Wiktionary
n. 1 (&lit a competition to determine who can urinate the furthest up a wall English) 2 (context idiomatic vulgar slang English) A pointless competition, dispute or conflict, often over some trivial matter.
Wikipedia
A pissing contest, or pissing match, is a game in which participants compete to see who can urinate the highest, the farthest, or the most accurately. Although the practice is often associated with adolescent boys, women have been known to play the game, and there are literary depictions of adults competing in it. Since the 1940s the term has been used as a slang idiomatic phrase describing contests that are "futile or purposeless", especially if waged in a "conspicuously aggressive manner". As a metaphor it is used figuratively to characterise ego-driven battling in a pejorative or facetious manner that is often considered vulgar. The image of two people urinating on each other has also been offered as a source of the phrase.
Usage examples of "pissing contest".
You've already had another pissing contest with Brett, haven't you?
Our job is not to get into a pissing contest with them unless they come down from the mountains.
Now, we can continue this pissing contest or you can agree to cooperate.
After what he suspected was a brief, and futile, pissing contest with Saetan, Luthvian walked away.
Grip-to-grip and eye-to-eye, they were engaged in something more than a pissing contest, but hopefully less than mortal combat.
There will be no interagency pissing contest over rules of engagement and we won't have to bring in any inflammatory old, hostile statements you made in the van about what sort of person she was.