There are times when you should raise even if you do not have the best cards at the moment. You are hoping that your opponent will fold a better hand, but even if he calls, you will still have a chance of improving to beat him.
Example: You hold AsKs
On the Turn, the board is Js-5c-6d-Qs
Lets assume you know your opponent has Jc-9c, and he is currently ahead of you with a pair, but you have many outs. Any A, K, T or spade gives you the winning hand. There are a total of 44 unknown cards (52 cards in the deck, minus 2 in your hand, minus 2 in your opponents hand, minus 4 on the board), and 18 cards will win the pot for you (3 A's, 3 K's, 4 T's, and 8 remaining spades, note that a ninth Spade has already been counted). This means you will win 18 out of 44 times. Normally we would use 46 unknown cards on the Turn, but in this case we are assuming we know our opponents hand.
To make this demonstration simpler, lets assume that if you do not make your hand on the River, you will simply fold, and that if you do hit your hand on the River, your opponent will call your bet half the time.
On the Turn, the pot contains 5 big bets, and your opponent bets into you, thus making it 6 big bets in the pot.
If you call here, you expect to hit your hand 18 out of 44 times, and make 6.5 big bets when you do win (remember we assumed that if you hit your hand on the River that your opponent will call you half the time but will not pay you off the other half of the time). You also expect to lose 1 big bet 26 out of 44 times. The expected value in this case would be 2.07 big bets to make this call, so it is worthwhile to at least play and stay in the hand to see the River card.
Expected Value of Calling = (18/44 x 6.5 big bets) + (26/44 x -1 big bet) = 2.07 big bets
Instead of calling, you could consider raising. Suppose if you raise, there is a 20% chance that your opponent folds right there on the spot, with the Q on the Turn, that is not altogether unlikely. If he calls you on the Turn, you realize he is definitely going to call again on the River if you do not hit your hand (so you cannot bluff on the River), but he will not call if you do hit your hand since the combination of your raise on the Turn and the scary board will now be too much for him. Now is it better to raise or just call?
Expected Value of Semi-Bluff Raising = (20% x 6 big bets) + (80% x 18/44 x 7 big bets) + (80% x 26/44 x -2 big bets) =2.55 big bets
The EV of the semi-bluff raising play is greater than the EV of calling with these numbers that we used, which means we should raise instead of just call.
But what if you had estimated his folding percentage incorrectly? What if instead of having a 20% chance that he folds on the Turn, this guy will actually never fold. Well now you have cost yourself money with a raise in this spot, because you are more likely to lose than win and you have put more money in the pot.
Expected Value of Semi-Bluff Raising if your opponent will never fold on the Turn = (0% x 6 big bets) + (100% x 18/44 x 7 big bets) + (100% x 26/44 x -2 big bets) = +1.68 big bets
With that adjustment, it is clear that a semi-bluff raise against this opponent is not a good idea, as it lowers your expectancy from 2.07 from calling down to 1.68. This is the main reason why the semi-bluff can sometimes be a useless concept in the lower limit games. Since the players in the low limit games are much more likely to call than players in the middle or higher limit games, players who use the semi-bluff raise too often in the low limit games will find that they are costing themselves money by making this play.
The math shows the breakeven point of the semi-bluff is for your opponent to fold 9% of the time. At that rate, your expected value of the semi-bluff raise would be 2.07, which was the same as just calling.
EV against a player who folds 9% of the time: (9% x 6) + (91% x 18/44 x 7) + (91% x 26/44 x -2) = 2.07
As you can see by this demonstration, whether a semi-bluff raise is correct or not depends on the frequency that your opponent will fold a made hand. This is a nice example of how combining the mathematical side of the brain with the social side of the brain can result in a correct analysis. If we just used the math side, we are still at a loss as to whether or not a raise is correct since we do not have an accurate assessment of his folding percentages. If we just used the social side, we are at a loss as to whether or not a raise is correct since we do not have an accurate assessment of the value of a possible fold compared to the negative value of a call by the opponent and losing more money when we lose. It is only when we combine both sides of the brain that we can make it all work.
Unless you are very good with math, or an idiot savant like the character that Dustin Hoffman played in Rain Man, you will not be able to do the math in your head. Even if you understand the concept, it is completely irrational to think anyone can do these calculations in the heat of the battle. But it is still useful to play with the spreadsheet and the math so you can have some idea of certain situations when you are at the table. In this case, you will notice that it takes the opponent's folding rate to be only 9% of the time for a semi-bluff to be a breakeven play when you have a 18 out of 44 chance of winning and an EV of 2.07 big bets when calling. Anything higher than a 9% folding rate makes the semi-bluff a positive expectancy play. Since most players will fold more than 9% of the time in a situation like this, you can keep in mind the strategy of the semi-bluff Turn raise when you have so many outs, and the opponent may only have second pair.
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