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subject to individual will or judgment without restriction; contingent solely upon one’s discretion:
an arbitrary decision.
decided by a judge or arbiter rather than by a law or statute.
having unlimited power; uncontrolled or unrestricted by law; despotic; tyrannical:
an arbitrary government.
capricious; unreasonable; unsupported:
an arbitrary demand for payment.
Mathematics. undetermined; not assigned a specific value:
an arbitrary constant.
arbitraries, Printing. (in Britain) (def 9).
Contemporary Examples

The truth is that Judd is really just picking an arbitrary number since there is no script.
Inside Sony’s ‘Pineapple Express 2 Drama’: Leaked Emails Reveal Fight Over Stoner Comedy Sequel William Boot December 20, 2014

arbitrary benchmarks cheat kids out of a fulfilling education.
James Dyson: America’s School Science Crisis James Dyson September 30, 2010

Having a shared cause (even an arbitrary one) is a monumental aspect of the meaningful life.
Why Favoritism Is Virtuous: The Case Against Fairness Stephen T. Asma December 6, 2012

It comes down to this: A command of standard English rules benefits a life trajectory even if the rules are arbitrary.
The Problem With Weird Al’s ‘Word Crimes’ Video John McWhorter July 22, 2014

Often, the reasons for being recommended for testing are arbitrary.
Roma Children Face Segregation In EU Schools Amana Fontanella-Khan March 7, 2014

Historical Examples

Up and down are arbitrary or relative terms after all, in the universe.
Punch or the London Charivari, Vol. 109, September 21, 1895 Various

Why, she was called in the Institution, Harriet Beadle—an arbitrary name, of course.
Little Dorrit Charles Dickens

arbitrary and merely personal judgments I have, however, endeavoured to avoid.
A Commentary to Kant’s ‘Critique of Pure Reason’ Norman Kemp Smith

He was arbitrary, but he was not so arbitrary as his instructions.
The Atlantic Monthly, Vol. 10, No. 58, August, 1862 Various

Each is absolutely without any arbitrary influence, any influence not natural or sympathetic or co-operative, upon the other.
The Will to Doubt Alfred H. Lloyd

adjective
founded on or subject to personal whims, prejudices, etc; capricious
having only relative application or relevance; not absolute
(of a government, ruler, etc) despotic or dictatorial
(maths) not representing any specific value: an arbitrary constant
(law) (esp of a penalty or punishment) not laid down by statute; within the court’s discretion
adj.

early 15c., “deciding by one’s own discretion,” from Old French arbitraire (14c.) or directly from Latin arbitrarius “depending on the will, uncertain,” from arbiter (see arbiter). The original meaning gradually descended to “capricious” and “despotic” (1640s). Related: Arbitrarily; arbitrariness.