-13
1067:2006-07-13
1012:References
971:Likelihood
933:psychology
674:Properties
311:food chain
302:12, and 2
1006:consensus
986:Such as:
952:economics
856:scissors
758:The term
640:¬
636:⟹
623:∧
593:∀
570:¬
566:⟹
553:∧
523:∀
488:is even.
409:¬
405:⟹
392:∧
362:∀
230:¬
227:∧
215:∧
185:∃
145:⟹
132:∧
102:∀
94:¬
1338:Category
1282:59390606
1187:12110887
834:scissors
493:defeated
445:to mean
1313:Bibcode
1295:Entropy
1195:4348391
1167:Bibcode
1129:, then
1024:Wolves
721:, then
345:feed on
315:feed on
1280:
1193:
1185:
1159:Nature
1113:, and
1038:
1002:vector
945:tastes
870:paper
837:paper
783:; and
745:Cycles
713:, and
703:mother
699:right-
688:connex
1303:arXiv
1278:S2CID
1258:(PDF)
1191:S2CID
943:, or
842:rock
800:cycle
695:left-
486:a + c
470:b R c
466:a R b
462:a + b
458:a R b
298:6, 6
268:a R b
1183:PMID
1036:ISBN
939:(or
831:rock
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476:and
468:and
1321:doi
1270:doi
1175:doi
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280:b
276:b
272:a
264:R
248:.
245:)
242:c
239:R
236:a
233:(
224:c
221:R
218:b
212:b
209:R
206:a
203::
200:c
197:,
194:b
191:,
188:a
163:.
159:)
155:c
152:R
149:a
141:c
138:R
135:b
129:b
126:R
123:a
120::
117:c
114:,
111:b
108:,
105:a
98:(
74:R
20:.
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