Editing Go (game) (section)

== Analyses of the game ==
{{See also|Go and mathematics}}
In formal [[game theory]] terms, Go is a non-chance, [[combinatorial game]] with [[perfect information]]. Informally that means there are no dice used (and decisions or moves create discrete outcome vectors rather than probability distributions), the underlying math is combinatorial, and all moves (via single vertex analysis) are visible to both players (unlike some card games where some information is hidden). Perfect information also implies sequence—players can theoretically know about all past moves.

Other game theoretical taxonomy elements include the facts
*Go is bounded by a finite number of moves and every game must end with a victor or a tie (although ties are very rare); 
*the strategy is associative because every strategy is a function of board position; 
*the format is non-cooperative (that is, it's not a team sport); 
*positions are extensible, and so can be represented by board position trees; 
*the game is [[zero-sum]] because player choices do not increase resources available, the rewards in the game are fixed and if one player wins, the other loses, and the utility function is restricted (in the sense of win/lose);
*however, ratings, monetary rewards, national and personal pride and other factors can extend utility functions, but generally not to the extent of removing the win/lose restriction, although [[affine transformation]]s can theoretically add non-zero and complex utility aspects even to two player games.<ref>{{cite book |first=Michael |last=Maschler |title=Game Theory |publisher=Cambridge University Press |year=2013 |isbn=978-1-107-00548-8}} {{page needed|date=May 2014}}</ref>

In the endgame, it can often happen that the state of the board consists of several subpositions that do not interact with the others. The whole board position can then be considered as a mathematical sum, or composition, of the individual subpositions.{{sfn|Moews|1996|p=259}} It is this property of Go endgames that led [[John Horton Conway]] to the discovery of [[surreal number]]s.<ref name="O'Connor Robertson Conway">{{citation | url = http://www-history.mcs.st-andrews.ac.uk/Biographies/Conway.html | title = Conway Biography | last1 = O'Connor | first1 = J.J. | last2 = Robertson | first2 = E.F. | access-date = 2008-01-24}}</ref>

In [[combinatorial game theory]] terms, Go is a [[Zero-sum game|zero-sum]], [[perfect information|perfect-information]], [[partisan game|partisan]], [[Deterministic system (mathematics)|deterministic]] [[strategy game]], putting it in the same class as chess, [[draughts]] (checkers), and [[Reversi]] (Othello).

The game emphasizes the importance of balance on multiple levels: to secure an area of the board, it is good to play moves close together; however, to cover the largest area, one needs to spread out, perhaps leaving weaknesses that can be exploited. Playing too ''low'' (close to the edge) secures insufficient territory and influence, yet playing too ''high'' (far from the edge) allows the opponent to invade. Decisions in one part of the board may be influenced by an apparently unrelated situation in a distant part of the board (for example, ladders can be broken by stones at an arbitrary distance away). Plays made early in the game can shape the nature of conflict a hundred moves later.

The [[game complexity]] of Go is such that describing even elementary strategy fills many introductory books. In fact, numerical estimates show that the number of possible games of Go far exceeds [[Observable universe#Matter content|the number of atoms in the observable universe]].{{efn|1=It has been said that the number of board positions is at most 3<sup>361</sup> (about 10<sup>172</sup>) since each position can be white, black, or vacant. Ignoring (illegal) suicide moves, there are at least 361! games (about 10<sup>768</sup>) since every permutation of the 361 points corresponds to a game. See [[Go and mathematics]] for more details, which includes much larger estimates.<br>This estimate, however, is inexact for two reasons: first, both contestants usually agree to end the game long before every point has been played; second, after a capture it may happen that an already played point is played again, even repetitively so in the case of a kō-battle.}}

Go also contributed to the development of [[combinatorial game theory]] (with Go infinitesimals<ref>{{Cite web|url=https://senseis.xmp.net/?GoInfinitesimals|title=Go Infinitesimals at Sensei's Library|website=senseis.xmp.net}}</ref> being a specific example of its use in Go).