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Subject: Informal experiment: how easy to find "the optimal disk placement" in various positions? rss

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Russ Williams
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I have often read people say that the disk placement phase of Fjords is too automatic and that the best move is obvious, or easy to calculate, or something to that effect.

But I feel that often it is not at all clear what the best disk placement is, and it's an interesting decision.

(Until all borders are settled of course: then the players just fill in their unambiguous territories.)

I propose an experiment (similar to one I did a couple months ago for Kingdom Builder here).

Here's a photo from the gallery of a game that has just finished the first phase, and players are about to start placing disks:



Let's say the houses are numbered like this for each color:

W3
B2 W2
B3

W1
B4
B1 W4

Two questions:

1. If you're black and have the first disk placement, which house (B1, B2, B3, B4) do you grow from and in which direction: NW, NE, W, E, SW or SE?

2. If you're white and have the first disk placement, which house (W1, W2, W3, W4) do you grow from and in which direction: NW, NE, W, E, SW or SE?

Please DO NOT post your 2 answers as a comment here! Geekmail them to me. After a week or so (depending on response rate) I'll summarize the results. Thanks! Hopefully this will be interesting.
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Darren Mac
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I have two related comments:

1 - I agree - the placement of fields does NOT seem obvious to me

2 - I usually lose this game

This is a neat question, and I am curious to see your results!
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Russ Williams
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darrenmac wrote:
This is a neat question, and I am curious to see your results!

Cool - don't forget to send me your answers!
 
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Russ Williams
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My response rate has seriously slowed down to a trickle, so if anyone is thinking of participating but hasn't done so yet, please send me your response soon, because I think I'll post a report within the next few days.
 
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russ wrote:
My response rate has seriously slowed down to a trickle, so if anyone is thinking of participating but hasn't done so yet, please send me your response soon, because I think I'll post a report within the next few days.


Have you got an identical pair of moves yet?
 
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Russ Williams
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NateStraight wrote:
Have you got an identical pair of moves yet?


All shall be revealed in the fullness of time! ninja
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Russ Williams
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OK, I seem to have stopped receiving survey results, so here's the report!

I have answers from 15 people. Several people changed their minds and updated their answers after further pondering. (This alone indicates that the best move was not "obvious".) In such cases, I used only their last answers.

A few people sent illegal moves: a few crossed mountains (at least one respondent indicated that they forgot about that rule initially due to not playing recently, then noticed and updated their answer.) A couple gave locations off-map, perhaps due to confusion about the notation or something.

Anyway, the answers were as follows (I omit user names out of a probably exaggerated sense of privacy, but feel free to reveal yourselves and discuss your reasoning!):

B4-SE W1-SW
B3-E W2-SE
B3-E W2-SE
B3-E W1-SW
B3-SE W1-SE
B3-SE W1-NE
B3-SW W1-NE
B3-SW W2-SE
B3-E W2-SE
B4-SE W4-W
B3-E W2-SE
B3-E W1-NE
B4-SE W4-E
B4-SE W4-NE
B3-E W1-NE


The black moves sorted by popularity:
B3-E x7
B4-SE x4
B3-SE x2

The white moves sorted by popularity:
W2-SE x5
W1-NE x4
W1-SW x2
W4-W x1

Map showing the frequency of votes for a given hex, for the black move and the white move:


It's a frequent cliche that the best move for your opponent is also the best move for you. So it's perhaps not surprising that there were 2 hexes identified as good for both black and white.

The "7-5" hex (B3-E = W2-SE) was the most popular first play for both players. Nonetheless, fewer than half the people picked it.

Interesting that the "4-1" hex (B4-SE = W4-W) was considered much more urgent for black than for white.

The only clear consensus was that (not surprisingly) the houses near the outer rim with no enemy houses nearby (B1, B2, W3) were not considered good starting areas. Those hexes are not urgent; they can be claimed later.

===

Apologies for any errors (corrections welcome).

Thanks to all who sent responses! It was interesting to see people's answers and sometimes people gave some reasoning for their answers. Perhaps my favorite, given the results, was:
Quote:
I'm no Fjords expert, by any stretch, but here are my answers:

(1) As Black, I'd move first from house B3, and I'd move East.

(2) As White, I'd move first from house W2, and I'd move Southeast.

i.e. this non-expert respondent selected both of the most popular moves.


Special thanks to David desJardins, who corresponded with me and put more than usual effort into solving the position, and also discussed the problem with his colleague Elwyn Berlekamp (coauthor of Winning Ways and Mathematical Go), who became interested in Fjords and might try proving a solution to the position.
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Scott Petersen
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Thanks for running this, Russ! I said both should go to the "22" hex in the southeast. For black, it completely neutralizes white and for white, it gives black two hexes to which it must respond.

The "15" hex can be directly responded to by either player, so I didn't see it as a priority. I am notoriously terrible at ply analysis games though.
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Russ Williams
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David suggested putting a map with the possible playable hexes numbered to make strategy discussion easier, so here we go:


(Hmm, whoops - Ignore that #3 is not actually playable on the first move, and that there is no #11... I'm not going to redo it!)

EDITED TO ADD: See better map in David's comment after this comment. Use it for any further discussion, not this comment's crappy map.

E.g. obviously bad first moves for either player are hexes:
1 2
9 10 12 13
17 18 19
24

Move 22 seems attractive because black can get 3 certain points locked up easily that way.

Black 15 is sente and easily countered by white 16, but then white has only 1 route into all the center territory, which is harder to analyze than that southeast area.

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George Leach
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I'm intrigued to learn that David DesJardins knows Edwin Berlekamp. I'd be intrigued to learn what Berlekamp's solution throws up. I must admit that I'm less positive about the game's depth since this survey rather than more positive like you seem to be Russ. Some interesting analysis there though.
 
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David desJardins
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I wanted to label all of the playable hexes, for analysis, so I hope no one minds if I substitute my own image:

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David desJardins
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I voted for hex 26 for both Black and White, my vote was confused a little bit as I changed my mind a few times. I think there are actually 4 votes for Black 26 and also four votes for White 26, some of the "illegal" votes seem to have confused east and west. There also seem to be many votes (probably more) for Black 10 and White 10.

White 10 is, in my opinion, quite a weak move, despite getting many votes, because it's gote (doesn't force a response). White 10 protects 5 spaces of territory, but it doesn't create any new threats, and so Black is free to make forcing moves elsewhere on the board, or else to make a large gote move of his own (i.e., Black 26).

White 26 is also gote, but it's worth much more, because it immediately gains a net of 4.5 points compared to letting Black play 26. (Black loses 22 and 25 which become dame, loses 29 which becomes White territory, and loses firm control of 18, which will become dame if White plays 22 first.) This is a large swing compared to almost anything else on the board.

After White 26, no side has anything better than Black 10, White 5, Black 19, White 20, Black 9, White 12, Black 11. Now White has 6 points of territory (1,2,6,7,24,29) and Black has 4 points of territory (3,8,27,28) and 2 half-points (14,18). The half-points will be split, so White has net +1 territory, and there are an odd number of dame, so they will be split evenly, so White wins 15-14.

If Black moves first, then the Black 10 opening looks like a stronger move than White 10 did, because it's sente (it threatens Black 5 which takes away several points from White). So it's tempting to open with Black 10, and if White plays 5, then Black can play 26.

But it turns out that playing 26 is so valuable that White can ignore the threat of Black 5, and play White 26. After the line Black 10, White 26, Black 5, White 20, Black 19, White 12, Black 9, White 11, we see that White has 2 points of territory (24,29) and one half-point (7), while Black has 2 points of territory (27,28) and three half-points (3,14,18). The next four moves will be on the half-points, yielding one additional net point of territory for Black (either Black +2 and White +1, or Black +1 and White 0). So Black will win by 1 point, 15-14.

Compare the line Black 10, White 5, Black 26, White 20, Black 19, White 12, Black 9, White 11. Now White has 5 points of territory (1,2,6,7,24) and no half-points, while Black has 6 points of territory (18,22,25,27,28,29) and two half-points (3,14). The next two moves will yield one additional point of territory for Black, so a net +2 territory. Since Black moves next, and there are an odd number of dame, Black will also get one extra dame, so Black wins by 3 points, 16-13.

However, if Black opens with 26, then White has nothing better than White 20, Black 19, White 10, Black 9, White 12, Black 11. Now White has 6 points of territory (1,2,5,6,7,24) and no half-points, while Black has 8 points of territory (3,8,18,22,25,27,28,29) and one half-point (14). White will contest 14 next, but Black is ahead by 2 in territory, and also gets the odd point of dame, so wins 16-13.

Therefore, I believe that Black 26 is the unique best opening for Black (the only move which ensures a 16-13 win), and White 26 is the unique best opening for White (the only move which ensures a 15-14 win).
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David desJardins
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Jugular wrote:
I'm intrigued to learn that David DesJardins knows Edwin Berlekamp.


Berlekamp was my dissertation advisor at Berkeley. I've known him for over 25 years. I showed him this position, he's interested in the game in general, but mostly from a more abstract view, I'm not sure he's going to give a definite answer or calculation for this specific position.
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George Leach
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Some interesting analysis there David. I think there are a few errors in your description and possibly in the analysis. You keep mentioning White 11 before White controls 12. I assumed you had to get there 'by land' so to speak. This obviously then affects the analysis as black has a more defensible area of territory. Forgive me if I have the rules wrong I've only ever read reviews about the game.
 
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David desJardins
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Jugular wrote:
Some interesting analysis there David. I think there are a few errors in your description and possibly in the analysis. You keep mentioning White 11 before White controls 12. I assumed you had to get there 'by land' so to speak. This obviously then affects the analysis as black has a more defensible area of territory. Forgive me if I have the rules wrong I've only ever read reviews about the game.


I think every time I mention White 11 it is after White 12.
 
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George Leach
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Perhaps I misread, it's quite tough to keep track of the moves when having to scroll as well.
 
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David desJardins
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I suggest printing out the map and consulting it beside the discussion. That also lets you view it in higher resolution.
 
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Russ Williams
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I think there is one typo:
Quote:
After White 26, no side has anything better than Black 10, White 5, Black 20, White 19,

should be:
Quote:
After White 26, no side has anything better than Black 10, White 5, Black 19, White 20,

right?

But yeah, very interesting analysis to read! Thanks very much for that post. I too was thinking the southeast corner was the urgent place due to the clear concrete swing of points there, whereas many other places either don't really decisively block or threaten, or they are easily responded to without making a big point swing.

PS: Thanks for making the better numbered map.
 
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David desJardins
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russ wrote:
I think there is one typo:


Yes, fixed.
 
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Martin Jackson
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Fjords » Forums » Strategy
Re: Informal experiment: how easy to find "the optimal disk placement" in various positions?
I think I'm guilty of invalid moves, I'm afraid the orientation of the board wasn't obvious.

So my vote for B3 SouthEast should actually be East. (B10 on the latest map)
My W1 SW is still the same move. (W20 on the latest map)

I chose these because they seemed like the biggest forcing moves.

The hex between the houses in the bottom right was a close second for me, but as David has said, that's a non-forcing move, so seems less urgent.
 
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David desJardins
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moik wrote:
My W1 SW is still the same move. (W20 on the latest map).


The thing is, if White plays 20 and Black responds with 19, or if Black plays 19 and White responds with 20, then you end up in the same position, and it's a very quiet position (low temperature, little incentive to move there). So moving down there means, either your opponent responds and you haven't gained anything, or maybe your opponent finds a better move. Little upside for you.
 
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P. Mihalarias
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Well, this has my vote for thread-of-the-year!

Though I don't have anything to add to the discussion of this particular problem, I have been pondering some related questions re. general solvability in Fjords that I would like to put to any experts who may be reading.

First, it is pleasing to note that the results posted here so far seem mirror my own experience of the game, which is to say that early optimization in the farm-placement phase typically involves nontrivial decisions. (Or maybe it just feels that way to those of us who are not mathematicians! cry)

In any case, I'd like to understand the parameterization effects of Fjord's variable board on its complexity. For instance, if you vary the number of available tiles and farms (as when you double the resources when playing the MegaFjords variant that uses two sets), does this necessarily increase the decision complexity in the early farm-placment phase?

More precisely, assuming each player has inexhaustible field markers: is there a ratio of tiles:farms that generalizes the complexity class of the early farm-placement game , e.g., as PSPACE- or even EXPTIME-complete? Or, does the fact that players always start with incomplete knowledge of the map configuration, regardless of the number of tiles used, mean that it is impossible to generalize completeness?

I ask this because my instinct suggests that, if the resource ratios are tuned just right, the tree of potential interactions in Fjords is as computationally rich as Go. But, alas, I lack the math skills to prove or disprove it.



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Russ Williams
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Tamburlain wrote:
Or, does the fact that players always start with incomplete knowledge of the map configuration, regardless of the number of tiles used, mean that it is impossible to generalize completeness?

Hmm, interesting question; indeed most such computational complexity results about games which I remember seeing seem to be about perfect-information games. But perhaps that's only due to tradition and the math being easier / more elegant/interesting...? I'm not sure there's any a priori reason one couldn't also consider how much computation is necessary to solve (in a weaker/different sense like maximizing a player's probability of winning) a game with luck of the draw like Fjords.
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P. Mihalarias
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russ wrote:

Hmm, interesting question; indeed most such computational complexity results about games which I remember seeing seem to be about perfect-information games. But perhaps that's only due to tradition and the math being easier / more elegant/interesting...? I'm not sure there's any a priori reason one couldn't also consider how much computation is necessary to solve (in a weaker/different sense like maximizing a player's probability of winning) a game with luck of the draw like Fjords.


Yes, good point, and I should've better articulated that I am speaking of the second phase of the game, which should yield perfect information, even though the map may be highly asymmetric and the starting positions highly partisan.

The problem, at least as I understand it, is that there is no clearly suitable way to generalize the crucial second phase, say as an arbitrarily large n-by-n homogenous map. With ingenious effort and unlimited tiles, there very well may be a way to create a tessellated Fjords map, and that would solve the generalizability problem, but of course then we'd no longer be playing Fjords. Unfortunately, as far as I know, generalizability is required in order to determine the complexity class of any game. (And again, by "game" in this context I always mean Fjord's 2nd phase.) Calculating the decision complexity with respect to any given map, such as the one you've posted here, should be straightforward, assuming one had access to right hardware. But we'd never know if the results were generalizable.

Let me back up. One often hears criticism of the second phase of the game for being trivially solvable. The rejoinder to this criticism is that during the first phase of the game, a good Fjords player seeks to influence the solvability of the second phase by strategic tile and farm placement (to the extent that luck of the draw allows). From a player's perspective, the whole goal of the first phase of Fjords is to render the second phase solvable in a sense, but in a highly partisan manner.

From a game designer's perspective (or in my case just from game tinker's POV) it would be interesting to understand the complexity class of the second phase, as it presumably constrains the degree to which players can influence solvability and partisanship in the first phase. For instance, even players who have never used the Mega-Fjords variant seem to intuit that a larger number of tiles and a few extra farms naturally make the game feel more Go-like during the second phase, more strategic. But what does this really mean? Does the game flip into EXPTIME-completeness? Or is this intuition just an illusion? To put it to the test, the idea would be to dial the tile:farm ratio up and down to explore any parametric effects on complexity.

But... if measuring computational complexity is a nonstarter due to unclear generalization, then I am stymied as to what other metric, if any, could be used instead? Hence, me sad.
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David desJardins
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Obviously, there's only a finite number of Fjords maps you can generate, just as there are only a finite number of Chess or Go positions. So in principle you could just build a table and look up the answer to any one. But in any of these games it's easy to imagine scaling to larger positions, just by adding more tiles, using a bigger board, etc. This is what people do when they talk about the computational complexity of a game. And it's clear that solving large Fjords positions perfectly is going to have high complexity, just as, e.g., Go does. Indeed I believe Elwyn Berlekamp was interested in constructing generalized Fjords positions that have similar characteristics to Go positions. I don't know what progress he's made on that, though.
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