The Game of No-Limit Hold 'em

 Like all variations of poker, no-limit hold 'em looks like a card game. But it's not, really.

No-limit hold 'em is actually a game of wagering based on imperfect information that uses cards to construct the situations for wagering. Players make bets and call bets based on their estimate that their hand (which they see) will, in the end, be better than their opponent's hand (which they can't see). To make an informed estimate, they have to take four factors into account:

1.    The likelihood that their hand will improve as more cards are dealt, which is pretty much a straight mathematical exercise.

2.    An estimate of the hand their opponent may hold, which is an exercise in inductive reasoning, based on hands he has held in the past, his general style of play, and the bets he has made thus far.

3.    The likelihood their opponent's hand will improve, another mathematical exercise, but complicated by the fact that their opponent's hand is not known for sure.

4.    The money odds being offered by the pot.

When a good no-limit hold 'em player plays a hand, he looks at his cards, looks at his opponents, considers the betting, and makes an educated guess whether to check, bet or call, raise or fold. In many hold 'em hands, one factor becomes so important that the other factors don't require much thought. For example:

1.    A player holds a hand so strong that he doesn't really care what his opponents have.

2.    A player holds a hand so weak that he thinks he's sure to lose a showdown.

3.    The pot odds are so large that he can play the hand with almost any holding.

Don't make the mistake, however, of assuming that even these hands are easy to play. In no-limit hold 'em, there are no trivial hands. Since you don't have to show your cards down to win, under the right circumstances any hand can be a winner.

Game theory

At the heart of non-cooperative game theory as it applies to playing poker is the concept that if you occasionally bluff, people will have to pay you off on your good hands. Game the­oretical analysis focuses on determining the correct bluffing frequency to maximize profits. In game theory models, when two actions have the same expected value, some randomizing method is often used to determine which one to choose. This is a reasonable theory for a mathematical model, but in re­ality, no two situations are ever the same. There is always information swaying you in one direction or another, and there is always an event history which includes plays that have been made on previous hands.

A good player bluffs based on:

1. Who he is playing against. Some opponents are more easily bluffed] than others.

2.      How strong he thinks his opponents hands are.

of his

bluff

3.     What he has represented up to this point about the strength own hand.

4.      Whether the opponent is in a calling mode. It is usually harder to a loser, although some may be easier to bluff because they don't to chase and lose again as they did on the last hand they played.

5.      His chances of winning without bluffing.

The amount that will he gained by a successful bluff 
 

7. The probability that his opponent will counter with a raise-bluff or a c h ec k-raise- b I u if.

S. The cards in his hand that his opponent would need to make a good hand.

9.   His perceived table image. If he bluffs too much, other players will tend to call him down.

10. Situational reasons, a) The opponent is short of money and doesn't have any more to rebuv. b) The opponent doesn't want to call with a weak hand and look stupid in front of a friend or backer who is watch­ing. c) In tournaments, it is easier to bluff right before a tournament break, immediately after the limits have been raised, or when there is a money increase for the next payout spot and the opponent is short on chips.

The same considerations apply when deciding whether to call with a weak hand if you think an opponent might be bluffing. Some players don't bluff in certain situations. For example, there are players who never raise- bluff. Other players must be called down with marginal hands due to the psychological state they are in.

When contemplating if you should call the final bet of a hand where you can only beat a bluff, you may make a pot-odds calculation. For ex­ample, in a limit game when you are getting 7 to 1 odds on a call, you might ask yourself ¡1 this opponent would try to bluff more than one time in eight if he didn't have a good hand. The answer will invari­ably be yes, but you must take into account that in many cases he is more than a 7 to 1 favorite to have a better hand than yours, based on the bet­ting at previous points of the hand.

SOME COOPERATIVE GAME THEORY STRATEGIES

1.     Not quitting a game unless there are players to take your seat. If you hit-and-run, other players will do this in the future, and you will create an atmosphere in which the game is always in danger of breaking up.

2.  Not antagonizing losing players. Even though it may make them play worse at this moment, they may not want to play in future poker games.

Not obsessively getting the best seat position. If you are jumping around the table, continually competing with oth­er players lor best position on live ones, you will make the game into a circus and cause the live ones to feel uncomfortable. 4. Compromising on which games to play and what stakes to play for. If you arc a winning player and always have to have your way, people will avoid playing with you. 5. Not trying to beat a catalyst for as much money as possible in a session. Although it may be best to maximize your profit in the short term, you don't want to do anything that will cause this player to stop playing on a regular basis . Not trying to convince a live one to quit when you aren't able to play. Some winning players arc so afraid that someone else will win money when they aren't in the game, that they try to get the live ones to play only with them. This is very insulting to every­body. Weak players have often made money by being successful businessmen. They are not stupid people. They just aren't experts at poker

Poker game theory

Game theory is a branch of mathematics used to ana­lyze competitive situations that depend not only on one's own choices, but also on the choices made by others. Some people have the mistaken perception that the ability to compute game theory and probability calculations separates expert players from the rest.

Quantitative knowledge of game theory is not necessary to become an expert poker player. Howev­er, qualitative concepts of game theory provide a framework for analyzing expert play and ideas. Similarly, you do not need to be an expert in psychology, although we can use psychology to explain how an expert poker player analyzes himself and his opponents.

There are two kinds of game theory, and both are relevant for the professional poker play­er. The first is cooperative game theory, which involves situations where more than one person has a common goal. In the case ol poker, the goal is to keep a good game going for an extended period of time. The second kind is non-cooperative game theory, which we apply to maximize profits in an individual hand of poker.

MISUSING MATHEMATICS

I have heard people say, "I had to call with a lousy hand because I was getting 5 to 1 on my money," not taking into account that they were up against five opponents with better hands than they had. For example, if you hold SV 2V in hold em, it might be right to call against one player if a lot of extra money was magically added to the pot. Against five players, however, you may lose with a small (lush to a bigger (lush or make two pair and be up against three Fives. There are more ways to lose when you are up against more hands.

Players sometimes say they called because of the implied odds, which takes into account action they were going to get if they made their hand. Often, they had no reason to expect they were going to get that kind of ac­tion. Also, they don't take into account that if they made the hand, it still might not end up being the winner. It is usually right to fold if it was® borderline decision even after accounting for the implied odds. I lowever.

when you analyze the situation, you sometimes realize that your opponent probably isn't going to be able to call your raise when you make the hand. If you come to the conclusion that he may be too weak to call a raise even now, then you should raise with your draw as a back-up way of winning the hand.

When players get beat with the best hand, they sometimes spout math­ematical nonsense about how big a favorite they were. For example, when they lose with Ace-King against King-Jack in hold'em, you may hear them complain that they got beat by a three-outer and were a 45 to 3 favorite. In reality, they were between a 2.3 to I favorite and a 3.1 to 1 favorite, depending on the suits.

Story 1: In 1992, I was playing in the best S75-S 1 50 hold'em game in which I had ever played. One player was playing like such a maniac that he raised before the (lop every time it was his turn. If it was two bets to him, he made it three bets. If it was three bets to him, he made it four bets, which was the cap. After a few rounds, the other players knew not to raise with their good hands. They would limp in, try to trap a few callers, and wait for the maniac's raise and then they would reraise. A well-known poker writer was sitting to the maniacs right. For seven consecutive rounds, the poker writer called in the small blind only to be knocked out before the flop when the maniac raised and some other player reraised. The maniac went broke after the seventh round of this, so I don't know how long this would have gone on. I was surprised that the poker writer, a supposedly winning player, had so little table feel. He must have played using strictly memorized guidelines. He would look at his hand, let's say 10V 4V or 04 84, and after consulting his mental chart, he would decide that the hand was worth a call in the small blind. Then, when it came back to him, he would decide the hand did not qualify for two more bets.

Story 2: In 1975, while 1 was attending college, I played dealer's choice with college students, locals, and a philosophy professor. The minimum buy-in was $20, but towards the end of the night, players would typically have a couple of hundred in front of them. One night, the professor and I had been killing the game, and we each had around $3000 in front of us.

The current game was pot-limit, seven-card stud hi-lo. I raised the bring-in $7, making the pot S28 after two of the other six players called on third street. A bet of $28 on fourth street got the pot down to heads- iip between the professor and me. The pot was now $84. 1 bet $84 on fifth street and the professor called, making the pot $252. He bet $252 on sixth street. I hesitated and then called. On the river, he bet $750 and I raised the rest of his chips.