The Combinatorial Game Theory of Well-Tempered Scoring Games

The Com binatorial Game Theory of W ell-temp ered Scoring Games Will Johnson Octob er 30, 2018 Abstract W e consider the class of “w ell-temp ered” in teger-v alued scoring games, which ha ve the property that the parit y of the length of the game is independent of the line of pla y . W e consider disjunctiv e sums of these games, and develop a theory for them analogous to the standard theory of disjunctiv e sums of normal-play partizan games. W e show that the monoid of w ell-temp ered scoring games mo dulo indistinguishabilit y is cancellative but not a group, and w e describ e its structure in terms of the group of normal-pla y partizan games. W e also classify Bo olean-v alued w ell-temp ered scoring games, showing that there are exactly seven ty , up to equiv alence. 1 In tro duction The standard theory of com binatorial game theory fo cuses on partizan games play ed under the normal play rule . In these games, t wo pla y ers tak e turns mo ving un til one pla yer is unable to. This play er loses. A substan tial amoun t of theoretical w ork has gone in to these games, as evidenced b y the b o oks [6], [3], [17] and [18]. In con trast, the earliest pap ers on partizan com binatorial games, by Milnor [16] and Hanner [12], fo cused on what are no w called sc oring games , which include games lik e Go and Dots-and-Boxes to v arying degrees. A scoring game sp eciﬁes an arbitrary numerical score dep ending on the ending p osition of the game, rather than on the identit y of the ﬁnal play er. One play er seeks to maximize this score while the other seeks to minimize it. The pap ers of Milnor and Hanner fo cused on scoring games having the prop erty that there is alwa ys an incentiv e to mov e. In other words, play ers w ould alwa ys prefer moving to passing. With this assumption, games mo dulo equiv alence form a nice ab elian group, one that is closely related to the normal-pla y partizan games. This assumption ma y seem limiting, but it is natural in games like Go, where there is no compulsion to mo ve. Later w ork by Ettinger [7] and [8] considered the general class of scoring games. In this case, games mo dulo equiv alence form a commutativ e monoid, but not a group. Ettinger was in terested in the question of whether this monoid was cancellative, but he seems to ha ve reac hed no deﬁnite conclusion. 1 The present pap er fo cuses on a limited class of scoring games, whic h we call wel l- temp er e d b ecause they are somewhat analogous to the well-tempered games of Grossman and Siegel [10]. A well-tempered scoring game is one in which the last play er to mov e is predetermined, and do es not dep end on the course of the game. In an o dd-temp ered game, the ﬁrst pla yer to mo ve will also be the last, while in an even-tempered game, the ﬁrst play er will nev er b e the last. These are exactly the scoring games which are trivial when pla yed as partizan games by the normal-play rule. Nevertheless, w e sho w that our class of games, lik e the class considered by Milnor, is closely related to the standard theory of partizan normal-pla y games. Our original motiv ation was the game To Knot or Not to Knot of [13]. F ew other games seem to ﬁt into this framework, so we hav e inv en ted a few. W e refer the reader to § 8 for examples. The outline of this pap er is as follows. In Section 2, we deﬁne precisely what we mean b y a well-tempered scoring game, how we add them, and what it means for t wo to be equiv alent. In § 3, we pro v e basic facts with these notions, sho wing that a certain class of sp ecial games (those having the prop ert y of Milnor and Hanner at even levels) form a well- b eha v ed ab elian group. T o a general game G , we can asso ciate tw o sp ecial games G + and G − , whic h c haracterize G . In fact, we show that a general game G has a double iden tity , acting as G + when Left is the ﬁnal pla yer, and as G − when Right is the ﬁnal play er. In Section 4, we consider v ariants of disjunctiv e addition. Sp eciﬁcally , we v ary the manner in whic h the ﬁnal scores of the summands are com bined into a total score. W e provide additional motiv ation for our earlier deﬁnition of equiv alence, generalize the results of § 3, and characterize the pairs ( G + , G − ) which can o ccur. In § 5, w e show ho w our class of games is closely related to and characterized b y the standard theory of normal-play partizan games. W e use this corresp ondence to give canonical forms for (sp ecial) games in § 6. In § 7 w e discuss the theory of { 0 , 1 } -v alued games, whic h is necessary to analyze games lik e To Knot or Not to Knot . W e show that there are seven t y { 0 , 1 } -v alued games mo dulo equiv alence, but inﬁnitely man y three-v alued games. W e close in § 8 with some examples of w ell-temp ered scoring games, including small dictionaries for them. W e assume the reader is familiar with the basic notations and con ven tions of Winning Ways [3] c hapters 1-2, or On Numb ers and Games [6] chapters 7-10. W e assume that the reader is capable of making basic inductiv e argumen ts of the sort found in c hapter 1 of ONA G . W e do not assume that the reader is familiar with thermograph y , atomic w eights, Norton multiplication, or the theory of loopy games, though we men tion them o ccasionally in passing. W e will refer to the normal-pla y partizan games of the standard theory as “partizan games.” W e will use C and B for the relations of “less than or fuzzy to” and “greater than or fuzzy to.” F or partizan games, = and ≡ will denote equiv alence and identit y , but for scoring games we will use ≈ and =, resp ectively . 2 Deﬁnitions W e are solely concerned with the following class of scoring games: 2 Deﬁnition 2.1. A n ev en-temp ered game is an inte ger or a p air { G L | G R } wher e { G L } and { G R } ar e ﬁnite nonempty sets of o dd-temp er e d games. 1 A n o dd-temp ered game is a p air { G L | G R } wher e { G L } and { G R } ar e ﬁnite nonempty sets of even-temp er e d games. A w ell-temp ered game is an even-temp er e d game or an o dd-temp er e d game. W e call the elemen ts of { G L } and { G R } the left and right options of G . If G is an in teger, w e deﬁne its options to b e the elemen ts of ∅ . If G is a well-tempered game, w e let π ( G ) = 0 if G is ev en-temp ered and π ( G ) = 1 if G o dd-temp ered. In what follo ws, all “games” will b e well-tempered scoring games, unless sp eciﬁed oth- erwise. W e view { G L | G R } as a game b et ween tw o play ers, Left and Righ t. Left can mo ve to any left option G L and Right can mov e to any right option G R . Once the game reac hes an integer, this v alue b ecomes the ﬁnal score. W e assume the Left is maximizing the score while Right is minimizing it. F or example, the game { 0 | 1 } is an o dd-temp ered game that will last exactly one turn. If Left go es ﬁrst, she will get 0 p oints, but if Righ t go es ﬁrst, Left will get 1 p oin t. On the other hand, in { 2 | − 2 } , whichev er play er go es ﬁrst will get tw o p oin ts. The o dd-temp ered game { 2 , 3 |{ 1 | 1 || 2 | 2 }} = { 2 , 3 |{{ 1 | 1 }|{ 2 | 2 }}} will last one turn if Left go es ﬁrst, or three turns if Right go es ﬁrst. Left can c ho ose to either receive 2 or 3 p oin ts. If Righ t go es ﬁrst, the ﬁnal score will necessarily b e 1, b ecause play will pro ceed { 2 , 3 |{ 1 | 1 || 2 | 2 }} → { 1 | 1 || 2 | 2 } → { 1 | 1 } → 1 since the play ers mov e in alternation. With these conv en tions, it is natural to deﬁne Deﬁnition 2.2. The left outcome and right outcome of G , denote d L( G ) and R( G ) , r esp e c- tively, ar e r e cursively deﬁne d as fol lows: L( G ) = ( G if G is an inte ger max G L R( G L ) otherwise R( G ) = ( G if G is an inte ger min G R L( G R ) otherwise Her e max G L R( G L ) denotes the maximum value of R( G L ) as G L r anges over the left options of G , and similarly min G R L( G R ) is the minimum as G R r anges over the right options of G . We deﬁne the outcome of G to b e the p air (L( G ) , R( G )) . The left and righ t outcomes of G are the scores that result under p erfect pla y when Left or Right mov es ﬁrst . In a well-tempered game, it also makes sense to discuss the play er who mo ves last , so we can mak e the follo wing deﬁnition: 1 W e are using the standard notational abbreviations of combinatorial game theory , where G L and G R are v ariables ranging ov er the left and right options of a game G . W e will use this kind of abbreviation later in Deﬁnitions 2.5 and 2.7 to sa v e space. 3 Deﬁnition 2.3. The left ﬁnal outcome of G , denote d Lf ( G ) , is L( G ) when G is o dd and R( G ) when G is even. Similarly, the righ t ﬁnal outcome of G , denote d Rf ( G ) , is R( G ) when G is o dd and L( G ) when G is even. In other w ords, the left or righ t ﬁnal outcome of G is the outcome when Left or Righ t is the ﬁnal pla yer, rather than the ﬁrst pla yer for ordinary outcomes. Note that (L( G ) , R( G ) , π ( G )) carries the same information as (Lf ( G ) , Rf ( G ) , π ( G )). Deﬁnition 2.4. L et G b e a wel l-temp er e d sc oring game. A subgame of G is G or a sub game of an option of G . If S is a set of numb ers, we say that G takes v alues in S or is S -v alued if every sub game of G that is an inte ger is an element of S . We denote the class of S -value d wel l-temp er e d sc oring games W S . Note that W Z is the class of all well-tempered scoring games. Also, every game has ﬁnitely many subgames, and in particular is in W S for some ﬁnite S . The basic op eration we would lik e to apply to our games is the disjunctiv e sum, in which w e pla y tw o games in parallel, and add the ﬁnal scores. W e will ﬁrst deﬁne a sligh tly more general op eration. Deﬁnition 2.5. L et S 1 , S 2 ⊆ f , and let f : S 1 × S 2 → Z b e a function. Then let the extension of f to games b e the function ˜ f : W S 1 × W S 2 → W Z deﬁne d r e cursively by ˜ f ( G 1 , G 2 ) = ( f ( G 1 , G 2 ) if G 1 and G 2 ar e inte gers { ˜ f ( G L 1 , G 2 ) , ˜ f ( G 1 , G L 2 ) | ˜ f ( G R 1 , G 2 ) , ˜ f ( G 1 , G R 2 ) } otherwise The game ˜ f ( G, H ) is obtained by pla ying G and H in parallel, and com bining the ﬁnal scores using f . A t eac h turn, the curren t pla yer chooses one of G or H to mo ve in. So in some sense, ˜ f is like the disjunctiv e sum of normal-play partizan games. In the case where S 1 , S 2 = Z and f ( x, y ) = x + y , we denote ˜ f ( G, H ) by G + H . W e call this the disjunctive sum of the scoring games G and H . Similarly , if f : S 1 × S 2 × · · · S n → Z is a function, then there is an extension to games ˜ f : W S 1 × · · · × W S n → W Z Note that ˜ f ( G 1 , . . . , G n ) is indeed a well-tempered game, with parity given as follows: π ( ˜ f ( G 1 , . . . , G n )) ≡ π ( G 1 ) + · · · + π ( G n ) ( mo d 2) . So in particular, π ( G + H ) ≡ π ( G ) + π ( H ) (mo d 2). W e mainly care ab out f whic h are order-preserving in the following sense: Deﬁnition 2.6. L et f : S 1 × · · · × S n → Z b e a function. Then we say that f is order- preserving if f ( x 0 1 , . . . , x 0 n ) ≥ f ( x 1 , . . . , x n ) whenever x 0 i ≥ x i for al l i . If f is or der-pr eserving, we c al l the extension to games ˜ f a generalized disjunctive op eration . 4 Another basic operation on games is negation, which reverses the scores and interc hanges the roles of the play ers: Deﬁnition 2.7. If G is an inte ger, let − G b e the ne gative of G in the usual sense. Otherwise, deﬁne − G by − G = {− G R | − G L } . W e let G − H denote G + ( − H ). The main obstacle to directly applying classic combinatorial game theory to scoring games is that a game lik e {− 1 | 1 } would b e equiv alent to 0 in the standard theory , but this fails in the theory of scoring games. The problem centers around games that are “cold” in the sense that there is no incen tive to mo ve. W e can measure this coldness as follo ws: Deﬁnition 2.8. L et G b e a wel l-temp er e d game. F or i = 0 , 1 , let gap i ( G ) b e the supr emum of R( H ) − L( H ) as H r anges over the sub games of G with π ( H ) = i . We c al l gap 0 ( G ) the ev en gap of G and gap 1 ( G ) the o dd gap of G . Note that if G is an integer, gap 0 ( G ) = 0 and gap 1 ( G ) = −∞ . Since ev ery game has at least one subgame that is an integer, gap 0 ( G ) ≥ 0 for ev ery G . Moreo v er, every game that isn’t an in teger satisﬁes gap 1 ( G ) > −∞ . Also note that if H is a subgame of G , then gap i ( H ) ≤ gap i ( G ). Another useful op eration on games is heating. Deﬁnition 2.9. If G is a game and t is an inte ger, then R t G ( G heated by t ) is the game deﬁne d inductively by Z t G = ( G if G ∈ Z n t + R t G L | − t + R t G R o otherwise The eﬀect of this operation is to giv e ev ery pla yer a b onus of t p oints for each mo ve she makes. W e allo w t to b e negative, in which case we refer to this op eration as c o oling . Note that our co oling op eration is m uch simpler than the co oling op eration of the standard theory . One of the central ideas b ehind combinatorial game theory is to consider games mo dulo equiv alence. F or us, the appropriate deﬁnition of equiv alence is the following: Deﬁnition 2.10. L et G , H b e games. We say that G and H ar e equiv alent , denote d G ≈ H , if L( G + X ) = L( H + X ) and R( G + X ) = R( H + X ) for every game X . Similarly, we say that G & H if L( G + X ) ≥ L( H + X ) and R( G + X ) ≥ R( H + X ) for every game X . We deﬁne G . H ⇐ ⇒ H & G . The relation ≈ is clearly an equiv alence relation, and & is a preorder whic h induces a partial order on W Z / ≈ . T o motiv ate Deﬁnition 2.12, w e need a basic fact: Theorem 2.11. L et G, H b e games. If G . H , then π ( G ) = π ( H ) . 5 Pr o of. Let K b e an y game, and consider the game K + {− N | N } for N a large integer. Clearly , neither pla yer w ants to mov e in {− N | N } , and if N  0, b oth pla yers will mak e it their top priority to not mov e in {− N | N } . Consequently , the ﬁnal play er will b e forced to mo ve in {− N | N } and will pay a hea vy p enalty . So for K arbitrary , lim N → + ∞ Lf ( K + {− N | N } ) = −∞ lim N → + ∞ Rf ( K + {− N | N } ) = + ∞ . No w supp ose that G . H and π ( G ) 6 = π ( H ). If G is ev en and H is o dd, then Rf ( G + {− N | N } ) = R( G + {− N | N } ) ≤ R( H + {− N | N } ) = Lf ( G + {− N | N } ) for all N , con tradicting the fact that the left hand side go es to + ∞ while the right hand side go es to −∞ . If G is o dd and H is ev en, then Rf ( G + {− N | N } ) = L( G + {− N | N } ) ≤ L( H + {− N | N } ) = Lf ( H + {− N | N } ) whic h again con tradicts the limiting b eha vior as N → + ∞ . Deﬁnition 2.12. If G, H ar e games, we let G ≈ + H if π ( G ) = π ( H ) and Lf ( G + X ) = Lf ( H + X ) for every X . Similarly, we let G & + H if π ( G ) = π ( H ) and Lf ( G + X ) ≥ Lf ( H + X ) for every H . Deﬁne G . + H if H & + G . Deﬁne ≈ − , . − , and & − analo gously, using Rf ( G + X ) and Rf ( H + X ) inste ad of Lf ( G + X ) and Lf ( H + X ) . We c al l ≈ + and ≈ − left-equiv alence and right-equiv alence , r esp e ctively. Note that if G . + H and G . − H , then G . H . The conv erse follo ws by Theorem 2.11, and analogous statements hold for ≈ and ≈ ± . 6 3 Games mo dulo equiv alence 3.1 Basic F acts Lemma 3.1. L et G, H , K ∈ W Z , and s, t ∈ Z . Then we have the fol lowing identities (not e quivalenc es): G + H = H + G (1) ( G + H ) + K = G + ( H + K ) (2) − ( G + H ) = ( − G ) + ( − H ) (3) − ( − G ) = G (4) G + 0 = G (5) Z 0 G = G (6) Z t 0 = 0 (7) Z t ( G + H ) = Z G + Z t H (8) Z t ( − G ) = − Z t G (9) Z t Z s G = Z t + s G (10) Pr o of. All of these identities ha ve easy inductive pro ofs akin to those found in Chapter 1 of [6]. T o giv e an example, we pro ve (10). In the base case where G is an integer, (10) is trivial. Otherwise, Z t Z s G = ( t + Z t  Z s G  L ! | − t + Z t  Z s G  R !) =  t + Z t  s + Z s G L  | − t + Z t  − s + Z s G R  =  t +  Z t s + Z t Z s G L  | − t +  − Z t s + Z t Z s G R  =  ( t + s ) + Z t Z s G L | − ( t + s ) + Z t Z s G R  =  ( t + s ) + Z t + s G L | − ( t + s ) + Z t + s G R  = Z t + s G where the p en ultimate equality follo ws by induction, and we ha ve also used (2), (3), (8), and (9). 7 Lemma 3.2. If G 1 , G 2 , . . . ar e games, t ∈ Z , and ˜ f is a gener alize d disjunctive op er ation, then π ( G 1 + G 2 ) ≡ π ( G 1 ) + π ( H 2 ) (mo d 2) (11) π ( − G 1 ) = π ( G 1 ) (12) π  Z t G 1  = π ( G 1 ) (13) π ( ˜ f ( G 1 , G 2 , . . . , G n )) ≡ n X i =1 π ( G i ) (mo d 2) (14) Again the pro ofs are easy inductions, so w e omit them. Lemma 3.3. L et G b e a game, n, t ∈ Z , i ∈ { 0 , 1 } , and f , g b e or der-pr eserving. Then L( − G ) = − R( G ) (15) R( − G ) = − L( G ) (16) L( G + n ) = L( G ) + n (17) L( ˜ f ( G )) = f (L( G )) (18) L( ˜ g ( n, G )) = g ( n, L( G )) (19) L  Z t G  = L( G ) + t · π ( G ) (20) R  Z t G  = R( G ) − t · π ( G ) (21) gap i  Z t G  = gap i ( G ) − 2 t · i (22) gap i ( − G ) = gap i ( G ) (23) (24) A lso, (17-19) hold with R inste ad of L , and (15-19) hold with Lf and Rf in plac e of L and R . Again, all the pro ofs are easy inductions, so we omit them. Lemma 3.4. L et  b e any of the fol lowing op er ations ≈ , & , . , ≈ ± , & ± , . ± , and let ˜  denote ≈ , & , . , ≈ ∓ , & ∓ , . ∓ , r esp e ctively. L et G, H , K b e games and supp ose that G  H . Then G + K  H + K (25) − H ˜  − G (26) Z t G  Z t H (27) A lso, if G i  H i for every i , then { G 1 , . . . , G n | G n +1 , . . . , G m }  { H 1 , . . . , H n | H n +1 , . . . , H m } (28) 8 Pr o of. F or example, supp ose that  is & + . If G & + H , then for an y game X , Lf (( G + K ) + X ) = Lf ( G + ( K + X )) ≥ Lf ( H + ( K + X )) = Lf (( H + K ) + X ) , so (25) holds. Similarly , Rf ( − H + X ) = Rf ( − ( H + ( − X ))) = − Lf ( H + ( − X )) ≥ − Lf ( G + ( − X )) = Rf ( − G + X ) , so − H & − − G and (26) holds. F or (27), note that Lf  Z t G + X  − Lf  Z t H + X  = Lf  Z t  G + Z − t X  − Lf  Z t  H + Z − t X  = Lf  G + Z − t X  − Lf  H + Z − t X  ≥ 0 using either (20) or (21) and the fact that G and H hav e the same parity . Th us (27) holds. F or the ﬁnal claim, one can show b y an easy induction on X that Lf ( { G 1 , . . . , G n | G n +1 , . . . , G m } + X ) ≥ Lf ( { H 1 , . . . , H n | H n +1 , . . . , H m } + X ) . W e lea ve this as an exercise to the reader. Ha ving sho wn the lemma for & + , the case of . + follo ws immediately , & − and . − follo w by symmetry , and the remaining ﬁve relations follow b ecause they are logical conjunctions of these four. An immediate consequence of this is that W Z / ≈ , W Z / ≈ + , and W Z / ≈ − are partially ordered monoids with addition induced b y + and partial orders induced b y & , & + , or & − resp ectiv ely . Also, the game-building op eration {· · · | · · · } is well-deﬁned on eac h of these monoids, as are heating, co oling, and negation. W e will see in Theorem 4.1 that generalized disjunctiv e op erations are also compatible with the nine relations. 3.2 Pla ying t wo games at once Lemma 3.5. If G is a game, then R( G − G ) ≥ 0 L( G − G ) ≤ 0 Pr o of. If Left goes second, she can resp ond to an y mo ve in G with the corresponding mo v e in − G , following this strategy till the game ends. This ensures her a score of at least 0, though she could p ossibly do b etter. So R( G − G ) ≥ 0. A similar argumen t sho ws L( G − G ) ≤ 0. In classic combinatorial game theory , a simple argument shows that G + H ≥ 0 if G ≥ 0 and H ≥ 0. The analogous result for scoring games is the follo wing confusing lemma, whic h encompasses ab out a dozen inequalities all prov en in analogous w ays: 9 Lemma 3.6. L et G, H b e wel l-temp er e d games. Then R( G + H ) ≥ R( G ) + R( H ) − max( π ( H ) · gap π ( G ) ( G ) , π ( G ) · gap π ( H ) ( H )) (29) L( G + H ) ≥ L( G ) + R( H ) − max( π ( H ) · gap 1 − π ( G ) ( G ) , (1 − π ( G )) · gap π ( H ) ( H )) (30) wher e i · n e quals n if i = 1 and 0 if i = 0 , even if n = −∞ . W e list the four cases of these equations for future reference: If G, H are ev en, then R( G + H ) ≥ R( G ) + R( H ) (31) L( G + H ) ≥ L( G ) + R( H ) − gap 0 ( H ) . (32) If G, H are o dd, then R( G + H ) ≥ R( G ) + R( H ) − max(gap 1 ( G ) , gap 1 ( H )) (33) L( G + H ) ≥ L( G ) + R( H ) − gap 0 ( G ) (34) If G is o dd and H is ev en, then R( G + H ) ≥ R( G ) + R( H ) − gap 0 ( H ) (35) L( G + H ) ≥ L( G ) + R( H ) (36) Finally , if G is even and H is o dd, then R( G + H ) ≥ R( G ) + R( H ) − gap 0 ( G ) (37) L( G + H ) ≥ L( G ) + R( H ) − max(gap 1 ( G ) , gap 1 ( H )) (38) Here we hav e used the fact that max(0 , gap 0 ( H )) = gap 0 ( H ) b ecause gap 0 ( H ) ≥ 0. Pr o of. W e prov e (29-30) together by induction on the complexit y of G and H . First supp ose that G is an integer. Then max( π ( H ) gap π ( G ) ( G ) , π ( G ) gap π ( H ) ( H )) = max( π ( H ) gap 0 ( G ) , 0 gap π ( H ) ( H )) = max( π ( H )0 , 0) = 0 . So (29) says R( G + H ) ≥ R( G ) + R( H ) , whic h holds b ecause R( n + H ) = n + R( H ) = R( n ) + R( H ) for any integer n , b y a v arian t of (17). So (29) holds in this case. A similar argument sho ws that (29) holds when H is an in teger. Supp ose neither G nor H is an in teger. Then R( G + H ) = L(( G + H ) R ) for some right option ( G + H ) R of G + H . The option ( G + H ) R is either of the form G R + H or G + H R . By symmetry , we can assume the former. Then R( G + H ) = L( G R + H ) ≥ L( G R )+R( H ) − max( π ( H ) gap 1 − π ( G R ) ( G R ) , (1 − π ( G R )) gap π ( H ) ( H )) 10 b y induction. But π ( G R ) = 1 − π ( G ), and gap i ( G R ) ≤ gap i ( G ), so R( G + H ) ≥ L( G R ) + R( H ) − max( π ( H ) gap π ( G ) ( G R ) , π ( G ) gap π ( H ) ( H )) ≥ L( G R ) + R( H ) − max( π ( H ) gap π ( G ) ( G ) , π ( G ) gap π ( H ) ( H )) . But since R( G ) = min G R L( G R ), we must ha ve L( G R ) ≥ R( G ). So R( G + H ) ≥ R( G ) + R( H ) − max ( π ( H ) gap π ( G ) ( G ) , π ( G ) gap π ( H ) ( H )) as desired. Thus (29) holds. Next, we show that (30) holds. First supp ose that G is an in teger. Then max( π ( H ) gap 1 − π ( G ) ( G ) , (1 − π ( G )) gap π ( H ) ( H )) = max( π ( H ) gap 1 ( G ) , gap π ( H ) ( H )) . If H is o dd-temp ered, this equals max( −∞ , gap 1 ( H )) = gap 1 ( H ). If H is even-tempered, this equals max(0 , gap 0 ( H )) = gap 0 ( H ). Either wa y , it equals gap π ( H ) ( H ). So we wan t to sho w that L( G + H ) ≥ L( G ) + R( H ) − gap π ( H ) ( H ) . But since G is an in teger, L( G + H ) = G + L( H ) = L( G ) + L( H ). So it remains to sho w that L( H ) − R( H ) ≥ − gap π ( H ) ( H ) whic h is true by deﬁnition of gap i ( H ). Finally , supp ose that G is not an integer. Then L( G ) = R( G L ) for some left option G L of G . As G L + H is a left option of G + H , L( G + H ) ≥ R( G L + H ) . By induction, it follo ws that L( G + H ) ≥ R( G L ) + R( H ) − max( π ( H ) gap π ( G L ) ( G L ) , π ( G L ) gap π ( H ) ( H )) = L( G ) + R( H ) − max( π ( H ) gap π ( G L ) ( G L ) , π ( G L ) gap π ( H ) ( H )) No w π ( G L ) = 1 − π ( G ) and gap i ( G L ) ≤ gap i ( G ), so L( G + H ) ≥ L( G ) + R( H ) − max( π ( H ) gap 1 − π ( G ) ( G L ) , (1 − π ( G )) gap π ( H ) ( H )) ≥ L( G ) + R( H ) − max( π ( H ) gap 1 − π ( G ) ( G ) , (1 − π ( G )) gap π ( H ) ( H )) as desired. By symmetry , w e also get the following: L( G + H ) ≤ L( G ) + L( H ) + max( π ( H ) gap π ( G ) ( G ) , π ( G ) gap π ( H ) ( H )) R( G + H ) ≤ R( G ) + L( H ) + max( π ( H ) gap 1 − π ( G ) ( G ) , (1 − π ( G )) gap π ( H ) ( H )) There are corresp onding duals of (31-38). W e will refer to these dual inequalities using a  . F or example, (37  ) says that if G is ev en and H is o dd, then L( G + H ) ≤ L( G ) + L( H ) + gap 0 ( G ) 11 3.3 Games with small gaps Using the inequalities of the previous section, w e can examine the in teraction of the gap functions with addition: Theorem 3.7. If G, H ar e games, then gap 0 ( G + H ) ≤ gap 0 ( G ) + gap 0 ( H ) and gap 1 ( G + H ) ≤ max(gap 1 ( G ) , gap 1 ( H )) . Pr o of. W e prov e each statemen t separately by induction. First consider gap 0 ( G + H ). If g = gap 0 ( G ) and h = gap 0 ( H ), then every option G 0 of G has gap 0 ( G 0 ) ≤ g and every option H 0 of H has gap 0 ( H 0 ) ≤ h . So by induction, every option of G + H has gap 0 ≤ g + h , and it remains to show that if G + H is even-tempered, then L( G + H ) − R( G + H ) ≥ − ( g + h ). If G and H are b oth ev en, then b y (32) L( G + H ) ≥ L( G ) + R( H ) − h and by (32  ) R( H + G ) ≤ R( H ) + L( G ) + g . Th us L( G + H ) − R( G + H ) ≥ − ( h + g ) as desired. Similarly , if G and H are b oth o dd, then by (34) and (34  ), L( G + H ) ≥ L( G ) + R( H ) − g R( H + G ) ≤ R( H ) + L( G ) + h so L( G + H ) − R( G + H ) ≥ − ( h + g ) as desired. If G and H hav e diﬀeren t parity , then G + H is not ev en and there is nothing to sho w. Next, supp ose that gap 1 ( G ) , gap 1 ( H ) ≤ m . W e w an t to sho w that gap 1 ( G + H ) ≤ m . Ev ery option of G and H also has gap 1 ≤ m , so b y induction, ev ery option of G + H has gap 1 ≤ m . So it remains to c heck that if G + H is o dd, then L( G + H ) − R( G + H ) ≥ − m . Without loss of generalit y , G is o dd and H is ev en. Then b y (38) and (36  ), L( H + G ) ≥ L( H ) + R( G ) − max(gap 1 ( H ) , gap 1 ( G )) ≥ L( H ) + R( G ) − m, R( G + H ) ≤ R( G ) + L( H ) Then L( G + H ) − R( G + H ) ≥ − m as desired. 12 Let I b e the class of games with gap 0 ( G ) = 0, and J be the class of games with gap 0 ( G ) = 0 and gap 1 ( G ) ≤ 2. By the previous theorem, I and J are closed under addition. They are also closed under negation, b y (23). Note that if G ∈ I , then R t G ∈ J for t  0, b y (22). The class J will b e imp ortan t in § 5, but for no w, w e fo cus on the nice prop erties of I . Lemma 3.8. If G ∈ I is an even game, then G & 0 if and only if R( G ) ≥ 0 . Pr o of. If G & 0 then R( G ) ≥ 0 trivially . Conv ersely , supp ose R( G ) ≥ 0. W e need to sho w that L( G + X ) ≥ L( X ) and R( G + X ) ≥ R( X ) for every X . If X is ev en, these follow by (32) applied to X and G , and (31) applied to G and X , resp ectiv ely . If X is o dd, these follo w from (36) applied to X and G , and (35) applied to G and X , resp ectiv ely . Theorem 3.9. If G, H ∈ I , then G − G ≈ 0 and the fol lowing ar e e quivalent: (a) G & H (b) G & − H (c) R( G − H ) ≥ 0 and π ( G ) = π ( H ) (d) G & + H (e) L( H − G ) ≤ 0 and π ( G ) = π ( H ) . Pr o of. Since G and − G are in I and π ( G ) = π ( − G ), G − G is also in I and is even-tempered. By Lemma 3.5, R( G − G ) ≥ 0. So b y Lemma 3.8, G − G & 0. By symmetry , G − G . 0, so G − G ≈ 0. Next we show that the ﬁve statements are equiv alen t. ( a ) = ⇒ ( b ) is easy , using Theorem 2.11. F or ( b ) = ⇒ ( c ), note that if G & − H then R( G − H ) & R( H − H ) = 0 , since H − H ≈ 0. Also, π ( G ) = π ( H ) by deﬁnition of & − . F or ( c ) = ⇒ ( a ), note that by Lemma 3.8, if R( G − H ) ≥ 0 and π ( G − H ) = 0, then G − H & 0. But then b y Lemma 3.4 and the fact that H − H ≈ 0, G = G + 0 ≈ G + ( H − H ) = H + ( G − H ) & H + 0 = H , so ( a ) holds. W e hav e shown that ( a ), ( b ), and ( c ) are equiv alent. The equiv alence of ( d ) and ( e ) follo ws by symmetry . In particular, we see that ev ery element of I is an in v ertible element of the monoid W Z / ≈ . Also, the relations ≈ and ≈ ± are all equiv alen t for games in I , as are the relations & and & ± . 13 3.4 Upsides and Do wnsides Next, w e turn to a characterization of the structure of W Z / ≈ and W Z / ≈ ± in terms of the group I / ≈ . The key step is the following lemma: Lemma 3.10. If G is even, and every option of G is in I , and L( G ) ≤ R( G ) , then G ≈ + R( G ) G ≈ − L( G ) Pr o of. Note that gap 0 ( G ) = R( G ) − L( G ) (39) b ecause the only subgame G 0 of G for which R( G 0 ) − L( G 0 ) could b e p ositiv e is G itself, as all options of G are in I . By symmetry , it suﬃces to prov e G ≈ + R( G ). W e need to show that if X is an y game, then Lf ( G + X ) ? = Lf (R( G ) + X ) = R( G ) + Lf ( X ) . If X is ev en, then b y (31) Lf ( G + X ) = R( G + X ) ≥ R( G ) + R( X ) = R( G ) + Lf ( X ) . (40) But by (32  ) and (39), R( X + G ) ≤ R( X ) + L( G ) + gap 0 ( G ) = R( X ) + L( G ) + R( G ) − L( G ) = R( X ) + R( G ) . So equality holds in (40), and Lf ( G + X ) = R( G ) + Lf ( X ) as desired. Otherwise, X is o dd. Then by (36) Lf ( G + X ) = L( X + G ) ≥ L( X ) + R( G ) = R( G ) + Lf ( X ) . (41) But by (35  ) and (39), L( X + G ) ≤ L( X ) + L( G ) + gap 0 ( G ) = L( X ) + L( G ) + (R( G ) − L( G )) = L( X ) + R( G ) . So equality holds in (41), and Lf ( G + X ) = R( G ) + Lf ( X ) as desired. This yields the follo wing key result: Theorem 3.11. If G is any game, then ther e exist games G + , G − ∈ I with G + ≈ + G ≈ − G − . If S ⊆ Z and G is S -value d, then G + and G − c an b e chosen to b e S -value d. A dditional ly, G + & G − . 14 Pr o of. By symmetry , w e only sho w the existence of G + . W e proceed b y induction on G + . If G is an in teger, then w e can take G + = G . Otherwise, by induction ev ery option of G is left-equiv alent to an S -v alued game in I . Let H b e the game obtained from G by replacing ev ery option of G with such a game. Then by Lemma 3.4 applied to ≈ + , G is left-equiv alent to H . Since H is an S -v alued game, we are done unless H fails to b e in I . But every option of H is in I , so this can only o ccur if H is ev en and L( H ) < R( H ). In this case, we hav e G ≈ + H ≈ + R( H ) b y the previous lemma. Since R( H ) ∈ S , w e can take the integer R( H ) to b e G + . F or the ﬁnal claim, note that G ≈ − G − . Therefore, by Lemma 3.4 − G ≈ + − ( G − ) and so G − G ≈ + ( G + − G − ). But by Lemma 3.5, R( G + − G − ) = Lf ( G + − G − ) = Lf ( G − G ) = R( G − G ) ≥ 0 , so G + & G − b y Theorem 3.9. Note that if H , K are tw o games in I , b oth left-equiv alent to G , then H ≈ + K so H ≈ K b y Theorem 3.9. Therefore the G ± of the previous theorem are determined uniquely up to equiv alence. Deﬁnition 3.12. If G is a wel l-temp er e d game, we let G + and G − b e the games of the pr evious the or em. These ar e deﬁne d only up to e quivalenc e. We c al l G + the upside of G and G − the downside of G , by analo gy with the the ory of lo opy games. Note that G + and G − dep end only on G up to ≈ , i.e., if G ≈ H then G ± ≈ H ± . Also, if G ∈ I , then G + and G − can b e taken to b e G . Let W = W Z . F or G a set of games, we use G , G + , and G − to denote the quotients G / ≈ , G / ≈ + and G / ≈ − . These ha ve natural structures as posets, coming from the relations . and . ± . If G is closed under addition, then these three quotients will also b e monoids. Using upsides and do wnsides, w e get a fairly complete description of W and W ± in terms of I . Theorem 3.13. The p osets I , I + , and I − ar e al l e qual, I is a p artial ly-or der e d ab elian gr oup, and the inclusions I = I ±  → W ± (42) ar e isomorphisms of p artial ly or der e d monoids. In p articular, W + and W − ar e gr oups. A lso, let I 2 = { ( x, y ) ∈ I × I : x ≥ y } . Make I 2 into a p artial ly or der e d monoid by taking the pr o duct or dering and c omp onentwise addition. Then the map W → I 2 induc e d by G 7→ ( G + , G − ) is an isomorphism of p artial ly or der e d monoids. Under this identiﬁc ation, the ne gative of ( x, y ) is ( − y , − x ) , and the image of I  → W → I 2 is the diagonal { ( x, x ) : x ∈ I } . 15 Pr o of. The fact that I , I + , and I − are equal stems from the fact that . − , . + , and . are all the same relation on I , by Theorem 3.9. The fact that I is a group follows from the fact that if G ∈ I , then − G ∈ I and G + ( − G ) ≈ 0 by Theorem 3.9. The inclusion I + → W + (sa y) is surjectiv e by Theorem 3.11. It is injective and strictly order-preserving, trivially . The remaining statements are equiv alent to the follo wing claims: • If G ∈ W , then G + & G − . This is part of Theorem 3.11. • If G ≈ H , then G + ≈ H + and G − ≈ H − . More generally , G & H ⇐ ⇒ G + & H + ∧ G − & H − . This follows from the facts that G & ± H ⇐ ⇒ G ± & ± H ± ⇐ ⇒ G ± & H p m. The ﬁrst ⇐ ⇒ is trivial and the second ⇐ ⇒ follo ws from Theorem 3.9. • If G, H ∈ I and G & H , then there exists a game K with K + ≈ G and K − ≈ H , i.e., K ≈ + G and K ≈ − H . W e defer the pro of of this to Theorem 4.6 in the next section. • If G, H are games, then ( G + H ) ± ≈ G ± + H ± . Since I is closed under addition, this amounts to sho wing that if G ≈ ± G ± and H ≈ ± H ± , then G + H ≈ ± G ± + H ± , which follows from Theorem 3.4. • If G is a game, then − G ± = ( − G ) ∓ . This similarly follo ws from Theorem 3.4. • If G ∈ I , then G + ≈ G − , which is clear by taking G + = G − = G . • If G + ≈ G − , then G is equiv alen t to a game in I . But if G + ≈ G − , then G ≈ − G − ≈ − G + , so G ≈ − G + . Since G ≈ + G + b y deﬁnition of G + , we hav e G ≈ G + ∈ I . Corollary 3.14. If G is a game, the fol lowing ar e e quivalent: (a) G is invertible, in the sense that G + H ≈ 0 for some H ∈ W . (b) G ≈ K for some K ∈ I . (c) G − G ≈ 0 Pr o of. The direction ( c ) = ⇒ ( a ) is trivial, and ( b ) = ⇒ ( c ) was part of Theorem 3.9. The direction ( a ) = ⇒ ( b ) follo ws from the general fact that if G is a partially-ordered ab elian group and we construct the monoid G 2 = { ( x, y ) ∈ G × G : x ≥ y } , then the inv ertible elements of G 2 are exactly the diagonal elements ( x, x ). T o describ e the structure of W Z , it remains to determine the structure of I . In § 5 w e will relate I to the group Pg of partizan normal-play games. 16 4 Other Disjunctiv e Op erations So far w e ha v e fo cused on the op eration of disjunctiv e addition G + H . In this section, we sho w that other generalized disjunctive op erations behav e like addition. F or example they are compatible with equiv alence, and they preserve membership in I . 4.1 Easy Inductions The following facts ha ve easy inductiv e pro ofs, whic h we lea ve as exercises to the reader: (a) If f : S → T is order-preserving and G is an S -v alued game, then R( ˜ f ( G )) = f (R( G )) and similarly for L( · ), Rf ( · ), and Lf ( · ). (b) If f , g : S 1 × · · · × S n → T are t wo order preserving functions and f ( x 1 , . . . , x n ) ≤ g ( x 1 , . . . , x n ) for all x i , then ˜ f ( G 1 , . . . , G n ) . ˜ g ( G 1 , . . . , G n ) . (c) If f : S 1 ×· · · × S n → T is a function (not necessarily order-preserving), σ is a p ermutation of { 1 , . . . , n } , and g : S σ (1) × · · · × S σ ( n ) → T is the function given b y g ( x σ (1) , . . . , x σ ( n ) ) = f ( x 1 , . . . , x n ) , then ˜ g ( G σ (1) , . . . , G σ ( n ) ) = ˜ f ( G 1 , . . . , G n ) (d) If f : S 1 × · · · × S n → T is a function, g : R 1 × · · · × R m → S j is another function, and h : S 1 × · · · × S j − 1 × R 1 × · · · × R m × S j +1 × · · · × S n → T is the function giv en by h ( x 1 , . . . , x j − 1 , y 1 , . . . , y m , x j +1 , . . . , x n ) = f ( x 1 , . . . , x j − 1 , g ( y 1 , . . . , y m ) , x j +1 , . . . , x n ) , then ˜ h ( G 1 , . . . , G j − 1 , H 1 , . . . , H m , G j +1 , . . . , G n ) = ˜ f ( G 1 , . . . , G j − 1 , ˜ g ( H 1 , . . . , H m ) , G j +1 , . . . , G n ) . In particular, if m = n = 1 we see that ] f ◦ g = ˜ f ◦ ˜ g . (e) If f : S 1 × · · · × S n × T is a function and g : ( − S 1 ) × · · · × ( − S n ) → ( − T ) is the function g ( x 1 , . . . , x n ) = − f ( − x 1 , . . . , − x n ) , then ˜ g ( G 1 , . . . , G n ) = − ˜ f ( − G 1 , . . . , − G n ) . (f ) If g : Z → Z is the constan t function 0, then ˜ g ( G ) ≈ 0 for π ( G ) = 0 and ˜ g ( G ) ≈ { 0 | 0 } for π ( G ) = 1. (g) If f : S 1 × · · · × S n is a function and ∗ is the game { 0 | 0 } , then ˜ f ( G 1 + ∗ , . . . , G n ) = ˜ f ( G 1 , . . . , G n ) + ∗ 17 4.2 Equiv alence and Generalized Disjunctiv e Op erations The following theorem generalizes to f of higher or lo w er arities, but we only prov e the case where f takes tw o argumen ts, for notational simplicity . Theorem 4.1. L et S 1 , S 2 ⊆ Z and f : S 1 × S 2 → Z b e or der pr eserving. L et G 1 , H 1 b e S 1 -value d games, and G 2 , H 2 b e S 2 -value d games. L et  b e one of the r elations & , . , ≈ , & ± , . ± , ≈ ± . Supp ose that G i  H i for i = 1 , 2 . Then ˜ f ( G 1 , G 2 )  ˜ f ( H 1 , H 2 ) , wher e ˜ f is the extension of f to games. Pr o of. Because all nine relations can b e deﬁned in terms of & + and & − , w e can assume  is & ± . By symmetry , we can tak e  = & + . Supp ose w e can sho w that for G, H , X appropriately-v alued, G & + H = ⇒ ˜ f ( G, X ) & + ˜ f ( H , X ) Then by symmetry (in terchanging the t wo argumen ts of f ), G & + H = ⇒ ˜ f ( X , G ) & + ˜ f ( X , H ) . Com bining these, w e get ˜ f ( G 1 , H 1 ) & + ˜ f ( G 1 , H 2 ) & + ˜ f ( G 2 , H 2 ) as desired. W e are reduced to showing that if f : S 1 × S 2 → Z is order-preserving and G & + H , then ˜ f ( G, X ) & + ˜ f ( H , X ) . Since only ﬁnitely many integers o ccur within G, H, X , we can assume without loss of gen- eralit y that S 1 and S 2 are ﬁnite, restricting the domain of f if necessary . W e need to show that for any game Y , Lf ( ˜ f ( G, X ) + Y ) ≥ Lf ( ˜ f ( H , X ) + Y ) . Supp ose not. Then there is some n suc h that Lf ( ˜ f ( G, X ) + Y ) < n Lf ( ˜ f ( H , X ) + Y ) ≥ n. Let δ m : Z → { 0 , 1 } b e the order-preserving function that sends x to 1 if x ≥ m and x to 0 if x < m . By Lemma 4.2 applied to the function g ( w , x, y ) = δ n ( f ( w , x ) + y ), there is a function h : S 2 × Z → Z such that δ n ( f ( w , x ) + y ) = δ 0 ( w + h ( x, y )) 18 for all ( w , x, y ) ∈ S 1 × S 2 × Z . Then by several applications of (c) and (d) of § 4.1, ˜ δ n ( ˜ f ( G, X ) + Y ) = ˜ δ 0 ( G + ˜ h ( X , Y )) ˜ δ n ( ˜ f ( H , X ) + Y ) = ˜ δ 0 ( H + ˜ h ( X , Y )) . But by (a) of § 4.1, it follo ws that 0 = δ n (Lf ( ˜ f ( G, X ) + Y )) = δ 0 (Lf ( G + ˜ h ( X , Y ))) 1 = δ n (Lf ( ˜ f ( H , X ) + Y )) = δ 0 (Lf ( H + ˜ h ( X , Y ))) . Then Lf ( G + ˜ h ( X , Y )) < Lf ( H + ˜ h ( X , Y )) , con tradicting G & + H . Lemma 4.2. L et S 1 , S 2 , S 3 b e subsets of Z , with S 1 ﬁnite. L et g : S 1 × S 2 × S 3 → { 0 , 1 } b e or der-pr eserving. Then ther e is a function h : S 2 × S 3 → { 0 , 1 } such that for every ( x, y , z ) ∈ S 1 × S 2 × S 3 , g ( x, y , z ) = 1 ⇐ ⇒ x + h ( y , z ) ≥ 0 Pr o of. Let M b e an integer greater than every element of S 1 . Replacing S 1 with S 1 ∪ { M } and extending g ( x, y , z ) by g ( M , y , z ) = 1 for all y , z , we can assume that for ev ery y , z , there is some x ∈ S 1 suc h that g ( x, y , z ) = 1. Let h ( y , z ) b e the negativ e of the smallest such x . Then g ( x, y , z ) = 1 if and only if x ≥ − h ( y , z ), as desired. F rom the theorem (and its generalization to higher arities), w e see that if f : S 1 × · · · × S n → T is order-preserving, then ˜ f induces well-deﬁned functions W S 1 × · · · × W S n → W T W + S 1 × · · · × W + S n → W + T W − S 1 × · · · × W − S n → W − T 4.3 In v ertible Games and Generalized Disjunctiv e Op erations Our next goal is to sho w that I is closed under generalized disjunctiv e op erations. In particular, letting I S = I ∩ W S , there is a well-deﬁned map I S 1 × · · · × I S n → I T . This map characterizes the full map W S 1 × · · · × W S n → W T , since by Theorem 4.1, ˜ f ( G 1 , . . . , G n ) ± ≈ ˜ f ( G ± 1 , . . . , G ± n ) assuming that the righ t hand side lies in I , as we shall prov e. W e ﬁrst pro ve something that is almost the desired result: 19 Lemma 4.3. L et f : S 1 × · · · × S n → Z b e or der-pr eserving, and for e ach i , let G i ∈ I S i . Then ˜ f ( G 1 , . . . , G n ) is e quivalent to a game in I . Pr o of. Since each G i has ﬁnitely man y subgames, we can assume without loss of generality that every S i is ﬁnite. Let M b e an integer greater in magnitude than ev ery elemen t of the range of f (which is ﬁnite, b ecause w e made the S i ﬁnite). Let κ : Z → Z b e the function κ ( x ) = ( 2 M x > 0 0 x ≤ 0 . Then since G i − G i ≈ 0, ˜ κ ( G i − G i ) ≈ ˜ κ (0) = 0 for each i , b y Theorem 4.1. It follows that ˜ κ ( G 1 − G 1 ) + · · · + ˜ κ ( G n − G n ) ≈ 0 . Consider the tw o functions S 1 × ( − S 1 ) × · · · × S n × ( − S n ) → Z given by h 1 ( x 1 , x 2 , . . . , x 2 n ) = f ( x 1 , x 3 , . . . , x 2 n − 1 ) − f ( − x 2 , − x 4 , . . . , − x 2 n ) h 2 ( x 1 , x 2 , . . . , x 2 n ) = κ ( x 1 + x 2 ) + κ ( x 3 + x 4 ) + · · · + κ ( x 2 n − 1 + x 2 n ) . It is straightforw ard to chec k that h 1 ≤ h 2 . Therefore, by (b) of § 4.1, ˜ h 1 ( G 1 , − G 1 , . . . , G n , − G n ) . ˜ h 2 ( G 1 , − G 1 , . . . , G n , − G n ) Ho wev er, ˜ h 1 ( G 1 , − G 1 , . . . , G n , − G n ) is nothing but ˜ f ( G 1 , G 2 , . . . , G n ) − ˜ f ( G 1 , . . . , G n ) and ˜ h 2 ( G 1 , − G 1 , . . . , G n , − G n ) is nothing but ˜ κ ( G 1 − G 1 ) + · · · + ˜ κ ( G n − G n ) ≈ 0 + · · · + 0 = 0 , b y several applications of (c), (d), and (e) of § 4.1. Th us ˜ f ( G 1 , G 2 , . . . , G n ) − ˜ f ( G 1 , . . . , G n ) . 0 Negating b oth sides, ˜ f ( G 1 , G 2 , . . . , G n ) − ˜ f ( G 1 , . . . , G n ) & 0 . Therefore ˜ f ( G 1 , . . . , G n ) is inv ertible, so b y Corollary 3.14, it is equiv alen t to a game in I . Our desired result follo ws easily: 20 Theorem 4.4. If f : S 1 × · · · × S n → Z is or der-pr eserving and G i ∈ I S i for e ach i , then ˜ f ( G 1 , . . . , G n ) ∈ I . Pr o of. By the previous lemma, ev ery subgame of ˜ f ( G 1 , . . . , G n ) is equiv alen t to a game in I . It therefore suﬃces to show that if H is a game and every subgame of H (including H ) is in vertible, then H ∈ I . If H 0 is any even-tempered subgame of H , then L( H 0 ) − R( H 0 ) ≥ 0 b ecause H 0 ≈ K for some K ∈ I , and L( K ) − R( K ) ≥ 0 b y deﬁnition of I . Another corollary of Theorem 4.1 is the following: Corollary 4.5. L et S and T b e ﬁnite subsets of Z , with | S | = | T | . L et f b e the or der- pr eserving bije ction fr om S to T . Then ˜ f : W S → W T induc es a bije ction W S → W T Pr o of. Let g : T → S b e the order-preserving function f − 1 . By (d) of § 4.1, ˜ f and ˜ g are in verses of eac h other. By Theorem 4.1, they induce w ell-deﬁned order-preserving maps from W S → W T and vice-versa. These t w o induced maps must b e in verses of eac h other. Because of this corollary , the study of S -v alued games is equiv alent to the study of { 1 , . . . , n } -v alued games, for n = | S | . 4.4 The P ossible Upsides and Do wnsides W e now complete the pro of of Theorem 3.13: Theorem 4.6. L et G, H b e games in I , with G & H . Then ther e is a K such that K ≈ + G and K ≈ − H , i.e., K + ≈ G and K − ≈ H . Mor e over, if G and H ar e in I S , then K c an b e chosen to b e in W S . Ther efor e the map W S → { ( x, y ) : x, y ∈ I S , x ≥ y } induc e d by G 7→ ( G + , G − ) is surje ctive. Pr o of. F or n ≥ 0, let Q n = {{ 0 | 0 }|{ n | n }} . One can chec k directly that Q n ≈ + n and Q n ≈ − 0 by Lemma 3.10. Let ∆ b e the game G − H . Then ∆ ∈ I and ∆ & 0. In particular, ∆ is ev en. Let ∆ 0 b e the game obtained from ∆ by replacing every integer subgame n with ( n if n ≤ 0 Q n if n > 0 21 Let f : Z → Z b e the function f ( n ) = min(0 , n ). Then ∆ 0 ≈ + ∆ and ∆ 0 ≈ − ˜ f (∆) b y an easy induction using the fact that Q n ≈ + n and Q n ≈ − 0. Let g : Z → Z b e the constant function 0. Since g ≥ f and ∆ & 0, 0 ≈ ˜ g (∆) & ˜ f (∆) & ˜ f (0) = 0 , b y (f ) and (b) of § 4.1 and Theorem 4.1. Therefore ˜ f (∆) ≈ 0. Since ∆ 0 ≈ + n and ∆ 0 ≈ − ˜ f (∆) ≈ 0, w e conclude that ∆ 0 + H ≈ + ∆ + H = ( G − H ) + H ≈ G ∆ 0 + H ≈ − 0 + H ≈ H Letting K 0 = ∆ 0 + H , K 0 has the desired upside and downside. It remains to make K 0 tak e v alues in S , if both G and H do. Supp ose G and H take v alues in S . Let r : Z → S b e the function that sends each x ∈ Z to the closest element of S , taking the lesser v alue in case of a tie. Clearly r is an order-preserving function that agrees with the iden tity on S . Thus ˜ r ( G ) = G and ˜ r ( H ) = H , and ˜ r ( K 0 ) is S -v alued. By Theorem 4.1, ˜ r ( K 0 ) ≈ + ˜ r ( G ) = G and ˜ r ( K 0 ) ≈ − ˜ r ( H ) = H , T aking K = ˜ r ( K 0 ), we get the desired game. If G, H ∈ I and G & H , we let G & H b e the game of the theorem, whose upside is G and whose do wnside is H . This is w ell-deﬁned up to equiv alence. Our notation is based on the analogy with the lo op y partizan theory . 4.5 Ev en and Odd Let ∗ b e the game { 0 | 0 } . If G is an S -v alued game, then so is G + ∗ . Indeed, letting f : S × { 0 } → S b e addition, G + ∗ = ˜ f ( G, ∗ ) ∈ W S . No w clearly ∗ ∈ I , so if G ∈ I S , then G + ∗ ∈ I S . As −∗ = ∗ , the game ∗ is its o wn in v erse. Therefore, the map G 7→ G + ∗ induces in volutions on W S and I S for any S . These in v olutions exc hange the even-tempered and o dd-temp ered games, establishing a bijection b et ween the ev en and o dd elements in each of W S and I S . (Note that w e can talk ab out the parit y of an elemen t of W S b y Theorem 2.11.) Deﬁnition 4.7. L et G b e a game. The even pro jection of G , denote d ev en( G ) , is G if G is even-temp er e d, and G + ∗ otherwise. This has a n umber of simple prop erties: Theorem 4.8. L et G, H b e games. 22 (a) If G is S -value d, then so is ev en( G ) . (b) If G ∈ I , then even( G ) ∈ I . (c) ev en( G ) is even-temp er e d and ev en(even( G )) = ev en( G ) . (d) ev en( − G ) = − even( G ) . (e) G & H if and only if even( G ) & ev en( H ) and π ( G ) = π ( H ) . In p articular, if G ≈ H then even( G ) ≈ even( H ) , so even induc es wel l-deﬁne d maps on W S and I S for any S . (f ) If ˜ f is a gener alize d disjunctive op er ation, then ˜ f (even( G 1 ) , . . . , ev en( G n )) = even( ˜ f ( G 1 , . . . , G n )) . In p articular, for the c ase of addition, ev en( G + H ) ≈ ev en( G ) + even( H ) . (g) ev en( G + ) ≈ ev en( G ) + and even( G − ) ≈ ev en( G ) − . Pr o of. W e already prov ed (a) and (b). (c) is trivial. (d) says that − G = − G when G is ev en-temp ered, and that ∗ − G = − ( ∗ + G ) when G is o dd-temp ered, b oth of which are trivial. The remaining three statements are pro ven as follo ws: (e) Suppose G & H . Then by Theorem 2.11, π ( G ) = π ( H ). It remains to see that if π ( G ) = π ( H ), then G & H ⇐ ⇒ even( G ) & even( H ). Let  be 0 if π ( G ) = π ( H ) = 0, and ∗ if π ( G ) = π ( H ) = 1. Then ev en( G ) = G +  and even( H ) = H +  . But clearly G & H ⇐ ⇒ G +  & H + , since  is in vertible. (f ) It follows by rep eated applications of (g) of § 4.1 that ˜ f (even( G 1 ) , . . . , ev en( G n )) = ˜ f ( G 1 , . . . , G n ) + ∗ + · · · + ∗ , where the num b er of ∗ ’s is the num b er of G i whic h are o dd-temp ered. Th us ev en( ˜ f (even( G 1 ) , . . . , ev en( G n ))) ≈ ev en( ˜ f ( G 1 , . . . , G n ) + ∗ + · · · + ∗ ) . (43) No w if X is any game, then even( X + ∗ ) ≈ X since even( X + ∗ ) is either X or X + ∗ + ∗ ≈ X . Th us ev en( ˜ f ( G 1 , . . . , G n ) + ∗ + · · · + ∗ ) ≈ even( ˜ f ( G 1 , . . . , G n )) . (44) But since ev ery ev en( G i ) is ev en-temp ered, ˜ f (even( G 1 ) , . . . , ev en( G n )) is also even- temp ered so ˜ f (even( G 1 , . . . , ev en( G n ))) = even( ˜ f (even( G 1 ) , . . . , ev en( G n ))) . (45) Com bining equations (43-45) gives the desired result. 23 (g) If G , G + , and G − are even-tempered, this sa ys that G + ≈ G + and G − ≈ G − . Otherwise, all three are o dd-temp ered, and this says ∗ + G ± ≈ ( G + ∗ ) ± whic h follo ws b ecause ( G + H ) ± ≈ G ± + H ± for any H (part of Theorem 3.13) and ∗ ± ≈ ∗ since ∗ ∈ I . W e summarize the results of § 4 in the follo wing analog of Theorem 3.13, whose pro of we lea ve as an exercise to the reader. Theorem 4.9. L et S ⊆ Z . L et I 0 S denote the even-temp er e d elements of I S . Then we have a natur al isomorphism of p osets W S ∼ = { ( x, y ) ∈ I 0 S : x ≥ y } × Z / 2 Z induc e d by G 7→ (even( G ) + , ev en( G ) − , π ( G )) If f : S k → S is or der-pr eserving, then the extension to games ˜ f : W k S → W S induc es maps ˜ f : W k S → W S ˜ f : ( I 0 S ) k → I 0 S . The r estriction to I 0 S determines the b ehavior on W S as fol lows: ev en( ˜ f ( G 1 , . . . , G k )) + ≈ ˜ f (even( G 1 ) + , . . . , ev en( G k ) + ) ev en( ˜ f ( G 1 , . . . , G k )) − ≈ ˜ f (even( G 1 ) − , . . . , ev en( G k ) − ) π ( ˜ f ( G 1 , . . . , G k )) ≡ π ( G 1 ) + · · · + π ( G k ) (mo d 2) Later when w e consider { 0 , 1 } -v alued games, this will allow us to restrict our atten tion to I 0 { 0 , 1 } , whic h has only eight elements, rather than considering W { 0 , 1 } , whic h has seven ty elemen ts. 5 Reduction to P artizan Games Returning to the general case of Z -v alued games and addition, the goal of this section is to giv e an explicit description of I Z (or equiv alently I 0 Z ) in terms of the classical combinatorial game theory of disjunctiv e sums of partizan games. Earlier we deﬁned J ⊆ I to b e the set of games G with gap 0 ( G ) = 0 and gap 1 ( G ) ≤ − 2. By Theorem 3.7, I and J are closed under addition. Since b oth consist of inv ertible games, 24 I and J are b oth subgroups of the monoid W . Now by (22) of Lemma 3.3, heating has the follo wing eﬀects on gap 0 and gap 1 : gap 0  Z t G  = gap 0 ( G ) gap 1  Z t G  = gap 1 ( G ) − 2 t. Therefore, I is preserv ed b y heating, J is preserv ed b y heating b y t ≥ 0, and if G is an y game in I , then R t G ∈ J for t  0. Because I is preserv ed b y heating, and heating is compatible with ≈ ± b y Lemma 3.4, Z t G ≈ ± Z t ( G ± ) ∈ I so  Z t G  ± ≈ Z t ( G ± ) . Th us heating is compatible with upsides and downsides. Also, heating induces monoid automorphisms on W and I . Note ho wev er that heating is not compatible with the map G 7→ even( G ) of the previous section, b ecause R t ∗ = { t | − t } 6≈ ∗ . 5.1 The Class K W e w ould lik e to deﬁne a class of games K that has the same relation to W that J has to I . Let K 0 b e the class of games G such that b oth G + and G − can be chosen to lie in J . Let K b e the class of games G suc h that ev ery subgame of G (including G itself ) lies in K 0 . Equiv alen tly , G ∈ K if ev ery option of G is in K and G ∈ K 0 . Theorem 5.1. (a) J = K ∩ I . In p articular, J ⊆ K . (b) K is close d under addition and ne gation. (c) If G is a game, then R t G ∈ K for t  0 . Also, if G ∈ K then R t G ∈ K for t ≥ 0 . (d) If S ⊆ Z , and max( S ) − min( S ) ≤ 2 , then W S ⊆ K . Pr o of. (a) If G ∈ J , then certainly G ∈ I . Also, for ev ery subgame H of G , H ∈ K 0 b ecause w e can tak e H + = H − = H ∈ J . So G ∈ K . Con versely , supp ose that G ∈ I ∩ K . If H is any o dd-temp ered subgame of G , then H ∈ K 0 ∩ I . Therefore H ≈ H + for some H + ∈ J , and L( H ) − R( H ) = L( H + ) − R( H + ) ≥ − 2. Since H w as arbitrary , gap 1 ( G ) ≥ − 2, and G ∈ J . 25 (b) Since J is closed under addition, and addition is compatible with taking upsides and do wnsides, K 0 is also closed under addition. Now let G, H ∈ K . By induction, ev ery option of G + H is in K . So G + H ∈ K as long as G + H ∈ K 0 , which follows from the fact that b oth G and H are in K 0 . The deﬁnitions of I , J , and K are symmetric, not distinguishing the tw o pla yers, so by symmetry all three classes are closed under negation. (c) The follo wing fact has an easy inductiv e pro of, which w e lea ve as an exercise to the reader: F act 5.2. If G is a game, and H is a sub game of R t G , then H = nt + R t G 0 for some sub game G 0 of G and some n ∈ Z . No w if H is an y game, then H + , H − ∈ I . As noted ab o ve, if w e heat games suﬃcien tly , they will lie in J . So for any H , w e ha v e  Z t H  ± = Z t ( H ± ) ∈ J for t  0. Th us R t H ∈ K 0 for t  0. Applying this to every subgame of G , w e see that for t  0, every subgame G 0 of G satisﬁes R t G 0 ∈ K 0 . No w let H b e any subgame of R t G . By the F act, H = nt + R t G 0 for some n ∈ Z and some subgame G 0 of G . Then nt ∈ Z ⊂ J ⊂ K 0 and R t G 0 ∈ K 0 . Then since K 0 is closed under addition, H = nt + R t G 0 ∈ K 0 . Since H was an arbitrary subgame of R t G , it follows that R t G ∈ K . A similar argument sho ws the second claim of (c). (d) First note that if G is any S -v alued game, then L( G ) , R( G ) ∈ S , so | L( G ) − R( G ) | ≤ | max( S ) − min( S ) | ≤ 2 . In particular, gap 1 ( G ) ≤ 2. It follows that I S ⊆ J . Then if G is in W S and H is any subgame of G , H + and H − can b e chosen to lie in I S ⊆ J . Therefore G ∈ K . 5.2 F rom Scoring Games to P artizan Games W e now turn to standard (normal-pla y , short, non-lo op y) partizan games. Deﬁnition 5.3. If G L 1 , . . . , G R 1 , . . . ar e p artizan games, let { G L 1 , . . . | G R 1 , . . . } + (46) { G L 1 , . . . | G R 1 , . . . } + (47) 26 denote the usual p artizan game { G L 1 , . . . | G R 1 , . . . } unless ther e is at le ast one inte ger n with G L i C n C G R j for every i, j , in which c ase we let (46) denote the lar gest such n and (47) denote the smal lest such n . These op erations sho w up in the deﬁnition of atomic weigh t in the theory of all-small partizan games, suggesting a p ossible connection to the present theory . One conv enient prop ert y of {·|·} ± is the following fact: Lemma 5.4. If n is an inte ger and { G L } , { G R } ar e sets of p artizan games, then n + { G L | G R } + = { n + G L | n + G R } + and n + { G L | G R } − = { n + G L | n + G R } − Pr o of. When the ca v eat that distinguishes { G L | G R } ± from { G L | G R } do esn’t apply , this is just the usual in teger a v oidance theorem, since { G L | G R } do es not equal an integer. In the other case, the largest (smallest) integer which lies b etw een the n + G L and the n + G R is n more than the largest (smallest) integer whic h lies b et ween the G L and the G R , so the result is clear. Lemma 5.5. L et G − = { G L | G R } − and H − = { H L | H R } − . Then G − + H − = { G L + H − , G − + H L | G R + H − , G − + H R } − (48) Pr o of. If { K L } and { K R } are sets of partizan games, then for n  0, { K L | K R } − + n = { K L + n | K R + n } − = { K L + n | K R + n } b y Lemma 5.4 and the simplicit y rule. Therefore, for n  0, ( G − + n ) + ( H − + n ) = { G L + n | G R + n } + { H L + n | H R + n } = { ( G L + n ) + ( H − + n ) , ( G − + n ) + ( H L + n ) | ( G R + n ) + ( H − + n ) , ( G − + n ) + ( H R + n ) } = { G L + H − + 2 n, G − + H L + 2 n | G R + H − + 2 n, G − + H R + 2 n } . Since 2 n  0, this equals { G L + H − + 2 n, G − + H L + 2 n | G R + H − + 2 n, G − + H R + 2 n } − = { G L + H − , G − + H L | G R + H − , G − + H R } − + 2 n W e conclude that for n  0, G − + H − + 2 n = { G L + H − , G − + H L | G R + H − , G − + H R } − + 2 n, and so (48) holds. 27 Our to ol for reducing J and K to the theory of partizan games will b e the following maps: Deﬁnition 5.6. L et Pg denote the class of short p artizan games, mo dulo the usual e quiva- lenc e. L et ψ , ψ + , ψ − b e the thr e e maps W → Pg deﬁne d r e cursively as fol lows: ψ ( n ) = ψ + ( n ) = ψ − ( n ) = n for n an inte ger, and otherwise, ψ ( { G L | G R } ) = { ψ ( G L ) | ψ ( G R ) } ψ + ( { G L | G R } ) = { ψ + ( G L ) | ψ + ( G R ) } + ψ − ( { G L | G R } ) = { ψ − ( G L ) | ψ − ( G R ) } − Her e the {· · · | · · · } on the left r efer to sc oring games, while the {· · · | · · · } on the right r efer to p artizan games. So for instance, ψ ( {− 1 | 1 } ) = 0 b ecause {− 1 | 1 } = 0 in the standard partizan theory . Also note that the scoring game 1 has no options, but ψ (1) = 1 = { 0 |} has a left option. Theorem 5.7. L et G, H b e wel l-temp er e d sc oring games. Then ψ ± ( G + H ) = ψ ± ( G ) + ψ ± ( H ) Pr o of. W e giv en an inductive pro of for the ψ + case. In the base case where G, H are integers, this is trivial. In the case where one of G or H is an in teger and the other is not, this is Lemma 5.4. In the case where G and H are b oth integer s, this is Lemma 5.5. Note that the analogous identit y fails for ψ , since ψ ( {− 2 | 2 } + 3) = ψ ( { 1 | 5 } ) = 2 6 = {− 2 | 2 } + 3 = ψ ( {− 2 | 2 } ) + ψ (3) . Ev en for ψ ± , these maps are discarding a lot of information, since ψ + ( {− 2 | 2 } ) = − 1 = ψ + ( {− 2 | 0 } ), although {− 2 | 2 } and {− 2 | 0 } are quite diﬀerent as scoring games. Nev ertheless, ψ preserves some information ab out outcomes. Lemma 5.8. L et G b e a wel l-temp er e d sc oring game. If G is even, then ψ ( G ) ≥ 0 if and only if R( G ) ≥ 0 . If G is o dd, then ψ ( G ) B 0 if and only if L( G ) ≥ 0 . (By symmetry, if G is even, then ψ ( G ) ≤ 0 if and only if L( G ) ≤ 0 , and if G is o dd, then ψ ( G ) C 0 if and only if R( G ) ≤ 0 .) Pr o of. W e pro ceed by induction. If G is an in teger n , then ψ ( G ) = n , which is ≥ 0 if and only if it is ≥ 0, as desired. If G is not an integer, but is ev en-temp ered, then ψ ( G ) ≥ 0 if and only if ev ery G R satisﬁes ψ ( G R ) B 0. By induction, this means that L( G R ) ≥ 0 for ev ery G R , whic h is the same as saying that R( G ) ≥ 0. Finally , suppose that G is o dd-temp ered. Then ψ ( G ) B 0 if and only if there exists a ψ ( G ) L = ψ ( G L ) ≥ 0. By induction, this is equiv alent to R( G L ) ≥ 0. So ψ ( G ) B 0 if and only if R( G L ) ≥ 0 for some G L . But this is equiv alent to L( G ) ≥ 0. 28 The main place where the maps ψ and ψ ± are useful is on J and K : Theorem 5.9. L et G ∈ J . Then (a) ψ ( G ) = ψ + ( G ) = ψ − ( G ) (b) If G is even-temp er e d and n ∈ Z , then n ≥ ψ ( G ) if and only if n ≥ L( G ) and n ≤ ψ ( G ) if and only if n ≤ R( G ) . (c) If G is o dd-temp er e d and n ∈ Z , then n B ψ ( G ) if and only if n ≥ R( G ) and n C ψ ( G ) if and only if n ≤ L( G ) . Pr o of. Let | G | denote the total n um b er of subgames of G . W e pro ve all three statements together, by induction on | G | . In the base case where | G | = 1, G is an integer, and all three statements are more or less trivial (the third one is v acuous). Next supp ose that we hav e prov en all three statemen ts for all games G ∈ J with | G | < m . W e ﬁrst prov e (a) for all games G ∈ J with | G | = m . Pro ving (a) amounts to sho wing that there is at most one integer b et w een the left and right options of { ψ ( G L ) | ψ ( G R ) } . If not, then there is some n such that ψ ( G L ) C n < n + 1 C ψ ( G R ) (49) for all G L and G R . Because | G L | < m and | G R | < m , we can apply (b) or (c) to these games b y induction. If G is ev en-temp ered, then by (c), R( G L ) ≤ n < n + 1 ≤ L( G R ) for all G L and G R . Therefore, L( G ) ≤ n < n + 1 ≤ R( G ) , so L( G ) < R( G ) and gap 0 ( G ) > 0, contradicting G ∈ I . Similarly , if G is o dd-temp ered, then by (b) and (49), R( G L ) < n < n + 1 < L( G R ) , for all G L and G R , so that gap 1 ( G ) > 2, con tradicting G ∈ J . Thus (a) holds for all G ∈ J with | G | = m . Next, we show that (b) holds for all ev en-temp ered G ∈ J with | G | = m . Note that | G − n | = | G | , so from (a), ψ ( G − n ) = ψ + ( G − n ) ψ ( G ) = ψ + ( G ) 29 Then by Theorem 5.7, we get ψ ( G − n ) = ψ ( G ) − ψ + ( n ) = ψ ( G ) − n So we are reduced to showing that 0 ≥ ψ ( G − n ) ⇐ ⇒ 0 ≥ L( G − n ) 0 ≤ ψ ( G − n ) ⇐ ⇒ 0 ≤ R( G − n ) So w e are reduced to the case n = 0. But this is just Lemma 5.8. The pro of of (c) is similar. Corollary 5.10. L et G, H ∈ J . Then ψ ( G + H ) = ψ ( G ) + ψ ( H ) ψ ( − G ) = − ψ ( G ) , and when π ( G ) = π ( H ) , G & H ⇐ ⇒ ψ ( G ) ≥ ψ ( H ) Pr o of. The ﬁrst claim follo ws from part (a) of the theorem together with Theorem 5.7. The second claim holds for an y game G . The third claim follows from the fact that G − H is ev en, and so G & H ⇐ ⇒ R( G − H ) ≥ 0 ⇐ ⇒ ψ ( G − H ) ≥ 0 ⇐ ⇒ ψ ( G ) − ψ ( H ) ≥ 0 ⇐ ⇒ ψ ( G ) ≥ ψ ( H ) using Theorem 3.9, Lemma 5.8 (which is the n = 0 case of part (b) of Theorem 5.9), and the ﬁrst tw o claims of this corollary . In particular, ψ induces a well-deﬁned map J → Pg , and this map is an order-preserving homomorphism of groups. Moreov er, if w e restrict to the ev en-temp ered elemen ts of J , it is injectiv e and strictly order-preserving. The kernel of the map J → Pg consists of the t wo elemen ts 0 and {− 1 | 1 } . The analog of Theorem 5.9 for K is Theorem 5.11. L et G ∈ K . Then ψ ± ( G ) = ψ ( G ± ) . In p articular, if G is even then n ≥ ψ + ( G ) ⇐ ⇒ n ≥ L( G + ) ⇐ ⇒ n & G + ⇐ ⇒ n & G n ≥ ψ − ( G ) ⇐ ⇒ n ≥ L( G − ) = L( G ) n ≤ ψ + ( G ) ⇐ ⇒ n ≤ R( G + ) = R( G ) n ≤ ψ − ( G ) ⇐ ⇒ n ≤ R( G − ) ⇐ ⇒ n . G − ⇐ ⇒ n . G. Similarly, if G is o dd then n B ψ − ( G ) ⇐ ⇒ n ≥ R( G − ) = R( G ) n C ψ + ( G ) ⇐ ⇒ n ≤ L( G + ) = L( G ) 30 Pr o of. Only the claim that ψ + ( G ) = ψ ( G + ) requires pro of, since everything else follo ws by symmetry , Theorem 5.9, or Theorem 3.13. W e pro ceed by induction on G . The case where G ∈ Z is clear. Otherwise, let G = { G L | G R } . Let H L and H R b e upsides of the options of G , c hosen to lie in J , and let H = { H L | H R } . Then G L ≈ + H L and G R ≈ + H R so by Lemma 3.4 G ≈ + H . W e consider the tw o cases H ∈ I and H / ∈ I separately . First supp ose that H ∈ I . Then H ≈ + G implies that H ≈ G + ∈ J , so L( H ) − R( H ) ≥ − 2 · π ( G ). Since ev ery option of H is in J , this shows that H ∈ J . Then b y Theorem 5.9 and Corollary 5.10, { ψ ( H L ) | ψ ( H R ) } + = { ψ + ( H L ) | ψ + ( H R ) } + = ψ + ( H ) = ψ ( H ) = ψ ( G + ) . But by induction, ψ ( H L ) = ψ + ( G L ) and ψ ( H R ) = ψ + ( H R ). So ψ + ( G ) = { ψ + ( G L ) | ψ + ( G R ) } + = { ψ + ( H L ) | ψ + ( H R ) } + = ψ ( G + ) as desired. Next, supp ose that H / ∈ I . Every option of H is in I , so the only wa y that H / ∈ I can o ccur is if H is even and L( H ) < R( H ). Then by Lemma 3.10, G ≈ + H ≈ + R( H ) . Therefore G + ≈ R( H ). Let n = R( H ). By induction and Theorem 5.9(a), ψ + ( G ) = { ψ + ( G L ) | ψ + ( G R ) } + = { ψ ( H L ) | ψ ( H R ) } + . T o sho w that ψ + ( G ) equals ψ ( G + ) = n , we need to show that n is the maxim um in teger x suc h that ψ ( H L ) C x C ψ ( H R ) for all H L and H R . By Theorem 5.9(c), this is equiv alent to R( H L ) ≤ x ≤ L( H R ) for all H L and H R , or equiv alen tly , L( H ) ≤ x ≤ R( H ). Clearly n = R( H ) is the greatest suc h x , using the fact that R( H ) > L( H ). 5.3 F rom P artizan Games to Scoring Games While K will b e useful § 7, for the rest of this section we fo cus on J . W e will ﬁnd an inv erse to the homomorphism J 0 → Pg , where J 0 is the even-tempered subgroup of J . Let φ 0 and φ 1 b e the maps from partizan games to w ell-temp ered scoring games deﬁned as follows: φ 0 ( G ) = ( n if G equals (is equiv alent to) an in teger n { φ 1 ( G L ) | φ 1 ( G R ) } otherwise φ 1 ( G ) = ( { n − 1 | n + 1 } if G equals (is equiv alent to) an in teger n { φ 0 ( G L ) | φ 0 ( G R ) } otherwise 31 F or example, φ 0 ( { 1 | 3 ∗} ) = 2, φ 0 ( { 1 | − 3 ∗} ) = { 0 | 2 || − 3 | − 3 } . Note that φ i ( G ) dep end on the form of G , not just its v alue, so w e do not get well-deﬁned maps Pg → W . Mo dulo equiv alence, how ev er, the φ i maps are an exact inv erse of ψ : Theorem 5.12. If G is a p artizan game and i = 0 or 1 , then φ i ( G ) is a wel l-temp er e d game, with π ( φ i ( G )) = i . Mor e over, φ i ( G ) ∈ J and ψ ( φ i ( G )) = G . Pr o of. The fact that φ i ( G ) is a well-tempered game of the correct parity follo ws b y an easy induction, as do es the fact that ψ ( φ i ( G )) = G . W e pro ve the remaining statement, that φ i ( G ) ∈ J , by induction on i . If G is equal in v alue to an in teger n , then φ i ( G ) is either n or { n − 1 | n + 1 } . Either wa y , it is in J . Otherwise, G do es not equal an integer. By induction, every option of φ i ( G ) is in J . It remains to show that L( φ 0 ( G )) ≥ R( φ 0 ( G )) (50) and L( φ 1 ( G )) ≥ R( φ 1 ( G )) − 2 (51) I claim that these inequalities hold strictly . Supp ose not. If (50) fails to hold strictly , then there exists some in teger n suc h that L( φ 0 ( G )) ≤ n ≤ R( φ 0 ( G )) Th us R( φ 1 ( G L )) ≤ n ≤ L( φ 1 ( G R )) for ev ery left option G L of G and ev ery righ t option G R of G . But b y induction, every φ 1 ( G L ) and φ 1 ( G R ) is in J , so Theorem 5.9 applies, giving G L = ψ ( φ 1 ( G L )) C n C ψ ( φ 1 ( G R )) = G R . By the simplicity rule, G equals an integer - the simplest n b et w een all the G L and G R . This con tradicts the assumption that G was not an integer, so (50) holds strictly . The argumen t for (51) is similar. Corollary 5.13. The p artial ly or der e d gr oups J and Pg × ( Z / 2 Z ) ar e isomorphic. The isomorphism J → Pg × ( Z / 2 Z ) is induc e d by G 7→ ( ψ ( G ) , π ( G )) , and its inverse is induc e d by ( H , i ) 7→ φ i ( H ) . Conse quently, the φ i maps ar e or der-pr eserving homomorphisms, in the sense that φ i ( G ) & φ i ( H ) ⇐ ⇒ G ≥ H and φ i + j ( G + H ) ≈ φ i ( G ) + φ j ( H ) This follows easily from Theorem 5.12 and Corollary 5.10. 32 5.4 Heating things up T o complete the description of I 0 , we need to ﬁnd the eﬀect of heating on Pg under the iden tiﬁcation of J 0 ∼ = Pg . If K ≥ 0 is a partizan game, we can recursiv ely deﬁne a function f K on partizan games b y f K ( G ) = { f K ( G L ) + K | f K ( G R ) − K } . It is easy to see that this map is a homomorphism and is exactly order preserving: f K ( G ) ≥ 0 if and only if G ≥ 0. Th us G = H implies that f K ( G ) = f K ( H ). In fact, this map is Norton m ultiplication by { K |} , or equiv alen tly , the ov erheating op eration R K { K |} . Because it is order- preserving, it is w ell-deﬁned on the quotient space Pg of partizan games modulo equiv alence. T aking K = 1 ∗ , we get Norton m ultiplication by { 1 ∗ |} , or equiv alen tly , the o verheating op eration G 7→ R 1 ∗ 1 G . Note that R 1 ∗ 1 n = n for n ∈ Z b ecause R 1 ∗ 1 1 = 1. As we noted ab o v e, J is closed under the op eration R t for t > 0, by (22) of Lemma 3.3. Theorem 5.14. L et G b e a wel l-temp er e d sc oring game in J . If G is even-temp er e d, then ψ  Z 1 G  = Z 1 ∗ 1 ψ ( G ) and if G is o dd-temp er e d, then ψ  Z 1 G  = ∗ + Z 1 ∗ 1 ψ ( G ) . Pr o of. Since ψ and φ are in v erses of each other, it suﬃces to sho w that if H is a partizan game, then φ 0  Z 1 ∗ 1 H  ≈ Z 1 φ 0 ( H ) (52) φ 1  ∗ + Z 1 ∗ 1 H  ≈ Z 1 φ 1 ( H ) (53) These tw o statemen ts are equiv alent, for ﬁxed H , b ecause φ 1  ∗ + Z 1 ∗ 1 H  = φ 1 ( ∗ ) + φ 0  Z 1 ∗ 1 H  = { 0 | 0 } + φ 0  Z 1 ∗ 1 H  Z 1 φ 1 ( H ) = Z 1 ( φ 1 (0) + φ 0 ( H )) = Z 1 ( {− 1 | 1 } + φ 0 ( H )) = { 0 | 0 } + Z 1 φ 0 ( H ) . W e pro ceed to pro ve (52-53) b y induction on H . If H is equal to an integer n , then R 1 ∗ 1 H = R 1 ∗ 1 n = n . Therefore R 1 ∗ 1 H also equals an integer, and φ 0  Z 1 ∗ 1 H  = φ 0 ( n ) = n. 33 But n = R 1 n = R 1 φ 0 ( n ) ≈ R 1 φ 0 ( H ), since n = H implies that φ 0 ( n ) ≈ φ 0 ( H ). This establishes (52) for H equal to an integer. When H do es not equal an in teger, neither do es R 1 ∗ 1 H . Then φ 0  Z 1 ∗ 1 H  = φ 0  Z 1 ∗ 1 H L + 1 ∗ | Z 1 ∗ 1 H R + 1 ∗  =  φ 1  Z 1 ∗ 1 H L + 1 ∗  | φ 1  Z 1 ∗ 1 H R + 1 ∗  ≈  φ 1  ∗ + Z 1 ∗ 1 H L  + φ 0 (1) | φ 1  ∗ + Z 1 ∗ 1 H R  + φ 0 (1)  ≈  Z 1 φ 1 ( H L ) + 1 | Z 1 φ 1 ( H R ) − 1  = Z 1 { φ 1 ( H L ) | φ 1 ( H R ) } = Z 1 φ 0 ( H ) pro ving (52). Here, the ﬁrst equality is the deﬁnition of R 1 ∗ 1 , the second follo ws b ecause R 1 ∗ 1 H do es not equal an integer, the third follows b ecause φ is a homomorphism, the fourth follo ws b y induction and the fact that φ 0 (1) = 1, the ﬁfth follows by deﬁnition of heating, and the sixth follo ws b ecause H do es not equal an integer. Consider the following families of well-tempered scoring games: F n = { G ∈ W : gap 0 ( G ) = 0 , gap 1 ( G ) ≤ 2 n } F 0 n = { G ∈ F n : π ( G ) = 0 } Then we hav e J = F 1 ⊂ F 2 ⊂ F 3 ⊂ · · · ⊂ I and in fact I = S n i =1 F i . By (22) of Lemma 3.3, R j is an isomorphism from F n + j → F n for every n, j . Also, by Theorem 3.7, eac h F n is closed under addition. Since they are also clearly closed under negation, it follows that the { F n } form a ﬁltration of subgroups of I . No w I 0 is the direct limit of the follo wing sequence of partially-ordered ab elian groups: F 0 1  → F 0 2  → F 0 3  → · · · where the maps are inclusions. But this is the top ro w of the following comm utative diagram, 34 whose vertical maps are isomorphisms: F 0 1 − − − → F 0 2 − − − → F 0 3 − − − → · · · id   y R 1   y R 2   y J 0 R 1 − − − → J 0 R 1 − − − → J 0 R 1 − − − → · · · ψ   y ψ   y ψ   y Pg R 1 ∗ 1 − − − → Pg R 1 ∗ 1 − − − → Pg R 1 ∗ 1 − − − → Consequen tly , as an abstract partially-ordered ab elian group, I 0 is isomorphic to the direct limit Pg R 1 ∗ 1 → Pg R 1 ∗ 1 → Pg R 1 ∗ 1 → · · · W e can do something similar with I , expressing it as a direct limit of copies of Pg × ( Z / 2 Z ). Com bined with Theorems 3.13 and 4.9, this gives us a complete description of W in terms of This in turn gives us a complete description of W in terms of Pg . T o reco v er right and left outcomes from this description, one can utilize the following facts (all of whic h ha ve b een previously prov en): • If G is even-tempered, then L  R t G  = L( G ). • If G is o dd-temp ered, then L  R t G  = L( G ) + t . • If G ∈ J is ev en-temp ered, then L( G ) = min { n ∈ Z : n ≥ ψ ( G ) } • If G ∈ J is o dd-temp ered, then L( G ) = max { n ∈ Z : n C ψ ( G ) } • If G is even-tempered, then L( G ) = Rf ( G ) = Rf ( G − ) = L( G − ). • If G is o dd-temp ered, then L( G ) = Lf ( G ) = Lf ( G + ) = L( G + ). • R( G ) = − L( − G ). 6 Canonical F orms for In v ertible Games In the standard partizan theory , ev ery equiv alence class of games has a canonical represen- tativ e, c haracterized by its lack of reversible and dominated mov es. In this section, we pro v e the same thing for well-tempered scoring games that are invertible . This section is included 35 mainly for completeness, and follows easily from the previous section, so the reader may w ant to skip it. W e ﬁrst observe that non-inv ertible w ell-temp ered scoring games need not hav e an y sort of canonical form. F or instance, { x | 0 || 1 | y } ≈ 1&0 for an y n um b ers x, y ∈ Z , and there is no sense in which any of these forms is minimal or canonical. Also, no simpler game is a form of 1&0. Probably the most canonical wa y of describing a general scoring game G is as G + & G − , using the canonical forms of G + and G − whic h we will describ e b elo w. Deﬁnition 6.1. L et G b e a wel l-temp er e d sc oring game. Then we say that a left option G L is reversible if ther e is some right option G LR of G L with G LR . G . We say that a right option G R is rev ersible if ther e is some left option G RL of G R with G RL & G . We say that a left option G L is dominated if some other left option G L 0 satisﬁes G L 0 & G L , and we say that a right option G R is dominated if some other right option G R 0 satisﬁes G R 0 . G R . If G ∈ I , we say that G is canonical if it has no r eversible or dominate d options, and the same is true for al l sub games. The rest of this section is a pro of that every inv ertible Z -v alued game has a unique form whic h is canonical. W e reduce this to the same fact for partizan games. There are probably simpler pro ofs. Lemma 6.2. If G 0 is an option of G , then the c orr esp onding option R t G 0 ± t of R t G is dominate d (r esp. r eversible) if and only if G 0 is dominate d (r esp. r eversible). Conse quently, G has dominate d (r esp. r eversible) options if and only if R t G do es. In p articular, if G ∈ I , then R t G is c anonic al if and only if G is. W e leav e the easy pro of as an exercise to the reader. Lemma 6.3. L et G b e a p artizan game in c anonic al form. Then φ 0 ( G ) and φ 1 ( G ) ar e c anonic al. Pr o of. This follo ws easily from induction, mo dulo the following fact: if G is not an integer (in v alue), but has the integer n as an option, then the option φ 1 ( n ) = { n − 1 | n + 1 } of φ 0 ( G ) is not rev ersible. Without loss of generality , n is a left option of G . Supp ose φ 1 ( n ) is rev ersible. Then n + 1 . φ 0 ( G ). Th us n + 1 ≤ G . Then n + 1 C G R for every right option of G . Since G is not an integer, G L C n + 1 cannot hold for all G L . So for some G L , n < n + 1 ≤ G L . Then n is a dominated left option of G , a contradiction. 36 Lemma 6.4. L et G b e a wel l-temp er e d sc oring game for which gap 0 ( G ) = 0 and gap 1 ( G ) ≤ 1 . If G is c anonic al, then ψ ( G ) is in c anonic al form (assuming that ψ ( n ) was chosen to b e in c anonic al form for al l inte gers n ). Pr o of. This theorem follows trivially b y induction, except that we need to verify one sp ecial case: if G is o dd-temp ered and n ∈ Z is an option of G , then ψ ( n ) is not a reversible option of ψ ( G ). Without loss of generality , n is a left option of G . W e need to show that no righ t option of ψ ( n ) = n rev erses Left’s mov e from ψ ( G ) to ψ ( n ). If n ≥ 0, then n has no right options, so supp ose n < 0. Since we are taking ψ ( n ) in canonical form, the only righ t option of n is n + 1. Thus, w e are reduced to proving that if G is o dd-temp ered and gap 1 ( G ) ≤ 1 and n is a left option of G , then n + 1 6≤ ψ ( G ). By Theorem 5.9, this amounts to showing that n + 1 ≥ R( G ). But otherwise, we w ould hav e L( G ) = n < n + 1 < R( G ) con tradicting gap 1 ( G ) ≤ 1. Lemma 6.5. If G ∈ I is c anonic al, and G ≈ n for some inte ger n , then G = n . Pr o of. Supp ose for the sak e of con tradiction that G is not an in teger. Then R( G ) = n , so there is some righ t option G R of G with L( G R ) = n . Since G R is o dd-temp ered, it is not an in teger, so there is some left option G RL of G R with R( G RL ) = n . As G RL − G ≈ G RL − n , R( G RL − G ) = R( G RL ) − n = 0. Then b y Theorem 3.8, G RL & G , and G has a rev ersible righ t option, a contradiction. Lemma 6.6. L et G b e a c anonic al wel l-temp er e d sc oring game with gap 0 ( G ) = 0 and gap 1 ( G ) ≤ 1 . Then φ π ( G ) ( ψ ( G )) = G (exactly, not up to e quivalenc e). Pr o of. W e pro ceed b y induction on G . If G is an integer n , then ψ ( G ) ≡ n , so φ π ( G ) ( ψ ( G )) is automatically n = G . Otherwise, if G is even-tempered, then b y induction φ 0 ( ψ ( G )) = { φ 1 ( ψ ( G ) L ) | φ 1 ( ψ ( G ) R ) } = { φ 1 ( ψ ( G L )) | φ 1 ( ψ ( G R )) } = { G L | G R } unless ψ ( G ) equals an in teger. But if ψ ( G ) = m for some m ∈ Z , then G ≈ φ 0 ( m ) = m . Then by Lemma 6.5, G = m , a con tradiction. Finally supp ose G is o dd-temp ered. Then again, by induction, φ 1 ( ψ ( G )) = { φ 0 ( ψ ( G L )) | φ 0 ( ψ ( G R )) } = { G L | G R } unless ψ ( G ) equals an integer. But if ψ ( G ) = m , then G ≈ φ 1 ( m ) = { m − 1 | m + 1 } , so R( G ) = R( { m − 1 | m + 1 } ) = m + 1 L( G ) = L( { m − 1 | m + 1 } ) = m − 1 and thus gap 1 ( G ) ≥ 2, a con tradiction. 37 Theorem 6.7. If G is invertible (mo dulo e quivalenc e), then ther e is a unique H ∈ I with H ≈ G and H c anonic al. Pr o of. By Corollary 3.14, we can assume without loss of generalit y that G ∈ I . By taking t suﬃcien tly large, we get R t G ∈ J . Let x b e the canonical form of ψ  R t G  . Then by Lemma 6.3, K = φ π ( G ) ( x ) is canonical. But since x = ψ  R t G  , K = φ π ( G ) ( x ) ≈ Z t G. Then by Lemma 6.2, R − t K is canonical. But it is in I and R − t K ≈ G , so w e hav e shown existence. F or uniqueness, supp ose that H and H 0 are tw o canonical games in I , with H ≈ H 0 but H 6 = H 0 . Then by taking t suﬃciently large, we can arrange for gap 1  Z t H  ≤ 1 ≥ gap 1  Z t H 0  Also, since R t is inv ertible, R t H 6 = R t H 0 and R t H ≈ R t H 0 . So without loss of generality , H and H 0 ha ve gap 1 ≤ 1. Then ψ ( H ) and ψ ( H 0 ) are b oth in canonical form by Lemma 6.4. Since they are canonical forms of equiv alent games, ψ ( H ) ≡ ψ ( H 0 ). Therefore, letting i = π ( H ) = π ( H 0 ) φ i ( ψ ( H )) = φ i ( ψ ( H 0 )) . But by Lemma 6.6, the tw o sides equal H and H 0 . 7 Bo olean-v alued Games In this section we fo cus on { 0 , 1 } -v alued scoring games, which we call Bo ole an-value d games . There are t wo interesting order-preserving functions { 0 , 1 } × { 0 , 1 } → { 0 , 1 } , namely logical or and logical and . W e denote these op erations, and their extensions to Bo olean-v alued games, by ∨ and ∧ , resp ectiv ely . Note that for i, j ∈ { 0 , 1 } , i ∨ j = max( i, j ) and i ∧ j = min( i, j ). One can view the tw o p ossible scores 0 and 1 as victory for Right and victory for Left, resp ectiv ely . In this in terpretation, G ∧ H and G ∨ H are comp ound games play ed as follo ws: G and H are play ed in parallel, like a disjunctive sum. If one play er wins b oth G and H , she wins the comp ound. If there is a tie, G ∨ H resolves it in fav or of Left, and G ∧ H resolves it in fav or of Right. Th us G ∨ H and G ∧ H can b e seen as biased disjunctiv e sums. By Theorem 4.9, ∨ and ∧ induce well-deﬁned order-preserving op erations W { 0 , 1 } × W { 0 , 1 } → W { 0 , 1 } whic h are c haracterized by their restrictions I 0 { 0 , 1 } × I 0 { 0 , 1 } → I 0 { 0 , 1 } 38 to even-tempered inv ertible games. Our goal is to determine the structure of I 0 { 0 , 1 } and the eﬀects of ∧ and ∨ on I 0 { 0 , 1 } . The main result in this direction is Theorem 7.7. 7.1 Classiﬁcation of Bo olean-v alued Games By Theorem 5.1(a) and (d), ev ery Bo olean-v alued game lives in K , and every element of I { 0 , 1 } is in J . Therefore, it is reasonable to apply the maps ψ ± and ψ to Bo olean-v alued games, and by Theorem 5.11, ψ ± ( G ) = ψ ( G ± ). Lemma 7.1. If G ∈ W 0 , 1 , then ψ ( G ) = ψ − ( G ) . Pr o of. Let G b e a minimal coun terexample. Then G is not an integer and ψ ( G ) = { ψ ( G L ) | ψ ( G R ) } 6 = ψ − ( G ) = { ψ ( G L ) | ψ ( G R ) } − . This can only happ en if some negative integer n satisﬁes ψ ( G L ) C n C ψ ( G R ) for all G L and G R . In this case, ψ − ( G ) will b e a negative integer n < 0. But this means that G − ≈ n or G − ≈ { n − 1 | n + 1 } , whic h are imp ossible since Lf ( G − ) ∈ { 0 , 1 } while the left outcomes of n and { n − 1 | n + 1 } are negative. It follows that ψ ( I { 0 , 1 } ) = ψ + ( W { 0 , 1 } ) = ψ − ( W { 0 , 1 } ) = ψ ( W { 0 , 1 } ) . (54) The ﬁrst tw o equalities hold on general principles, b y Theorems 4.9 and 5.11. As we noted in § 5, the ov erheating operation R 1 / 2 { 1 / 2 |} = R 1 / 2 1 deﬁned recursively on partizan games by Z 1 / 2 1 G = ( Z 1 / 2 1 G L + 1 2 | Z 1 / 2 1 G R − 1 2 ) is a well-deﬁned order-preserving homomorphism. Let S b e the following set of partizan games: S = { 0 , 1 4 , 3 8 , 1 2 , 1 2 + ∗ , 5 8 , 3 4 , 1 } . Theorem 7.2. The set ψ ( I { 0 , 1 } ) = ψ ( W { 0 , 1 } ) of Equation (54) is the fol lowing: ( Z 1 / 2 1 x : x ∈ S ) ∪ ( ∗ + Z 1 / 2 1 x : x ∈ S ) Mor e sp e ciﬁc al ly, { ψ ( G ) : G ∈ W { 0 , 1 } , π ( G ) = 0 } = ( Z 1 / 2 1 x : x ∈ S ) (55) 39 and { ψ ( G ) : G ∈ W { 0 , 1 } , π ( G ) = 1 } = ( ∗ + Z 1 / 2 1 x : x ∈ S ) (56) A lso, the two sets n R 1 / 2 1 x : x ∈ S o and n ∗ + R 1 / 2 1 x : x ∈ S o ar e disjoint. Pr o of. W e omit most of the details of the proof, since it consists entirely of calculations. One can directly c heck that if x ∈ S , then φ 0 Z 1 / 2 1 x ! and φ 1 ∗ + Z 1 / 2 1 x ! are { 0 , 1 } -v alued. So all the sp eciﬁed v alues o ccur as ψ ( G ) for v arious Bo olean-v alued games G . The fact that the tw o sets n R 1 / 2 1 x : x ∈ S o and n ∗ + R 1 / 2 1 x : x ∈ S o can b e c hec ked by examination. The only thing left is a pro of that only the sp eciﬁed v alues of ψ o ccur as v alues of Bo olean-v alued games. Letting X 0 = n R 1 / 2 1 x : x ∈ S o and X 1 = n ∗ + R 1 / 2 1 x : x ∈ S o , w e assert the following: Claim 7.3. If G is a p artizan game with mor e than zer o left options and mor e than zer o right options, and if every option of G is in X i , then G is (e quivalent to a game) in X 1 − i . Since the sets X i are ﬁnite, this can b e chec k ed directly . W e might as well assume that the left options of G are pairwise incomparable, and the same for the right options. This sp eeds up the v eriﬁcation, since the p osets X 0 and X 1 ha ve very few an tichains, as they are almost linearly ordered. This leav es less than t w o h undred cases to chec k. Another useful tric k is to prov e that for an y even-tempered game G with gap 0 ( G ) = 0 and gap 1 ( G ) ≤ 1, ψ Z 1 / 2 G ! ≈ Z 1 / 2 1 ψ ( G ) . Then, one can co ol all the Bo olean-v alued games by 1 / 2 (which leav es them in J ), and one is reduced to considering n um b ers. The simplicit y rule no w comes into play , and the veriﬁ- cation b ecomes even easier. There are some technical diﬃculties inv olved with applying the R − 1 / 2 op erator to o dd-temp ered games, since this yields half-integer-v alued scoring games. Nev ertheless, this approach w orks, but still degenerates into a case-b y-case exhaustion of the p ossibilities, so we do not pursue it here. Giv e Claim 7.3, it is easy to verify inductiv ely that ψ ( G ) ∈ X π ( G ) for any Bo olean-v alued game G , which is the remaining statemen t of the theorem. It follows from this theorem that I { 0 , 1 } is isomorphic as a p oset to the direct pro duct of S and Z / 2 Z , since R 1 / 2 1 is order-preserving. By directly considering the structure of the p oset, w e see that the num b er of pairs ( x, y ) ∈ S × S with x ≥ y is 35. Therefore there are exactly 35 even-tempered elements of W { 0 , 1 } and exactly 70 elemen ts total. In other words, there are 70 Bo olean-v alued games, up to equiv alence. 40 Deﬁnition 7.4. L et G b e a Bo ole an-value d wel l-temp er e d game. Then we deﬁne u + ( G ) and u − ( G ) to b e the unique elements of S such that ψ ± ( G ) = Z 1 / 2 1 u ± ( G ) if G is even, and ψ ± ( G ) = ∗ + Z 1 / 2 1 u ± ( G ) if G is o dd. If u + ( G ) = u − ( G ) , we denote the c ommon value by u ( G ) . We c al l u + ( G ) and u − ( G ) the u -v alues of the game G . Note that since ψ ± ( ∗ + G ) = ∗ + ψ ± ( G ), we hav e u ± (ev en( G )) = u ± ( G ). 7.2 Op erations on Bo olean-v alued Games T o ﬁnish our accoun t of Bo olean-v alued well-tempered games, we need to describe ho w u - v alues interact with the op erations ∧ and ∨ . By Theorem 4.9, ∧ and ∨ are determined b y what they do to I 0 { 0 , 1 } whic h we can iden tify with S . Therefore the follo wing deﬁnition mak es sense: Deﬁnition 7.5. L et ∪ and ∩ b e the binary op er ations on S such that for any G, H ∈ W { 0 , 1 } , u + ( G ∨ H ) = u + ( G ) ∪ u + ( H ) u − ( G ∨ H ) = u − ( G ) ∪ u − ( H ) u + ( G ∧ H ) = u + ( G ) ∩ u + ( H ) u − ( G ∧ H ) = u − ( G ) ∩ u − ( H ) The fact that these identities hold for o dd-temp ered G or H follows from the equiv alence u ± (ev en( G )) = u ± ( G ). Note that ∪ and ∩ inherit commutativit y and asso ciativit y from ∨ and ∧ . Theorem 7.6. F or any x, y ∈ S , x ∪ y is the maximum z ∈ S such that z ≤ x + y . Similarly, x ∩ y is the minimum element w ∈ S such that w ≥ x + y − 1 . Note that it is not a priori clear that suc h a z or w exist in general. Pr o of. By symmetry(!) w e only need to prov e that x ∪ y has the stated form. F or s ∈ S , let G s ∈ I (0 , 1) b e ev en-temp ered with u ± ( G s ) = s . F or x, y , z ∈ S , we need to show that z ≤ x ∪ y ⇐ ⇒ z ≤ x + y , or equiv alen tly , G z . ( G x ∨ G y ) ⇐ ⇒ z ≤ x + y . Since R 1 / 2 1 and ψ are strictly order-preserving homomorphisms, this is the same as sho wing that G z . ( G x ∨ G y ) ⇐ ⇒ G z ≤ G x + G y . (57) 41 Let f ( n ) = min( n, 1). Then f is the iden tity on { 0 , 1 } , f ( n ) ≤ n , and ˜ f ( G x + G y ) = G x ∨ G y . Th us ˜ f ( G z ) = G z and ( G x ∨ G y ) . G x + G y . Therefore G z . G x + G y = ⇒ ˜ f ( G z ) . ( G x ∨ G y ) = ⇒ G z . ( G x ∨ G y ) = ⇒ G z . G x + G y . This prov es (57). W e summarize all our results on Bo olean-v alued w ell-temp ered games in the following theorem: Theorem 7.7. L et S b e the fol lowing set of p artizan games: S = { 0 , 1 4 , 3 8 , 1 2 , 1 2 ∗ , 5 8 , 3 4 , 1 } F or every G ∈ W { 0 , 1 } we have elements u ± ( G ) ∈ S such that ψ ± ( G ) = Z 1 / 2 1 u ± ( G ) if G is even-temp er e d, and ψ ± ( G ) = ∗ + Z 1 / 2 1 u ± ( G ) if G is o dd-temp er e d, wher e ψ is the map of § 5. Mor e over W { 0 , 1 } is isomorphic to { ( x, y , i ) : x, y ∈ S, x ≥ y , i ∈ { 0 , 1 }} via the map induc e d by G 7→ ( u + ( G ) , u − ( G ) , π ( G )) . This map is or der-pr eserving, in the sense that if G and H ar e two elements of W { 0 , 1 } , then G . H if and only if u + ( G ) ≤ u + ( H ) , u − ( G ) ≤ u − ( H ) , and π ( G ) ≤ π ( H ) . If G, H ∈ W { 0 , 1 } then u ± ( G ∧ H ) = u ± ( G ) ∩ u ± ( H ) u ± ( G ∨ H ) = u ± ( G ) ∪ u ± ( H ) wher e x ∪ y is the gr e atest element of S less than or e qual to x + y , and x ∩ y is the minimum element of S gr e ater than or e qual to x + y − 1 . If G ∈ W { 0 , 1 } is even-temp er e d, then L( G ) = 0 ⇐ ⇒ u − ( G ) = 0 R( G ) = 1 ⇐ ⇒ u + ( G ) = 1 while if G is o dd-temp er e d, then L( G ) = 0 ⇐ ⇒ u + ( G ) ≤ 1 2 R( G ) = 1 ⇐ ⇒ u − ( G ) ≥ 1 2 . 42 Pr o of. All of this follo ws b y piecing together previous results of this section, except for the ﬁnal statement ab out outcomes. This can b e prov en from Theorem 5.11 and some facts ab out o verheating, or b y directly chec king the sixteen inv ertible games. (In fact, by monotonicity one only needs to c hec k it for the even-tempered games with u -v alues 1 4 and 3 4 , and the o dd-temp ered games with u -v alues 1 2 and 1 2 ∗ .) 7.3 The Num b er of S-v alued Games Let S b e a ﬁnite set of integers. By Corollary 4.5, the n um b er of S -v alued games, up to equiv alence, dep ends only on | S | . If S = ∅ , then there are no S -v alued games. If S = { 0 } , it is easy to see that the only games are 0 and { 0 | 0 } (for example, use Lemma 3.8 and the fact that any S -v alued game must b e in I ). Ab o v e we saw that the num b er of { 0 , 1 } -v alued games, up to equiv alence, is exactly seven ty . F or the remaining case, we ha ve the following theorem: Theorem 7.8. If S ⊆ Z has mor e than two elements, then I 0 S c ontains a subp oset isomorphic to the al l-smal l games. In p articular, ther e ar e inﬁnitely many wel l-temp er e d S -value d games. Pr o of. W e might as w ell tak e S = {− 1 , 0 , 1 } . Let G b e an y all-small game. Then φ 0 ( G ) and φ 1 ( G ) are in W {− 1 , 0 , 1 } b y an easy-induction. Since φ 0 is faithful, having ψ as its in v erse, there is a complete cop y of the partially ordered ab elian group of all-small games in I 0 {− 1 , 0 , 1 } . In summary , w e see that the size of W S is given as follo ws: | W S | =          0 | S | = 0 2 | S | = 1 70 | S | = 2 ∞ | S | ≥ 3 Since the group of all-small games contains a copy of Pg (via the embedding G 7→ G. ↑ = ˆ G ), it follo ws that the theory of S -v alued games is as complicated as the theory of short partizan games, for | S | ≥ 3. W e can get at the additive structure of the group of all-small games as follo ws: Deﬁnition 7.9. L et  b e the op er ation {− 1 , 0 , 1 } 2 → {− 1 , 0 , 1 } given by i  j = max( − 1 , min(1 , i + j )) , as wel l as the extension of this op er ation to {− 1 , 0 , 1 } -value d games. Theorem 7.10. If G, H ar e al l-smal l (p artizan) games, then φ 0 ( G + H ) ≈ φ 0 ( G )  φ 0 ( H ) W e leav e the pro of as an exercise to the reader - it is similar to the pro of of Theorem 7.6. 43 8 Examples V ery few naturally o ccurring scoring games are well-tempered. The games Triangles and Squares , Mercenar y Clobber , Celestial Clobber , and Rebel Hex were in ven ted b y the author to giv e examples of scoring games. The only pre-existing scoring games considered here are Hex , Misere Hex , and To Knot or Not to Knot . 8.1 T riangles and Squares The joke game of Brussel Sprouts is a v ariant of Conw a y and P aterson’s Sprouts , pla yed as follows: initially the b oard is full of small crosses. Play ers tak e turns connecting the lo ose ends of the crosses. Ev ery time you connect tw o lose ends, you add another cross in the middle of your new segment. Pla y contin ues un til someone is unable to mo v e; this p erson loses. A sample game is shown in Figure 1. Ev ery mov e in Brussel Sprouts preserves the total num b er of lo ose ends, but decreases the num b er of connected comp onents − the n umber of faces b y one. A t the end of the game, each face contains exactly one lo ose end, and there is one connected comp onen t. As there are initially 4 n lo ose ends, n connected comp onents, and one face, the total length of the game is 5 n − 2 mo ves. The game is pla yed by the normal play rule, so if n is o dd, the ﬁrst play er wins, and if n is ev en, the second pla yer wins, r e gar d less of how the players play . In particular, there is no strategy inv olv ed - this is the jok e of Brussel Sprouts. This mak es Brussel Sprouts friv olous as a normal-pla y partizan game, but w ell-suited for our purp oses. F urther information on Sprouts and Brussel Sprouts can b e found in Chapter 17 of [3]. W e deﬁne Triangles and Squares to b e pla yed exactly the same as Brussel Spr outs except that at the end, a score is assigned as follows: Left gets one p oin t for ev ery “triangLe” (3-sided region), and loses one p oint for ev ery “squaRe” (4-sided region). Here, we sa y that a region has n sides if there are n corners around its p erimeter, not counting the tw o by the unique lo ose end. W e include the exterior un b ounded region in our coun t, as if the game w ere pla y ed on a sphere. An example endgame is shown in Figure 2. Note that we ha ve in tro duced an asymmetry b et w een Left and Right, so the game is no longer impartial. P ositions in T riangles and Squares naturally decomp ose as sums of smaller p ositions, making the game amenable to our analysis. Indeed, each cell (face) of a p osition acts indep enden tly of the others. Figure 3 shows a few small p ositions and their v alues. Using these v alues of individuals cells, w e can calculate the v alue of a comp ound p osition, as demonstrated in Figure 5. Rather complicated v alues seem to o ccur in this game, suc h as the p osition in Figure 4. 8.2 W ell-temp ered Scored Clobb er Clobber , ﬁrst presented in [2], is an all-small partizan game pla yed b et ween Blac k and White using black and white c hec kers on a square grid. On y our turn, you can mov e one 44 Figure 1: A sample game of Br ussel Sprouts (or of Triangles and Squares ). The initial p osition is in the top left, and the successive p ositions are shown from left to righ t, top to b ottom. Because there were n = 2 crosses initially , the game lasts 5 n − 2 = 8 turns. 4 3 2 2 3 2 4 4 Figure 2: The num b er of “sides” of each region in the ﬁnal p osition of Figure 1. Since there are t wo “triangles” and three “squares,” the ﬁnal score is 2 − 3 p oin ts for Left, or 3 − 2 p oints for Right. 45 { 3 ∗ |∗} { 2 ∗ | − 1 ∗} − 1 1* {∗| − 3 ∗} − 2 ∗ { 2 ∗ | − 1 ∗} 0 { 2 ∗ | 1 ∗} 1 − 1 ∗ 2 ∗ { 1 ∗ | − 2 ∗} { 2 ∗ | 1 ∗} Figure 3: The v alues of some small p ositions of Triangles and Squares , from Left’s p oin t of view. Figure 4: This o ctagon has v alue {{ 2 | 0 } , { 2 || 1 ∗ | − 1 ∗}|{ 0 | − 3 } , { 1 ∗ | − 1 ∗ || − 3 }} 1 1 − 1 ∗ { 2 ∗ | − 1 ∗} { 2 ∗ | 1 ∗} { 2 ∗ | − 1 ∗} + { 2 ∗ | 1 ∗} + 1 + 1 − 1 ∗ = { 5 ∗ | 4 ∗ || 2 ∗ | 1 ∗} Figure 5: This p osition can b e decomp osed as the sum of its ﬁve regions (faces). Its v alue is the sum of the v alues of the ﬁve regions. 46 { 0 | − 1 } { 1 | 0 } − 1 1& − 1 {− 1 | 1 } & − 1 ∗ {− 1 | 0 } 1& − 1 − 1& − 2 { 2 |{ 1 | 3 || 0 | 2 }} & 0 2& − 2 { 0 | 2 || 0 | 2 } & 2& − 2 0& − 2 − 1& − 2 {{ 0 | 1 || − 1 | 0 }| − 1 } {− 2 | 0 || − 2 | 0 } Figure 6: Sample p ositions of Mercenar y Clobber , in which you get one p oin t for collecting an enemy piece and zero p oints for collecting one of y our own pieces. Scores are from Left = Blac k’s p oint of view. of your pieces on to an orthogonally adjacen t piece of the enemy color, which is remo ved from the game. Even tually , no pieces of opp osite colors are adjacen t, at whic h p oin t neither pla yer can mo v e and the game ends. The last pla y er able to mov e is the winner. A common starting p osition is a b oard full of alternating blac k and white chec k ers. One w a y to mak e this into a scoring game w ould b e to assign scores for captured pieces. This fails to mak e a wel l-temp er e d scoring game, so we do something diﬀerent. Sa y that a c heck er of color C is isolate d if it b elongs to a connected group of chec kers of color C , none of whic h is adjacen t to a c heck er of the opp osite color. In other w ords, a chec ker is isolated if it has no chance of b eing tak en b y an enem y chec k er. W e then add a new c ol le cting mov e, in which y ou can collect an isolated chec k er of either color, removing it from the b oard. With this rule, the game ends when no chec kers remain on the b oard. Ev ery mov e (clobb ering or collecting) decreases the total n umber of c hec kers b y exactly one. If the initial p osition has n c heck ers, the game will last exactly n mo v es. So this is a w ell-temp ered game. In Mer cenar y Clobber , you get one p oint for collecting an enem y chec ker, and zero p oin ts for collecting one of your own. Some example p ositions and their v alues are sho wn in Figure 6 In Celestial Clobber , the t wo pla y ers are GoLd and EaR th (Left and Righ t). Golden 47 { 1 | − 1 } 0 | 0 = ∗ 0& − 1 ∗ & { 0 | − 1 } ∗ & {− 1 | 0 } { 0 | 1 } {− 1 | 0 } { 0 | 1 || − 1 | 0 } 0& − 1 { 0 | 1 || 0 | 0 } & { 0 | 0 || − 1 | 0 } & 0& − 2 0 0& − 1 0& − 1 0& − 1 {− 1 | 0 || − 2 | − 1 } { 0 | 1 || − 1 | − 1 } Figure 7: Sample p ositions of Celestial Clobber , in which golden pieces are worth 1 p oin t and earthen pieces are worth none. Here, goLden pieces are denoted  while eaRthen pieces are denoted ⊕ . Scores are from Left = Gold’s p oint of view pieces are worth 1 p oint for who ev er collects them, and earthen pieces are worth zero p oin ts. Some example p ositions and their v alues are shown in Figure 7. 8.3 T o Knot or Not to Knot The game To Knot or Not to Knot , introduced in [13], is a game play ed using knot pseudo diagrams. In knot theory , it is common to represent three-dimensional knots with tw o- dimensional knot diagr ams , such as those in Figure 8a. Each crossing graphically expresses whic h strand is on top. In a knot pseudo diagr am , am biguous crossings are allow ed, in which it is unclear whic h strand is on top. Figure 8b sho ws tw o examples. Knot Pseudo diagrams were introduced b y Hanaki [11]. The motiv ation for studying pseu- do diagrams comes from fuzzy electron microscopy images of DNA strands. The motiv ation for the game of To Knot or Not to Knot is less clear: Tw o pla yers, King L e ar and Ursula (also known as Knotter and Unknotter , or Left and Right) start with a knot pseudo diagram in which ev ery crossing is unresolved. Such a pseudo diagram is called a knot shadow . They tak e turns alternately resolving an unresolved crossing. (This game is b est pla yed on a chalkboard.) Ev entually , ev ery crossing is resolved, 48 (a) knot diagrams * * * * (b) pseudo diagrams Figure 8: Knot diagrams and pseudo diagrams. In 8b, the in tersections mark ed as ∗ are am biguous, unresolved crossings, where we do not sp ecify whic h strand is on top. → → → = = Figure 9: A sample game of To Knot or Not to Knot . The top row shows the actual sequence of play , which tak es three mov es. The b ottom ro w shows why the Unknotter has w on. yielding a genuine knot diagram. The Unknotter wins if the knot is top ologically equiv alen t to the unknot, and the Knotter wins otherwise. A sample game is sho wn in Figure 9. The connection with combinatorial game theory comes from p ositions such as the one sho wn in Figure 10, whic h decomp ose as connected sum of smaller pseudo diagrams. F rom knot theory , we kno w that a connected sum of knots will b e unknotted if and only if ev ery summand is unknotted. 2 Th us the Knotter wins a connected sum of p ositions b y winning at least one of the summands. This shows that a connected sum of p ositions is strategically the same as the disjunctive or (the op eration ∨ of § 7). Since the Knotter is Left and the Unknotter is Right, we assign a v alue of 1 to a Knotter victory and a v alue of 0 to an Unknotter victory . With these con ven tions, Figure 11 shows the v alues of a few small p ositions. 2 See pages 99-104 of [1] for an outline of a pro of of this fact. ∨ ≈ ∨ Figure 10: The p osition on the left decomp oses as a connected sum of the three p ositions on the right. 49 { 0 | 0 } = ∗ 0 1 { 0 | 0 } = ∗ { 1 | 0 } 1&0 0 1&0 ∗ ∨ (1&0) = 1 ∗ & ∗ { 1 | 0 } ∨ (1&0) = 1 ∗ & { 1 | 0 } Figure 11: The v alues of a few p ositions of To Knot or Not to Knot , from the Knotter (= Left)’s p oint of view. One diﬃculty with To Knot or Not to Knot is the problem of determining which pla yer has won at the game’s end! Determining whether tw o knots are equiv alen t is no easy problem. How ev er, there is a certain class of knot diagrams whic h has an eﬃcien t test for the unknot. These are the r ational knots . A r ational tangle is a tangle (knot with loose strands) built up from the initial tangles of Figure 12a b y the op erations of adding twists to the b ottom or righ t sides, as in Figure 12b. 3 A rational knot is one obtained from a rational tangle by closing up the top strands and the b ottom strands, as in Figure 12c. Figure 13a shows the construction of a typical rational tangle, and Figure 13b shows the resulting knot. There is a simple wa y to classify rational knots using con tinued fractions; see [1] or [5] for details. Consequently , r ational knot shadows like those shown in Figure 14 b ecome pla yable games 4 : It turns out that there is a general rule for determining the v alue of a rational shado w. 5 3 It turns out that adding twists on the left or top is equiv alen t to adding t wists on the righ t or b ottom. 4 In the sense that at the game’s end, it is algorithmically p ossible to identify the winner. 5 Let [ n 1 , n 2 , . . . , n m ] denote the rational knot shadow obtained b y adding n 1 v ertical t wists, n 2 horizon tal t wists, n 3 v ertical twists, and so on. Then we hav e some reductions for simplifying the game: [0 , n + 1 , n 2 , . . . ] = ∗ + [0 , n, n 2 , . . . ] [ . . . , n m − 2 , n + 1 , 0] = ∗ + [ . . . , n m − 2 , n, 0] [0 , 0 , n 2 , . . . ] = [ n 2 , . . . ] [ . . . , n m − 2 , 0 , 0] = [ . . . , n m − 2 ] [ . . . , n k − 1 , 0 , n k +1 , . . . ] = [ . . . , n k − 1 + n k +1 , . . . ] [1 , n 1 , . . . ] = [1 + n 1 , . . . ] 50 (a) The base tangles T T T T T = ⇒ = ⇒ = ⇒ = ⇒ (b) Construction of new tangles T T (c) T urning a tangle in to a knot Figure 12: The construction of rational tangles. The rational tangles are the smallest class of tangles containing the t wo tangles of 12a and closed under the four op erations of 12b. A rational knot is one obtained from a rational tangle by closing oﬀ a tangle, as in 12c. See 13 for an example. Figure 13: An example construction of a rational knot. 51 = = Figure 14: Rational shado ws, on whic h w e can play TK ONTK . The only u -v alues that I ha ve seen in To Knot or Not to Knot are 0 , 1 and 1 2 ∗ . As noted ab o ve, To Knot or Not to Knot the motiv ation for considering well-tempered scoring games. The fact that 5 of the 8 p ossible u -v alues are missing is therefore disapp oin t- ing. To Knot or Not to Knot is an example of a Bo olean-v alued well-tempered scoring game that can b e play ed with any b o olean function f : { 0 , 1 } n → { 0 , 1 } . Thinking of f as a blac k b ox with n inputs and one output, w e let the tw o pla yers Left and Righ t take turns setting the v alues of the inputs to f . After n mo v es, the inputs are all determined, and then the output of f sp eciﬁes the winner. There is no stipulation that Left can only set inputs to 1 or that Righ t can only set inputs to 0, or that the play ers w ork through the inputs in any particular order. W e call this the Input-setting Game for f . To Knot or Not to Knot is an instance of this game where f is the function whic h takes the crossing information and outputs a bit indicating whether the ﬁnal knot is knotted. The scarcit y of v alues o ccurring in To Knot or Not to Knot led me to initially conjecture that all input-setting games can only inv olv e the u -v alues 0 , 1, and 1 2 ∗ . How ever, it turns out that every Bo olean-v alued game is equiv alen t to an input-setting game for some b o olean function f . 6 This leads me to suspect that the same holds within To Knot or [ . . . , n m − 1 , 1] = [ . . . , n m − 1 + 1] After applying these rules as muc h as p ossible, one reac hes a minimal form [ n 1 , n 2 , . . . , n m ]. Then if all of the n k for 1 < k < m are ev en, and n 1 + n m is o dd, then this p osition has v alue ∗ . Otherwise, it has v alue 1&0 or (1 + ∗ )& ∗ dep ending on whether n 1 + n 2 + · · · + n m is ev en or o dd. Note that this rule determines the v alue of rational shadows , and do es not apply to arbitrary rational pseudo diagr ams . See [14] for further information. 6 Consider Bill T aylor and Dan Ho ey’s game of Projective Hex [19], pla yed on the faces of a do- decahedron or the vertices of an icosahedron. The same strategy-stealing argument for Hex shows that do decahedral Projective Hex is a ﬁrst-play er win. By symmetry , there are no bad op ening mov es. W e can view this game as an input-setting game for some six-input binary function f ( x 1 , . . . , x 6 ). Let 52 (a) an 11 × 11 Hex b oard (b) a winning c hain for Black Figure 15: An 11 × 11 b oard of Hex, and an example of a winning chain for Black. W e will alw ays assume that Black is trying to connect the right and left sides, while White is trying to connect the top and b ottom sides. In practice, other b oard sizes are often used, such as 10 × 10 and 14 × 14. Not to Knot , since there are no obvious restrictions on which b o olean functions can b e realized by a pseudo diagram. 8.4 Hex, Misere Hex, Reb el Hex Hex is a well-kno wn game due to Piet Hein and John Nash. 7 Tw o play ers, Blac k and White, alternately place pieces on the vertices of the b oard sho wn in Figure 15a. Blac k wins b y forming an unbrok en chain of black pieces connecting the left and right sides (as in Figure 15b), while White wins by forming a c hain of white pieces connecting the top and b ottom sides. It turns out that the b oard cannot ﬁll up without one pla yer winning. 8 Therefore, ties are imp ossible. In Figure 16(a), Black can guaran tee a connection betw een the tw o blac k pieces. If White mo ves at one of the interv ening spaces, then Blac k can mo v e at the other and complete the connection. The conﬁguration of Figure 16(a) is called a bridge . Chains of bridges can pro vide guaranteed connections across long distances. F or example, in Figure 16(b), Blac k can guarantee victory . On the other hand, the connection of Figure 16(c) is a w eaker connection. With one more mo v e, Black can establish a bridge, and guarantee a connection b etw een her pieces. Ho wev er, White can also connect his pieces via a bridge with one mov e. Thus whic hever pla yer mov es next can control the connection. How ev er, when t wo of these conﬁgurations are place in parallel, as in Figure 16(d), the result is a safe connection for Blac k, b ecause g ( x 1 , . . . , x 7 ) = f ( x 1 , . . . , x 6 ) ⊕ x 7 , where ⊕ denotes exclusiv e-or. Then the input-setting game of g is a second-pla y er win. (This is not diﬃcult to show, given what w e just said ab out do decahedral Projective Hex .) By examining Theorem 7.7, one can v erify that an y o dd-tempered Boolean-v alued game that is a second-pla y er win must hav e u + = u − = 1 / 2. This establishes the existence of input-setting games with 1 / 2 as a u -v alue. Com bining this game with the p ositions app earing in Figure 11 via ∨ and ∧ , one can in fact get all 35 pairs of p ossible u -v alues, without leaving the realm of b o olean input-setting games. 7 A go o d source on Hex is Browne’s b o ok [4]. 8 This fact is related to the Brou w er ﬁxed-p oint theorem. See Gale’s article [9], for example. 53 (a) (b) (c) (d) Figure 16: Examples of connections of v arious strengths. if White in terferes with the top connection, Blac k can resp ond b y solidifying the b ottom connection, and vice v ersa. These wa ys of chaining p ositions together in series or parallel corresp ond to the op erations ∧ and ∨ of § 7, if we assign a v alue of 1 to a successful Black connection and a v alue of 0 to a successful White connection. Figure 17 sho ws a num b er of small p ositions and their v alues, with these conv en tions. The reader will notice that the only u -v alues which occur are 0, 1, and 1 2 ∗ , and that every v alue is in I . In other w ords, the only even-tempered v alues are 0 , 1 , and { 1 ∗ ||∗} and the only o dd-temp ered v alues are ∗ , 1 ∗ , and { 1 | 0 } , and every game is already in vertible. This follo ws from the fact that in Hex, moving never hurts you . Consequen tly , if G is a p osition, then L( G L ) ≥ L( G ) ≥ L( G R ) and R( G L ) ≥ R( G ) ≥ R( G R ) for an y left option G L and right option G R . This preven ts the v alues { 0 | 1 } and {∗| 1 ∗} from ever occurring, and these in turn are necessary to pro duce an y of the other missing v alues. A v ariant of Hex is Misere Hex , which is play ed under identical rules, except that the pla yer who connects his tw o sides loses . As in Hex, p ositions can b e chained together in series or parallel, and Section 7 b ecomes applicable. 9 Assigning a v alue of 1 to a p osition with a white connection, and 0 for a p osition with a blac k connection, some sample p ositions and their v alues are shown in Figure 18. Again, we notice a scarcity of v alues. Lagarias and Sleator [15] sho w that n × n Misere 9 It is unclear that these conﬁgurations o ccur in actual games of Misere Hex, how ever. 54 { 1 | 0 } 1 { 1 | 1 || 0 | 0 } { 1 | 1 || 0 | 0 } 1 { 1 | 0 } 1 1 Figure 17: Sample p ositions in Hex , from the p oint of view of Black = Left = Horizontal. Blac k victory is scored as 1, while White victory is scored as 0. An arro w on a piece indicates that the piece is connected to the edge in the sp eciﬁed direction. { 0 | 1 } 1 { 0 | 1 } ∨ { 0 | 1 } = 1 { 0 | 1 } ∧ { 0 | 1 } = 0 1 { 0 | 1 || 0 | 1 } = { 0 | 1 } + ∗ { 0 | 1 } Figure 18: Sample p ositions in Misere Hex , from the point of view of Black = Left = Horizon tal. Blac k victory is scored as 1, while White victory is scored as 0. An arro w on a piece indicates that the piece is connected to the edge in the sp eciﬁed direction. 55 * * * * (a) (b) (d) (c) Figure 19: Rules of Rebel Hex . Hex is a win for the ﬁrst play er when n is even, and for the second pla yer when n is o dd. Using their techniques, it is easy to show that more generally: • If G is an even-tempered p osition, then G cannot b e a second-play er win. • If G is an o dd-temp ered p osition, then G cannot b e a ﬁrst-pla yer win. • If G is any p osition, then G L & G + ∗ & G R for every G L and G R . • If G is any position, then u + ( G ) = u − ( G ) ∈ { 0 , 1 , 1 2 } . In other words, the only p ossible v alues are 0 , 1 , { 0 | 0 } , { 1 | 1 } , { 0 | 1 } , { 0 | 1 || 0 | 1 } . In fact, no new games can b e built from these without contradicting one of the preceding bullet p oin ts. T o get more in teresting v alues, we consider a third v ariant of Hex, Rebel Hex . This is pla yed the same as Hex, except for the follo wing additional rule: whenev er a piece is play ed directly b etw een tw o enemy pieces, it c hanges colors. F or example in Figure 19(a), if White pla ys a piece at the ∗ , it will immediately c hange colors to black. On the other hand, if Blac k plays at ∗ , her piece will remain black. In Figure 19(b), if either play er mo ves at ∗ , the piece will change colors. F or a color rev ersal to o ccur, the surrounding enemy pieces m ust b e directly adjacent and on opp osite sides. Thus in Figure 19(c), no rev ersal will o ccur if either pla yer mov es at ∗ . Moreo v er, pla ying a piece can never change the color of a piece already on the b oard, so in Figure 19(d), if Black mo ves at ∗ , the white piece remains white, although it is no w surrounded. With this new rule in place, new v alues can o ccur, as shown in Figure 20. F or example, w e can now ha v e quarter-connections and 3 / 8-connections. If the pla y ers pla y a thousand 56 { 0 | 1 } { 1 | 1 } { 1 | 0 } { 1 | 1 || 0 | 0 } { 0 | 1 || 0 | 0 } { 1 |{ 0 | 1 || 0 | 1 }} { 1 |{ 0 | 1 || 0 | 1 }||∗} 1 2 1 1 2 ∗ 1 2 ∗ 1 4 3 8 3 4 Figure 20: Sample p ositions in Rebel Hex , from the p oint of view of Black = Left = Horizon tal. Blac k victory is scored as 1, while White victory is scored as 0. An arro w on a piece indicates that the piece is connected to the edge in the sp eciﬁed direction. The p ositions on the b ottom ro w are assumed to b e along the b ottom edge of the b oard. Eac h p osition is given with its v alue and its co oled u -v alue. copies of Figure 21a in parallel and Blac k tries to win as many as p ossible, she will b e able to win ab out 250 of them. The signiﬁcance of these partial connections is exhibited when w e chain them together in series or parallel. F or example, White’s 3 4 -strength connection in Figure 21a ensures that White will win, regardless of who mov es next. But when four copies of this conﬁguration are placed in series, as in Figure 21b, the resulting connection has strength 3 4 ∩ 3 4 ∩ 3 4 ∩ 3 4 = 0 , and now Black is guaranteed a victory! 9 Conclusion Sadly , none of our theory seems to generalize to scoring games which are not well-tempered. Cen tral notions suc h as ≈ ± , upsides, and do wnsides only make sense in a setting where the iden tity of the ﬁnal play er is predetermined, whic h generally do es not o ccur. If G and H are well-tempered scoring games suc h that G ≈ H , it may w ell b e the case that G + X and H + X hav e diﬀeren t outcomes for some (non-w ell-temp ered) scoring game X . 10 One approac h to general scoring games might be to try heating them until L( H ) ≥ R( H ) is satisﬁed for all subgames H . There are a couple of problems with this approac h. First of 10 One case where this do es not o ccur is the follo wing: if G is well-tempered and G & 0, then L( G + X ) ≥ L( X ) and R( G + X ) ≥ R( X ) for all scoring games X , well-tempered or not. This follo ws b ecause the wel l-temp er e d game X = {− N | N } for N  0 provides a suﬃcient test game for comparing an arbitrary game to 0. 57 (a) (b) Figure 21: The v alue of 21a is ∗ + { 0 | 1 || 0 | 0 } = {{ 0 | 1 || 0 | 1 }| 0 } whic h is a win for White no matter which play er goes ﬁrst. Note that this is a u -v alue of 3 4 for White, or 1 4 for Black. On the other hand, 21b has u -v alue 1 for Black, or 0 for white, b ecause 1 4 + 1 4 + 1 4 + 1 4 = 1. Therefore 21b is a guaranteed win for Black. all, a game like {∗| 1 ∗} will not satisfy this prop ert y under any amoun t of heating, b ecause { 2 t | 0 || 1 | − 2 t } alwa ys has left outcome 0 and right outcome 1. A bigger problem is that after suﬃcien t co oling, the theory of scoring games is equiv alen t to the theory of misere all-small games. 11 Since the theory of misere all-small games is somewhat hop eless, co oling had b etter not b e a w ell-deﬁned op eration on scoring games! But if it is not, heating b ecomes a lossy op eration, and its usefulness diminishes. 9.1 Ac kno wledgmen ts This work was done in part during REU programs at the Universit y of W ashington in 2010 and 2011, and in part while supp orted by the NSF Graduate Research F ello wship Program in the Autumn of 2011. The author w ould lik e to thank Reb ecca Keeping, James Morrow, Neil McKay , and Ric hard Now ak o wski, with whom he discussed the conten t of this paper, as w ell as Allison Henrich, who introduced the author to To Knot or Not to Knot , whic h ultimately led to this in vestigation. References [1] Colin C. Adams. The Knot Bo ok: A n Elementary Intr o duction to the Mathematic al The ory of Knots . American Mathematical So ciet y , Providence, RI, 2004. 11 If G is a scoring game and N  0, then the left outcome of R − N G is appro ximately − N or appro ximately 0, according to whether Left loses or wins G pla y ed as a misere game with Left going ﬁrst. 58 [2] Mic hael H. Alb ert, J. P . Grossman, Ric hard J. Now ako wski, and David W olfe. An in tro duction to clobb er. INTEGERS: The Ele ctr onic Journal of Combinatorial Numb er The ory , 5(2), 2005. [3] Elwyn R. Berlek amp, John H. Con wa y , and Ric hard K. Guy . Winning Ways for your Mathematic al Plays . A K Peters, W ellesley , MA, 2nd edition, 2001. [4] Cameron Browne. Hex Str ate gy: Making the Right Conne ctions . A K Peters, Natick, MA, 2000. [5] John H. Conw a y . On en umeration of knots and links, and some of their algebraic prop erties. In J. Leech, editor, Pr o c e e dings of the c onfer enc e on Computational Pr oblems in Abstr act A lgebr a held at Oxfor d in 1967 , pages 329–358. Pergamon Press, 1967. [6] John H. Con w ay . On Numb ers and Games . A K Peter s, W ellesley , MA, 2nd edition, 2001. [7] J. Mark Ettinger. On the semigroup of p ositional games. Online at h ttp://citeseerx.ist.psu.edu/viewdo c/summary?doi=10.1.1.37.5278, 1996. [8] J. Mark Ettinger. A metric for p ositional games. Online at h ttp://citeseerx.ist.psu.edu/viewdo c/summary?doi=10.1.1.37.7498, 2000. [9] Da vid Gale. The game of hex and the brouw er ﬁxed-p oint theorem. The A meric an Mathematic al Monthly , 86(10):818–827, December 1979. [10] J. P . Grossman and Aaron N. Siegel. Reductions of partizan games. In Michael H. Alb ert and Richard J. Now ako wski, editors, Games of No Chanc e 3 , n um b er 56 in Mathematical Sciences Research Institute Publications, pages 427–445. Cam bridge Univ ersity Press, 2009. [11] Ry o Hanaki. Pseudo diagrams of knots, links and spatial graphs. Osaka Journal of Mathematics , 47(3):863–883, 2010. [12] Olof Hanner. Mean play of sums of p ositional games. Paciﬁc Journal of Mathematics , 9(1):81–99, 1959. [13] A. Henrich, N. MacNaugh ton, S. Naray an, O. Pec henik, R. Silv ersmith, and J. T ownsend. A midsummer knot’s dream. Col le ge Mathematics Journal , 42:126–134, 2011. [14] Will Johnson. The knotting-unknotting game play ed on sums of rational shado ws. Online at http://arxiv.org/abs/1107.2635, 2011. [15] Jeﬀrey Lagarias and Daniel Sleator. Who wins mis ` ere hex? In Elwyn R. Berlek amp and T om Ro dgers, editors, The Mathemagician and Pie d Puzzler: A Col le ction in T ribute to Martin Gar dner , pages 237–240. A K P eters, Natick, MA, 1999. 59 [16] John W. Milnor. Sums of p ositional games. In H. W. Kuhn and A. W. T uck er, editors, Contributions to the The ory of Games II , num b er 28 in Annals of Mathematics Studies, pages 291–301. Princeton Univ ersity Press, Princeton, NJ, 1953. [17] Ric hard J. No wak o wski, editor. Games of No Chanc e . Num b er 29 in Mathematical Sciences Research Institute Publications. Cambridge Universit y Press, 1996. [18] Ric hard J. No w ako wski, editor. Mor e Games of No Chanc e . Number 42 in Mathematical Sciences Research Institute Publications. Cambridge Universit y Press, 2002. [19] Bill T aylor. Pro jective hex. F orum p ost. Accessed online at h ttp://www.mathforum.org/kb/message.jspa?messageID=352866. 60

The Combinatorial Game Theory of Well-Tempered Scoring Games

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