Fuzzy game
In the world of Fuzzy game, there are endless aspects and details worth exploring. From its origins to its impact today, Fuzzy game has captured the attention of millions of people around the world. Whether through history, science, music, art or any other field, Fuzzy game continues to be a topic of interest to people of all ages and cultures. In this article, we will delve into the different aspects of Fuzzy game, exploring its many facets and analyzing its influence on today's society. From its beginnings to the present, Fuzzy game has left an indelible mark on history and will surely remain relevant for future generations.
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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]
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|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
- ^ Billot, Antoine (1998). "Elements of Fuzzy Game Theory". The Handbooks of Fuzzy Sets Series. Vol. 1. Boston, MA: Springer US. pp. 137–176. doi:10.1007/978-1-4615-5645-9_5. ISBN 9781461375838.