In
combinatorial game theory
Combinatorial game theory is a branch of mathematics and theoretical computer science that typically studies sequential games with perfect information. Research in this field has primarily focused on two-player games in which a ''position'' ev ...
, 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 type of play usually undertaken for entertainment or fun, and sometimes used as an educational tool. Many games are also considered to be work (such as professional players of spectator sports or video games) or art ...
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, 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 . 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
In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...
.
References
{{DEFAULTSORT:Fuzzy Game
Combinatorial game theory