• GNU Backgammon against its Murat Mutant: The first 1000 games

    From Axel Reichert@21:1/5 to All on Wed Oct 27 20:24:29 2021
    Hello,

    I gave it a try. Since I am programming a bot anyway (based on rules of thumb, for personal experimentation and gathering of statistical data, not as a serious contender to world class players), it was a rather simple
    exercise to implement a strategy as follows:

    1. Checker play: Do what GNU Backgammon does with checker play set to
    "Expert" level.

    2. Cube decisions:

    a) Double with more than 50 percent cubeless winning chances

    b) Take or pass according to GNU Backgammon's assessment ("World Class")

    c) Raccoon if beavered ("higher order" rodents forbidden)

    I hope this is more or less what Murat politely suggests us to do to
    become better backgammon players, supposedly on "alpha" level.

    I pitted this strategy against GNU Backgammon, of course likewise with
    "Expert" checker play and "World Class" cube decisions. So both
    opponents essentially did the same checker plays, only the cube
    strategies were different. Think of my bot as a mutant with some genetic changes in its doubling brain.

    The score after 1000 games (money session, Jacoby) is

    GNU Backgammon: 14182 (564 wins)
    Murat Mutant: 7582 (436 wins)

    Now the score is very high for "only" 1000 games (of course, since the
    mutant drives up the cube value), so some statistical checks were in
    order.

    The average game value was 21.764 and the variance was 19263.4, hence
    its standard deviation was a whopping 139. There were 34 games with at
    least 128 points, the most points won in a single game were 4096.

    Now with

    https://bkgm.com/rgb/rgb.cgi?view+709

    we can assess whether GNU Backgammon's win was already statistically significant with such a volatile/erratic opponent. We assume that the
    mutant's strategy is as good as GNU Backgammon's, so expect with 95 % probability that the absolute difference between the two players is
    smaller than

    2 * sigma * sqrt(1000) =

    2 * 139 * 32 =

    8778

    This is larger than 6600 (GNU Backgammon's net result), so the jury is
    still out. I will continue the run and keep you posted. In the mean
    time, statistical advice is very welcome.

    Best regards

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Wed Oct 27 14:33:59 2021
    On October 27, 2021 at 12:24:33 PM UTC-6, Axel Reichert wrote:

    I gave it a try.

    At last someone tried! Great news. Hopefully you will correct,
    refine and improve on the experiment to go beyond what I had
    suggested. Surely we all will learn new things either way.

    1. Checker play: Do what GNU Backgammon does with checker
    play set to "Expert" level.

    How long did the 1000 games take? Can you set it to the highest
    level from now on? If it would take too long to run each iteration,
    I would be willing to contribute CPU time.

    2. Cube decisions:

    a) Double with more than 50 percent cubeless winning chances

    Can we run this using both cybeless and cubeful winning chances?

    And also cubeless winning chances calculated by other bots for at
    least the opening and reply rolls?

    In my own experiments, for example, I double agressively after early
    62, 63, 64 and 21 depending on how they are played by either side.
    Did your mutant double immediately after gnubg opened with 63 and
    took the beaver?

    b) Take or pass according to GNU Backgammon's assessment
    ("World Class")

    I don't remember exactly what I had proposed years ago but can we
    make this based on more than 50 percent also (in light of what Paul
    is using in his other thread in a related discussion)?

    c) Raccoon if beavered ("higher order" rodents forbidden)

    Again can you run this either way? In my later experiments I tried to
    raccon also but I never did enough to publish the results since I lost
    overall interest.

    Can you explain why the limitation? How would it help the experiment?

    I hope this is more or less what Murat politely suggests us to do to
    become better backgammon players, supposedly on "alpha" level.

    Let's be clear that I didn't equate the two. I argued that an alpha-bg
    bot will handily beat the current bots and debunk all current dogmas.
    In my own experiments, I tried to play like how I best predicted that
    an alpha-bg bot may play but never claimed that I knew exactly how
    an alpha-bg bot will play.

    With that said, I would actually like to see your experiments be based
    on improved predictions than mine (i.e. closer to an alpha-bg bot) but
    I don't know how we could know that without having already access
    to an alpha-bg bot.

    The score after 1000 games (money session, Jacoby) is

    I never use Jacoby myself. Can you turn it off even if it may not make a difference in this kind of experiments?

    This is larger than 6600 (GNU Backgammon's net result), so the jury
    is still out. I will continue the run and keep you posted.

    Honestly, I couldn't make sense of your statistical calculations and/or conclusions but I am glad that the jury is still out and you will continue
    the "experiments" (not just "run" the same one longer) with my above corrections/suggestions.

    In the mean time, statistical advice is very welcome.

    As well as any other advice, I would add. Hopefully with efforts toward
    not just proving me wrong (which is okay also) but to learn new things.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Wed Oct 27 16:18:26 2021
    On October 27, 2021 at 12:24:33 PM UTC-6, Axel Reichert wrote:

    In the mean time, statistical advice is very welcome.

    Here are a few from me. When I analysed my experiments,
    I tabulated how many times which side started first, cubed
    first, cubed last, lost while holding the cube, last cube value,
    won by race, etc.

    I had found all these more or less relevant/meaningful to
    try learning from. You may want to keep similar statistics.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Timothy Chow@21:1/5 to Axel Reichert on Thu Oct 28 23:42:16 2021
    On 10/27/2021 2:24 PM, Axel Reichert wrote:
    The score after 1000 games (money session, Jacoby) is

    GNU Backgammon: 14182 (564 wins)
    Murat Mutant: 7582 (436 wins)

    [calculation omitted]

    This is larger than 6600 (GNU Backgammon's net result), so the jury is
    still out. I will continue the run and keep you posted. In the mean
    time, statistical advice is very welcome.

    This is an interesting experiment. I think it's going to be a little
    tricky to attach a number like "95% confidence" because we're dealing
    with two very different players. I mean, you could say that the null hypothesis is that GNU is the same as the mutant, and you could reject
    that hypothesis very quickly with high confidence, but that's not very interesting---we *know* that GNU is not the same as the mutant without collecting any statistics at all.

    Or your null hypothesis could be that the mutant's expected net score
    against GNU is zero. But rejecting this hypothesis with high confidence
    isn't too interesting either, since that wouldn't say anything about the *magnitude* of the difference between the two players, which is what you
    are really interested in.

    What I would suggest is to forget about 95% confidence and just plot a histogram of the results---for each possible game outcome (by which I
    just mean the score---e.g., a gammon with the cube on 2 is treated the
    same as a single win with the cube on 4), plot the number of games that
    have that outcome. Then just play enough games so that even the rarer
    outcomes are achieved more than a handful of times.

    ---
    Tim Chow

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Tim Chow on Thu Nov 4 10:27:41 2021
    On October 28, 2021 at 9:42:22 PM UTC-6, Tim Chow wrote:

    .... Then just play enough games so that even the rarer
    outcomes are achieved more than a handful of times.

    How many would be "enough" and should that number
    be decided beforehand? Otherwise, if one is not happy
    with the results after 5000 games (undeclared but
    initially considered enough), what prevent going on to
    6000 games and stoppig at that if the results prove a
    certain point (which may not be so again after 7000
    games)?

    Another suggestion I have is to make the mutant make
    cube decisions based on not just expected 50%+ winning
    chances but on other criteria such as "whether enough
    checker play is left in the game".

    I think those are the words I had used in one occasion
    when I was explaining (to Michael?) how I made cube
    decisions.

    Can a bot determine not just the complexity of a position
    but also expected number of turns left to play in the game,
    allowing tome enough for luck to swing (perhaps more
    than once) and more importantly for the checker skill to
    make a decisive difference??

    If any bot mutant is to be called a Murat mutant, it needs
    to be able to do this! (and my other similar strategies).

    Otherwise, just call it Jacoffsky's mutant...

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Thu Nov 4 19:29:25 2021
    MK <murat@compuplus.net> writes:

    How long did the 1000 games take?

    Over night.

    Can you set it to the highest level from now on?

    No. "Expert" is certainly strong enough checker play for this
    experiment, especially since both sides will have the same
    setting. "World Class" likewise is strong enough for cube decisions
    ("Expert" will not "see" market losers)

    Can we run this using both cybeless and cubeful winning chances?

    No. The percentages that I use for the doubling criterion of the
    mutant are given by GNU Backgammon as cubeless. Cubeful values are
    given with equities.

    And also cubeless winning chances calculated by other bots for at
    least the opening and reply rolls?

    No. I do not have other bots. All are strong enough for this kind of experiment. I neither care for the opening and reply rolls, just
    whether GWC is over the 0.5 threshold.

    Did your mutant double immediately after gnubg opened with 63 and
    took the beaver?

    No. According to "World Class", after GNU Backgammon splits with 62
    and 63 or runs with 64 (which is was "Expert" checker play does) the
    mutant is a slight underdog. Hence no double. But there are other
    "Expert" plays where "World Class" thinks the replier is favourite:

    21S, 41S, 51S, 43Z => Double by the mutant

    And of course, Beaver by GNU Backgammon, and depending on my settings,
    Raccoon by the mutant.

    can we make this based on more than 50 percent

    Perhaps I am willing to do a test with 0.618 (golden ratio, to throw in esoterics for good measure), which is between 0.6 and 0.625: Leaving gammons/backgammons aside, one could make a simplistic case for these
    numbers based on live/dead cube assumptions and the football analogy:

    https://bkgm.com/articles/Kleinman/FootballFields/index.html

    c) Raccoon if beavered ("higher order" rodents forbidden)

    Again can you run this either way?

    No. I never play with Raccoons, and in our club we had discussions of
    banning Beavers as well (they tend to attract the "wrong" players).

    Can you explain why the limitation? How would it help the experiment?

    More on this in a different thread.

    I never use Jacoby myself. Can you turn it off

    I always use Jacoby. This is crucial in chouettes, because without
    Jacoby the team might be bored to death while the captain, having
    slept during the cube decisions, tries to squeeze out a Gammon.

    So here are the results. The Null hypotheses in all cases was that the
    mutant's cubing is as strong as GNU Backgammon's.

    1. Jacoby, 0 Beavers allowed

    Histogram after 5000 games (Tim asked for this):

    | Frequency | Points |
    |-----------+--------|
    | 241 | 1 |
    | 2575 | 2 |
    | 1471 | 4 |
    | 34 | 6 |
    | 517 | 8 |
    | 8 | 12 |
    | 111 | 16 |
    | 2 | 24 |
    | 30 | 32 |
    | 9 | 64 |
    | 1 | 96 |
    | 1 | 256 |

    Average: 0.3418
    Variance: 48.289
    Maximum allowed lead: 983
    Actual lead: 1709

    Mutant plays worse!

    2. Jacoby, 1 Beaver allowed

    Histogram after 3000 games:

    | Frequency | Points |
    |-----------+--------|
    | 133 | 1 |
    | 489 | 2 |
    | 1366 | 4 |
    | 8 | 6 |
    | 648 | 8 |
    | 15 | 12 |
    | 183 | 16 |
    | 1 | 24 |
    | 116 | 32 |
    | 1 | 48 |
    | 20 | 64 |
    | 2 | 96 |
    | 13 | 128 |
    | 2 | 256 |
    | 1 | 512 |
    | 1 | 1024 |
    | 1 | 8192 |

    Average: -1.70367
    Variance: 23033.5
    Maximum allowed lead: 16625
    Actual lead: -5111

    Jury is still out!

    3. Jacoby, 2 Beavers allowed (= Raccoons)

    Histogram after 3000 games

    | Frequency | Points |
    |-----------+--------|
    | 152 | 1 |
    | 453 | 2 |
    | 283 | 4 |
    | 10 | 6 |
    | 1105 | 8 |
    | 4 | 12 |
    | 632 | 16 |
    | 17 | 24 |
    | 199 | 32 |
    | 3 | 48 |
    | 36 | 64 |
    | 3 | 96 |
    | 63 | 128 |
    | 24 | 256 |
    | 1 | 384 |
    | 6 | 512 |
    | 5 | 1024 |
    | 1 | 2048 |
    | 3 | 4096 |

    Average: 2.18933
    Variance: 21576.6
    Maximum allowed lead: 16091
    Actual lead: 6568

    Jury is still out!

    So in the cases with Beavers (1 or more allowed) the Null hypothesis
    could not be rejected.

    My gut feeling says that this is almost certainly not because your
    doubling strategy is competitive, but rather due to the St Peterburg
    Paradoxon. More of my thoughts on this in a different thread.

    Out of curiosity I tested some other doubling "strategies" (all with
    Jacoby and without Beavers, Null hypothesis as before):

    - Double with 50 % winning chances, always take

    This could be rejected after 2000 games.

    - Always double if legal, take according to GNU Backgammon "World
    Class"

    This could be rejected after 1000 games.

    - Always double if legal, always take

    This could be rejected after 200 games.

    Note that even this last, maniac strategy needed 200 games to get
    rejected with 95 % certainty! This means that you should be extremely suspicious regarding results from a mere 100 games, especially if the
    strategy tends to drive the cube up, up and away.

    That's it, I will not spend more time on things like this, there are
    far more interesting questions out there, that I want to discuss in a
    different thread.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to Timothy Chow on Thu Nov 4 20:09:00 2021
    Timothy Chow <tchow12000@yahoo.com> writes:

    plot a histogram of the results

    Done, see my other post in this thread.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Sat Nov 6 00:00:25 2021
    On November 4, 2021 at 12:29:29 PM UTC-6, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    How long did the 1000 games take?

    Over night.

    So, it sounds like we can run numerous quick experiments
    and even longer ones within a reasonable amount of time.

    For the questions that answered "no", (asked for my reasons
    and answered for your reasons), I guess there is no point in
    wasting time by discussing further but thanks for answering.

    Did your mutant double immediately after gnubg opened
    with 63 and took the beaver?

    No. According to "World Class", after GNU Backgammon splits with
    62 and 63 or runs with 64 (which is was "Expert" checker play does)
    the mutant is a slight underdog. Hence no double. But there are other "Expert" plays where "World Class" thinks the replier is favourite:

    This one is important and that's why I asked if you could use
    opening equity calculations by other bots (like TD-Gammon).
    I thought the opening book was user editable, no? Also, bots
    like XG++ split the 64 and slot the 21. If you want to imitate
    Murat's experiments, you need to do these. Otherwise, it's an
    experiment of your own and has nothing to do with what I had
    suggested (as the thread title implies).

    can we make this based on more than 50 percent

    Perhaps I am willing to do a test with 0.618 (golden ratio, to throw
    in esoterics for good measure), which is between 0.6 and 0.625

    Okay, this is good news. Just try whatever numbers you fancy
    but try some different things. In fact, I had asked/suggested
    many times in the past that any arbitrary and/or calculated
    constants used by the bots should be made user selectable
    variables in the settings or in a editable config file. I will be
    curiously waiting to see what you come up with 0.618, etc...

    I wish I could volunteer to help with CPU time and such but
    you won't share your utility.

    Again can you run this either way?

    No. I never play with Raccoons, and in our club we had discussions
    of banning Beavers as well (they tend to attract the "wrong" players).

    But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
    call "cube skill"...?

    This is not human play at your club. This an experiment about the
    "cube skill" thing. Beavers may attract the wrong players at your
    club but bots are asexual... ;)

    I'm grinning from ear to ear though, at your idea of banning Beavers.
    Go ahead. In fact, I have been advocating banning the doubling cube
    altogether and I now have raised hopes that with your help it may
    come to that sooner than later...

    Can you explain why the limitation? How would it help the experiment?
    More on this in a different thread.

    I'll look for it.

    I never use Jacoby myself. Can you turn it off

    I always use Jacoby. This is crucial in chouettes, because without
    Jacoby the team might be bored to death while the captain, having
    slept during the cube decisions, tries to squeeze out a Gammon.

    This has nothing to do with boring chouettes but fine with me. When
    the common complaint that the cube gets too high, I don't think we
    need to worry too much about games ending without a cube action.

    My gut feeling says that this is almost certainly not because your
    doubling strategy is competitive, but rather due to the St Peterburg Paradoxon. More of my thoughts on this in a different thread.

    I'll look for it.

    Out of curiosity I tested some other doubling "strategies" (all with
    Jacoby and without Beavers, Null hypothesis as before):

    - Double with 50 % winning chances, always take

    Should be at least 51% and no point in always taking without any
    chace of winning left. Meaningless test.

    - Always double if legal, take according to GNU Backgammon "World
    Class"

    Always doubling without any chace of winning is also pointless.
    So, another meaningless test.

    - Always double if legal, always take

    This could be rejected after 200 games.

    Why are you even wasting time with these??

    Note that even this last, maniac strategy needed 200 games to get
    rejected with 95 % certainty! This means that you should be extremely suspicious regarding results from a mere 100 games, especially if the strategy tends to drive the cube up, up and away.

    If you are referring to my 100 games, I can't very well play 1000
    games in one night and that's exactly why we are making bots
    to play "long enough" sessions.

    As for "jacking up the cube" (the old popular expression), I always
    used the saying "It takes two to tango". If the bot's "extraterrestrial
    cube skill" dances along, how can you blame for using my own
    "human cube skill" to my advantage??

    That's it, I will not spend more time on things like this

    Sounds like the results went against your expectations? Oh well,
    at least you tried by doing more than anyone else has done yet.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Sat Nov 6 10:23:05 2021
    MK <murat@compuplus.net> writes:

    On November 4, 2021 at 12:29:29 PM UTC-6, Axel Reichert wrote:

    No. According to "World Class", after GNU Backgammon splits with 62
    and 63 or runs with 64 (which is was "Expert" checker play does) the
    mutant is a slight underdog. Hence no double. But there are other
    "Expert" plays where "World Class" thinks the replier is favourite:

    This one is important and that's why I asked if you could use opening
    equity calculations by other bots (like TD-Gammon). I thought the
    opening book was user editable, no? Also, bots like XG++ split the 64
    and slot the 21. If you want to imitate Murat's experiments, you need
    to do these. Otherwise, it's an experiment of your own and has nothing
    to do with what I had suggested (as the thread title implies).

    To my knowledge, GNU Backgammon has no opening book, it just plays
    according to the current settings, be that Expert (for speed reasons) or
    World Class (too slow for these mass runs). According to XG's opening
    book, the replier has an advantage only after 41S (with or without
    Jacoby). My understanding is that you double after all 6x splits and
    maybe for other opening rolls, I don't know. My mutant doubles after
    1x splits and 43Z. In all these cases the winning chances of the replier
    should be between and 49.48 % and 50.16 %. So in my opinion it does not
    matter in which of these cases you raise the stakes, it is too early
    anyway, because you are foregoing the possibility to double your
    opponent out. This is the value of cube ownership, see the football
    field analogy by Danny Kleinman.

    any arbitrary and/or calculated constants used by the bots should be
    made user selectable variables in the settings or in a editable config
    file. I will be curiously waiting to see what you come up with 0.618,
    etc...

    Which is also "arbitrary". If you start to come up with ideas about
    gammonish positions and "play left in the game" I should start to get
    worried, because then you would reinvent the (mathematical) concepts of
    equity and volatility. (-;

    No. I never play with Raccoons, and in our club we had discussions of
    banning Beavers as well (they tend to attract the "wrong" players).

    But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
    call "cube skill"...?

    The interesting question I try to research here is not any "maniac" cube strategy, but whether a Petersburg Paradoxon occurs in backgammon with unlimited cube (depending on number of beavers allowed). If yes, I think
    we will have an interesting (but mostly philosophical) discussion about
    skill.

    But as soon as the cube is capped (e.g., in match play), no Peterburg
    Paradoxon can occur and a "maniac" cube strategy will hurt the maniac
    and thus prove his inferior cube handling. It follows immediately that
    skill is involved.

    - Double with 50 % winning chances, always take

    Should be at least 51%

    Why not 50.000001 %? I should have written (and have implemented) "more
    than 50 % winning chances".

    Why are you even wasting time with these??

    See above. My interest is the Peterburg Paradoxon, not the cube
    strategy.

    Note that even this last, maniac strategy needed 200 games to get
    rejected with 95 % certainty! This means that you should be extremely
    suspicious regarding results from a mere 100 games, especially if the
    strategy tends to drive the cube up, up and away.

    If you are referring to my 100 games, I can't very well play 1000
    games in one night and that's exactly why we are making bots
    to play "long enough" sessions.

    Sure. But for precisely this reason you should be cautious with claims
    that your "unorthodox" doubling strategies are superior.

    Sounds like the results went against your expectations?

    No, but they indicate that another approach might make more sense: If a Petersburg Paradoxon occurs, it no surprise that strategies jacking up
    the cube can not be proven any more to be worse (or better). So in that
    case you might be claiming superior cube skill but ignoring the pink
    elephant in the room (by the way, I liked your Commodore story).

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Thu Nov 11 14:28:20 2021
    On November 6, 2021 at 3:23:08 AM UTC-6, Axel Reichert wrote:

    To my knowledge, GNU Backgammon has no opening book
    .....
    According to XG's opening book

    Ah, okay, I got them confused.

    the replier has an advantage only after 41S

    How about we go by the Gnubg rollouts from this link:

    https://bkgm.com/openings/rollouts.html

    (with or without Jacoby).

    Jacoby never interested me beyond wondering what do bots
    play differently other than using different doubling windows?
    I really don't even care to know anything about it.

    My understanding is that you double after all 6x splits and
    maybe for other opening rolls, I don't know.

    Yes 62, 63, 63 because they give me 2 direct 1 indirect shots,
    (ignoring my 1 point), and I like how the games develop then.

    Lately I added 41 if slotted but without raccoon for now since
    it gives me only 1 direct 1 indirect shots.

    And of course, I play certains ways on the 2nd and 3rd rolls
    also, as well as the rest of the games.

    I just looked at some TD-Gammon articles and noticed that
    63 was one of the rolls v1 had "misplayed" (24/21 13/7) and
    found it quite interesting. Perhaps it did that not because the
    24/21 13/7 was a good play but 24/18 13/10 ended up being
    worse based on how it played against itself.

    In other word, the "styles" or "strategies" of players also matter
    in determining whether certain rolls and moves are conducive
    to them... Maybe that's why I still love this game.

    My mutant doubles after 1x splits and 43Z. In all these cases
    the winning chances of the replier should be between and
    49.48 % and 50.16 %.

    I don't understand the S's, Z's, etc. after the rolls but I can agree
    with going by the winning percentages because it's easier to
    program those into the bot. I don't understand why the 49.48%
    to 50.16% range but according to the summary table at the link
    above 41, 43, 64, 32 result in < 50% winning chances.

    So in my opinion it does not matter in which of these cases
    you raise the stakes, it is too early anyway, because you are
    foregoing the possibility to double your opponent out.

    Well the "too early" part is exactly my point for a different reason,
    which is that you can't calculate cubeful equities until towards the
    end of the game. I thought the common teaching of "cube skill"
    was that it was better used to maximize your winning and not
    necessarity to double your opponent out.

    Which is also "arbitrary". If you start to come up with ideas about
    gammonish positions and "play left in the game" I should start to
    get worried, because then you would reinvent the (mathematical)
    concepts of equity and volatility. (-;

    I don't know what to say other than you may as well start getting
    worried, not because I want to reinvent any such "mathematical
    concepts", but to debunk them alltogether.

    But aren't Raccoons, Rats, Bats, Cats, etc. all part of what you all
    call "cube skill"...?

    The interesting question I try to research here is not any "maniac"
    cube strategy, but whether a Petersburg Paradoxon occurs in
    backgammon with unlimited cube

    I didn't invent doubling, nor beavering, raccooning, etc. not do I mind
    your calling my or any other strategy "maniac" as long at it results in
    winning more.

    And I have no idea what Petersburg Paradox has anything to do with
    the subject where more than just the probabilites of luck, i.e. "skill" is involved, especially with some people arguing more for skill than luck
    in backgammon.

    ... If yes, I think we will have an interesting (but mostly philosophical)
    discussion about skill.

    The only practical thing you can do about the "cube skill fantasy" is
    just counting the potatoes and living with the result. I think it's you
    guys who are making more out of it than what it is.

    Should be at least 51%

    Why not 50.000001 %?

    Sure, as long as you have enough potatoes to count... My argument
    had initially started based on the fact that winning the opening roll
    gives an advantage (according to the link above +.0393 on average).

    Checker+cube skills being equal, the player who will win more opening
    rolls will win more. You can't argue against this simple statistics fact.

    What I'm interested in is to find out if luck+checker skills are equal,
    how much does the so called "cube skill" matter after 4 billion games?

    Why are you even wasting time with these??

    See above. My interest is the Peterburg Paradoxon, not the cube
    strategy.

    In that case we're just wasting time. That is, I am anyway... :(

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Fri Nov 12 20:22:42 2021
    MK <murat@compuplus.net> writes:

    On November 6, 2021 at 3:23:08 AM UTC-6, Axel Reichert wrote:

    How about we go by the Gnubg rollouts

    As I wrote, this will not matter. Your thinking seems to be that as soon
    as you are even a tiny favourite, you should raise the stakes. This is
    correct if you cannot get redoubled. But if the opponent has the
    opportunity to redouble, he could perhaps double you out. So by giving
    the cube (giving it away!) you forgo your chances of doubling out the
    opponent and thus make it harder for you. You have move the goal posts,
    and this is a huge cost for you, hence doubling early with only a tiny advantage is very wrong. So it does not matter whether you have 49.89 % according to bot A or 50.03 % according to bot B, since this difference
    will be dwarfed by the difference between having access to the cube or
    not.

    I don't understand the S's, Z's, etc. after the rolls

    https://bkgm.com/articles/Keith/nactation.html

    I don't understand why the 49.48% to 50.16% range

    These are the winning chances after said rolls according to

    http://www.extremegammon.com/OB/Opening_in_unlimited_game.html

    I thought the common teaching of "cube skill" was that it was better
    used to maximize your winning and not necessarity to double your
    opponent out.

    Precisely. And because you give away the powerful weapon of the cube you
    should not double to early, even if you are a favourite. Please read

    https://bkgm.com/articles/Kleinman/FootballFields/index.html

    not because I want to reinvent any such "mathematical concepts", but
    to debunk them alltogether.

    I know. But it won't be easy. (-:

    not do I mind your calling my or any other strategy "maniac" as long
    at it results in winning more.

    And I have no idea what Petersburg Paradox has anything to do with the subject

    See below.

    winning the opening roll gives an advantage (according to the link
    above +.0393 on average).

    Sure, but this is not enough to give the weapon away.

    Checker+cube skills being equal, the player who will win more opening
    rolls will win more.

    Sure. But in the long run no player will win the opening roll more often
    than the other.

    What I'm interested in is to find out if luck+checker skills are equal,
    how much does the so called "cube skill" matter after 4 billion games?

    This can be answered as long as you can put numbers on the value of a
    position. If you run into a Petersburg Paradox you cannot do this any
    more, so at that point discussions about the pros and cons of particular
    cube strategies become meaningless, because there are no numbers to
    compare. Now if your cube strategy turns backgammon into a Petersburg
    Paradox than you can neither claim that your cubing is better than the
    bot's nor could someone else claim that it is worse than the bot's. It
    cannot be proven any more.

    If, on the other hand your cube strategy does not turn backgammon into a Petersburg Paradox (e.g., because it is too timid for this, or because
    it is prevented by rules, be it match play, a cap on the cube in money sessions, forbidding beavers, ...), then there exists a number for the
    value of a position, and so immediately one can debunk one strategy or
    the other, even if it takes lots of games.

    I think without beavers (as shown in one of my previous posts in this
    thread) the case against the "mutant" strategy was already settled after
    5000 games. With one beaver or, even more so, with raccoons allowed, it
    was not yet settled, because due to the high cubes the scores were
    wildly fluctuating and I was expecting an awfully high number of games
    needed to settle the question. Hence I decided to first run a "stability
    check" using Markov chain reasoning fed with the data accumulated from
    10000 games. By now I am quite sure that with raccoons forbidden
    backgammon does not turn into a Peterburg Paradox, whereas with raccoons allowed it probably does.

    This would mean that the mutant strategy can be debunked in the
    beavers-only case by spending more CPU time. And it would also mean that
    with raccoons it does not make sense to spend more CPU time, because it
    will just produce random noise. If so, there is an easy explanation for
    your success against the bot's in a particular session of only 100 games
    (not your mistake, of course, I do understand the reasons).

    Best regards

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Frank Berger@21:1/5 to All on Fri Nov 12 15:14:56 2021
    I don't expect to convince anyone that the cube has a value, but I found this article https://plus.maths.org/content/os/issue15/features/doubling/index very convincing. As Janowsji writes in a comment the model is to simplistic as it treats BG as a
    continous model, but I think it illustrates the facts excellently. As soon as you include non steady stuff it get's complicated.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to Frank Berger on Sat Nov 13 11:02:15 2021
    Frank Berger <bgblitz59@googlemail.com> writes:

    I don't expect to convince anyone that the cube has a value

    (-:

    https://plus.maths.org/content/os/issue15/features/doubling/index

    very convincing

    Yes, linked on bkgm.com, where I have read it long ago. Probably to probabilistic for some target audience ...

    This is why I like the football/tug of war analogy: Somewhere above the
    60 mark you feel strong enough to shift your own target from 80 to 100,
    because then you still have 40 to go (from 60 to 100), whereas the taker
    has to tug you from 60 to 20 (live cube, no (back)gammons). With a dead
    cube you shift your own target from 75 to 100, so you should ensure that
    you have less to go than the taker (from x to 0, not from x to 25,
    because the cube is dead), which is the case at x=50, obviously.

    All this is certainly more or less trivial for you, but as you can see
    from this thread I still have not given up explaining. Even elementary
    cube theory is not intuitive for beginners (as can be seen when trying
    to teach them the 25 % take point for the dead cube).

    So one could characterize my mutant's doubling "strategy" as treating
    himself the cube as dead but hoping that the opponent treats it as very
    much alive. (-;

    Greeting from the Isar to the Rhine!

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Sat Nov 13 17:14:53 2021
    On November 12, 2021 at 12:22:44 PM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    How about we go by the Gnubg rollouts

    As I wrote, this will not matter. Your thinking seems to be
    that as soon as you are even a tiny favourite, you should
    raise the stakes. This is correct if you cannot get redoubled.
    .....

    In your experiment, the mutant plays normally after the first
    cube. In my proposal, the mutant never drops except when
    it has no chance of winning. Thus, it tries to turn the games
    into "cubeless" games as much as it can, by forcing then to
    be played out to the last roll. I propose that this will counter
    the value of cube ownership. This has been one of my many
    arguments for 20 years.

    So it does not matter whether you have 49.89 % according to
    bot A or 50.03 % according to bot B

    I didn't put much importance on this. I only proposed using
    the numbers from that link since Gnubg has no opening book.

    since this difference will be dwarfed by the difference between
    having access to the cube or not.

    That's where we differ. I'm not so sure about the value of cube
    ownership and I believe it can be overcome or even surpassed
    by the strategy I proposed above.

    I don't understand the S's, Z's, etc. after the rolls

    https://bkgm.com/articles/Keith/nactation.html

    I guessed so but wasn't sure because I had the impression that
    it was more complicated. I never even took a casual look at it,
    as part of my trying to not keep my bg brain uninfected and to
    not mutate into one of you flock. This seems to be a harmless
    clever shorthand but I probably will keep avoid using it.

    I don't understand why the 49.48% to 50.16% range

    These are the winning chances after said rolls according to ...

    Okay. One thing I find very interesting, intrigueing is tha XGR++
    slots with opening 41, as was TD-G v1 doing. At the time it was
    considered "misplayed" by the human experts, along with the
    other "bad" opening roll 63. I find it curious that the mother of
    all later bg bots, and the only one without human bias! would
    misplay the two worst opening rolls. I wonder if time may prove
    otherwise...?

    I thought the common teaching of "cube skill" was that it was
    better used to maximize your winning and not necessarity to
    double your opponent out.

    Precisely. And because you give away the powerful weapon of
    the cube you should not double to early, even if you are a favourite.
    Please read https://bkgm.com/articles/Kleinman/FootballFields

    This and its variations, as well as other analogies have been used
    many times here in the past. To me, resorting to such analogies
    only shows a person's inability make his mathematical argument
    strictly related to bg alone.

    I myself sometimes use analogies but not to sustitute facts, such
    as likening your guys' elaborate yet inapplicable equity, skill, etc. calculations to pre-Copernican astronomy when they had refined
    their formulas to calculate and predict some planets' retrograde
    movements exactly, even though planets never moved backwards.

    So now, let's assume Jeffrey Epstein is playing against Mocky,
    for stakes high enough for Mocky but peanuts for Jeffy. I and
    John Wayne are advising him, looking over his shoulder. When
    Mocky rolls and opening 63 and splits, I urge him to immediately
    double. He does. Mocky beavers. Jeffy raccoons. Mocky now has
    "the powerful weapon of the cube" ownership.

    Later, Mocky get a chance to redouble. Oops. But Duke says:
    "Damn to torpedos! Full speed ahead!". After all, losing a few
    million bucks would only be a mosquito byte for Jeffy... So, in
    short, it's all relative and unverified, unproven.

    not because I want to reinvent any such "mathematical concepts",
    but to debunk them alltogether.

    I know. But it won't be easy. (-:

    Yes, but I stuck to my guns (and my puns) as a Lone Ranger for 20
    years. Lately I feel like I'm finally getting some traction. If you can
    hang in there, in the end you may get some credit as Tonto. :)

    And I have no idea what Petersburg Paradox has anything to do with
    the subject

    See below.

    winning the opening roll gives an advantage (according to the link
    above +.0393 on average).

    Sure, but this is not enough to give the weapon away.

    How do you know? Have you tested and verified how much is the
    weapon worth?

    What I'm interested in is to find out if luck+checker skills are equal,
    how much does the so called "cube skill" matter after 4 billion games?

    This can be answered as long as you can put numbers on the value
    of a position. If you run into a Petersburg Paradox you cannot do this
    any more, so at that point discussions about the pros and cons of
    particular cube strategies become meaningless, because there are no
    numbers to compare. Now if your cube strategy turns backgammon
    into a Petersburg Paradox than you can neither claim that your cubing
    is better than the bot's nor could someone else claim that it is worse
    than the bot's. It cannot be proven any more.

    My argument goes back to the stage before the "numbers" are
    calculated. I propose that even the cubeless equity calculations
    after TD-G v.1 are human biased and inaccurate by an unknown
    magnitute. Thus, cube double/take points, etc. calculated based
    on those equities are also inaccurate, in addition to being plain
    wrong for other reasons and to being partially inapplicable to bg.

    Have you read my discussions with Chow about HypestGammon?

    It's a variant that I created to isolate the cube skill, in oder to test, quantify and define it. Since it's played with only one die, there is
    zero checker skill involved. It's pure cube skill game.

    Chow claimed he could calculate the equities for all possible
    positions and shared his findings. Since there are only a small
    number possible positions, even with desktop CPU power, we
    could create an alpha-bot that would be trained through "cubeful
    self-play" and then we could compare the calculated equities to
    the statistical equities, in order to see if the formulas used in the calculations were accurate. For reasons/excuses, questionable
    to me, he never finished the experiment.

    If the numbers matches, it wouldn't necessarily prove anything
    about "real bg" but if they didn't match, it would mean that further,
    more complex expriments would be needed and be worth doing
    to test the accuracy of Jackoffsky, etc. formulas used in "real bg".

    My opening 63 proposal was an alternative, a substite test that
    we could do with limited CPU power, just to get a glipse of what
    may be a much bigger problem to investigate. And this is where
    you came into the picture, offering to do an experiments but I
    don't think it was what I had proposed to begin with and then
    further deviated into something else completely.

    If, on the other hand your cube strategy does not turn backgammon
    into a Petersburg Paradox (e.g., because it is too timid for this, or
    because it is prevented by rules, be it match play, a cap on the cube
    in money sessions, forbidding beavers, ...), then there exists a number
    for the value of a position, and so immediately one can debunk one
    strategy or the other, even if it takes lots of games.

    As I had said, I don't think the Petersburg Paradox applies here
    and don't understand your arguments related to it. But I think I
    understand part of what you are saying above.

    Yes, after the opening 63 and split, for example, we need to
    play lots of games, i.e. the proverbial 4 billion games, and just
    count potatoes... But the mutant needs to play as I descibed
    above, trying to minimize and ideally surpass any value of cube
    ownership.

    Deciding most games by checker play is very inportant to my
    proposed experiment. In fact, I had dome my own personal
    experiment by playing against the bot with me making the worst
    first turn move in each game but to play normally after that, and
    especially trying to recover from the huge blunder.

    See the first experiment (with real-time Youtube videos and all) at:

    https://www.montanaonline.net/backgammon/xg.php

    I was surprised that I wasn't totally decimated by the bot as I had
    expected to happen.

    What that experiment proved to me that checker errors early in the
    games are not as important as errors in late stages of games.

    I propose that the same applies to cube errors also and that they
    should be rated on a sliding scale of some sort. (This doesn't mean
    that I acknowledge their accuracy but just saying that even as they
    are wrong, they should be still wrongly calculated by differently.)

    This would mean that the mutant strategy can be debunked in the
    beavers-only case by spending more CPU time....

    But your mutant is not mutated enough to match the experiment
    I had proposed. Cubing early aggressively at 50%+, only to drop it
    later according to Jackoffsky's take/drop points is meaningless...

    If so, there is an easy explanation for your success against the
    bot's in a particular session of only 100 games
    (not your mistake, of course, I do understand the reasons).

    I don't know what you are insinating but this is experiment is apart
    from my a few dozen short sessions of 100 games.

    If there is any link at all, it may be that I raised the question and
    proposed various experiments indeed because I myself had no
    clear explanation (which I admitted openly in the past) for my
    "success". And also, even if I had, the overall total number of
    games I had played wasn't statistically sicnificant enough for
    most of you. And since I couldn't play 4 billion games myself,
    I proposed that we make the bots do it instead.

    From my stand, I too do understand the reasons for your resistance
    or reluctance, and I do give you credit for having tried this much,
    (which is more than anyone else has done), and I hope that you will
    keep doing more logical experiments. Because, you know, it's just a
    matter of time...

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Sat Nov 13 18:11:49 2021
    On November 13, 2021 at 3:02:18 AM UTC-7, Axel Reichert wrote:

    Frank Berger <bgbl...@googlemail.com> writes:

    I don't expect to convince anyone that the cube has a value

    All this is certainly more or less trivial for you, but as you
    can see from this thread I still have not given up explaining.

    The feeling of frustration is mutual. To me, you guys sound
    like pre-Copernican astronomers showing me all kinds of
    elaborate formulas and trying to impress me with how so
    exactly you can predict retrograde movements of planets.

    And I still have not given up explaining to you guys that your
    calculations have no value applicable to reality because
    planets don't travel backwards.

    Interesting how you validate each other so eagerly. I guess
    misery likes company...

    Even elementary cube theory is not intuitive for beginners
    (as can be seen when trying to teach them the 25 % take
    point for the dead cube).

    Maybe it's just difficult to "teach" (convince of) something
    that doesn't add up..?

    Also, your "cube hypothesis" must have self-verified itself
    into "cube theory" without the use of any empirical data,
    test or experiment. Convincing a small number of mentally
    ill gamblers that they will win more by doubling/taking at
    certain calculated equities, and then "observing" that they
    all try to play like that but only the ones most capable of
    it (i.e. achieving low PR's) win more, is not enough to make
    your cube hypothesis a cube theory.

    So one could characterize my mutant's doubling "strategy"
    as treating himself the cube as dead but hoping that the
    opponent treats it as very much alive. (-;

    Well enough with the clarification that it is "your" mutant
    alone and not mine nor anyone else's. So, yes, you should
    be given full credit for "your mutant's" silly doubling strategy.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Sun Nov 14 12:32:46 2021
    MK <murat@compuplus.net> writes:

    In your experiment, the mutant plays normally after the first cube. In
    my proposal, the mutant never drops except when it has no chance of
    winning.

    So you always double with winning chances > 50 % and always take with
    winning chances > 0 %?

    My mutant did essentially this (beavers forbidden) and got
    thrashed. Your comment one week ago was "Meaningless test". Note that it
    does not matter much whether I use "> 0 %" (your suggestion) or ">= 0 %"
    (my test, = "always take"). I admit that beavers will make a difference
    (if only ending up as Petersburg paradox, probably like your Epstein
    thought experiment, in which the deeper pockets will trivially win).

    How do you know? Have you tested and verified how much is the weapon
    worth?

    Would about this proposition?

    https://www.bkgm.com/rgb/rgb.cgi?view+838

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Nasti Chestikov@21:1/5 to All on Sun Nov 14 09:07:18 2021
    On Sunday, 14 November 2021 at 02:11:40 UTC, MK wrote:
    On November 13, 2021 at 3:02:18 AM UTC-7, Axel Reichert wrote:

    Frank Berger <bgbl...@googlemail.com> writes:

    I don't expect to convince anyone that the cube has a value
    All this is certainly more or less trivial for you, but as you
    can see from this thread I still have not given up explaining.
    The feeling of frustration is mutual. To me, you guys sound
    like pre-Copernican astronomers showing me all kinds of
    elaborate formulas and trying to impress me with how so
    exactly you can predict retrograde movements of planets.

    And I still have not given up explaining to you guys that your
    calculations have no value applicable to reality because
    planets don't travel backwards.

    Interesting how you validate each other so eagerly. I guess
    misery likes company...
    Even elementary cube theory is not intuitive for beginners
    (as can be seen when trying to teach them the 25 % take
    point for the dead cube).
    Maybe it's just difficult to "teach" (convince of) something
    that doesn't add up..?

    Also, your "cube hypothesis" must have self-verified itself
    into "cube theory" without the use of any empirical data,
    test or experiment. Convincing a small number of mentally
    ill gamblers that they will win more by doubling/taking at
    certain calculated equities, and then "observing" that they
    all try to play like that but only the ones most capable of
    it (i.e. achieving low PR's) win more, is not enough to make
    your cube hypothesis a cube theory.
    So one could characterize my mutant's doubling "strategy"
    as treating himself the cube as dead but hoping that the
    opponent treats it as very much alive. (-;
    Well enough with the clarification that it is "your" mutant
    alone and not mine nor anyone else's. So, yes, you should
    be given full credit for "your mutant's" silly doubling strategy.

    MK

    What these inbred cocksuckers can't grasp is the chances of GnuDung rolling a 4-5 are 2-in-36......but the chances of rolling a 4-5 when it's *exactly* the roll it needs is.....?

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Nasti Chestikov on Sun Nov 14 14:01:26 2021
    On November 14, 2021 at 10:07:19 AM UTC-7, Nasti Chestikov wrote:

    ... the chances of GnuDung rolling a 4-5 are 2-in-36......but the
    chances of rolling a 4-5 when it's *exactly* the roll it needs is.....?

    Unrelated to your point, this reminds me of my own ancients
    arguments that the luck calculations need to be progressive
    through the stages of games and proportionate to positions
    complexities.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Sun Nov 14 13:54:05 2021
    On November 14, 2021 at 4:32:49 AM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    In your experiment, the mutant plays normally after the first cube. In
    my proposal, the mutant never drops except when it has no chance of
    winning.

    So you always double with winning chances > 50 % and always take with
    winning chances > 0 %?

    This is where things get complicated. The answer may depend on
    what we are trying to achieve with our experiment.

    The goal of my 63 experiment was not to provide a comprehensive
    answer by proving an alternative cube theory but just to poke a hole
    in what you call a "cube theory". So, we can try different ways to see
    if one achieves this.

    My proposals weren't revelations to me. I'm not attached to any of
    them and I'm willing to emend. I'm trying to come up with something
    from my own way of cube strategy which is progressive, "depending
    on the amount of checker play left in the game".

    The paper linked by Frank acknowledges and explains that a little
    but preceeds as if not anyway. I realize "chance of winning" doesn't
    seem to work too well for explaining myself. Maybe another word
    like "hope" would work better? You double and take as long as you
    still have "hope of winning". At the start of the game your hopes
    are high (d/t point is low), towards the end of the game your hopes
    are low (d/t point is high).

    If I understand it correctly, this is similar to the difference between
    live and dead cube points(??) but more precise. If your math phd's
    can come up with a formula for this, more power to you. In the past,
    I had suggested many variants of cubefull bg, anywhere from raising
    the cube by an arbitrary number as in poker (i.e. not just doubling it
    but raising it by 100 or 1,500 etc.) to rasing the cube by fractional
    values like 3.5 or 10.62 or 0.29 etc.

    In this latter one, if played with ultimately precise cube skill, the cube ("stakes") would be raised at each and every turn, by the exact equity,
    by both sides, until the last roll. This would take all the gambling fun
    out of doubling and cubefull backgammon.

    Since we can't change the numbers 2, 4, 8... on the cube, we need to
    adjust the d/t points instead according to "how much hope we still
    have left" or "how much checker play is still left in the game". Since
    I'm not a math phd tempted to nail everything with a math hammer,
    I would gladly settle for being a simple potato counter and run an
    experiment with 4 billion trials to see what works best.

    How do you know? Have you tested and verified how much is the weapon
    worth?

    Would about this proposition?
    https://www.bkgm.com/rgb/rgb.cgi?view+838

    11 can't be an opening roll unless playing by the middle-eastern rules
    but even ignoring that, the argument relies on calculations using the
    results of previous calculations in a circular manner. If the previous calculations are biased and inaccurate, then the ensuing ones will
    necessarily be so as well.

    The question is: how much are you willing to buy back the "weapon"
    you previously sold? If I may use a non-numerical analogy here, this
    is like not being able to buy your own house back for the same price
    in a raising real estate market but onlyif/because you have no means
    to control/effect the market. In the "cube market" of backgammon,
    however, you can manipulate the market by checker play, which is
    "the other weapon"...

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Mon Nov 15 20:29:15 2021
    MK <murat@compuplus.net> writes:

    On November 14, 2021 at 4:32:49 AM UTC-7, Axel Reichert wrote:

    You double and take as long as you still have "hope of winning". At
    the start of the game your hopes are high (d/t point is low), towards
    the end of the game your hopes are low (d/t point is high).

    Standard cube theory has the concepts of dead/life cube and volatility
    for this.

    If I understand it correctly, this is similar to the difference
    between live and dead cube points(??) but more precise.

    Less. Quantify hope. (-:

    But your thinking reminds me on a player in our club who was eager to
    take the most desperate positions if only the volatility was sky-high.

    Since we can't change the numbers 2, 4, 8... on the cube

    This is what

    https://bkgm.com/rgb/rgb.cgi?view+429

    is about. Imagine a tripling cube (3, 9, 27, ...) and you end up with a Petersburg paradox. Which is why I am so eager to find out whether
    aggressive cube strategy, beavers, or raccoons have the same effect as
    the tripling cube. And because of the high volatility this cannot be
    done by just running long sessions with the bot (they would be too
    long), but we need to have a surrogate model (Markov chains), which is
    fed with the data from shorter sessions with the bot. The surrogate
    model can then easily be run a billion times. This is what I am doing.

    "how much checker play is still left in the game". Since I'm not a
    math phd tempted to nail everything with a math hammer, I would gladly
    settle for being a simple potato counter

    How would this potato counter look like? We need to quantify things, not because we like our math hammer, but because otherwise we cannot test hypotheses.

    In the "cube market" of backgammon, however, you can manipulate the
    market by checker play, which is "the other weapon"...

    With both sides playing their checkers like the bot, this weapon is
    cancelled with the argument from symmetry. So my experiment leaves just
    the cube skill in the game, as desired.

    Best regards

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Tue Nov 16 10:58:23 2021
    On November 15, 2021 at 12:29:18 PM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    You double and take as long as you still have "hope of winning". At
    the start of the game your hopes are high (d/t point is low), towards
    the end of the game your hopes are low (d/t point is high).

    Standard cube theory has the concepts of dead/life cube and
    volatility for this.

    Yes, but the range is not wide enough. Also, I maintain that there
    is no such thing as a "standard cube theory".

    If I understand it correctly, this is similar to the difference
    between live and dead cube points(??) but more precise.

    Less. Quantify hope. (-:

    I meant more precise than live/dead cube points. Did you mean
    the same by saying less in the opposite direction?

    I could try to use other words like "expectation", etc. but I don't
    think I can quantify. Isn't that what we are trying to do, with me
    arguing that you can't quantify cube skill..?

    But your thinking reminds me on a player in our club who was
    eager to take the most desperate positions if only the volatility
    was sky-high.

    Why not? Especially against completely predictable opponents
    like bots? Their predictability allows you to "steer" them through
    tactical checker moves. People like Chow bring up repeatedly
    that bots can be beaten by forcing them into backgames, etc.
    It may be harder to do with unpredictabe human opponents but
    arguably it can be done.

    Since we can't change the numbers 2, 4, 8... on the cube

    This is what
    https://bkgm.com/rgb/rgb.cgi?view+429
    is about.

    Were you curious enough to look up the RGB thread that it was
    extracted from? I did. And wow! Hundreds of long and detailed
    articles written by perhaps 40-50 different participants, many of
    the apparently mathematicians. They are broken into 5 pages. I
    spent a couple of hours last night and I could only read hald of
    one page. I'll keep reading and will post on this subject again.

    If I were you, I wouldn't just rely on posts hand-picked by bkgm.

    Imagine a tripling cube (3, 9, 27, ...) and you end up with a
    Petersburg paradox. Which is why I am so eager to find out
    whether aggressive cube strategy, beavers, or raccoons have
    the same effect as the tripling cube.

    Regardless of my opinion on Petersburg paradox in backgammon,
    what will the result of your experiment mean regarding what you
    call "standard cube theory"? You need to state this ahead of time,
    not make it fit retroactively.

    And because of the high volatility this cannot be done by just
    running long sessions with the bot (they would be too long),

    I don't understand what high volatility has to do with it but still, I
    think the hard way is the only way.

    but we need to have a surrogate model (Markov chains), which is
    fed with the data from shorter sessions with the bot. The surrogate
    model can then easily be run a billion times. This is what I am doing.

    This to me is like saying that a high resolution poster would require
    to much work and resources, thus you will take a snapshot and blow
    it up to poster size. I say it will end up very grainy, blurry.

    "how much checker play is still left in the game". Since I'm not a
    math phd tempted to nail everything with a math hammer, I would
    gladly settle for being a simple potato counter

    How would this potato counter look like? We need to quantify things,
    not because we like our math hammer, but because otherwise we
    cannot test hypotheses.

    You just run long enough sessions and tally the potatoes. I'm not
    trying to make less of math but just saying that complex math is
    not always necessary and can even be counter productive. If you
    can run 10,000 games overnight, we should be able to tackle this
    without questionable substitutions.

    BTW: did you mean that you can test hypotheses by quantifying
    things with math?

    In the "cube market" of backgammon, however, you can manipulate
    the market by checker play, which is "the other weapon"...

    With both sides playing their checkers like the bot, this weapon is
    cancelled with the argument from symmetry. So my experiment
    leaves just the cube skill in the game, as desired.

    Well enough as far as your specific experiment. But you can do
    other experiments with bots playing the cube moves the same
    and the mutant bot making maniac checker moves... :) We are
    anly using bots in the experiment because we can't use humans.
    We just need to wait until we can have an alpha-bot to see it
    decimate all current bots, as well as humans (including even me:),
    by making maniac cube and checkers plays.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Wed Nov 17 20:10:35 2021
    MK <murat@compuplus.net> writes:

    On November 15, 2021 at 12:29:18 PM UTC-7, Axel Reichert wrote:

    Were you curious enough to look up the RGB thread that it was
    extracted from? I did. And wow! Hundreds of long and detailed articles written by perhaps 40-50 different participants, many of the
    apparently mathematicians.

    Thanks for the hint, I might take a look.

    what will the result of your experiment mean

    If raccoons turn out to end up as Petersburg paradox, it would just be an incentive for some clubs (mine, for example) to have them forbidden in
    order to keep backgammon a mind sport, not a gambling amusement. Same
    for beavers (if Petersburg kicks in there as well). If not, then maniac
    cube strategies (contradicting standard cube theory) can be dismissed by investing CPU time, be it "real" sessions or Markov chain runs.

    By the way, in 10000 games with 1 beaver allowed, double > 0.5 and
    take > 0.0 the mutant lost 62117 against gnubg's 84870. In one game the
    cube reached 4096 (gnubg's limit), so I checked this game manually from
    the session file. In fact gnubg held the cube after beavering the
    mutant's redouble to 2048. The game turned around, so gnubg would have
    had a redouble, take to 8192. The game turned such that the mutant was
    over 50 percent again, so another redouble beavered by gnubg. Cube now
    at 32768 and a win for gnubg. So the final result would have been rather
    113542 versus 62117 for gnubg, which is quite a margin.

    But before declaring victory over the mutant's strategy, I need to
    ensure that the expectation settles, of which I was not yet sure after 5 billion games (Markov chain runs).

    And because of the high volatility this cannot be done by just
    running long sessions with the bot (they would be too long),

    I don't understand what high volatility has to do with it

    The higher the volatility (the one of the whole process, not the
    volatility of an individual position), the longer it takes until the law
    of large numbers kicks in.

    just run long enough sessions

    From my Markov chain runs it is quite certain that even several million
    games are not enough. I will not spent half a year of CPU time if smart surrogate methods yield a robust result much quicker. "There nothing
    more practical than a good theory."

    did you mean that you can test hypotheses by quantifying things with
    math?

    Sure. This is called simulation, my field of expertise for 25 years.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Timothy Chow@21:1/5 to Axel Reichert on Wed Nov 17 22:48:37 2021
    On 11/17/2021 2:10 PM, Axel Reichert wrote:
    If raccoons turn out to end up as Petersburg paradox, it would just be an incentive for some clubs (mine, for example) to have them forbidden in
    order to keep backgammon a mind sport, not a gambling amusement. Same
    for beavers (if Petersburg kicks in there as well). If not, then maniac
    cube strategies (contradicting standard cube theory) can be dismissed by investing CPU time, be it "real" sessions or Markov chain runs.

    I still find this line of reasoning peculiar. The St. Petersburg
    is an arcane mathematical oddity; why would anyone care about it in
    the context of a practical decision (like the rules for a club)?
    The positions with undefined equity already demonstrate that the
    paradox arises with ordinary backgammon, but apparently that does
    not dissuade your club from allowing money games. So why would
    showing that the paradox arises with raccoons dissuade your club from
    allowing raccoons? I don't follow the logic.

    Again, if the problem is that money games smell of gambling, then don't
    play money games. What's wrong with that simple logic?

    I can see that you might not want raccoons or even beavers if it
    causes the cube to get so high *in practice* that it starts to
    have negative effects on how people behave (maybe they lose more
    money than they can afford). But that again has nothing to do with
    the St. Petersburg paradox.

    ---
    Tim Chow

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Thu Nov 18 02:20:50 2021
    On November 17, 2021 at 12:10:38 PM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    Were you curious enough to look up the RGB thread that it was
    extracted from? I did. And wow! Hundreds of long and detailed
    articles written by perhaps 40-50 different participants, many
    of the apparently mathematicians.

    Thanks for the hint, I might take a look.

    It's not a hint. It's what anyone capable of independent thinking
    should be doing. I believe that bkgm.com is maintained by Tom
    Keith who is apparently another one of the dime a dozen math
    authorities in gamble-gammon. I have nothing against him and
    I benefoted from some of his articles but I have to assume that
    he is biased as one of the butt sniffing mutts pack of mentally
    ill gamblers. His web site only includes selected articles without
    opposing arguments, even is they were taken from RGB which is
    an open, unmoderated forum. "The real world".

    I estimated the number of articles in that thread which is more
    likely to be around 80-90 but from a few dozen participants for
    sure. You should read them. Most, if not all, of the articles are
    from mathematicians with some of them dissenting at times.
    Like David Ullrich. Personally, I kind of liked quite a few of the
    things that he wrote in RGB. You can't find a single reference
    to him in bkgm.com or bgonline.org. You should ask why??

    what will the result of your experiment mean

    If raccoons turn out to end up as Petersburg paradox, it would
    just be an incentive for some clubs (mine, for example) to have
    them forbidden in order to keep backgammon a mind sport, not
    a gambling amusement.

    Can you reword that in terms of what it means about "cube skill"?

    Would it support "cube skill" or not, regardless of whether you
    prefer to call it a "theorie" or a "hypothesis"?

    Same for beavers (if Petersburg kicks in there as well). If not, then
    maniac cube strategies (contradicting standard cube theory) can
    be dismissed by investing CPU time, be it "real" sessions or Markov
    chain runs.

    I had never seen the term "maniac" used in this context before and
    thought it was your own coining until I just ran into it in an article
    by Gary Wong, (whom I detest as one of the biggest assholes in
    gamble-gammon world), who had also estimated that by 2048 (or
    so??) we should have enough CPU power to solve calculate all
    cubeless equities in 3 minutes...

    By the way, in 10000 games with 1 beaver allowed, double > 0.5
    and take > 0.0 the mutant lost 62117 against gnubg's 84870.

    What's important here is how that compares to what would be
    expected. Did you use your high math to calculate a prediction
    before you started? Would you have expected that the mutant
    would lose by ten fold, twenty fold, etc...? You haven't. And the
    above numbers are incredibly good towards proving that the
    so-called "cube skill" is bullshit. When you refine your doubling
    and especially taking points from >0 to more logical/practical
    one (such as deriving from my tests), you will see the the mutant
    will decimate gnubg...

    The test we are doing now is a preliminary one to see if there is
    any sense of going further and your numbers above are 10 times
    more than what anyine would have expected to say "yes".

    In one game the cube reached 4096 (gnubg's limit), so I checked
    this game manually from the session file. In fact gnubg held the
    cube after beavering the mutant's redouble to 2048.... blah blah

    Was gnubg wrong to hold the cube? Then argue for it why. And
    you may convince yourself against yourself...

    But before declaring victory over the mutant's strategy, I need to
    ensure that the expectation settles, of which I was not yet sure
    after 5 billion games (Markov chain runs).

    Take your time, I've got the beer chilling... ;)

    I don't understand what high volatility has to do with it

    The higher the volatility (the one of the whole process, not the
    volatility of an individual position), the longer it takes until the
    law of large numbers kicks in.

    Okay, thanks for clarifying.

    just run long enough sessions

    From my Markov chain runs it is quite certain that even several
    million games are not enough.

    That's funny. You substitute Markov Chain for real trials and then
    turn around to extrapolate how many real trials you woul have
    needed based on your Markov Chain runs..? Too much jacking
    off going on in the gamble-gammon world for my taste.. :(

    I will not spent half a year of CPU time if smart surrogate
    methods yield a robust result much quicker.

    I suggested that we could distrubute the needed CPU time and
    offered to contribute to it myself. What's you problem with that?!

    "There nothing more practical than a good theory."

    Is that a quote from the bible?? Hallelujah! Amen!

    did you mean that you can test hypotheses by quantifying
    things with math?

    Sure. This is called simulation, my field of expertise for 25 years.

    Okay, we'll just make a note of this and remeber it when time comes.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to Timothy Chow on Thu Nov 18 19:49:09 2021
    Timothy Chow <tchow12000@yahoo.com> writes:

    The positions with undefined equity already demonstrate that the
    paradox arises with ordinary backgammon, but apparently that does not dissuade your club from allowing money games. So why would showing
    that the paradox arises with raccoons dissuade your club from allowing raccoons?

    The "eternal redoubling" position is, as you state, an oddity, so that,
    if it ever comes up during practical play, it would be more of a reason
    to celebrate this rare event than frown upon the Petersburg paradox.
    (-:

    But there is in my opinion a more fundamental thing: It should be hardly possible to deliberately strive for this kind of positions by employing particular checker play "techniques". Whereas my "maniac" cube strategy
    is comparatively simple to implement and would be annoying throughout
    the evening.

    So the one has the taste of lucky occurrence, while the other has the
    smell of abuse (from the subjective view of someone who wants to ensure
    the "mind sport" aspect).

    Best regards

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Thu Nov 18 21:51:13 2021
    MK <murat@compuplus.net> writes:

    On November 17, 2021 at 12:10:38 PM UTC-7, Axel Reichert wrote:

    If raccoons turn out to end up as Petersburg paradox [...]

    Can you reword that in terms of what it means about "cube skill"?

    I have tried to explain this before. Without existing expected value I
    do not think there is a way to quantify the difference between cube
    strategies (Big O notation, anyone?). As long as you have an expected
    value, it is possible to dismiss particular cube strategies as
    inferior.

    By the way, in 10000 games with 1 beaver allowed, double > 0.5 and
    take > 0.0 the mutant lost 62117 against gnubg's 84870.

    What's important here is how that compares to what would be
    expected. Did you use your high math to calculate a prediction before
    you started? Would you have expected that the mutant would lose by ten
    fold, twenty fold, etc...? You haven't.

    As in my other trials the Null hypothesis was that the mutant cubes as
    good as gnubg. Assuming a normal distribution of the cube value (which
    is not quite correct, since it should be closer to a geometrical
    distribution) the lead should be between -13296 and +13296 with 95 % probability. However, gnubg's lead was nearly twice as high. But see
    below.

    And the above numbers are incredibly good towards proving that the
    so-called "cube skill" is bullshit.

    Why so? I am eagerly waiting for your irrefutable reasons.

    When you refine your doubling and especially taking points from >0 to
    more logical/practical one (such as deriving from my tests), you will
    see the the mutant will decimate gnubg...

    Show the results once you have them. I am eagerly waiting for your statistically significant experiments.

    gnubg held the cube after beavering the mutant's redouble to 2048

    Was gnubg wrong to hold the cube?

    With "held" I meant "was possessing". It could not redouble any
    more. Like I wrote, without the hard-wired limit of 4096, it would have redoubled a couple of rolls later according to standard cube theory.

    But before declaring victory over the mutant's strategy, I need to
    ensure that the expectation settles, of which I was not yet sure
    after 5 billion games (Markov chain runs).

    Take your time, I've got the beer chilling... ;)

    Good, me too. (-:

    So in the meantime I have arrived at 100e9 games, see the results from
    beyond 10000 games here:

    | Games | Avg. game value | Avg. advantage for gnubg (PPG) | |-----------------+-----------------+--------------------------------|
    | 20.000 | 28 | 4.22185 |
    | 50.000 | 28 | 5.79926 |
    | 100.000 | 119 | 93.25437 |
    | 200.000 | 34 | 1.63743 |
    | 500.000 | 45 | -0.795082 |
    | 1.000.000 | 44 | 2.359327 |
    | 2.000.000 | 83 | -37.00945 |
    | 5.000.000 | 41 | 5.281775 |
    | 10.000.000 | 44 | 8.318696 |
    | 20.000.000 | 73 | 35.509407 |
    | 50.000.000 | 49 | 10.718045 |
    | 100.000.000 | 79 | 36.462276 |
    | 200.000.000 | 103 | -26.76603 |
    | 500.000.000 | 57 | 6.1772475 |
    | 1.000.000.000 | 71 | 21.412815 |
    | 2.000.000.000 | 70 | 8.383391 |
    | 5.000.000.000 | 63 | 14.327376 |
    | 10.000.000.000 | 125 | 74.427704 |
    | 20.000.000.000 | 73 | 5.6380033 |
    | 50.000.000.000 | 71 | 3.6864755 |
    | 100.000.000.000 | 79 | 23.954538 |

    As you can see, GNU Backgammon was leading most of the time, on
    average with an advantage amounting to roughly 1/5th of the average
    game value (this is a lot!).

    However, if you plot the second column over the (logarithmic) number
    of games, then to me it seems that the average value DOES NOT SETTLE,
    BUT KEEPS ON INCREASING. My interpretation of all this is that you
    have successfully entered Petersburg country, so your cube strategy
    cannot be dismissed easily (big O notation, anyone? Perhaps this could
    be applied in this context), because no expected value
    exists. However, with all due respect, you will almost certainly not
    be able to back your strategy with cash in a real money session, since
    the highest cube I encountered was roughly 500e9. Even at a modest
    stake of 1 Euro/point this is a fairly ambitious amount of money.

    It goes without saying that (if I am right about the beavers) allowing
    raccoons will end up in Petersburg country as well.

    Without beavers, it is a different story: From my experiments with the
    more conservatively taking mutant ("take like gnubg") I am very sure
    that for the current mutant's cube strategy there exists an expected
    value, and the strategy will fail miserably. But I will test this and
    report on it. I do not believe that even further restrictive measures
    (such as capping the cube at 64, "beginner-friendly") are called for
    to get out of Petersburg country.

    "There nothing more practical than a good theory."

    Is that a quote from the bible?

    No. https://en.wikiquote.org/wiki/Kurt_Lewin

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Timothy Chow@21:1/5 to All on Thu Nov 18 21:46:03 2021
    On 11/18/2021 5:20 AM, MK wrote:
    Like David Ullrich. Personally, I kind of liked quite a few of the
    things that he wrote in RGB. You can't find a single reference
    to him in bkgm.com or bgonline.org. You should ask why??

    He is mentioned on bkgm.com a few times.

    https://www.bkgm.com/rgb/rgb.cgi?view+983 https://www.bkgm.com/rgb/rgb.cgi?view+1254 https://bkgm.com/rgb/rgb.cgi?view+1310 https://bkgm.com/articles/Zare/UndefinedEquity/index.html

    ---
    Tim Chow

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Fri Nov 19 01:05:36 2021
    On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    Can you reword that in terms of what it means about "cube skill"?

    I have tried to explain this before. Without existing expected value I
    do not think there is a way to quantify the difference between cube strategies (Big O notation, anyone?). As long as you have an expected
    value, it is possible to dismiss particular cube strategies as inferior.

    I can't speak your language and was asking to reworded in simpler
    English but maybe it's not easy for you to do either.

    Let me try another way. If you ran sessions with different "cube
    strategies", (BTW: I like that you are using these words), against
    the "standard cube theory" that gnubg uses, can you compare if
    some do better than others? If they all give similar results, maybe
    this will be the best argument that show that the "standard cube
    theory" us flawed (or in my exaggerated words, that the "cube
    skill" is mostly bullshit).

    By the way, in 10000 games with 1 beaver allowed, double > 0.5
    and take > 0.0 the mutant lost 62117 against gnubg's 84870.

    What's important here is how that compares to what would be
    expected. Did you use your high math to calculate a prediction
    before you started? Would you have expected that the mutant
    would lose by ten fold, twenty fold, etc...? You haven't.

    As in my other trials the Null hypothesis was that the mutant cubes
    as good as gnubg. Assuming a normal distribution of the cube value
    (which is not quite correct, since it should be closer to a geometrical distribution) the lead should be between -13296 and +13296 with 95 % probability. However, gnubg's lead was nearly twice as high. But see
    below.

    All this sounds like unnecessarily overcomplicating things. If more
    cube skill wins more, then a defying, almost nonexistent cube skill
    should lose incomparably more. Didn't you have a "gut feeling", an
    rough guess about how the results would come out? I did. :)

    And the above numbers are incredibly good towards proving that
    the so-called "cube skill" is bullshit.

    Why so? I am eagerly waiting for your irrefutable reasons.

    Before running the experiment, if you took bets from gamblers here
    on what would be gnubg's lead, I would've bet that they would've bet
    that it would be much more than merely twice.

    I don't think either that you are comfortable enough with obtaining
    the same results if you ran another 10,000 games. Gnubg may win
    by more but may also lose this time.

    When you refine your doubling and especially taking points from
    0 to more logical/practical one (such as deriving from my tests),
    you will see the the mutant will decimate gnubg...

    Okay, two things here. Let me first correct myself that of all people,
    I shouldn't have said things like "more logical/practical take points
    than > 0" because I am the one arguing against a "certain cube skill"
    to begin with. :( I don't know what was I thinking. Unless proven, I
    don't think there is any way to claim that a take point of > 10% will
    win more than a take point of > 0% or 20% or 50%. It's much more
    complicated than that for me anyway.

    Show the results once you have them. I am eagerly waiting for your statistically significant experiments.

    I already said no human can live long enough to satisfy you guy's
    requirement for statistically significant. All I can offer is that I'm
    fairly confident that, if given the opportunity to play an observed
    100 games long demonstration session, I can substancially
    duplicate the results of my previous several dozens.

    gnubg held the cube after beavering the mutant's redouble to 2048

    Was gnubg wrong to hold the cube?

    With "held" I meant "was possessing". It could not redouble any
    more. Like I wrote, without the hard-wired limit of 4096,

    Okay, I understand now.

    it would have redoubled a couple of rolls later according to standard
    cube theory.

    But even then, there is no way to know what else would ensue, no?
    How can you tell that mutant wouldn't take and win or redouble?
    Especially in a long enough run, if the situation arose multiple times
    with different results for either side, maybe mutant would win more?

    ... My interpretation of all this is that you have successfully entered Petersburg country, so your cube strategy cannot be dismissed
    easily (big O notation, anyone? Perhaps this could be applied in this context), because no expected value exists.

    I suppose this is good news for my argument even if the experiment
    wasn't exactly what I had proposed?

    However, with all due respect, you will almost certainly not be
    able to back your strategy with cash in a real money session,

    Why not? Wouldn't anyone who believed in math do so??

    since the highest cube I encountered was roughly 500e9. Even
    at a modest stake of 1 Euro/point this is a fairly ambitious
    amount of money.

    What if I have the money? Wouldn't you if you had the money?
    I bet Zare would.

    "There nothing more practical than a good theory."

    Is that a quote from the bible?

    No. https://en.wikiquote.org/wiki/Kurt_Lewin

    There it says: "A business man once stated that there is nothing
    so practical as a good theory". Regardless, it sound like a cheap
    slogan, if not an oxymoron, to me.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Tim Chow on Fri Nov 19 01:12:32 2021
    On November 18, 2021 at 7:46:07 PM UTC-7, Tim Chow wrote:

    On 11/18/2021 5:20 AM, MK wrote:

    Like David Ullrich. Personally, I kind of liked quite a few of the
    things that he wrote in RGB. You can't find a single reference
    to him in bkgm.com or bgonline.org. You should ask why??

    He is mentioned on bkgm.com a few times. https://www.bkgm.com/rgb/rgb.cgi?view+983 https://www.bkgm.com/rgb/rgb.cgi?view+1254 https://bkgm.com/rgb/rgb.cgi?view+1310 https://bkgm.com/articles/Zare/UndefinedEquity/index.html

    Thanks for looking up. I'm not very familiar with how to search
    that site. The first three are unrelated to math, just about rules.

    Zare gives him credit by saying "See the rec.games.backgammon
    archive. I’d like to thank Chris Yep and David Ullrich for their work
    and helpful discussions" but there is nothing from him at the link.

    Anyway, I value and encourage dissent in any area. I don't think
    there much in rgb anymore. When everyone agrees, you get dogma.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Fri Nov 19 01:46:58 2021
    On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:

    Show the results once you have them. I am eagerly
    waiting for your statistically significant experiments.

    Whether statistically significant or not, I just played a
    100-games session today, trying to refine my strategy,
    to play more carefully and consistently, accepting XG's
    raccons this time around, and the results were:

    Errors: checker = 9.19 / cube = 32.95 / overall = 13.26
    Wins: expected = -302 / effective = +182 / difference = +484

    The results for the previous three similar sessions were:

    Errors: checker = 18.56 / cube = 25.21 / overall = 19.55
    Wins: expected = -114 / effective = +7 / difference = +121

    Errors: checker = 13.89 / cube = 31.09 / overall = 16.91
    Wins: expected = -79 / effective = -56 / difference = +23

    Errors: checker = 8.97 / cube = 29.39 / overall = 12.80
    Wins: expected = -140 / effective = +56 / difference = +196

    Now that I have become interested again, I'll try to play
    more sessions just trying my "new and improved strategy".

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Mon Nov 22 19:07:19 2021
    MK <murat@compuplus.net> writes:

    accepting XG's raccons

    So this means you beavered XG's doubles. According (roughly) to what
    criterion? Being above (estimated) 40 % winning chances? Or always?

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Mon Nov 22 19:38:55 2021
    MK <murat@compuplus.net> writes:

    On November 18, 2021 at 1:51:16 PM UTC-7, Axel Reichert wrote:

    I can't speak your language and was asking to reworded in simpler
    English but maybe it's not easy for you to do either.

    Right. Some complex concepts require a certain language, which is why mathematician have invented formula notation.

    If you ran sessions with different "cube strategies" [...], can you
    compare if some do better than others?

    This is what I did. And by now I also have ideas what to do in the cases
    where we end up with a Petersburg paradox (that is what the "big O"
    notation referred to).

    I shouldn't have said things like "more logical/practical take points
    than > 0" because I am the one arguing against a "certain cube skill"

    Well, a very good test to check whether strategic elements are involved
    (and, by the way, also used in German court rulings about games of luck
    versus games of skill) is to compare against a random player. So you
    could roll a dice for every cube decision (1, 2: pass, 3, 4: take, 5, 6:
    beaver and 1, 2, 3: double, 4, 5, 6: roll). For checker play, copy what
    the bot would do. If you think this foolish, then apparently you believe
    in cube skill. (-;

    I already said no human can live long enough to satisfy you guy's
    requirement for statistically significant.

    Of course you can. But the length of the trial depends on the
    volatility. Once we are "in Petersburg country", a more or less trivial strategy is to get the cube to a high level, win one of these games and
    protect the lead from then on by passing.

    My interpretation of all this is that you have successfully entered
    Petersburg country, so your cube strategy cannot be dismissed easily
    (big O notation, anyone? Perhaps this could be applied in this
    context), because no expected value exists.

    I suppose this is good news for my argument even if the experiment
    wasn't exactly what I had proposed?

    Not too fast. Let us put it this way: In Petersburg country we need to
    employ other techniques (big O notation, analytical approaches) to
    dismiss foolish strategies. I might have a 66 in my dice cup ...

    More to come!

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to Axel Reichert on Mon Nov 22 23:21:43 2021
    Axel Reichert <mail@axel-reichert.de> writes:

    by now I also have ideas what to do in the cases where we end up with
    a Petersburg paradox

    [...]

    In Petersburg country we need to employ other techniques (big O
    notation, analytical approaches) to dismiss foolish strategies.

    [...]

    More to come!

    So here we go. All of this is on a maths level for (almost) adults, so
    the later high school years.

    Probabilities for cube decisions with the mutant doubling above 50 and
    taking above 0 percent winning chances, gathered from 10000 games played against gnubg:

    | Cube decision | Probability | |---------------------------------+-------------|
    | gnubg doubles, mutant takes | 0.1530 |
    | gnubg doubles, mutant beavers | 0.0000 | |---------------------------------+-------------|
    | gnubg redoubles, mutant takes | 0.5447 |
    | gnubg redoubles, mutant beavers | 0.0000 | |---------------------------------+-------------|
    | mutant doubles, gnubg takes | 0.1955 |
    | mutant doubles, gnubg beavers | 0.6454 | |---------------------------------+-------------|
    | mutant redoubles, gnubg takes | 0.1754 |
    | mutant redoubles, gnubg beavers | 0.1439 |

    Over-doubling and over-redoubling cube

    So with respect to "centered" cube action, gnubg wields a "doubling"
    cube with an average value of 2, whereas the mutant wields an
    "over-doubling" cube with an average value of

    M = (0.1955*2 + 0.6454*2*2^b) / (0.1955 + 0.6454)
    = 3.5350 (for b = 1, which stands for 1 beaver allowed)

    Likewise, with respect to "owned" cube action, gnubg wields a
    "redoubling" cube with an average value of 2, whereas the mutant wields
    an "over-redoubling" cube with an average value of

    M = (0.1754*2 + 0.1439*2*2^b) / (0.1754 + 0.1439)
    = 2.9013 (for b = 1)

    Case 1: Mutant starts cubing

    Let's assume that the mutant cubes first (this is far more likely than
    gnubg doubling first, around 84 % versus 15 %, see the above table).

    Then let us assess the probability of an odd number of cubings:

    p(n = 1) = (0.1955 + 0.6454) * (1 - 0.5447)

    p(n = 3) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439) * (1 - 0.5447)

    ...

    p(n = 2k-1) = (0.1955 + 0.6454) * (0.5447 * (0.1754 + 0.1439))^(k-1) * (1 - 0.5447)

    = 0.3745 * 0.1739^(k-1)

    (k = 1, 2, 3, ...)

    How high (on average) is the cube for an odd number of cubings?

    c(n = 1) = 3.5350

    c(n = 3) = 3.5350 * 2 * 2.9013

    c(n = 5) = 3.5350 * 2 * 2.9013 * 2 * 2.9013

    ...

    c(n = 2k-1) = 3.5350 * (2 * 2.9013)^(k-1)

    = 3.5350 * 5.8026^(k-1)

    (k = 1, 2, 3, ...)

    Now the same for an even number, with the mutant starting the cubing:

    p(n = 2) = (0.1955 + 0.6454) * 0.5447 * (1 - 0.1754 - 0.1439)

    p(n = 4) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439)
    * 0.5447 * (1 - 0.1754 - 0.1439)

    p(n = 6) = (0.1955 + 0.6454) * 0.5447 * (0.1754 + 0.1439)
    * 0.5447 * (0.1754 + 0.1439)
    * 0.5447 * (1 - 0.1754 - 0.1439)

    ...

    p(n = 2k) = (0.1955 + 0.6454) * 0.5447 * ((0.1754 + 0.1439)*0.5447)^(k-1)
    * (1 - 0.1754 - 0.1439)

    = 0.3118 * 0.1739^(k-1)

    (k = 1, 2, 3, ...)

    How high (on average) is the cube for an even number of cubings?

    c(n = 2) = 3.5350 * 2

    c(n = 4) = 3.5350 * 2 * 2.9013 * 2

    c(n = 6) = 3.5350 * 2 * 2.9013 * 2 * 2.9013 * 2

    ....

    c(n = 2k) = 3.5350 * 2 * (2.9013 * 2)^(k-1)

    = 7.0700 * 5.8026^(k-1)

    (k = 1, 2, 3, ...)

    With respect to the existence of the expected value, we can neglect
    whether a single game, a gammon, or a backgammon was won. This will be
    relevant only for assessing the gains of one cube strategy over the
    other (postponed to a later article).

    E(n = 2k-1) = p(n = 2k-1) * c(n = 2k-1)

    = 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1)

    = 1.3239 * 1.0092^(k-1)

    E(n = 2k) = p(n = 2k) * c(n = 2k)

    = 0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1)

    = 2.2044 * 1.0092^(k-1)

    In sum we have:

    E = E(n = 2k-1) + E(n = 2k)

    = 1.3239 * 1.0092^(k-1) + 2.2044 * 1.0092^(k-1)

    = 3.5283 * 1.0092^(k-1)

    This is a geometrical series, and a divergent one, since

    1.0092 >= 1

    Hence the mutant cube strategy ends up as a Petersburg paradox, if
    beavers are allowed. But it is close.

    With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
    With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

    Case 2: GNU Backgammon starts cubing

    For the much rarer case that gnubg cubes first (around 15 % versus 84 %
    for the mutant cubing first) the mathematics is exactly the same. If you
    do this boring exercise (left for the eager reader), it turns out that
    the crucial terms (the ones with the exponent k-1) have exactly the same
    factor in front of them. So the verdict is the same:

    With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
    With 1 beaver allowed, we get 1.0092. Close, but still Petersburg.
    With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

    So my next task will be to use these numbers to calculate the expected
    value including gammons and backgammon. This will need some distinctions between the various cases (cube ownership, even/odd number of cubings,
    ...), but most likely I will have time for these in the next couple of
    days.

    Stay tuned!

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Tue Nov 23 22:50:31 2021
    On November 22, 2021 at 11:09:43 AM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:
    accepting XG's raccons

    So this means you beavered XG's doubles. According
    (roughly) to what criterion? Being above (estimated)
    40 % winning chances? Or always?

    Oops, I misspoke. Sorry. I meant I raccooned XG's
    beavers. I hardly ever beaver and of I do, it's often
    irrational almost self-destructive gambling (which
    also applies to some of my plain takes but usually
    only if I'm way ahead). :(

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Wed Nov 24 00:01:57 2021
    On November 22, 2021 at 11:38:57 AM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    ... If you think this foolish, then apparently you believe in cube skill. (-;

    When I say "cube skill is bullshit", it's an exaggeration but only to
    mean that "cube skill" is exaggerated. Whenever needed, I have
    always clarified that I acknowledge a cube skill ranging from
    barely defensible early in the game to undeniable late in the game.

    I wonder if it's even misleading to call it "cube skill" when it's really
    based on the skill of estimating one's winning chances at any given
    point during the game and that what I said above applies to equity
    calculations in early/late positions in the game.

    Talking about foolish, while reading old articles in RGB, I saw that
    I had even done an experiment over 10 years ago against Gnubg,
    doubling at my first opportunity, without any other conditions, to
    see if I could overcome the cost of giving away the cube. It looks
    like I had some "colorful" discussions with old RGB regulars who
    were much more numerous back then and who all liked me. :))

    I had even suggested experiments to quantify the skill level of
    random checker play while arguing that FIBS formula was too
    arbitrary and inaccurate. It was never done and many thought
    it was a ridiculous idead but when I insisted, Ullrich at least had
    obliged to estimate that random play could achieve 1700 rating... :)

    I already said no human can live long enough to satisfy you guy's
    requirement for statistically significant.

    Of course you can. But the length of the trial depends on the volatility. Once we are "in Petersburg country", a more or less trivial strategy is
    to get the cube to a high level, win one of these games and protect the
    lead from then on by passing.

    Are you suggesting that this is what I have been doing? In my last
    session the highest I won was a gammon with cube at 32 on the
    59th game. On the 70th game I lost with cube at 16. Without them
    I would still be +134. Even though I was never under, during the first
    10 games I had gained +21 vs the last 10 games only +7. So, you
    may have something there... ;)

    Bu this is not about me at all. I just shared my experience similar
    to your experiment that's all. Just ignore all of my stuff if you wish.
    Let's see what will statistically sugnificant bot vs bot experiments
    reveal.

    I suppose this is good news for my argument even if the experiment
    wasn't exactly what I had proposed?

    Not too fast. Let us put it this way: In Petersburg country we need to
    employ other techniques (big O notation, analytical approaches) to
    dismiss foolish strategies. I might have a 66 in my dice cup ...

    I'm still trying to understand how this "Petersburg country" thing is
    relevant here? Are we eliminating all (cube) skill so that what is left
    is just probabilities? And then going into a gambling frenzy from
    there to "Petersburg country"??

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Wed Nov 24 00:50:51 2021
    On November 22, 2021 at 3:21:45 PM UTC-7, Axel Reichert wrote:

    E(n = 2k-1) = p(n = 2k-1) * c(n = 2k-1)
    = 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1)
    = 1.3239 * 1.0092^(k-1)
    E(n = 2k) = p(n = 2k) * c(n = 2k)
    = 0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1)
    = 2.2044 * 1.0092^(k-1)
    This is a geometrical series, and a divergent one, since
    1.0092 >= 1
    With 0 beavers allowed, we get 0.6957. No Petersburg paradox.
    With 1 beaver allowed, we get 1.0092. Close, but still Petersburg.
    With 2 beavers allowed, we get 1.6363. Clearly Petersburg.

    Looks like a lot of "smoke and maths" to me... :)

    So my next task will be to use these numbers to calculate the
    expected value including gammons and backgammon. This
    will need some distinctions between the various cases (cube
    ownership, even/odd number of cubings, ...), but most likely I
    will have time for these in the next couple of days.
    Stay tuned!

    Do you or anyone else have any predictions as to what the results
    will be?

    Or are you trying to make happen certain results to prove one side
    of the argument?

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Thu Dec 2 17:32:53 2021
    On November 22, 2021 at 3:21:45 PM UTC-7, Axel Reichert wrote:

    So my next task will be to use these numbers to calculate the
    expected value including gammons and backgammon. This
    will need some distinctions between the various cases (cube
    ownership, even/odd number of cubings, ...), but most likely I
    will have time for these in the next couple of days.
    Stay tuned!

    It has been 10 or "a couple of 5" days since... I hope you are
    still intending to do or working on this. I don't know about the
    others but I'm all ears, impatiently waiting to hear about it.

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Sat Dec 4 22:27:48 2021
    MK <murat@compuplus.net> writes:

    On November 22, 2021 at 3:21:45 PM UTC-7, Axel Reichert wrote:

    So my next task will be to use these numbers to calculate the
    expected value including gammons and backgammon. This
    will need some distinctions between the various cases (cube
    ownership, even/odd number of cubings, ...), but most likely I
    will have time for these in the next couple of days.
    Stay tuned!

    It has been 10 or "a couple of 5" days since... I hope you are
    still intending to do or working on this. I don't know about the
    others but I'm all ears, impatiently waiting to hear about it.

    Yes, I am making progress and think my approach works.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Sat Dec 4 17:10:59 2021
    On December 4, 2021 at 2:27:49 PM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    It has been 10 or "a couple of 5" days since... I hope you are
    still intending to do or working on this. I don't know about the
    others but I'm all ears, impatiently waiting to hear about it.

    Yes, I am making progress and think my approach works.

    Even as I asked for it twice, you never offered any predictions
    nor committed to what will your results mean either way. Can
    we at least observe your progress for the sake of openness?

    To me, it looks like your are working on a secret receipe in a
    dark room, tweaking and calculating, tweaking and calculating
    and that you will let us know if/when you finally fabricate your
    desired results...

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to murat@compuplus.net on Mon Dec 6 19:47:10 2021
    MK <murat@compuplus.net> writes:

    Even as I asked for it twice, you never offered any predictions nor
    committed to what will your results mean either way.

    Mathematical proofs are different in nature from statistical hypothesis testing. Pythagoras did not need to state a hypothesis about the sides
    of a triangle before testing it, he proved it. For eternity.

    Since my approach is analytical, there is no need for hypothesizing
    either. If it works, fine, if not, the jury is still out.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Thu Dec 9 21:52:05 2021
    On December 6, 2021 at 11:47:12 AM UTC-7, Axel Reichert wrote:

    MK <mu...@compuplus.net> writes:

    Even as I asked for it twice, you never offered any predictions
    nor committed to what will your results mean either way.

    Mathematical proofs are different in nature from statistical
    hypothesis testing. Pythagoras did not need to state a
    hypothesis about the sides of a triangle before testing it,

    Your equating the two is laughable but let's see what you come
    up with. Any attempt is better than dogmatic complacency.

    Since my approach is analytical, there is no need for
    hypothesizing either. If it works, fine, if not, the jury is still out.

    Shouldn't I have a right to ask you to define the "works" before
    you declare if it works or not? Also, who is the jury? You?

    And, regardless of all that, what harm would it do to allow us
    to observe your progress as you are fabricating your "proof"?

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Axel Reichert@21:1/5 to Axel Reichert on Sat Apr 16 11:36:17 2022
    Axel Reichert <mail@axel-reichert.de> writes:

    So my next task will be to use these numbers to calculate the expected
    value including gammons and backgammon. This will need some
    distinctions between the various cases (cube ownership, even/odd
    number of cubings, ...)

    We need some further results from my older posting and some results
    omitted there back then for brevity:

    Case 1: Mutant starts an odd number n of cubings

    Probability of n cubings
    p(n = 2k-1) = 0.3745 * 0.1739^(k-1)

    Cube value after n cubings
    c(n = 2k-1) = 3.5350 * 5.8026^(k-1)

    Case 2: Mutant starts an even number n of cubings

    Probability of n cubings
    p(n = 2k) = 0.3118 * 0.1739^(k-1)

    Cube value after n cubings
    c(n = 2k) = 7.0700 * 5.8026^(k-1)

    Case 3: GNU Backgammon starts an odd number n of cubings

    p(n = 2k-1) = 0.1041 * 0.1739^(k-1)
    c(n = 2k-1) = 2 * 5.8026^(k-1)

    Case 4: GNU Backgammon starts an even number n of cubings

    p(n = 2k) = 0.0222 * 0.1739^(k-1)
    c(n = 2k) = 5.8026 * 5.8026^(k-1)

    We further need the game values (without cube value factored in) from
    the session of 10000 games. The games without any cubing have been
    omitted to simplify things: With one of the player so cube-happy, this
    is an occurence in the per mille range. The following table lists the
    fraction of games (single, gammon, backgammon) won/lost by GNU
    Backgammon depending on final cube ownership:

    | Cube owner | Win 1 | Win 2 | Win 3 | Lose 1 | Lose 2 | Lose 3 | |----------------+--------+--------+--------+--------+--------+--------|
    | GNU Backgammon | 0.0230 | 0.0210 | 0.0006 | 0.2981 | 0.1076 | 0.0050 |
    | Mutant | 0.4427 | 0.1422 | 0.0057 | 0.0901 | 0.0000 | 0.0000 |

    So the game values from GNU Backgammon's point of view are

    0.0230*1 + 0.0210*2 + 0.0006*3 - 0.2981*1 - 0.1076*2 - 0.0050*3 =

    -0.4615

    if GNU Backgammon holds the cube at the end and

    0.4427*1 + 0.1422*2 + 0.0057*3 - 0.0901*1 - 0.0000*2 - 0.0000*3 =

    0.6541

    if the mutant holds the cube at the end.

    Note that if the mutant starts cubing and we have an odd number of
    cubings, then GNU Backgammon owns the cube. For an even number of
    cubings, the mutant owns the cube. Vice versa if GNU Backgamman starts
    cubing.

    With this in place, we can calculate the expected "points per game" for
    GNU Backgammon:

    Case 1 and 2: Mutant starts cubing

    l_m = p(n = 2k-1) * c(n = 2k-1) * e(n = 2k-1) +
    p(n = 2k) * c(n = 2k) * e(n = 2k)

    = 0.3745 * 0.1739^(k-1) * 3.5350 * 5.8026^(k-1) * (-0.4615) +
    0.3118 * 0.1739^(k-1) * 7.0700 * 5.8026^(k-1) * 0.6541

    = 1.3239 * 1.0092^(k-1) * (-0.4615) + 2.2044 * 1.0092^(k-1) * 0.6541

    = 0.8309 * 1.0092^(k-1)

    Case 3 and 4: GNU Backgammon starts cubing

    l_g = p(n = 2k-1) * c(n = 2k-1) * e(n = 2k-1) +
    p(n = 2k) * c(n = 2k) * e(n = 2k)

    = 0.1041 * 0.1739^(k-1) * 2 * 5.8026^(k-1) * 0.6541 +
    0.0222 * 0.1739^(k-1) * 5.8026 * 5.8026^(k-1) * (-0.4615)

    = -0.0594 * 1.0092^(k-1)

    Since the mutant starts cubing in 0.8409 of the games and GNU Backgammon
    starts cubing in 0.1530 of the games, we have

    l = 0.8409 * l_m + 0.1530 * l_g

    = 0.8409 * 0.8309 * 1.0092^(k-1) + 0.1530 * (-0.0594) * 1.0092^(k-1)

    = 0.6896 * 1.0092^(k-1)

    To summarize: Like for the expected value of a single game (as shown previously), we have a Petersburg Paradox occuring for the lead of GNU Backgammon in a session of such games, so the expected value of this
    lead does not exist (base of the exponential term > 1 for the math
    people, "oscillations too wild" for the non-math people).

    But let's assume for the sake of argument that my session of 10000 games
    with this cube strategy was rather an outlier, and that the "real" base
    of the exponential term is slightly smaller than 1, even if one beaver
    is allowed. Then the mutant strategy of doubling above 50 % GWC and
    taking above 0 % GWC (checker play like GNU Backgammon) will result in
    losing almost 0.7 (cube-normalized) points per game against GNU
    Backgammon.

    Case closed.

    Axel

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From MK@21:1/5 to Axel Reichert on Mon Apr 18 02:38:33 2022
    On April 16, 2022 at 3:36:20 AM UTC-6, Axel Reichert wrote:

    Axel Reichert <ma...@axel-reichert.de> writes:

    Case closed.

    Just for the record that I'm not ignoring this post in this thread.

    I have some things to say about it all but I'm standing aside for
    the moment in order to not be rude by getting ahead of the math
    PHD's here. My prediction is that we won't hear a peep from any
    of them, for the reasons even you should be able to guess, but I
    will patiently wait a to see... (Before slapping you some more ;)

    MK

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Nasti Chestikov@21:1/5 to All on Wed Aug 24 08:43:40 2022
    On Monday, 18 April 2022 at 10:38:34 UTC+1, MK wrote:
    My prediction is that we won't hear a peep from any
    of them, for the reasons even you should be able to guess, but I
    will patiently wait a to see... (Before slapping you some more ;)

    MK

    Yeah, cocksucking GnuDung fanboys soon went quiet.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)