Game Theory Basics - Department of Mathematics

1.6. COMBINATORIAL GAMES, IN PARTICULAR IMPARTIAL GAMES 5Lemma 1.1 The nim position m,n is winning if and only if m ≠ n, otherwise losing, forall m,n ≥ 0.This lemma applies also when m = 0 or n = 0, and thus includes the cases that one orboth heap sizes are zero (meaning only one heap or no heap at all).With three or more heaps, nim becomes more difficult. For example, it is not immediatelyclear if, say, positions 1,4,5 or 2,3,6 are winning or losing positions.⇒ At this point, you should try exercise 1.1(a) on page 20.1.6 Combinatorial games, in particular impartial gamesThe games we study in this chapter have, like nim, the following properties:1. There are just two players.2. There are several, usually finitely many, positions, and sometimes a particular startingposition.3. There are clearly defined rules that specify the moves that either player can makefrom a given position to the possible new positions, which are called the options ofthat position.4. The two players move alternately, in the game as a whole.5. In the normal play convention a player unable to move loses.6. The rules are such that play will always come to an end because some player will beunable to move. This is called the ending condition. So there can be no games whichare drawn by repetition of moves.7. Both players know what is going on, so there is perfect information.8. There are no chance moves such as rolling dice or shuffling cards.9. The game is impartial, that is, the possible moves of a player only depend on theposition but not on the player.As a negation of condition 5, there is also the misère play convention where a playerunable to move wins. In the surrealist (and unsettling) movie “Last year at Marienbad”by Alain Resnais from 1962, misère nim is played, several times, with rows of matchesof sizes 1,3,5,7. If you have a chance, try to watch that movie and spot when the otherplayer (not the guy who brought the matches) makes a mistake! Note that this is misèrenim, not nim, but you will be able to find out how to play it once you know how to playnim. (For games other than nim, normal play and misère versions are typically not sosimilar.)In contrast to condition 9, games where the available moves depend on the player (asin chess where one player can only move white pieces and the other only black pieces)are called partisan games. Much of combinatorial game theory is about partisan games,which we do not consider to keep matters simple.