General concepts

Points of Play

Tournament Play

Should You Wait?

Suppose you are playing in a hold’em or lowball tournament and find yourself down to a very short stack. Say you have just enough to call the big blind. And there is just one or two hands before you will be forced to blind it. 

I have frequently been asked in this situation if it is correct to loosen up your calling requirements. Many players have suggested to me that since you have to blind it anyway on the next hand, you should play any hand that is better than the average random hand you will get on your blind. 

Others say that the upcoming hand makes no difference.  It should not influence you to take the worst of it on the present hand.

While mulling over this problem, I realized that a simply stated mathematical question can be to solved to give some insight into this dilemma.

Suppose you have $ 100 in your pocket with no way of getting more for the time being. Meanwhile, in five minutes, you will be forced to bet this $ 100 with someone on a sports event or some such thing. 

You can only get out of this bet if you lose the $ 100 before he shows up.  (However, if you have increased your bankroll, your bet still be exactly $ 100.)

The question is what should you do if someone else offers to bet $ 100 on something right now?

If you take him up on it, you will either wind up with $ 200 ($ 100 of which you will have to bet back in five minutes, so that your final bankroll will be either $ 100 or $ 300 ) or you will go broke, which eliminates your obligation to make the second bet.

Before deciding whether to take the first bet or whether to pass it and just wait to make the second bet, we have to know the chances of winning each bet. 

Call your chances of winning the first bet “X” and your chances of winning the second bet (if you get the chance to make it) “Y.”

To solve this problem we must calculate the expected value of your bankroll if you take the first bet, and also if you pass it. The second option is easy to figure. 

If you pass the first bet, and only make the second bet, the expected value of your bankroll is simply $ 200 Y. What about if you do take the first bet? 

Now you will wind up with either $ 0 (chances of this are [1-X], $ 300 (chances of this are XY), or $ 100 (chances of this are X [1-Y]. So the total expected value of your bankroll if you take the first bet is:

0(1-X) + 300XY + 100X [1-Y]

This reduces to:

200XY + 100X

the expected value of your bankroll if you accept the first $ 100 bet and then make the second bet if you win it. Notice that both options give you the exact same expected value if.

200Y = 200XY +100X

Solving for X:

200Y = X(200Y +100)

     X =        200Y     
               200Y + 100
        =        2Y    
              2Y + 1

So if the probability of winning the first bet is greater than:

        2Y   
    2Y + 1

where Y is the probability of winning the second bet, then you should accept it.

Let’s look at some examples. If your chances of winning the upcoming bet is 40 percent, then you should gamble on a preceding bet if your chances are (.8)/(1.8) which is about 45 percent. 

If your upcoming bet is ten percent, the preceding bet should have a probability of (.2)/(1/2) or about 17 percent to make it worthwhile.

These figures show that if your imminent bet does indeed have little chance, it is right to make an earlier bet with less the worst of it.

However, this earlier bet must be more than just slightly better than the upcoming one. (Notice that if the upcoming bet is exactly 50 percent, then the probability of winning the first bet must be greater than:

     2(.5)
2(.5) + 1

which is one half. This agrees, of course, with common sense.)

What may not agree with common sense is what you do if your upcoming bet has a greater than 50 percent chance.  Now taking the first bet will jeopardize your opportunity to make this second good bet. 

Does this mean you should pass up the first bet even if it has a little bit the best of it? Using the same formula, we see, for example, that if the second bet has a 60 percent chance of winning, the chances for the first bet must be greater than (1.2)/ (2.2) or about 55 percent.

If the second bet has an 80 percent chance, the first bet must be (1.6)/ (2.6) or about 62 percent. So it is true that it might be right to pass up a bet with a little bit the best of if the upcoming bet has a lot the best of it. 

However, you should never pass up a bet as a more than 2-to-1 favourite (if the second bet is 100 percent, the first bet by this formula must merely be above (200%)/ (300%)).

These results have great significance, not just to tournament play either. If you are one of those people who constantly risk going broke with a little bit the best of it even when there are better bets just over the horizon, this chapter may make you want to think again.

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