By Tommy Angelo
We have limited the size of your bet at these extremes: bet more than they can call profitably, but less than an amount which will chase suddenly course the pot. Now, we need to determine the exact implementation size.
You want to bet to maximize your profits. Grossly, one could say that your winnings are equal to the size of the errors of your opponents multiplied by the chance they have to make this mistake.
'Value of the error', we mean the average amount that your opponent will lose by making the mistake. Let's say that your opponent can "breaker even" (sum zero, no loss, no gain) by calling a bet of $ 100 (and enjoy calling a sum less than $ 100). If you bet $ 101, your opponent made a mistake in calling, but this is just a small mistake. The value of this error is less than $ 1 (lower because a few times, your opponent will hit the card he would run and win this dollar).
On the other hand, if your opponent calls a bet of $ 1000, it will be a huge mistake. Do some math to see what size will be his mistake.
Suppose that you wager $ 100 into a $ 200 pot and your opponent is underdog to 3 against 1. Ignore future updates at the moment. If your opponent calls, on average, it goes 'breaker even.
$ 0 = (¼) ($300) + (¾) ($-100)
Now, suppose that you wager $ 150 and your opponent calls. On average, your opponent will lose $ 25.
-$ 25 = (¼) ($350) + (¾)-$150
If you bet $ 200 and your opponent calls, it will lose on average $ 50.
-$50 = (¼) ($400) + (¾) ($-200)
If you bet $ 600 and your opponent calls, on average, it will lose $ 250.
$250 = (¼) ($800) + (¾) ($-600)
Therefore, when you bet $ 50 more than the 'break even', he will lose $ 25. When you wager $ 100, it will lose $ 50. When you wager $ 500 more, it will lose $ 250.
In general, the value of the error of your opponent will be proportional to the amount exceeding the "break-even point" (sum = 0).
This is the most important concept and this is why we are going to repeat it. The sum expected from the loss of your opponent (and thus your gain) is proportional to the excess of the bet amount (called) over the "break-even point" and not the total bet. If a $ 500 bet is "break-even", you therefore double so your profits by betting $ 600 instead of $ 550. (A bogus conclusion would be to say that it is always better to wager $ 600 instead of $ 550 in this situation because you necessarily double your profits. Your opponent will have to call twice more often $ $550 600 so this update is profitable, but in practice, it will never happen).
The value of the error of your opponent is only only half of your gain expected. To obtain your maximum gain, you need to multiply the value of its error by the chances he has to make. Again, upgrade all in will give your opponent the opportunity to make a bigger mistake, but if your opponent n is never stupid enough to make the call, you will never win anything.
Suppose you are almost sure that your opponent has a the flush draw and a $ 100 bet would be break-even for him. You have to choose between a bet of $ 150, $ 200 or $ 500.
You feel that your opponent will call a bet of 150$ 70% of the time, a bet of $ 200 about 40% and made $ 500 approximately 5% of the time. To find the best, you'll have to multiply the size of their error by the chances that your opponent has to make this mistake:
$ 35 =(150$-100$) (0.70)
$ 40 =(200$-100$) (0.40)
$ 20 =(500$-100$) (0.05)
The best bet is $ 200. This is not what is called more often, but it is she who gives a better profit.
Bet the amount that will maximize your winnings; the value of the error of your opponent multiplied by the chances he has to commit.
Possible gain and wide range of handsIn the last example, you have maximized your winnings against a single opponent and one hand. If your opponent can have one or more prints, you must maximize your gains against all its range (range) of possible hands. A few times, this will mean that you let your opponent run profitably cards.
On the other hand, if your opponent may have a circulation of 4 cards only (outs), a random 8 outs or even 15 outs, which will maximize your profits will be one that will allow him to call correctly with 15 outs but not if it has a circulation of 4 or 8 outs.
Suppose that your opponent has one or two prints; one which will take 25% (4: 1) chances to get out and the other 50% (2 for 1) opportunities to get out. You assume that it will have its draw at 4 to 1 75% of the time and its draw at 2 for 1 25% of the time.
Again, for simplicity's sake, assume that there will be no placing on the River (we will adjust to the river later). The pot is $ 1,000.
The "break-even" for the 2 for 1 draw is a bet of $ 1,000 ($2,000 for a $ 1000). The point of break-even for the draw to 4 against 1 is a $ 333 ($ 1333 for $ 333).
You consider two updates: $ 1500 and $ 500. If you bet $ 1500, you are almost certain that your opponent will fold, even with a draw (which will be the correct game for him). If you bet $ 500, you are almost certain that your opponent will call with its two draws (correctly with two draws, but incorrectly with the 4: 1 draw).
If you bet $ 1500, you win the pot and nothing more. We will call this zero point and we will allocate the value $ 0. You don't win anything the error of your opponent, but you do not lose anything by making it profitable calling.
If you bet $ 500, you win something because the incorrect call on a 4: 1 draw but you lose on the 2 for 1 draw because the call will be correct. The value of the error of your opponent by calling with a 4: 1 draw is $ 100.
$100 = (0.20) ($1500) + (0.80) (-$500)
The value of your error in leaving your opponent correctly call a draw of 2 to 1 is $ 167
-$ 167 = (1/3) ($1500) + (2/3) (-500$)
Even if you make a few times a mistake leaving your opponent correctly call with a draw, your opponent will make a bigger mistake by improperly calling the rest of the time. Overall, in this situation, you maximize your winnings with smaller updates.
Choose the size of your bets to maximize your overall winnings even if it means a few times that your opponents will call your bets correctly on prints
We ignore the possible bets on the river. In reality, the fact that your opponents may have one or more prints will mean that their implied odds will be better than the pot odds. Moreover, in relying on the previous rule, you will need to wager a larger amount than you would if you knew your opponent draw. So, you want to bet more than $ 500 to make sure that the call on the 4: 1 draw is a mistake.