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[[Combinatorial game theory]] has several ways of measuring '''game complexity'''. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity. | |||
==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 including illegal positions or 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: the number of 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. | |||
However, 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 infinite. | |||
=== Decision trees === | |||
The next two measures use the idea of a '''decision tree'''. A decision tree is a subtree of the game tree, with each position labelled with "player A wins", "player B wins" or "drawn", 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 with player A to move 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. And correspondingly for positions with B to move.) | |||
==== 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 '''game-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 we would have to evaluate in a [[minimax]] search to determine the value of the initial position. | |||
It's hard even to estimate the game-tree complexity, but for some games a reasonable lower bound can be given by raising the game's average [[branching factor]] to the power of the number of [[Ply (chess)|plies]] in an average game, or: | |||
<math>GTC \geq b^d</math> | |||
=== Computational complexity === | |||
The '''[[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.) | |||
==Example: tic-tac-toe== | |||
For [[tic-tac-toe]], a simple upper bound for the size of the state space is 3<sup>9</sup> = 19,683. (There are three states for each cell and 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. And when rotations and reflections of positions are considered identical, there are only 765 essentially different positions. | |||
A simple upper bound for the size of the game tree is 9! = 362,880. (There are nine positions for the first move, eight for the second, and so on.) This includes illegal games that continue after one side has won. A more careful count 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 this game can be solved in [[DSPACE]](''mn'') by searching the entire game tree. This places it in the important complexity class [[PSPACE]]. With some more work it can be shown to be [[PSPACE-complete]].<ref name="Reisch1980">{{cite journal | author = Stefan Reisch | title = Gobang ist PSPACE-vollstandig (Gomoku is PSPACE-complete) | journal = Acta Informatica | volume = 13 | page = 5966 | year = 1980}}</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. 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. | |||
{| class="wikitable sortable" cellpadding="5" | |||
|- | |||
!Game | |||
!Board size | |||
(positions) | |||
!State-space complexity | |||
(as [[logarithm|log]] to base 10) | |||
!Game-tree complexity | |||
(as [[logarithm|log]] to base 10) | |||
!Average game length | |||
([[Ply (game theory)|plies]]) | |||
!Branching factor | |||
!Ref | |||
![[Complexity class]] of suitable [[generalized game]] | |||
|- | |||
|[[Tic-tac-toe]] | |||
|align="right"|9 | |||
|align="right"|3 | |||
|align="right"|5 | |||
|align="right"|9 | |||
|align="right"|4 | |||
|align="right"| | |||
|[[PSPACE-complete]]<ref name="Reisch1980"/> | |||
|- | |||
|[[Sim (pencil game)|Sim]] | |||
|align="right"|15 | |||
|align="right"|3 | |||
|align="right"|8 | |||
|align="right"|14 | |||
|align="right"|3.7 | |||
|align="right"| | |||
|[[PSPACE-complete]]<ref>[http://portal.acm.org/citation.cfm?id=728124 Wolfgang Slany: The Complexity of Graph Ramsey Games]</ref> | |||
|- | |||
|[[Pentomino]]es | |||
|align="right"|64 | |||
|align="right"|12 | |||
|align="right"|18 | |||
|align="right"|10 | |||
|align="right"|75 | |||
|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> | |||
|?, but in [[PSPACE]] | |||
|- | |||
|[[Kalah]] <ref> See van den Herik et al for rules. </ref> | |||
|align="right"|14 | |||
|align="right"|13 | |||
|align="right"|18 | |||
|align="right"| | |||
|align="right"| | |||
|align="right"|<ref name="GamesSolved"/> | |||
|Generalization is unclear | |||
|- | |||
|[[Connect Four]] | |||
|align="right"|42 | |||
|align="right"|13 | |||
|align="right"|21 | |||
|align="right"|36 | |||
|align="right"|4 | |||
|align="right"|<ref>{{cite web | title = John's Connect Four Playground | author = John Tromp | year = 2010 | url = http://www.cwi.nl/~tromp/c4/c4.html}}</ref> <ref name="Allis1994"/> | |||
|?, but in [[PSPACE]] | |||
|- | |||
|[[Domineering]] (8 × 8) | |||
|align="right"|64 | |||
|align="right"|15 | |||
|align="right"|27 | |||
|align="right"|30 | |||
|align="right"|8 | |||
|align="right"|<ref name="GamesSolved"/> | |||
|?, but in [[PSPACE]]; in [[P (complexity class)|P]] for certain dimensions<ref> {{cite paper| authors=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> | |||
|- | |||
|[[Congkak]] | |||
|align="right"|14 | |||
|align="right"|15 | |||
|align="right"|33 | |||
|align="right"| | |||
|align="right"| | |||
|align="right"|<ref name="GamesSolved">{{cite journal | title= Games solved: Now and in the future | author = H. J. van den Herik | author2 = J. W. H. M. Uiterwijk | author3 = J. van Rijswijck | year = 2002 | journal = Artificial Intelligence | volume = 134 | issue=1–2 | pages=277–311 | doi= 10.1016/S0004-3702(01)00152-7}} </ref> | |||
| | |||
|- | |||
|[[English draughts|English draughts (8x8) (checkers)]] | |||
|align="right"|32 | |||
|align="right"|20 or 18 | |||
|align="right"|31 | |||
|align="right"|70 | |||
|align="right"|2.8 | |||
|align="right"|<ref>{{cite journal | author = Jonathan Schaeffer et al. | title = Checkers is Solved | journal = Science | date = July 6, 2007 | doi=10.1126/science.1144079 | volume=317 |issue= 5844 |pages= 1518–1522 | pmid=17641166 }}</ref> or <ref name="Allis1994"/> | |||
|[[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> | |||
|- | |||
|[[Oware|Awari]]<ref>See Allis 1994 for rules</ref> | |||
|align="right"|12 | |||
|align="right"|12 | |||
|align="right"|32 | |||
|align="right"|60 | |||
|align="right"|3.5 | |||
|align="right"|<ref name="Allis1994"/> | |||
|Generalization is unclear | |||
|- | |||
|[[Qubic]] | |||
|align="right"|64 | |||
|align="right"|30 | |||
|align="right"|34 | |||
|align="right"|20 | |||
|align="right"|54.2 | |||
|align="right"|<ref name=Allis1994/> | |||
|[[PSPACE-complete]]<ref name="Reisch1980"/> | |||
|- | |||
|[[Fanorona]] | |||
|align="right"|45 | |||
|align="right"|21 | |||
|align="right"|46 | |||
|align="right"|44 | |||
|align="right"|11 | |||
|align="right"|<ref name="Schadd2008">{{cite journal| author = M.P.D. Schadd, M.H.M. Winands, J.W.H.M. Uiterwijk, H.J. van den Herik and M.H.J. Bergsma | year = 2008 | title = Best Play in Fanorona leads to Draw | journal = [[New Mathematics and Natural Computation]] | volume = 4 |issue = 3 | pages = 369–387| url = https://dke.maastrichtuniversity.nl/m.winands/documents/Fanorona.pdf| doi = 10.1142/S1793005708001124}}</ref> | |||
|?, but in [[EXPTIME]] | |||
|- | |||
|[[Nine Men's Morris]] | |||
|align="right"|24 | |||
|align="right"|10 | |||
|align="right"|50 | |||
|align="right"|50 | |||
|align="right"|10 | |||
|align="right"|<ref name="Allis1994"/> | |||
|?, but in [[EXPTIME]] | |||
|- | |||
|[[International draughts|International draughts (10x10)]] | |||
|align="right"|50 | |||
|align="right"|30 | |||
|align="right"|54 | |||
|align="right"|90 | |||
|align="right"|4 | |||
|align="right"|<ref name="Allis1994"/> | |||
|[[EXPTIME-complete]]<ref name="robson1984"/> | |||
|- | |||
|[[Chinese checkers]] (2 sets) | |||
|align="right"|121 | |||
|align="right"|23 | |||
|align="right"| | |||
|align="right"| | |||
|align="right"| | |||
|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|arxiv=0803.1245|url = http://emis.ams.org}}</ref> | |||
|[[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|author=Takumi Kasai, Akeo Adachi, and Shigeki Iwata|doi=10.1137/0208046|issue=4}} Proves completeness of the generalization to arbitrary graphs.</ref> | |||
|- | |||
|[[Chinese checkers]] (6 sets) | |||
|align="right"|121 | |||
|align="right"|78 | |||
|align="right"| | |||
|align="right"| | |||
|align="right"| | |||
|align="right"|<ref name=Bell_Halma/> | |||
|[[EXPTIME]]-complete <ref name=pebble/> | |||
|- | |||
|[[Lines of Action]] | |||
|align="right"|64 | |||
|align="right"|23 | |||
|align="right"|64 | |||
|align="right"|44 | |||
|align="right"|29 | |||
|align="right"|<ref name="Winands2004">{{cite thesis | author = Mark H.M. Winands | year = 2004 | title = Informed Search in Complex Games | degree = Ph.D. |publisher= Maastricht University, Maastricht, The Netherlands | isbn = 90-5278-429-9 | url = https://dke.maastrichtuniversity.nl/m.winands/documents/informed_search.pdf}}</ref> | |||
|?, but in [[EXPTIME]] | |||
|- | |||
|[[Reversi]] (Othello) | |||
|align="right"|64 | |||
|align="right"|28 | |||
|align="right"|58 | |||
|align="right"|58 | |||
|align="right"|10 | |||
|align="right"|<ref name="Allis1994"/> | |||
|[[PSPACE-complete]]<ref>{{cite journal | author = S. Iwata and T. Kasai | title = The Othello game on an n*n board is PSPACE-complete | journal = Theor. Comp. Sci. | issue = 2 | year = 1994 | pages = 329–340 | volume = 123 | doi = 10.1016/0304-3975(94)90131-7}}</ref> | |||
|- | |||
|[[OnTop (board game)|OnTop]] (2p base game) | |||
|align="right"|72 | |||
|align="right"|88 | |||
|align="right"|62 | |||
|align="right"|31 | |||
|align="right"|23.77 | |||
|align="right"|<ref name="OnTopComputer">{{cite thesis | title=Analysis and Implementation of the Game OnTop | | |||
url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Briesemeister_Thesis.pdf | | |||
author = Robert Briesemeister | year=2009 | publisher = Maastricht University, Dept of Knowledge Engineering }} </ref> | |||
| | |||
|- | |||
|[[Hex (board game)|Hex (11x11)]] | |||
|align="right"|121 | |||
|align="right"|57 | |||
|align="right"|98 | |||
|align="right"|40 | |||
|align="right"|280 | |||
|align="right"|<ref name="GamesSolved"/> | |||
|[[PSPACE-complete]]<ref>{{cite journal | author = Stefan Reisch | title = Hex ist PSPACE-vollständig (Hex is PSPACE-complete) | journal = Acta Inf. | issue = 15 | year = 1981 | pages = 167–191}}</ref> | |||
|- | |||
|[[Gomoku]] (15x15, freestyle) | |||
|align="right"|225 | |||
|align="right"|105 | |||
|align="right"|70 | |||
|align="right"|30 | |||
|align="right"|210 | |||
|align="right"|<ref name="Allis1994"/> | |||
|[[PSPACE-complete]]<ref name="Reisch1980"/> | |||
|- | |||
|[[Go (game)|Go (9x9)]] | |||
|align="right"|81 | |||
|align="right"|38 | |||
|align="right"| | |||
|align="right"|45 | |||
|align="right"| | |||
|align="right"|<ref name="cwi">{{cite web | title = Combinatorics of Go | author = John Tromp and Gunnar Farnebäck | year = 2007 | url = http://www.cwi.nl/~tromp/go/gostate.ps}} This paper derives the bounds 48<log(log(''N''))<171 on the number of possible games ''N''.</ref> <ref name="Allis1994"/> | |||
|[[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> | |||
|- | |||
|[[Chess]] | |||
|align="right"|64 | |||
|align="right"|47 | |||
|align="right"|123 | |||
|align="right"|80 | |||
|align="right"|35 | |||
|align="right"|<ref name="Shannon1950">The size of the state space and game tree for chess were first estimated in {{cite journal | author = [[Claude Shannon]] | title = Programming a Computer for Playing Chess | journal = Philosophical Magazine | volume = 41 | issue = 314 | year = 1950 | url = http://archive.computerhistory.org/projects/chess/related_materials/text/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon/2-0%20and%202-1.Programming_a_computer_for_playing_chess.shannon.062303002.pdf}} Shannon gave estimates of 10<sup>43</sup> and 10<sup>120</sup> respectively, smaller than the upper bound in the table, | |||
which is detailed in [[Shannon number]].</ref> | |||
|[[EXPTIME-complete ]] (without 50-move drawing rule) <ref name="Fraenkel1981">{{cite journal | author = [[Aviezri Fraenkel]] and D. Lichtenstein | title = Computing a perfect strategy for n×n chess requires time exponential in n | journal = J. Comb. Th. A | issue = 31 | year = 1981 | pages = 199–214}}</ref> | |||
|- | |||
|[[Connect6]] | |||
|align="right"|361 | |||
|align="right"|172 | |||
|align="right"|140 | |||
|align="right"|30 | |||
|align="right"|46000 | |||
|align="right"|<ref name=EnhancePNConn6> {{cite journal | title=2009 Chinese Control and Decision Conference | doi=10.1109/CCDC.2009.5191963 | chapter=Enhancements of proof number search in connect6 | year=2009 | last1=Chang-Ming Xu | last2=Ma | first2=Z.M. | last3=Jun-Jie Tao | last4=Xin-He Xu | isbn=978-1-4244-2722-2 | pages=4525 }} </ref> | |||
|[[PSPACE|PSPACE-complete]]<ref>[http://portal.acm.org/citation.cfm?id=1290250 On the fairness and complexity of generalized k-in-a-row games<!-- Bot generated title -->]</ref> | |||
|- | |||
|[[Backgammon]] | |||
|align="right"|28 | |||
|align="right"|20 | |||
|align="right"|144 | |||
|align="right"|50-60 | |||
|align="right"|250 | |||
|align="right"|<ref>http://books.nips.cc/papers/txt/nips04/0259.txt</ref> | |||
|Generalization is unclear | |||
|- | |||
|[[Xiangqi]] | |||
|align="right"|90 | |||
|align="right"|48 | |||
|align="right"|150 | |||
|align="right"|95 | |||
|align="right"|38 | |||
|align="right"|<ref name="Allis1994">{{cite thesis | author = [[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 | author = Shi-Jim Yen, Jr-Chang Chen, Tai-Ning Yang, and 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}}</ref> | |||
|?, believed to be [[EXPTIME-complete]] | |||
|- | |||
|[[Abalone (board game)|Abalone]] | |||
|align="right"|61 | |||
|align="right"|25 | |||
|align="right"|154 | |||
|align="right"|87 | |||
|align="right"|60 | |||
|align="right"|<ref name=PasChor>{{cite web|last=Chorus|first=Pascal|title=Implementing a Computer Player for Abalone Using Alpha-Beta and Monte-Carlo Search|url=https://project.dke.maastrichtuniversity.nl/games/files/msc/pcreport.pdf|publisher=Dept of Knowledge Engineering, Maastricht University|accessdate=29 March 2012}}</ref> | |||
|?, but in [[EXPTIME]] | |||
|- | |||
|[[Havannah]] | |||
|align="right"|271 | |||
|align="right"|127 | |||
|align="right"|157 | |||
|align="right"|66 | |||
|align="right"|240 | |||
|align="right"|<ref name="GamesSolved"/> <ref>{{cite web|last=Joosten|first=B|title=Creating a Havannah Playing Agent|url=https://project.dke.maastrichtuniversity.nl/games/files/bsc/bscHavannah.pdf|accessdate=29 March 2012}}</ref> | |||
|?, but in [[PSPACE]] | |||
|- | |||
|[[Quoridor]] | |||
|align="right"|81 | |||
|align="right"|42 | |||
|align="right"|162 | |||
|align="right"|91 | |||
|align="right"|60 | |||
|align="right"|<ref name=MasterQuor> {{cite thesis | author = Lisa Glendenning | title = Mastering Quoridor | year = 2005 | department = Computer Science | degree = B.Sc. | publisher = [[University of New Mexico]] | month = May | url = http://hyperion.cs.washington.edu/attachments/15/glendenning_ugrad_thesis.pdf }}</ref> | |||
|?, but in [[PSPACE]] | |||
|- | |||
|[[Carcassonne (board game)|Carcassonne]] (2p base game) | |||
|align="right"|72 | |||
|align="right"|>40 | |||
|align="right"|195 | |||
|align="right"|71 | |||
|align="right"|55 | |||
|align="right"|<ref name="CarcassoneComputer">{{cite thesis | title=Implementing a Computer Player for Carcassonne | | |||
url = https://project.dke.maastrichtuniversity.nl/games/files/msc/MasterThesisCarcassonne.pdf | | |||
author = Cathleen Heyden | year=2009 | publisher = Maastricht University, Dept of Knowledge Engineering }} </ref> | |||
|Generalization is unclear | |||
|- | |||
|[[Game of the Amazons|Amazons (10x10)]] | |||
|align="right"|100 | |||
|align="right"|40 | |||
|align="right"|212 | |||
|align="right"|84 | |||
|align="right"|374 or 299<ref>The lower branching factor is for the second player.</ref> | |||
|align="right"|<ref name="Kloetzer_themonte-carlo">{{cite journal | author = Julien Kloetzer | author2 = Hiroyuki Iida | author3 = Bruno Bouzy | title = The Monte-Carlo Approach in Amazons | year = 2007 | url = http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.79.7640}} </ref> <ref name="Hengens_thesis">{{cite paper | 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 | title = Amazons is PSPACE-complete | date = 2005-02-02 | eprint = cs.CC/0502013 | class = cs.CC}}</ref> | |||
|- | |||
|[[Go (game)|Go (13x13)]] | |||
|align="right"|169 | |||
|align="right"|79 | |||
|align="right"| | |||
|align="right"|90 | |||
|align="right"| | |||
|align="right"|<ref name="Allis1994"/> <ref name="cwi" /> | |||
|[[EXPTIME-complete]]<ref name="Robson1983" /> | |||
|- | |||
|[[Shogi]] | |||
|align="right"|81 | |||
|align="right"|71 | |||
|align="right"|226 | |||
|align="right"|115 | |||
|align="right"|92 | |||
|align="right"|<ref name="Hsu2004"/> <ref> {{cite journal | title = Computer shogi | doi = 10.1016/S0004-3702(01)00157-6 | journal = Artificial Intelligence | volume=134 | issue=1–2 |date=january 2002 | pages=121–144 | author= Hiroyuki Iida, Makoto Sakuta, [[Jeff Rollason]] }} </ref> | |||
|[[EXPTIME-complete]]<ref>{{cite journal | author = H. Adachi, H. Kamekawa, and S. Iwata | title = Shogi on n × n board is complete in exponential time | journal = Trans. IEICE | volume= J70-D | pages = 1843–1852 | year = 1987}}</ref> | |||
|- | |||
|[[Arimaa]] | |||
|align="right"|64 | |||
|align="right"|43 | |||
|align="right"|402 | |||
|align="right"|92 | |||
|align="right"|17281 | |||
|align="right"|<ref name="Cox2006">{{cite web | title = Analysis and Implementation of the Game Arimaa | author = Christ-Jan Cox | year = 2006 | url = https://project.dke.maastrichtuniversity.nl/games/files/msc/Cox_thesis1.pdf}}</ref> <ref name="Wu2011">{{cite web | title = Move Ranking and Evaluation in the Game of Arimaa | author = David Jian Wu | year = 2011 | url = http://icosahedral.net/downloads/djwuthesis.pdf}}</ref> <ref name="Haskin2006">{{cite web | title = A Look at the Arimaa Branching Factor | author = Brian Haskin | year = 2006 | url = http://arimaa.janzert.com/bf_study/}}</ref> | |||
|?, but in [[EXPTIME]] | |||
|- | |||
|[[Go (game)|Go (19x19)]] | |||
|align="right"|361 | |||
|align="right"|171 | |||
|align="right"|360 | |||
|align="right"|150 | |||
|align="right"|250 | |||
|align="right"|<ref name="cwi" /> <ref name="Allis1994"/> | |||
|[[EXPTIME-complete]]<ref name="Robson1983" /> | |||
|- | |||
|[[Stratego]] | |||
|align="right"|92 | |||
|align="right"|115 | |||
|align="right"|535 | |||
|align="right"|381 | |||
|align="right"|21.739 | |||
|align="right"|<ref name="ArtsStratego"> {{cite thesis | author = A.F.C. Arts | title = Competitive Play in Stratego | year = 2010 | url =https://project.dke.maastrichtuniversity.nl/games/files/msc/Arts_thesis.pdf | university = Maastricht }}</ref> | |||
| | |||
|- | |||
|[[Computer bridge|Double dummy bridge]]<ref>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]], which motivates including it in the list. 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 the last 4 plies are always forced moves with branching factor 1.</ref> | |||
|align="right"|(52) | |||
|align="right"|<17 | |||
|align="right"|<40 | |||
|align="right"|52 | |||
|align="right"|5.6 | |||
|align="right"| | |||
| | |||
|} | |||
==See also== | |||
*[[Go and mathematics ]] | |||
*[[Solved game]] | |||
*[[Shannon number]] | |||
*[[List of NP-complete problems#Games and puzzles|list of NP-complete games and puzzles]] | |||
*[[List of PSPACE-complete problems#Games and puzzles|list of PSPACE-complete games and puzzles]] | |||
==Notes and references== | |||
<references/> | |||
==External links== | |||
* [[David Eppstein]]'s [http://www.ics.uci.edu/~eppstein/cgt/hard.html Computational Complexity of Games and Puzzles] | |||
[[Category:Combinatorial game theory]] | |||
[[Category:Game theory]] |
Revision as of 12:30, 24 January 2014
Combinatorial game theory has several ways of measuring game complexity. This article describes five of them: state-space complexity, game tree size, decision complexity, game-tree complexity, and computational complexity.
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.[1]
When this is too hard to calculate, an upper bound can often be computed by including illegal positions or 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: the number of 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.
However, 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 infinite.
Decision trees
The next two measures use the idea of a decision tree. A decision tree is a subtree of the game tree, with each position labelled with "player A wins", "player B wins" or "drawn", 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 with player A to move 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. And correspondingly for positions with B to move.)
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 game-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.[1] A full-width tree includes all nodes at each depth.
This is an estimate of the number of positions we would have to evaluate in a minimax search to determine the value of the initial position.
It's hard even to estimate the game-tree complexity, but for some games a reasonable lower bound can be given by raising the game's average branching factor to the power of the number of plies in an average game, or:
Computational complexity
The computational complexity of a game describes the 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 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.)
Example: tic-tac-toe
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 cell and 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. And when rotations and reflections of positions are considered identical, there are only 765 essentially different positions.
A simple upper bound for the size of the game tree is 9! = 362,880. (There are nine positions for the first move, eight for the second, and so on.) This includes illegal games that continue after one side has won. A more careful count 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. A natural generalization is to 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 this game can be solved in DSPACE(mn) by searching the entire game tree. This places it in the important complexity class PSPACE. With some more work it can be shown to be PSPACE-complete.[2]
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. 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.
Game | Board size
(positions) |
State-space complexity
(as log to base 10) |
Game-tree complexity
(as log to base 10) |
Average game length
(plies) |
Branching factor | Ref | Complexity class of suitable generalized game |
---|---|---|---|---|---|---|---|
Tic-tac-toe | 9 | 3 | 5 | 9 | 4 | PSPACE-complete[2] | |
Sim | 15 | 3 | 8 | 14 | 3.7 | PSPACE-complete[3] | |
Pentominoes | 64 | 12 | 18 | 10 | 75 | [4] [5] | ?, but in PSPACE |
Kalah [6] | 14 | 13 | 18 | [4] | Generalization is unclear | ||
Connect Four | 42 | 13 | 21 | 36 | 4 | [7] [1] | ?, but in PSPACE |
Domineering (8 × 8) | 64 | 15 | 27 | 30 | 8 | [4] | ?, but in PSPACE; in P for certain dimensions[8] |
Congkak | 14 | 15 | 33 | [4] | |||
English draughts (8x8) (checkers) | 32 | 20 or 18 | 31 | 70 | 2.8 | [9] or [1] | EXPTIME-complete[10] |
Awari[11] | 12 | 12 | 32 | 60 | 3.5 | [1] | Generalization is unclear |
Qubic | 64 | 30 | 34 | 20 | 54.2 | [1] | PSPACE-complete[2] |
Fanorona | 45 | 21 | 46 | 44 | 11 | [12] | ?, but in EXPTIME |
Nine Men's Morris | 24 | 10 | 50 | 50 | 10 | [1] | ?, but in EXPTIME |
International draughts (10x10) | 50 | 30 | 54 | 90 | 4 | [1] | EXPTIME-complete[10] |
Chinese checkers (2 sets) | 121 | 23 | [13] | EXPTIME-complete [14] | |||
Chinese checkers (6 sets) | 121 | 78 | [13] | EXPTIME-complete [14] | |||
Lines of Action | 64 | 23 | 64 | 44 | 29 | [15] | ?, but in EXPTIME |
Reversi (Othello) | 64 | 28 | 58 | 58 | 10 | [1] | PSPACE-complete[16] |
OnTop (2p base game) | 72 | 88 | 62 | 31 | 23.77 | [17] | |
Hex (11x11) | 121 | 57 | 98 | 40 | 280 | [4] | PSPACE-complete[18] |
Gomoku (15x15, freestyle) | 225 | 105 | 70 | 30 | 210 | [1] | PSPACE-complete[2] |
Go (9x9) | 81 | 38 | 45 | [19] [1] | EXPTIME-complete[20] | ||
Chess | 64 | 47 | 123 | 80 | 35 | [21] | EXPTIME-complete (without 50-move drawing rule) [22] |
Connect6 | 361 | 172 | 140 | 30 | 46000 | [23] | PSPACE-complete[24] |
Backgammon | 28 | 20 | 144 | 50-60 | 250 | [25] | Generalization is unclear |
Xiangqi | 90 | 48 | 150 | 95 | 38 | [1] [26] | ?, believed to be EXPTIME-complete |
Abalone | 61 | 25 | 154 | 87 | 60 | [27] | ?, but in EXPTIME |
Havannah | 271 | 127 | 157 | 66 | 240 | [4] [28] | ?, but in PSPACE |
Quoridor | 81 | 42 | 162 | 91 | 60 | [29] | ?, but in PSPACE |
Carcassonne (2p base game) | 72 | >40 | 195 | 71 | 55 | [30] | Generalization is unclear |
Amazons (10x10) | 100 | 40 | 212 | 84 | 374 or 299[31] | [32] [33] | PSPACE-complete[34] |
Go (13x13) | 169 | 79 | 90 | [1] [19] | EXPTIME-complete[20] | ||
Shogi | 81 | 71 | 226 | 115 | 92 | [26] [35] | EXPTIME-complete[36] |
Arimaa | 64 | 43 | 402 | 92 | 17281 | [37] [38] [39] | ?, but in EXPTIME |
Go (19x19) | 361 | 171 | 360 | 150 | 250 | [19] [1] | EXPTIME-complete[20] |
Stratego | 92 | 115 | 535 | 381 | 21.739 | [40] | |
Double dummy bridge[41] | (52) | <17 | <40 | 52 | 5.6 |
See also
- Go and mathematics
- Solved game
- Shannon number
- list of NP-complete games and puzzles
- list of PSPACE-complete games and puzzles
Notes and references
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 1.12 1.13 Template:Cite thesis
- ↑ 2.0 2.1 2.2 2.3 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Wolfgang Slany: The Complexity of Graph Ramsey Games
- ↑ 4.0 4.1 4.2 4.3 4.4 4.5 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Hilarie K. Orman: Pentominoes: A First Player Win in Games of no chance, MSRI Publications – Volume 29, 1996, pages 339-344. Online: pdf.
- ↑ See van den Herik et al for rules.
- ↑ Template:Cite web
- ↑ Template:Cite paper
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 10.0 10.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ See Allis 1994 for rules
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 13.0 13.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 14.0 14.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Proves completeness of the generalization to arbitrary graphs. - ↑ Template:Cite thesis
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite thesis
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ 19.0 19.1 19.2 Template:Cite web This paper derives the bounds 48<log(log(N))<171 on the number of possible games N.
- ↑ 20.0 20.1 20.2 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - ↑ The size of the state space and game tree for chess were first estimated in One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang Shannon gave estimates of 1043 and 10120 respectively, smaller than the upper bound in the table, which is detailed in Shannon number. - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ On the fairness and complexity of generalized k-in-a-row games
- ↑ http://books.nips.cc/papers/txt/nips04/0259.txt
- ↑ 26.0 26.1 One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ Template:Cite thesis
- ↑ Template:Cite thesis
- ↑ The lower branching factor is for the second player.
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite paper
- ↑ Template:Cite arxiv
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ Template:Cite web
- ↑ Template:Cite thesis
- ↑ 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, which motivates including it in the list. 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 the last 4 plies are always forced moves with branching factor 1.