Making slow progress on my digest of Axel's experiment
because I want to give a good background on my various
past arguments leading up to it and as it feels tedious, it
is easy for my to get side tracked on some specific topics.

For long years, I thought that mathematicians, gamblers,
bot developers, etc. were making more of backgammon
than what it really is. Today, I started thinking on this again
while doing yard work for several hours and kept wondering
if the complexity of backgammon is exaggerated because
it's harder to create a top level backgammon bot than bots
for chess and other games, simply due to its high branching
factor??

I started searching and reading about the topic... This Wiki
gives a decent overall idea about it and even has a table to
compare the complexity of tens of games.

https://en.wikipedia.org/wiki/Game_complexity

We know that backgammon has a higher average branching
factor (250?) than chess (35?) but chess is considered much
more complex (harder).

While trying to understand and compare the various factors
of complexity of games, I felt like it was as difficult to compare
economic factors of various countries, which can't give a good
picture by themselves or even by a few at a time, but all need
to be considered in retation to one another.

All of a sudden the word "inflation" rang in my head! Can the
high braching factor in backgammon lead to an inflation of
positions and decisions, each with a lower value as a unit
than the less numerous ones in chess (like currency exchange
rates)...??

I hope that my anology works and makes sense to at least
some of you because at this moment I don't know how else
to communicate my thoughts except by analogies. Maybe we
will find a common language later.

Who cares if there are so many more ways to get from Seattle
to New York in backgammon-land than in chess-land..? It just
makes it easier to recover from making the wrong turn in the
backgammon-land.

Who cares if the train ticket in backgammon-land cost much
more than in chess-land..? You are paying with "inflated"
backgammon dollars vs more valuable chess dollars.

Why should it be harder for a human to play backgammon
than chess because it is harder for bots to play backgammon
than chess..!?

Especially when the bots' early game equity esyimates are
totally bogus and worthless for anything. See my last XG
experiment with making the worst move at my first turn:

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

Because of the dice and luck, in backgammon no checker
or cube decision is as consequential as in chess or other
similar "skill games". In fact, depending on your opponents
next roll, your huge blunder may end up being a gift from
the sky. This is true only for that game but can be true for
many other similar moves, at a considerable ratio even in
four billion trials...

Frankly, I'm just beyond amazed that more humans can't
beat the bots decisively in meaningfully long sessions. Is
anyone even honestly trying?? (Not trying to play exactly
like bots, with a low PR, of course. You need to try playing
like me..:))

Okay, I think this should be enough to get the discussion
rolling, if any of you find this topic interesting.

MK

On June 8, 2022 at 2:58:43 AM UTC-6, wind said:

> I started searching and reading about the topic... This Wiki
> gives a decent overall idea about it and even has a table to
> compare the complexity of tens of games.
> https://en.wikipedia.org/wiki/Game_complexity

Still thinking about this afterwards, I noticed that the only
game of strategy there played with dice was backgammon

https://en.wikipedia.org/wiki/Backgammon

and tried to think of other such games.

I found out that a game we used the play as youngsters is
commonly called "ludo",

https://en.wikipedia.org/wiki/Ludo_(board_game)

as one of many variants of an old dice game called "pachisi"

https://en.wikipedia.org/wiki/Pachisi

which are apparently considered "complex race games", i.e.
requiring some skill, with backgammon variants "multiplex
race games", i.e. requiring more skill, and snakeandladders
type games "simple race games", i.e. requiring no skill.

Unfortunately, I couldn't fins anything about the branching
factor, etc. in ludo/pachisi. I wonder if any of our resident
mathematicians can figure it out based of number of pieces,
squares/points, possible positions, etc...??

I also remembered that some years ago even a chess variant
played with dice was talked about here but I don't remember
much about what the topics were, nor do I know about how it
is played. Are there people here with knowledge and interest
about it to contribute comments?

MK

On 6/12/2022 2:56 AM, MK wrote:
> I also remembered that some years ago even a chess variant
> played with dice was talked about here but I don't remember
> much about what the topics were, nor do I know about how it
> is played. Are there people here with knowledge and interest
> about it to contribute comments?

Here's one version of dice chess.

http://math.lfc.edu/~yuen/dicechess/index.html

---
Tim Chow

On Sunday, 12 June 2022 at 07:56:32 UTC+1, MK wrote:
>
> I also remembered that some years ago even a chess variant
> played with dice was talked about here but I don't remember
> much about what the topics were, nor do I know about how it
> is played. Are there people here with knowledge and interest
> about it to contribute comments?
>
> MK

https://www.tesco.com/groceries/en-GB/products/273767481

I played this a lot as a boy, oddly similar to backgammon.

On June 12, 2022 at 5:48:50 AM UTC-6, Tim Chow wrote:

> On 6/12/2022 2:56 AM, MK wrote:

>> Are there people here with knowledge and interest
>> about it to contribute comments?

> Here's one version of dice chess.
> http://math.lfc.edu/~yuen/dicechess/index.html

I was hoping for more personal experience type
comments than links which I can find myself but
maybe they will come later.

Since that site is Java based, I'll have to try it on
my other/test computer but I don't know what I
can get out of it for my not being a chess player
beyond knowing how to play and having played
at most a hundred times or so in my life. That's
why I'm seeking comments from decent/good
players.

Looking for other sites about it, I found the Wiki

https://en.wikipedia.org/wiki/List_of_chess_variants

and I was totally astounded that there are over 2000
chess variants, excluding who know how many more
"trivial" ones not worth mentioning. Wow! But I wonder
who, if any people, play any of those variants? Are they
perhaps invented just for the sake of it? Too see how
much more complex/difficult variants one can create
or in the opposite, to come up with ones easier to play?

Apparently there are even about a dozen variants of
"dice chess"... Again, who play these? Has anyone
here ever played "dice chess"?

I also searched if dice chess was played for gambling
but coudn't find anything. Nor could I find anything on
if it's played with a doubling cube, etc. Surely it wouldn't
be too hard for some gamblemathicians to concoct a
jackoffski equity formulas for chess positions.

Is dice chess harder or easier than regular chess?

If easier, then I would've expected that it would attract
some gamblers who aren't smart enough to play plain
chess but who want to brag about being smart enough
to gamble playing "games of skill"...

Is dice chess harder or easier than gamblegammon?

If either, why gamblechess isn't as popular? Especially
if harder, why don't we see any gamblegammon giants,
who may be too good for gamblegammon, step up to a
higher challenge of "skill gambling"...??

With chess being so much popular than backgammon,
gamblechess giants would have such a large pool of
potential preys.

MK

On June 12, 2022 at 8:25:17 AM UTC-6, Nasti Chestikov wrote:

> https://www.tesco.com/groceries/en-GB/products/273767481
> I played this a lot as a boy, oddly similar to backgammon.

Yes, I thought so too. There are backgammon variants
with both players moving in the same direction, etc. It
looks like such backgammon variants with more players
racing fewer pieces to run them off the board to safety.

Even though we knew it required some skill to win more,
we never made much of it beyond a friendly/family game
perhaps because the spirit wasn't to prove one-on-one
superiority of skill, etc.

I wonder though, if it's played by adults, with only two
players or two teams of two players, and if taken more
seriously, would we discover that it may actually require
much more skill than we think it does...?

I would be really curious to see some of the common
game complexity formulas applied to it, in order to see
where it ranks among race games requiring skill.

I have a feeling that by discovering that such games
require more skill than expected, we may discover that
the skill assumed to be needed in backgammon may be
less than what is exaggerated...??

MK

On 6/12/2022 9:11 PM, MK wrote:
> Apparently there are even about a dozen variants of
> "dice chess"... Again, who play these? Has anyone
> here ever played "dice chess"?

The version of dice chess that I linked to was created by
David Yuen, who used to be a consultant for my company. He
popularized it at my company and many people played it. The
reason he invented it was to "level the playing field" somewhat.
Two players whose skill level in chess is very different won't
have much fun playing because the stronger player will always
win easily. In dice chess the weaker player always has a chance.

I played it myself a few times but did not seriously attempt to
become skillful at it.

> Is dice chess harder or easier than regular chess?
>
> If easier, then I would've expected that it would attract
> some gamblers who aren't smart enough to play plain
> chess but who want to brag about being smart enough
> to gamble playing "games of skill"...

I don't know exactly what "harder" or "easier" means. As for
popularity---there are very few board games that become popular.
Probably every year, hundreds if not thousands of new board games
are invented, and many of them are probably (by some measure) just
as good as popular board games, but they don't become popular,
because there are so many other factors that drive popularity than
just the intrinsic qualities of the game.

---
Tim Chow

On 6/13/2022 3:55 PM, peps...@gmail.com wrote:
> On Monday, June 13, 2022 at 1:50:53 PM UTC+1, Tim Chow wrote:
>> On 6/12/2022 9:11 PM, MK wrote:
>>> Apparently there are even about a dozen variants of
>>> "dice chess"... Again, who play these? Has anyone
>>> here ever played "dice chess"?
>> The version of dice chess that I linked to was created by
>> David Yuen, who used to be a consultant for my company.
>
> Wow! You have a company!
> Why not plug it here?

I think it was C. S. Lewis who said that the word "my" can have a
range of connotations, ranging from "my boots" to "my phone number"
to "my neighborhood" to "my daughter" to "my boss" to "my country"
to "my God."

---
Tim Chow

On Monday, 13 June 2022 at 20:55:34 UTC+1, peps...@gmail.com wrote:

> Wow! You have a company!
> Why not plug it here?
>
> I might invest.
>
> Paul

You want to invest in a guy who drives fast cars recklessly around the US in the hope of getting pulled by the cops and, I believe the phrase is, "getting content" for his YouTube channel?

On 6/14/2022 1:15 PM, Nasti Chestikov wrote:
> On Monday, 13 June 2022 at 20:55:34 UTC+1, peps...@gmail.com wrote:
>
>> Wow! You have a company!
>> Why not plug it here?
>>
>> I might invest.
>>
>> Paul
>
> You want to invest in a guy who drives fast cars recklessly around the US in the hope of getting pulled by the cops and, I believe the phrase is, "getting content" for his YouTube channel?

Don't forget the big bucks I'm raking in because of Eraser.

---
Tim Chow

On Thursday, 16 June 2022 at 03:04:32 UTC+1, Tim Chow wrote:

> Don't forget the big bucks I'm raking in because of Eraser.
>
> ---
> Tim Chow

Yeah, I'm big enough to apologise for that one, the guy concerned has disappeared up his own backside so obviously not you.

How did Gumball 3000 go?

On June 13, 2022 at 6:50:53 AM UTC-6, Tim Chow wrote:

> The version of dice chess that I linked to was created
> by David Yuen ..... popularized it at my company and
> many people played it. The reason he invented it was
> to "level the playing field" somewhat. ..... In dice chess
> the weaker player always has a chance.

Thanks fro this interesting info. If you feel like like it,
can you tell more about what people tought about it,
if it's still played, etc. especially "by what factor did it
level the field"? (in terms of ELO, for example)

But what I would really like to further discuss is "his
reason" which seems to be the generally accepted
about difficulty of games, i.e. injecting elements of
luck makes them easier.

However, I wonder if the opposite may also be true?
Can adding luck to an easy skill game, (i.e. playing a
simple, solved, skill game like tic-tac-toe with dice),
make it actually harder, more complex/difficult? Any
thoughts??

Similarly, can eliminating luck/dice from games like
backgammon also make them even harder, more
complex/difficult?

I don't know but I would guess that some people must
have already invented diceless backgammon variants.
Does anyone know about any existing ones?

A variant closest to chess would be played by moving
one piece per turn but I wonder if it would be playable.
Any ideas??

I probably wouldn't like such backgammon variants
any more than chess that I don't like or play but I find
these subjects interesting to discuss anyway.

MK

MK <murat@compuplus.net> writes:

> Can adding luck to an easy skill game, (i.e. playing a simple, solved,
> skill game like tic-tac-toe with dice), make it actually harder, more
> complex/difficult? Any thoughts??

Not if the full game tree is known (tic-tac-toe) to all players,
IMHO. For chess (game tree not fully known) I assume that more exotic
parts of the game tree might get visited more often (e.g. weird
openings). No thoughts about the complexity.

> Similarly, can eliminating luck/dice from games like backgammon also
> make them even harder, more complex/difficult?

Depends on the choices that are left. Vice versa, just inventing a game
with a huge will not necessarily make this interesting.

Axel

MK <murat@compuplus.net> writes:

> I was totally astounded that there are over 2000 chess variants,
> excluding who know how many more "trivial" ones not worth
> mentioning. Wow! But I wonder who, if any people, play any of those
> variants? Are they perhaps invented just for the sake of it? Too see
> how much more complex/difficult variants one can create or in the
> opposite, to come up with ones easier to play?

When a colleague told me about a tournament of

https://en.wikipedia.org/wiki/Fischer_random_chess

I read about it a bit. The bottom line of Fischer's idea for me is to
(at least initially) compensate for the smaller (compared to backgammon)
branching factor and quickly get into uncharted terrain. To have 960
different starting positions is roughly similar to what backgammon has
after the reply roll.

Axel

MK <murat@compuplus.net> writes:

> I found out that a game we used the play as youngsters is
> commonly called "ludo",
>
> https://en.wikipedia.org/wiki/Ludo_(board_game)
>
> as one of many variants of an old dice game called "pachisi"

Kids here in Germany probably all know

https://en.wikipedia.org/wiki/Mensch_%C3%A4rgere_Dich_nicht

, another variant. When I learned backgammon and the neural nets playing
it, I was wondering whether adults tend to underestimate Ludo et al.'s
complexity and whether an AI would show us how to play this REALLY well.
I never explored this further, though.

> I couldn't fins anything about the branching factor, etc. in
> ludo/pachisi. I wonder if any of our resident mathematicians can
> figure it out based of number of pieces, squares/points, possible
> positions, etc...??

The branching factor should be rather small (referring to the German
variant): On average you have 2.5 pieces and one of 6 numbers on the
die, which makes for 15 as a positional branching factor. 2.5 moves to
consider. Very, very roughly, of course.

Axel

MK <murat@compuplus.net> writes:

> Why should it be harder for a human to play backgammon than chess
> because it is harder for bots to play backgammon than chess..!?

Because the mathematical difficulty of the game are independent of
whether hard- and software or wetware is used.

> Because of the dice and luck, in backgammon no checker or cube
> decision is as consequential as in chess

Unless you are doubled out. Sure, backgammon has swings even with
competent players acting, different from chess.

Axel

On 6/18/2022 9:36 PM, MK wrote:
> Thanks fro this interesting info. If you feel like like it,
> can you tell more about what people tought about it,
> if it's still played, etc. especially "by what factor did it
> level the field"? (in terms of ELO, for example)

The purpose was to create a social, rather than a competitive,
game. So of course we did not try to compute Elo ratings.

> But what I would really like to further discuss is "his
> reason" which seems to be the generally accepted
> about difficulty of games, i.e. injecting elements of
> luck makes them easier.

I don't know what "easier" means.

Here's one possible definition. Take the best player in the
world; call that player A. Find someone, B, who can consistently
achieve a score of 25% against A. Then find someone, C, who can
consistently achieve a score of 25% against B. Continue until
you can't find any more levels. The number of levels is the
"difficulty" of the game.

Is that what you mean?

Here's a different possible definition. The difficulty of a game
is measured in terms of how much effort needs to be expended to
find the best move in a given position. This is not the same as
the previous definition. It could be that the difference in quality
between the best move and the worst move in every position is tiny,
and that the game is dominated by luck. But if you really want to
find the very best move, it will require enormous effort.

Yet another definition is that the difficulty of a game is measured
in terms of how much effort needs to be expended to find *some*
reasonable move. We can invent a "difficult" game in this sense
just by making the rules extremely complicated, so that almost
nobody can understand them, and only a few world experts even have
the ability to find a move that obeys the rules. I don't think that
this is what you have in mind, but even if we don't go to this
extreme, it's important to recognize that you can make it "difficult"
to play a game just by making the rules inscrutable.

---
Tim Chow

Timothy Chow <tchow12000@yahoo.com> wrote:
>On 6/18/2022 9:36 PM, MK wrote:
> ...
>> But what I would really like to further discuss is "his
>> reason" which seems to be the generally accepted
>> about difficulty of games, i.e. injecting elements of
>> luck makes them easier.
>
>I don't know what "easier" means.
> ...

Related:
| ...
| In any competition including academic tests, athletic
| events, and company management where there is an element of
| luck that causes performances to be an imperfect measure of
| ability, there is an important difference between
| competitions among people with high ability and
| competitions among people of lesser ability. If four work
| friends play a round of golf and one player is much better
| than the others, the winner is determined mostly by
| ability. If four of the top golfers in the world play a
| round of golf, the winner is determined mostly by luck.
| This is the paradox of luck and skill: the more skilled the
| competitors are, the more the outcome is determined by
| luck.
| ...
<https://mindmatters.ai/2019/08/the-paradox-of-luck-and-skill/>
(Written by Econ Prof, Gary Smith)

--bks

On 6/19/2022 3:55 AM, Axel Reichert wrote:
> When a colleague told me about a tournament of
>
> https://en.wikipedia.org/wiki/Fischer_random_chess
>
> I read about it a bit. The bottom line of Fischer's idea for me is to
> (at least initially) compensate for the smaller (compared to backgammon)
> branching factor and quickly get into uncharted terrain. To have 960
> different starting positions is roughly similar to what backgammon has
> after the reply roll.

Chess960 tournaments are still occasionally held at high levels.
I think there are two ideas behind it. The first is to combat the
intense memorization of "opening books" that dominates top-level
chess nowadays. The second is to generate interesting positions
that would never occur in ordinary chess.

It's hard for a new game to catch on, so I think that's the main reason
Chess960 hasn't caught on more. A secondary reason is that Chess960 is
still basically chess, so the best Chess960 players are pretty much the
best chess players. If it were different enough that the top Chess960
players were different from the top chess players, then maybe that would
help increase its popularity among spectators.

---
Tim Chow

On Sunday, June 19, 2022 at 2:19:15 PM UTC+1, Tim Chow wrote:
> On 6/19/2022 3:55 AM, Axel Reichert wrote:
> > When a colleague told me about a tournament of
> >
> > https://en.wikipedia.org/wiki/Fischer_random_chess
> >
> > I read about it a bit. The bottom line of Fischer's idea for me is to
> > (at least initially) compensate for the smaller (compared to backgammon)
> > branching factor and quickly get into uncharted terrain. To have 960
> > different starting positions is roughly similar to what backgammon has
> > after the reply roll.
> Chess960 tournaments are still occasionally held at high levels.
> I think there are two ideas behind it. The first is to combat the
> intense memorization of "opening books" that dominates top-level
> chess nowadays. The second is to generate interesting positions
> that would never occur in ordinary chess.
>
> It's hard for a new game to catch on, so I think that's the main reason
> Chess960 hasn't caught on more. A secondary reason is that Chess960 is
> still basically chess, so the best Chess960 players are pretty much the
> best chess players. If it were different enough that the top Chess960
> players were different from the top chess players, then maybe that would
> help increase its popularity among spectators.
>
> ---
> Tim Chow

In pro darts, they tried a new version where where there was a segment nearer the centre of
the board than the triple region but further from the centre than the 25-scoring outer ring.
In this region, the scores were multiplied by 4, rather than 3.
Not only did this version fail to catch on, but, at the highest levels, players often ignored the possiblity
of hitting quadruples and simply ignored those sectors.
Would you argue that this darts-with-quadruples hasn't caught on more, mainly because it's hard
for a new game to catch on, and that a secondary reason is that the game is basically just darts so the
best darts-with-quadruples players are pretty much the best darts players?
If you would not make (or agree to) this argument, I might be tempted to criticise you for your glaring
failure to maintain a consistent standard when comparing different games.

Also there was a rule trialled in Challenger pro tennis tournaments where serves that hit the net were
treated normally, and not given a let.
What is your theory on why this version failed to catch on?

Paul

On June 19, 2022 at 12:58:44 AM UTC-6, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:
>> Can adding luck to an easy skill game, (i.e. playing a simple,
>> solved, skill game like tic-tac-toe with dice), make it actually
>> harder, more complex/difficult? Any thoughts??

> Not if the full game tree is known (tic-tac-toe) to all players,
> IMHO. For chess (game tree not fully known) I assume that
> more exotic parts of the game tree might get visited more
> often (e.g. weird openings). No thoughts about the complexity.

The "weird openings" would result mostly from "forced moves"
which may be really bad or losing moves beyond just weird and
thus may require more skill to survive those? I'm just bouncing
ideas around...

>> can eliminating luck/dice from games like backgammon
>> also make them even harder, more complex/difficult?

> Depends on the choices that are left. Vice versa, just inventing
> a game with a huge will not necessarily make this interesting.

I didn't understand what you mean here. Can you elaborate?

MK

On June 19, 2022 at 1:55:35 AM UTC-6, Axel Reichert wrote:

> https://en.wikipedia.org/wiki/Fischer_random_chess

> I read about it a bit. The bottom line of Fischer's idea for
> me is to (at least initially) compensate for the smaller
> (compared to backgammon) branching factor and quickly
> get into uncharted terrain. To have 960 different starting
> positions is roughly similar to what backgammon has
> after the reply roll.

I don't think you can call various initial setups as branching.

Black and white pieces are always mirrored. Games haven't
started yet and no branchings have occured yet. In fact, all
of those 960 starting positions can be considered individual
chess variants with substancially similar branching factors.

MK

On June 19, 2022 at 2:12:11 AM UTC-6, Axel Reichert wrote:

> MK <mu...@compuplus.net> writes:
>> https://en.wikipedia.org/wiki/Ludo_(board_game)

> Kids here in Germany probably all know
> https://en.wikipedia.org/wiki/Mensch_%C3%A4rgere_Dich_nicht

In Turkish, it was called "Kızma birader" ("Don't get angry, brother")..

> When I learned backgammon and the neural nets playing it,
> I was wondering whether adults tend to underestimate Ludo
> et al.'s complexity and whether an AI would show us how to
> play this REALLY well. I never explored this further, though.

Me too. So maybe it's time we explore a little. There are online
Ludo servers and surely there must be Ludo apps/bots. I will
try to play some when I find time.

> The branching factor should be rather small (referring to the
> German variant): On average you have 2.5 pieces and one of
> 6 numbers on the die, which makes for 15 as a positional
> branching factor. 2.5 moves to consider. Very, very roughly,
> of course.

Can you expand on how you came up with these? There are 16
men and 40 squares to land on. As kids, we mostly tried to run
one man at a time, happily hopping from square to square. But
what if the optimum strategy is to run all your men at the same
time. Almost half of the 40 squares would be occupied, with a
lot of men constantly sent back home.

What if only two players or two teams of two players control 8
men each, on either opposite or adjacent "home" positions?

I can't offer any branching numbers, etc. but I have the feeling
that it can get pretty darn complicated...

MK

On June 19, 2022 at 2:20:15 AM UTC-6, Axel Reichert wrote:

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

>> Why should it be harder for a human to play
>> backgammon than chess because it is harder
>> for bots to play backgammon than chess..!?

> Because the mathematical difficulty of the
> game are independent of whether hard- and
> software or wetware is used.

I would expect a better answer/comment than
this from you, if you were to make an effort to
understand the question before indogtrainedly
preaching from the book... :(

MK

On June 19, 2022 at 7:12:46 AM UTC-6, Tim Chow wrote:

> I don't know what "easier" means.

How about "requiring less skill"?

> Here's one possible definition. Take the best player in
> the world; call that player A. Find someone, B, who can
> consistently achieve a score of 25% against A. Then find
> someone, C, who can consistently achieve a score of 25%
> against B. Continue until you can't find any more levels.
> The number of levels is the "difficulty" of the game.
> Is that what you mean?

I wasn't trying to "mean" anything. I was just inquiring.

This makes better sense to me than the approach used
in the Wiki page I previously gave a link to. Being based
on actual results, it seems more realistic than the ones
based on theoretical/mathematical extrapolations.

Did you come up with this? If so, is 25% a random pick?
Or is this actually used to rank players in some games?

Does anyone know how the player levels in backgammon
bots are determined? Using a similar method? Elsewise?

I suppose the number of levels would be proportionate
to the amounts of skill needed in various games. Also,
they would be more precise and predictable in higher
skill games.

It seems like chess has 16 levels while gamblegammon
has 4 tournament brackets: small, medium, large, giant. :)

I think this says enough about how much skill is really
needed in backgammon/gamblegammon...

> Here's a different possible definition. The difficulty of a
> game is measured in terms of how much effort needs
> to be expended to find the best move in a given position.

I stopped reading this definition here because unless the
game is solved, there's no way to find all the "best" moves.

I like the previous method better as the most practical for
popular games with enough players to deserve rankings.

There is no point in trying to measure the complexity of
some obscure games/variants with "inscrutable" rules
that are only played rarely and/or by handfuls of people.

MK