SET's and MET's for checker and cube play in backgammon and gamblegammon

[MK]

I believe that winning best of three or of five shorter
matches would be harder to win than a single longer
match, but I have no idea by how much, since there
have been no experiments done on the subject.

After writing a couple of paragraps about it in this:

https://groups.google.com/g/rec.games.backgammon/c/vFOtjr8HYt8/m/JaTv4S1-AAAJ

I decided to try doing my own experiments but found
them impossible to do with neither Ex-G nor Noo-BG.

A - First, let's talk about classic (cubeless) backgammon.

The most common match length is a 5-pointer. A real
challenge is settled by a best of three 5-pointers (or in
the alternative by a single 7-pointer if time is limited).

There is no way to test if winning a single 7-pointer or
even a 9-pointer would be as hard as winning a best of
three 5-pointers using Ex-G or Noo-BG because they do
offer a cubeless variant of gamblegammon but do not
truely implement backgammon, (i.e. allowing doubles
in the opening rolls, without 3-point wins, etc.)

An AI bot, (at least as AI as Ex-G and Noo-BG), called
Palamedes can play classic backgammon (and several
variants of it) and can be downloaded from:

https://nikpapa.com/Palamedes/index.html

Unfortunately, it can only play one game at a time and
lacks the bot-vs-bot play capability. Thus, I can't use it
for my experiment either.

However, I applaud Nikolaos Papahristou's efforts to
develop such a bot and hope that he can find time to
further improve it. I will offer a couple of suggestions
here, while introducing the new concepts of SET's and
MET's, (Set Equity Tables and Match Equity table), for
checker play decisions.

In a best of three match, a match really becomes a set
of games and then several sets constitute a match. For
now, I am borrowing "table" from gamblegammon but
I have no clear idea on how adjusting checker decisions
according to score can be implemented in bots, either
using tables or otherwise. This is an area where humans
can exert a decisive superiority on currently existing bots.

During the first set of a best of three, the SET would be
the same for both sides.

Afterwords, the winner of the first set will have a great
advantage and thus we would need two different SET's
for leading and trailing players during the second set.

A tie-breaker set can use the same SET as the first one.

To carry the concept even further, in a best of five match
there would be even more SET's for intermediary scores.

A MET will become a higher level modifier table of SET's.

I hope that such a bot will arrive soon for us all to enjoy.
Until then, if a human player (i.e. Murat :) claims that he
can adjust his checker play according to the score, there
is no way to prove or disprove it...

B - Now, let's talk about the cubeful gamblegammon.

In gamblegammon, we would need SET's and MET's not
only for checker decisions (as in backgammon), but also
for cube decisions.

Since it's probably impossible to implement checker play
SET's and MET's in bots any time soon, let's just ignore it
for now and focus on cube play.

Fortunately, in gamblegammon there are already MET's
for cube play but unfortunately, (even without debating
their accuracy), they won't work in "best of N" situations
because initial, intermediary and final sets would all need
different SET's and then what we call MET today would
become a MET of the SET's.

BTW!: this brings to mind the feature suggestion of a few
years' ago that Noo-BG should allow a different MET for
each player, (which would allow comparing MET's), but it
was never implemented. I would have liked to use it to see
how a "mutant MET" would measure up, for example... ;)

Back to our subject. So again, currently there is no way to
test if winning a cubeful 25-pointer would be as hard as
winning best of three 13-pointers, (or best of five shorter
9-pointers), using neither Ex-G nor Noo-BG.

I guess nobody else is as anxious as I am to challenge the
super-human bots to best of three or best of five matches.

Consequently, it is unlikely that there will be any incentive
to improve the bots for that purpose any day soon... :(

C - Some related spontaneous thoughts.

The fact that MET's to adjust cube play to score have been
feasible to implement, even if for simple matches, (i.e. best
of one), while nobody has never before even talked about
the idea of similarly adjusting checker play to score, shows
how the so called "cube skill" is simpler and secondary to
checker skill, as well as being "external" to the game itself.

It would be very interesting to hear any suggestions about
how the implemention of checker play SET's and MET's can
be approached, assuming that it would be possible with our
current technology and knowledge??

MK

[Philippe Michel]

On 2023-02-17, MK <mu...@compuplus.net> wrote:

> I will offer a couple of suggestions
> here, while introducing the new concepts of SET's and
> MET's, (Set Equity Tables and Match Equity table), for
> checker play decisions.

The "new concept of SET" has been known as Pascal's triangle for almost
400 years. But its use is limited to settle unfinished sets (history
reports that Pascal discovered it when working out this problem for a
gambler friend).

As far as play decisions go, this is irrelevant. Whatever the set score,
your goal is to win the match you are playing.

What make backgammon METs and their influence on play more complex is
that points inside a match can be won by lots, not only one by one.

> I believe that winning best of three or of five shorter
> matches would be harder to win than a single longer
> match, but I have no idea by how much, since there
> have been no experiments done on the subject.

> There is no way to test if winning a single 7-pointer or
> even a 9-pointer would be as hard as winning a best of
> three 5-pointers using Ex-G or Noo-BG because they do
> offer a cubeless variant of gamblegammon but do not
> truely implement backgammon, (i.e. allowing doubles
> in the opening rolls, without 3-point wins, etc.)

There is no way to test matches played on non-implemented rules,
obviously, but in GNUbg there are some hints in the unbalanced jac050
and jac100 METs on how it may work out in modern backgammon.

There are 3 ways to win a best of three contest: WW, WLW, LWW. If your
winning chance in a single match is p, that amounts to p*p + p*(1-p)*p +
(1-p)*p*p, or 3*p^2-2*p^3.

A quick sanity check confirms that it is 0, 0.5 and 1 if p is 0, 0.5 and 1.

The math for best of five is similar but a bit more complex.

The jac100 tables suggests that a player 100 Elo stronger would win a 5
points match 56.1% of them time. For a best of 3 5-pointers that would
imply a 59.1 winning rate.

The same table gives 57.4%, 58.5% and 59.4% for a single 7-, 9- or
11-points match, so best of 3 5-pointers would be about as selective as
a single 11-points match.

The result may be slightly different if the skill difference is higher
or lower or if the jac tables are not that accurate after all.

[MK]

On February 23, 2023 at 2:31:10 PM UTC-7, Philippe Michel wrote:

> On 2023-02-17, MK <mu...@compuplus.net> wrote:

>> here, while introducing the new concepts of SET's
>> and MET's, (Set Equity Tables and Match Equity
>> table), for checker play decisions.

> The "new concept of SET" has been known as
> Pascal's triangle for almost 400 years. But its
> use is limited to settle unfinished sets (history
> reports that Pascal discovered it when working
> out this problem for a gambler friend).

I don't think you are grasping the depth of what I'm
talking about but I do appreciate your participating
in the discussion and I will make twice the effort to
better explain in more detail.

Honestly, I had never heard of Pascal's triangle as I
have no interest in game/gambling theories beyond
backgammon. After reading about it, I can see that
I wasn't missing anything and that it doesn't apply to
my subject here.

It may be useful for binary decisions such as coin
tosses or cube actions but not for multiple-choice
decisions like checker moves.

> As far as play decisions go, this is irrelevant.
> Whatever the set score, your goal is to win the
> match you are playing.

This is not true at all. Even since the Jellyfish days,
adjusting one's checker play to the opponent's skill
level and score in simple (i.e. best of one) matches
has been talked about but never discussed in depth.

It's accepted that humans may be capable to doing
this, developing bots capable of the same seemed
beyond possible as ultimately complex.

Here, I'm going beyond scoring by simple matches
and proposing a "best of N" scoring system similar
to tennis. With this the challenge of developing bots
capable of it becomes exponentially more difficult
because it will require not a single-tier but a two-tier
adjusting system which is what I'm intruducing here
and I'll explain more further down.

> What make backgammon METs and their influence
> on play more complex is that points inside a match
> can be won by lots, not only one by one.

Cube actions aren't "complex" but the consequences
can be more drastic. A classic 5-point backgammon
match can't be shorter than 3 games (i.e. one player
winning 2 gammons and 1 single game vs. zero) but
even a 25-point gamblegammon match can be over
in just 1 game with the cube at 32 or a gammon and
the cube at 16. That's why I keep saying that the cube
magnifies luck (without increasing the so-called cube
skill but decreasing the checker skill).

Also, as I said before, I have no clear idea at all about
how bots can be made to adjust checker decisions. It
may be done through AI training for match play, which
would require immense amounts of computing power,
of through some magically ingenious system similar
to MET's. I personally don't think it can be done using
"tables" but I use the word to transition from a concept
and terminology that you are all familiar with to a new
concept that I don't know what words to use for. Once
anyone of you understand what I'm trying to explain,
feel free to make suggestions...

>> I believe that winning best of three or of five shorter
>> matches would be harder to win than a single longer
>> match, but I have no idea by how much, since there
>> have been no experiments done on the subject.

> ... in GNUbg there are some hints in the unbalanced

> jac050 and jac100 METs on how it may work out in
> modern backgammon.

They won't help at all since my subject here has nothing
to do with skill differences between players (which may
be discussed as a separate subject and/or in relation to
my subject here about "score differences").

> There are 3 ways to win a best of three contest: WW,
> WLW, LWW. If your winning chance in a single match
> is p, that amounts to p*p + p*(1-p)*p + (1-p)*p*p, or
> 3*p^2-2*p^3.
> A quick sanity check confirms that it is 0, 0.5 and 1 if
> p is 0, 0.5 and 1.
> The math for best of five is similar but a bit more complex.

Whether for best of three or for best of five, your math is
only good enough for simple, binary decisions based on
a single condition, such a cube action based on the equity
of a position but won't work for checker decisions when
each "p" can be the result of multiple choices.

> The jac100 tables suggests that a player 100 Elo stronger
> would win a 5 points match 56.1% of them time. For a best
> of 3 5-pointers that would imply a 59.1 winning rate.
>
> The same table gives 57.4%, 58.5% and 59.4% for a single
> 7-, 9- or 11-points match, so best of 3 5-pointers would be
> about as selective as a single 11-points match.
>
> The result may be slightly different if the skill difference is
> higher or lower or if the jac tables are not that accurate
> after all.

In addition that skill difference being irrelevant here, what
you are failing to see is that the weights of all 5-pointers in
a best of three or a best of five sets are not the same.

In your WW, WLW, LWW combinations, after the first win
(regardless of which side wins), the losing side must not
lose again! while the winning side can afford to lose next
and then win again.

I will concede that you guys understand gambling better
than I do but you don't understand bakgammon (not even
gamblegammon) as well as you understand gambling.

The "strategies" to maximize your chances of winning and
to minimize your chances of losing are not the same.

Thus, for the second 5-pointer in a best of three match,
you need two different strategies for the two sides, (as
implemented in bots using "tables" or however else).

I hope that I don't come accross as putting you down for
not understanding what I'm talking about and I hope that
you and others will try to continue this discussion as best
as you all can.

If it may make you all feel better, let me tell you that I don't
think even giants like mocky, micky, sticky can't wrap their
heads around this subject or even if they can understand it,
they can't apply it (i.e. adjust their checker plays according
to the score in a simple best of one match, let alone in a
best of three or best of five matches).

Currently we have no ways of settling such claims but we
can have them try their bests at it, put the results in a time
capsule, bury it and wait until we have "AI enough" bots to
verify. I'm willing to bet everything I own that the results
will prove that no gamblegammon giant had even a clue
about it as of the year 2023 of our good lord... ;)

MK

[MK]

On February 25, 2023 at 3:11:02 AM UTC-7, MK wrote:

> Also, as I said before, I have no clear idea at all about
> how bots can be made to adjust checker decisions. It
> may be done through AI training for match play, which
> would require immense amounts of computing power,
> of through some magically ingenious system similar
> to MET's. I personally don't think it can be done using
> "tables" but I use the word to transition from a concept
> and terminology that you are all familiar with to a new
> concept that I don't know what words to use for. Once
> anyone of you understand what I'm trying to explain,
> feel free to make suggestions...

Perhaps an illustration using your own concepts may
help. Let's take one the of common subjects of your
position discussions: "pay now or later".

Depending on the score, (strictly focusing on score and
ignoring skill difference, etc. for now), it may make sense
to pick one of a number of legal moves as the best one
to pay now, or to pay later. But how do you make a bot
to make that decision?

One way could be to "color code" the moves according
to their "nature", (i.e. having different equities for various
strategies). In other words, a move at a certain position
may be marked as "pay now", (if that's what the strategy
prescribes), or "pay later", (if that's what another strategy
prescribes), depending on whether the player is trying to
maximize his wins or trying to minimize his losses, let's
say as the player who has won or lost the first 5-pointer
in a set of best of three match.

Okay, I think I have done my best effort to explain myself.
Anyone who hasn't understood what I mean by now, will
never understand. I'll let it be whatever it is...

MK