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A set of dice is intransitive (or nontransitive) if it contains X>2 dice, X1, X2, and X3... with the property that X1 rolls higher than X2 more than half the time, and X2 rolls higher than X3 etc... more than half the time, but where it is not true that X1 rolls higher than Xn more than half the time.
The terms "transitive game" or "intransitive game" are not used in game theory. A prototypical example of an intransitive game is the game rock, paper, scissors. In probabilistic games like Penney's game, the violation of transitivity results in a more subtle way, and is often presented as a probability paradox.
Intransitive dice demonstrate that the relation "die X rolls a higher number than die Y more than half the time" need not be transitive. In psychology , intransitivity often occurs in a person's system of values (or preferences , or tastes ), potentially leading to unresolvable conflicts.
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In grammar, an intransitive verb is a verb, aside from an auxiliary verb, whose context does not entail a transitive object. That lack of an object distinguishes intransitive verbs from transitive verbs , which entail one or more objects.
In contrast, the verb iru can be used as an auxiliary with non-volitional intransitive verbs to simply indicate a state of being, without the element of volition. Iru also has a basic meaning of 'be' or 'stay', but is used with animate subjects. In this use as an auxiliary verb, iru appears in the form -te-iru.
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The genitive case can be used both for the direct objects of transitive verbs and the subjects of unaccusative verbs while the sentence is being negated. This grammatical case is not allowed to be used for the NPs in the subject position of transitive and transitive [clarification needed] verbs, and with unergative verbs. [15]