Game complexity: Difference between revisions

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{{Short description|Notion in combinatorial game theory}}

{{Use mdy dates|cs1-dates=ly|date=May 2023}}

[[Combinatorial game theory]] has several ways of [[Computational complexity theory|measuring]]measures '''game [[complexity]]'''. Thisin articleseveral describes five of themways: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.

#State-space complexity (the number of legal game positions from the initial position)

#Game tree size (total number of possible games)

#Decision complexity (number of leaf nodes in the smallest decision tree for initial position)

#Game-tree complexity (number of leaf nodes in the smallest full-width decision tree for initial position)

#Computational complexity (asymptotic difficulty of a game as it grows arbitrarily large)

These measures involve understanding the game positions, possible outcomes, and [[Computational complexity theory|computational complexity]] of various game scenarios.

==Measures of game complexity==

=== State-space complexity ===

The '''state-space complexity''' of a game is the number of legal game positions reachable from the initial position of the game.<ref name="Allis1994"/>

When this is too hard to calculate, an [[upper bound]] can often be computed by also counting (some) illegal positions, meaning (positions that can never arise in the course of a game).

=== Game tree size ===

The '''game tree size''' is the total number of possible games that can be played:. This is the number of [[Leaf nodes#Terminology|leaf nodes]] in the [[game tree]] rooted at the game's initial position.

The game tree is typically vastly larger than the state -space because the same positions can occur in many games by making moves in a different order (for example, in a [[tic-tac-toe]] game with two X and one O on the board, this position could have been reached in two different ways depending on where the first X was placed). An upper bound for the size of the game tree can sometimes be computed by simplifying the game in a way that only increases the size of the game tree (for example, by allowing illegal moves) until it becomes tractable.

For games where the number of moves is not limited (for example by the size of the board, or by a rule about repetition of position) the game tree is generally infinite.

=== Decision trees ===

TheA next two measures use the idea of a ''[[decision tree]]'', which is a subtree of the game tree, with each position labelled with "player A wins", "player B wins", or "drawndraw", if that position can be proved to have that value (assuming best play by both sides) by examining only other positions in the graph. (Terminal positions can be labelled directly; a position withdirectly—with player A to move, a position can be labelled "player A wins" if any successor position is a win for A, or labelled; "player B wins" if all successor positions are wins for B,; or labelled "draw" if all successor positions are either drawn or wins for B. (With Andplayer correspondinglyB forto positionsmove, withcorresponding Bpositions toare movemarked similarly.)

The following two methods of measuring game complexity use decision trees:

==== Decision complexity ====

'''Decision complexity''' of a game is the number of leaf nodes in the smallest decision tree that establishes the value of the initial position.

==== Game-tree complexity ====

The '''gameGame-tree complexity''' of a game is the number of leaf nodes in the smallest ''full-width'' decision tree that establishes the value of the initial position.<ref name="Allis1994"/> A full-width tree includes all nodes at each depth. This is an estimate of the number of positions one would have to evaluate in a [[minimax]] search to determine the value of the initial position.

It is hard even to estimate the game-tree complexity, but for some games an approximation can be given by <math>GTC \geq b^d</math>, where {{Mvar|b}} is the game's average [[branching factor]] and ''{{Mvar|d}}'' is the number of [[Ply (chess)|plies]] in an average game.

This is an estimate of the number of positions one would have to evaluate in a [[minimax]] search to determine the value of the initial position.

It is hard even to estimate the game-tree complexity, but for some games an approximation can be given by raising the game's average [[branching factor]] ''b'' to the power of the number of [[Ply (chess)|plies]] ''d'' in an average game, or:

<math>GTC \geq b^d</math>.

=== Computational complexity ===

The '''[[Computational complexity theory|''computational complexity'']]''' of a game describes the [[Asymptotic analysis|asymptotic]] difficulty of a game as it grows arbitrarily large, expressed in [[big O notation]] or as membership in a [[complexity class]]. This concept doesn't apply to particular games, but rather to games that have been [[generalized game|generalized]] so they can be made arbitrarily large, typically by playing them on an ''n''-by-''n'' board. (From the point of view of computational complexity, a game on a fixed size of board is a finite problem that can be solved in O(1), for example by a look-up table from positions to the best move in each position.)

The asymptotic complexity is defined by the most efficient algorithm for solving the game (in terms of whatever [[computational resource]] one is considering). algorithm for solving the game; theThe most common complexity measure, ([[computation time]]), is always lower-bounded by the logarithm of the asymptotic state-space complexity, since a solution algorithm must work for every possible state of the game. It will be upper-bounded by the complexity of any particular algorithm that works for the family of games. Similar remarks apply to the second-most commonly used complexity measure, the amount of [[DSPACE|space]] or [[computer memory]] used by the computation. It is not obvious that there is any lower bound on the space complexity for a typical game, because the algorithm need not store game states; however many games of interest are known to be [[PSPACE-hard]], and it follows that their space complexity will be lower-bounded by the logarithm of the asymptotic state-space complexity as well (technically the bound is only a polynomial in this quantity; but it is usually known to be linear).

* The [[depth-first search|depth-first]] [[minimax strategy]] will use [[computation time]] proportional to the game's tree-complexity, (since it must explore the whole tree), and an amount of memory polynomial in the logarithm of the tree-complexity, (since the algorithm must always store one node of the tree at each possible move-depth, and the number of nodes at the highest move-depth is precisely the tree-complexity).

* [[Backward induction]] will use both memory and time proportional to the state-space complexity, as it must compute and record the correct move for each possible position.

==Example: tic-tac-toe (noughts and crosses)==

For [[tic-tac-toe]], a simple upper bound for the size of the state space is 3⁹ = 19,683. (There are three states for each cellof andthe nine cells.) This count includes many illegal positions, such as a position with five crosses and no noughts, or a position in which both players have a row of three. A more careful count, removing these illegal positions, gives 5,478.<ref>{{Cite web|url=https://math.stackexchange.com/questions/485752/tictactoe-state-space-choose-calculation|title=combinatorics - TicTacToe State Space Choose Calculation|website=Mathematics Stack Exchange|access-date=2020-04-08}}</ref><ref>{{Citationcite web|last=T|first=Brian|title=Btsan/generate_tictactoe|website=[[GitHub]] |date=2018-10-20|url=https://github.com/Btsan/generate_tictactoe|access-date=2020-04-08}}</ref> And when rotations and reflections of positions are considered identical, there are only 765 essentially different positions.

To bound the game tree, there are 9 possible initial moves, 8 possible responses, and so on, so that there are at most 9! or 362,880 total games. However, games may take less than 9 moves to resolve, and an exact enumeration gives 255,168 possible games. When rotations and reflections of positions are considered the same, there are only 26,830 possible games.

The computational complexity of tic-tac-toe depends on how it is [[generalized game|generalized]]. A natural generalization is to [[m,n,k-game|''m'',''n'',''k''-games]]: played on an ''m'' by ''n'' board with winner being the first player to get ''k'' in a row. It is immediately clear that thisThis game can be solved in [[DSPACE]](''mn'') by searching the entire game tree. This places it in the important complexity class [[PSPACE]]. With; somewith more work, it can be shown to be [[PSPACE-complete]].<ref name="Reisch1980b">{{cite journal | author = Stefan Reisch | title = Gobang ist PSPACE-vollständig (Gobang is PSPACE-complete) | journal = Acta Informatica | volume = 13 | issue = 1 | pages = 59–66 | year = 1980 | doi=10.1007/bf00288536| s2cid = 21455572}}</ref>

==Complexities of some well-known games==

Due to the large size of game complexities, this table gives the ceiling of their [[logarithm]] to base 10. (In other words, the number of digits). All of the following numbers should be considered with caution: seemingly- minor changes to the rules of a game can change the numbers (which are often rough estimates anyway) by tremendous factors, which might easily be much greater than the numbers shown.

Note: ordered by game tree size

{{sticky header}}

{| class="wikitable sortable"

{| class="wikitable sortable sticky-header"

|-

!Game

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|style="text-align:right;"|3.7

|style="text-align:right;"|

|[[PSPACE-complete]]<ref>{{cite conference

|[[PSPACE-complete]]<ref>{{cite book|url=http://dl.acm.org/citation.cfm?id=647481.728124|title=The Complexity of Graph Ramsey Games|first=Wolfgang|last=Slany|date=26 October 2000|publisher=Springer-Verlag|pages=186–203|access-date=12 April 2018|via=dl.acm.org|isbn=9783540430803}}</ref>

| last = Slany | first = Wolfgang

| editor1-last = Marsland | editor1-first = T. Anthony

| editor2-last = Frank | editor2-first = Ian

| contribution = The complexity of graph Ramsey games

| doi = 10.1007/3-540-45579-5_12

| pages = 186–203

| publisher = Springer

| series = Lecture Notes in Computer Science

| title = Computers and Games, Second International Conference, CG 2000, Hamamatsu, Japan, October 26-28, 2000, Revised Papers

| volume = 2063

| year = 2000}}</ref>

|-

|[[Pentomino]]es

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|style="text-align:right;"|10

|style="text-align:right;"|75

|style="text-align:right;"|<ref name="GamesSolved"/><ref>{{cite conference

|style="text-align:right;"|<ref name="GamesSolved"/><ref>Hilarie K. Orman: ''Pentominoes: A First Player Win'' in ''[http://www.msri.org/publications/books/Book29/contents.html Games of no chance]'', MSRI Publications – Volume 29, 1996, pages 339-344. Online: [http://www.msri.org/publications/books/Book29/files/orman.pdf pdf].</ref>

| last = Orman | first = Hilarie K.

| editor-last = Nowakowski | editor-first = Richard J.

| contribution = Pentominoes: a first player win

| contribution-url = https://www.msri.org/publications/books/Book29/files/orman.pdf

| isbn = 0-521-57411-0

| mr = 1427975

| pages = 339–344

| publisher = Cambridge University Press

| series = Mathematical Sciences Research Institute Publications

| title = Games of No Chance: Papers from the Combinatorial Games Workshop held in Berkeley, CA, July 11–21, 1994

| volume = 29

| year = 1996}}</ref>

| {{dunno}}, but in [[PSPACE]]

|-

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|style="text-align:right;"|8

|style="text-align:right;"|<ref name="GamesSolved"/>

| {{dunno}}, but in [[PSPACE]]; in [[P (complexity class)|P]] for certain dimensions<ref>{{citationcite |author=Michael Lachmann |author2=Cristopher Moore |author3=Ivan Rapaport |title=Who wins domineering on rectangular boards? |volume=MSRI Combinatorial Game Theory Research Workshop |date=July 2000}}</ref>conference

| last1 = Lachmann | first1 = Michael

| last2 = Moore | first2 = Cristopher

| last3 = Rapaport | first3 = Ivan

| editor-last = Nowakowski | editor-first = Richard

| contribution = Who wins Domineering on rectangular boards?

| isbn = 0-521-80832-4

| mr = 1973019

| pages = 307–315

| publisher = Cambridge University Press

| series = Mathematical Sciences Research Institute Publications

| title = More Games of No Chance: Proceedings of the 2nd Combinatorial Games Theory Workshop held in Berkeley, CA, July 24–28, 2000

| volume = 42

| year = 2002}}</ref>

|-

|[[Congkak]]

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|style="text-align:right;"|70

|style="text-align:right;"|2.8

|style="text-align:right;"|<ref name="Allis1994"/><ref>{{cite journal | author = Jonathan Schaeffer| title = Checkers is Solved | journal = Science | date = July 6, 2007 | doi=10.1126/science.1144079 | volume=317 |issue= 5844 |pages= 1518–1522 | pmid=17641166 |display-authors=etal|bibcode=2007Sci...317.1518S | s2cid = 10274228 | doi-access= free }}</ref><ref>Schaeffer,{{cite Jonathanjournal
| (2007).last "Game= over:Schaeffer Black| tofirst play= and draw in checkers". ''[[ICGA Journal]]''. '''30''' (4): 187–197. [[CiteSeerX (identifier)Jonathan
|CiteSeerX]] 10.1.1.154.255.doi [[Doi= (identifier)|doi]]:10.3233/ICG-2007-30402.</ref>

| issue = 4

| journal = [[ICGA Journal]]

| pages = 187–197

| title = Game over: Black to play and draw in checkers

| url = https://ticc.uvt.nl/icga/journal/contents/Schaeffer07-01-08.pdf

| archive-url = https://web.archive.org/web/20160403093928/https://ticc.uvt.nl/icga/journal/contents/Schaeffer07-01-08.pdf

| archive-date = 2016-04-03

| url-status = dead

| volume = 30

| year = 2007}}</ref>

|[[EXPTIME-complete]]<ref name="robson1984">{{cite journal | author = J. M. Robson | title = N by N checkers is Exptime complete | journal = [[SIAM Journal on Computing]] | volume = 13 | issue = 2 | pages = 252–267 | year = 1984 | doi = 10.1137/0213018}}</ref>

|-

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|[[PSPACE-complete]]<ref name="Reisch1980"/>

|-

|[[Computer bridge|Double dummy bridge]]{{refn|group=nb|'''Double dummy bridge''' (i.e., double dummy problems in the context of [[contract bridge]]) is not a proper board game but has a similar game tree, and is studied in [[computer bridge]]. The bridge table can be regarded as having one slot for each player and trick to play a card in, which corresponds to board size 52. Game-tree complexity is a very weak upper bound: 13! to the power of 4 players regardless of legality. State-space complexity is for one given deal; likewise regardless of legality but with many transpositions eliminated. Note that theThe last 4 plies are always forced moves with branching factor 1.}}

|style="text-align:right;"|(52)

|style="text-align:right;"|<17

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|style="text-align:right;"|5.6

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|align="left"|PSPACE-complete<ref>{{cite conference

|align="left"|PSPACE-complete<ref>{{Cite book|last1=Bonnet|first1=Édouard|last2=Jamain|first2=Florian|last3=Saffidine|first3=Abdallah|date=2013-08-03|title=On the complexity of trick-taking card games|url=http://dl.acm.org/citation.cfm?id=2540128.2540199|publisher=AAAI Press|pages=482–488|isbn=9781577356332}}</ref>

| last1 = Bonnet | first1 = Edouard

| last2 = Jamain | first2 = Florian

| last3 = Saffidine | first3 = Abdallah

| editor-last = Rossi | editor-first = Francesca

| contribution = On the complexity of trick-taking card games

| contribution-url = https://www.aaai.org/ocs/index.php/IJCAI/IJCAI13/paper/view/6920

| pages = 482–488

| publisher = IJCAI/AAAI

| title = IJCAI 2013, Proceedings of the 23rd International Joint Conference on Artificial Intelligence, Beijing, China, August 3-9, 2013

| year = 2013}}</ref>

|-

|[[Fanorona]]

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|style="text-align:right;"|180

|style="text-align:right;"|<ref name=Bell_Halma>{{cite journal|author=G.I. Bell|title=The Shortest Game of Chinese Checkers and Related Problems|journal=Integers|year=2009|volume=9|doi=10.1515/INTEG.2009.003|arxiv=0803.1245|bibcode=2008arXiv0803.1245B|s2cid=17141575}}</ref>

|[[EXPTIME]]-complete<ref name=pebble>{{cite journal

|[[EXPTIME]]-complete <ref name=pebble>{{cite journal|title=Classes of Pebble Games and Complete Problems|journal= SIAM Journal on Computing| volume = 8| year = 1979 |pages= 574–586|author1=Takumi Kasai |author2=Akeo Adachi |author3=Shigeki Iwata |doi=10.1137/0208046|issue=4}} Proves completeness of the generalization to arbitrary graphs.</ref>

| last1 = Kasai | first1 = Takumi

| last2 = Adachi | first2 = Akeo

| last3 = Iwata | first3 = Shigeki

| doi = 10.1137/0208046

| issue = 4

| journal = SIAM Journal on Computing

| mr = 573848

| pages = 574–586

| title = Classes of pebble games and complete problems

| volume = 8

| year = 1979}} Proves completeness of the generalization to arbitrary graphs.</ref>

|-

|[[Chinese checkers]] (6 sets)

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|style="text-align:right;"|600

|style="text-align:right;"|<ref name=Bell_Halma/>

|[[EXPTIME]]-complete <ref name=pebble/>

|-

|[[Reversi]] (Othello)

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|style="text-align:right;"|10

|style="text-align:right;"|<ref name="Allis1994"/>

|[[PSPACE-complete]]<ref>{{cite journal

|[[PSPACE-complete]]<ref>{{cite journal |author1=S. Iwata |author2=T. Kasai | title = The Othello game on an n*n board is PSPACE-complete | journal = Theor. Comput. Sci. | issue = 2 | year = 1994 | pages = 329–340 | volume = 123 | doi = 10.1016/0304-3975(94)90131-7| doi-access = free }}</ref>

| last1 = Iwata | first1 = Shigeki

| last2 = Kasai | first2 = Takumi

| doi = 10.1016/0304-3975(94)90131-7 | doi-access = free

| issue = 2

| journal = Theoretical Computer Science

| mr = 1256205

| pages = 329–340

| title = The Othello game on an <math>n\times n</math> board is PSPACE-complete

| volume = 123

| year = 1994}}</ref>

|-

|OnTop (2p base game)

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|style="text-align:right;"|<ref name="GamesSolved"/>

|[[PSPACE-complete]]<ref name="Reisch1980">{{cite journal | author = Stefan Reisch | title = Hex ist PSPACE-vollständig (Hex is PSPACE-complete) | journal = Acta Inform. | issue = 15 | year = 1981 | pages = 167–191}}</ref>

|-

|[[Chess]]

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}} Shannon gave estimates of 10⁴³ and 10¹²⁰ respectively, smaller than the upper bound in the table,

which is detailed in [[Shannon number]].</ref>

|[[EXPTIME-complete]] (without [[Fifty-move rule|50-move drawing rule]])<ref name="Fraenkel1981">{{Citation |author1=Aviezri Fraenkel |author-link=Aviezri Fraenkel |author2=D. Lichtenstein | title = Computing a perfect strategy for n×n chess requires time exponential in n |cite journal = J. Combin. Theory Ser. A | volume = 31 | year = 1981 | pages = 199–214 | doi = 10.1016/0097-3165(81)90016-9 | issue = 2| doi-access = free }}</ref>

| last1 = Fraenkel | first1 = Aviezri S. |author1-link = Aviezri Fraenkel

| last2 = Lichtenstein | first2 = David

| doi = 10.1016/0097-3165(81)90016-9 | doi-access = free

| issue = 2

| journal = [[Journal of Combinatorial Theory, Series A]]

| mr = 629595

| pages = 199–214

| title = Computing a perfect strategy for <math>n\times n</math> chess requires time exponential in <math>n</math>

| volume = 31

| year = 1981}}</ref>

|-

|[[Bejeweled (video game)|Bejeweled]] and [[Candy Crush Saga|Candy Crush]] (8x8)

|style="text-align:right;"|64

|style="text-align:right;"|<50

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|style="text-align:right;"|

|style="text-align:right;"|70

|style="text-align:right;"|<ref name="Gual14">{{cite arXiv |author1=L. Gualà |author2=S. Leucci |author3=E. Natale | title = Bejeweled, Candy Crush and other Match-Three Games are (NP-)Hard | eprint= 1403.5830| year = 2014|class=cs.CC }}</ref>conference

| last1 = Gualà | first1 = Luciano

| last2 = Leucci | first2 = Stefano

| last3 = Natale | first3 = Emanuele

| arxiv = 1403.5830

| contribution = Bejeweled, Candy Crush and other match-three games are (NP-)hard

| doi = 10.1109/CIG.2014.6932866

| pages = 1–8

| publisher = IEEE

| title = 2014 IEEE Conference on Computational Intelligence and Games, CIG 2014, Dortmund, Germany, August 26-29, 2014

| year = 2014}}</ref>

|[[NP-hard]]

|-

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|style="text-align:right;"|95

|style="text-align:right;"|38

|style="text-align:right;"|<ref name="Allis1994">{{cite thesis | author = Victor Allis | author-link = Victor Allis | year = 1994 | title = Searching for Solutions in Games and Artificial Intelligence | degree = Ph.D. |publisher= University of Limburg, Maastricht, The Netherlands | isbn = 90-900748-8-0 | url = https://project.dke.maastrichtuniversity.nl/games/files/phd/SearchingForSolutions.pdf}}</ref><ref name="Hsu2004">{{cite journal | author1 = Shi-Jim Yen, Jr-Chang Chen | author2 = Tai-Ning Yang | author3 = Shun-Chin Hsu | title = Computer Chinese Chess | date = March 2004 | journal = International Computer Games Association Journal | volume = 27 | issue = 1 | pages = 3–18 | url = http://www.csie.ndhu.edu.tw/~sjyen/Papers/2004CCC.pdf | url-status = dead | archive-url = https://web.archive.org/web/20070614111609/http://www.csie.ndhu.edu.tw/~sjyen/Papers/2004CCC.pdf | archive-date = 2007-06-14 | doi=10.3233/ICG-2004-27102 | s2cid = 10336286 }}</ref><ref name="Donghwi">{{cite arXiv | author = Donghwi Park | title = Space-state complexity of Korean chess and Chinese chess | eprint= 1507.06401| year = 2015| class = math.GM }}</ref>

| {{dunno}}, believed to be [[EXPTIME-complete]]

|-

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|[[PSPACE-hard]], and in [[EXPTIME]]

|-

|[[Havannah (board game)|Havannah]]

|style="text-align:right;"|271

|style="text-align:right;"|127

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|style="text-align:right;"|452

|style="text-align:right;"|<ref name="Thesis_Moesker">{{cite thesis | title=TWIXTTxixt: THEORYTheory, ANALYSISAnalysis, ANDand IMPLEMENTATIONImplementation |

url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Thesis_Moesker.pdf |

author = Kevin Moesker | year=2009 | publisher = Maastricht University, Faculty of Humanities and Sciences of Maastricht University }}</ref>

|style="text-align:right;"|

|-

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|style="text-align:right;"|84

|style="text-align:right;"|374 or 299<ref>The lower branching factor is for the second player.</ref>

|style="text-align:right;"|<ref name="Kloetzer_themonte-carlo">{{cite conference

|style="text-align:right;"|<ref name="Kloetzer_themonte-carlo">{{citation |author=Julien Kloetzer |author2=Hiroyuki Iida |author3=Bruno Bouzy |title=The Monte-Carlo Approach in Amazons |year=2007 |citeseerx=10.1.1.79.7640}}</ref><ref name="Hengens_thesis">{{cite web |author=P. P. L. M. Hensgens |title=A Knowledge-Based Approach of the Game of Amazons |year=2001 |publisher=Universiteit Maastricht, Institute for Knowledge and Agent Technology |url=https://project.dke.maastrichtuniversity.nl/games/files/msc/Hensgens_thesis.pdf}}</ref>

| last1 = Kloetzer | first1 = Julien

|[[PSPACE-complete]]<ref>{{cite arXiv | author = R. A. Hearn | title = Amazons is PSPACE-complete | date = 2005-02-02 | eprint = cs.CC/0502013 }}</ref>

| last2 = Iida | first2 = Hiroyuki

| last3 = Bouzy | first3 = Bruno

| contribution = The Monte-Carlo approach in Amazons

| contribution-url = https://helios2.mi.parisdescartes.fr/~Bouzy/publications/KIB-MCAmazons-CGW07.pdf

| pages = 185–192

| title = Computer Games Workshop, Amsterdam, the Netherlands, 15-17 June 2007

| year = 2007}}</ref><ref name="Hengens_thesis">{{cite web |author=P. P. L. M. Hensgens |title=A Knowledge-Based Approach of the Game of Amazons |year=2001 |publisher=Universiteit Maastricht, Institute for Knowledge and Agent Technology |url=https://project.dke.maastrichtuniversity.nl/games/files/msc/Hensgens_thesis.pdf}}</ref>

|[[PSPACE-complete]]<ref>{{cite arXiv | author = R. A. Hearn | author-link=Bob Hearn | title = Amazons is PSPACE-complete | date = 2005-02-02 | eprint = cs.CC/0502013 }}</ref>

|-

|[[Shogi]]

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|style="text-align:right;"| <ref name="Schadd2010">{{cite thesis | author = F.C. Schadd | title = Monte-Carlo Search Techniques in the Modern Board Game Thurn and Taxis | year = 2009 | url =https://project.dke.maastrichtuniversity.nl/games/files/msc/Fschadd_thesis.pdf | publisher = Maastricht University| archive-url = https://web.archive.org/web/20210114164554/https://project.dke.maastrichtuniversity.nl/games/files/msc/Fschadd_thesis.pdf | archive-date = 2021-01-14 }}</ref>

|

|-

|[[Go (game)|Go (19x19)]]

|style="text-align:right;"|361

|style="text-align:right;"|170

|style="text-align:right;"|360505

|style="text-align:right;"|150211

|style="text-align:right;"|250

|style="text-align:right;"|<ref name="Allis1994"/><ref name="cwi">{{cite web | title = Combinatorics of Go |author1=John Tromp |author2=Gunnar Farnebäck | year = 2007 | url = https://tromp.github.io/go/gostate.ps}} This paper derives the bounds 48&lt;log(log(''N''))&lt;171 on the number of possible games ''N''.</ref><ref name="Tromp2016">{{cite web | title=Number of legal Go positions | author=John Tromp | year=2016 | url=https://tromp.github.io/go/legal.html}}</ref>

<ref>{{Cite web|url=https://homepages.cwi.nl/~aeb/go/misc/gostat.html|title = Statistics on the length of a go game}}</ref>

|[[EXPTIME-complete]]<ref name="Robson1983">{{Cite book | author = J. M. Robson | chapter = The complexity of Go | title = Information Processing; Proceedings of IFIP Congress | year = 1983 | pages = 413–417}}</ref>

|[[EXPTIME-complete]] (without the [[superko rule]])<ref name="Robson1983">{{Cite book | author = J. M. Robson | chapter = The complexity of Go | title = Information Processing; Proceedings of IFIP Congress | year = 1983 | pages = 413–417}}</ref>

|-

|[[Arimaa]]

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|style="text-align:right;"|infinite

|style="text-align:right;"|<ref>{{cite arXiv | author = CDA Evans and Joel David Hamkins | title = Transfinite game values in infinite chess | year = 2014| class = math.LO | eprint = 1302.4377 }}</ref>

|{{Unknown}}, but mate-in-''n'' is decidable<ref name="Brumleve2012">{{cite journal | author = Stefan Reisch, Joel David Hamkins, and Phillipp Schlicht | title = The mate-in-n problem of infinite chess is decidable | journal = Conference on Computability in Europe | year = 2012 | pages = 78–88 | arxiv = 1201.5597 | url = https://arxiv.org/pdf/1201.5597.pdf}}</ref>

|-

|[[Magic: The Gathering]]

Line 463 ⟶ 576:

|style="text-align:right;"|

|style="text-align:right;"|

|style="text-align:right;"|<ref name="Churchill2020">{{cite journalarXiv | author = Alex Churchill, Stella Biderman, and Austin Herrick | title = Magic: the Gathering is Turing Complete | journal = Fun with Algorithms | year = 2020 | arxivclass = 1904cs.09828AI | urleprint = https://arxiv.org/pdf/1904.09828.pdf }}</ref>

|AH-hard<ref name="Biderman">{{cite arXiv | author = Stella Biderman | title = Magic: the Gathering is as Hard as Arithmetic | year = 2020 | class = cs.AI | eprint = 2003.05119 <!-- | url = https://arxiv.org/pdf/2003.05119.pdf -->}}</ref>

|-

|[[Wordle]]

|style="text-align:right;"|5

|5

|style="text-align:right;"|4.113 (12,972)

|12,972

|style="text-align:right;"|

|

|style="text-align:right;"|6

|6

|style="text-align:right;"|

|

|style="text-align:right;"|<ref>{{Citecite journalarXiv |lastlast1=Lokshtanov |firstfirst1=Daniel |last2=Subercaseaux |first2=Bernardo |date=2022-05-14 |title=Wordle is NP-hard |urlclass=http://arxivcs.org/abs/2203.16713CC |journaleprint=arXiv:2203.16713 [cs]}}</ref>

|[[NP-hardness|NP-hard]], unknown if [[PSPACE-complete]] with parametization.

|}

Line 496 ⟶ 609:

[[Category:Combinatorial game theory]]

[[Category:Game theory]]