10-27-2001, 12:27 AM
Let us begin to answer the question I posed regarding your EV in various positions in a typical 15-30 game with a 10 and 15 blind. I stipulated that you are a very good player against mediocrities and asked what your Ev would be in first position (UTG), on the button, and two to the right of the button.
Lately I have been criticized for merely using my judgement for questions such as these when simulators and documentators are available. I responded that I am not using merely judgement but also logic and math. I will now demonstrate what I meant by examining the EV of a good player when he is under the gun.
First notice that this player's TOTAL Ev will be about $30 per hour. That is about $7 per round. Or about 80 cents per hand. Clearly however, he doesn't average 80 cents in each position. He must do way better than this on the button and way worse in the blinds. But the sum of his Evs in each position should be about $7.
Now common sense tells you that even world champions could not beat even easy games if they were always in the blind positions. In other words, these positions must have negative Ev. On the other hand, that negative Ev must be less than the dollar amounts of the blinds. Since you could fold every single hand and lose that much. Without elaborating, it should be clear that the blind positions lose (Ev wise) something in the neighborhood of half of what they put in to start with. Let's call it $13 between the two of them. Add that to the $7 a good player expects to make overall per round, and that means that the other eight positions in a ten handed game will make about $20 total or an average of about $2.50 per hand.
This 2.50 per hand does not take into account the difference in positions. The button should gobble up lots more of this $20 than UTG will. In fact will UTG show a positive expectation at all? Some posters postulated that it wouldn't. Is that reasonable? Actually no. Simple logic disproves that notion, for a good player, because if it was true, then it would become correct to bring the EV up to zero by folding every hand. Since that is obviously wrong it is easy to see that UTG has to have a positive expectation, even if only because you play nothing but aces, kings, queens and AK suited.
Now suppose you played, (and raised with) only those four hands. What would that do to your EV? Lets look at aces. Mike M. made the ridiculous comment that I was implying that no hand has an EV higher than the total of the blinds. The comment shows an ignorance of the FTOP. Because with rare exceptions, any hand that will get calls from players who wouldn't call if they knew what you had, adds that much more to your profits over and above the size of the blinds. Aces are the ultimate example. Everyone should fold if they saw your hand, thus giving you an automatic profit of $25. Their incorrect calls of your raise merely add to that. It is difficult to say exacxtly how much aces are worth under the gun since it depends a lot on what hands you will get called with, but it is clear that it is at least fifty bucks in almost all games.
Kings are almost as good. It is rare that another player will have a hand that will eat into the kings EV. So it gains from almost all calls. It is true that kings can get outrun by lone aces but they don't lose a lot when that happens. Let's say their EV is $40, Queens are a bit more likely to get called by hands that they don't like. Still, I don't think they would opt to steal the blinds without a fight. Say their Ev is $30.
We are now conservatively estimating that the three big pairs gives us an average Ev of $40 UTG. It may be more like $60 in some games. And you get this $40 average three times in 221 hands. That is 55 cents right there. Jacks are at least fifteen bucks one in 221 or seven cents more. AK suited is at least fifteen bucks every 330 hands for another nickel. These big hands alone give UTG an EV of at least 67 cents. Throw in my estimate of $8 for AK offsuit once every 110 hands and we are up to 75 cents. Add in a little for two suited high cards, medium pairs and AQ and we are right around a buck. (Keep in mind that the worst of the hands we play UTG may be break even or lose a few pennies but are still right to play to maximize what we make from the other hands.)
Bottom line is that the final answer must certainly be between $.75 and $1.50. It can't be less than that because just the big pairs are worth at least that much, and it can't be better then that because that $20 is skewed to the later positions. Common sense says the last three are worth more than ten bucks. (More on this later) Thus, the first five are worth an average of two bucks with UTG probably being least of all. I'd say $1.25.
Lately I have been criticized for merely using my judgement for questions such as these when simulators and documentators are available. I responded that I am not using merely judgement but also logic and math. I will now demonstrate what I meant by examining the EV of a good player when he is under the gun.
First notice that this player's TOTAL Ev will be about $30 per hour. That is about $7 per round. Or about 80 cents per hand. Clearly however, he doesn't average 80 cents in each position. He must do way better than this on the button and way worse in the blinds. But the sum of his Evs in each position should be about $7.
Now common sense tells you that even world champions could not beat even easy games if they were always in the blind positions. In other words, these positions must have negative Ev. On the other hand, that negative Ev must be less than the dollar amounts of the blinds. Since you could fold every single hand and lose that much. Without elaborating, it should be clear that the blind positions lose (Ev wise) something in the neighborhood of half of what they put in to start with. Let's call it $13 between the two of them. Add that to the $7 a good player expects to make overall per round, and that means that the other eight positions in a ten handed game will make about $20 total or an average of about $2.50 per hand.
This 2.50 per hand does not take into account the difference in positions. The button should gobble up lots more of this $20 than UTG will. In fact will UTG show a positive expectation at all? Some posters postulated that it wouldn't. Is that reasonable? Actually no. Simple logic disproves that notion, for a good player, because if it was true, then it would become correct to bring the EV up to zero by folding every hand. Since that is obviously wrong it is easy to see that UTG has to have a positive expectation, even if only because you play nothing but aces, kings, queens and AK suited.
Now suppose you played, (and raised with) only those four hands. What would that do to your EV? Lets look at aces. Mike M. made the ridiculous comment that I was implying that no hand has an EV higher than the total of the blinds. The comment shows an ignorance of the FTOP. Because with rare exceptions, any hand that will get calls from players who wouldn't call if they knew what you had, adds that much more to your profits over and above the size of the blinds. Aces are the ultimate example. Everyone should fold if they saw your hand, thus giving you an automatic profit of $25. Their incorrect calls of your raise merely add to that. It is difficult to say exacxtly how much aces are worth under the gun since it depends a lot on what hands you will get called with, but it is clear that it is at least fifty bucks in almost all games.
Kings are almost as good. It is rare that another player will have a hand that will eat into the kings EV. So it gains from almost all calls. It is true that kings can get outrun by lone aces but they don't lose a lot when that happens. Let's say their EV is $40, Queens are a bit more likely to get called by hands that they don't like. Still, I don't think they would opt to steal the blinds without a fight. Say their Ev is $30.
We are now conservatively estimating that the three big pairs gives us an average Ev of $40 UTG. It may be more like $60 in some games. And you get this $40 average three times in 221 hands. That is 55 cents right there. Jacks are at least fifteen bucks one in 221 or seven cents more. AK suited is at least fifteen bucks every 330 hands for another nickel. These big hands alone give UTG an EV of at least 67 cents. Throw in my estimate of $8 for AK offsuit once every 110 hands and we are up to 75 cents. Add in a little for two suited high cards, medium pairs and AQ and we are right around a buck. (Keep in mind that the worst of the hands we play UTG may be break even or lose a few pennies but are still right to play to maximize what we make from the other hands.)
Bottom line is that the final answer must certainly be between $.75 and $1.50. It can't be less than that because just the big pairs are worth at least that much, and it can't be better then that because that $20 is skewed to the later positions. Common sense says the last three are worth more than ten bucks. (More on this later) Thus, the first five are worth an average of two bucks with UTG probably being least of all. I'd say $1.25.