Analysis of Howard's example
Tom Weideman
analysis here on rgp:
The poker game
--------------
Two players are going to the river with a pot size of N bets. Player A is
the current leader, Player B has a 50% chance of catching a card that beats
Player A's current hand, and Player A has a 50% chance of catching a card
that beats Player B's improved hand.
The special condition
---------------------
Assume that the pot is infinite in size (N->infinity), so that neither
player can fold if they have any hand that can beat any other hand (i.e.
only Player B can fold, when he doesn't improve).
The objection
-------------
Using an infinite pot as a means of assuring certain behavior on the part of
the players is not a good approach for estimating play in large pots,
because some fold frequencies that are 0% in infinite pots become 100% in
finite pots as a means of exploiting the "approximately optimal" strategy.
I suggested just imposing the rule that a player is not allowed to fold
unless he can't possibly win, rather than talking about N->infinity.
The easing of constraints
-------------------------
Now let the pot size be finite: N. Howard quotes Jimmy Warren that the
correct bluffing frequency for Player A when he fails to improve is
1/(4*N+2). This can be used to compute all other action frequencies.
------------------
Okay, that's the history. But now it has taken a weird turn. Howard's
analysis is based on the same "can't fold unless you can't win" rule that I
suggested, EVEN THOUGH the game is now being solved for finite N. This
seems a bit odd, since Sklansky's original reason for making N infinite was
to get around having to impose this rule, so that (presumably) the solution
would be close to that for large N. Well, this solution works for any N,
but ONLY if we don't allow people to fold! So this is even FURTHER from
real-life poker than Sklansky's example, especially when N is small.
Here's where the problem occurs. From Howard's "Analysis of 1 bet pot"
post:
> If Player B uses strategy 1 from the previous post, he will win the pot 25%
> of the time. This makes Player A's pot equity .75 bets. But Player A will
> bet with the best hand 1/3rd of the time and get called 1/6th of the time.
> He will also bluff 1/6th of the time and get caught 1/12th of the time.
> This makes his river betting equity (.333/2)-(.167/2) = .083. And it makes
> his total equity .750 + .083 = .833.
Why on earth would Player B use strategy 1 from the previous post? To
refresh everyone's memory, Player B HAD TO CALL if check-raised (ostensibly
because the pot was infinite), so he was indifferent to betting or checking
when Player A checked to him. But now the pot is finite. In fact, it is
SMALL. With Player A announcing that he will never check-raise bluff,
Player B's best course of action when checked to is to bet and fold if
check-raised. He is NOT indifferent to his choices - this is definitely his
best strategy. Of the times he is checked to, he wins 1 extra bet 2/3 of
the time and loses 1 bet 1/3 of the time.
Obviously other strategies (like perhaps allowing Player A check-raise
bluff) need to be included in order to truly reach the optimal solution for
real-life poker. The point is that there are precious few situations in
real poker that can be easily solved on the back of an envelope. Typically
it requires some fairly contrived circumstances (such as Sklansky's infinite
pot or Howard's subsequent calculation) for the math to be done so easily.
That's just fine (and very instructive), but one has to keep track of the
limitations of this approach, or some incorrect conclusions can be drawn.
I have some other comments regarding Howard's recent interesting posts about
poker math (no, my comments aren't mean, and I hope this post didn't come
across that way), but they will have to wait until after I wrap some
presents and revel in the merriment of the season with my family for a day
or so.
Tom Weideman
Howard Lederer
"Tom Weideman" <zwi...@attbi.com> wrote in message
news:BA2E214B.2268E%zwi...@attbi.com...
> Okay, that's the history. But now it has taken a weird turn. Howard's
> analysis is based on the same "can't fold unless you can't win" rule that
I
> suggested, EVEN THOUGH the game is now being solved for finite N. This
> seems a bit odd, since Sklansky's original reason for making N infinite
was
> to get around having to impose this rule, so that (presumably) the
solution
> would be close to that for large N. Well, this solution works for any N,
> but ONLY if we don't allow people to fold! So this is even FURTHER from
> real-life poker than Sklansky's example, especially when N is small.
You did not carefully read my description of Player B's two options in my
post on the 9.5 bet hand. He can fold every time Player A bets or he can
call every time. Of course he could mix it up, but that will not influence
his equity and it would make the math difficult. What I do not allow Player
B to do, is ever bet after Player A checks. I admit that this takes a
possibly profitable option away from Player B. But, I wanted to simplify
the problem so I could get to the general poker observations I made in my
posts. Both Jimmy and David raised the possibility that Player B might be
able to recoup some of his lost river equity by betting sometimes after
Player A checks. I plan on looking into the proper betting strategy for
Player B after a check and Player A's counter strategy. I do not know
whether Player B can bet profitably, but I plan to try to figure it out.
The bigger question is whether Player B employing a betting strategy changes
Player A's proper betting frequencies on the initial action. I look forward
to tackling this.
I do think that when looking at problems like this, you have to limit the
number of bluff-rebluff-rerebluff options. The return of poker knowledge
you get can't be justified by the massive increase in complexity if you
don't limit theses options. Limiting the last betting round to bet or check
followed by call or fold may have been an oversimplification, but it served
its purpose for the points I wanted to make.
Howard Lederer
Barbara Yoon
>> Two players are going to the river with a pot size of N bets. Player A
>> is the current leader, Player B has a 50% chance of catching a card
>> that beats Player A's current hand, and Player A has a 50% chance
>> of catching a card that beats Player B's improved hand.
>>
>> for Player A when he fails to improve is 1/(4*N+2).
>>
>> rule that I suggested, EVEN THOUGH the game is now being solved
>> around having to impose this rule, so that (presumably) the solution
>> would be close to that for large N. Well, this solution works for any N,
>> but ONLY if we don't allow people to fold! So this is even FURTHER
>> from real-life poker than Sklansky's example, especially when N is small.
Ooopps.....I didn't see Tom's post here before I posted similar reservations
in earlier thread...
Howard Lederer:
> You did not carefully read my description of Player B's two options...
> He can fold every time Player A bets or he can call every time. Of course
> he could mix it up, but that will not influence his equity and it would make
> the math difficult. What I do not allow Player B to do, is ever bet after
> Player A checks. I admit that this takes a possibly profitable option away
> from Player B. But, I wanted to simplify the problem so I could get to the
> general poker observations I made in my posts. Both Jimmy and David
> raised the possibility that Player B might be able to recoup some of his
> lost river equity by betting sometimes after Player A checks. I plan on
> looking into the proper betting strategy for Player B after a check and
> Player A's counter strategy. I do not know whether Player B can bet
> profitably, but I plan to try to figure it out. The bigger question is whether
> Player B employing a betting strategy changes Player A's proper betting
> frequencies on the initial action. I look forward to tackling this.
Tom Weideman:
> Here's where the problem occurs. From Howard's "Analysis of 1 bet
> pot" post:
>
>>> If Player B uses strategy 1 from the previous post, he will win
>>> the pot 25% of the time. This makes Player A's pot equity .75 bets.
>>> But Player A will bet with the best hand 1/3rd of the time and get
>>> called 1/6th of the time. He will also bluff 1/6th of the time and
>>> get caught 1/12th of the time. This makes his river betting equity
>>> (.333/2)-(.167/2) = .083. ...his total equity .750 + .083 = .833.
>
> Why on earth would Player B use strategy 1 from the previous post?
> To refresh everyone's memory, Player B HAD TO CALL if check-raised
> (ostensibly because the pot was infinite), so he was indifferent to betting
> or checking when Player A checked to him. But now the pot is finite.
> In fact, it is SMALL. With Player A announcing that he will never
> check-raise bluff, Player B's best course of action when checked to
> is to bet and fold if check-raised. He is NOT indifferent to his choices
> - this is definitely his best strategy. Of the times he is checked to, he wins
> 1 extra bet 2/3 of the time and loses 1 bet 1/3 of the time. Obviously
> other strategies (like perhaps allowing Player A check-raise bluff) need to
> be included in order to truly reach the optimal solution for real-life poker.
> The point is that there are precious few situations in real poker that can be
> easily solved on the back of an envelope. Typically it requires some fairly
> contrived circumstances (such as Sklansky's infinite pot or Howard's
> subsequent calculation) for the math to be done so easily. That's just fine
> (and very instructive), but one has to keep track of the limitations of this
> approach, or some incorrect conclusions can be drawn.
Yes.....my question too....."With Player A announcing that he will never
check-raise bluff, Player B's best course of action when checked to
is to bet and fold if check-raised" -- so there's no point in Player A
check-raising if Player B knows to never call such any such check-raise
.....and so Player A just 'check-calling' would yield same here.....OK?!
tadperry
Thank you very much for posting regarding a question of mine, and one I
assume was shared by others. Also, please consider this a "peace offering"
of sorts for this Holiday Season.
Happy Holidays to you and everyone on RGP.
Tad Perry
tadperry
news:aub4l8$61b$1...@bob.news.rcn.net...
Guys and gals,
I think the problem here is the fact that when it comes to poker the
psychology dominates the math.
As Tom says, for contrived circumstances the math can be straightforward and
easy, and while that can be instructive regarding the general case, you have
to keep in mind that it differs from the general case considerably.
For instance, even including layers of complexity regarding bluffing and
re-bluffing, consider if a psychologist were to find the following result in
terms of human psychology and pattern recognition: a human upon seeing the
same pattern 7 times in a row is very highly likely to assume the pattern
results in trial 8 even though aware that he or she knows nothing for
certain.
Well, that would just totally put a wrench in these mathematical formulas
and it would be very clear what to do.
Ultimately the answers lies in the nature of human psychology. There may be
math formulas that give the answers to questions such as "how much should I
bluff", but they may include terms to express odd facts such as the above
(one I pulled out of a hat, but which may be true in some form).
The time to bluff is whenever it will work. Depending on the psychology of
your opponent, their attitudes about you and the game and so on, this might
work every time and it might work never, or your ability to do this lies
somewhere in between the two extremes.
Obviously, your ability to bluff goes down if you're bluffing too frequently
and goes up if you're bluffing too infrequently.
For many opponents, you can display a bluffing frequency that is far less
than optimal, but which *strikes* the opposition as being too much. They
then turn into police and you then exploit them because they do not alter
their approach as you feed them the bare minimum number of bluffs they need
to be happy and reap the rewards.
This is but one example of psychology dominating the numbers. Although this
approach is actually quite effective, there are other approaches that I
think are even better, but which are also considered along the same lines of
human psychology.
Regards,
Tad Perry
Howard Lederer
> check-raise bluff, Player B's best course of action when checked to
> is to bet and fold if check-raised" -- so there's no point in Player A
> check-raising if Player B knows to never call such any such check-raise
> .....and so Player A just 'check-calling' would yield same here.....OK?!
As I state in my reply to Tom, Player A NEVER gets the chance to check
raise. He checks with the intention of check raising often enough to
discourage Player B from betting. The previous post I refer to is not the
thread from a couple of weeks ago, but the analysis of the 9.5 bet hand. I
guess I was unclear. I hope this clears it up for you. I still stand by my
math for the simple problem I tried to tackle. I admit that the math might
break down if Player B ever decides to bet.
Howard Lederer
Howard Lederer
"tadperry" <tadp...@attbi.com> wrote in message
news:sw9O9.499120$NH2.34392@sccrnsc01...
>
> Guys and gals,
>
> I think the problem here is the fact that when it comes to poker the
> psychology dominates the math.
> As Tom says, for contrived circumstances the math can be straightforward
and
> easy, and while that can be instructive regarding the general case, you
have
> to keep in mind that it differs from the general case considerably.
I couldn't agree more. I play poker for a living. I do not solve problems
about it for a living. But I do feel that trying to understand the math of
these situations frees me up to play instinctively, while still remaining
true to the basic math of the situation.
Howard Lederer
tadperry
news:aybO9.157559$6k.29...@news1.west.cox.net...
I believe you've found the very best use for the information.
tvp
Tom Weideman
You are approaching the study in a way that I have done for several years -
start with a simple example, figure out what ways it doesn't reflect real
poker, and then complicate the problem slightly by including one of those
ways and solve again. Keep bootstrapping until a reasonable model is built,
solving piecewise-tractable problems along the way. But you have strayed in
this case, because you actually haven't solved anything - you've just thrown
together a few things that look like they fit together. I'll see if I can
explain...
When doing this bootstrap method, it's important that you construct an
actual well-defined game. You'll have to abridge the actual rules of poker
somewhat in the process, but that's okay so long as the game itself is
self-consistent (you'll just have to be careful about drawing conclusions
about actual poker from this abridged game, and the more changes you make
from the actual game, the less these conclusions can be trusted).
In the example at hand, it's fine to require that Player B cannot bet if
checked to, and can only call or fold to a bet. But when you do this, the
optimal solution will certainly not include a "check raise" option for
player A. I guess you could insist as one of the "rules" of your
abbreviated game that whatever frequency Player A chooses to check when he
misses, he must check half that often when he hits. But inventing such
rules are dangerous unless you have good reason to do so. You might make
the argument that you got this "rule" from an earlier, simpler incarnation
of the game you are solving, but the problem is that THE vital criterion
that brought this part of strategy into being (the pot size is infinity, so
no one can fold a hand that might win) has now been relaxed.
When no one could fold a possible winner, the check/raise-to-check/call
ratio was found to be 1:2, because Player B would have to call if raised, so
he was risking 2 bets to win 1. But now if he were to bet and get
check-raised, he would only be getting 4-to-1 on his call (not
infinity-to-1), so he might very well decide to fold (very few people
check-raise bluff the river more often than 1 out of 5 times). In addition,
the check-raise bluff is not even one of Player A's stated options. Okay,
I'm straying a bit, but my point is that Sklansky's infinite pot criterion,
while a bit sloppy, was intended to take away ALL bluffing options while at
the same time removing most folding options. As soon as the condition is
relaxed, many more things come out of the woodwork, and most importantly for
this example, the check-raise frequency intended to make Player B
indifferent to betting when checked to is no longer valid. Who knows what
inserting it my fiat will do to the overall solution?
Okay, so to reiterate: It's perfectly fine to truncate problems, and you
could make the case that you have done so here, by saying you inserted the
rule that Player B cannot bet when checked to, AND Player A must check half
as many made hands as unmade hands. Fine. But as I said, the farther you
get from the rules of the actual game you are studying, the less you can
trust your conclusions. IMO, you have actually gotten farther from reality
than the original "infinite pot" problem, not closer. I mean, what if (for
example) for some values of N you find that you should (almost) never
check-raise? (I am not saying this is the case, but you wouldn't have a way
of finding this out with this game.) Then tying the check-raise frequency
to the frequency that you check an unimproved hand (by the 2:1 ratio) might
lead to some really bad conclusions.
> I do think that when looking at problems like this, you have to limit the
> number of bluff-rebluff-rerebluff options. The return of poker knowledge you
> get can't be justified by the massive increase in complexity if you don't
> limit theses options. Limiting the last betting round to bet or check
> followed by call or fold may have been an oversimplification, but it served
> its purpose for the points I wanted to make.
As I discussed above, I agree 100%. Perhaps you don't realize it, but you
have done more than than just curtail the allowed actions - you have linked
the frequency of one action (checking when unimproved) to another action
(checking when improved), and you have done so using an answer from a
simpler problem whose circumstances don't apply here. Of course, as the pot
size grows, this won't be such a problem (since it will approach the simpler
problem for which this ratio was intended), but it could be a significant
error for situations like N<=1 (big bet poker).
The simple conclusions you appear to have drawn so far can also be drawn
from simpler but more consistent ("safer") versions of poker. I'm just
trying to point out that trying to go further in the manner that you have
begun will probably not lead to the results you want.
Tom Weideman
Barbara Yoon
>> Two players are going to the river with a pot size of N bets. Player A
>> is the current leader, Player B has a 50% chance of catching a card
>> that beats Player A's current hand, and Player A has a 50% chance
>> of catching a card that beats Player B's improved hand.
>>
>> for Player A when he fails to improve is 1/(4*N+2).
>>
>> analysis is based on the same "can't fold unless you can't win" rule
>> that I suggested, EVEN THOUGH the game is now being solved for
>> finite N. This seems a bit odd, since Sklansky's original reason for
>> making N infinite was to get around having to impose this rule, so that
>> (presumably) the solution would be close to that for large N. Well, this
>> solution works for any N, but ONLY if we don't allow people to fold!
>> So this is even FURTHER from real-life poker than Sklansky's example,
>> especially when N is small.
Howard Lederer:
> You did not carefully read my description of Player B's two options...
> He can fold every time Player A bets or he can call every time.
> Of course he could mix it up, but that will not influence his equity
> and it would make the math difficult. What I do not allow Player B
> to do, is ever bet after Player A checks.
But, Howard and Tom.....if Player B is not allowed to initiate any bet,
then doesn't the whole question, in the "N = 1" case, simply boil down
to the basic game theory situation where Player A should bet on ALL
of his best hands, and bluff on half of his worst hands -- wouldn't this
strategy be better than that from Jimmy Warren's formula...?!
Howard Lederer:
> I admit that this takes a possibly profitable option away from Player B.
> But, I wanted to simplify the problem so I could get to the general poker
> observations I made in my posts. Both Jimmy and David raised the
> possibility that Player B might be able to recoup some of his lost river
> equity by betting sometimes after Player A checks. I plan on looking
> into the proper betting strategy for Player B after a check and Player A's
> counter strategy. I do not know whether Player B can bet profitably,
> but I plan to try to figure it out.
Howard.....you might find that Player A's basic advantage in this 'game'
is just too overwhelming for any betting at all by Player B...
Barbara Yoon
> Two players are going to the river with a pot size of N bets. Player A
> is the current leader, Player B has a 50% chance of catching a card
> that beats Player A's current hand, and Player A has a 50% chance
> of catching a card that beats Player B's improved hand. Howard
> quotes Jimmy Warren that the correct bluffing frequency for Player A
> when he fails to improve is 1/(4*N+2).
Howard Lederer:
> Player B's two options... He can fold every time Player A bets or he
> can call every time. Of course he could mix it up, but that will not
> influence his equity and it would make the math difficult. What I do
> not allow Player B to do, is ever bet after Player A checks.
me:
> But, Howard and Tom.....if Player B is not allowed to initiate any bet,
> then doesn't the whole question, in the "N = 1" case, simply boil down
> to the basic game theory situation where Player A should bet on ALL
> of his best hands, and bluff on half of his worst hands -- wouldn't this
> strategy be better than that from Jimmy Warren's formula...?!
Howard Lederer: alternative strategy:
>>>>> 1/6 Bluff 1/4
>>>>> 2/6 Check after missing 1/4
>>>>> 1/6 Check raise -
>>>>> 2/6 Bet after making 2/4
>>>>> --- ---
>>>>> 6/6 4/4
>>>>>
>>>>> ...equity .833. ...equity .875. (...OK?!)
Howard Lederer:
> I admit that this takes a possibly profitable option away from Player B.
> But, I wanted to simplify the problem so I could get to the general poker
> observations I made in my posts. Both Jimmy and David raised the
> possibility that Player B might be able to recoup some of his lost river
> equity by betting sometimes after Player A checks. I plan on looking
> into the proper betting strategy for Player B after a check and Player A's
> counter strategy. I do not know whether Player B can bet profitably,
> but I plan to try to figure it out.
me:
Tom Weideman
> I have some other comments regarding Howard's recent interesting posts about
> poker math (no, my comments aren't mean, and I hope this post didn't come
> across that way), but they will have to wait until after I wrap some
> presents and revel in the merriment of the season with my family for a day
> or so.
--------------
In the "Analysis of 9.5 bet pot" thread, Howard wrote:
> I find it instructive to think about this simple hand. I am often surprised
> to see how seldom you should bluff in certain situations. In a real life
> hand like this, I would find it hard to imagine a player actually bluffing
> only 1/40 times. This tells me that folding on the end in limit, when you
> have chance to win, is a losing proposition. I am sure that in general
> poker players bluff way too often in limit. But, conversely players fold
> too much, thus justifying the high bluffing frequencies. Tackling a problem
> like this gives me renewed conviction that my stubborn ways on the river are
> correct.
This conclusion actually employs some faulty reasoning, though it may not be
clear because the study of this game was perhaps not as detailed as it could
have been. You can't conclude whether you should call more or call less on
the river based on the fraction of total hands a player bluffs. For
example, let's say we stipulate that the optimal fraction of hands that a
player should bluff in an actual poker game (not the example given) is 1/40.
If this person actually bluffs only 1/80, it still may be correct to call
him more often than someone that bluffs less than the optimal amount (say
1/20). The point I'm trying to make is that the correct ratio to consider
when deciding whether or not to call is not the number of bluffs to the
total number of hands. It is not a good idea to try to get a sense of
whether someone bluffs too much or too little from such a comparison. Doing
so may actually make you more stubborn than you should be or not stubborn
enough, depending upon the opponent. I'm sure you see what I am talking
about.
> Will spending the time to solve this problem prepare me for when this exact
> same hand comes up in the future? No, but it will make it more likely that
> my on-the-fly decisions at the table will be correct when a similar hand
> does come up.
Agreed. I know several people who have approached poker this way for many
years (and they have gone a lot further and deeper with the analysis than
you appear to have, and not just in game theory). This is not to say that
they aren't knowledgeable about other poker ideas outside this realm, such
as psychology and tells, either. Many of them do not play regularly in LV,
nor do they have the bankroll to play ultra-high, so they are entirely off
your radar screen. I think you would be very surprised by the amount of
poker knowledge and at-table talent lurks in the poker world outside the
Bellagio top section. I've tried to express this before in threads about
"the best players in the world" and such, but I am virtually always shunned
for this opinion.
> I recently had an interesting
> discussion with Gus Hansen (the winner of the $10,000 Bellagio tournament).
> He firmly believes that poker is in its infancy. He thinks that the play,
> even in the biggest games, isn't even close to perfect. I know that he is a
> deep thinker, and he is going to try to revolutionize the game in the
> process.
I'm a bit surprised that you found this position "interesting". Perhaps I
have misread your meaning, but I thought it was pretty obvious that poker is
still extremely young as a game. One way to gauge this is what I expressed
in another post (granted it's only an opinion): A computer that plays
optimally would easily beat even the best human players today. As the game
evolves and humans develop a better understanding of it by constructing a
theoretical framework for it (this would allow them to make assessments for
situations too difficult to compute, much like opening theory does for
chess), play by the informed will get closer to optimal, and that computer
will not win as easily (though obviously it will still win). When/if this
stage is reached, I think it is entirely possible that correct poker play
will look entirely different than it does today, in the same way that the
Evans gambit has virtually vanished from top-level chess play, even though
it was quite a popular opening in the late 19th / early 20th centuries.
It's possible I have this entirely wrong, but the parallels with chess are
quite compelling. The obsolete gambits popular around 1900 were very
effective at exploiting the play of unsophisticated opponents present at the
time, and were therefore considered to be powerful weapons. Once these
problems were solved and the solutions became common knowledge, players with
the white pieces had to develop new weapons, which were also eventually
solved, and so on. While I don't think we are at the very beginning in
poker, we have only developed, solved, and discarded very few weapons, and
we likely have much further to go.
-----------
In the "Analysis of 1 bet pot" thread, Howard wrote:
> I have found David's problem to be quite ingenious. It informs us on so
> many levels, many of which, I'm sure, were unintended by David himself.
You really need to get out more, heh. David is a smart fellow, and he likes
to look at game theory, but he admits he is too lazy to get into the
nitty-gritty (whether that means doing algebra or learning to program a
computer for analysis), and therefore devises examples that can be solved
with a few simple rules of thumb. While information can be derived from
these examples, there is much more to the subject than this, and a great
deal more that can help someone with their at-table play.
As far ahead of the general poker public as he is, David is actually pretty
far behind the curve compared to both the literature (you'd be surprised to
find out how much about poker has been solved long ago) and several poker
player/analysts I know (who have extended the theory beyond what is in the
literature). Most of these people keep a low profile, as they are players
(some pros, some not), and knowledge is money. To my knowledge, although
several have played fairly high on a consistent basis, none of these folks
have played in the biggest of games (maybe someday they will have the
bankroll and inclination to join in - I'm not sure which of these is
actually the greater barrier), though I wouldn't be surprised if there are
other "cells" of analysts I am not aware of that already play very big. As
I said, there just isn't much incentive to share information, so it's hard
to know for sure.
Tom Weideman
![Mark B [Diputsur]'s profile photo](http://lh3.googleusercontent.com/a-/ALV-UjW1ukZesLHTvLryryGaLhoUTsW-4I5_N8bKNVoFKGU5T0OxggP6=s40-c)
Mark B [Diputsur]
HOw many threads did this one need? For the sake of completion... enjoy