Simple games and magic squares

JOURNAL OF COMBINATORIAL THEORY, Series A 71, 67-88 (1995)

Simple Games and Magic Squares

A L A N T A Y L O R AND W I L L I A M Z W I C K E R *

Department of Mathematics, Union College, Schenectady, New York 12308

Communicated by the Managing Editors

Received August 17, 1993

For each integer k ~> 3, we introduce a simple game [:]k built from a k x k "strongly rigid" magic square. These games are not weighted, yet come very close to being weighted, and thus they provide a uniform sequence of counterexamples to several conjectures that have arisen over the past three decades in the fields of threshold logic, hypergraphs, reliability systems, and simple games. In particular, we show that D~ is k - 1 asummable but not k asummable (thus strengthening and simplifying an often-referenced, but unpublished, result of R. O. Winder) and that a certain variant of [~k is monotonic, strong, proper, has an acyclic "group" desirability relation, and yet is not weighted (thus strengthening a result of E. Einy and answering a question of B. Peleg). © 1995 Academic Press, Inc.

1. I N T R O D U C T I O N

A simple game G consists of a pa i r (P, W) in which P is a set (finite, for this pape r ) and W is a col lect ion of subsets of P. A simple game can mode l any vot ing system tha t pits a single a l ternat ive (such as a bill) aga ins t the s ta tus quo: P becomes the set of voters , a subset ) o f P is cal led a coalition and is said to be winning if a bill passes when precisely those voters in X vote "aye" (and to be losing otherwise) , and W becomes the col lect ion of winning coal i t ions. U n d e r this identif icat ion, G is a weighted game if it co r r e sponds to a weighted vot ing system: there exists some weight funct ion

w: P ~ R and quo ta q e R such tha t the winning coal i t ions are those satisfying w(X)>~ q, where w(X) is sho r thand for Z { w ( r ) [ r e X}.

There are at least three reasons to s tudy classes of games tha t p rope r ly include the weighted ones. Firs t , a t t empts to find var ious combina to r i a l cha rac te r i za t ions of weightedness p roduce na tu ra l var iants tha t are weaker than weightedness. Second, des i rabi l i ty order ings (which make precise the idea of one vo te r or coa l i t ion being more influential than ano the r ) have

* The authors were supported by NSF Grant DMS 9101830. Some results in this paper were circulated in previous versions titled "Threshold Hypergraphs, Simple Voting Games and Magic Squares" and '% Remarkable Class of Simple Games."

67 0097-3165/95 $12.00

Copyright © 1995 by Academic Press, Inc. All rights of reproduction in any form reserved.

68 TAYLOR AND ZWICKER

certain regularities of behavior for weighted games, but also for some non- weighted ones. Finally, it is possible to take the definition of "weighting" and soften it directly, as is done elsewhere [TZl ; TZ2]. Ideally, each of the natural game classes that arises can be equivalently described in terms of a form of trade robustness (a combinatorial characterization discussed later in this paper), in terms of a regularity of a desirability order and in terms of the existence of some pseudo or local weighting; this is the approach taken in [ TZ 1; TZ2 ].

Of course, to distinguish among these classes requires a supply of counterexamples--games that satisfy certain properties close to weightedness yet fail to satisfy some others. In this paper we use a certain type of magic square to construct an infinite sequence of such games. The magic square games appear to constitute an almost universal source of counterexamples, yielding some results that are new (e.g., Theorem 5.10, Corollary 5.4), some that are improved (Theorem 3.3), and some new proofs of known results. One advantage of the magic square games is that they unify various ideas, since they serve so many purposes. Another is that they are transparent; it is easy to see why they have certain properties, and this makes it possible to introduce variations.

In Section 2 we introduce the notion of strongly rigid magic squares and construct the games [] k. Section 3 discuss the properties of k-trade robustness and k-asummability, and then uses the [] k to prove that these properties form a strict hierarchy. Section 4 defines the group desirability order <L and discusses Einy's work and Peleg's question on its acyclicity. In Section 5 we explore conditions under which acyclicity of <L fails to carry over to the dual game and produce a variant of [] k whose constant sum extension answers Peleg's question. Other applications of r-q~, not in this paper, show that it bears a simple example of an "SSA" pseudo- weighting (see [TZl ]) and is highly "locally weighted" (see [TZ2].)

It is important to note that while some of the work we build on was done by those studying simple games, isomorphic structures, similar notions, and related questions have arisen in other fields. In particular, investigators in threshold logic (see [M] , a general reference) consider switching functions f : {0, 1 } ' ~ {0, 1}. If we let P = {1, 2, 3 .... , n}, then any coalition X c P can be identified with its characteristic vector, x, of 0's and l's that has a 1 in t h e j th place if and only i f j~X. Now any collection W of coalitions can be identified with the switching function f for which

f (x ) = 1 if and only if X~ W. A threshold function then corresponds to a weighted game.

Winder's work in threshold logic is particularly relevant to this paper. Also, both graphs and hypergraphs (see [J; Gc; HIP]) and reliability systems [R] are, essentially, simple games. Many game theorists seem to be unaware of the work in the threshold logic, yet at least some of those

SIMPLE GAMES AND MAGIC SQUARES 69

doing voting theory (e.g., I F ] ) are aware of work done on systems of linear inequalities in the early days of linear programming ( [ G n ] , for example), as are those in threshold logic, and Winder [Wi] knew of Shapley's work on simple games. The history of cross-fertilization is patchy, then, but possibilities for future interaction would seem to be great, because the different fields offer distinct perspectives, yet often ask similar questions. (For example, the vicinal order considered by graph theorists, e.g., in [ CH] , is a desirability order.)

2. STRONGLY RIGID MAGIC SQUARES AND THEIR ASSOCIATED GAMES

A magic square [A ] is a k x k matrix M of integers for which there is a constant p (called the magic sum) such that the sum of every row is p and the sum of every column is p. We will say that a magic square is strongly rigid if each set S of entries that sums to p is either a row or a column.

Note that strong rigidity forces us to abandon the traditional requirement that the entries of a magic square be exactly the integers from 1 through k 2. Also note that if M is any magic square then the effect of first permuting M's rows, then its columns (and possibly following this by a reflection about the diagonal), is to create a new magic square built from the same entries as M. Sometimes, there also exist magic rearrangements of M's entries that cannot be achieved via such permutations. If not, we will say M is rigid. It is easy to see that strong rigidity implies rigidity.

LEMMA 2.1. For each integer k>~3, there exists a strongly rigid k x k magic square.

Proof. We show this for k = 4 below; the construction for k = 3, 5, 6, ... follows similarly. We begin by presenting a 4 x 4 matrix ~ , each entry of which is itself a 4 x 4 matrix A i'j of numbers:

3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3

O 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0


0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

0 1 1 1 1 0 1 1 1 1 0 1 1 1 1 0

3 0 0 0 0 3 0 0 0 0 3 0 0 0 0 3

Note that ~ itself possesses a higher-order form of magic in that the matrix sum of any row or column of Jg is equal to the matrix

3 3 3 3

3 3 3 3

3 3 3 3

3 3 3 3 ,

which we will denote as T. Also, we claim that J is strongly rigid in the sense that any set J of

Ai° 's that sums to T must be a row or a column of ~ . For example, assume that ~9 ~ sums to T and that

1 0 1 1

1 0 1 1 1 3 , 2 0 3 0 0

1 0 1 1

is an element of ~ . Then each other element of ~ must have a 0 in the 3, 2 position, and so each must come from J/t's third row or second column. But if even one of these comes from Jd's third row, say,

1 1 1 0

1 1 1 0 A3,4~ 0 0 0 3

1 1 1 0

then the remaining elements of ~ must have 0's in both their 3, 2 and 3, 4 positions, which forces them to be the remaining two entries from ~ ' s third row. Similarly, if some second element of ~ came from ~ ' s second

S I M P L E G A M E S A N D M A G I C S Q U A R E S 71

column, then 5 p would be forced to contain exactly the members of ~ ' s second column. It is straightforward to check that these examples reflect the general situation.

To change ~ into a matrix M of numbers, we need a function f that converts each matrix entry A i'J of ~ into a corresponding numerical entry Mi, j of M. We wish f to have a certain preservation property: whenever we start with a set S of M's entries, apply f - 1 to each entry, perform matrix addition on the Aid's that arise, and then apply f to the result, we get as the final answer the sum of S's entries. Clearly, this property o f f guarantees that M inherits Jg's magic and strong rigidity.

The desired f takes an A ~'j, "scans" it row by row from left to right, and interprets the resulting digit string as an integer written in base R, where R is some sufficiently large base, as chosen below. For example, f ( A 3, 2) = 1011101103001011R (which equals the sum 1R ~5 + 0 R 14 n t- - . . -J-

3R 6 -+- . . . + 1R 1 + 1.) Note that M's magic sum is 3333333333333333R. The only way that f can fail to have the needed preservation property is

if "carrying" takes place when adding the elements of some set S of M's entries. Such a set has at most 16 elements, and the largest digit appearing in any of its elements is a 3, so by choosing R to be 49 we ensure that M is both magic and strongly rigid.

We will refer to the M above as M4; Mk is constructed similarly, except that A ~'j has, as its i,j entry, the integer k - 1, with O's in the remainder of the ith row and j t h column, and l's elsewhere. A value of k 3 - k 2 will be more than large enough for R. |

The A ~'j appear to have some other interesting properties. For example, they span the vector space, under matrix addition and scalar multiplica- tion, of all k x k magic squares and Mk, in particular, is a simple linear combination of them. However, they are not linearly independent; their dependencies are exactly those that must be satisfied by the entries of every ordinary k x k magic square by virtue of its magic.

Next, we use Mk to build a simple game [~k = (P~, W~). We let the elements of Pk be the k 2 (distinct) elements of Mk, and we endow Pk with a "rough weight" function rw: P~ ~ N. In this case, rw is the identity function. Our "rough quota," rq, will be the magic sum, p, of Mk.

Finally, Wk is defined by setting, for each coalition X of P~,:

(1) I f r w ( X ) > r q , XE WI,, and

(2) If rw(X) < rq, X¢ Wk, and

(3) If rw(JO = rq then, by strong rigidity, X is a row (in which case we set X s WI,), or is a column (and we set X¢ Wk).

More generally, we define any simple game G to be roughly weighted if it can be endowed with a rough weight function and a rough quota that


satisfy (1) and (2) above (replacing Wk with the W of G), with no com- ment about coalitions X satisfying rw(J0 =rq. In [M, (7.5.1)] the term "pseudo-threshold function" is used for the same concept. A game G = (P, W) is said to be monotonic provided that whenever Xe W and Y~ X, YE W. It is clear that any roughly weighted game with strictly positive weights is monotonic. It is common in game theory to require simple games to be monotonic, but this is less natural for the other fields mentioned.

To summarize, we have proved the following.

LEMMA 2.2. For each integer k ) 3 , the game [5 k on Mk & monotonic and roughly weighted, and the only coalitions weighted exactly rq are Mk's rows (which are winning) and columns (which are losing.)

3. TRADE ROBUSTNESS AND ASUMMABILITY

In 1961, Elgot [ Eli characterized threshold switching functions via the "asummability" property (the proof is also in [ M] ). Recently, the authors used a related property, "trade robustness," to characterize weighted simple games. (The proof in [TZ3] is more constructive and can be modified to yield a method for producing weights, as in [TZ4].) We are grateful to the referee, in response to a previous version of the current paper, for drawing our attention to the earlier work in threshold logic.

Suppose that X1, ]k'~2, Y1, and Y2 are coalitions satisfying the following (where ~ denotes "disjoint union"):

X1 =A1 ~ B1,

X z = A 2 ~ B 2,

Y1 = A 1 ~ B2,

] f2 = A2 ~ BI"

That is, X~ and X2 can, by exchanging (possibly empty) subsets B~ for B2, be transformed into Y1 and I12. We will say that the X~, X2 sequence has been transformed into the Y~, Y2 sequence via a 2-trade. (For the sake of intuition, the reader might prefer to think of B1 and B 2 as being disjoint; this does not affect possible transformations.) Similarly, a sequence of such 2-trades among the coalitions appearing in some k-sequence X of coalitions (possibly including repeats) and having the effect of eventually transforming it into another k-sequence ~/, is called a k-trade. (An entirely equivalent development could be founded instead on the idea of a one-way transfer, in which X~ = A 1 ~ B, X 2 = )(2, Y~ = A ~, and Y2 = A z ~ B; the


change would yield a modest gain in simplicity with, perhaps, a modest loss of motivation, particularly for the monotonic, k = 2 case.)

LEMMA 3.1. I f Y{' and c~ are both k-sequences of coalitions, then the following are equivalent:

(1) Some k-trade transforms Y" into q/.

(2) There exists partitions Xi= U {Zi, i l 1 <~j <.k} for each i<~k, such that Yi= U {Z,.j 1 1 <~i<<.k} for each j<.k (we allow some of the Z~,j to be empty.)

(3) For each q e P, I{il q~J;-i}l = I{il q~ I'-,}1 (this can be expressed in terms of the "degree sequence" of the vertices of a hypergraph.)

(4) I f the X~ and Ye are replaced by their respective characteristic vec- tors, then Z{xi ] 1 <~i<~k} = Z { y i l1 <<.i<<.k}.

Part two can also be phrased in terms of the existence of a matrix partition, [Z~.i], for which any two sets lying in a common row or column are required to be disjoint, such that Xe is the union of the ith row, and Y~ the union of the ith column, for each i ~< k. The proof of 3.1 is straightforward if the X~ are pairwise disjoint, but overlap makes the details of the general case a bit messy. Here is a sketch.

Proof (1) ~ (2) By induction on the number of 2-trades that effect the k-trade. Assume that a sequence o f j + 1 2-trades transforms 5f into ~/, that the first j of these alone transforms f into d#, and that the ( j + 1)th 2-trade is

Um =A1 v~ B1,

U n = A 2 w B 2 ,

Y m = A 1 ~ B2,

Yn = A2 ~ B1.

By induction, assume that [Zi, i ] is a matrix partition whose rows union to the X~ and whose columns union to the Ui. To modify [Zi, i ] so that its rows union to the X,. and its columns union to the Yt, perform the following switch for each p ~ BI: for some unique integer r, we have that p ~ Zr, m" We delete p from Zr, m and add p to Zr, , . After the analogous switch is done for each p ~ B2, the new matrix has the desired property.

The proof that (2) ~ (3), and that (3) and (4) are equivalent, is easy. For (3) ~ (1), assume that the equal degree sequence condition in (3) holds and define the quantity Q to be the sum, over all q ~ P, of

I{i I q E ~ . - Y,}I+ I{i] q~ Y,- xi} I.


To show that Q can be reduced to zero through a sequence of 2-trades, it suffices to show that there exists some single 2-trade that reduces Q's value while preserving the truth of the equal degree sequence condition. If Q > 0, simply identify some q, i, and j such that q e X i - Yi and q e Y j - X j and have the trade move q out of Xi and into Xj (here {q} is "traded" for the null set, a one-way transfer). This reduces Q's value by 2. |

Now we define a simple game G = (P, W) to be k-trade robust if it is impossible for a k-trade to transform a sequence of k winning coalitions into a sequence of k losing coalitions. If this definition is made cumulative (required to hold for each j ~<k) and rephrased in the language of threshold logic (using part 4 of the lemma above), then it becomes the definition of k asummable. We will use both terms, since it is convenient to have both cumulative and non-cumulative versions. We say G is trade robust, or asummable, if it is k-trade robust for every k ~> 2.

TI-mOI~M 3.2. [Elgot]. A simple game is weighted if and only i f it is asummable (trade robust).

The question of whether any bounded amount of trade robustness implies weightedness was settled by Winder in his 1962 Ph.D. thesis [Wi]. He produces, for each k/> 3, a game (actually, a switching function) that is k asummable, but not re(k) asummable (hence, not weighted). Here, rc is a function satisfying rc(k)>k and having a fairly complex description. Winder's proof is lengthly and has not appeared, except in his thesis. His result seems to leave open the question of whether k asummability ever implies j asummability for some j > k.

We provide a short proof that the properties of k asummability form a strict hierarchy.

THEOREM 3.3. For each integer k >~ 3, there exists a simple game that is k - 1 asummable, but not k asummable.

This is an immediate corollary of the following.

THEOREM 3.4. The magic square game [] k is j-trade robust i f and only i f j is a not a multiple o fk .

First, we prove the following 1emma.

LEMMA 3.5. Let G be any roughly weighted game, with rough weight function rw and rough quota rq. Then for any pair Y(, cy that witnesses a failure o f j-trade robustness, rw(Xi)= rw(Yi) = rq for each i <~ j.


Proof From the fact that ~Y arises from ~ via a j-trade, it follows immediately that Z{rw(X~)]i~j} = Z{rw( Yi) li <<.j} (see part (3) of 3.1.) But each X~ is winning, so each term in the first sum is at least rq. Similarly, each term in the second sum is at most rq, and the desired con- clusion follows. |

Proof of 3.4. First, we observe that []k fails to be k-trade robust, because there is a k-trade that transforms its (winning) rows into its (losing) columns. This trade may be thought of as the reflection of Mk about its main diagonal: the ruth row trades its nth element for the ruth element of the n th row. This trade can be copied i times (each row appears i times in f , etc.) so that [] k is not j-trade robust for j = k, 2k, 3k ....

Next, we show that the only failures of trade robustness are essentially the same as these multiples of the diagonal reflection. Assume that 2~, ~/ witness some failure of D~ to be j-trade robust for some j. By 3.3, above, each X; and Yi has rough weight equal to the magic sum p, so, by strong rigidity, each of these coalitions is either a row or a column. But each X~ is winning, so each must be a row, and each Y~ must similarly be a column. Now suppose that some row appears exactly r times (r >~ 1) in 5~. Then, in order for ~ to absorb the r occurrences of the sth member of this row among the coalitions in ~r, the sth column must appear exactly r times in Y/. Since this is true for each s, ~/consists of the k columns of Mk, each appearing exactly r times. So j is a multiple of k, as desired. (Also each row must appear exactly r times in X.) |

4. DESIRABILITY, ACYCLICITY, AND CYCLIC TRADES

A "desirability relation" represents a way to make precise the idea that a particular voting system may give one voter, or group of voters, more influence than another. For example, if G = (P, W) is a simple game, the individual desirability relation is defined by setting, for each q, r ~ P,

q < i r if there exists some coalition X c P - {q, r}

such that Xw {q} ¢ Wbut X u {r} e W.

This relation goes back at least to Isbell [ I ] , where the transitivity of < r is (essentially) established. Consequently, whenever < i has a cycle (a sequence q~<iq2<i"'" <lqn<rql) it has a 2-cycle, ql<iq2<iql. Weighted games have an acyclic <i , but so do such fairly simple non- weighted games as the rule once used to select rabbis in portions of eastern Europe (see [Pg3] ) or the 1982 procedure for amending the Canadian constitution (see [TZ5] or IT] ) .


Lapidot [ L ] introduced the simplest extension of ( i to a relation on coalitions: for each Y, Z c P we define

Y < L Z if there exists some coalition X c P - ( Y w Z )

such that X ~ Y ~ W but X u Z e W;

this is referred to as the group (or coalitional) desirability relation, although there are different relations that may have a competing claim for this name [TZ1] . The group desirability relation is always acyclic for a weighted game, but it need not be transitive (see IMP, Pgl, Pg2, Ei]). It detects non-weightedness more effectively than does <z; for example, the Canadian constitution game has a 2-cycle in <L [TZ5] .

Peleg studied <L and asked whether its acyclicity implies weightedness. Einy [Ei] constructed a monotonic counterexample that is strong (whenever a coalition X is losing, P - X is winning) but fails to have the property that whenever a coalition X is winning, P - X is losing (it fails to be proper). In his example, [P[ = 9 and winning coalitions have as few as three elements. This suggests that his example may be related to the magic square game [] 3, which shares all these properties. He also constructs a second example, with [P[ = 10, that is proper but not strong and observes (2.3 of [Ei ] ) that any monotonic game for which <L has no 2-cycles must be either strong or proper.

Peleg's new question (private communication) is:

Peleg's Question. If G is a monotonic constant sum (simultaneously strong and proper) game with an acyclic <L, must G be weighted?

We show, in the next section, that the answer is "no," but a very similar question has a positive answer; the nature of these answers suggests that the question is more interesting than it might first appear. Before starting the argument, it is important to note that our <L (also, < / ) is the "existential and strict" version (see [TZ1]) , rather than the " % " used in, for example, [Ei] . This % comes with its own strict version, which is slightly different from <L, and which rules out 2-cycles by definition. However, there is a direct translation between results involving the two versions (for example, <L is acyclic if and only if % is complete and its strict version is acyclic), and the questions we address are equivalent to those considered in [ Ei].

One reason for favoring the existential version of <L is that it allows us to replace acyclicity with a type of trade robustness that is equivalent. Recall that when a k-trade transforms the sequence ~f of coalitions into the sequence ~ , points originally in X~ may ultimately end up in any or all of the Yj. If we demand instead that points transferred out of X; end up only in Y~+I (or in Y1 if i = k) then the result is a cyclical pattern of transfer, as expressed more precisely in the following diagram:


y"

X I - - A s ~ B 1

X2 = A 2 ~ B 2 X3--A 3 ~ B 3

Xk=Ak ~ B~

Y I = A 1 ~ B~ Y2 = A2 ~ B1 Y 3 - ~ A 3 ~) B 2

Yk = Ak ~ B~_~

In this case, we say that a k-cycle trade has occurred. Note that we have characterized such a trade in terms analogous to the "matrix partition" clause (2) in Lemma 3.1 (in effect, a cycle trade requires that certain of the sets in the matrix [Z/ , : ] be empty.) It is worth mentioning for what follows that neither of the other clauses of 3.1 appear to have simple restrictions equivalent to k-cycle trade transformations.

Remark 4.1. In fact, even the idea of sequential trades between pairs of coalitions must be approached with caution in this case, for while a k-cycle trade may sometimes be achievable via a sequence of transfers in which X1 transfers some points to X2, then the newly enlarged X2 transfers some points to X 3, etc., until the enlarged X k transfers some points to the (shrunken) Z1, not every such sequential transfer corresponds to a k-cycle trade (because this sequential transfer allows X1 to transfer points to ~3 via X 2, while a cyclic trade only allows this to happen for points that are elements of X2.)

A simple game is k-cycle trade robust if no k-cycle trade transforms a sequence of winning coalitions into a sequence of losing coalitions, and is cycle trade robust if it is k-cycle trade robust for all integers k >~ 2.

THEOREM 4.2. I f G is any simple game, then

(1) G is k-cycle trade robust i f and only i f <L has no k-cycles, and cycle trade robust i f and only i f < r is acyclic.

(2) G is 2-trade robust i f and only i f it is 2-cycle trade robust.

The proof is left to the reader.

How can one tinker with a given simple game G to make it be strong or proper? We next consider two ways to build games from G and continue the approach taken in [Ei] by exploring key properties that are preserved by these constructions. Let G = (P, W) be a simple game. The dual game, G u, is defined by Ga= (P, Wd), where X ~ W a if and only if P - X q ~ W. For a monotonic game G, we may think of the dual in terms of voting; the winning coalitions of G a are the one that can block passage of a bill in G. See [S] for more on the dual.


THEOREM 4.3. Let G be any simple game. Then

(1) Gad= G.

(2) G is monotonic i f and only if G a is monotonic.

(3) G is strong if and only if G a is proper (hence G is proper if and only i f G a is strong) and, i f G is constant sum, then G = G a.

(4) G is roughly weighted via rw, rq if and only i f G a is roughly weighted via rw, X - r q (where ~ is the sum o f the rough weights o f all elements o f P), and the corresponding equivalence holds for true weightedness in place o f rough weightedness.

(5) For every integer k>~2, G is k-trade robust i f and only i f G a is k-trade robust.

Proof Parts (1)-(4) are straightforward. Recall that for k = 2 , trades and cycle trades are the same, so a consequence of (5) is that whenever <L has no 2-cycles in G, it has no 2-cycles in Ga; this special case of (5) appears in [Ei], in the form, "if ~< is complete in G, it is complete in Gd. ''

The proof of (5) and the reason that the preservation argument does not extend to cycle trades are of interest. Suppose that the pair Y', o~ of k-sequences of coalitions witness the failure of some form of trade robustness in G a, so that each X i ~ W a, each Yi ~ La = ~ ( P ) - W a, and some form of trading transforms one sequence into the other. Then the sequences ~, Y" (where ~ denotes the sequence of complements, P - Zi, of sets in ~ ) possibly witness a parallel failure of the same type of robustness in G, since, by the definition of G a, each P - Y~ is winning in G and each P - Xi is losing. To complete this line of reasoning would require showing that the same form of trading that transforms ~ into Yg can transform ~ into ~.

This last step seems doable, since any single transfer of points B from one coalition to another corresponds to a transfer of the same points from the complement of the second coalition to the complement of the first:

If X1 = A ~ B, )(2 = C, Y1 = A, and Y2 = C ~ B,

then Y I = A ~ B ~ B , Y 2 = C ~ B , X I = A ~ B , a n d

X2=C~Bt~B.

Thus any sequence of transfers that transforms 5f into ~ corresponds to a "complementary" sequence of transfers transforming ~ into ~, and this establishes (5). |

However, if the original sequence created a cycle trade, the transform of the complements need not be obtainable via a cycle trade (even though the complementary sequence of transfers may follow in a cycle--see Remark 4.1.) This begins to explain why the preservation of cycle trade robustness is more subtle; we take it up in the next section.


If G = (P, W) is a monotonic simple game which is either strong but not proper or proper but not strong, then there is simple construction due to Von Neumann and Morgenstern ( [VM]; also discussed in [ Ei]). The constant sum extension, G C, of G is defined by G ~ = (pc, We), where * is chosen to be any object satisfying * $P , p c = P w {*}, and in the case that G is strong and not proper, W ~ is defined by

X ~ W c if and only if either • ~ X and X e W,

or • $ X a n d P - X ( ~ W.

If G is proper and not strong, W c is defined similarly, with the underlined sections exchanged; for the remainder of the discussion we will work, with no loss of generality, with the first (italicized) case. Intuitively, G c is constructed by gluing together G, which is G~'s "reduced game" (see, for example, [E i ] ) and G d, which is G~'s "subgame" (or vice versa).

THEOREM 4.4. I f G = (P, W) is any monotonic simple game that is strong but not proper (or vice versa), then

(1) I f G is monotonic and strong but not proper (or vice versa) then G c is monotonic and constant sum (i.e., both strong and proper),

(2) G is roughly weighted i f and only if G c is roughly weighted (with uniform methods, below, for modifying rw and rq), and the corresponding equivalence holds for true weightedness,

(3) For every k >12, G is k-trade robust i f and only i f G ~ is k-trade robust, and

(4) For k>~ 3, k-cycle trade robustness of G c implies k-cycle trade robustness of G and G ~. (The authors do not know whether simultaneous k-cycle trade robustness of G and of G a implies that of GL )

Proof Part (1) is standard. For (2), if G ~ is roughly weighted with function rw: pc__, R and rough quota rq, then it is easy to see that G is roughly weighted with weight function r w l P (the restriction of rw to domain P) and rough quota equal to r q - r w ( * ) . If G is roughly weighted via rw and rq, let ~ denote the sum of rw's values over all q ~ P. It is straightforward to check that if we extend the rw function by assigning rw( . ) = Z - 2 r q and use a rough quota of 2 ; - r q , this function and value witness that G ~ is roughly weighted. If we wish to establish the same result for true weightedness we need only choose some quota which is not equal to the total weight of any coalition (this can be done) and use the same argument.

For the proof of (3), we first introduce some notation: whenever X c pc we will write _~ to denote J f n P, so that either X = X'w { ,} or X = 2. Now suppose that X, ~J witnesses a failure of k-trade robustness of G c. By the

582a/71/1-6


degree sequence criterion, Lemma 3.1(4), the number of values of i for which X~= X-~w {,} equals the number for which Yi= ~ w {*}, so without loss of generality we may assume that 5f and q¢ are as in the first array below:

Winning in G C

Xlw {*} :

2m+1

Losing in G C

Y1 k-3 { * }

Here, 21 ..... Xrn E W, and P - 2m +1,-.., P -- Xk • W~ and Y1 ..... rrn ¢ W~ and P - Ym +1,---, P - - Yk e W. We claim that the k-sequences depicted in the second array, below, show that G is not k-trade robust:

Winning in G

21

P - ~'m+l

p- ,~

Losing in G

rl

% P - X m + I

P-2k

To check the degrees, choose any q s P . Suppose that r = ]{i<~m[q~}[, s=l{i>/m+llq~P-Iri} , t=[{i~m]q~Yi}[, and u = [{ i ~> m + 1 ] q ~ P - 2 i}. Then we must show r + s = t + u. But from the first array, we know that [ { i ~ < k [ q ~ } ] = [ { i ~ < k [ q s Y i } ] . Thus r + ( ( k - m ) - u ) = t + ( ( k - m ) - s ) , so r+s=t+u, as desired.

This completes the proof of (3), as the converse is trivial. As for (4), it is immediate that a failure of k cycle trade robustness in G or G a induces a like failure in G c. |

5. T w o ANSWERS TO PELEG'S QUESTION

A tempting approach to Peleg's question is to start with a game that is cycle trade robust but not weighted and take the constant sum extension. It turns out, however, that the simpler examples of such games have cycles


in the dual (an example appears in [ Ei], where this fact is exploited to conclude that a certain game is non-weighted). Apparantly, such examples fail to be trade robust in a particularly strong way. In particular, this happens with the magic square games, so we need to modify [] k before taking the extension.

Recall that one equivalent definition of "k-trade" (see 3.1(2)) entails a matrix [ Zi.i ] of subsets of P satisfying that any two of the Z;,j are disjoint, provided that they lie in some common row or column. We will say that such a matrix is totally disjoint if every pair of Zi, y are disjoint, without proviso. If some totally disjoint matrix [Zi, j ] witnesses the failure of the game G = (P, W) to be k-trade robust (because the union of each row is winning, and the union of each column is losing) then we will say that it witnesses a totally disjoint failure of k-trade robustness. If G has no such totally disjoint failures for any k, then G will be said to be weakly weighted. In the k-trade via the diagonal flip of Mk, the Zi.j are distinct, single- element sets, so [] k has a totally disjoint failure of k-trade robustness.

TH~O~M 5.1. I f the simple game G has a totally disjoint failure of k-trade robustness then G a has a k-cycle in < L.

An immediate consequence of 5.1 is that a variant of Peleg's question has a positive answer.

COROLLARY 5.2. I f G is a constant sum simple game with an acyclic <L, then G is weakly weighted.

Proof of 5.2. If G were not weakly weighted, then G d would have a cycle in <L' As G is constant sum, however, G = G d, so G would have a cycle in <L. |

This result helps explain why the original question is both difficult and interesting. Another consequence of 5.1 is the following.

COROLLARY 5.3. The game []~ is k - 1 asummable and has a k-cycle in <L.

Proof of 5.3. The asummability follows immediately from Theorem 3.4 and part (5) of Theorem 4.3, while the k-cycle in <z follows from 5.1. |

COROLLARY 5.4. For no k does k asummability imply cycle trade robustness.

To complete this chain of results, we note the following.

COROLLARY 5.5. The game Vq k is cycle trade robust, strong and not proper (and, of course, monotonic and not weighted).


Proof of 5.5. Since [] k is roughly weighted with a rough quota rq < ½S and has sets X with rq < rw(X)< ½ Z, it follows that it is strong and not proper. Any failure of cycle trade robustness would have to convert Mk's rows into its columns (see the proof of 3.4). But the nature of a cycle trade (refer to the diagram in the definition) is such as to imply that Y~+ 1 ~ X~ w Xg+ l, and no column of a k x k matrix (k ~> 3) is contained in the union of two rows. |

COROLLARY 5.6. Cycle trade robustness does not imply 3-trade robustness.

The proof of 5.1 follows easily from the following lemma, which came as a surprise to the authors.

LEMMA 5.7. Let [a id] be a k x k matrix of distinct points. Then there is a k-cycle trade that transforms the k "column complements" of the matrix into the k "row complements."

Proof A "column complement" is the set C of all the ai, y not in the column C. As the ai,; are distinct, C is equal to the union of the other k - 1 columns (similarly for a row complement, R). We first provide a picture proof for the case k = 4:

O • • • 0 O O 0

o ~m • • • • • •

o • • • • • • •

o • • • • • • •

• o • • • • • •

• 0 • • 0 0 O O C2 R2

• o ~m • • • • •

• o • • • • • •

• • 0 • • • • •

• • 0 • • • • • C3 • • O • 0 0 0 0 Ks

• • 0 ~m • • • •

~m • • Q • • • •

• • • o • • • • c~ • • • o • • • • R4

• • • 0 0 0 0 0

SIMPLE GAMES A N D M A G I C SQUARES 83

Each column complement is depicted via an array, in which the elements of B~ (the part of C~ that gets transferred to the next set down, becoming part of/~i+ 1--see the diagram preceding 4.1) are represented as O's, the elements of Ag (the part of Ci that does not transfer and becomes part of R~) are represented as • ' s , and the points not in C; are represented as ©'s. The same method is used for the row complements--for example, since the 2, 1 position of/~4 = A4 k) B 3 is filled above with a • , a2, 1 is an element of B 3 .

Note that the small symbol %" sits, in the case of C4, at the upper left- hand corner of a simple configuration: a 4 x 3 rectangle divided into an upper triangle of six • ' s surmounting a lower triangle of six • ' s . The same pattern holds for the other Ci, provided you view the matrix as wrapping around (left edge pasted to right, and top to bottom) as in the edge identification diagram for a torus. As we move from C1 to C2 to C3, etc., the "c" shifts one unit along the diagonal. A similar pattern describes the partition of t h e / ~ into Ai and B~+I, and the same idea can now be used to construct versions of this picture for other k values. |

Proof of 5.1. Assume that [Zg,: ] is a matrix of sets witnessing a totally disjoint failure of the simple game G = (P, W) to be k-trade robust. We will use the word "row" (or "column") ambiguously, to refer to a row (or column) of [Z~,:] or to the union of all the sets in the row (or column) of this matrix. Let S denote the set

P - U {z,,j I

Since [Z~.,j ] is totally disjoint, each column complement is the union of S with the other k - 1 columns (similarly for row complements); note that this does require total disjointness. Hence, in G a the union of S with any k - 1 columns is winning, while the union of S with any k - 1 rows is losing.

Now we can interpret the diagram from the preceding proof as repre- senting a failure of k-cycle trade robustness for G a. Each location in the array now represents a set Zi, j rather than a single point, B; becomes the union of the Zi,]s having positions marked by • ' s and Ae becomes the union of S with the Z~,fs having positions marked by • ' s . The total disjointness of [Z~,j] guarantees that A~ is disjoint from both B~ and Bi+l, as required in a cycle trade. |

We now turn to the task of modifying U] k so as to force a large measure of overlap in any matrix witnessing a failure of k-trade robustness; the resulting game will be weakly weighted. Let [k] 2 denote {( i , j )] l<~i , j<.k}. Given (i,j) and (m,n)~[k] 2, we will write (i,j) # (m, n) provided that both iCm and j ¢ n , so the (i,j) and (m, n)


entries of a matrix lie neither in a common row, nor in a common column. We will use D k × k to denote the collection of all two-element subsets { (i, j ) , (m, n)} of [ k ] 2 satisfying that (i,j) # (m, n). Now suppose that xeDk×k, x = {a, b}, a = (i,j), and b = (m, n). Let s~ be some very small positive real number, which we will refer to as a "sliver." We will subtract the amount sx from each of the entries M i j and Mm, n of the original k x k magic square Mk. The new game will now include one player with rough weight equal to s~, and two others whose rough weights are the reduced values of Mi, j and M . . . . respectively, so that either of the reduced entries must join with Sx to bring it back up to its original rough weight. This will force the sets Zi, i and Zm, n to share the element sx, destroying total disjointness. Of course, we need to do this carefully for each x e Dk×k.

Description of the "Slivered" Version, [] ~k of [] k

Choose an indexed collection 5 ~ = {s~ Ix e Dk×k} satisfying all the conditions below:

(1) Each sx is a positive real number.

(2) XSP < 0.1.

(3) The set 5 p is linearly independent over Q, so that no two linear combinations of elements of 5 p are equal unless they use precisely the same coefficients from Q.

For each ( i , j ) e [ k ] 2, let MT, j=Mi, j - Z { s ~ l ( i , j ) ex} , where Mi, j is the i ,j entry of our original magic square Mk, and let Z i j = {M,5 } u {s~ I ( i , j )ex}.

We are now ready to define [] ~k. We set

P~k= { M i j l ( i , j ) e [k ] 2} u {sx [xeDk×k},

and endow PS k with the identity function as a rough weight function and the original magic sum, p, of Mk as the rough quota. We refer to the Mi_ j as major elements and to the s x as slivers. We now define W~k by setting, for any coalition X c P~k, X e WS~ if rw(X) > p and X¢ W~k if rw(X) <p , so that [] 'k is roughly weighted. If r w ( X ) = p , then we set X e W ~ if X is a row of [Z; , j ] and X ¢ WSk if Xis a column of [Z; , j ] . As we show below, the slivering process preserves a key feature of Mk, so there is no other X to consider.

LEMMA 5.8. The matrix [ Zi, j ] is strongly rigid, in the sense that the only subsets X of PS~ satisfying S X = p are the rows and columns of [Zi, j].


Proof If X c W k satisfies Z X = p , let X = J~l k..)J~2, where XI contains only major elements and X2 contains only slivers. As S X 2 < 0.1, the dif- ference between ZX1 and p must be less than 0,1. Then by strong rigidity of Mk, the elements of X1 must be precisely the major elements arising from a row or a column. For example, assume that X~ consists of the first row of the Mi. j (any other example proceeds similarly.) Then

S X 2 = p - [M~,~ +M~,2 + . - . + M~,k].

Now if we let Srowl be the set of slivers appearing in row 1 of [Ze, j ] then we also have

SSrowl = p - [ M ~ + M~, 2 + .. . + M~,k], so that

SSrowl = SX2, from which the linear independence of the slivers over Q tells us that SrowX = X2. Hence X is the first row of [Z,, j] . |

LEMMA 5.9. The game •~k is roughly weighted, monotonic, strong and not proper, k - 1 asummable, not k-trade robust, cycle trade robust, and weakly weighted.

Proof All but the last part is clear, or proceeds as in the proof for [] k. For weak weightedness, it suffices to prove that any witness, [Be, j ] , to a failure of k-trade robustness is not totally disjoint. Each row of such a [Bu] must have rough weight of exactly p (Lemma 3.5) and must be winning, so its union must equal the union of some row of [Zed], with similar comments for the columns of [B i j ] . But once some complete row of Mm, n appears as a subset of some row of [Bid ], each column of the Mi7 j must appear as a subset of a column of [Be, j ] , and so each column of [Zi, j ] appears as a subset of some column of [Be, j ] . This prevents [Be, j ] from being totally disjoint. |

THEOREM 5.10. Let [] Sk be the slivered k x k magic square game (k >~ 3) and D*x c be its constant sum extension. Then [~skc is monotonic, roughly weighted, constant sum, cycle trade robust ( <z is acyclic), and not k-trade robust (hence not weighted). Thus, acyclicity of <r does not imply weightedness, even for games that are monotonic and constant sum.

Proof All but acyclicity follow immediately from Theorem 4.4 and Lemma 5.9. It remains to show that [] SkC is cycle trade robust. Assume, by way of contradiction, that the diagram below represents a failure of cycle trade robustness for [] ~k C.


Winning in [] ~S

X1 =A1 w B 1

X2----A 2 U9 9 2

X 3 = A 3 ~ B3

X ~ = A ~ w B ~

Losing in []~k ~

Y1 =A1 w B~ Y2 = A2 ~ B1

Y 3 = A 3 ~ B 2

z

Y ~ = A ~ v B ~ 1

From the above, I7,.+ 1 c X; w X,.+ 1; we will contradict this containment. In the claim below, [Zi, j] refers to the specific matrix mentioned in DSk's construction•

CLAIM. (i) For each X i with , s X i , 2 i is some row, Rj ( o f [ Z i , ] ] ) .

(ii) For each X , with * C X , , X , is the complement o f some column.

(iii) For each Yi with * ~ Yi, Yi is some column, Cj.

(iv) For each Y , with • q~ Y, , Y , is the complement o f some column.

Proo f For the claim, recall that Lemma 3.5 guarantees that each Xi and Y/has rough weight exactly equal to [] skc's rough quota, and from the proof of 4.4(2), this equals

(X for [] ~) - (the rough quota for [] ~k) = ( kp - S S ) - p = ( k - 1 ) p - XS ,

while r w ( , ) = (X for [ ]~k)-2( the rough quota for [ ] ' k ) = ( k - 2 ) p - X S , where p is M~'s magic sum. Now consider (ii) of the claim. Since • ¢ X,, and rw(Xn ) = (k - 1 ) p - XS , rw(P~k - Xn) = (kp - £ S ) - ( (k - 1 ) p - S S ) =p, so by strong rigidity P S k - X , is a row or a column of [ Z U ]. But as X n is winning in [] ~k ~, P~k- X~ is losing in []~k, so P~k- Xn is a column of [ Z# ], from which X , is the complement of a column, as desired. Parts (i), (iii), and (iv) are established similarly.

We can be certain that • ¢Xi for at least one i, for otherwise we contradict [] Sk's cycle trade robustness.

Case 1. Assume that • is an element of none of the X~, Then • is an element of none of the Y~, so each X~ is as in part (ii) of the above claim, while each I1,. is as in part (iv). Then a piece of the trading array looks like:

Winning in []s S

xi=C. X i + 1 ~- Cj

Losing in []~k ~

Yi + 1 ~ R n


We will show that Yi+ ~ ¢ Xi w X~+ 2, contradicting the above. If m = j , let u<<.k with u ¢ n . Then M2, me Y~+I, but M2,,,¢X,.uX~+~. If m e j , let x = {(v, m), (w , j ) } , where v, w, and n are distinct. Then s~e Y~+I, but s , ¢ X~uJ(~+~. Note that this case establishes the cycle trade robustness of [] ~k a.

Case 2. Assume that * is an element of some, but not all, of the X~. Then a piece of the trading array may be assumed to look like:

Winning in [~%c

X i + 1 = Cj

Losing in DSk c

V/+l=&or C.w {*}

Case 2A. Assume that Y i+ l= /~ , in the above. Let u ~< k with u ¢ n and u ¢ m. Then M 2 j e Y~ + 1, but M ~ , j (~ Xi vo X i + i.

Case 2B. Assume that Y i + l = C , w { , } in the above. I f j = n , let u ~ k with uCm. Then M , d ~ Y i + 1, but M~,j(~X~voX~+ 1. If j C n , let u, v<<.k such that u, v, and m are all distinct, and set x = {(u, j ) , (v, n)}. Then SxS I5/+1, but s x C Z i u X i + ~. |

Note added in proof In his unpublished Ph.D. thesis (Syracuse, 1961), Gablemen constructed the game g] k for k = 3, 4, and 5, based on a different construction of what we call here strongly rigid magic squares.

REFERENCES

[A] W.S.A.VDREWS, "Magic Squares and Cubes," The Open Court, Chicago, 1908. [CH] V. CHVATAL AND P. L. HAMMER, Aggregation of inequalities in integer programming,

Ann. Discrete Math. 1 (1977), 145-162. [C] O. COGIS, Ferrers digraphs and threshold graphs, Discrete Math. 38 (1982), 33-46. [EiJ E. EINY, The desirability relation of simple games, Math. Soc. Sci. 10 (1985), 155-168. [EIJ C . C . ELGOT, "Truth Functions Realizable by Single Threshold Organs," AIEE

Conference Paper 60-1311 (Oct. 1960), revised Nov. 1960; Switching Circuit Theory and Logical Design, Sept. (1961), 341-345.

[F] P. FISHBURN, "The Theory of Social Choice," Princeton Univ. Press, Princeton, NJ, 1973.

[FH] S. FOLDES AND P. L. HAMMER, The Dilworth number of a graph, Ann. Discrete Math. 2 (1978), 211-219.

[Gn] A . J . GOLDMAN, Resolution and separation theorems for polyhedral convex sets, in "Linear Inequalities and Related Systems" (H. W. Kuhn and A. W. Tucker, Eds.), Princeton Univ. Press, Princeton, NJ, 1956.

[ Gc] M.C. GOLUMBIC, "Algorithmic Graph Theory and Perfect Graphs," Academic Press, New York, 1980.


[HIP]

[i]

[J]

[L]

[MP]

[M]

[ Pgl ]

[Pg2]

[Pg3]

[Pd] [R]

IS]

[T] [ TZ1 ]

[TZ2]

[TZ3]

[TZ4] [ TZ5 ]

[VM]

[We]

[Wi]

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Simple games and magic squares

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