dead ends in misere play: the misere monoid of canonical

Rebecca Milley, Gabriel Renault
To cite this version:
CANONICAL NUMBERS. Discrete Mathematics, Elsevier, 2013, 313, pp.2223-2231. <hal-00985737>
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arXiv:1212.6435v2 [math.CO] 16 Apr 2013
Rebecca Milley
Dept. of Mathematics and Statistics, Dalhousie University, Halifax, Canada, B3H 4R2
Gabriel Renault
Univ. Bordeaux, LaBRI, UMR5800, F-33400 Talence, France
CNRS, LaBRI, UMR5800, F-33400 Talence, France
We find the misère monoids of normal-play canonical-form integer and
non-integer numbers. These come as consequences of more general results
for the universe of dead-ending games. Left and right ends have previously
been defined as games in which Left or Right, respectively, have no moves;
here we define a dead left (right) end to be a left (right) end whose options
are all left (right) ends, and we define a dead-ending game to be one in which
all end followers are dead. We find the monoids and partial orders of dead
ends, integers, and all numbers, and construct an infinite family of games
that are equivalent to zero in the dead-ending universe.
Combinatorial game; Partizan; Misère; Monoids; Dead-ending.
1. Introduction
In many combinatorial games (two-player games of perfect information
and no chance), players take turns placing pieces on a board according to
some set of rules. Usually these rules imply that the board spaces available
to a player on his or her turn are a subset of those available on the previous
turn; games such as domineering, col, snort, hex, and nogo, among
many others, fit this description. What sorts of properties do these placement
games have over other games? What can be said about their game trees? In
contrast to a game like maze or konane, placement games have the property
that a player cannot ‘open up’ moves for him or herself, or for the opponent;
in particular, if a player has no available moves at some position of the game
then they will have no moves in any follower1 of that position. This particular
property, which we call dead-ending, is the focus of the present paper.
A left end is a game position with no options for the player we call Left,
and a right end is a position with no options for the player Right. A game
with no options for either player (called the zero game) is both a left end and
a right end. We can thus define dead-ending games as follows.
Definition 1. A left (right) end is a dead end if every follower is also a left
(right) end. A game G is called dead-ending if every end follower of G is a
dead end. The set of all dead-ending games is denoted E.
In addition to the games listed above, many well-studied positions from
normal-play game theory have the dead-ending property: integers are dead
ends, and non-integer numbers, all-small games, and all hackenbush positions are dead-ending. The set of all dead-ending games is thus a meaningful
(and large) universe to consider. Restricted universes are of particular interest to those studying misère games (where the last player to move loses),
since the restrictions may reintroduce some of the algebraic structure that is
lost in general misère play. The dead-ending universe is a natural extension
of the dicot2 universe, which has been the focus of recent research in misère
game theory (see [2],[3], and [5], for example).
In this paper we establish some basic results for dead-ending games and
demonstrate that several significant subuniverses (ends, integers, and noninteger numbers) have many of the ‘nice’ algebraic properties that are missing
from general misère play. More specifically, we find the misère monoids of
these sets of games, and determine the associated partial orders. The concept
of a misère monoid, and other prerequisite material, is discussed in Section
By follower we mean any position that can be reached from a given game position, by
alternating or non-alternating play, including the original position itself.
In a dicot game (called all-small in normal play), the position and every follower
satisfies the property that either both players have a move or the game is over. These
games are trivially dead-ending, as no follower can be a non-zero end.
1.1. General misère background
By convention, the players Left and Right are female and male, respectively. Under normal play, the first player unable to move in a game loses;
the less-studied and less-structured ending condition known as misère play
declares that the first player unable to move is the winner. Games or positions are defined in terms of their options: G = {GL | GR }, where GL is the
set of positions GL to which Left can move in one turn, and similarly for GR .
The simplest game is the zero game, 0 = {· | ·}, where the dot indicates an
empty set of options.
In both play conventions, the outcome classes next (N ), previous (P),
left (L), and right (R) are partially ordered as shown in Figure 1, with
Left preferring moves toward the top and Right preferring moves toward
the bottom. We use N − , P − , L− , and R− to denote the outcome classes
under misère play. We also use the outcome functions o− (G) and o+ (G) to
distinguish between the misère and normal outcomes, respectively.
Figure 1: The partial order of outcome classes.
Many definitions from normal-play game theory3 are used without modification for misère games, including birthday, disjunctive sum, equality, and
inequality. Thus, for misère games,
G = H if o− (G + X) = o− (H + X) for all games X,
G ≥ H if o− (G + X) ≥ o− (H + X) for all games X.
In normal-play, the negative of a game is defined recursively as −G =
{−GR |−GL }, and is so-called because G + (−G) = 0 for all games G. Under
general misère play, however, this property holds only if G is the zero game
[6]. To avoid confusion and inappropriate cancellation, we write G instead
of −G and refer to this game as the conjugate of G.
A complete overview of normal-play game theory can be found in [1].
In normal-play games, there is an easy test for equality: G = 0 if and
only if o+ (G) = P, and so G = H if and only if o+ (G − H) = P. In
misère-play, no such test exists. Equality of misère games is difficult to prove
and is, moreover, relatively rare: for example, besides {· | ·} itself, there are
no games equal to the zero game under misère play [6]. As with normalplay, we can reduce a misère game to a unique canonical form by eliminating
dominated options and bypassing reversible ones [10], but instances of domination (inequality) and reversibility are much less common under misère play.
Plambeck (see [8] and [9], for example) introduced a partial solution to these
challenges: modify the definitions of equality and inequality by restricting
the game universe. Specifically, given a set of games U , misère equivalence
(modulo U ) is defined by
G ≡ H (mod U ) if o− (G + X) = o− (H + X) for all games X ∈ U,
while misère inequality (modulo U ) is defined by
G ≧ H (mod U ) if o− (G + X) ≥ o− (H + X) for all games X ∈ U.
We use the words equivalent and indistinguishable interchangeably, and if
G 6≡ H (mod U ) then G and H are said to be distinguishable. If G 6≧ H and
G 6≦ H then G and H are incomparable, and we write G||H. In this paper
we use the symbol to indicate strict modular inequality. The set U is often
called the universe.
Given a universe U , we can determine the equivalence classes under ≡
(mod U ) and form the quotient semi-group U / ≡. This quotient, together
with the tetra-partition of elements into the sets P − , N − , R− , and L− , is
called the misère monoid of the set U , denoted MU . It is usually desirable to
have the set of games U closed under disjunctive sum; when a set of games
is not already thus closed, we consider its closure or set of all sums of those
Plambeck and Siegel [9] used the monoid approach with much success in
analyzing impartial misère games. Allen [2, 3] extended the idea to partizan
game theory, investigating monoids within the dicot universe. Most recently,
McKay, Milley, and Nowakowski [5] determined the monoid for a subset of
the dicot universe corresponding to instances of hackenbush, and Milley,
Nowakowski, and Ottaway [7] found the monoid for ends in the universe
of alternating (or ‘consecutive-move-ban’) games. In this paper we make a
significant contribution to the literature by determining the misère monoid
of all normal-play canonical-form numbers. We find this to be the same as
the monoid of the closure of dead ends.
In Section 2 we establish some basic properties of the dead-ending universe. In Section 3 we analyze ends in the dead-ending universe, which
includes all integers in normal-play canonical form. In Section 4 we extend
this analysis to non-integer numbers and find that the monoid of all numbers is equivalent to the monoid of integers. We also determine the partial
orders of these subuniverses (modulo the subuniverse as well as modulo E),
and establish invertibility of the elements (modulo E). Finally, in Section 5
we discuss other dead-ending games, in the context of equivalency to zero
modulo the dead-ending universe.
2. Preliminary results
We begin with some immediate consequences of the definition of deadending. Lemmas 1 and 2 show that the universe E of dead-ending games
is ‘closed’ in two important respects: it is closed under followers and closed
under disjucntive sum.
Lemma 1. If G is dead-ending then every follower of G is dead-ending.
Proof. If H is a follower of G, then every follower of H is also a follower of
G; thus if G satisfies the definition of dead-ending, then so does H.
Lemma 2. If G and H are dead-ending then G + H is dead-ending.
Proof. Any follower of G + H is of the form G′ + H ′ where G′ and H ′ are
(not necessarily proper) followers of G and H, respectively. If G′ + H ′ is a
left end, then both G′ and H ′ are left ends, which must be dead, since G and
H are dead-ending. Thus, any right options G′ R and H ′ R are left ends, and
so all options G′ R + H ′ and G′ + H ′ R of G′ + H ′ are left ends. A symmetric
argument holds if G′ + H ′ is a right end, and so G + H is dead-ending.
Under misère play, Left trivially wins any left end playing first. In general,
Left may or may not win a left end playing second; for example, the game
{· | 1} is a left end in N − . If a (non-zero) left end is dead, however, then it
is a win for Left playing first or second.
Lemma 3. If G 6= 0 is a dead left end then G ∈ L− , and if G 6= 0 is a dead
right end then G ∈ R− .
Proof. A left end is always in L− or N − . If G is a dead left end then any
right option GR is also a left end, so Right has no good first move. Similarly,
a dead right end is in R− .
The following lemma, which describes a sufficiency condition for invertibility, applies generally to any misère universe. In the present paper we
apply it to various subsets of dead-ending games.
Lemma 4. Let U be any game universe closed under conjugation, and let
S ⊆ U be a set of games closed under followers. If G + G + X ∈ L− ∪ N −
for every game G ∈ S and every left end X ∈ U , then G + G ≡ 0 (mod U )
for every G ∈ S.
Proof. Let S be a set of games with the given conditions. Since U is closed
under conjugation, by symmetry we also have G + G + X ∈ R− ∪ N − for
every G ∈ S and every right end X ∈ U.
Let G be any game in S and assume inductively that H + H ≡ 0 (mod
U ) for every follower H of G. Let K be any game in U , and suppose Left
wins K. We must show that Left can win G + G + K. Left should follow
her usual strategy in K; if Right plays in G or G to, say, GR + G + K ′ , with
K ′ ∈ L− ∪ P − , then Left copies his move and wins as the second player on
GR + G + K ′ = GR + GR + K ′ ≡ 0 + K ′ , by induction. Otherwise, once
Left runs out of moves in K, say at a left end K ′′ , she wins playing next on
G + G + K ′′ by assumption.
The main argument above begins with the phrase ‘suppose Left wins K’.
This appears to be ambiguous, or incomplete; does Left win K playing first
or playing second? The implied assumption with such a statement is that
the argument to follow holds for both cases.
In subsequent sections we refer to the two game functions below, which
are well-defined for our purposes — namely, for numbers and ends.
Definition 2. The left-length of a game G, denoted l(G), is the minimum
number of consecutive left moves required for Left to reach zero in G. The
right-length r(G) of G is the minimum number of consecutive right moves
required for Right to reach zero in G.
In general, left- and right-length are well-defined if G has a non-alternating
path to zero for Left or Right, respectively, and if the shortest of such
paths is never dominated by another option. The latter condition ensures
l(G) = l(G′ ) when G ≡ G′ . As suggested above, both of these conditions
are met if G is a (normal-play) canonical-form number or if G is an end
in E. If l(G) and l(H) are both well-defined then l(G + H) is defined and
l(G + H) = l(G) + l(H). Similarly, when right-length is defined for G and
H, we have r(G + H) = r(G) + r(H).
3. Integers and other dead ends
Let n denote the game {n − 1 | ·}, where 0 = 0 = {· | ·}. That is,
n is the position with the same game tree as the integer n in normal-play
canonical form. In this paper the term ‘integer’ will always refer to such
a position. Note that we should distinguish between the game n and the
number n, since, among other shortcomings, n 6> n − 1 in general misère
play. One property that does hold in both normal and misère play is that the
disjunctive sum of positive integers n and m is the integer n + m, although
this is not generally true (in misère games) if one of n or m is negative and
the other positive. In this section we prove that the restricted universe of
integers under misère play has much of the structure enjoyed by normal-play
An integer is an example of a dead end: if n > 0 then Right has no move
in n and no move in any follower of n. Similarly, if n < 0 then n is a dead
left end. Thus, the following results for ends in the dead-ending universe are
true for all integers, modulo E.
Our first result shows that when all games in a sum are dead ends, the
outcome is completely determined by the left- and right-lengths of the games.
Lemma 5. If G is a dead right end and
o− (G + H) = L−
 −
H is a dead left end, then
if l(G) = r(H),
if l(G) < r(H),
if l(G) > r(H).
Proof. Each player has no choice but to play in their own game, and so the
winner will be the player who can run out of moves first.
We use Lemma 5 to prove the following theorem, which demonstrates the
invertibility of all ends in E. In particular, this shows that every integer has
an additive inverse modulo E.
Theorem 6. If G is a dead end then G + G ≡ 0 (mod E).
Proof. Assume without loss of generality that G 6= 0 is a dead right end.
Since every follower of a dead end is also a dead end, Lemma 4 applies,
with S the set of all dead left and right ends. It therefore suffices to show
G + G + X ∈ L− ∪ N − for any left end X in E. We have l(G) = r(G) and
r(X) ≥ 0, so l(G) ≤ r(G) + r(X) = r(G + X), which gives G + G + X ∈
L− ∪ N − by Lemma 5.
Corollary 7. If n is an integer then n + n ≡ 0 (mod E).
Note that equivalency in E implies equivalency in all subuniverses of E;
thus in the universe of integers alone, every game has an inverse.
Lemma 5 shows that when playing a sum of dead ends, both players
aim to exhaust their own options as quickly as possible. This suggests that
options with longer paths to zero will be dominated by shorter paths; in
particular, we have that integers are totally ordered among dead ends, as
established in Theorem 8 below. Note that this ordering only holds in the
subuniverse of the closure4 of dead ends, and not in the whole universe E.
In fact, we see immediately in Theorem 9 that distinct integers are pairwise
incomparable modulo E, just as they are in the general misère universe.
In the following arguments we frequently use the fact that, when H ∈ U
has an additive inverse modulo U , G H (mod U ) if and only if G + H 0.
Theorem 8. If n < m ∈ Z then n m modulo the closure of dead ends.
Proof. By Corollary 7, it suffices to show n + m 0 (equivalently, k 0
for any negative integer k), modulo the closure of dead ends. Let X be
any game in the closure of dead ends; then X = Y + Z where Y is a dead
right end and Z is a dead left end. Suppose Left wins X playing first; then
by Lemma 5, l(Y ) ≤ r(Z). We need to show Left wins k + X, so that
o− (k + X) ≥ o− (X). Since k is a negative integer, r(k) is defined and
r(k) = −k > 0. Thus l(Y ) ≤ r(Z) < r(Z) + r(k) = r(Z + k), which gives
k + Y + Z = k + X ∈ L− ∪ N − , by Lemma 5 .
Since the sum of a dead left end and a dead right end may not be a dead end (or any
end at all), the set of dead ends is not closed under disjunctive sum; thus the universe we
consider is the closure of dead ends, as defined in Section 1.1.
In general, G ≥ H under misère play implies G ≥ H under normal play
[10]; Theorem 8 shows this is not always the case for misère inequality modulo
a restricted universe.
Theorem 9. If n 6= m ∈ Z then n||m (mod E).
Proof. Assume n > m. Then we have n 6≧ m (mod E), because n +m ∈ R−
while m + m ≡ 0 ∈ N − . It remains to show n 6≦ m.
Define a family of games λk by
λ1 = {0 | − 1}, λk = {0 | λk−1 }.
Note that n + λn ∈ L− , since Left wins playing first or second by ignoring
λn and forcing Right to play there, bringing the game to −1 with either Left
or Right to play next.
If n > m ≥ 0 then m + λn is in P − or R− : Left loses as soon as she plays
in λn , and so plays only in m, but (moving first) she will run out of moves
in m before λn is brought to −1. Thus n 6≦ m in this case, since Left can
win n + λn but not m + λn .
If m < 0 then let k = −m − 1 and take X = k + λn+k . As above,
n + k + λn+k ∈ L− . However, m + k + λn+k ≡ −1 + λn+k ∈ N − since each
player can move to a position from which the opponent is forced to move to
zero. In this situation we see Left prefers n over m, so again n 6≦ m.
Theorem 9 tells us that, modulo E, the games {0, 1 | ·} and {0 | ·} are
distinguishable, as the option to 0 does not in general dominate the option
to 1. Thus, in the dead-ending universe, there exist ends that are not integers.
However, if we restrict ourselves to the subuniverse of dead ends, then the
ordering given in Theorem 8 implies that every end reduces to an integer.
This fact is presented in the following lemma.
Lemma 10. If G is a dead end then G ≡ n, modulo the closure of dead
ends, where n = l(G) if G is a right end and n = −r(G) if G is a left end.
Proof. Let G be a dead right end (the argument for left ends is symmetric).
Assume by induction that every option GLi of G (necessarily a dead right
end) is equivalent to the integer l(GLi ). Modulo dead ends, by Theorem 8,
these left options are totally ordered; thus G = {GL1 | ·} for GL1 with smallest
left-length. Then G is the canonical form of the integer l(GL1 ) + 1 = l(G).
Lemma 10 shows that the closure of dead ends has precisely the same
monoid as the set of canonical-form integers. The game of domineering
on 1 × n and n × 1 strips is an instance of these universes. The results of
this section allow us to completely describe the monoid, which we present in
Theorem 11. By N we mean the set of natural numbers, including zero.
Theorem 11. Under the mapping n 7→ αn , the misère monoid of the set of
normal-play canonical-form integers is
MZ = h1, α, α−1 | α · α−1 = 1i ∼
= (Z, +),
with outcome partition
N − = {0}, L− = {α−n | n ∈ N}, R− = {αn | n ∈ N},
and total ordering
αn αm ⇔ n < m.
4. Numbers
4.1. The monoid of Q2 .
We say that a game a is a non-integer number in a universe U if it is
equivalent, modulo U , to the normal-play canonical form of a (non-integer)
dyadic rational:
m − 1 m + 1
a= j =
2j 2j
with j > 0 and m odd. The set of all integer and non-integer (combinatorial
game) numbers is thus the set of dyadic rationals, which we denote by Q2 .
As we did for integers in the previous section, we now determine the outcome
of a general sum of dyadic rationals and thereby describe the misère monoid
of the closure of numbers.
Note that the sum of two non-integer numbers (even if both are positive)
is not necessarily another number. For example, in general misère play,
1 + 1/2 = {1/2, 1 | 2} =
6 3/2 implies that 1/2 + 1/2 = {1/2 | 1 + 1/2} =
6 1.
We will see that, unlike integers, the set of dyadic rationals is not closed under
disjunctive sum even when restricted to the dead-ending universe; however,
closure does occur when we restrict to numbers alone.
Lemma 14 below — analogous to Lemma 5 of the previous section —
shows that the outcome of a sum of numbers is determined by the left- and
right-lengths of the individual numbers. To prove this, we require Lemma 13,
which establishes a relationship between the left- or right-lengths of numbers
and their options; and to prove Lemma 13, we need the following proposition.
Proposition 12. If a ∈ Q2 \ Z then at least one of aRL and aLR exists, and
either aL = aRL or aR = aLR .
Proof. Let a = m/2j with j > 0 and m odd. If m ≡ 1 (mod 4) then
m−1 m+3 m+1
m−1 R m+1
a =
, a =
= j−1 =
j−1 j−1
so aL = aRL . Otherwise, m ≡ 3 (mod 4) and then
m−3 m+1 m−1
a =
2j−1 2j−1
so aR = aLR .
Note that if a > 0 is a dyadic rational then l(a) = 1 + l(aL ), and if
a < 0 is a dyadic rational then r(a) = 1 + r(aR ). We also have the following
inequalities for left-lengths of right options and right-lengths of left options,
when a is a non-integer dyadic rational.
Lemma 13. If a ∈ Q2 \ Z is positive then l(aR ) ≤ l(a); if a is negative then
r(aL ) ≤ r(a).
Proof. Assume a > 0 (the argument for a < 0 is symmetric). Since a is in
canonical form, both aL and aR are positive numbers. If aL = aRL then
l(aR ) = 1 + l(aRL ) = 1 + l(aL ) = l(a). Otherwise aR = aLR , by Proposition
12; then aL is not an integer because aLR exists, so by induction we obtain
l(aR ) = l(aLR ) ≤ l(aL ) = l(a) − 1 < l(a).
We can now determine the outcome of a general sum of numbers, both
integer and non-integer.
Lemma 14. If {ai }1≤i≤n and P
{bi }1≤i≤m arePsets of positive and negative
numbers, respectively, with k = ni=1 l(ai ) − m
i=1 r(bi ), then
!  L− if k < 0
N − if k = 0
ai +
bi =
 −
if k > 0.
Proof. Let G = ni=1 ai + m
i=1 bi . All followers of G are also of this form,
so assume the result holds for every proper follower of G. Suppose k < 0. If
n = 0 then Left will run out of moves first because Left cannot move last in
any negative number. So assume n > 0. Left moving first can move in an
ai to reduce k by one (since l(ai L ) = l(ai ) − 1), which is a left-win position
by induction. If Right moves first in an ai then k does not increase, since
l(ai R ) ≤ l(ai ) by Lemma 13, so the position is a left-win by induction; if
Right moves first in a bi then k does increase by one, but Left can respond in
an ai (since n > 0) to bring k down again, leaving another left-win position,
by induction. Thus G ∈ L− if k < 0.
The argument for k > 0 is symmetric. If k = 0 then either G = 0 is
trivially next-win, or both n and m are at least 1 and both players have a
good first move to change k in their favour.
Lemma 14 shows that in general misère play, the outcome of a sum of
numbers is completely determined by the left-lengths and right-lengths of the
positive and negative components, respectively. From this we can conclude
that, modulo the closure of canonical-form numbers, a positive number a is
equivalent to every other number with left-length l(a). In particular, every
positive number a is equivalent to the integer l(a). This is Corollary 15
below; together with Theorem 18, it will allow us to describe the monoid of
canonical-form numbers.
Corollary 15. If a is a number, then
if a ≥ 0,
−r(a) if a < 0.
As an example, the dyadic rational 1/2 is equivalent to l(1/2) = 1, and
−3/4 ≡ −r(−3/4) = −2, modulo Q2 . Note that these equivalencies do not
hold in the larger universe of E; indeed, as we see in section 4.2, if a 6= b are
numbers then a 6≡ b (mod E).
We see then that the closure of numbers is isomorphic to the closure of just
integers; when restricted to numbers alone, every non-integer is equivalent
to an integer. Thus the misère monoid of numbers, given below, is the same
monoid presented in Theorem 11. The partial order of the set of numbers,
modulo E, is described in Section 4.2.
Theorem 16. Under the mapping a 7→ αn , where n = l(G) if a ≥ 0 and
n = −r(G) if a < 0, the misère monoid of the set of canonical-form dyadic
rationals is
MQ2 = h1, α, α−1 | α · α−1 = 1i ∼
= (Z, +),
with outcome partition
N − = {0}, L− = {α−n | n ∈ N}, R− = {αn | n ∈ N}.
As with integers, some of the structure found in the number universe is
also present in the larger universe E. We end this subsection with a proof
that all numbers — not just integers — are invertible in the universe of deadending games. We require the following lemma, an extension of Lemma 14.
Lemma 17. If {ai }1≤i≤n P
and {bi }1≤i≤m
P are sets of positive and negative
numbers, respectively, and ni=1 l(ai ) − m
i=1 r(bi ) < 0, then
ai +
b i + X = L− ,
for any left end X ∈ E.
Proof. The argument from Theorem 14 works again, since if Right uses his
turn to play in X then Left responds with a move in a1 to decrease k by 1,
which is a win for Left by induction.
Theorem 18. If a ∈ Q2 then a + a ≡ 0 (mod E).
Proof. Without loss of generality we can assume a is positive. Since every
follower of a number is also a number, we can use Lemma 4. That is, it
suffices to show a + a + X ∈ L− ∪ N − for any left end X ∈ E. If X = 0 this
is true by Lemma 14. If X 6= 0 then we claim a + a + X ∈ L− ; assume this
holds for all followers of a. Left can win playing first on a + a + X by moving
to aL , since l(aL ) − r(a) = l(aL ) − l(a) < 0 implies aL + a + X ∈ L− by
Lemma 17. If Right plays first in X then again Left wins by moving a to aL ;
if Right plays first in a then Left copies in a and wins on aL + aL + X ∈ L−
by induction.
Theorem 18 shows that in dead-ending games like domineering, hackenbush, etc., any position corresponding to a normal-play canonical-form
number has an additive inverse under misère play. So, for example, the positions in Figure 2 would cancel each other in a game of misère hackenbush.
Figure 2: Normal-play canonical forms of 1/2 (left) and −1/2 (right) in hackenbush.
4.2. The partial order of numbers inside E.
In section 3 we found that all integers were incomparable in the deadending universe. We will see now that non-integer numbers are a bit more
cooperative; although not totally ordered, we do have a nice characterization
of the partial order of numbers in the universe E. First note that any two
distinct numbers are distinguishable modulo E; this is an immediate corollary
of the following theorem of [4], which extends a result of [10] referenced
Theorem 19. [4] If G ≧ H (mod E) then G ≥ H in normal play.
Corollary 20. If a, b ∈ Q2 and a 6= b then a 6≡ b (mod E).
Theorem 19 says that if a ≧ b (mod E) then a ≥ b as real numbers
(or as normal-play games). The converse is clearly not true for integers,
by Theorem 9; it is also not true for non-integers, since 1/2 − 1/2 ∈ N −
while 3/4 − 1/2 ∈ R− (which the reader can verify), so that 1/2 6≦ 3/4
(mod E). Theorem 23 shows that the additional stipulation l(a) ≤ l(b) is
sufficient for a ≧ b (mod E). To prove this result we need the following
lemmas. As before, non-bolded symbols represent actual numbers, so that
‘a < b’ indicates inequality of a and b as rational numbers (or as normal-play
games), and aL means the rational number corresponding to the left-option
of the game a in canonical form. Recall that if x = {xL |xR } is in (normalplay) canonical form then x is the simplest number (i.e., the number with
smallest birthday) such that xL < x < xR . Thus, if xL < x, y < xR and
x 6= y then x is nn y.
Lemma 21. If a and b are positive numbers such that aL < b < a, then
l(aL ) < l(b).
Proof. We have aL < b < a < aR , so a must be simpler than b. Thus
bL ≥ aL , since otherwise bL < aL < b < bR would imply that b is simpler
than aL , which is simpler than a. Now, if bL = aL then l(aL ) = l(bL ) =
l(b) − 1 < l(b), and if bL > aL then by induction aL < bL < b < a gives
l(aL ) < l(bL ) = l(b) − 1 < l(b).
Lemma 21 is used to prove Lemma 22 below, which is needed for the proof
of Theorem 23. Note that in the following two arguments we frequently use
the fact that, if a ≧ b (mod E), then Left wins on the position a + b + X
whenever she wins X ∈ E.
Lemma 22. If a and b are positive numbers such that aL < b < a, then
a b (mod E).
Proof. Note that b ∈
6 Z since there are no integers between aL and a if a is
in canonical form. We must show that Left wins a + b + X whenever she
wins X ∈ E.
Case 1: bR = a.
Left can win a + b + X by playing her winning strategy on X. If Right moves
in a + b to aR + b + X ′ , then Left responds to aR + bR + X ′ = aR + a + X ′ ,
which she wins by induction since aRL ≤ aL (see Proposition 12) gives
aRL < a < aR . If Right moves to a + b + X ′ = bR + b + X ′ , with
X ′ ∈ L− ∪ P − (since Left is playing her winning strategy in X), then Left’s
response depends on whether bRL = bL or bLR = bR : if the former, Left
moves to bRL + b + X ′ = bL + bL + X ′ ≡ X ′ (mod E); if the latter then Left
moves to bR + bL + X ′ = bR + bLR + X ′ = bR + bR + X ′ ≡ X ′ . In either
case Left wins as the previous player on X ′ ∈ L− ∪ P − .
When Left runs out of moves in X, she moves to aL + b + X. By Lemma
21 we know l(aL ) < l(b), and this gives aL + b + X ∈ L− by Lemma 17.
Case 2: bR 6= a.
Note that bR cannot be greater than a, since aL < b < a < aR implies a is
simpler than b, while bL < b < a < bR would imply that b is simpler than
a. So bR < a, and together with aL < b < bR this gives aL < bR < a, which
shows a bR (mod E) by induction. Similarly bRL ≤ bL < b < bR implies
bR b (mod E), by Case 1. Then by transitivity we have a b (mod E).
With Lemma 22 we can now prove Theorem 23 below. The symmetric
result for negative numbers also holds.
Theorem 23. If a and b are positive numbers such that a > b and l(a) ≤
l(b), then a b (mod E).
Proof. By Corollary 20 we have a 6≡ b (mod E), and so it suffices to show
a ≧ b (mod E). Again we have b 6∈ Z. Since a > b, if b > aL then Lemma 22
gives a b (mod E) as required. So assume b ≤ aL . Again let X ∈ E be a
game which Left wins playing first; we must show Left wins a+b+X playing
first. Left should follow her winning strategy from X. If Right plays to
a+bL +X ′ , where X ′ ∈ L− ∪P − , then Left responds with aL +bL +X ′ , which
she wins by induction: bL < b ≤ aL and l(bL ) = l(b) − 1 ≥ l(a) − 1 = l(aL )
implies aL bL (mod E).
If Right plays to aR + b + X ′ (assuming this move exists — that is,
assuming a 6∈ Z) then Left’s response is aRL + b + X ′ , if aRL > b, or aR +
bR + X ′ if aRL ≤ b. In the first case Left wins by induction because aRL > b
and l(aRL ) = l(aR ) − 1 ≤ l(a) − 1 < l(b) implies aRL b (mod E). In the
latter case, note firstly that in fact aRL 6= b, since we have already seen that as
games they have different left-lengths. Then we see aRL < b < a < aR < aRR ,
which shows aR must be simpler than b. This gives bR ≤ aR , as otherwise
bL < b < a < aR < bR would imply that b is simpler than aR . If bR = aR
then bR = aR , and if bR < aR then we can apply Lemma 22 to conclude that
aR bR (mod E). In either case, Left wins aR +bR +X ′ , with X ′ ∈ L− ∪P − ,
as the second player.
Finally, if Left runs out of moves in X then she moves to aL + b + X ′′
where X ′′ is a dead left end; then Left wins by Lemma 17 because l(aL ) <
l(a) ≤ l(b) = r(b).
Corollary 24. For positive numbers a, b ∈ Q2 , a b (mod E) if and only
if a > b and l(a) ≤ l(b).
Proof. We need only prove the converse of Theorem 23. Suppose a > b and
l(a) > l(b); then by [4] it cannot be that a ≦ b (mod E), so we need only
show a 6≧ b (mod E). We have b + b ∈ N − , while a + b ∈ R− , since in
isolation the latter sum is equivalent to the positive integer l(a) − l(b), by
Theorem 16. Thus a 6≧ b (mod E).
To completely describe the partial order of numbers within E, it remains
to consider the comparability of a and b when a ≥ 0 and b < 0 (or, symmetrically, when a > 0 and b ≤ 0). As before we cannot have a ≦ b (mod E),
and the same argument as above (b + b ∈ N − and a + b ∈ R− ) shows a 6≧ b
(mod E) . The results of this section are summarized below.
Theorem 25. The partial order of Q2 , modulo E, is given by
a ≡ b (mod E)
a b (mod E)
a||b (mod E)
if a = b;
if 0 < a < b and l(a) ≤ l(b),
or b < a < 0 and r(b) ≤ r(a);
5. Zeros in the dead-ending universe
We have found that integer and non-integer numbers, as well as all ends,
satisfy G + G ≡ 0 (mod E). It is not the case that every game in E has an
additive inverse; for example, ∗ + ∗ 6≡ 0 (mod E), although the equivalence
does hold in the dicot universe D ⊂ E. Likewise, many familiar ‘all-small’
games from normal-play, which have inverses in the dicot universe5 , are not
invertible here.
The following lemma describes an infinite family of games that are not
invertible in the universe of dead-ending games.
Lemma 26. If G = {n1 , . . . , nk | n1 , . . . , nk } with each ni ∈ N, then G +
G 6≡ 0 (mod E).
Proof. Let X = {n1 , . . . , nk | ·} ∈ R− . Note that G = G. We describe a
winning strategy for Left playing second in the game G+G+X = G+G+X.
Right has no first move in X, so Right’s move is of the form G + ni + X.
Left can respond by moving X to ni , leaving G + 0. Now Right plays in G
to a nonpositive integer, which as a right end must be in L− or N − .
We conclude with an infinite family of games that are equivalent to zero
in the dead-ending universe, which are not all of the form G + G for some G.
These games are illustrated in Figure 3.
The games ↑= {0 | ∗}, ↓= {∗ | 0}, and all other day-2 misère dicots with the exception
of ∗2 = {0, ∗ | 0, ∗}, are shown to be invertible modulo D in the first author’s thesis (in
progress), using Lemma 4.
Theorem 27. If G is a dead-ending game such that every GL is a left end
with an option to zero and every GR is a right end with an option to zero,
then G ≡ 0 (mod E).
Proof. Let X be any game in E and suppose Left wins X. Then Left wins
G + X by following her strategy in X. If Right plays in G then he moves to
GR + X ′ from a position G + X ′ with X ′ ∈ L− ∪ P − ; Left can respond to
0 + X ′ and win as the second player. If both players ignore G then eventually
Left runs out of moves in X and plays to GL + X ′′ , where X ′′ is a left end.
But GL is also a left end, so the sum is a left-win by Lemma 3.
G L1 GR1
Figure 3: An infinite family of games equivalent to zero modulo E (i ≥ 1, j ≥ 1).
6. Future directions
From this first initial investigation, the universe of dead-ending games
appears to be filled with potential for successful misère analysis. It includes
as subuniverses many of the games that have already excited interest among
combinatorial game theorists; we hope some of the techniques of the present
paper can be fruitfully applied to these subuniverses.
A natural extension of this work would include analysis of specific games,
such as nogo, col, snort, etc., in the context of the dead-ending universe.
It would also be interesting to consider other properties, besides dead-ending,
of the ‘placement games’ described in our opening paragraph.
7. Acknowledgments
The authors greatly appreciate the comments and suggestions provided
by Paul Dorbec, Neil McKay, Richard Nowakowski, Aaron Siegel, and Éric
8. References
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[5] N.A. McKay, R. Milley, R.J. Nowakowski, Misère-play hackenbush
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