Third Hand High

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Basically, you go third hand high to cooperate with your partner in getting the best bargain you can out of a hand. Let me take this very common position:
8 7 4
Q 9 6 5 K 10 3
A J 2

Your partner lays down the 5 of spades on opening lead. If you get uptight about "wasting" your K, if you feel you'd like to hold onto it, maybe the three will suffice, but anyway, you'll play the 10, then you can see that the declaring side gets two spade tricks when they're only entitled to one. What! You've told your partner never to lead from a queen in an unbid suit? Oh, come on. Your partner has to lead something, and your task is not to follow some rigid rules but to make the most productive use of your cards. It's certainly possible that your partner would have given away a trick in any of the other three suits, or maybe in two of the other three suits, and this one looked as promising as anything. Or for that matter, maybe in none of the other three suits, this also being a safe lead. You don't want to turn it into a disaster. Your king would not be wasted going up here, of course, but on the contrary would be the most productive play you can make.
You might note that this would work as well if your partner led from a jack:
8 7 4
J 6 5 2 K 9 2
A Q 10

Oh, he would go and do that, huh! Leading into declarer's A Q, just handing him a painless finesse for which he takes no risk, sweats not a drop. But the finesse was on all the time, and though declarers have been known to eschew finesses that were on, those aren't the ones you wanna get uptight about. You'll note that though your side has given declarer a painless finesse, your partner will now control the third round of the suit -- if you go up. And if you can't spare your king, declarer wins with the 10, and the finesse against the K is still on! Allowing three spade tricks to declarer.
Yes, last week, you held off and a declarer eventually lost all his entries and gave you a trick with the K. Yes, yes, of course. Not only are there occasional inept declarers, but there are bound to be situations where it would be better to hang onto your king. But you've gotta attribute some acumen to your opponent. Opponents may make all sorts of silly mistakes, but when you count on a mistake though you have the tickets to pick up a good score on your own steam, I think you'll find that the opponents don't make those convenient mistakes so often you need them. On top of which, there's a partner to impress and keep happy.
So unless there's a positive reason for not going up -- and I've offered some reasons here -- you'll by and large find it worthwhile going up third hand.