Game Theory Basics - Department of Mathematics

1.9. EQUIVALENT GAMES 11It is an easy exercise to show that in sums of games, games can be replaced by equivalentgames, resulting in an equivalent sum. That is, for all games G,H,J,G ≡ H =⇒ G + J ≡ H + J. (1.5)Note that (1.5) is not merely a re-statement of definition 1.4, because equivalence of thegames G + J and H + J means more than just that the games are either both winning orboth losing (see the comments before lemma 1.9 below).Lemma 1.6 (The copycat principle) G + G ≡ ∗0 for any impartial game G.Proof. Given G, we assume by induction that the claim holds for all simpler games G ′ .Any option of G + G is of the form G ′ + G for an option G ′ of G. This is winning bymoving to the game G ′ +G ′ which is losing, by inductive assumption. So G+G is indeeda losing game, and therefore equivalent to ∗0 by lemma 1.5.We now come back to the issue of inverse elements in abstract groups, mentionedat the end of section 1.8. If we identify equivalent games, then the addition + of gamesdefines indeed a group operation. The neutral element is ∗0, or any equivalent game (thatis, a losing game).The inverse of a game G, written as the negative −G, fulfilsG + (−G) ≡ ∗0. (1.6)Lemma 1.6 shows that for an impartial game, −G is simply G itself.Side remark: For games that are not impartial, that is, partisan games, −G existsalso. It is G but with the roles of the two players exchanged, so that whatever movewas available to player I is now available to player II and vice versa. As an example,consider the game checkers (with the rule that whoever can no longer make a move loses),and let G be a certain configuration of pieces on the checkerboard. Then −G is thesame configuration with the white and black pieces interchanged. Then in the game G +(−G), player II (who can move the black pieces, say), can also play “copycat”. Namely,if player I makes a move in either G or −G with a white piece, then player II copiesthat move with a black piece on the other board (−G or G, respectively). Consequently,player II always has a move available and will win the game, so that G+(−G) is indeed alosing game for the starting player I, that is, G + (−G) ≡ ∗0. However, we only considerimpartial games, where −G = G.The following condition is very useful to prove that two games are equivalent.Lemma 1.7 Two impartial games G,H are equivalent if and only if G + H ≡ ∗0.Proof. If G ≡ H, then by (1.5) and lemma 1.6, G+H ≡ H +H ≡ ∗0. Conversely, G+H ≡∗0 implies G ≡ G + H + H ≡ ∗0 + H ≡ H.Sometimes, we want to prove equivalence inductively, where the following observationis useful.