Editing Fuzzy game

{{Short description|Type of combinatorial game}}
{{refimprove|date=July 2018}}
In [[combinatorial game theory]], a '''fuzzy game''' is a game which is incomparable with the [[zero game]]: it is not greater than 0, which would be a win for Left; nor less than 0 which would be a win for Right; nor equal to 0 which would be a win for the second player to move.  It is therefore a first-player win.<ref name="Billot1998">{{cite book |last1=Billot |first1=Antoine |title=The Handbooks of Fuzzy Sets Series |volume=1 |date=1998 |publisher=Springer US |location=Boston, MA |isbn=9781461375838 |pages=137–176 |language=en |chapter=Elements of Fuzzy Game Theory|doi=10.1007/978-1-4615-5645-9_5 }}</ref>

==Classification of games==
In combinatorial game theory, there are four types of game. If we denote players as Left and Right, and G be a [[Surreal_number#Games|game]] with some value, we have the following types of game:

1. Left win: G > 0 
:No matter which player goes first, Left wins.
2. Right win: G < 0
:No matter which player goes first, Right wins.
3. Second player win: G = 0
:The first player (Left or Right) has no moves, and thus loses. 
4. First player win: G ║ 0 (G is fuzzy with 0)
:The first player (Left or Right) wins.

Using standard Dedekind-section game notation, {L|R}, where L is the list of [[undominated]] moves for Left and R is the list of undominated moves for Right, a fuzzy game is a game where all moves in L are strictly non-negative, and all moves in R are strictly non-positive.

==Examples==
One example is the fuzzy game [[star (game theory)|* = {0|0}]], which is a [[first-player win]], since whoever moves first can move to a second player win, namely the [[zero game]].  An example of a fuzzy game would be a normal game of [[Nim]] where only one heap remained where that heap includes more than one object.

Another example is the fuzzy game {1|-1}.  Left could move to 1, which is a win for Left, while Right could move to -1, which is a win for Right; again this is a first-player win.  

In [[Blue-Red-Green Hackenbush]], if there is only a green edge touching the ground, it is a fuzzy game because the first player may take it and win (everything else disappears).

No fuzzy game can be a [[surreal number]].

== References ==
{{reflist}}

{{DEFAULTSORT:Fuzzy Game}}
[[Category:Combinatorial game theory]]