Math discussion about preflop odds from epic O8 thread

So there won’t be further digression about this in the WSOPE Omaha Eight or Better thread, I will give some details here:

There is a lot of confusion surrounding preflop pot odds. Some players think that at each betting interval in a poker hand, you should (at least) call if you have positive equity if the betting stopped at that point. This is frequently not true.

Let’s say you are the big blind in $100-$200 limit hold’em, with blinds of $50 and $100. Your opponent on the button raises to $200. You are getting 3:1 preflop odds on your call. (Even though that $100 big blind is sitting in front of you, you should really think of it as part of the main pot, sitting with your opponent’s $200. You need to put $100 more into the pot to call and try to win the $300 already in the pot.)

Every hand in limit hold’em is less than a 3:1 underdog on average against the raising range of the player on the button. Even lowly 7-2 offsuit is around a 2:1 underdog against a good hand like AK suited. But if there is betting on later streets, our opponent won’t keep putting in 3 units for every 1 of ours. The rest of the money goes in dollar for dollar. We need to fold our bad hands before the flop because we will often be getting less than 1.5:1 odds by the time all the betting is done. (If we call and see a flop with 7-2 offsuit and we simply call our opponent down whenever we make a pair on the flop, we would be putting in 6 units to win 8, which is 1.33:1 odds.)

Some people will argue that hold’em postflop betting is too complex and the real issue is that we don’t always know whether to call or fold after the flop, turn, or river. A game that has easier play decisions but provides good examples showing that your initial odds don’t necessarily indicate your action is Razz.

So let’s play a razz hand where we don’t have the extra calling incentive of being the bring-in. We can assume there is no bring in or that a third player with the high card brings it in and folds to a raise. You have no money invested except for your ante, which as in the hold’em example really isn’t yours anymore. If there are normal sized antes in the pot and you start off with three cards to a 10 against a raiser who has three cards to a 7 or better, you would be +EV if you called your opponent’s raise and we could stop the betting there and run the cards out.

However, if the betting is allowed to continue and you catch equal with your opponent (let’s say you each catch a good one and a bad one on the next two streets), you will be in a situation where it will be right to call him on sixth street if you both catch comparable cards again, even though you are still behind. It was probably right to call him on fourth and fifth street also, even though on each of these three streets you have been calling dollar for dollar with an inferior hand. Your calls are mathematically correct because you are getting good enough pot odds from the money already in the pot.

Overall, it is a losing play to cold call a competent opponent who raises with three low cards in razz when you have three to a ten. The initial call of the opening raise set you on a course of throwing good money after bad.

Another example I gave in the other thread was betting on a football game when your team is down 7 points after the first quarter. Assume that we can determine that right odds are 3:1 against the team that is behind winning the game. If you bet someone $100 to win $300, you will break even with him in the long run. Now let’s make it similar to betting streets in poker where we add the stipulation that after the end of each of the next two quarters either bettor has the option of making an even money bet of $100 and the other person may take the bet or concede all previous bets if he wishes. You aren’t getting full 3:1 odds anymore.

In the football game and in poker, the underdog will be in bad shape more often than the favorite at subsequent betting intervals. Money added to the bet is done so at even money, which lessens the odds.

What is happening in all of these examples is that there are reverse implied odds. That is, we can foresee extra bets at a worse price than the original odds.

So is the moral of the story that we shouldn’t ever call with a hand unless it rates to give us preflop odds that are much better than the price we are getting? The answer is a resounding no. Some hands can go from big underdogs to big favorites when we catch the right cards. Small pairs, suited connectors, and Ace-suited hands are types of drawing hands that usually become overwhelming favorites and potential big money makers when the draws come in.

We will call an extra betting unit in the big blind in limit hold’em with a small pair, and we will usually call an initial preflop raise with a small pair in no-limit hold’em. We call even though we are more than a 4:1 underdog against our opponent if he holds a bigger pair. Our plan will usually be to fold to his continuation bet if we don’t flop a set. Sometimes we will be folding the best hand because it is too hard to discern that he didn’t start with a bigger pair and he didn’t connect with the flop. So why do we call when we know we aren’t getting the correct preflop odds? We call because of the implied odds we get from the times we flop a set and go on to win a big pot.