A staring contest (or blinking contest) is a game in which two people stare into each other's eyes and attempt to maintain eye contact for a longer period than their opponent. The game ends when one participant blinks or looks away.
(Wikipedia, 'Staring'.)
So the Republicans were the first to flinch in the schoolyard game the American government has been reduced to recently.
The notion of the loneliness of the tumbleweed in the U.S. West is captured by the song "Tumbleweed," by Douglas Van Arsdale (made famous by Joan Baez):
I feel like a lonesome tumbleweed/rolling across an open plain,/I feel like something nobody needs/I feel my life drifting away,/drifting away -
I feel like a broken wagon wheel/when I can't hop a slow-moving train/Think I know how a coyote feels/when he's howling just to/ease the pain, since he's been away.
Lord, I feel like rolling,/rolling along, so keep your big/wind blowing till all my natural/days are gone -/till my days are all gone.
I'm just a lonesome tumbleweed/turning end over end./Once I pulled all my roots free/I became a slave to the wind,/a slave to the wind.
So it is a sad and lonely feeling (according to the badge namers at Stack Exchange) when you ask a question and few people see it and no one responds to it.
Interesting tumbleweed fact: Although tumbleweeds of various plant families are common in parts of the United States (some of them native to North America), one of the largest and in some places most prevalent species west of the Mississippi River is not native to the New World; rather, it is a Eurasian species also known as the Russian Thistle (Kali tragus) and (perhaps most evocatively) as the "wind witch."
Wikipedia's general article on tumbleweeds ends with a discussion of the symbolism of the plant that seems relevant to the current discussion:
The tumbleweed's association with the Western film genre has led to a highly symbolic meaning in visual media. It has come to represent locations that are desolate, dry, and often humorless, with few or no occupants. A common use is when characters encounter a long abandoned or dismal-looking place: a tumbleweed will be seen rolling past, often accompanied by the sound of a dry, hollow wind. This is sometimes used for comic effect in locations where tumbleweeds are not expected, but the emptiness is obvious.
As with the sound of crickets, tumbleweeds can also be shown to emphasize an awkward silence after a bad joke or a character otherwise making an absurd declaration, with the aforementioned sound of wind and the plant rolling past in the background.
The awkward silence memorialized by Stack Exchange's tumbleweed badge is the emptiness of the page where the question has been posted but no one has answered it, commented on it, or voted on it for a full week. Bury me not on the lone prairie.
Best Answer
The phrase “wouldn't blink twice” sounds to me like a conflation of a “wouldn't blink” idiom with a “wouldn't think twice” idiom.
“Wouldn't blink”, in forms like “wouldn't blink an eye” and “wouldn't bat an eye/eyelash/eyelid” connote not reacting or responding, ie, failing to show surprise.
“Wouldn't think twice” means not having to give something a second thought; ie, not having to reconsider something, because it is so obvious or clear in the first place.
Edit: Following on from Sven Yargs' comment, some Google ngrams results may be of interest. First, a few caveats:
• Google ngrams records the 't parts of contractions wouldn't and didn't as separate elements. A search for ' t is satisfied by wouldn't, couldn't, shouldn't, didn't, don't, etc.
• Near the end of Google's ngrams info page, it says “we only consider ngrams that occur in at least 40 books”.
• Books linked below the graph mentioned below include a few dozen results for ' t blink twice even though a page label says there was only one result.
With those caveats understood, the general trend according to the Google ngrams graph for ' t blink,' t think twice,' t blink twice, is that overall frequency of ' t blink is about the same as that of ' t think twice; both forms have been occurring since the early 1800's; each occurs several thousand times as frequently as ' t blink twice. That is, ' t blink twice negative contractions are significantly uncommon (a few dozen occurrences, most within the past 25 years) compared to ' t blink or ' t think twice negative contractions.