Intransitivity

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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...
, intransitivity (sometimes called nontransitivity) is a property of
binary relation In mathematics, a binary relation associates elements of one set, called the ''domain'', with elements of another set, called the ''codomain''. A binary relation over Set (mathematics), sets and is a new set of ordered pairs consisting of elem ...
s that are not
transitive relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
s. This may include any relation that is not transitive, or the stronger property of antitransitivity, which describes a relation that is never transitive.

# Intransitivity

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 if it is not transitive, that is, (if the relation in question is named $R$) $\lnot\left(\forall a, b, c: a R b \land b R c \implies a R c\right).$ This statement is equivalent to $\exists a,b,c : a R b \land b R c \land \lnot(a R c).$ For instance, in the
food chain A food chain is a linear network of links in a food web A food web is the natural interconnection of food chains and a graphical representation of what-eats-what in an ecological community. Another name for food web is Consumer-resource sy ...

, wolves feed on deer, and deer feed on grass, but wolves do not feed on grass. Thus, the relation among life forms is intransitive, in this sense. Another example that does not involve preference loops arises in
freemasonry Freemasonry or Masonry refers to fraternal organisations that trace their origins to the local guilds of stonemasons Stonemasonry or stonecraft is the creation of building A building, or edifice, is a structure with a roof and walls ...

: 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.

# Antitransitivity

Often the term is used to refer to the stronger property of antitransitivity. In the example above, the 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. A relation is if this never occurs at all, i.e. $\forall a, b, c: a R b \land b R c \implies \lnot (a R c).$ Many authors use the term to mean . An example of an antitransitive relation: the ''defeated'' relation in
knockout tournament Image:SixteenPlayerSingleEliminationTournamentBracket.svg, upright=1.3, The tournament has been completed until the semifinals, the winner of Lisa-Ernie match and Andrew-Robert match will play in the final. A single-elimination, knockout, or sudde ...
s. 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. By transposition, each of the following formulas is equivalent to antitransitivity of ''R'': $\begin &\forall a, b, c: a R b \land a R c \implies \lnot (b R c) \\$ &\forall a, b, c: a R c \land b R c \implies \lnot (a R b) \end

## Properties

* An antitransitive relation is always
irreflexive In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...
. * An antitransitive relation on a set of ≥4 elements is never connex. On a 3-element set, the depicted cycle has both properties. * An irreflexive and left- (or right-) unique relation is always anti-transitive. An example of the former is the ''mother'' relation. If ''A'' is the mother of ''B'', and ''B'' the mother of ''C'', then ''A'' cannot be the mother of ''C''. * If a relation ''R'' is antitransitive, so is each subset of ''R''.

# Cycles

The term 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 Rock paper scissors (also known by other orderings of the three items, with "rock" sometimes being called "stone", or as roshambo or ro-sham-bo) is a hand game intransitive diceA set of dice is intransitive (or nontransitive) if it contains three dice, ''A'', ''B'', and ''C'', with the property that ''A'' rolls higher than ''B'' more than half the time, and ''B'' rolls higher than ''C'' more than half the time, but it is ...

; intransitive machines; and
Penney's game Penney's game, named after its inventor Walter Penney, is a Binary numeral system, binary (head/tail) sequence generating game between two players. Player A selects a sequence of heads and tails (of length 3 or larger), and shows this sequence to p ...
are examples. Real combative relations of competing species, strategies of individual animals, and fights of remote-controlled vehicles in BattleBots shows ("robot Darwinism")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 In mathematics, a homogeneous binary relation ''R'' over a set (mathematics), set ''X'' is reflexive if it relates every element of ''X'' to itself. An example of a reflexive relation is the relation "equality (mathematics), is equal to" on the se ...
, 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 ) is known as an . Notice that a cycle is neither necessary nor sufficient for a binary relation to be not transitive. For example, an
equivalence relation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
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 Rock paper scissors (also known by other orderings of the three items, with "rock" sometimes being called "stone", or as roshambo or ro-sham-bo) is a hand game

Occurrences in preferences

* Intransitivity can occur under
majority rule Majority rule is a decision rule that selects alternatives which have a majority A majority, also called a simple majority to distinguish it from similar terms (see the "Related terms" section below), is the greater part, or more than half, ...
, in probabilistic outcomes of
game theory Game theory is the study of mathematical model A mathematical model is a description of a system A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole. ...
, and in the
Condorcet voting A Condorcet method (; ) is an election method that elects the candidate who wins a majority rule, majority of the vote in every head-to-head election against each of the other candidates, that is, a candidate preferred by more voters than any othe ...
method in which ranking several candidates can produce a loop of preference when the weights are compared (see voting paradox). * Intransitive dice demonstrate that probabilities are not necessarily transitive. * In psychology, intransitivity often occurs in a person's Value system, system of values (or preferences, or Taste (sociology), tastes), potentially leading to unresolvable conflicts. * Analogously, in economics intransitivity can occur in a consumer's Preference (economics), preferences. This may lead to consumer behaviour that does not conform to perfect Rationality#Economics, economic rationality. In recent years, economists and philosophers have questioned whether violations of transitivity must necessarily lead to 'irrational behaviour' (see Anand (1993)).

# Likelihood

It has been suggested that Condorcet method, Condorcet voting 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 conservative. In such cases intransitivity reduces to a broader equation of numbers of people and the weights of their units of measure in assessing candidates. Such as: * 30% favor 60/40 weighting between social consciousness and fiscal conservatism * 50% favor 50/50 weighting between social consciousness and fiscal conservatism * 20% favor a 40/60 weighting between social consciousness and fiscal conservatism While each voter may not assess the units of measure identically, the trend then becomes a single Probability vector, vector on which the consensus agrees is a preferred balance of candidate criteria.