Abstract The general behavior of combinatorial scoring games is not well-understood. In this paper, we focus on a special class of “well-tempered” scoringgames. By analogy with Grossman and Siegel’s notion of even- and odd-temperednormal play games, we declare a dicot scoring game to be even-tempered if all itsoptions are odd-tempered, and odd-tempered if it is not atomic and all its optionsare even-tempered. Games of either sort are called well-tempered. These show upnaturally when analyzing one of the “knot games” introduced by Henrich et al. Weconsider disjunctive sums of well-tempered scoring games, and develop a theory forthem analogous to the standard theory of disjunctive sums of normal play partizangames. We isolate a special class of inversive well-tempered games which behave likenormal play partizan games or like the “Milnor games” considered by Milnor andHanner. In particular, inversive games (modulo equivalence) form a partially orderedabelian group, and there is an effective description of the partial order. Moreover, thefull monoid of well-tempered scoring games (modulo equivalence) admits a completedescription in terms of the group of inversive games. We also describe several exam-ples of well-tempered scoring games and provide dictionaries listing the values ofsome small positions in two of these games.
Keywords Combinatorial games · Scoring games · Milnor games
Mathematics Subject Classification (2000) 91A46
W. Johnson (B)Mathematics, Computer Science, University of California,Berkeley, Canadae-mail: [email protected]
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1 Introduction
1.1 Scoring games
Research in combinatorial game theory has largely focused on games in which thewinner is determined by who moves last, according to either the normal rule or themisère rule. Many score-based games, such as Go and Dots-and-Boxes, do not fallinto these categories. In these scoring games, the objective is to maximize a numericalscore. Attempts by combinatorial game theorists to analyze the class of scoring gameshave been scarce and isolated; in fact there seems to be no agreement on how toprecisely define this class.
The most general class of scoring games considered thus far is the class of “scoringplay games” defined by Stewart (2013). Although Stewart’s games have canonicalforms [Section 3 of Stewart (2013)], they behave rather poorly on the whole. Forexample, almost no games have additive inverses [Theorem 16 of Stewart (2013)].One can also show that two scoring play games G and H can only be equivalent if theyare equivalent as misère partizan games (forgetting all scoring data).1 Consequently,Stewart’s class of games is as complicated as the class of misère partizan games.
To avoid these problems, we will work with a restricted class of Stewart’s games.Specifically, we will follow Milnor (1953), Hanner (1959), and Ettinger (1996), andassume all our scoring games are dicot. A game is dicot if there are no positions inwhich exactly one of the two players can move. Consequently, positions fall into twosorts: non-atomic positions in which both players can move, and atomic positions inwhich neither player can move and the game has ended. The class of (dicot) scoringgames can be defined as follows:
Definition 1 A scoring game G is either a real number (in which case we say G isatomic) or an ordered pair (L,R) where L and R are finite nonempty sets of scoringgames. If G is non-atomic, we refer to the elements of L and R as the left options andright options of G, respectively. Otherwise, we declare that G has no options. We usethe notation
〈A, B, C, . . . |D, E, F, . . .〉to denote the pair
({A, B, C, . . .}, {D, E, F, . . .}).As in standard combinatorial game theory, we think of a scoring game G as a
position in a game played between two players named “Left” and “Right”. If G isnon-atomic, then Left can move from G to any of the left options of G and Right canmove from G to any of the right options of G. If G = x ∈ R is atomic, then the gamehas ended with Left winning by x points. For example,
1 If G is one of Stewart’s scoring play games, then the outcome of the game G+{|x3|{0|0|−x2}+{x |−x |}}for x ∈ R, x >> 0 is the same as the misère outcome of G. Indeed, Right can only safely move to{0|0| − x2} + {x | − x |} after exhausting his options in G, and if he does so, he will win.
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〈1| − 1〉
is a game that ends after one move; whichever player moves first gains a point. Similarly
〈−1|1〉
is a game in which the first player loses a point. The game
G = 〈〈1| − 1〉|〈1| − 1〉〉
is a game that lasts two moves. Whichever player makes the second move gains onepoint. To avoid nesting too many angle brackets, we will use the abbreviated notationdiscussed in Chapter 10 of Conway (2001), using double and triple vertical bars || and|||. For example, the aforementioned game G would be abbreviated as 〈1|− 1||1|− 1〉in this notation.
Our games do not keep track of intermediate scores at non-atomic (non-terminal)positions. Such data is strategically irrelevant in our setting.
In this paper, all games will be finite. We use angle brackets 〈·|·〉 rather than curlybraces {·|·} to highlight the fact that our scoring games are not “games” in the tra-ditional sense of Berlekamp et al. (2001) or Conway (2001). Except for the use ofangle brackets, we are mostly following Mark Ettinger’s notation. In Fraser Stewart’snotation, an atomic game x would be denoted x or {.|x |.}, while a non-atomic game〈A, B, C, . . . |D, E, F, . . .〉 would be denoted {A, B, C, . . . |0|D, E, F, . . .}. Stewartrefers to atomic games as “termination vertices.”
We assume that play alternates between Left and Right, and the game ends oncean atomic position is reached. The outcome of a game under perfect play is thereforedefined as follows:
Definition 2 If G is a scoring game, the left outcome L(G) and right outcome R(G)
are the real numbers defined recursively as follows:
L(G) ={
G if G is atomicmaxGL R(GL) if not
R(G) ={
G if G is atomicminG R L(G R) if not
where GL and G R are variables ranging over the left and right options of G.
Thus L(G) or R(G) is the final score of the game G under perfect play when thefirst player is Left or Right, respectively. Our notation matches Ettinger except wewrite L(G) and R(G) instead of L(G) and R(G) [see Definition 3 in Ettinger (1996)].In Stewart’s notation, L(G) and R(G) are denoted GSL
F and GS RF and called the “Left
final score” and “Right final score” [Definition 7 of Stewart (2013)].One could also define what it means for Left or Right to win, lose, or tie, but we
have no use for these notions.
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Addition and negation are defined as follows:
G + H ={
G + H in the usual sense as numbers if G and Hare atomic〈GL + H, G + H L |G R + H, G + H R〉 otherwise
−G ={−G in the usual sense if G is atomic
〈−(G R)| − (GL)〉 otherwise
We are using the usual shorthand symbols GL and G R for typical left and right optionsof G, in the same manner as Conway (2001) and Berlekamp et al. (2001). As a noteof clarification, if G is non-atomic and x ∈ R is atomic, then x has no options,so
G + x = 〈GL + x |G R + x〉.
Addition can be described as playing two games in parallel, and adding the final scores.Negation corresponds to reversing the roles of the two players. Addition of scoringgames is associative and commutative, and the atomic game 0 is the additive identity.We let G − H denote G + (−H).
Addition of games is a variant of the disjunctive sum operation of standard com-binatorial game theory. It is the same as Ettinger’s game addition “+” and Stewart’sdisjunctive sum “+�” Ettinger (1996), Stewart (2013).
Note that the symbol “+” refers to both addition of games and addition of numbers.There is no risk of confusion, because the former generalizes the latter.
We say that G is a subgame of H if either G = H or G is a subgame of an optionof H .
The following table compares our notation with that of other authors. The lastcolumn is by analogy.
There is one way in which our notation is non-standard, namely, our use of the symbol= for identity (equality) of games rather than for equivalence (indistinguishability)of games, which we in turn denote by ∼. This was motivated by the desire to avoidlugging around countless ≡’s in all our theorems. Also, this prevents us from usingnotation directly opposite to that of Ettinger (1996). Our use of ∼ for equivalence istaken from Milnor (1953).
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1.2 Prior work on dicot scoring games
The earliest work on scoring games was done by Milnor (1953), who considered thefollowing class of games:
Definition 3 A scoring game G is a Milnor game if L(H) ≥ R(H) for all subgamesH of G.
Milnor proved that the set of Milnor games is closed under addition. He also showedthat Milnor games modulo ∼M form a group, where ∼M denotes the natural equiva-lence relation on Milnor games. In more detail, G ∼M H means that for every Milnorgame X ,
L(G + X) = L(H + X) and R(G + X) = R(H + X).
This relation [denoted ∼ in Milnor (1953)], is the coarsest equivalence relation onMilnor games which is compatible with addition and the functions L(·) and R(·).This definition of ∼M is difficult to use in practice, but Milnor provided the followingalternate characterization: G ∼M H holds if and only if R(G − H) = L(G − H) = 0.
Olof Hanner investigated the class of Milnor games further, inventing thermographyand mean-value theory Hanner (1959) before they were rediscovered in the normalplay partizan setting. In hindsight, the results of Milnor and Hanner can be explainedby noting that Milnor games are essentially the same as normal play partizan gamesmodulo infinitesimals.
The first person to consider the full class of (dicot) scoring games was Ettinger(1996, 2000) during the 1990s. This case seems to be far more mysterious than thespecial case of Milnor games. Let ∼E denote the natural equivalence relation onscoring games.2 Ettinger observed that the monoid M of scoring games modulo ∼E isnot a group, and found a homomorphism from M to the group of dicot (i.e., all-small)normal play partizan games. He also found an algorithm for testing whether a scoringgame G is invertible modulo ∼E .
Unfortunately, Ettinger left many questions open:
1. How close is M to being a group? Ettinger posed the question of whether M iscancellative, but never answered it.
2. Is there a reasonable algorithm which takes two scoring games G and H anddetermines whether G ∼E H? Ettinger found one in the case where H = 0.
3. Do scoring games have canonical forms? Ettinger showed that this is false on anaive level. For example, the scoring games 〈0|1||1|0〉 and 〈0|1||1|1〉 are equiva-lent and equally complicated, but no simpler scoring game is equivalent to them.
2 In other words, G ∼E H if and only if
L(G + X) = L(H + X) and R(G + X) = R(H + X)
for all scoring games X . Even if G and H are both Milnor games, this doesn’t necessarily agree with∼M . For example, 0 and 〈0|0〉 are equivalent as Milnor games, but 0 �∼E 〈0|0〉 because 0 + 〈−1|1〉 and〈0|0〉 + 〈−1|1〉 have different outcomes. Note 〈−1|1〉 is not a Milnor game. Ettinger denotes the relation∼E with ≡, which we avoid since it seems to clash with the notational conventions of Conway (2001) andBerlekamp et al. (2001).
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Nevertheless, it may still be possible to give each equivalence class a canonicalname of some sort. For an example of what this might look like, see Remark 1 inSect. 1.4 of this paper.
1.3 Fixed-length and well-tempered scoring games
Rather than directly attacking these problems, this paper focuses on another restricteduniverse of scoring games, one that behaves almost as well as Milnor games. In thisrestricted setting, the analogs of the three questions above are solvable.
Definition 4 Let G be a scoring game. We say that G has fixed-length 0 if G is atomic.For n ≥ 0, we say that G has fixed-length n +1 if G is non-atomic and all options of Ghave fixed-length n. We say that G is a fixed-length scoring game if it has fixed-lengthn for some n.
In other words, G is a fixed-length game if all lines of play, including the non-alternating ones, have the same length. Equivalently, all leaves of the game tree ofG have the same depth. For example, 〈0||0|0〉 is not fixed-length, because its leftoption is length 0 while its right option is length 1.
A simple example of a fixed-length scoring game is Dots-and-Boxes played withoutbonus moves. Players take turns adding length-1 segments to a rectangular grid of dots,until all segments have been filled in. Whenever a player makes a move completingthe fourth wall of a 1 × 1 box, she receives one point (and her turn ends). We call thisgame Modboxes. A few small positions are analyzed in Sect. 2.
Another example of a fixed-length scoring game can be obtained by taking BrusselSprouts and assigning a score based on the ending configuration [see Chapter 17 ofBerlekamp et al. (2001)]. For example, we could give Left one point for every three-sided region in the final configuration, and give Right one point for every four-sidedregion. The fact that Brussel Sprouts is fixed-length is an exercise in graph theory, andthe reason why normal play Brussel Sprouts is trivial.
More fixed-length examples can be obtained by taking Hex or Misère Hex andassigning a final score of 1 or 0 according to which player wins. This can also bedone in the game To Knot or Not to Knot of Henrich et al. (2011). The class of such{0, 1}-valued games will be considered in a subsequent paper.
For technical reasons, we will work with a slightly more general class of scoringgames. 3
Definition 5 Let G be a scoring game. We say that G is even-tempered if all optionsof G are odd-tempered. We say that G is odd-tempered if G is non-atomic and alloptions of G are even-tempered. In particular, atomic games are vacuously even-tempered. We say that G is well-tempered if G is even-tempered or odd-tempered.4
3 The additional generality will not affect the notion of equivalence or the resulting monoid, by Proposition 1.Working with well-tempered games rather than fixed-length games is necessary to get canonical forms.4 We use the terminology “well-tempered” due to the similarities between our definition and Grossman andSiegel’s definition of well-tempered normal play games Grossman (2009). There is no direct connectionbetween these concepts, however.
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For G well-tempered, we let π(G) ∈ Z/2Z be 0 if G is even-tempered, and 1 if G isodd-tempered.
The definition of π(G) makes sense because of the easily verified fact that no gameis both even-tempered and odd-tempered.
This definition can be thought of as “fixed-length mod 2.” A game is even-temperedor odd-tempered if all leaves of the game tree are at an even depth or an odd depth,respectively. One easily checks that fixed-length games are well-tempered—in fact agame of fixed length n is even-tempered or odd-tempered if n is even or odd, respec-tively.
Neil McKay suggested the following example of a playable game which is well-tempered but not fixed-length. The game is played like Domineering, except with 3×1and 1 × 1 tiles rather than dominoes. One player places 3 × 1 and 1 × 1 tiles onto asquare grid, while the other places 1 × 3 and 1 × 1 tiles. A player receives one pointfor placing a large tile and zero points for placing a small tile. Any position with anodd or even number of empty squares is odd-tempered or even-tempered, respectively.We call this game Unitromineering. A few small positions are analyzed in Sect. 2.The game can be seen as a variant of Blanco and Fraenkel’s Tr(i)omineering (2006).
We let W denote the set of well-tempered scoring games, and F denote the set offixed-length scoring games. An easy exercise shows that both W and F are closedunder addition and negation. In proving this, one sees that π(G + H) = π(G)+π(H)
for G, H ∈ W .
Definition 6 We say that G, H ∈ W are equivalent, denoted G ∼ H , if
L(G + X) = L(H + X) and R(G + X) = R(H + X)
for every X ∈ W . Similarly, we say that G � H and H � G if
L(G + X) ≥ L(H + X) and R(G + X) ≥ R(H + X)
for every X ∈ W .
Equivalence is the coarsest equivalence relation on W compatible with the additionand negation operations and the outcome functions L(·) and R(·). From the point ofview of Plambeck (2009), equivalence is the canonical “indistinguishability” relationfor the restricted universe of well-tempered scoring games.
For the remainder of this paper, all “games” will be well-tempered scoring games,unless specified otherwise. In particular, we drop the word “scoring.” When we needto refer to the classic normal play partizan games of ONAG and Winning Ways, wewill call them “normal play games” and use curly braces rather than angle brackets.
1.4 Statement of results
We now state the main results of this paper.
Definition 7 A game G is inversive if L(H) ≥ R(H) for every even-temperedsubgame H of G. Let I ⊆ W denote the set of inversive games.
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We will eventually see (Corollary 2 in Sect. 6) that a game G is invertible, mod-ulo equivalence, if and only if it is equivalent to an inversive game, hence the term“inversive.”
Theorem 1 Let G and H be inversive games.
(a) G + H and −G are inversive.(b) G − G ∼ 0. In particular, G is invertible modulo equivalence, with inverse given
by −G.(c) G � H if and only if both π(G) = π(H) and R(G − H) ≥ 0.
Thus I is closed under addition, and forms an abelian group modulo ∼. By part (c) wehave an efficient method for comparing inversive games. Although we won’t proveit, one can also show that inversive games have canonical forms and a mean-valuetheory. Thus, they behave very much like Milnor games or normal play games.
To consider more general games in W , we will use two idempotent maps from Wonto I.
Definition 8 For G ∈ W , we define the game G+ recursively as follows. If G isatomic, then G+ = G. Otherwise, let H be the game
H = 〈(GL)+|(G R)+〉.
If H is even-tempered and L(H) < R(H), let G+ be the atomic game R(H). Other-wise, let G+ be H . The game G− is defined symmetrically, so that −(G−) = (−G)+.We call G+ and G− the upside and downside of G, respectively.
Less formally, the upside of a game G can be computed by the following algorithm:
1. Replace all options of G with their upsides, recursively.2. If G is inversive, return G.3. Otherwise, return the atomic game R(G).
One can verify inductively that G+ and G− are inversive for arbitrary G. In the casewhere G is inversive, G+ = G− = G. Furthermore, π(G) = π(G+) = π(G−) forany G.
(a) G � H if and only if G+ � H+ and G− � H−.(b) (G + H)+ ∼ G+ + H+ and (G + H)− ∼ G− + H−.(c) If G is even-tempered, then L(G) = L(G−) and R(G) = R(G+).
(c’) If G is odd-tempered, then L(G) = L(G+) and R(G) = R(G−).
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Combining part (a) of this theorem with part (c) of Theorem 1, we get a reasonablealgorithm for comparing arbitrary games.
In a well-tempered game, we can always predict who will make the final move,once we know who starts. It therefore makes sense to talk about the player who “getsthe last word” in the game. Parts (c) and (c’) of Theorem 2 can be restated as follows:
When the player who “gets the last word” is Left or Right, a game may bereplaced with its upside or downside, respectively.
If A and B are two inversive games, we let A&B denote a game with upsideequivalent to A and downside equivalent to B. By Theorem 2(a), the game A&B isuniquely determined up to equivalence, if it exists. The following theorem gives acriterion for existence:
Theorem 3 Let A, B be inversive games. Then A&B exists if and only if A � B.
All games are of the form A&B, and Theorem 2 can be interpreted as a set of rulesfor working with expressions of the form A&B:
(A&B) � (A′&B ′) ⇐⇒ (A � A′ and B � B ′)(A&B) + (A′&B ′) = (A + A′)&(B + B ′)
Our terminology “upside/downside” and notation A&B is inspired by the theory ofloopy normal play games, which appears to behave rather similarly (see Chapter 11of Berlekamp et al. (2001)).
Remark 1 Well-tempered scoring games do not have canonical forms. For example〈0|1||2|0〉 and 〈1|1||2|2〉 are equivalent to each other, but no strictly simpler gameis equivalent to both. However, inversive games do have canonical forms, so we cangive each game G a canonical description of the form A&B where A, B ∈ I are incanonical form. Then each game in W has a canonical description, and equivalentgames have equal canonical descriptions. For example, the two games above bothhave canonical description 2&1.
We can summarize the structure of the class of well-tempered scoring games inthe following abstract way. Let W and I denote the quotients of W and I moduloequivalence. Then by Theorem 2, the map G → (G+, G−) induces an embedding
W → I × I
which is an injective, strictly order-preserving homomorphism. By Theorem 3, theimage of this map is the set of pairs (x, y) such that x ≥ y. Thus W can be identifiedwith the submonoid
{(x, y) ∈ I2 : x ≥ y} ⊂ I × I.
In particular, the monoid W is cancellative, and the structure of W is entirely deter-mined by I.
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Fig. 1 A small Modboxes dictionary, written in Strings-and-Coins notation
Fig. 2 A position which breaksinto pieces
2 Example games
Recall from Sect. 1.3 the game Modboxes, that is, Dots-and-Boxes without bonusmoves. Figure 1 lists the values of some small positions, depicted using the dualStrings-and-Coins point of view (see Chapter 16 of Berlekamp et al. (2001)). For eachposition, we have listed a value of the form G&H , where G is the upside and H isthe downside. If G ∼ H , we have written G rather than G&G. We use ∗ and n∗ todenote the games 〈0|0〉 and 〈n|n〉.
Using this dictionary, we evaluate the Modboxes position of Fig. 2. The followingtable lists the upside and downside of each component. By Theorem 2(b), we candetermine the upside and downside (modulo equivalence) of the total by adding thecolumns.
The last row of the table may be verified as follows. First of all,
〈2|0〉 + 〈1| − 1〉 = 〈3|1||1| − 1〉
by the definition of addition. However, this game is equivalent to 1 by the criterion ofTheorem 1(c), since
is even-tempered and has left outcome and right outcome zero. So the total of theupsides is equivalent to
1 + 1 + 〈2| − 2〉 + ∗
From the definitions, this is exactly
〈〈4|4〉, 〈4|0〉|〈4|0〉, 〈0|0〉〉.
This becomes 〈4|4||0|0〉 = 〈4 ∗ |∗〉 after deleting dominated moves (which is legal).The sum of the downsides follows by symmetry.
Now we can conclude that the total value of Fig. 2 is
〈4 ∗ |∗〉&〈∗| − 4∗〉
What is the final score if Left goes first? The position happens to be even-tempered, soTheorem 2(c) applies, and we may replace the game with its downside. Alternatively,we know that if Left goes first, then Right “gets the last word,” and so the game actslike its downside 〈∗| − 4∗〉. But the left outcome of 〈∗| − 4∗〉 = 〈0|0|| − 4| − 4〉 isclearly zero. We conclude therefore that if Left goes first in Fig. 2, under perfect playthe final score will be zero. By symmetry, the same thing happens if Right goes first.Note however that the position of Fig. 2 is not equivalent to the zero position. Forexample, if we played two copies of Fig. 2 in parallel, the result would have downside
〈∗| − 4∗〉 + 〈∗| − 4∗〉
which happens to be equivalent to −4. In particular, if Left went first, Left would loseby four points.
The values observed in Fig. 1 seem to be typical for Modboxes. In fact, everyModboxes position I have analyzed has a value which can be expressed as a linearcombination of 1, 1& − 1, ∗, 〈1| − 1〉, and 〈2| − 2〉. The switches ∗, 〈1| − 1〉, and〈2| − 2〉 are their own negatives, so multiple copies of these cancel out in pairs. Onthe other hand, the sum of n copies of 1& − 1 is n& − n. So it seems likely thatin a large position of Modboxes which decomposes into many small positions, thevalue of the total can be approximated by N& − N for some large integer N . The sole
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Fig. 3 An assortment of Unitromineering positions. Dots represent 1 × 1 boxes
effect of a position like N& − N is to reward N points to whichever player has “thelast word” in the game. This yields the following heuristic: a position in Modboxestends to be a first-player win if it is odd-tempered, and a second-player win if it iseven-tempered.
We next consider the game Unitromineering of Sect. 1.3. Recall that this is likeDomineering (or Tromineering) except that the tile sizes are 1 × 3, 3 × 1, and 1 × 1.Either player can place 1 × 1 tiles, and so the game only ends when the board iscompletely full. However, points are rewarded for placing the larger tiles.
The values of some assorted positions are shown in Fig. 3. For compactness, wedenote the positions using the graph notation of Chapter 10 of Conway (2001). Weassume that Left places vertical tiles while Right places horizontal tiles. Unlike thecase of Modboxes, all the positions seem to be inversive. However, more complicatedinversive games occur, such as the one in the bottom right corner of Fig. 3.
We should note that there are positions in which the optimal move is to play a smalltile, even when a large tile could be played instead. For example, in the ‘C’-shapedposition at the top right corner of Fig. 3, there are two places where Right can placea large horizontal tile. However, the optimal move is to place a small tile along theleft edge, blocking Left from placing a larger tile on the next turn. If such positionsdid not arise, Unitromineering would be a disguised version of Blanco and Fraenkel’sTromineering (2006).
3 Preliminary lemmas
We remind the reader that all games are assumed to be well-tempered scoring games,unless otherwise stated.
The following basic fact will be used frequently in later sections.
Lemma 1 If G ∈ W and x ∈ R, then L(G+x) = L(G)+x and R(G+x) = R(G)+x.If x, y ∈ R, then x � y if and only if x ≥ y as numbers.
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Proof The first statement has a simple inductive proof and is intuitively clear, and thesecond follows easily from the first. ��Lemma 2 If G is a game, then
R(G − G) ≥ 0
L(G − G) ≤ 0
Proof This is proved in exactly the same way that G − G ≥ 0 is proved in classiccombinatorial game theory. The second player can ensure a score of at least zero bymirroring the moves of her opponent. ��
As we stressed in the introduction, a well-tempered scoring game G has a dualpersonality, acting like its upside G+ when Left “gets the last word,” i.e., when Leftmakes the final move of the game; and like its downside G− when Right “gets the lastword.” We can make this more precise as follows.
Definition 9 If G is a game, let Le(G) and Re(G) be the real numbers defined asfollows:
(Le(G), Re(G)) ={
(L(G), R(G)) when G is odd-tempered(R(G), L(G)) when G is even-tempered
We call Le(G) the left-ending outcome of G, and Re(G) the right-ending outcomeof G.
The left- or right-ending outcome of G is the outcome when Left or Right is theplayer who will make the last move of the game, rather than the first. With thisdefinition, Theorem 2(c-c’) can be stated more concisely as follows, using the factthat π(G+) = π(G−) = π(G):
Le(G) = Le(G+)
Re(G) = Re(G−)
In an earlier version of this paper, left-ending and right-ending outcomes werecalled “left final” and “right final” outcomes. We have changed terminology to avoidconfusion with Fraser Stewart’s “Left final score” and “Right final score,” whichcorrespond to our L(G) and R(G) rather than to our Le(G) and Re(G).
Lemma 3 Let G, H be games. If G � H, then π(G) = π(H).
Proof For N ∈ R, let JN denote the game 〈−N |N 〉. Let X be any game, and considerthe game X + JN for N � 0. Because neither player wants to move in JN , the effectof JN is to heavily penalize whoever makes the last move of the game. Consequently,
limN→+∞ Le(X + JN ) = −∞
limN→+∞ Re(X + JN ) = +∞.
Now suppose that G � H and π(G) �= π(H). If G is even-tempered and H isodd-tempered, then
which again contradicts the limiting behavior as N → +∞. ��Definition 10 If G, H are games, we say that G and H are left-equivalent, denotedG ∼+ H , if π(G) = π(H) and if Le(G + X) = Le(H + X) for every X ∈ W .Similarly, we say G �+ H or H �+ G if π(G) = π(H) and if Le(G + X) ≥Le(H + X) for every game X .
Define ∼− (right-equivalence), �−, and �− analogously, using Re(G + X) andRe(H + X) instead of Le(G + X) and Le(H + X).
The relations ∼+ and ∼− should be thought of as even coarser variants of equivalence,obtained by choosing to only look at left-ending outcomes or right-ending outcomes,respectively. In other words, G and H are left-equivalent if and only if they are indis-tinguishable in settings where Left “gets the last word.” We will eventually see thatG ∼+ H is true if and only G and H have equivalent upsides, while G ∼− H is trueif and only if G and H have equivalent downsides. Similarly, G �+ H is equivalentto G+ � H+, and G �− H is equivalent to G− � H−. These facts will be seen inthe proof of Theorem 2(a) at the end of Sect. 5.
Lemma 4 Two games G and H are equivalent if and only if they are both left-equivalent and right-equivalent. Similarly, G � H if and only if G �+ H andG �− H.
Proof For any game X , the triples
(π(X), L(X), R(X)) and (π(X), Le(X), Re(X))
carry the same information. In particular, taking the discrete partial order on Z/2Z
and the product partial order on Z/2Z × R × R,
(π(G + X), L(G + X), R(G + X)
)≥
(π(H + X), L(H + X), R(H + X)
)(1)
if and only if
(π(G + X), Le(G + X), Re(G + X)
)≥
(π(H + X), Le(H + X), Re(H + X)
). (2)
But (2) holds for all X if and only if G �+ H and G �− H , while (1) holds for allX if and only if G � H , by Lemma 3. This handles the case of � and �±; the caseof ∼ and ∼± is handled similarly. ��
Note that the relations ∼± and �± are compatible with addition, for the same reasonthat � and ∼ are. They are also compatible with the operation of game-building, inthe following sense:
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Well-tempered Scoring Games
Lemma 5 Let �� be one of ∼, ∼±, �, or �±. If Gi �� Hi for every i , then
We leave the proof as an exercise to the reader.Having stated the previous lemma, now is as good a time as any to tie up a loose end.
Recall that F denotes the class of fixed-length games. For G, H ∈ F , let G ∼F Hindicate that
L(G + X) = L(H + X) and R(G + X) = R(H + X)
for all X ∈ F . So ∼F is the coarsest equivalence relation on F compatible withaddition and outcomes.
Proposition 1 (a) Every well-tempered game G is equivalent (in the sense of ∼) to afixed-length one.
(b) The relations ∼ and ∼F agree on fixed-length games. That is, ∼F is the restrictionof ∼ to fixed-length games.
(c) The indistinguishability quotients W/ ∼ and F/ ∼F are isomorphic as partiallyordered groups.
Proof Let Z be the fixed-length game 〈0|0〉+〈0|0〉. By using Theorem 1(c) one checksthat Z ∼ 0. The proof of Theorem 1(c) will not rely on this proposition. Note thatif H is any game of fixed-length n, then H + Z ∼ H and H + Z has fixed-lengthn + 2.
We prove part (a) by induction on G. If G is atomic, then G is already fixed-length.Otherwise, each option of G is equivalent to a fixed-length game by induction. UsingLemma 5, we may assume that all options of G are fixed-length. If all options of Gnow have the same length, we are done. Otherwise, the lengths of the options differ byeven numbers, because G is well-tempered. Adding copies of Z to the shorter optionsof G, we can arrange for the options of G to have the same length, proving (a).
We originally defined G ∼ H to mean that G + X and H + X have the sameoutcome for all well-tempered X . By part (a), we could restrict X to range over fixed-length games in the definition, proving part (b). Part (c) follows immediately from (a)and (b). ��
The upshot is that our decision to work with well-tempered games rather thanfixed-length games has no effect on the final theory.
4 Inversive games
4.1 Heating to convert to Milnor games
Definition 11 Let G be a well-tempered game. For i = 0, 1, let γi (G) be the supre-mum of R(H) − L(H) as H ranges over the subgames of G with π(H) = i . We callγ0(G) the even gap of G and γ1(G) the odd gap of G.
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W. Johnson
These functions measure the extent to which the even- and odd-tempered positionsof G fail to be “hot” or “exciting.” Note that if G is atomic, then γ0(G) = 0 andγ1(G) = −∞. Since every game has at least one subgame that is atomic, γ0(G) ≥ 0for every G. Moreover, every non-atomic game G satisfies γ1(G) > −∞. Also notethat if H is a subgame of G, then γi (H) ≤ γi (G). A game is inversive (Definition 7)if and only if its even gap is zero.
For α ∈ R, let Iα denote the set of games G with γ0(G) = 0 and γ1(G) ≤ α. ThenI0 is exactly the set of well-tempered Milnor games (see Definition 3), while the setI of inversive games is
I =⋃α∈R
Iα.
Our first order of business is proving that I is closed under addition. By the sentenceafter Equation (4) of Milnor (1953), the set of Milnor games is closed under addition.Consequently, I0 is closed under addition. The rest of I can be related to I0 by thefollowing operation.
Definition 12 If G is a game and t is a number, then G heated by t , denoted∫ t G, is
the game defined inductively byt∫
G =⎧⎨⎩
G if G is atomic⟨t +
t∫GL
∣∣∣∣ − t +t∫
G R⟩
otherwise
This definition is obviously analogous to the classical notion of heating a normal playgame. We allow t to be negative, however. As in the classical case, the effect of heatinga game G by a quantity t is to reward any move in G with a bonus of t points. When Gis being played in combination with other games, heating G increases the incentivesfor the players to move in G rather than the other components.
This operation has a number of easily verified properties, whose proofs we omit.First of all, heating is compatible with addition, in the sense that
∫ t(G + H) =
(∫ t G) + (
∫ t H). Additionally, heating has predictable effects on outcomes: if G iseven-tempered, then
L
⎛⎝
t∫G
⎞⎠ = L(G) and R
⎛⎝
t∫G
⎞⎠ = R(G),
while if G is odd-tempered, then
L
⎛⎝
t∫G
⎞⎠ = L(G) + t and R
⎛⎝
t∫G
⎞⎠ = R(G)−t.
Using these, one can show that heating affects gaps in the following manner:
γ0
⎛⎝
t∫G
⎞⎠ = γ0(G) and γ1
⎛⎝
t∫G
⎞⎠ = γ1(G) − 2t.
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Well-tempered Scoring Games
Consequently,
G ∈ Iα ⇐⇒α/2∫
G ∈ I0.
It follows that for each α ∈ R, Iα is closed under addition. Indeed, if G, H ∈ Iα , then∫ α/2 G and∫ α/2 H are both in I0. Since I0 is closed under addition,
α/2∫(G + H) =
⎛⎝
α/2∫G
⎞⎠ +
⎛⎝
α/2∫H
⎞⎠ ∈ I0.
Consequently G + H ∈ Iα .Now since I is the union of the increasing chain I0 ⊆ I1 ⊆ I2 ⊆ · · · , it follows
that I itself is closed under addition.
4.2 Comparing inversive games
Lemma 6 If G is inversive and even-tempered, then G � 0 if and only if R(G) ≥ 0.
Proof Let S be the set of even-tempered inversive games with R(G) ≥ 0. Note thefollowing two properties of S:
– If G ∈ S, then L(G) ≥ 0. Indeed, L(G) ≥ R(G) by definition of inversive, andR(G) ≥ 0 by definition of S.
– If G ∈ S, then every right option G R of G has a left option G RL which is in S.Indeed, suppose G R is a right option of some G ∈ S. Then G R is odd-tempered,hence non-atomic, and furthermore
L(G R) ≥ R(G) ≥ 0.
Consequently, there must be some G RL with R(G RL) = L(G R) ≥ 0. Now G RL
is even-tempered and inversive, and consequently in S.
In light of these two properties, we see that every game in S is left-0-safe in thesense of Ettinger (Definition 5 of Ettinger (1996)). By Proposition 7 of Ettinger (1996),every game in S is � 0. Conversely, if G � 0, then R(G) ≥ R(0) = 0 by definitionof �. ��
We may now complete the proof of Theorem 1.
Proof of Theorem 1 Part (a) was proven above in Sect. 4.1, except for the fact that I isclosed under negation. This follows from the easy observation that γi (G) = γi (−G)
for any G ∈ W and i ∈ {0, 1}.For part (b), suppose that G is inversive. Then G − G is inversive by part (a), while
R(G − G) ≥ 0 by Lemma 2. As G − G is even-tempered, G − G � 0 by Lemma 6.By symmetry, G − G � 0, and we conclude that G − G ∼ 0.
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W. Johnson
We now turn to part (c). Let G, H be inversive games. If G � H , then by Lemma 3,π(G) = π(H), and by definition of �, R(G − H) ≥ R(H − H). But by Lemma 2,R(H − H) ≥ 0, so that R(G − H) ≥ 0.
Conversely, suppose that π(G) = π(H) and R(G − H) ≥ 0. Then G − H iseven-tempered, and inversive by Theorem 1(a). By Lemma 6, G − H � 0. Using thefact that H + (−H) ∼ 0, we see that
G ∼ G + H + (−H) = (G − H) + H � 0 + H = H.
��In fact, we see from this argument that for inversive games, left-equivalence and
right-equivalence agree with ordinary equivalence:
Lemma 7 If G, H are inversive, then
G � H ⇐⇒ G �+ H ⇐⇒ G �− H
and consequently
G ∼ H ⇐⇒ G ∼+ H ⇐⇒ G ∼− H
Proof By symmetry, we only need to show that G � H is equivalent to G �+ H . Onedirection of the implication is Lemma 4. For the other direction, suppose G �+ H .Then π(G) = π(H) by definition. Also,
R(G − H) = Le(G − H) ≥ Le(H − H) = R(H − H) ≥ 0,
so G − H � 0 by Theorem 1(c). ��
5 General games
The next lemma explains the mysterious operations G → G+ and G → G− fromDefinition 8.
Lemma 8 If G is even-tempered, L(G) ≤ R(G), and every option of G is inversive,then
G ∼+ R(G)
G ∼− L(G).
Proof We will make heavy use of Lemma 1.First consider the special case where R(G) = L(G) = x for some x ∈ R. In
particular, G is inversive. Then R(G − x) = R(G) − x = 0 ≥ 0, so G � x byTheorem 1(c). By symmetry G � x , so G ∼ x and in particular G ∼+ x = R(G)
and G ∼− x = L(G).Now consider the general case. By symmetry, it suffices to prove that G ∼+ R(G).
Let δ = R(G)−L(G) ≥ 0. Let H = 〈GL +δ|G R〉. Then L(H) = L(G)+δ = R(G),
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Well-tempered Scoring Games
while R(H) = R(G). All options of H are inversive, H is even-tempered, and L(H) =R(G) = R(H). Therefore, H ∼ R(G) by the special case considered above. Sinceδ � 0, we have H � G by Lemma 5. Thus R(G) ∼ H � G, and so G �+ R(G).
It remains to show that G �+ R(G). To do this, it suffices to prove the followingtwo statements:
(a) If K is odd-tempered and L(K + R(G)) ≥ 0, then L(K + G) ≥ 0.(b) If K is even-tempered and R(K + R(G)) ≥ 0, then R(K + G) ≥ 0.
We prove these by induction on K . The inductive derivation of (a) from (b) is straight-forward. Statement (b) is proven as follows. If K is atomic, then (b) follows fromLemma 1. Otherwise, we need to consider the right options of K + G, proving thateach one has nonnegative left outcome. The options of the form K R +G are handled by(a), since L(K R) ≥ R(K ) ≥ − R(G), which in turn implies that L(K R + R(G)) ≥ 0.The options of the form K + G R are also of the form K + H R since G and H havethe same right options. Since we assumed R(K + R(G)) ≥ 0 and since H ∼ R(G),we have R(K + H) ≥ 0. Therefore
L(K + G R) = L(K + H R) ≥ R(K + H) ≥ 0
as needed.We can explain the proof that G �+ R(G) more intuitively as follows: Left can
pretend that G is either the number R(G) (when she is selecting her own moves) or thegame H (when responding to a move by Right), since H and G have the same rightoptions. When considering left-ending outcomes, we may assume that any positionthat Right moves to is odd-tempered. In particular he cannot move to an atomic gameplus G. Consequently, Left is never compelled to move in G, which is the only potentialflaw in her strategy of treating G as R(G) when deciding on her own moves. ��Corollary 1 Let G be any game. Then G+ ∼+ G and G− ∼− G.
Proof If G is atomic, then G+ = G, so certainly G+ ∼+ G. Otherwise, let H =〈(GL)+|(G R)+〉. By induction, each option of H is left-equivalent to an option ofG, so by Lemma 5, G ∼+ H . If H is inversive, then G+ := H and we are done.If not, then H is even-tempered, L(H) < R(H), every option of H is inversive,and G+ is defined to be R(H). By Lemma 8 applied to H , H ∼+ R(H). ThereforeG ∼+ H ∼+ R(H) = G+ and we are done. The proof that G− ∼− G is similar. ��
We can now prove Theorem 2.
Proof of Theorem 2 (a) Since G ∼+ G+ and H ∼+ H+, it follows that G �+ H ifand only if G+ �+ H+. On the other hand, G+ and H+ are inversive, so by Lemma 7,this holds if and only if G+ � H+. Similarly, G �− H if and only if G− � H−. But� is the logical conjunction of �+ and �− by Lemma 4.(b) Because ∼+ is compatible with addition,
G+ + H+ ∼+ G + H ∼+ (G + H)+.
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W. Johnson
Both G+ + H+ and (G + H)+ are inversive, so by Lemma 7, they are equivalent. Asimilar argument holds for downsides.(c-c’) Since G− ∼− G ∼+ G+, we have Le(G) = Le(G+) and Re(G) = Re(G−)
by definition of ∼±. Expressing Le(·) and Re(·) in terms of L(·) and R(·) gives thedesired claims. ��
6 The range of upsides and downsides
The goal of this section is to prove Theorem 3, which says that A&B exists if and onlyif A � B. One direction of the theorem is easy:
by Lemma 2. Then by the criterion of Theorem 1(c), it follows that G+ � G−. ��This immediately gives us the following corollary, which motivates the terminologyinversive:
Corollary 2 A game G is invertible (modulo equivalence) if and only if G is equivalentto an inversive game. In the case where G is invertible, an inverse is given by −G.
Proof Recall that W and I denote the classes W and I modulo equivalence, and thatthe map G → (G+, G−) yields an injective monoid homomorphism W → I × I byTheorem 2. We have just seen that the image of this homomorphism sits inside thesubmonoid {(x, y) : x ≥ y} of I × I. The only invertible elements of this monoidare of the form (x, x), because I is a partially ordered group. So if G is invertiblemodulo ∼, then G+ ∼ G−. In this case, G ∼+ G+ and G ∼− G− ∼− G+, so thatG ∼± G+. Thus G is equivalent to the inversive game G+, by Lemma 4. Conversely,every inversive game is invertible by Theorem 1(b). Theorem 1(b) also shows thatwhen G is inversive, −G is an inverse. ��
To prove the other direction of Theorem 3, we will use the following tool. If f :R → R is a weakly order-preserving function, we let f∗ : W → W be the functiondefined recursively as follows:
f∗(G) ={
f (G) if G is atomic〈 f∗(GL)| f∗(G R)〉 otherwise
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Well-tempered Scoring Games
In other words, f∗(G) is obtained by taking all the numbers that occur in G andapplying f to them. For example,
f∗(〈15|〈2|3||1|8〉〉) = 〈 f (15)|〈 f (2)| f (3)|| f (1)| f (8)〉〉.
Easy inductive arguments show that
R( f∗(G)) = f (R(G)) and L( f∗(G)) = f (L(G))
for any game G, and also that f∗(G) is inversive whenever G is.We now prove the remaining direction of Theorem 3.
Lemma 9 Let X � 0 be an inversive game. Then there is some G ∈ W such thatG ∼+ X and G ∼− 0.
Proof Let g : R → R be the weakly-increasing function g(x) = min(x, 0). Forα ∈ R, let
Qα ={ 〈0|0||α|α〉 α > 0
α otherwise
Note that this is even-tempered for any α. Applying the definition of upsides anddownsides, we see that Q+
α = α and Q−α = g(α) for any α. Thus Qα ∼+ α and
Qα ∼− g(α).Let G be the game obtained from X by systematically replacing every atomic game
α with the even-tempered game Qα . Because Qα ∼+ α for every α, G ∼+ X byLemma 5. Similarly, because Qα ∼− g(α) for every α, G ∼− g∗(X) by Lemma 5.
Now since X � 0, L(X) ≥ 0 and R(X) ≥ 0. Thus
L(g∗(X)) = g(L(X)) = 0
and
R(g∗(X)) = g(R(X)) = 0.
But g∗(X) is inversive and even-tempered because X is. Therefore, g∗(X) ∼ 0 byTheorem 1(c). Consequently G ∼+ X and G ∼− g∗(X) ∼ 0, as desired. ��Proposition 3 Let A � B be inversive games. Then there is a game G with A ∼+G ∼− B.
Proof Let D = A − B. Then D is inversive and D � 0. Therefore, there is someH with H ∼+ D and H ∼− 0. Let G = H + B. Then G ∼+ D + B ∼ A, andG ∼− 0 + B = B. ��Combined with Proposition 2, this completes the proof of Theorem 3.
As an example of the above process, consider the problem of finding a game G suchthat G+ ∼ 〈2|2〉 and G− ∼ 〈1|0〉. Such a game must exist by Proposition 3. Followingthe recipe outlined in the above proofs, we first apply Lemma 9 to the difference
It turns out we can delete dominated moves, so this is essentially the same thing as
〈2|2||1|1〉
Now we replace each atom α with Qα , yielding the monstrosity
〈Q2|Q2||Q1|Q1〉.
Following the proof of Proposition 3, we now add this back to 〈1|0〉, yielding a gameG which is not worth writing down, but has the desired property that G ∼+ 〈2|2〉 andG ∼− 〈1|0〉.
In this particular case, there are simpler solutions to the problem; for example
〈1 + Q1|Q2〉
is a solution by Lemma 5.
7 Further directions
7.1 Boolean-valued games
The original example of a well-tempered scoring game which motivated this wholeline of inquiry was the game To Knot or Not to Knot (TKontK) of the paperA Midsummer Knot’s Dream Henrich et al. (2011). Like Hex or Misère Hex, we canmake this game into a scoring game by assigning a final score of 0 or 1 according towhich player wins. We call these kind of {0, 1}-valued well-tempered scoring gamesboolean-valued (well-tempered scoring) games. A sum of two boolean-valued gamesis not necessarily boolean-valued. In order to remain within the universe of boolean-valued games, the following operation is more appropriate:
G ∨ H ={
max(G, H) if G and Hare both atomic〈GL ∨ H, G ∨ H L |G R ∨ H, G ∨ H R〉 if not
For various reasons, this operation shows up in TKontK. It also shows up in Hex,when connections are chained together in parallel. The dual operation ∧, defined usinga minimum instead of a maximum, similarly shows up in Hex when connections arechained together in series.5 One can show that the operations ∨ and ∧ are compatiblewith ∼, so that ∨ and ∧ give well-defined operations on the class of boolean-valuedgames modulo equivalence. Moreover, one can show that up to equivalence, there areexactly 70 boolean-valued games, and a complete description of this seventy-elementset can be given. These matters will be discussed further in a subsequent paper, whichwill also consider the application to TKontK as well as an abstract generalization of
5 The operations ∨ and ∧ have nothing to do with the union and join operations discussed in Chapters 9-10of Berlekamp et al. (2001) or Chapter 14 of Conway (2001).
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Well-tempered Scoring Games
TKontK. The resulting theory tells us nothing useful about Hex, but one can constructvariants of Hex in which the theory becomes relevant.
7.2 Integer-valued games
If one is willing to restrict to the case of integer-valued scores, then a complete descrip-tion of scoring games modulo equivalence can be given in terms of the classical theoryof normal play partizan games. Let WZ denote the class of integer-valued games. Inother words, a scoring game G is in WZ if and only if every atom occurring in Gis in Z. The class WZ is closed under addition and negation, and the results of thispaper are essentially all true in this restricted setting. Let Pg denote the set of classicalfinite normal play partizan games, modulo the usual equivalence relation. Letting WZ
denote WZ modulo equivalence, it can be shown that WZ is isomorphic as a partiallyordered abelian group to the one obtained from Pg by formally inverting the Nortonoverheating operation
∫ 1∗1 . This allows us to describe Z-valued scoring games in terms
of classical partizan games. It also shows that the class W of well-tempered gameshas roughly the same complexity as Pg.
7.3 The outlook towards general dicot scoring games
Unfortunately, none of the methods of this paper seem to generalize to scoring gameswhich are not well-tempered.
Central notions such as ∼±, upsides, and downsides only make sense in a well-tempered setting. Even though we were considering a special class of scoring games,our monoid W is not necessarily a submonoid of the monoid of scoring games (moduloequivalence), because ∼ and ∼E may not agree on scoring games.
One way to explain the good behavior of inversive games is to note that theyare obtained from Milnor games by applying the heating operator
∫ t with t < 0,an operation we might call “cooling.” Because heating and cooling are invertibleoperations compatible with equivalence (∼), inversive games inherit the genteel man-ners of Milnor games. While the definition of heating and cooling (Definition 12)makes sense outside of the well-tempered context, certain key properties are lost. Inparticular, cooling ceases to be compatible with equivalence (∼E ). For example, ifG = 〈0, 〈0|0〉|0, 〈0|0〉〉, then G + G ∼E 0, but
∫ −1(G + G) is not equivalent to zero,
and has left outcome −1. Even worse, for any scoring game H , the limiting outcomeof
∫ −n H as n → +∞ is determined by the misère outcome of H . On the other hand,one can show that heating by a positive number is still compatible with ∼E .
Although the methods of this paper do not extend to the general case of dicot scoringgames, our results provide a partial glimpse into the world of such games, and providehints of what to expect in general.
Acknowledgments This work was done in part during REU programs at the University of Washingtonin 2010 and 2011, and in part while supported by the NSF Graduate Research Fellowship Program (GrantNo. DGE 1106400) in 2011–2013. The author would like to thank Rebecca Keeping, James Morrow, NeilMcKay, and Richard Nowakowski, with whom he discussed the content of this paper, as well as Allison
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W. Johnson
Henrich, who introduced the author to the game of To Knot or Not to Knot, which initiated this entireinvestigation. The author would also like to thank the anonymous referees for their constructive feedback.
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