761:
805:, 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.
896:
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
833:
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
825:
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
990:
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
680:
524:
184:
448:
269:
764:
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
529:
798:
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.
335:: 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.
773:
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:
358:
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.
100:
95:
73:
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
675:{\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}}}
1092:
977:. Economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).
368:
994:
In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units of measure in assessing candidates.
1354:
191:
1067:
1050:
1254:
1229:
510:. 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.
834:
does not defeat paper. Finally, it is also true that no option defeats itself. This information can be depicted in a table:
28:
821:
Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, an
301:
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
1016:
826:
that does not have any cycles. In particular, by virtue of being antitransitive the relation is not transitive.
921:
method in which ranking several candidates can produce a loop of preference when the weights are compared (see
1096:
966:
830:
787:
709:
514:
1218:
1265:
1323:
1177:
822:
810:
705:
691:
507:
348:
58:
54:
1011:
While each voter may not assess the units of measure identically, the trend then becomes a single
1313:
1288:
1201:
1012:
970:
1193:
1046:
955:
928:
900:
Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.
791:
1219:
Leutwyler, K. (2000). Mating
Lizards Play a Game of Rock-Paper-Scissors. Scientific American.
897:
defeats y, and y defeats z, then x does not defeat z. Hence the relation is antitransitive.
1331:
1280:
1185:
1168:(2002). "Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors".
1165:
987:
918:
795:
1255:
Bar-Hillel, M., & Margalit, A. (1988). How vicious are cycles of intransitive choice?
1071:
760:
698:
324:, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the
50:
1327:
1181:
922:
80:
17:
1348:
910:
1292:
1205:
947:
1007:
20% favor a 40/60 weighting between social consciousness and fiscal conservatism
974:
914:
802:
332:
38:
1284:
1004:
50% favor 50/50 weighting between social consciousness and fiscal conservatism
1001:
30% favor 60/40 weighting between social consciousness and fiscal conservatism
951:
943:
321:
712:) unique relation is always anti-transitive. An example of the former is the
962:
1197:
179:{\displaystyle \lnot \left(\forall a,b,c:aRb\land bRc\implies aRc\right).}
27:
This article is about intransitivity in mathematics. For other uses, see
1189:
77:
if it is not transitive, that is, (if the relation in question is named
1336:
1301:
517:, each of the following formulas is equivalent to antitransitivity of
1230:
Atherton, K. D. (2013). A brief history of the demise of battle bots.
1318:
759:
331:
Another example that does not involve preference loops arises in
1043:
Wild Health: Lessons in
Natural Wellness from the Animal Kingdom
801:
Assuming no option is preferred to itself i.e. the relation is
443:{\displaystyle \forall a,b,c:aRb\land bRc\implies \lnot (aRc).}
57:. This may include any relation that is not transitive, or the
701:. On a 3-element set, the depicted cycle has both properties.
1266:"Complexity and intransitivity in technological development"
697:
An antitransitive relation on a set of ≥4 elements is never
264:{\displaystyle \exists a,b,c:aRb\land bRc\land \lnot (aRc).}
328:
relation among life forms is intransitive, in this sense.
1164:
Kerr, Benjamin; Riley, Margaret A.; Feldman, Marcus W.;
65:, which describes a relation that is never transitive.
293:. This relation is intransitive since, for example, 2
1045:(paperback ed.). Houghton Mifflin. p. 141.
1019:
agrees is a preferred balance of candidate criteria.
527:
371:
194:
103:
83:
502:
A second example of an antitransitive relation: the
1273:Journal of Systems Science and Systems Engineering
674:
442:
263:
178:
89:
958:), potentially leading to unresolvable conflicts.
1302:"Intransitivity in Theory and in the Real World"
939:more than half the time" need not be transitive.
8:
946:, intransitivity often occurs in a person's
649:
645:
579:
575:
418:
414:
158:
154:
1335:
1317:
1247:Foundations of Rational Choice Under Risk
965:intransitivity can occur in a consumer's
528:
526:
370:
193:
102:
82:
836:
747:is antitransitive, so is each subset of
495:is even, or vice-versa. In either case,
1028:
808:Therefore such a preference loop (or
690:An antitransitive relation is always
7:
931:demonstrate that the relation "die
273:For example, consider the relation
1249:. Oxford: Oxford University Press.
1150:by left uniqueness, contradicting
650:
603:
580:
533:
419:
372:
365:if this never occurs at all, i.e.
240:
195:
112:
104:
34:Property of mathematical relations
25:
973:that does not conform to perfect
188:This statement is equivalent to
1264:Klimenko, Alexander Y. (2014).
935:rolls a higher number than die
913:, in probabilistic outcomes of
909:Intransitivity can occur under
1355:Properties of binary relations
1068:"Guide to Logic, Relations II"
665:
653:
646:
595:
583:
576:
434:
422:
415:
297:6 (2 is a divisor of 6) and 6
255:
243:
155:
1:
29:Intransitive (disambiguation)
1300:Klimenko, Alexander (2015).
475:is odd, is intransitive. If
305:(see below); for example, 2
986:It has been suggested that
467:on the integers, such that
320:As another example, in the
1371:
904:Occurrences in preferences
463:For example, the relation
452:Many authors use the term
354:In the example above, the
277:on the integers such that
26:
1285:10.1007/s11518-014-5245-x
1039:in fact eat grass – see
736:cannot be the mother of
347:is used to refer to the
1257:Theory and Decision, 24
1166:Bohannan, Brendan J. M.
18:Antitransitive relation
1093:"IntransitiveRelation"
766:
676:
444:
265:
180:
91:
1041:Engel, Cindy (2003).
831:rock, paper, scissors
788:Rock, paper, scissors
763:
677:
445:
351:of antitransitivity.
266:
181:
92:
1128:would hold for some
975:economic rationality
823:equivalence relation
525:
508:knockout tournaments
369:
192:
101:
81:
55:transitive relations
1328:2015Entrp..17.4364K
1190:10.1038/nature00823
1182:2002Natur.418..171K
969:. This may lead to
783:C is preferred to A
780:B is preferred to C
777:A is preferred to B
704:An irreflexive and
49:) is a property of
971:consumer behaviour
767:
672:
670:
440:
261:
176:
87:
45:(sometimes called
1337:10.3390/e17064364
1312:(12): 4364–4412.
1245:Anand, P (1993).
1176:(6894): 171–174.
1154:by irreflexivity.
929:Intransitive dice
894:
893:
814:) is known as an
792:intransitive dice
720:is the mother of
491:are both odd and
349:stronger property
285:is a multiple of
90:{\displaystyle R}
59:stronger property
16:(Redirected from
1362:
1341:
1339:
1321:
1296:
1270:
1250:
1232:
1227:
1221:
1216:
1210:
1209:
1161:
1155:
1149:
1114:
1108:
1107:
1105:
1104:
1095:. Archived from
1089:
1083:
1082:
1080:
1079:
1070:. Archived from
1064:
1058:
1056:
1033:
988:Condorcet voting
961:Analogously, in
948:system of values
919:Condorcet voting
837:
681:
679:
678:
673:
671:
601:
531:
458:antitransitivity
449:
447:
446:
441:
339:Antitransitivity
289:or a divisor of
270:
268:
267:
262:
185:
183:
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96:
94:
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88:
63:antitransitivity
51:binary relations
21:
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1239:Further reading
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906:
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687:
669:
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599:
598:
523:
522:
471:if and only if
367:
366:
343:Often the term
341:
281:if and only if
190:
189:
111:
107:
99:
98:
79:
78:
71:
47:nontransitivity
35:
32:
23:
22:
15:
12:
11:
5:
1368:
1366:
1358:
1357:
1347:
1346:
1343:
1342:
1297:
1279:(2): 128–152.
1261:
1252:
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1234:
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1156:
1109:
1084:
1059:
1051:
1027:
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1024:
1021:
1009:
1008:
1005:
1002:
991:conservative.
983:
980:
979:
978:
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940:
926:
923:voting paradox
905:
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868:
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850:
849:
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843:
840:
817:
816:intransitivity
813:
785:
784:
781:
778:
772:
771:intransitivity
757:
754:
753:
752:
743:If a relation
741:
728:the mother of
702:
695:
686:
683:
667:
664:
661:
658:
655:
652:
648:
644:
641:
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635:
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556:
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547:
544:
541:
538:
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532:
530:
483:, then either
459:
455:
454:intransitivity
439:
436:
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430:
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421:
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413:
410:
407:
404:
401:
398:
395:
392:
389:
386:
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364:
363:antitransitive
361:A relation is
357:
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337:
327:
304:
303:antitransitive
260:
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69:Intransitivity
67:
43:intransitivity
33:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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1356:
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1260:
1259:(2), 119-145.
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1215:
1212:
1207:
1203:
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1191:
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1157:
1153:
1148:
1144:
1139:
1135:
1131:
1127:
1123:
1119:
1113:
1110:
1099:on 2016-03-03
1098:
1094:
1088:
1085:
1074:on 2008-09-16
1073:
1069:
1063:
1060:
1054:
1052:0-618-34068-8
1048:
1044:
1038:
1032:
1029:
1022:
1020:
1018:
1015:on which the
1014:
1006:
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976:
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945:
941:
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930:
927:
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920:
917:, and in the
916:
912:
911:majority rule
908:
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827:
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799:
797:
796:Penney's game
793:
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774:
770:
762:
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750:
746:
742:
739:
735:
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723:
719:
716:relation. If
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515:transposition
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74:
68:
66:
64:
60:
56:
53:that are not
52:
48:
44:
40:
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1276:
1272:
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1225:
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1169:
1159:
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1142:
1137:
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1129:
1125:
1121:
1117:
1112:
1101:. Retrieved
1097:the original
1087:
1076:. Retrieved
1072:the original
1062:
1042:
1036:
1031:
1010:
996:
993:
985:
936:
932:
899:
895:
829:The game of
828:
820:
807:
800:
786:
768:
748:
744:
737:
733:
729:
725:
721:
717:
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518:
512:
506:relation in
503:
501:
496:
492:
488:
484:
480:
476:
472:
468:
464:
462:
451:
360:
353:
345:intransitive
342:
330:
319:
317:12 as well.
314:
310:
306:
298:
294:
290:
286:
282:
278:
274:
272:
187:
75:intransitive
72:
62:
46:
42:
36:
967:preferences
952:preferences
915:game theory
803:irreflexive
692:irreflexive
333:freemasonry
39:mathematics
1319:1507.03169
1103:2006-07-13
1078:2006-07-13
1023:References
982:Likelihood
944:psychology
685:Properties
322:food chain
313:12, and 2
1017:consensus
997:Such as:
963:economics
867:scissors
769:The term
651:¬
647:⟹
634:∧
604:∀
581:¬
577:⟹
564:∧
534:∀
499:is even.
420:¬
416:⟹
403:∧
373:∀
241:¬
238:∧
226:∧
196:∃
156:⟹
143:∧
113:∀
105:¬
1349:Category
1293:59390606
1198:12110887
845:scissors
504:defeated
456:to mean
1324:Bibcode
1306:Entropy
1206:4348391
1178:Bibcode
1140:, then
1035:Wolves
732:, then
356:feed on
326:feed on
1291:
1204:
1196:
1170:Nature
1124:, and
1049:
1013:vector
956:tastes
881:paper
848:paper
794:; and
756:Cycles
724:, and
714:mother
710:right-
699:connex
1314:arXiv
1289:S2CID
1269:(PDF)
1202:S2CID
954:, or
853:rock
811:cycle
706:left-
497:a + c
481:b R c
477:a R b
473:a + b
469:a R b
309:6, 6
279:a R b
1194:PMID
1047:ISBN
950:(or
842:rock
708:(or
487:and
479:and
1332:doi
1281:doi
1186:doi
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1152:aRb
1126:aRc
1122:bRc
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513:By
61:of
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394:a
391::
388:c
385:,
382:b
379:,
376:a
315:R
311:R
307:R
299:R
295:R
291:b
287:b
283:a
275:R
259:.
256:)
253:c
250:R
247:a
244:(
235:c
232:R
229:b
223:b
220:R
217:a
214::
211:c
208:,
205:b
202:,
199:a
174:.
170:)
166:c
163:R
160:a
152:c
149:R
146:b
140:b
137:R
134:a
131::
128:c
125:,
122:b
119:,
116:a
109:(
85:R
31:.
20:)
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