PRODUCT
Copyright (c) 2006 Nick Bentley
rules adaptation by João Neto & Bill TaylorProduct is played on an empty 5x5 hexagonal board:
- GROUP - A connected chain of friendly stones.
- TURN - Initially, Black drops one stone of either color, then for the remaining turns players drop two stones of either color.
- GOAL - After the board is full, wins the player with the higher product between its two largest groups.
- If both products are equal, wins the player with less stones on board (note: draws are impossible).
An example Black's turn. Right now he has 21 points (7 stones for his largest group, and 3 stones for the 2nd largest. He decided to connect the largest group with the isolated stone (moves [1]). With this move he gets 30 points.
However that was a bad move. Now White simply plays two black stones at [2] connecting all black stones into a single group. So, the actual score for Black is not zero (his 2nd largest group does not exist, ie, is made of zero stones).
It will be impossible for Black to create another isolated group without White's help.
This is a game designed by Nick. Me and Bill just proposed two rules to improve gameplay and reduce the game's playtime: (a) added the 12* move sequence, and (b) removed draws by adding the 'less stones' criteria. This two addons were used when the game was included in the Portuguese Tournament of Board Games.
Here's a post by Nick talking about the game. The game was posted at r.g.a newsgroup in Dec 2008.
There's a variant called Alpha with the following rules: A player's score is the product of the size of the largest group of his color and the number of distinct groups of his color in the final position. The player with the highest score wins. In case of a tie, White wins.
Here's the original newsgroup text (in case Google decides to close the archive):
nicko...@gmail.com
Dec 8, 2008, 12:23:08 AM12/8/08
to
Here's a little game-design challenge to anyone who wants to think
about an interesting problem. I'm not going to pose the problem
rigorously, because I'm lazy. Nonetheless:
There's an argument to be made in favor of the idea that an ideal game
is *not* drawless. The idea is that two perfect players should play
to a draw in an ideal game, and that any other outcome is both an
unseemly asymmetry, and a violation of the principle of fairness, to
be expunged. So, what would draws in an ideal game be like? Perhaps
an ideal game would have exactly one drawn state.
So, my challenge to you is to design such a game.
One approach is to pick a drawn state, based on some set of criteria
that I can't quite fathom yet, and then ask yourself, "What game would
have *this* for its only draw?" Ideally, you'd want there to be only
one possible game that could sensibly have that state for a draw. But
there might be better ways of going about it.
I'm curious to see what gameplay would be like for a game with a
single known drawn. Conservative players might play toward the draw,
before one breaks away when he sees an opportunity for advantage.
Having a single known draw might increase the clarity in the game,
because players could use knowledge of the draw as a kind of guide to
play, as in the "conservative" play example mentioned above. But I
could be completely wrong.
On the other hand, having a single drawn state may not be a very
worthy goal. Maybe the real goal should be to have a game where draws
are possible, but where the draw ratio in actual play achieves a
theoretical non-zero minimum. Basically, a game where a draw can only
occur between 2 perfect players, and no one else.
Is it possible to design a game where draws are possible, but nobody
except perfect players can ever get to it? I suppose it requires that
nobody but perfect players even be able to imagine a drawn state.
This seems a ridiculous notion, but sometime the greatest insights can
come from dwelling on ridiculous notions. You'd need some set of rules
from which you can prove that draws are possible, but for which you
can provably only find the draws by being able to search the entire
game tree.
Also, Bill Taylor, if you read this: I've noticed that you sprinkle
your posts with design challenges. It might be a cool idea to compile
a list of such challenges and then offer little Erdos prizes for those
who solve them. Might be a way to stimulate more non-propaganda
discussion on this forum. I wish there were more traffic here. Is
there someplace where all the designers go to discuss abstract game
design?
nicko...@gmail.com
Dec 8, 2008, 12:38:34 AM12/8/08
to
replying to my own post, you *could* theoretically have a game where
only perfect players can ever find the drawn position. the drawn
position would have to be defined as some function of every node and
connection in the game tree itself. That way, only perfect players
who know the whole game tree could figure out what the drawn position
is. But is it possible to design a non-stupid game with this
property?
Torben Ægidius Mogensen
Dec 8, 2008, 10:10:52 AM12/8/08
to
nicko...@gmail.com writes:
> There's an argument to be made in favor of the idea that an ideal game
> is *not* drawless. The idea is that two perfect players should play
> to a draw in an ideal game, and that any other outcome is both an
> unseemly asymmetry, and a violation of the principle of fairness, to
> be expunged. So, what would draws in an ideal game be like? Perhaps
> an ideal game would have exactly one drawn state.
Having exactly one drawn state doesn't really matter. Consider this
game:
The board is an odd number of spaces in a line. A token is placed in
the middle. Each player starts with N coins. Players alternate paying
a number of coins to move the token that many spaces in their
direction. You can pass (only) when you have no coins. The game ends
when both players have no coins. The winner is the one who has the
token closest to himself. If it is in the middle, it is a draw.
The game obviously has only one drawn position: The token in the
middle and both players have 0 coins. But this is the only possible
outcome of the game, so it is not interesting.
A variant of this is fairly well known and nontrivial: Both players
secretly bid any number of coins and the winner gets to move the token
one space in their direction (on a tie the token doesn't move). Both
pay their bids. If both bid 0, the game ends. The winner is the one
who has the token closest to himself. If it is in the middle, it is a
draw.
Unlike the above, this is not a total-information game, so the notion
of perfect play doesn't exist. So we really need to restrict
ourselces to total-information games,
The interesting properties for a total-information game with draw
would be:
1. No game can go on forever (there is a maximum number of moves in a
game).
2. With perfect play, no player can obtain more than a draw.
3. Perfect play is not obvious.
The first is usually fairly easy to decide, the second is well-defined
but can be hard to prove. For example, Chess is AFAIK still not known
to lead to a draw with perfect play.
The third property is not well-defined, as "obvious" is a subjective
property. A reasobable way of defining non-obviousness is
computational complexity: If the problem of finding an optimal move is
computationally hard, then it is non-obvious.
Usually, for a game to be computationally hard, there has to be a
large number of legal (and not obviously bad) moves in each turn and
the number of turns in a game has to be high. Go is a prime example:
You have an average of around 100 possible moves per turn (though many
are obviously bad) and there number of turns is around 100.
Connection games (Hex or similar) can be modified to be like this in
the following way: There are not enough tokens to fill the board.
While a completely filled board is a win by one side or the other in
Hex, Y, etc., partially filled boards can be tied (neither side has a
connection). A plausible Hex variant could be: Use an NxN board,
where N is odd. Give each player (N^2-1)/2 tokens. This leaves
exactly one space empty, and both players get the same number of
moves.
Torben
Harald Korneliussen
Dec 8, 2008, 1:22:19 PM12/8/08
to
On Dec 8, 11:10 am, torb...@pc-003.diku.dk (Torben Ægidius Mogensen)
wrote:
> A plausible Hex variant could be: Use an NxN board,
> where N is odd. Give each player (N^2-1)/2 tokens. This leaves
> exactly one space empty, and both players get the same number of
> moves.
>
> Torben
Since 5x5 and 7x7 are solved, it may be possible to check with brute
force whether this is enough to ensure a tie with perfect play.
nicko...@gmail.com
Dec 8, 2008, 3:49:45 PM12/8/08
to
Cool. Another version of the "Hex with draws" idea, which might reduce
the number of draws further, and perhaps the actual draw ratio, is to
play hex on a board with an odd number of cells and designate the
central hex off limits. The central hex is a nice choice because it
is both the only unique cell on the board, and because filling it
often gives one an advantage, players would opt away from the draw
often (assuming that such a game is drawn with perfect play; I don't
actually know). Also, then you you wouldn't have to count out a
certain number of pieces for each side before the game beings, which
is more convenient.
The hex variants are good, but they aren't provably best when it comes
to minimizing # of draws or draw-ratios, in any way that I can see.
it would be nice to find games that do.
Bill Taylor
Dec 16, 2008, 5:05:57 AM12/16/08
to
Well.
I'm finally beginning to catch up on backlogged articles/emails.
nickobe...@gmail.com wrote:
> There's an argument to be made in favor of the idea that
> an ideal game is *not* drawless. The idea is that two
> perfect players should play to a draw in an ideal game,
> and that any other outcome is both an unseemly asymmetry,
> and a violation of the principle of fairness, to be expunged.
Indeed so! And in the world of Go, that is very nearly the case.
Because the game is fine-grained in terms of final outcome,
(the final score on the board is about as likely to be anything
between plus and minus 30, between roughly-equal players),
this leaves scope for...
(a) considerable tinkering with "komi", (numerical compensation
for the second-moving player); and
(b) the opportunity to allow for (fairly rare) draws by means
of getting the komi "exactly right".
It is interesting to note that there may be two somewhat
different komis. One is the "perfect single figure", known
only to God, that would make a draw from exactly perfect play;
and the other is "practical komi" which results in about
equal numbers of wins and losses for either colour.
These two figures are almost certainly not the same,
due to the fallibility of real human players.
Strangely, the Go world seems to go out of its way to AVOID ALL
possibility of draws, which is odd, and not at all like
oriental culture in general, where concensus and "market share"
tend to dominate thinking. It is OC much more like American
culture, where in commerce and sport, "winner-take-all" views
tend to dominate, as witness the extreme lengths in baseball,
basketball and gridiron football to expunge draws.
And yet in Go-scoring ideals it tends to be the other way round!
Odd. And IMHO an opportunity badly missed. It would be great
to arrange komi so that it would allow draws infrequently but
significantly. Indeed, I often think that for amateur players,
where the scores tend to fluctuate more randomly, it would be
a great idea to use a "thick komi" that allows a wider interval
of board-score results to be counted as draws. But this is OT.
> So, what would draws in an ideal game be like?
The above remarks suggest it might be similar to Go-like games,
that is, games where there is more than just a win/loss outcome,
but a range of "scored" outcomes. There aren't many of these,
but there are some. Games like "Amazons", where at game end
the loser has run out of legal moves, but the winner still
has some (again, say, anything from one to twenty) left over.
So, that would be a start. Alas, there aren't too many others,
though perhaps some could be artificially adapted. For example,
in a connection game, where one player can win so easily that,
at game's end, he could claim victory by allowing his opponent
some number of "free moves", and still win.
It's not much of an idea, but perhaps worth thinking about.
> Perhaps an ideal game would have exactly one drawn state.
Could be.
> So, my challenge to you is to design such a game.
I'm working on it! No luck yet, however.
> I'm curious to see what gameplay would be like for a game with a
> single known draw.Conservative players might play toward the draw,
> before one breaks away when he sees an opportunity for advantage.
Indeed! I suspect that would be quite common. It could even
lead to a "chicken"-like element in games, which I have spoken
of before here. Both players getting closer and closer to
explosive instability, till one cracks and commits to something.
> Having a single known draw might increase the clarity in the game,
> because players could use knowledge of the draw as a kind of guide
> to play, as in the "conservative" play example mentioned above.
> But I could be completely wrong.
No; I think these are all good observations.
> Is it possible to design a game where draws are possible,
> but nobody except perfect players can ever get to it?
That would actually be a good design specification.
> I suppose it requires that nobody but perfect players
> even be able to imagine a drawn state.
Oh no, not at all! It could be very easy for anyone with basic
game knowledge to *check* that a certain position satisfies
the (maybe even unique) drawing conditions in the rules; but still
hugely difficult to actually achieve it. Indeed there are hints
of NP-not-equal-P-ness to this. (Don't read that aloud to fast!)
Hard to find but easy to verify. That's essentially P =/= NP.
> Also, Bill Taylor, if you read this:
Hello there Nick! :-)
> I've noticed that you sprinkle your posts with design challenges.
Well, I'm glad you think so, because it's a laudable goal.
But I'm surprised you think so, because I can't think of
any particular design challenges I've sent out here!
Please recall and tell us of some, it sounds fun.
I know of many I've *worked on*, but mostly they're kept to
games that Joao Neto and myself play privately. Some are
design goals I've been very pleased indeed to work on, and
even pleased-er to have solved! But I don't recall any on rga.
OC, it's become a bit dangerous to put out general ideas here,
because of a certain you-know-who, who shall remain nameless,
but has a distressing tendency to claim every good idea as
one of his own, and belittle all those which he can't! ;-)
So until this poisonous viper changes his ways, or (better)
just buggers off, your general goal of co-operative construction
might be unattainable!
> It might be a cool idea to compile a list of such challenges
Well great! But I don't recall any. Can YOU start a list for us?
> Is there someplace where all the designers go to discuss
> abstract game design?
THIS is the place! It has been a great place up till recently,
but now the waters have become poisoned, as happens all too
often on Usenet, alas. Pity.
-- Wearied William.
nicko...@gmail.com
Dec 16, 2008, 11:47:49 PM12/16/08
to
> > So, what would draws in an ideal game be like?
>
> The above remarks suggest it might be similar to Go-like games,
> that is, games where there is more than just a win/loss outcome,
> but a range of "scored" outcomes. There aren't many of these,
> but there are some. Games like "Amazons", where at game end
> the loser has run out of legal moves, but the winner still
> has some (again, say, anything from one to twenty) left over.
If we're looking for a "points" game where the spread is typically
huge (and therefore draws minimized), there is one that I play often
that fits the criterion well. The rules:
1. board=hexhex
2. two colors of stones. let's call them red and blue.
3. on each turn, player fills a cell with *either* available color
4. game ends when board is full
5. player 1 wins if the product of the sizes of the two largest red
groups is larger than than the same product for blue groups. If there
is only one group for a color, the final score for that color is
zero.
Because the final score is a product the scores can and do swing
wildly. I've seen (and participated in) spreads greater than 2000
points. One of the greatest game experiences of my life came while
playing this game, when the difference in final scores ended up being
2, something I've not come close to achieving before or since. The
game obviously works better played online, where the computer tallies
the score for you.
You can step through the moves in a recently completed game here:
http://gc1.iggamecenter.com/gm.php?gid=17&sid=48567&place=0&lang=en
(note in the version that you see there, there are no ties, because
player 1 wins in the event of an equal score. But if you change it a
bit so that equal scores are defined as a tie, the game is a good
example of what we're looking for). There's more than one drawn state
in that game, though, so it's not really what I'm looking for.
>
> > I suppose it requires that nobody but perfect players
> > even be able to imagine a drawn state.
>
> Oh no, not at all! It could be very easy for anyone with basic
> game knowledge to *check* that a certain position satisfies
> the (maybe even unique) drawing conditions in the rules; but still
> hugely difficult to actually achieve it. Indeed there are hints
> of NP-not-equal-P-ness to this. (Don't read that aloud to fast!)
>
> Hard to find but easy to verify. That's essentially P =/= NP.
ahh yes. That is a helpful analogy. Gives me a new way to think
about it.
>
> > I've noticed that you sprinkle your posts with design challenges.
>
> Well, I'm glad you think so, because it's a laudable goal.
> But I'm surprised you think so, because I can't think of
> any particular design challenges I've sent out here!
> Please recall and tell us of some, it sounds fun.
One that I remember offhand: a *local* YMMV pattern. I've given that
one some serious thought, without much progress. I don't have time at
the moment to go search through the archives for others, but I'm
pretty sure there are others.
>
> > It might be a cool idea to compile a list of such challenges
>
> Well great! But I don't recall any. Can YOU start a list for us?
Yes! Maybe not at this very moment, but certainly in the next week or
so. So now off the top of my head, we have: 1) a local YMMV pattern;
2) a game with a single drawn state that only perfect players can
reach; and 3) the "analogue" connection game that you guys are
discussing in the other thread (about which I have fruitlessly
wondered on many occasions).
> THIS is the place! It has been a great place up till recently,
> but now the waters have become poisoned, as happens all too
> often on Usenet, alas. Pity.
This is not the first forum that's gone wanky as a result of a-certain-
someone's presence. Nor is it the second. There are vivid,
entertaining accounts of past difficulties across the web, if you want
a chuckle.
Bill Taylor
Dec 19, 2008, 5:24:39 AM12/19/08
to
> If we're looking for a "points" game where the spread is typically
> huge (and therefore draws minimized), there is one that I play often
> that fits the criterion well. The rules:
>
> 1. board=hexhex
> 2. two colors of stones. let's call them red and blue.
> 3. on each turn, player fills a cell with *either* available color
> 4. game ends when board is full
> 5. player 1 wins if the product of the sizes of the two largest red
> groups is larger than than the same product for blue groups. If there
> is only one group for a color, the final score for that color is zero.
That's a rather cool game! And, I suspect, one of the few which
is actually *better* on a small board than a big one; or rather,
more skillful anyway. Because on really small boards there
would be a definite subgoal conflict; where the player tries to
keep his groups big to score well, but also keep them small enough
to ensure there are at least two.
Anyway, games where the opponent can move or play *your* pieces
as well as his own are always fun and spritzy. Chameleon is like
this.
And the genral idea can be modified a little, especially for those who
object to an overtly arithmetical element. The sum of the sizes of
the top two is simpler, but still with 0 total score for one group.
Even cooler with respect to mixed goals:- win with the biggest
NUMBER of separate groups, with ties going to the biggest single
group! That would be diabolical.
All of these to be played with the "both can play either" rule.
> This is not the first forum that's gone wanky as a result of a-certain-
> someone's presence. Nor is it the second. There are vivid,
> entertaining accounts of past difficulties across the web, if you want
> a chuckle.
Wow. VERY interesting indeed. But a google search doesn't
seem to help. Can you tell us what groups to look in, please?
-- Battling Bill