Harrington advocates in his book that you should "value bet" in a situation where you hold a reasonable hand and believe it is ahead due to the betting paterns but that your opponent will call with a worse hand.

Is this always correct though?

Perhaps you have made a 3-4BB raise from the button with AJ and your opponent on the BB calls leaving you heads up.

The flop comes down JavaScript 7h 6h

Opponent bets/ You bet 3/4 pot / Opponent calls

Turn 3s

Opponent checks/ You bet 3/4 pot / Opponent calls

River Qs

Opponent checks...

In this situation I would usually expect I am ahead and check to see. Harrington would say you should place perhaps a 1/2 pot value bet. He doesn't say what you then do if you face a raise. What if your opponent has AA or trips and since you have bet every other street checks into you again? Or perhaps he was holding A7 of spades, called on 2nd pair and has infact hit a backdoor flush draw, or was calling on AQ and now hit - again he checks believing you will bet into him.

Given that this is referring to tournaments, even if your opponent calls 50% of the time, folds 45% of the time and raises 5% of the time, do people think it is worth gaining an extra 100 chips here and there, if 1 time out of 20 you are raised all in and forced to make a decision on your whole tournament?

The flush would have me a little worried, but operating on the basis that if he's bad enough to call my bets with Qx then he's not good enough to check raise the river with a Q I might well bet it, though it would depend on what info I had on the guy. In the absence of any information I may well check. Overall state of play in the tournament would also be a factor.

Good point PJ, a reraise is more often than not going to what it appears to be (a sign you are beaten) so it wasn't your pot.

But I'm playing like a donkey today so far so what do I know

I think this is a difficult decision, and I think that a lot depends on the opponent and how loose/tight you think he is. Overall, though, I think I'd tend towards checking here.

First, what hands might he call (but not reraise) a preflop raise from the button and then check/call thereafter?

Recognizing that an opening raise from the button might not indicate great strength, the range of calling hands therefore doesn't need to be tremendously strong, either. Still, I wouldn't expect to see my opponent have a pocket pair of TT or better. I would expect those to be reraising hands on at least one of the earlier streets, especially given the two flush draws presented by the turn. I also wouldn't expect to see a hand like KJ because the opponent should have lead out on the flop with such a hand (if he's not willing to lead with those cards, then why call the preflop raise in the first place?). Finally, I wouldn't expect to see a hand like AQo or AsKs because the post-flop betting doesn't make much sense for those kind of cards.

What makes sense to me is if the opponent has a hand like QJo, JTo, KhQh, or a middle pair (like 88 or 99). You win some of these matchups and you lose others. However, if those are reasonable possibilities, then I guess you can do some math to figure out the percentage of times that your opponent will have one of those combinations, add together the percentages where you win, and then determine if it is more likely than not that you are ahead.

I'm not good enough to do that kind of math in my head at the table. However, the combination of (1) an overcard; (2) a backdoor (if improbable) flush on the board; and (3) the 0% likelihood of getting called by someone who had only a busted heart draw would all lead me to check behind the BB on the river.