40% isn't the Generals, that's 0%. 40% is more like the Redskins.
The generals don't score 0 points...they just (almost) always lose.
If the Libertarians became a "major player" as a 3rd party, all that would likely mean is that they and the Republicans would split 50% of the vote, give or take 10%. Thus leaving the Democrats with the other 50% (give or take).
So while they would be a "major player" compared to today where if they got over 5% of the total vote it'd be
astounding, they'd still be essentially the Washington Generals....a team that pretty much ALWAYS loses.
I you have a viable third party, you only need 34% to win. In any race other than President, anyway. In a Presidential race, 3 candidates getting about 1/3 throws it to the House.
The problem with the 34% notion is the assumption that you have three parties that relatively evenly split the voting populatoin.
Going off Haymarket's (and seemingly your) logic, it's mostly Republicans voters that would flock to the Libertarian party (see haymarkets whole "courage" notion).
So for the Libertarians to get from the low single digits they're at today to your suggested "34%", the reality is...based on the logic used to suggest they could possibly do such a thing...that extra 25 to 30% is being siphoned primarily off the Republicans instead of the Democrats.
So to get to that 34% number, it's far more likely that the vast majority of times that would mean it'd look like:
34% libertarian, 16% Republican, 50% democrat.
Then it would 34%, 33%, 33% with fill in parties as you'd like.
THEORITICALLY, you're idea in a general sense (not specific to libertarians, or to haymarkets theory on it) is correct. A third party candidate COULD concievably win IF they found some way to siphon off votes from both sides in a relatively even split.
However, I don't think any of the three parties list above could reasonably do that, and I don't think any long stand established "party" of some sort would be able to do it. I could, on a HUGE off chance, see a cult of personality type of individual with significant financial backing possibly pushing ahead as an independent of sorts and winning on a national scale.
But the far more likely scenario is that if a "third party" becomes a "major player" then all it's going to do is assure that two of the three "major players" are really just seat fillers. To continue the basketball analogy, you change it from a game of one and one between Byrd and Magic into a game of 1 on 2 between Jordan and two high school kids. Sure, they're in the same game as Jordan....but being in the game doesn't mean they're ever likely to win.