Talk:Game complexity

message: I've added a comment to Talk:Connect6 and have modified Connect6. The number 140 for the log game-tree complexity of Connect6 given on that page is uncited and based on a questionable assumption. If one makes a different (and perhaps more reasonable) assumption one gets 70 (in other words, the same as that for Gomoku). I'll update this page to say "70-140?" for Connect6, but perhaps it should simply be left as "?". kfgauss (talk) 20:57, 13 January 2008 (UTC)
replies: text: The game length on this page, which was listed as 200 and uncited, made no sense (note that the entire board would be filled by that point). I've updated it to 15-30?, which is what is mentioned on Connect6, based on analogy with published numbers for Gomoku. A quick check of some games by top players on Little Golem suggests this range is at least in the right ballpark. kfgauss (talk) 21:09, 13 January 2008 (UTC)
thread_title:  Connect6 game-tree complexity 

message: What about Arimaa?
Does anybody have a clue about that one?
132.231.54.1 21:15, 1 Feb 2005 (UTC)
thread_title:  Other board games 

message: I would like to see a reference to the numbers in the complexity table, i did some calculations for the othello game and i'm getting lower values that the ones show in the table.
Abulmo (talk) 14:29, 25 November 2013 (UTC)
thread_title:  Where the numbers in the table come from 

message: I think it would be most clarifying to make the complexity of various board games directly comparable by adding columns to the table for the average branching factors and average moves per game (for a rationally played game).  It may not be easy to retrieve this information for some entries, however.  --BadSanta
: I agree. Anyone know any numbers/calculations? Give me formulas and I can calculate. 70.111.251.203 19:09, 21 February 2006 (UTC)
thread_title:  Average Branching Factors and Average Moves Per Game 

message: An editor suggested that generalized Connect6 might be EXPTIME-complete. I removed this, because it's almost certainly false: if I understand the rules of Connect6 correctly, since pieces never move or capture once placed, a game can last no more  moves than there are spaces on the board, and that's a polynomial function of the board size. It may well be PSPACE-complete, using similar methods to those used to show that generalized Gomoku is PSPACE-complete, but I think we need a reference. Gdr 01:16, 3 January 2006 (UTC)
thread_title:  Complexity class for Connect6 

message: Shouldn't this article be under combinatorial game theory and not just regular game theory?70.111.251.203 03:05, 7 February 2006 (UTC)
replies: text: Sure. I've changed it, but in future, Be bold. ··gracefool |☺ 04:35, 7 February 2006 (UTC)
thread_title:  Under which category? 

message: Although certain game exceed other games in complexity by state space and/or game tree, this is largely do to the size of the boards, numbers of pieces, etc. Thus the table is misleading; the game of Chess has a board size of 8x8, thus 64 squares, Go has a 19x19 board, with 361 intersections. Go is over 5 times as large, while the game complexity is comparatively only approximately 3 times as complex (4 if   you're generous, doesn't matter).  {correction - the numbers for Go were horrendously wrong when you were looking at the comparison.  The actual numbers for Go are 10 to the 172 for state space and 10 to the 766 for game tree.} Because of the lack of a 1:1 ratio, the games can't be compared on equal ground. The game of chess is actually more complex for it's size. I'm not sure how pieces play into this. Here is a table:
Game             	log(State space)	log(Game tree)	Complexity class of suitable generalized game	Board size (number of squares/intersections for a piece)
Tic-tac-toe      	3               	5             	PSPACE-complete                              	9
Nine Men's Morris	10              	50            	?                                            	?
Awari            	12              	32            	?                                            	?
Pentominoes      	12              	18            	?                                            	?
Connect Four     	14              	21            	?                                            	42
American checkers	21              	31            	EXPTIME-complete                             	64 or 32?
Lines of Action  	24              	56            	?                                            	64?
Othello          	28              	58            	PSPACE-complete                              	64
Backgammon       	20              	144           	?                                            	24?
Chess            	46              	123           	EXPTIME-complete                             	64
Chess960         	46              	123           	EXPTIME-complete                             	64
Xiangqi          	75              	150           	EXPTIME-complete                             	90
Shogi            	71              	226           	EXPTIME-complete                             	81
Go               	172             	360           	EXPTIME-complete                             	361
Connect6         	172?            	140?          	?                                            	any size depending on board size, but if on 19x19, then 361, can be larger though
Arimaa           	54              	440           	?                                            	64
This chart actually shows how Shogi and Arimaa are the most complex games per space. Shogi because of both numbers, and Arimaa only because of the Game Tree. Does anyone know where that 440 number for Arimaa's Game Tree came from? It doesn't make much since considering how you can choose the initial setup of pieces, but there are superior setups, and almost always people put all their rabbits in the back row. 
70.111.251.203 14:08, 7 February 2006 (UTC)
replies: text: Maybe there is a relationship that should be made between the state space and the game tree too. 70.111.251.203 13:34, 9 February 2006 (UTC)
replies: text: not sure where Arimaa gets a mention in that discussion but a key feature is to look at the strength of computer playing programs the single critical piece makes chess a lot simpler strategically.Tetron76 (talk) 23:05, 27 February 2011 (UTC)
thread_title:  Chart of Complexities 

message: There are a lot of different variants of poker, and the complexities vary depending on the number of players; maybe we should create a separate page just for the poker variants? Ben Standeven 20:31, 9 March 2006 (UTC)
replies: text: If you have the numbers, put them on here for now. No point in making a seperate page if there is very little information on it. 128.6.175.59 21:09, 13 April 2006 (UTC)
replies: text:  Problem is, I've tried repeatedly to add a small texas hold'em poker section, but a certain activist editor keeps removing it without explanation. After three thwarted attempts I've given up. I'm not going to waste my life on endless edit wars. Oh well, it's wikipedia's loss. People will have to go to other websites for the info. 21 August 2010  —Preceding unsigned comment added by 98.216.12.97 (talk) 17:15, 21 August 2010 (UTC)
replies: text: I believe the scope of the table is combinatorial games, and most are games of perfect information. Poker is an interesting game, but I don't believe it is a combinatorial game (or if so, would pose problems if mapped out on a game tree). It is also not a game of perfect information (although some other games in the list aren't either, such as backgammon). Can we define a scope of the table, and add it either before the table, or in the article's introduction?—LithiumFlash (talk) 15:42, 7 April 2017 (UTC)
thread_title: Poker

message: I removed the question mark from the Arimaa average game length of 35.  That is in fact the average length of rated games between humans on arimaa.com, as calculated from the publicly available database of games played there.  Games involving two computers are slightly longer on average because sometimes neither computer can come up with a plan.  Games between expert humans (rated over 1900) are also longer on average, about 45 moves per game.  However, compared to chess, top-level Arimaa is intrinsically a slightly shorter game.  The greater length of Arimaa games occurs because agreed draws are forbidden and resignation is highly discouraged on arimaa.com.  If resignation and agreed draws were forbidden in chess, the average expert game of chess would probably be well over 45 moves.  --Fritzlein 17:53, 19 June 2006 (UTC)
thread_title:  Moves per game in Arimaa 

message: I added references to the article, then removed claims about complexity classes that I couldn't find a reference for (magic fingers, xiangqi, connect6) and then removed entries from the table where no information was being provided (magic fingers, seven-card stud poker, contract bridge, mahjong).  I also took removed Chess960 since from a complexity point of view it's much the same as ordinary chess.
I was slightly surprised that I couldn't find a proof that xiangqi is EXPTIME-complete.  Am I missing it?  Gdr 17:19, 3 May 2007 (UTC)
thread_title:  Tidying 

message: This article starts Game complexity is a measure of the complexity of a game. Does this sentence say anything at all. Also, does complexity here have anything to do with the everyday meaning of the word? --Apoc2400 06:56, 29 June 2007 (UTC)
replies: text: "Complexity" is being used in two different technical meanings in this article. Firstly, in "state complexity" and "tree complexity" it's just a fancy way of saying "size" or "count". Second, in "computational complexity" it has the meaning from the field of complexity theory, which means, very roughly, "the asymptotic rate at which computational resources need to increase as problem instances grow in size". Gdr 12:43, 30 November 2007 (UTC)
thread_title:  The first sentence 

message: I corrected the figures for the state space complexity for Chinese Checkers, which were too high. There are 121 cells, and there may be 20 to 60 pieces on board, as the number of players varies from 2 through 6. For 2 players, the maximum number of positions possible is 121C20 = (121!)/(101!.20!) = 3.5e22. For 6 players, 121C60 = (121!)/(60!.61!) = 1.9e35. -- Brhaspati\talk/contribs 06:54, 2 December 2007 (UTC)
thread_title:  chinese checkers 

message: I've added a table entry for Amazons, which is PSPACE-complete (reference provided in the Amazons article). I could not find the true state space complexity of the game anywhere, but I have upper-bounded it as follows: 100 squares on the board, of which 4 are white queens, 4 are black queens and 92 are free of queens. This can be done in (100!)/(92!4!4!) ways. Each of the 92 queen-free squares can either have an arrow or be empty, causing another factor 2^92. The upper bound is the product of these two numbers, evaluating to 6.45e40. Some of these positions may not be achievable in the course of a game, hence the number is an upper bound on the real state space complexity. If someone gets a real reference, please update the entry. -- Brhaspati\talk/contribs 07:03, 2 December 2007 (UTC)
replies: text: You ought to divide by 8 for the symmetries of the board. Gdr 12:18, 2 December 2007 (UTC)
replies: text: Yup, you're right. New upper bound is 8.06e39. Updating article. -- Brhaspati\talk/contribs 14:48, 2 December 2007 (UTC)
thread_title:  Amazons 

message: An anonymous editor added an entry for Irensei, assigning it a state-space complexity of "?171", a game-tree complexity of "360", and an average game length of "80". No reference was given, so I removed it for now. We could put it back if there were a reference. Gdr 12:18, 2 December 2007 (UTC)
thread_title:  Irensei 

message: I'm surprised Dots is not on here.  Does anyone have any info on this, regardless of grid size?  you know that 2x2 grid is won by 1st player.  So thats a start.  —Preceding unsigned comment added by Doedicurus (talk • contribs) 05:17, 25 January 2008 (UTC)
replies: text: You mean dots and boxes, right? Winning Ways shows that it is NP-hard. No completeness result is known (I would guess it's PSPACE-complete, but that's only a guess). Gdr 14:14, 26 April 2008 (UTC)
thread_title:  Dots 

message: The article says that game tree complexity is the smallest full-width decision tree, which is a sub-tree of the game tree.  However, the game tree article says that game tree complexity is the number of leaf nodes of the entire game tree.  Are they saying the same thing?  It seems like this article says that game tree complexity concerns only a sub-tree while the game tree article says game tree complexity concerns the entire game tree.  —Preceding unsigned comment added by MengTzu622 (talk • contribs) 00:31, 4 August 2008 (UTC)
replies: text: The definition in this article (the smallest full-width decision tree) is the one given in Victor Allis's thesis. The definition in game tree is wrong. It's my mistake, so I will fix it. Gdr 02:02, 26 September 2008 (UTC)
thread_title:  Question about Game Tree Complexity 

message: "The game tree size is  total number of possible games that can be played: it's the number of leaf nodes in the game tree"
These are not the same. 217.229.13.214 (talk)  —Preceding undated comment added 12:04, 15 August 2010 (UTC)
replies: text: NOTE: This appears to have since been answered in the above section (at a date after both this post and the original post of the previous section). -SColombo (talk) 06:38, 28 March 2012 (UTC)
thread_title:  Question about Game Tree Complexity 

message: The claimed complexity for International Draughts, EXPTIME-complete, is incorrect. Or rather, the cited paper is not about International Draughts; it is about Checkers. The rules are sufficiently different that the results do not apply to International Draughts. As an unbounded 2-player game, it ought to be EXPTIME-complete as well, but I know of no proof. The only result I know of is that determining whether an individual move is legal is NP-complete: is-it-np-hard-to-play-international-draughts-correctly Bobhearn (talk) 06:51, 17 January 2011 (UTC)
thread_title:  International Draughts 

message: What do people think of adding average branching factor as a column in the table?  I think it says a lot about the complexity of a Game Tree beyond just what comes out in the final number of game tree complexity. (Granted, the table is a bit crowded, so it might take a little wikitext-juggling to make it fit).
Thoughts?
-SColombo (talk) 06:40, 28 March 2012 (UTC)
thread_title:  Average [[branching factor]] 

message: There should be a single ref. column to include all references per row. Of course, this ref. column would be to the far right of the complexity table, as the references would apply to the entire row. Having multiple ref. columns down the middle of the table is clutter. KyuuA4 (Talk:キュウ) 15:23, 29 March 2012 (UTC)
thread_title:  Single Ref column 

message: contrary to claim in Game_complexity#Complexities_of_some_well-known_games section, 3 by 3 tic-tac-toe is not PSPACE-complete. As per the associated link, it is more complex similar "first n in a row" games like gomoku that might be pspace complete. 76.119.30.87 (talk) 06:34, 12 February 2015 (UTC)
replies: text: You did read the "suitable generalized game" in the heading of the column that gives the complexity, right? —David Eppstein (talk) 06:37, 12 February 2015 (UTC)
thread_title:  tic-tac-toe not PSPACE complete 

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thread_title:  External links modified 

message: The citation failed verification - I can find no complexity data in that source. The branching factor cannot possibly be 280 asstated - the board only has 121 playable spaces, so I've corrected that to reflact the 40 ply game: the average number of vacant spaces is (121+(121-81))=101.  Now the game tree size is 121!*120!*...82! = 121!/81! = 1.6*1080, so I've also substituted that number.  If anyone can cite a source for the 40 ply average game length, we're home.  In my experience, games are a little longer, maybe 40-50 ply, but my games are not a valid source.Sbalfour (talk) 02:19, 19 January 2017 (UTC)
thread_title:  Game tree complexity for Hex 

message: Has anybody scrutinized the complexity of Hearthstone already?--Powerlts (talk) 19:11, 26 May 2017 (UTC)
replies: text: In a manner of speaking, yes - Programming a Hearthstone agent using Monte Carlo Tree Search Jelleecat (talk) 20:34, 12 January 2019 (UTC)
replies: text: I actually thought I was including this, so I'll do so now - Monte Carlo Tree Search experiments in Hearthstone the point is still valid! Jelleecat (talk) 20:49, 12 January 2019 (UTC)
replies: text:  Wikicoding hint: for external links a single pair of square brackets is enough. --CiaPan (talk) 06:42, 14 January 2019 (UTC)
replies: text:  You're a doll, thank you! Am I allowed to edit those into single square brackets? ;) Jelleecat (talk) 02:09, 18 January 2019 (UTC)
replies: text:  Well done! :) --CiaPan (talk) 11:44, 18 January 2019 (UTC)
thread_title:  Hearthstone 

message: I'm kinda surprised to see Beheweled and Candy Crush in the page. However I wonder if the following differences between the two games have been considered:
1. Bejeweled is the one that uses 8x8 grids. Candy Crush uses 9x9 (royalgames.com version).
2. Bejeweled uses 7 colors (except diamond mine mode in Bejeweled 3), while Candy Crush uses up to 6 colors.
User670839245 (talk) 23:28, 7 July 2017 (UTC)
replies: text: These differences are irrelevant to the analysis on this page, because they are abstracted away as variables. Stellaathena (talk) 06:08, 20 November 2021 (UTC)
thread_title:  Bejeweled and Candy Crush 

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thread_title:  External links modified 

message: Schaeffer released a correction to an older estimation of Checker's/English draughts' game tree complexity at 10^31 and established that the true value was ~10^40. I corrected that in the game-tree complexity table with the source here:
https://icga.org/icga/journal/contents/Schaeffer07-01-08.pdf
There's some indication however that the same is true of International draughts and that the game-tree complexity is actually much higher. As high as 10^92.
Has anyone done anymore research on this, and is there newer literature on the subject either acknowledging or "debunking" the error?
source:
https://laatste.info/bb3/viewtopic.php?t=7817
archived source:
https://archive.is/GkfT1
T0afer (talk) 14:44, 30 August 2021 (UTC)
thread_title:  International Draughts much more complex than the listed 10^54 value? 

message: Currently the specific games table contains estimates for various games including Magic the Gathering at the bottom. The claim that Magic the Gathering has infinite/unbounded state-space or gametree complexity is unfounded, obviously incorrect (with a finite number of cards each with finite numbers of targets, the decision-space is obviously finite - except in the sense that repetitions may be allowed, but in this sense a game like chess should also then be called 'infinite'; even videogames, with a much larger decision-space than Magic the Gathering, still have finite though enormous gametrees simply by discretization of time into frames and of space into pixels), and not supported or claimed by the research linked. The papers linked analyse the computational complexity of generalizing Magic the Gathering (which they find to be as hard as arithmetic), not the size of the game in practice as it exists in the real-world (which is clearly finite as the number of cards is finite).  — Preceding unsigned comment added by 2A00:23C8:218D:FA01:1D9A:1ADE:E33D:6 (talk) 20:17, 12 February 2022 (UTC)
thread_title:  Magic The Gathering - sketchy claims regarding gametree size 

message: Hello. The article seems quite perfect to me, but I have a minor question regarding adding Scrabble to the table. 
"The game [of Scrabble] is sold in 121 countries and is available in more than 30 languages; approximately 150 million sets have been sold worldwide, and roughly one-third of American and half of British homes have a Scrabble set."
So it is really popular.
However, this article tells us nothing about Scrabble, and nobody noticed Scrabble even on the discussion page.
I just wonder if there are some specific reasons for that. Maybe it's huge amount of legal board positions, or total number of possible games, or the number of possible words which can be formed at every move etc.
2A00:1858:1047:9BEF:74A6:7DAD:2EB6:38AD (talk) 13:43, 15 July 2023 (UTC)
thread_title:  Why is there no Scrabble in the main table?