Star (game theory)

A combinatorial game has a positive and negative player; which player moves first is left ambiguous. The combinatorial game 0, or { | }, leaves no options and is a second-player win. Likewise, a combinatorial game is won (assuming optimal play) by the second player if and only if its value is 0. Therefore, a game of value ∗, which is a first-player win, is neither positive nor negative. However, ∗ is not the only possible value for a first-player win game (see nimbers). Star does have the property that the sum , has value 0, because the first-player's only move is to the game ∗, which the second-player will win. ==Example of a value-∗ game==

Nim, with one pile and one piece, has value ∗. The first player will remove the piece, and the second player will lose. A single-pile Nim game with one pile of n pieces (also a first-player win) is defined to have value ∗n. The numbers ∗z for integers z form an infinite field of characteristic 2, when addition is defined in the context of combinatorial games and multiplication is given a more complex definition. ==See also==