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.
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
game
A game is a structured form of play (activity), play, usually undertaken for enjoyment, entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator s ...
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, , 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
* = , which is a
first-player win
In combinatorial game theory, a two-player deterministic perfect information turn-based game is a first-player-win if with perfect play the first player to move can always force a win. Similarly, a game is second-player-win if with perfect play the ...
, 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
Nim is a mathematical two player game.
Nim or NIM may also refer to:
* Nim (programming language)
* Nim Chimpsky, a signing chimpanzee Acronyms
* Network Installation Manager, an IBM framework
* Nuclear Instrumentation Module
* Negative index met ...
where only one heap remained where that heap includes more than one object.
Another example is the fuzzy game . 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
Hackenbush is a two-player game invented by mathematician John Horton Conway. It may be played on any configuration of colored line segments connected to one another by their endpoints and to a "ground" line.
Gameplay
The game starts with the pl ...
, 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
{{DEFAULTSORT:Fuzzy Game
Combinatorial game theory