A new concept in game complexity, from me: "branching factor inflation".

[MK]

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

[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

[Timothy Chow]

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

[Nasti Chestikov]

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.

[MK]

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

[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

[Timothy Chow]

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

[peps...@gmail.com]

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 might invest.

Paul

[Timothy Chow]

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

[Nasti Chestikov]

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?

[Timothy Chow]

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

---
Tim Chow

[Nasti Chestikov]

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?

[MK]

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

[Axel Reichert]

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.

> 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

[Axel Reichert]

MK <mu...@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

[Axel Reichert]

MK <mu...@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

[Axel Reichert]

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.

> 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

[Timothy Chow]

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

[Bradley K. Sherman]

Timothy Chow <tchow...@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

[Timothy Chow]

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

[peps...@gmail.com]

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

[MK]

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

[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

[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

[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

[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

[MK]

On June 19, 2022 at 7:19:15 AM UTC-6, Tim Chow wrote:

> A secondary reason is that Chess960 is still
> basically chess, so the best Chess960 players
> are pretty much the best chess players.

This made me look at game complexity (i.e. skill
required vs luck) from yet a different perspective:
"how long a game lasts on the average".

Wiki says: "10 to 60 minutes for casual games,
10 minutes (fast chess) to 6 hours or more for
tournament games" both for regular chess and
chess960.

For backgammon it says "5 to 60 minutes".

For Ludo, Mensch ärgere Dich nicht, etc. it says
"about 30 minutes".

I don't know how they came up with these but
they seem reasonable to me.

In chess, the low 10 minutes is for the variant
"fast chess" or if a huge skill difference between
the players leads to a quick defeat.

In backgammon, the low 5 minutes is also and
only for a variant "gamblegammon" played with
cube. No cubeless backgammon game will be
over in 5 minutes even with big skill differences
between the players or if played fast by equally
skilled players.

Just the act of rolling the dice uses up time in
Ludo also which is probably why it doesn't end
in 5 to 10 minutes even when played between
unskilled children.

Clearly gamblegammon with the cube is more
of a game of luck than even Ludo...

MK

[Timothy Chow]

On 6/21/2022 9:57 PM, MK wrote:
> 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?

I did not come up with this. I don't know who first came up with
it, but it dates back at least to Bill Robertie in 1992.

https://web.archive.org/web/20190112044437/https://thegammonpress.com/comparing-games-skill-chance/

The concept is also discussed in the book "Characteristics of Games,"
by Elias, Garfield, and Gutschera, where they call this measure the
"length of the skill chain." The number 25% is a random pick; it's
what Robertie used, but Elias et al. use 40%. Actual ranking of players
is always done by something like the Elo system, in every game that I
am aware of.

Elias et al. make some simple but important observations. The first
is that the length of the skill chain depends not just on the game
itself, but the community of players. Tic-tac-toe has a longer skill
chain among kids than among adults. For games like chess, the skill
chain may change over time as people learn more about the game and
the size of the playing population changes.

Another point is that the win percentage of Player A over Player B
depends on the duration of a playing session. The more skillful
player is going to have a better chance of winning a best-of-5 match
than a single game. So if Game 1 has a longer skill chain than Game 2,
we may be able to change that by replacing Game 2 by best-of-5 matches
of Game 2. This again shows that the length of the skill chain is not
an intrinsic property of the game, but depends on "external" factors
such as how long people are willing to spend on a playing session.

---
Tim Chow

[Timothy Chow]

On 6/21/2022 10:39 PM, MK wrote:
> This made me look at game complexity (i.e. skill
> required vs luck) from yet a different perspective:
> "how long a game lasts on the average".

In the book, "Characteristics of Games" by Elias, Garfield, and
Gutschera, the duration of a game is the very first characteristic
of a game that they consider. This is not an accidental choice,
since often the very first question a new player has when offered
the opportunity to play a game is, "How long will the game last?"

I think you'll find a lot of interesting stuff in this book.

---
Tim Chow

[Axel Reichert]

b...@panix.com (Bradley K. Sherman) writes:

> 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)

Add-on: If two clueless backgammon players compete, the outcome is also
determined mostly by luck.

Best regards

Axel

[Axel Reichert]

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

[Ludo, Pachisi, ...]

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

I imagine that the "terrain", as perceived by an optimizer, is very
flat, meaning that the equity difference between the (few) possible
moves will be tiny, similar to long races such as this:

The score (after 0 games) is: gnubg 0, axel 0
Move number 1: axel to play 11

GNU Backgammon Position ID: sG+3AACwb7cAAA
Match ID : cIkEAAAAAAAA
+24-23-22-21-20-19------18-17-16-15-14-13-+ O: gnubg
| O O | | O O O O | 0 points
| O O | | O O O |
| O | | O |
| O | | |
| O | | |
| |BAR| |v (Cube: 1)
| X | | |
| X | | |
| X | | X |
| X X | | X X X | Rolled 11
| X X | | X X X X | 0 points
+-1--2--3--4--5--6-------7--8--9-10-11-12-+ X: axel
Pip counts: O 106, X 106

There are more than 40 moves within 0.01 equity of the best move.

Best regards

Axel

[MK]

On August 13, 2022 at 7:21:13 AM UTC-6, Axel Reichert wrote:

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

> [Ludo, Pachisi, ...]

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

> I imagine that the "terrain", as perceived by an
> optimizer, is very flat, meaning that the equity
> difference between the (few) possible moves
> will be tiny, similar to long races such as this:

I will argue the exact opposite but let's talk about
your backgammon example first.

> GNU Backgammon Position ID: sG+3AACwb7cAAA
> Match ID : cIkEAAAAAAAA

> There are more than 40 moves within 0.01 equity
> of the best move.

Thank you, thank you, for this example perfectly
supporting my argument about "branching factor
inflation" in backgammon.

I categorically question the accuracy of all equity
estimates but 0.01 here meaning tiny/negligeable
difference suits me just fine.

There you have not 40 branches but 40 twigs and
your opponent has also 40 twigs... I wonder how
many possible positions there will be until the end
of this game. A few hundreds? A few thousands?
Gods forbid, even more?

Now, on to Ludo. As I mentioned before, there are
40 squares on the board with the possibility of all
16 men being on the board.

1) Considering that all 4 players are opponents of
one another, this is would be a very crowded board.

2) Each player may adopt a strategy of having 1, 2,
3 or 4 of his men on the board. Thus, every other
player needs to adjust his own strategy according
to the combination of startegies adopted by the
other 3 players.

3) You can't stack your pieces to make points to
save your blots or block your opponents. Every
piece on the board is a blot, ripe for picking by
any of the 3 opponents.

4) After making your move, you have to wait 3 turns
until it's your turn again. Unlike once in backgammon,
3 times the chance of getting it.

5) Even if you have to choose between moving one
of two pieces to either follow one opponent of to
jump ahead of another opponent will be much more
than a tiny equity difference.

6) I'm sure we could come up with many more
arguments for the complexity of Ludo...

I totally understand your obsession with cube skill
theories, equities accurate to several decimals, etc.
in gamblegammon and your need to exaggerate the
skill level needed in order to make yourself feel better
about not being just an addicted gambler settling for
an inferior skill game than chess, etc. because that's
all you can handle...

I bet it would really upset you if Ludo proved to need
more skill than gamblegammon but why should you
even worry about it? It's also a game of luck played
by rolling a die. You can concoct a "doubling cube
theory for Ludo" using tinyly accurate equities and
then just switch to gambling with gambleludo... ;)

MK

[Axel Reichert]

Just because equity differences between moves are tiny (think "greedy"
bear-off versus optimal bear-off) does not mean that the game is a
difficult and skillful game.

Imagine a toin coss experiment where prior to the toss the player has to
skillfully select (by visual examination) one of 11 imperfect coins
(yielding heads with 0.495, 0.496, ..., 0.504, 0.505 probability,
respectively). Then the coin is tossed, head wins.

Skillful? Maybe.
Dominated by luck? For sure.
Boring? You can bet.

Most stages of Ludo to me fit this picture.

Axel

[MK]

On August 28, 2022 at 4:25:09 AM UTC-6, Axel Reichert wrote:

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

>> I bet it would really upset you if Ludo proved to
>> need more skill than gamblegammon

> Just because equity differences between moves are tiny
> (think "greedy" bear-off versus optimal bear-off) does not
> mean that the game is a difficult and skillful game.

I don't see what are you trying to say with this comment.

In the previous posts, regarding Ludo you said: "the equity
difference between the (few) possible moves will be tiny";
and I responded: "I will argue the exact opposite".

Later in my reply, talking about equity difference in Ludo, in
paragraph #5, I said: "Even if you have to choose between

moving one of two pieces to either follow one opponent
of to jump ahead of another opponent will be much more
than a tiny equity difference."

> Imagine a toin coss experiment where.....

No need to imagine coin tossess. You had already given
an actual gamblegammon example using a position with

more than 40 moves within 0.01 equity of the best move.

Now let's apply your questions/answers to that example:

> Skillful? Maybe.
> Dominated by luck? For sure.
> Boring? You can bet.

I totally agree that these are true about gamblegammon
way too often. And I would also say more often than Ludo.

I was trying to argue that unlike in gamblegammon where
positions with more than 40 moves within 0.01 equity of
the best move are common, (making it a "Maybe skillful",
"Dominated by luck" and "Boring" game), in Ludo no such
"inflation of inconsequential decisions" would occur and
any equity differences between available moves wouldn't
be as "tiny" (i.e. 0.01) as gamblegammon.

> Most stages of Ludo to me fit this picture.

To each his opinion...

MK

[Axel Reichert]

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

> On August 28, 2022 at 4:25:09 AM UTC-6, Axel Reichert wrote:
>
>> Skillful? Maybe.
>> Dominated by luck? For sure.
>> Boring? You can bet.
>
> I totally agree that these are true about gamblegammon
> way too often. And I would also say more often than Ludo.

Then please try your luck in rec.games.board, just around the corner.

Axel

[MK]

You made many empty, unnecessary, unproductive
comments today. Trying to discuss things with you
is turning into a waste of time... :(

MK

[Tim Chow]

On Wednesday, June 22, 2022 at 8:40:48 AM UTC-4, Tim Chow wrote:
> On 6/21/2022 9:57 PM, MK wrote:
> > 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?
> I did not come up with this. I don't know who first came up with
> it, but it dates back at least to Bill Robertie in 1992.
>
> https://web.archive.org/web/20190112044437/https://thegammonpress.com/comparing-games-skill-chance/

I learned from Ken Regan that the idea of measuring the "depth" of
a game in this way seems to be due to Laszlo Mero, who described
the concept in his 1990 book, "Ways of Thinking."

---
Tim Chow

[MK]

On September 19, 2022 at 7:23:42 PM UTC-6, Tim Chow wrote:

> On June 22, 2022 at 8:40:48 AM UTC-4, Tim Chow wrote:

>> it, but it dates back at least to Bill Robertie in 1992.

> I learned from Ken Regan that the idea of measuring the "depth" of
> a game in this way seems to be due to Laszlo Mero, who described
> the concept in his 1990 book, "Ways of Thinking."

Good to see the credit go to the right person.

Do you have anything new to say on the topic, by chance?
I'm disappointed that nothing more came out of it.

I looked for Ludo software but couldn't find anything other
than Android apps and no online servers, nothing to take
seriously anyway. It seems nobody plays Ludo competitively.
I guess it will stay at this unless we stumble onto something
more interesting about it in the future... :(

MK