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           London lowball is an excellent game for studying mathematical models that illustrate key principles of pot-limit play. The reasons are the last card dealt to each player is not visible to the others (as opposed to games with common cards like hold’em poker and Omaha), and it can be clear with one card to come that a hand is presently behind, but will certainly win if it improves. We will look at two related ideas in action. First, proper bluffing frequency; second, the leverage on the end that a drawing hand possesses.

            How often should a drawing hand bluff if it busts out? Naturally, this is a decision with strong psychological elements. But we should be aware that mathematics can dictate the proper frequency if those elements are removed, and the participants in the pot play in a “correct” manner. By “correct” we mean at random and with the proper frequency of calling and folding.
            Game theory says to determine the proper frequency of bluffing, we should make the decisions on the last betting round of either always calling or always folding yield the same result. We shall assume the player will always bet exactly what is in the pot. Then by bluffing exactly half as often as the frequency of betting with a winning hand , it will not matter in the long run how the opponent reacts to our bet. He will get the same result. 

             Let us use a concrete example to illustrate this principle. Suppose there is a grand in the pot going into seventh street. Player A has the best hand right now, and player B will have the best hand in all cases where he improves on the last card. At the river player A should of course check. Suppose player B has ten winning cards he can hit. If B bluffs on five of the cards where he misses, then B is bluffing the precise amount that no matter what poker strategy player A adopts, the outcome will always be the same. If A always folds, he will not lose any additional money. If A always calls, he will lose $10,000 on the hands where B has hit, but gain back $10,000 on the five hands where B has run a bluff ($5,000 in B’s bluffs, plus another $5,000 that was originally in the pot). Another way to look at the situation is that when player B bets on the end using the proper game strategy of bluffing half as often as having the goods, then Player A getting 2-1 on the money will break even by always calling. Note that B is doing well on these fifteen hands despite bluffing, profiting by winning an equivalent amount to the main pot. A good poker player is going to do better than in our mathematical model. He will vary from the dictates of game theory by noticing whether his opponent is normally a caller or a folder, whether he has a tell when bluffing, and so forth.

          Here is another model, to tell us when the player with the best made should check going into the last card to deprive the opponent of the extra leverage gained from the bigger pot size.
          Suppose two players are contesting a pot with a grand in it. As previously, Player A is leading with one card to come, and Player B is trying to draw out. Assume there is enough money left for a bet and raise, or a bet now and a bet on the end. For a grand in the pot, this amount would be four thousand apiece. How much the best of it does Player A need to have for it to be correct to bet at the point there is one card to come? Remember, the bet now triples the amount that B can bet on the end. Is betting a good idea when you are increasing the leverage for the drawing hand on the end?
           Let us look at where the break-even point between betting and checking is located. That point is when B has exactly a third of the available cards with which to improve. We illustrate this by seeing what happens in our thousand dollar pot when there are 30 unknown cards, B draws out with 10 of them, and A has his hand stand up with the other 20. (For most games, there would be more unknown cards than this, but the ratios are the same, and these numbers are easier to work with.) 

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B will bet on his 10 winning cards, and 5 of his losing cards. In the case where A checks with one card to come, and always calls on the end (note that always folding yields the same result), A is going to break even on his action. A makes a profit of $2,000 x5=$10,000 on the five hands where B bluffs, plus $1,000 x15=$15,000 on the fifteen poker hands where B busts out and does not attempt to bluff. Thus A shows a total profit of $25,000 on all the pots that he wins. On the ten pots that A loses, he is out 10 x $1,000=$10,000. Thus, we see that A, by checking with one card to come, will win $15,000 on the long-term average, according to game theory. B will win $2,000 for each hand he completes and lose $1,000 for each hand he loses. Thus, he also wins 415,000. The total $30,000 is the original money in the pot on Card Six.

              How does A do by betting the pot with one card to come? Assume that B calls the $1,000 bet at this point, and that A calls B’s $3, 000 bet on the end (as we know, folding on the end yields the identical result). Player A will make a profit of $5,000x 5=$25,000 on the five pots where B bluffs, plus $2,000 x 15=$30,000 on the fifteen pots where B does not bluff. Thus A shows a total profit of $ 55,000 on all the pots that he wins. On the ten pots that A loses, he is out 10 x $4,000. So by betting with one card to came, Player A’s profits are again $15,000.
              The preceding discussion to prove out premise is quite involved, but the premise itself is easy to grasp. According to game theory, the break-even point between betting and checking in the situation we have outlined is when the drawing hand is a 2-1 underdog. If the leading hand is a greater than 2-1 favorite with one card to come, it is correct for him to bet. If the leading hand is less than a 2-1 favorite, it is correct for him to check. Note that when A checks, B must not bet, or A will reraise him all-in, which deprives B of any leverage on the end, leaving him a dog as a result.

               We should mention the cases when A definitely should bet, to make everything crystal clear. Anytime you are the favorite and have the best hand, be sure to bet if you can thereby get the opponent all-in before the last card. Now the drawing hand does not have any implied odds, and the made hand extracts the maximum from the situation. Second, do not try to apply this mathematical principle to games where the last card is dealt faceup, giving the made hand important information about when to call and when to fold should the drawing hand bet on the end. The drawing hand normally has less leverage in such a situation.
         We should apply the information in our discussion to find out how big an underdog the drawing hand must be to fold when the opponent bets the pot with one card to come. He does not have to go strictly by the pot odds, as the implied odds help him because of his betting leverage on the end. When the opponent bets the size of the pot, you are getting 2-1 on the money for a call. So if you can bring your implied odds up to make you no more than a 2-1 underdog, you should call. As we have seen, the implied odds say the power of the drawing hand enhances the actual odds by a factor of fifty percent. So a player with a two-ninths chance to win the pot is actually a three-ninths contender for the money. Three-ninths chance is the same as saying a 2-1 underdog. This means any time the drawing hand has two chances out of nine to win the pot, it is correct to call a pot-size bet. As we have said, this applies only when the draw can bet the size of the pot on the end.

          So the important pot-limit concepts presented in this section are:

  1. On the end a drawing hand should bluff half as often as it bets with a winning hand, assuming the opponent is playing correctly according to poker game theory.
  2. The earning power of a drawing hand is increased by fifty percent by being able to bet on the end. This means:
    1. Any time the made hand is a favorite, he should make the drawing hand go all-in when he can do so.
    2. With one card to come, if the made hand cannot get the drawing hand all-in (or close to it), the made hand should bet only when he is at least a 2-1 favorite.
    3. With one card to come, the drawing hand should call any time he is no worse than a 7-2 underdog, provided he has the option of a pot-size bet on the end.

       The same analysis cannot be applied as rigorously to no-limit, because we have been assuming a pot-size bet. Suffice t say, if you intend to over-bet the pot size each time, and want to play according to game theory, then you should bluff more frequently. Similarly, if you intend to underbet the pot size each time, you should bluff less frequently. 


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