Composition of Simple Games

Rupert FreemanDepartment of MathematicsThe University of Auckland

Supervisor: Arkadii Slinko

A dissertation submitted in partial fulfillment of the requirements for the degree of BSc(Hons)in Mathematics, The University of Auckland, 2012.

Abstract

In many voting situations, voting is not direct but via an indirect intermediate vote. In an earlypaper on the theory of simple games, Shapley [7] introduced a notion of compound simplegames, as a way of putting games together to make bigger games. Beimel, Tassa and Wein-reb [2] used a partial case of this composition to characterise ideal weighted threshold secretsharing access structures.

Very little work has been done on understanding such compositions. I study this operationusing game-theoretic techniques which allowed me to derive several essential properties. Themain results focus on conditions for compositions of complete or weighted games to also becomplete or weighted.

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Contents

Abstract 1

1 Introduction 3

2 Preliminaries 5

3 Definition and Properties of the Composition 8

4 Compositions of Complete Games 12

5 Composition of Weighted Majority Games 16

6 Conclusions 19

References 20

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Chapter 1

Introduction

In 2008, Beimel, Tassa and Weinreb [2] suggested a new kind of description of ideal weightedthreshold access structures, which was used by Farras and Padro, [5], in their 2010 papercharacterising these structures. The description revolved around a method of composition ofaccess structures. The idea is that any access structure is either irreducible, or can be split intoa main access structure together with a secondary one, which is substituted into the main one.If we can understand the irreducible structures and the way they can be put together, we canobtain a complete description of ideal weighted threshold access structures.

First however we must understand the operation of composition. For example, there is noguarantee that the composition of two weighted threshold access structures will be weighted.Thus we can not say which structures can be combined and in what ways, to produce a morecomplex weighted access structure. This dissertation answers this question and others via apurely game-theoretic approach.

To take the composition, G = G1◦G2 via a player g ∈ G1, is to say that winning coalitionsin G either take the form of a winning coalition in G1 which does not include player g, or acoalition in G1 that would be winning in G1 with g, together with a winning coalition of G2.Thus we can think of g as a collective member in the game G1.

We may consider games since every access structure arises as the winning coalitions ofsome simple game. Further, every weighted threshold access structure can be desribed as aweighted voting game. I begin by reminding the reader of basic properties of simple games,with emphasis on desirability relations, completeness, and weightedness of games.

In Chapter 3 I define and motivate the composition G ◦g H of two simple games G andH over a player g ∈ G. I include elementary properties and definitions which are used insubsequent sections, and examples of composition and decomposition of games.

Chpater 4 focuses on compositions of complete games. We are able to obtain several niceresults by restricting to this subclass of simple games. In particular, I show that the compositionof two complete games is complete if and only if the composition is performed over one of the

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4 CHAPTER 1. INTRODUCTION

least desirable players of the main game. I also prove that every complete game can be uniquelyexpressed as a composition of irreducible complete games.

Finally I derive a condition for when the composition of two weighted majority games willitself be weighted. It is not enough to simply compose over the weakest player. However,we are guaranteed that the composition will be weighted provided that the main game can bewritten so that the player we compose over can be given weight one in an integral system ofweights.

Chapter 2

Preliminaries

Although the main motivation for this research comes from secret sharing, this dissertation issolely game-theoretic. For a broad introduction to secret sharing, the reader is directed to [1]or [8].

Definition 1. A simple game is a pair G = (PG,WG), where PG is a set of players andWG ⊆ 2PG is a non-empty set of subsets (coalitions) which satisfy the monotonicity condition:

if X ∈ WG and X ⊆ Y, then Y ∈ WG

Coalitons from WG are called winning coalitions of G, the others are called losing coaltions.

For the rest of this dissertation a simple game will be referred to as just a game.

Definition 2. A coalition X ∈ WG is a minimal winning coalition if X\{g} is losing for allg ∈ X .

By monotonicity, WG is determined completely by the set WminG of minimal winning coali-tions. Thus, when describing a game it is necessary only to specify the players and the set ofminimal winning coalitions. We say that a game with a unique minimal winning coalition is anoligarchy.

Definition 3. A player which does not belong to any minimal winning coalitions is calleddummy. A player g such that {g} is winning is said to be a passer, and a player h such thatPG\{h} is losing is said to be a blocker. A player who is both a passer and a blocker is calleda dictator.

As a typical example of a game we have the United Nations Security Council, which con-tains 5 permanent members and 10 non-permanent members. Disregarding the possibility ofabstentions, a resolution requires the support of all 5 permanent members and at least 4 non-permanent members to pass. It is easy to see that the permanent members are blockers. Thereare no passers or dummies.

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6 CHAPTER 2. PRELIMINARIES

Definition 4. A game G is called a weighted majority game if there exist nonnegative weightsw1, ..., wn and a real number q, called the quota, such that

X ∈ WG ⇔�

j∈Xwj ≥ q

We denote the game [q;w1, ..., wn].

By convention, we usually arrange the weights so that w1 ≥ ... ≥ wn. I will often refer tothe players in a weighted majority game as simply 1, ..., n with associated weight wi for playeri, where no confusion will arise.

For a coalition X in a game G, I denote�

j∈X wj simply by wG(S), or w(S) if there is noambiguity.

The UN Security Council can be expressed as a weighted majority game. Give each per-manent member weight 7 and each non-permanent member weight 1. Then the game is written[39; 7, ..., 7, 1, ..., 1].

Definition 5. We say that a sequence of coalitions

(X1, ..., Xk;Y1, ..., Yk) (2.1)

is a trading transform if the players from coalitions X1, ..., Xk can be rearranged to formcoalitions Y1, ..., Yk.

Trading transforms have an important link to weighted majority games. For a proof of thefollowing two theorems, see [9].

Theorem 1. A game G is a weighted majority game if and only if there is no integer k suchthat there exists a trading transform (X1, ..., Xk;Y1, ..., Yk) such that Xi is winning for all1 ≤ i ≤ k and Yi is losing for all 1 ≤ i ≤ k.

This property is also called trade robustness. We say that any trading transform (2.1) withX1, ...,Kk all winning and Y1, ..., Yk all losing is a certificate of non-weightedness.

It is helpful to have a comparative notion of the desirability of players as coalition partners.This will be crucial to later discussions.

Definition 6. The desirability relation ”�” on a coalitional game G = (PG,WG) is definedas follows:

• i �G j if for all U ⊂ PG\{i, j}, U ∪ i ∈ WG =⇒ U ∪ j ∈ WG. We say that j is moredesirable than i.

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• i ∼G j if for all U ⊂ PG\{i, j}, U ∪ i ∈ WG ⇐⇒ U ∪ j ∈ WG. We say that j and iare equally desirable.

• i ≺G j if i � j but i ∼ j does not hold. We say j is strictly more desirable than i.

• If neither i � j nor j � i holds then we say that i and j are incomparable.

Continuing the example of the United Nations Security Council, it is obvious that everypermanent member is strictly more desirable than every non-permanent member. If a gameG has no two players incomparable, we say that G is complete. This property also has acharacterization in terms of trading transforms.

Theorem 2. A game G is complete if and only if there exists no certificate of non-weightednessof the form

(X ∪ {x}, Y ∪ {y};X ∪ {y}, Y ∪ {x}).

That is, for any two winning coalitions it is impossible to swap one player from each andhave both coalitions become losing. This property is called swap robustness. Such a tradingtransform, if it exists, is called a certificate of incompleteness.

For complete games, we may partition players into desirability classes. Two players i andj are in the same class if and only if i ∼ j.

Trade robustness implies swap robustness, so every weighted majority game is complete.This is apparent in another way; if wi ≤ wj then i � j.

Example 1. Consider the weighted majority game [4; 3, 2, 2, 2]. No one player can form awinning coalition on their own, but any two players constitute a winning coalition. Thereforeall players are equally desirable. Thus it is not generally true that if wi < wj then i ≺ j.

Definition 7. Let G = (PG,WG) be a game and let Y ⊆ PG. We define

Wsg = {X ⊆ Y C : X ∈ WG}Wrg = {X ⊆ Y C : X ∪ Y ∈ W}

Then we say that GY = (Y C ,Wsg) is the subgame induced by Y on G, and GY = (Y C ,Wrg)is the reduced game induced by Y on G.

So a subgame takes a subset of players and restricts the game to that subset, assuming thatall other players vote no. A reduced game takes a subset and restricts to that subset, supposingthat all other players vote yes.

Definition 8. Two games G = (PG,WG) and H = (PH ,WH) are isomorphic if there exists abijection f : PG → PH such that X ∈ WG if and only if f(X) ∈ WH . We write G ∼= H .

Letting G be the weighted majority game in the example above, and H = [4; 2, 2, 2, 2], wehave that G ∼= H . The choice of weights to represent a weighted majority game is not unique.

Chapter 3

Definition and Properties of theComposition

Suppose that a key member of the board of a large company resigns. It may be sensible toreplace him by a group of experts, under the following structure. The group of experts caststheir own vote to determine how they should cast their collective vote. A representative isthen able to attend the board meeting and cast the vote which was previously allocated to theretired member. The question is: is there a way to re-allocate votes so that the experts canvote individually at the board meeting, but the net result of voting is the same as if they hadundertaken the two step process outlined? As we will see, there is a restrictive condition on theboard’s voting structure which means that, in general, this is not possible.

I now formally define the composition.

Definition 9. Let G and H be two games such that PG and PH are disjoint. Define the com-position C = G ◦g H via some player g ∈ PG by PC = (PG\{g}) ∪ PH and

Wmin

C = {X ⊆ PC : X ∈ WminG } ∪ {X ⊆ PC : (X ∩ PG) ∪ {g} ∈ WminG and X ∩ PH ∈ WminH }

So we are substituting the game H into a single player g of G. Winning coalitions of G thatdo not contain g remain winning in C. Winning coalitions in G that contain g are winning inC only if g is replaced by a winning coalition from H . This is consistent with the hypotheticalscenario outlined above.

Observe that if |G| = 1, then G ◦g H ∼= H , and if |H| = 1 then G ◦g H ∼= G.

Definition 10. A game C is irreducible if there do not exist games G and H , with min(|PG|, |PH |) >1, such that C ∼= G ◦g H for some g ∈ PG. Otherwise we say C is reducible.

For a game C = G ◦g H , |PC | = |PG| + |PH | − 1. Thus if |PC | ≤ 2, C is irreducible,since if C is reducible we have |PG|+ |PH | ≤ 3, so one of the games contains only one player.

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Proposition 1. Let G1, G2, G3 be simple games defined on disjoint sets of players, and letg1 ∈ PG1 and g2 ∈ PG2 . Then

(G1 ◦g1 G2) ◦g2 G3 ∼= G1 ◦g1 (G2 ◦g2 G3)

Proof. Let X be a winning coalition in G1 ◦g1 (G2 ◦g2 G3). Then X has one of the followingforms:

(i) X1(ii) (X1\{g1}) ∪X2

(iii) (X1\{g1}) ∪ (X2\{g2}) ∪X3

Whrere X1, X2, X3 are winning coalitions in G1, G2 and G3 respectively. Each of these iswinning in (G1 ◦g1 G2) ◦g2 G3. The first two types describe winning coalitions in (G1 ◦g1 G2),while the third describes a winning coalition in (G1 ◦g1 G2) but for the absence of player g2,together with a winning coalition in G3. Note that these are the only forms that a winningcoalition in (G1 ◦g1 G2) ◦g3 G3 may take. So the two games are isomorphic.

Proposition 2. Let G and H be games defined on disjoint sets of players, and g ∈ PG. ThenC = G ◦g H has no dummies if and only if G and H have no dummies.

Proof. Suppose C has no dummies. Let g �= x ∈ PG. Then x ∈ PC , so x is in some minimalwinning coalition of C. Hence x is in some minimal winning coalition of G. Also note if gwere dummy in G, then all y ∈ PH would be dummy in C, so g is not dummy in G. Hence Gcontains no dummies. Let h ∈ H . Then h is a member of some minimal winning coalition inC. Therefore h is a member of some minimal winning coalition in H . So H has no dummies.Now suppose G and H have no dummies. Let c ∈ C. If c is in no minimal winning coalitionof C then it is in no minimal winning coalition of G or H . But G and H have no dummies. Soc is not dummy.

It would be nice to show that all games can be uniquely decomposed as a composition ofirreducibles. However, this is not the case. Consider C, with

PC = {1, 2, 3, 4, 5}W

min

C = {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5}

Then C ∼= G ◦g H , where G is defined on players {1, 2, g} and has minimal winning coalition{1, 2, g} and H is defined on players {3, 4, 5} and has minimal winning coalitions {3, 4}, {4, 5},and {3, 5}. G and H are irreducible.

We also have that C ∼= I ◦i J , where I is defined on players {3, 4, 5, i} with minimalwinning coalitions {3, 4, i}, {3, 5, i} and {4, 5, i}. J is defined on {1, 2} with minimal winning

10 CHAPTER 3. DEFINITION AND PROPERTIES OF THE COMPOSITION

coalition {1, 2}. I is not irreducible; indeed the game C has additional structure that I will notlook into, but note that J is irreducible, and J �∼= H . Thus the decomposition of C intoirreducible games is not unique.

As we will see in the next chapter, things are better if we consider only complete games.

Proposition 3. Let G be a game without dummies which contains a passing player, g. Then Gis reducible. Specifically, G = G1 ◦h G2 where G1 is a one out of two simple majority gamedefined on g and h, and G2 is defined on players of G\{g}, with minimal winning coalitionsall those minimal winning coalitions of G which do not contain g.

Proof. Minimal winning coalitions of G1 ◦h G2 are those consisting of g alone, or a minimalwinning coalition of G2, which are minimal winning coalitions of G not containing g. Theseare precisely the minimal winning coalitions of G.

Proposition 4. Let G be a game without dummies which contains a blocking player, g. ThenG is reducible. Specifically, G = G1 ◦h G2 where G1 is a two out of two simple majority gamedefined on g and h, and G2 is defined on players of G\{g} with minimal winning coalitions

W = {X\{g} : X ∈ WminG }

Proof. Minimal winning coalitions of G1 ◦h G2 are coalitions consisting of g together witha minimal winning coalition of G2. But minimal winning coalitions of G2 are precisely theminimal winning coalitions of G, minus player g. So minimal winning coalitions of G1 ◦h G2are precisely the minimal winning coalitions of G.

So if a game G contains passing or blocking players, we may decompose as in Proposition3 or Proposition 4 to remove one passer or blocker at a time, leaving a game with no passingor blocking players.

I modify the following three definitions from [2]

Definition 11. Let G be a game. Let Y,X ⊂ PG be two distinct sets of players. Then we definethe restriction of G induced by Y on X , GX,Y := (G(X∪Y )C )Y . That is, the game on playersfrom X with minimal winning coalitions

{Z ⊆ X : Y ∪ Z ∈ WminG }

.

Definition 12. Let G be a game and let S ⊂ PG. Given Y ⊆ S, if Y is not winning but SC ∪Yis winning, then Y is an S-cooperative set.

Definition 13. Let G be a game with no dummies, passers or blockers and let ∅ �= S ⊂ PGsuch that |S| ≥ 2. If for any two S-cooperative sets Y1, Y2 ⊆ S, GSC ,Y1 = GSC ,Y2 , then S isa strong set of players.

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Theorem 3. Let G be a game without passers, blockers or dummies. Then G is a compositionof two games, each defined on a smaller set of players, if and only if there exists S ⊂ PG suchthat S is a strong set of players. In particular, G = G1 ◦g G2, where G1 is defined on S plusan extra player, g, and G2 is defined on SC .

Proof. ⇐: Suppose S ⊂ PG is a strong set. Define G1 = (S ∪ {g},WG1), such that WminG1 ={Y ∪ {g} : Y is an S-cooperative set} ∪ {X ⊆ S : X ∈ Wmin

G} . Let G2 = GSC ,Y for some

S-cooperative set Y . All choices of Y are equivalent since S is a strong set.⇒: Suppose G is of the form G1 ◦g G2. Then it is obvious that G1\{g} is a strong set as

long as |G1\{g}| ≥ 2. Suppose that |G1\{g}| = 1. But then |PG1 | = 2, so each player in G1will be either a dummy, passer or blocker. Thus, writing PG1 = {g, h}, we have that h will beone of a dummy, passer or blocker in G, which contradicts the condition of the theorem.

Chapter 4

Compositions of Complete Games

In this chapter I consider games of the form C = G ◦g H , where g ∈ G and G and Hare complete. I will show that C is complete if and only if g is a member of the weakestdesirability class of G. We will also see that the class of complete games may be viewed as asemigroup under composition, and that an arbitrary complete game may be uniquely expressedas a composition of irreducible complete games.

First I consider the simple case where G is an oligarchy.

Proposition 5. Let G and H be complete games, and g ∈ G. If G is an oligarchy, then G◦g His complete.

Proof. Suppose G is an oligarchy. Let X ⊆ PG be the unique minimal winning coalition. Ifg �∈ X then g is dummy, so h is dummy in G ◦g H for all h ∈ H . So G ◦g H is completesince it is just G with some extra dummy players. So suppose g is not dummy in G ie. g ∈ X .Then all winning coalitions in G ◦g H take the form X\{g} ∪ Y for some Y ∈ WH . Let Y1and Y2 be arbitrary winning coalitions in H , and let Z1 and Z2 be obtained from Y1 and Y2 byswapping a pair of players. Since H is complete, note that either Z1 or Z2 is winning in H . Soany certificate of incompleteness for G ◦g H would take the form:

(X\{g} ∪ Y1, X\{g} ∪ Y1;X\{g} ∪ Z1, X\{g} ∪ Z2)

But Z1 or Z2 is winning in H , so one of the right hand side coalitions is winning in G◦gH .Hence no such certificate of incompleteness exists.

The case where H is an oligarchy cannot be summarised so easily, and I will not completelyresolve it. I now establish the first main result of this chapter.

Lemma 1. Suppose G and H are complete games without dummies, and suppose that neitheris an oligarchy. Let g ∈ PG and suppose that H contains at least one player who is not apasser, say v . If C := G ◦g H is complete, then g is a member of the weakest desirability classof G.

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Proof. Suppose there exists g� ∈ G such that g� ≺ g. Then there exist minimal winningcoalitions U1 and U2 for G: U1 containing g but not g� such that (U1\{g} ∪ {g�}) is losing,and U2 containing g�, which may or may not contain g. We are guaranteed the existence of U2since g� is not a dummy. Let V1 and V2 be two distinct minimal winning coalitions for H , withv ∈ V1 but v �∈ V2. Such coalitions must exist since v is not a dummy or a passer in H . Fromthis, we know that U1\{g} ∪ {g�} ∪ V1\{v} is losing in C, since it is a losing coalition in Gtogether with a losing coalition in H . Suppose g ∈ U2. Then U2\{g�, g} ∪ V2 ∪ {v} is losingin C since U2\{g�} is losing in G. So we have the certificate of incompleteness for C:

(U1\{g} ∪ V1, U2\{g} ∪ V2; U1\{g} ∪ {g�} ∪ V1\{v}, U2\{g�, g} ∪ V2 ∪ {v})

Now suppose g �∈ U2. Then U1\{g} ∪ {g�} ∪ V1\{v} is losing in C by the same reasoning asabove. And U2\{g�}∪{v} is losing in C since U2\{g�} is losing in G. This gives the certificateof incompleteness for C:

(U1\{g} ∪ V1, U2; U1\{g} ∪ {g�} ∪ V1\{v}, U2\{g�} ∪ {v})

Lemma 2. Suppose G and H are complete games. If g ∈ PG is a member of the weakestdesirability class of G, then G ◦g H is complete.

Proof. Suppose g is a member of the weakest desirability class. If g is dummy then G ◦g His clearly complete, so suppose g is not dummy. Denote two winning coalitions in G ◦g H asW1 = X1 ∪Y1 and W2 = X2 ∪Y2, where Xi ∪ {g} is winning in G, and Yi is minimal winingin H , or empty if Xi is winning in G1. If we swap a member of X1 with a member of X2,then one of W1 or W2 is still winning, by the completeness of G. If we swap a member of X1,say x for a member of Y2, say y, then we have that X2 ∪ {x} is winning in G since g � x,so X2 ∪ {x} ∪ Y2\{y} is winning in G ◦g H (or vice-versa, swapping a member of X2 fora member of Y1). Swapping a member of Y1 for a member of Y2 (if both are non-empty anddistinct) likewise leaves one coalition winning, by the completeness of H .

Combining Lemma 1 and Lemma 2, we get the following theorem.

Theorem 4. Let G and H be complete games without dummies and suppose that neither is anoligarchy. Suppose further that H contains at least one player who is not a passer. Let g ∈ G.Then G ◦g H is complete if and only if g is a member of the weakest desirability class of G.

Proposition 6. Let C = G ◦g H be complete. Then G and H are complete. Further, we havethe following desirability relations in C:

(i) h �C g� for all g� ∈ G\{g}, h ∈ H and h ≺C g� if h is not a passer in H .(ii) g1 �C g2 if and only if g1 �G g2

14 CHAPTER 4. COMPOSITIONS OF COMPLETE GAMES

(iii) h1 �C h2 if and only if h1 �H h2Proof. That G and H are complete is obvious. To prove (i), let X be a minimal winningcoalition in G◦gH that contains h. Then X = Y ∪Z for some Y such that Y ∪{g} is minimalwinning in G, and Z minimal winning in H such that h ∈ Z. Now consider replacing h byg� in X . We have X\{h} ∪ {g�} = Y ∪ {g�} ∪ Z\{h}. But since C is complete, we know

g �G g�. So Y ∪g� is winning in G. Thus X\{h}∪{g�} is winning in C. So h �C g�. Supposeh is not a passer in H . Take a coalition U such that U is minimal winning in G, and g� ∈ U .Then U is winning in C. But U\{g�} ∪ h is losing, since h is not a passer in H . So g� �∼C h,thus h ≺C g�. This proves (i). Parts (ii) and (iii) are obvious.

Lemma 3. Let X be a reducible complete simple game. Suppose that X ∼= G1 ◦g1 H1 ∼=G2 ◦g2 H2, where G1 and G2 are irreducible. Then G1 ∼= G2.Proof. We may suppose without loss of generality that G1 contains less or equal players thanG2.

Suppose that G1\{g1} ⊂ G2. I claim that G2 is then reducible as G1 ◦g1 Y , for some Y .It suffices to show that G1\{g1} is a strong set in G2. Let K1,K2 be (G1\{g1})-cooperativesets in G2 and let Y = G2\(G1\{g1}). Then K1 ∪ Y and K2 ∪ Y are winning in G2. HenceK1 ∪H1 and K2 ∪H1 are winning in X , so K1 and K2 are G1\{g1}-cooperative sets in X .Then they induce the same restriction on X\(G1\{g1}). Now, suppose J ⊂ Y is such thatK1 ∪ J is winning in G2. Thus K1 ∪ J is winning in X , where if g2 ∈ J we replace it bya winning coalition of H2. Hence K2 ∪ J is winning in X , again replacing g2 by a winningcoalition of H2 is necesary. By the definition of the composition, this means that K2 ∪ J iswinning in G2, or else it would not be winning in X . By symmetry, if K2 ∪ J is winning inG2 then K1 ∪ J is winning in G2. Hence K1 and K2 induce the same restriction on G2. SoG1\{g1} is a strong set of G2, so G2 is reducible.

Now suppose that G1\{g1} �⊂ G2. Since X is complete, G1\{g1} must contain all playersfrom the first k desirability classes, for some k ∈ N, and some from the (k + 1)-th class, say asuch players. Similarly, G2\{g2} must contain all players from the first k desirability classes,and some from the (k + 1)-th class, say b such players, where a ≤ b and there is at least oneplayer from G1\{g1} not in G2\{g2}. If a = b, then G1 ∼= G2, since all players in the samedesirability class are exactly equivalent, and we are done. Suppose a < b. Then we have thatG1\{g} ∼= H , where H ⊂ G2\{g2} consists of all players from the first k desirability classesplus a players from the (k + 1)-th desirability class who are also in G2\{g2}. So H ∪ {g1} ∼=G1, hence X = H ∪ {g1} ◦g1 Z for some complete game Z. So by the above argument wehave that G2 ∼= (H ∪ {g1}) ◦g1 Y for some Y , since (H ∪ {g1})\{g1} = H ⊂ G2\{g2}. SoG2 is reducible, a contradiction.

Given a reducible simple game C, we can find a unique (up to isomorphism) G1 suchthat C ∼= G1 ◦g1 H for some g1 ∈ G1. Repeating this process as necessary to H , we haveC ∼= G1 ◦g1 G2 ◦g2 ... ◦gk Gk where Gi is irreducible for 1 ≤ i ≤ k.

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Theorem 5. Let G be the set of all complete games. Then G, equipped with the operation ofcomposition over a member of the weakest class, forms a semigroup with identity. Every G ∈ Gcan be expressed uniquely as a product of irreducible elements in this semigroup.

Proof. We know already that the composition is associative. We need only check that it is welldefined. Let G,H ∈ G and let g and g� both be members of the weakest desirability class ofG. Since g and g� are equivalent and interchangeable in any coalition, G ◦g H ∼= G ◦g� H , socomposition over a member of the weakest desirability class is well-defined. Define e = [1; 1],the game with only one player. Then for any complete game (or indeed any game) G, we have

G ◦g e ∼= Gand e ◦G ∼= G

Unique decomposition into irreducibles follows from the remarks preceding the statement ofthe theorem.

Chapter 5

Composition of Weighted MajorityGames

We have seen that the operation of composition behaves quite nicely in the context of completegames. Weighted majority games, however, form a much smaller subclass of simple games.This chapter is motivated by the question: under what conditions is the composition of twoweighted majority games itself weighted?

As in the case of complete games, we first consider the case where G is an oligarchy.

Proposition 7. Suppose G and H are weighted voting games. If G is an oligarchy then G◦gHis a weighted voting game.

Proof. Let U be the unique minimal winning coalition for G. If g �∈ U then G ◦g H has aunique minimal winning coalition, so is a weighted voting game. So suppose g ∈ U . Thenall minimal winning coalitions take the form U\{g} ∪ Vi where each Vi is a distinct minimalwinning coalition for H . Taking any collection of such minimal winning coalitions, the onlyplayers available to swap are those in the Vi’s. But since H is a weighted voting game, it isimpossible to make all such coalitions losing. So at least one minimal winning coalition forG ◦g H remains winning.

Once again if H is an oligarchy then the situation is more difficult. This will be evidentsoon.

It should be noted that for G,H weighted, g ∈ G and C = G◦gH , it is necessary that g bea member of the weakest desirability class of G. Were this not the case, C would not even becomplete, by the results of the last chapter. However, with some experimentation it is quicklyapparent that this is not a sufficient condition.

Example 2. Let G = [7; 3, 3, 2, 2, 2, 2] and let H = [2; 1, 1, 1]. Say that the two players ofweight 3 in G are players of type A, the players of weight 2 in G are type B, and the players

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in H are type C. Let g ∈ G be one of the players of type B. Then we have the certificate ofnonweightedness for G ◦g H:

(ABC2, ABC2;A2C,B2C3).

Each coalition of the type ABC2 is certainly winning in G ◦g H , since AB2 is winning in Gand we have replaced one of the players of type B by two of type C - a winning coalition inH . But notice that both coalitions on the right are losing in G ◦g H . A2C is losing since A2 islosing in G and C is losing in H . B2C3 is losing since C3 is winning in H , but B2∪{g} = B3is losing in G.

The proofs in this section require a different approach to the trading transform methodsused so far. Given any weighted majority game, we can easily choose the weights to be ra-tional numbers, and thus integers. In this work, I define a minimal integer representation of aweighted majority game to be a system of integral weights such that no other system of integralweights allocates a lower weight to the lowest weighted player. For example, any system ofintegral weights which allocates some player the weight 1 is a minimal system of weights. Theweighting of game G in the example above is minimal, since there is no way to represent Gby giving some player weight 1. This is not obvious, but it will follow as a consequence ofTheorem 6.

For the needs of this paper I define the minimum integer weight of a weighted voting gameG to be the weight of the lowest weighted player in a minimal integer representation. In theexample above, G has minimum integer weight 2 and H has minimum integer weight 1.

Definition 14. Let G be a weighted majority game. We say that G is homogeneous if thereexists a system of weights such that every minimal winning coalition has the same weight.

For example, G = [6; 4, 3, 1, 1, 1] is homogeneous, however it is not a homogeneous rep-resentation. A homogeneous representation of G is [5; 3, 2, 1, 1, 1]. Ostmann, [6], proved thatevery homogeneous game has minimum integer weight one. In light of this, I will say that agame with minimum integer weight one is almost homogeneous. So homogeneous games area subset of almost homogeneous games.

I now show that a composition C = G ◦g H of weighted majority games G and H isweighted if G is almost homogeneous, provided g is one of the least desirable players of G.

Definition 15. Let G be a game. Let B ⊂ PG. Define a subset A ⊂ PG\B to be useful to B ifthere exists L ⊂ B losing such that L ∪A is winning. Otherwise say that A is not useful to B.

Lemma 4. Let G = X ◦g Y and let U ⊂ Y . Suppose too that g ∈ PX is not dummy. Then Uis useful to X\{g} if and only if U is winning in Y .

Proof. Suppose U is winning in Y . Then, since g ∈ X is non-dummy, there exists a coalitionL ⊂ X such that L is losing in X but L∪ {g} is winning. Hence L is losing in G but L∪U is

18 CHAPTER 5. COMPOSITION OF WEIGHTED MAJORITY GAMES

winning in G. So U is useful to X\{g}. Now suppose U is losing in Y . Then, by the definitionof the composition, there is no losing coalition L of X such that L∪U is winning in G. HenceU is not useful to X\{g}.

Lemma 5. Let G be an almost homogeneous game and let [q;w1, w2, ..., wn] be a minimalinteger representation for G. That is, wn = 1. Let m ∈ N. Then Gmk := [mq − m +1;mw1,mw2, ...,mwn−1, k] ∼= G for all real numbers k such that 1 ≤ k ≤ m.

Proof. Let k ∈ R with 1 ≤ k ≤ m. Suppose S = {a1, ..., ai} ⊆ {1, .., n} is winning in G. Ifn �∈ S then wa1+...+wai ≥ q. Thus mwa1+...+mwai ≥ mq ≥ mq−m+1. Suppose n ∈ S,and without loss of generality write ai = n. Then wa1+...+wai = wa1+...+wai−1+1 ≥ q, somwa1+...+mwai−1+1 ≥ mq−m+1. But k ≥ 1, so mwa1+...+mwai−1+k ≥ mq−m+1.So in either case, S is winning in Gm

k.

Next suppose S is winning in Gmk

, and that n �∈ S. Then mwa1+...+mwai ≥ mq−m+1.Dividing by m gives wa1 + ...+ wai ≥ q − 1 + 1m . And since all weights are integers, this isequivalent to wa1 + ...+wai ≥ q− 1+ 1 = q, so S is winning in G. Now suppose n ∈ S, andwithout loss of generality that ai = n, so wai = 1. Then mwa1+...+mwai−1+k ≥ mq−m+1.As before, division by m gives wa1 + ... + wai−1 +

k

m≥ q. Since k

m≤ 1 = wn, we have

wa1 + ...+ wai ≥ q, so S is winning in G.

Theorem 6. Let G and H be weighted majority games and suppose that G is almost homo-geneous. Let g ∈ G be a member of the weakest desirability class. Then C = G ◦g H is aweighted majority game.

Proof. Take a representation of G such that wg = 1 and all weights are integer. Since g is amember of the weakest desirability class, we know that such a weighting exists. Set the quotain H to be 1, and scale the weights accordingly. We don’t need the weights to be integer. Let Wbe the sum of all player weights in H under this weighting. Now consider the game G�W+1�

W,

as defined in Lemma 5. Now we may substitute the game H , with the new system of weights,for player g directly into G�W+1�

W. Then any losing coalition in H has weight less than 1, so

will not be useful to G\{g}, but any winning coalition has weight between 1 and W , so willexert exactly the same influence as player g alone, by Lemma 5.

From Theorem 6 it is clear that the game G from Example 2 is not almost homogeneous.If there were a way to give some player weight 1 in an integral system of weights, then G ◦g Hwould be weighted, which the example showed it is not. So we know that G has minimuminteger weight 2.

It remains an open question to prove the converse of Theorem 6. That is, if G and H areweighted majority games with g ∈ G a member of the weakest class, and G ◦g H is weighted,then is G almost homogeneous?

Chapter 6

Conclusions

I have shown three main results. Firstly, using a simple trading transform argument, that acomposition of complete games is complete if and only if the composition is made via theweakest player in the main game. By showing associativity of the composition, I next provedthat complete games form a semigroup, and further that each element of this semigroup may berepresented uniquely as a composition of irreducible elements. In the final section I consideredproperties of weighted majority games to show that a composition of two weighted games willbe weighted if the first game has lowest integer weight one. The next question is whether wecan formulate a converse to this statement.

Likewise, it is unclear whether these results can assist in research on homogeneous games.Certainly, one place to start would be to ascertain if the set of homogeneous games is indeedclosed under the operation of composition. Another question to be answered is to properlyinvestigate the decomposition of arbitrary games.

This work also suggests a link between compositions of weighted games and the morewell-researched area of finding minimal integer representations for weighted majority games.For example, compositions can be used to show that certain games do not have a minimuminteger weight of one. It may be possible to extend these results.

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References

[1] Beimel, A., Secret-Sharing Schemes: A Survey, Ben-Gurion University, Beer-Sheva, Is-rael.

[2] Beimel, A., Tassa, T., Weinreb, E., Characterizing ideal weighted threshold secret shar-ing, SIAM J. Discrete Math, 22(1), 360-397, 2008.

[3] Carreras, F., and Freixas, J. Complete simple games, Mathematical Social Sciences, Vol-ume 32, Issue 2 1996, pp139-155.

[4] de Keijzer, B. A Survey on the Computation of Power Indices, Delft University of Tech-nology

[5] Farras, O., and Padro, C. Ideal hierarchical secret sharing schemes, Theory of Cryptog-raphy Vol. 5978, pp219-236, 2010.

[6] Ostmann, A., On the Minimal Representation of Homogeneous Games, InternationalJournal of Game Theory, Volume 16, Issue 1; pp69-81, 1984.

[7] Shapley, L. S., Simple Games: An Outline of the Descriptive Theory, Behav. Sci., 1962Jan; pp59-66.

[8] Stinson, R., An explication of secret sharing schemes, Designs, Codes and Cryptography,Volume 2, Issue 4 1992, pp357-390.

[9] Taylor, A., and Zwicker, W., Simple Games: Desirability Relations, Trading, Pseu-doweightings, Princeton University Press, 1999.

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