Fuzzy game
Encyclopedia
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.
1. Left win: G > 0
2. Right win: G < 0
3. Second player win: G = 0
4. First player win: G ║ 0 (G is fuzzy with 0)
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.
. 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
.
Combinatorial game theory
Combinatorial game theory is a branch of applied mathematics and theoretical computer science that studies sequential games with perfect information, that is, two-player games which have a position in which the players take turns changing in defined ways or moves to achieve a defined winning...
, a fuzzy game is a game which is incomparable with the zero game
Zero game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero...
: 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 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 * = {0, which is a first-player win, since whoever moves first can move to a second player win, namely the zero gameZero game
In combinatorial game theory, the zero game is the game where neither player has any legal options. Therefore, the first player automatically loses, and it is a second-player win. The zero game has a Sprague–Grundy value of zero...
. An example of a fuzzy game would be a normal game of Nim
Nim
Nim is a mathematical game of strategy in which two players take turns removing objects from distinct heaps. On each turn, a player must remove at least one object, and may remove any number of objects provided they all come from the same heap....
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
Surreal number
In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number...
.