Star (game Theory)

Star (game Theory)

In combinatorial game theory, star, written as or , is the value given to the game where both players have only the option of moving to the zero game. Star may also be denoted as the surreal form {0|0}. This game is an unconditional first-player win.

Star, as defined by John Conway in Winning Ways for your Mathematical Plays, is a value, but not a number in the traditional sense. Star is not zero, but neither positive nor negative, and is therefore said to be fuzzy and confused with (a fourth alternative that means neither "less than", "equal to", nor "greater than") 0. It is less than all positive rational numbers, and greater than all negative rationals. Since the rationals are dense in the reals, this also makes * greater than any negative real, and less than any positive real.

Games other than {0 | 0} may have value *. For example, the game, where the values are nimbers, has value * despite each player having more options than simply moving to 0.