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Subject: Number of pieces over time rss

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David Ploog
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As I mentioned before, I'm not just writing guides for some games (more will come) but I am also trying to collect some overarching ideas. There's probably no need to rehash the perpetual "what's depth" and "strategy vs tactics" debates, but I thought some of you might like the following graphs (updated 2019/06/03):



Let me explain: Each graph shows the number of pieces/stones of one side on the vertical axis; the horizontal axis is the number of turns. One could average over many games. Instead, I, naturally and lazily, simply took games which I claim are representative.

Why I'm doing this: in my grand scheme, properties of (abstract/combinatorial) games come in three layers:
(a) immediate properties: inferred directly from the rules.
(b) heuristic properties: these require experience with the game.
(c) subjective properties: these depend on taste.

Examples of (a) are: permanence in Christian's sense, temperature (absence/presence of cold phases), drawlessness, possibility of repeated positions, the above graphs. Examples of (b) are patterns or shapes, tactics and strategies. Examples of (c) are simplicity, theme/narrative, clarity.

I think that looking at the function number of white/black stones over time says something about a game. For example, you can spot pure placement games at a glance. You see whether a game has an initial position. You can see captures.

I wouldn't read too much into these graphs but I think they're fun to look at. Here are the nine categories I show above:
Constant: self-explaining. Do you know other examples apart from Halma, Agon and similar racing games?
Fixed growth: Pure placement (traditionally one stone/turn; no captures). I picked Hex as the classical example.
Fast growth: by this I mean that during the game, the number of pieces added can go up (i.e. the curve is monotonously increasing and has inflection points). Christian's Symple, Sygo, Scware all do this.
Volatile growth: You see what I mean when you look at the curve. Are the examples besided Reversi? Note that "growth" is fitting because the number of black+white stones does always go up.
Fixed decay: Initial position (usually full board), and each turn removes exactly one stone. Games like the traditional Konane do the same, with possible longer captures; I am not sure whether that's worth drawing.
Slow decay: I mean games with an initial position, and possible capture of at most one piece/turn. Such as Chess, LOA etc.
Fast decay: As above, but several pieces can be captured per turn. E.g. Draughts/Dameo. I picked Fanorona because it is adorably violent.
Volatile: I picked Hex Oust. The number tends to go up, but there can be spikes. (In Hex Oust, it is possible to win after being reduced to a single stone.)
Placement with capture:I am not happy with the graph itself, but I wanted to show what such a game could look like... (This is a 9x9 Go match between Go Seigen and Miyamoto Naoki.)

As usual, I have some questions to you (edited 2019/06/12 -- thanks for replies!):

(1) Are you aware of other games with distinctively different graphs?
(2) I need a name for these graphs! What about "deployment function"?

Thanks for reading!
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Stephen Tavener
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dpeggie wrote:
(5) Is there concensus whether it's "Othello" or "Reversi"?
Reversi is the original name of the game, published at the end of the 19th century. Othello is a modern trademark when the game was revived in the 1970s. Functionally, Reversi has two opening positions (adjacent or diagonal), Othello standardises on diagonal only.
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christian freeling
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It's a nice angle for categorisation for sure!

dpeggie wrote:
As usual, I have some questions to you:

(1) Are you aware of other games with distinctively different graphs?
I'd naturally be interested in Storisende and Emergo. The former because it features growth and decline (as does Mark Steere's Oust by the way) and the latter because it may intertwine growth in the number of men with decline in the number of columns.

P.S.
Emergo has another weird property in this context. There may be games where one player grows linear while the other must capture. This results in a 'shadowpiece' that must be entered last as one column that may consist of several men, so the graphs of both players will in such a case be very different.
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Russ Williams
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Fun; I like this kind of thing.

dpeggie wrote:
(1) Are you aware of other games with distinctively different graphs?
(2) Should I draw Go? (This is like pure placement, with infrequent capture.)
Go seems covered (at least loosely) by your "Volatile Growth" category.

Quote:
(3) Should I draw Shogi?
Shogi seems to answer your question 1: it's a different type ("Roughly constant"?).

Quote:
(4) There must be interesting games with linear decay (one per turn)...
E.g. Quantum Leap.

A related idea is to graph the total number of pieces (instead of the number of pieces of one player). Then e.g. Shogi would be exactly constant, Othello would be far less volatile (almost exactly linear except for occasional "pass" turns), etc. It might be interesting to see which games have the same growth under both views (e.g. Omega = strictly linear growth of total pieces and of one player's pieces) and which don't, etc.
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Cameron Browne
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dpeggie wrote:
(4) There must be interesting games with linear decay (one per turn)...
Clobber
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Very nice! Here's a thought and a possible suggestion that might provide further insight into the properties of games: Plot two lines on each graph - one for the winner and one for the loser, in two different colors, like green for the winner and red for the loser. Then we can see the correlation between material and victory. In most games, more pieces is an advantage, but that isn't the case in all games. We could also see the correlation between increases and decreases for one player and decreases and increases for the other.

dpeggie wrote:
(1) Are you aware of other games with distinctively different graphs?
One that came to mind is Pueblo, which in the basic version is just linear growth, but in the "construction/deconstruction" version, it's a period of linear growth followed by a period of linear decay. Which suggests to me that there may be some other linear decay games out there. (Question 4) Nim comes to mind, but it doesn't really count.

There are also games like Blokus and Blokus 3D, which could be considered linear growth, but since the pieces are of different sizes - which have an impact on victory - it might be considered a somewhat variable growth. And these and similar games have a "place of all your pieces" goal, which puts a cap on the growth (unlike Hex).

Quote:
(2) Should I draw Go? (This is like pure placement, with infrequent capture.)
(3) Should I draw Shogi?
Maybe. These have some subtle differences from the growth/decay in other games, which might be interesting to track and compare.
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Cameron Browne
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How about Discrete or Stepwise, e.g. Roulette:



Assuming the chips count as pieces.
 
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Daniel Piovezan
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Ha! I knew it was Fanorona when I saw the third graph!
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Michael Van Biesbrouck
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Are there any abstract games with a sawtooth pattern? You would lose pieces and then get reinforcements later. (Slowly gaining and then losing a bunch would be Go.)

It might be worthwhile to show two lines, one for each player. Reversi would should a linear increase in total pieces with flips making opposed changes. Go would show that pieces aren't lost from both players at the same time. The hypothetical reinforcement game might show that both players get pieces together.
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Julien Griffon
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dpeggie wrote:

Constant: self-explaining. Do you know other examples apart from Halma, Agon and similar racing games?
Linear growth: traditionally one stone/turn; no captures. I picked Hex as the classical example.

Ringo starts with a Linear growth phase possibly followed by a Constant phase.
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Russ Williams
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DVONN would be an interesting one. In the initial placement phase, it's pure linear growth, then in the jumping phase, occasional drops occur typically (but not necessarily).

TZAAR (if played with an initial placement phase) is similar, but then in the main part after placement, the opponent loses a piece and then the opponent or the active player loses a piece. (Or not, as occasionally a player might pass their second action.)

---

How to define "number of pieces" in Trax? If number of physical pieces, then it's growing stair-step-style (by one or more each turn). If number of routes belonging to a player, it's typically growing, but in a volatile way. Similarly for other similar games with player-assigned colors/regions created across tiles, like Palago.

---

Some games don't have player pieces at all, e.g. ZÈRTZ. There are rings being removed over time, and balls being added and removed.
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Nine Men's Morris and similar games (Kensington), start at zero, have linear growth for a while, and then captures start to happen while growth is still possible - but there is often a maximum. I guess this might count as the same as "volatile growth".

mlvanbie wrote:
Are there any abstract games with a sawtooth pattern? You would lose pieces and then get reinforcements later. (Slowly gaining and then losing a bunch would be Go.)
I can't think of any abstract games that have a sawtooth pattern, but I do know of some Euro-games that do. You spend some time building up your position, and then you deplete your pieces to take "profits", or victory points, and you might do this multiple times over the course of the game. If you include the various "resource engine" games, the accumulation and spending of resources would probably give a sawtooth pattern in most cases.

Now all someone needs to do is design an abstract game like that!
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David Ploog
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Many thanks for the replies! They'll certainly improve things!

In particular, the idea to show the function for both players and/or for one player and total number of pieces are interesting.

Some bits that occurred to me while I read your replies:

There is a recorded game of Mark Steere's Tanbo with >1000 moves; this must have cycles of captures and re-growth (make no mistake, Tanbo is a slow, viscious, tedious and boring game).
Sowing games (Aware, Kalaha, Buku etc.) are also very volatile, although "is in a hole on my side" is not quite the same as "my piece", which is why I didn't attempt to draw one.

Plot two lines on each graph - one for the winner and one for the loser, in two different colors, like green for the winner and red for the loser. Then we can see the correlation between material and victory. In most games, more pieces is an advantage, but that isn't the case in all games. We could also see the correlation between increases and decreases for one player and decreases and increases for the other.

Phil Fleischmann wrote:
Very nice! Here's a thought and a possible suggestion that might provide further insight into the properties of games: Plot two lines on each graph - one for the winner and one for the loser, in two different colors, like green for the winner and red for the loser. Then we can see the correlation between material and victory. In most games, more pieces is an advantage, but that isn't the case in all games. We could also see the correlation between increases and decreases for one player and decreases and increases for the other.
This is true but it goes in scope way beyond what I am trying to (and realistically can) convey: my idea is just that different games (rather: subgenres of games) have different pieces/turn functions. This is entirely unrelated to strategies, winning etc. and just a consequence of the rules. This is why I get by just picking one reaonable game.
If you want to make an analysis about material (dis)advantage and winning/losing, then you need to look at very many games, and think about deeper properties of games.

As one example how this can be tricky: in Go, captures are sometimes crucial (cutting stones, life & death) but generally not very important. Capturing ten stones in the midgame (which is rare among good players) would look impressive on the graph, but not mean much at all. Of course, Go may be special in this regard...
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David Bush
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russ wrote:

Quote:
(3) Should I draw Shogi?
Shogi seems to answer your question 1: it's a different type ("Roughly constant"?). :)
That depends on whether you distinguish between pieces in play and pieces on the board. The number of pieces that are involved in the game, or in play, remains constant.

EDIT: I should clarify between pieces in hand, which in Shogi means pieces you have captured which you may return to the board, and pieces in reserve such as Go stones in a bowl. The former I call pieces in play, and the latter I do not.
 
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David Bush
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dpeggie wrote:

(1) Are you aware of other games with distinctively different graphs?
Trax is always increasing, usually one tile per move, but with frequent cascade moves that add multiple tiles in a single move. That differs from your volatile increasing, since the count never goes down.

Hextris starts at zero, then pieces are added, then the board count could go up or down, peaking around mid game, then going down towards the end. There have to be other games that do this.
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Russ Williams
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twixter wrote:
russ wrote:

Quote:
(3) Should I draw Shogi?
Shogi seems to answer your question 1: it's a different type ("Roughly constant"?).
That depends on whether you distinguish between pieces in play and pieces on the board. The number of pieces that are involved in the game, or in play, remains constant.
Yes, I noted that later in my comment from which you're quoting! ("A related idea is to graph the total number of pieces (instead of the number of pieces of one player). Then e.g. Shogi would be exactly constant,")

The OP was talking about the number of pieces of one player ("Each graph shows the number of pieces/stones of one side").
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David Ploog
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So I have updated the section thanks to your feedback. This is very welcome!

The first posting has the graphs; now nine. Someone mentioned Hex Oust, that was a very good idea. I am not convinced by the graph for Go (this is a funky 9x9 game with Go Seigen playing white).

If someone is interested in more details, send me a message. The section sits at 2.5 pages.

Edit: last time I totally forgot to update the text; this is now done, too. I can use this apology to highlight an English question:

I am looking for a good word or phrase to describe these graphs. Technically, they're the "number of [black] pieces as a function of turncount". Now, that's unwield and I need something shorter. Would "deployment function" work? Or "piece count"?
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