One thing you will notice if you have gotten this far with your hand is that you have a lot of company. This is also where you will suffer through a seemingly endless series of bad beats. You should plan on playing much more timidly here, especially if any scare card has fallen with all but your strongest hands. You do not want to get into a raising war on the river vs four or five opponents unless you have something very close to the nuts. Your pair of kings with an Ace kicker was good on the flop but if you did not improve beyond that then do not be surprised to be beaten by some of the strangest two pair (and beyond) that you've ever seen.
Against One or Two Opponents
However when you are up against only one or two opponents on the river it is also important to bet your made one pair hands on the river if you have been driving the betting the rest of the hand. This is because your opponents at a typical "no foldem holdem" game will call the final river bet with as little as Ace-high. If you are raised then you should almost always throw away all one pair hands (including overpairs). The exception to this is against a tricky opponent or one that has shown that they can make a river-bluff raise in the past.
Against Three or More Opponents
If you are up against three or more opponents at the river then you may want to purposefully show weakness by checking. For example if you started with AKo and flopped a pair of kings and have betting the whole way and then check the river into four callers, someone with a king and a worse kicker will often bet for you and you will save yourself calling an extra bet by the two pair river chasers when they raise this person and also gain additional bets by overcalling when no one is willing to raise the new bettor.
Try to see all the possible straights, flushes, etc. on the board. Consciously do that because when you are focused on your hand it is sometimes very easy to miss a straight or flush that just appeared. Especially if you are focused on your opponent having a similar (but smaller) hand than your own. Be especially careful if you make a straight and a third flush card has also fallen even if that third flush card was a backdoor flush.
Beware of Excessive Overcalling on the River
If you've followed the advice here you'll be getting to the river with at least fairly strong hands (and many times with the absolute highest hands [the nuts!]). You should almost always call a single bet on the river if you have any reasonable hand at all. It can be a huge mathematical mistake to fold on the river if you are up against one or two opponents. Most of your folding should be before the flop, on the flop and on the turn. If you paid for the turn you should call with any reasonably strong hand. The exception to this is where there are multiple callers to a bet on the river. Your odds of winning with an average hand go down exponentially the more callers there are. Calling a bet after more than one person has already called is called "overcalling" and you need more strength to call each overcall--and if you have enough strength to be the fourth overcaller you should raise.
Don't get too upset if you lose a made hand (AA, top two pair, etc.) on the river. If you got your money in during the hand with the best of it that's all that really matters. In the long run you make money by making good decisions and in the short run you are largely at the mercy of luck. Keep making the right decisions! The rest will take care of itself.
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