Editing Intransitivity (section)

== Cycles ==
[[File:Three-part cycle diagram.png|alt=Cycle diagram|thumb|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 1.]]
The term {{em|intransitivity}} 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:

* A is preferred to B
* B is preferred to C
* C is preferred to A

[[Rock, paper, scissors]]; [[intransitive dice]]; and [[Penney's game]] are examples. Real combative relations of competing species,<ref>{{Cite journal | doi=10.1038/nature00823| pmid=12110887| title=Local dispersal promotes biodiversity in a real-life game of rock–paper–scissors| journal=Nature| volume=418| issue=6894| pages=171–174| year=2002| last1=Kerr| first1=Benjamin| last2=Riley| first2=Margaret A.| last3=Feldman| first3=Marcus W.| last4=Bohannan| first4=Brendan J. M.| bibcode=2002Natur.418..171K| s2cid=4348391|authorlink4=Brendan Bohannan}}</ref> strategies of individual animals,<ref>[http://www.scientificamerican.com/article/mating-lizards-play-a-gam/ Leutwyler, K. (2000). Mating Lizards Play a Game of Rock-Paper-Scissors. Scientific American.]</ref> and fights of remote-controlled vehicles in BattleBots shows ("robot Darwinism")<ref>[http://www.popsci.com/technology/article/2013-06/elaborate-history-how-wedges-ruined-battlebots Atherton, K. D. (2013). A brief history of the demise of battle bots.]</ref> can  be cyclic as well.

Assuming no option is preferred to itself i.e. the relation is [[irreflexive]], 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.

Therefore such a preference loop (or {{em|[[Cycle (graph theory)|cycle]]}}) is known as an {{em|intransitivity}}.

Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive.  For example, an [[equivalence relation]] 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 that does not have any cycles.  In particular, by virtue of being antitransitive the relation is not transitive.

The game of [[rock, paper, scissors]] 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 does not defeat paper.  Finally, it is also true that no option defeats itself. This information can be depicted in a table:
{| class="wikitable" style="text-align:center;
! !! rock !! scissors !! paper
|-
! rock
| 0 || 1 || 0
|-
! scissors
| 0 || 0 || 1
|-
! paper
| 1 || 0 || 0
|}

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 defeats y, and y defeats z, then x does not defeat z.  Hence the relation is antitransitive.

Thus, a cycle is neither necessary nor sufficient for a binary relation to be antitransitive.