Game Theory Basics - Department of Mathematics

1.8. SUMS OF GAMES 7principle of induction that says: if a property P(n) is true for n = 0, and if the propertyP(n) implies P(n + 1), then it is true for all natural numbers.We use the following notation for nim heaps. If G is a single nim heap with n chips,n ≥ 0, then we denote this game by ∗n. This game is completely specified by its options,and they are:options of ∗n : ∗0, ∗1, ∗2,..., ∗(n − 1). (1.1)Note that ∗0 is the empty heap with no chips, which allows no moves. It is invisible whenplaying nim, but it is useful to have a notation for it because it defines the most basiclosing position. (In combinatorial game theory, the game with no moves, which is theempty nim heap ∗0, is often simply denoted as 0.)We could use (1.1) as the definition of ∗n; for example, the game ∗4 is defined by itsoptions ∗0,∗1,∗2,∗3. It is very important to include ∗0 in that list of options, because itmeans that ∗4 has a winning move. Condition (1.1) is a recursive definition of the game∗n, because its options are also defined by reference to such games ∗k , for numbers ksmaller than n. This game fulfils the ending condition because the heap gets successivelysmaller in any sequence of moves.If G is a game and H is a game reachable by one or more successive moves fromthe starting position of G, then the game H is called simpler than G. We will often provea property of games inductively, using the assumption that the property applies to allsimpler games. An example is the – already stated and rather obvious – property that oneof the two players can force a win. (Note that this applies to games where winning orlosing are the only two outcomes for a player, as implied by the “normal play” conventionin 5 above.)Lemma 1.2 In any game G, either the starting player I can force a win, or player II canforce a win.Proof. When the game has no moves, player I loses and player II wins. Now assumethat G does have options, which are simpler games. By inductive assumption, in each ofthese games one of the two players can force a win. If, in all of them, the starting player(which is player II in G) can force a win, then she will win in G by playing accordingly.Otherwise, at least one of the starting moves in G leads to a game G ′ where the secondmovingplayer in G ′ (which is player I in G) can force a win, and by making that move,player I will force a win in G.If in G, player I can force a win, its starting position is a winning position, and wecall G a winning game. If player II can force a win, G starts with a losing position, andwe call G a losing game.1.8 Sums of gamesWe continue our discussion of nim. Suppose the starting position has heap sizes 1,5,5.Then the obvious good move is to option 5,5, which is losing.