Computing the nucleolus of cyclic permutation gamesTamás Solymosi ∗

Budapest University of Economic Sciences, Hungary

T. E. S. Raghavan †

University of Illinois at Chicago, USA

Stef Tijs ‡

Tilburg University, The Netherlands

November 6, 2001

Abstract The question of `fairly' allocating costs or bene�ts among the participants of a

joint enterprise is frequently answered by determining the nucleolus of a related cooperative

game. We discuss this issue in connection with multi-person decision situations in which

�nding the best course of joint action for a group of participants can be modeled by the

classic assignment optimization problem. The related transferable utility cooperative games

are called permutation games. We consider a large and important subclass, and show how

the nucleolus of an n-player cyclic permutation game can be computed by a known O(n4)

algorithm directly from the underlying data. We also demonstrate that this approach might

not work if for the group of all participants the optimal course of action is not given by a

permutation consisting of a single cycle.

Keywords: Game theory, Combinatorial optimization games, Nucleolus, Computation.

∗Department of Operations Research, Budapest University of Economic Sciences and Public Administration,1828 Budapest, Pf. 489, Hungary, e-mail: tamas.solymosi@opkut.bke.hu. Supported by the Foundation forthe Hungarian Higher Education and Research (AMFK) and by the Hungarian Scienti�c Research Fund(OTKA T030945). Corresponding author.

†Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, 851 S.Morgan, Chicago, IL 60607, USA, e-mail: ter@uic.edu. Supported by NSF Grant DMS 970-4951.

‡Department of Econometrics and Operations Research, Tilburg University, P.O. Box 90153, 5000 LETilburg, The Netherlands, e-mail: tijs@kub.nl.

1

1 Introduction

In situations when individuals can mutually bene�t from working together, the question of

how to distribute the proceeds of cooperation among the participants is a fundamental one.

Without due incentives the individuals will not participate in the joint enterprise, and the

opportunity for bene�ting from cooperation will be lost for all parties. Cooperative games are

designed to model these problems and solutions are proposed to answer such questions. For

some applications and further reference in this matter see, for example, the papers (Lucas,

1981), (Young, 1985), (Tijs and Driessen, 1986), and (Legros, 1986), or the books (Driessen,

1988) and (Curiel, 1997).

To obtain a game theoretic model which summarizes the relevant information about the

situation, one has to quantify the bene�ts of cooperation for the possible coalitions of the

individuals. Having built the game, the general solution concepts for cooperative games can

be applied and the solutions be interpreted for the particular situation. For this purpose, the

Shapley-value and the nucleolus are the most popular solution concepts, probably because

both of them propose a unique way of dividing the bene�ts, furthermore, both proposed

distributions meet certain natural standards of equity and fairness. In this paper, we apply

the nucleolus concept to a class of cooperative games which are closely connected to the classic

assignment optimization problem.

Permutation games were introduced by Tijs et al. (1984). They proved that permutation

games � like many other types of cooperative games related to mathematical programming

problems, cf. (Curiel, 1997) � are totally balanced. Another proof of this was given by Curiel

and Tijs (1986). They established relations between permutation games and assignment games

� a class of games introduced by Shapley and Shubik (1972) in connection with certain two-

sided matching market situations �, and used these relations in translating Shapley and

Shubik's results on the cores of assignment games to permutation games. This connection was

further explored by Quint (1996). He showed that the entire core of a permutation game can

be obtained from the core of a related assignment game.

In this paper we focus on the nucleolus of permutation games. The nucleolus is a point-

valued solution for transferable utility cooperative games suggested by Schmeidler (1969). It

is always in the core, whenever the core is nonempty. We establish further links between core-

payo�s of a special type of permutation games and the associated assignment games. The

2

subclass of the so-called cyclic permutation games is rich enough to contain not just those

permutation games in which the grand coalition is essential (i.e., its breaking up into smaller

coalitions is disadvantageous for all participants), but also all assignment games. We show

how the e�cient assignment nucleolus algorithm of Solymosi and Raghavan (1994) can be

used to compute the nucleolus of cyclic permutation games. We also demonstrate that this

approach does not work in the non-cyclic case.

The organization of the paper is as follows. We recall the necessary de�nitions in the

next section. Section 3 summarizes the results on the core of permutation games we need for

the nucleolus algorithm. Cyclic permutation games are introduced and their key structural

properties are discussed in section 4. Section 5 contains the characterization of the nucleolus

that allows its e�cient computation in the cyclic case. The examples in section 6 demonstrate

that in non-cyclic permutation games our approach does not work.

2 De�nitions and preliminaries

A transferable utility cooperative game on the nonempty �nite set N of players is de�ned by

a coalitional function V : 2N −→ R satisfying V (∅) = 0. The function V quanti�es the worth

of every coalition S ⊆ N . We shall denote by

P := {S ⊆ N : S 6= ∅, N}

the collection of proper coalitions.

A subcollection S of P is called balanced, if there are positive weights γS , S ∈ S, such that∑

S∈S,S3i γS = 1 for all i ∈ N . A balanced collection is said to be minimal if it contains no

proper balanced subcollection. It is well known (cf. (Shapley, 1967)) that a balanced collection

is minimal if and only if the system of balancing weights is unique.

The game (N,V ) is called superadditive, if S ∩ T = ∅ implies V (S ∪ T ) ≥ V (S) + V (T )

for all S, T ⊆ N ; balanced, if V (N) ≥ ∑S∈S γSV (S) for every minimal balanced collection

S with its unique weights γS , S ∈ S; and totally balanced, if every subgame (i.e., the game

obtained by restricting the player set to a coalition and the coalitional function to the power

set of that coalition) is balanced. Note that totally balanced games are superadditive.

Given a game (N, V ), a payo� allocation x ∈ RN is called e�cient, if x(N) = V (N);

individually rational, if xi = x({i}) ≥ V ({i}) for all i ∈ N ; coalitionally rational, if x(S) ≥

3

V (S) for all S ⊆ N ; where, by the standard notation, x(S) =∑

i∈S xi if S 6= ∅, and x(∅) = 0.

We denote by X (N,V ) the imputation set (i.e., the set of e�cient and individually rational

payo�s), and by C(N,V ) the core (i.e., the set of e�cient and coalitionally rational payo�s)

of the game (N, V ). It is well known (cf. (Shapley, 1967)) that for transferable utility games

balancedness and nonemptiness of the core are equivalent properties.

Given a game (N, V ), the excess e(S, x) := V (S) − x(S) is the usual measure of gain (or

loss if negative) to coalition S ⊆ N if its members depart from allocation x ∈ RN in order to

form their own coalition. Note that e(∅, x) = 0 for all x ∈ RN , and

C(N, V ) = {x ∈ RN : e(N, x) = 0, e(S, x) ≤ 0 ∀S ∈ P},

i.e., the core is the set of e�cient allocations which yield nonpositive excess for all coalitions.

The nucleolus was introduced by Schmeidler (1969), who proved that it is a single-element

subset of the imputation set for all games with a nonempty imputation set. We shall use the

following alternative de�nition that was given in (Maschler et al., 1979), because its iterative

nature excellently serves computational purposes. Since we deal only with balanced games in

this paper and for balanced games the nucleolus is in the core, we initiate the procedure with

the core (instead of the imputation set).

Let X0 := C(N, V ) and Σ0 := P. For r = 1, . . . , % de�ne recursively

εr := minx∈Xr−1 maxS∈Σr−1 e(S, x),

Xr := {x ∈ Xr−1 : maxS∈Σr−1 e(S, x) = εr},Σr := {S ∈ Σr−1 : minx∈Xr e(S, x) = maxx∈Xr e(S, x)},Σr := Σr−1 \ Σr,

(1)

where % is the �rst value of r for which Σr = ∅. The �nal set X% is the nucleolus N (N,V ) of

the balanced game (N,V ). We refer to the unique vector in X% as the nucleolus allocation.

On the player set N , games V and W are called strategically equivalent, if there exist α > 0

and b ∈ RN such that W (S) = αV (S) + b(S) for all S ⊆ N . Observe that both the core and

the nucleolus are covariant with transformations which yield strategically equivalent games.

More precisely, if the set F equals the core or the nucleolus of the game V , then for any α > 0

and b ∈ RN the transformed set αF + b equals the core or the nucleolus, respectively, of the

transformed game αV + b.

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3 The core of permutation games

Given two �nite sets S and T , we call µ ⊆ S×T an (S, T )-assignment, if it is a bijection from

some S′ ⊆ S to some T ′ ⊆ T such that |S′| = |T ′| = min(|S|, |T |). Trivially, µ = ∅ if S = ∅ or

T = ∅. We shall write (i, j) ∈ µ as well as µ(i) = j. We denote by Π(S,T ) the set of all (S, T )-

assignments. Obviously, Π(S,T ) = {∅} if S = ∅ or T = ∅. An (S, S)-assignment is called an

S-permutation. We denote by ΠS the set of all S-permutations. We call an S-permutation σ

cyclic, if for any proper subset S′ of S, the restriction of σ to S′ (i.e., σ ∩ (S′ × S′)) is not an

S′-permutation. A cycle of an S-permutation σ is the restriction of σ to a subset S′ of S, if

the restriction is a cyclic S′-permutation. Clearly, every permutation is the union of its cycles.

A game (N, V ) is a permutation game if there exists a square matrix A = [aij ]i∈N,j∈N such

that

V (S) = maxσ∈ΠS

∑

i∈S

aiσ(i) ∀S ⊆ N. (2)

Note that ΠS is nonempty and �nite for all S ⊆ N , so V is well de�ned. Moreover, V (∅) =

0. Naturally, many di�erent matrices induce the same game. Given a �xed matrix A, we

denote by Π∗S the set of optimal solutions to (2). An element of Π∗S , typically denoted by σS ,

will be called a maximal S-permutation in the matrix A. Note that permutation games are

superadditive.

Certain nonnegative permutation games are traditionally of special interest. A game

(N, V ) is an assignment game if there is a partition N = I ∪ J , I ∩ J = ∅, of the player

set, and there exists a nonnegative matrix B = [bij ]i∈I,j∈J such that

V (S) = maxµ∈Π(S∩I,S∩J)

∑

(i,j)∈µ

bij ∀S ⊆ N.

It was observed by Curiel and Tijs (1986) that the assignment game induced by the matrix

B = [bij ]i∈I,j∈J is also a permutation game (on the same player set N = I ∪ J) induced by

the matrix A = [aij ]i∈N,j∈N with aij = bij ∀ (i, j) ∈ I × J , and aij = 0 otherwise.

A kind of reverse embedding was suggested by Shapley (cf. (Tijs et al., 1984)). Take a

permutation game that is induced by a nonnegative matrix. Since each permutation game is

strategically equivalent to one which is induced by a nonnegative matrix (just add the same

constant to every element of a row or column), this assumption is not restrictive. The idea

is to duplicate the players and consider the assignment game that is induced by the same

5

matrix; then �nd an allocation in the assignment game by letting the duplicated players act

independently; �nally, derive an allocation for the permutation game by merging the payo�s

of the related players. In this way an n-player (nonnegatively induced) permutation game can

be embedded in an (n + n)-player assignment game. This connection has been proven useful

not just for showing the nonemptiness of the core of permutation games in (Tijs et al., 1984)

and (Curiel and Tijs, 1986), but also for analyzing its structure in (Quint, 1996). As we shall

demonstrate, it can be fruitfully exploited in �nding the nucleolus too.

The (total) balancedness of permutation games was �rst proved by Tijs et al. (1984).

We brie�y recall the alternative proof given by Curiel and Tijs (1986), because it leads to

the representation of the core we need. Let (N, V ) be a permutation game induced by the

matrix A, and let π denote a �xed maximal N -permutation, i.e., V (N) =∑

i∈N aiπ(i). It

follows from the well-known Birkho�-von Neumann theorem on doubly stochastic matrices

(cf., e.g., (Bapat and Raghavan, 1998)) that π de�nes an optimal solution to the following

linear program by letting zij = 1 if π(i) = j, and zij = 0 otherwise:

(P )

max∑

i∈N

∑j∈N aijzij

s.t.∑

j∈N zij = 1 ∀ i ∈ N∑

i∈N zij = 1 ∀ j ∈ N

zij ≥ 0 ∀ i, j ∈ N.

Consider the dual linear program:

(D)min

∑i∈N ui +

∑j∈N vj

s.t. ui + vj ≥ aij ∀ i, j ∈ N,

and the set D of optimal solutions to problem (D). It follows from linear programming duality

that D is nonempty. Furthermore, if we denote by

eij(u, v) := aij − ui − vj

the slack at (u, v) ∈ RN × RN in the dual constraint related to (i, j) ∈ N ×N , and by

H := N ×N \ π

the set of o�-π pairs of players, then by complementary slackness we get

D = {(u, v) ∈ RN × RN : eij(u, v) = 0 ∀ (i, j) ∈ π, eij(u, v) ≤ 0 ∀ (i, j) ∈ H}.

6

Let the mapping m : RN ×RN −→ RN be de�ned by m(u, v) := u+ v. Curiel and Tijs (1986)

showed that m(D) ⊆ C, consequently C is nonempty. The reverse inclusion m(D) ⊇ C was

proved by Quint (1996). The representation C = m(D) of the core will be fundamental for

our investigation of the nucleolus.

Let us remark that if the underlying matrix A is nonnegative, the = constraints in problem

(P ) can be relaxed to ≤ constraints, implying nonnegativity for the variables in the dual

problem (D). Thus, for the core of a (nonnegatively induced) permutation game we have

C = m(D+), where

D+ = {(u, v) ∈ D : ui ≥ 0, vj ≥ 0 ∀ i, j ∈ N}.

Since by the classic result of Shapley and Shubik (1972), D+ is precisely the core of the

assignment game induced by A, we get that there is an exact correspondence between the

cores of the two related games.

4 Cyclic permutation games

From this point on, we concentrate on a subclass of permutation games which is interesting

from both the application and the structural points of view. It contains all permutation

games in which the grand coalition N is essential, i.e., V (N) > V (S)+V (N \S) for all proper

coalitions S. Such games can serve as models for situations when there is an incentive for

all coalitions to cooperate with the rest of the players, so the grand coalition is the rational

outcome of the coalition formation process. In contrast, assignment games are also cyclic

permutation games, but in them the grand coalition (like most of the coalitions) is never

essential.

We call a permutation game (N, V ) cyclic, if there exists a cyclic N -permutation which is

maximal in the underlying matrix. In order to simplify the notation, from now on we shall

assume that N = {1, . . . , n}, and the cyclic N -permutation

π(i) :=

i + 1 if i 6= n

1 if i = n

is maximal in the underlying matrix A, i.e., V (N) =∑

i∈N aiπ(i).

7

Since the �xed cyclic permutation π induces a kind of local `linear ordering' of the players,

for i, j ∈ N we de�ne the interval from i to j as

[i, j] :=

{i, π(i), . . . , πk(i) = j} if i 6= j and 1 ≤ k ≤ n− 1

{i} if i = j.

For example, [1, 2] = {1, 2} because π(1) = 2, but [2, 1] = N because π(2) = 3, . . . , πn−1(2) =

1. Two intervals are called neighboring if their union is again an interval. It is clear that any

coalition is the union of non-neighboring intervals which are maximal for inclusion. These

intervals are called the components, their union the interval-partition of the coalition.

We associate with the pair (i, j) of players the interval coalition [j, i], because (i, j) can be

augmented with pairs in π to get a cyclic [j, i]-permutation. Indeed, the mapping

σij(i) := j, and σij(h) := π(h) ∀h ∈ [j, i] \ {i}.

is a cyclic [j, i]-permutation. (This is not the case for [i, j], unless i = j, so reversing the order

of the players to get the endpoints of the related interval is crucial.) Observe that (i, j) is an

o�-π pair (i.e., (i, j) ∈ H) if and only if the related interval [j, i] is a proper coalition. Let us

denote by

H = {[j, i] : (i, j) ∈ H}

the set of proper intervals. For [j, i] ∈ H the associated σij may or may not be maximal. Let

tij := V ([j, i])−∑

(p,q)∈σij

apq. (3)

denote its `de�ciency'. In our discussion the collection

H∗ := {[j, i] ∈ H : tij = 0}

will be of particular importance.

We say that a collection T ⊆ H reproduces S ∈ P (modulo N) if every player in S is

contained in exactly one more interval from T than any player not in S. Such a collection

appears in the following lemma, which states that the intervals related to the (o�-π) pairs of

a permutation reproduce the underlying coalition (modulo N).

Lemma 1 In a cyclic permutation game, for any S ∈ P and σ ∈ ΠS, the collection {[j, i] :

(i, j) ∈ σ \ π} reproduces S (modulo N).

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Proof Since [j, i] = N if (i, j) ∈ π, we only need to show that the collection {[j, i] : (i, j) ∈ σ}reproduces S (modulo N).

To this end, let us �rst assume that σ is cyclic, i.e., S = {σh(i) : h = 0, . . . , s−1} with some

�xed i ∈ S and s = |S|. Take the intervals [σh+1(i), σh(i)] in the reverse order h = s−1, . . . , 0,

so the last player in an interval is the same as the �rst player in the subsequent interval. Thus,

if we account the players in the order given by σ−1 for the intervals and by π for the players,

we get a list that starts with i = σs(i); ends in i = σ0(i); and � except the only double

appearence of the other players in S � contains distinct subsequent players. It follows that

the players of S are listed exactly one more time than the players not in S.

In case σ is not cyclic on S, we apply the above argument to each subcoalition of S on

which the restriction of σ is cyclic. Since the decomposition of σ into its cycles induces a

partition of S, the lemma follows. ¤

As an example, let N = {1, . . . , 8}, S = {1, 2, 4, 5, 6} and σ = {(1, 2), (2, 5), (5, 4), (4, 6),

(6, 1)}. The collection of the related intervals listed in the reverse order is {[1, 6], [6, 4], [4, 5],

[5, 2], [2, 1]}. Each of the players 1, 2, 4, 5 and 6 appears in exactly four intervals, while each

of the players 3, 7 and 8 only in three. Both numbers decrease by one if we omit [2, 1] = N

related to the only in-π pair (1, 2).

Above we considered intervals related to (S, S)-assignments. Now we look at intervals

arising from (S, π(S))-assignments. A sequence of an even number of distinct pairs of players

is said to be a loop, if it is of the form [(i1, j1), (i1, j2), (i2, j2), (i2, j3), . . . , (ik, jk), (ik, jk+1)]

such that k ≥ 2, jk+1 = j1 and π(ih) = jh+1 ∀h = 1, . . . , k. Since every second pair must be

in π, the sequence of the o�-π pairs [(ih, jh) : h = 1, . . . , k] � that de�nes a bijection from

I = {i1, . . . , ik} to π(I) = {j1, . . . , jk} that is disjoint from π � completely determines the

whole loop. In other words, a set of entries in the underlying matrix forms a loop if and only

if each row and each column contains exactly zero or two entries from this set, and in the

latter case one entry is from π.

We call a minimal balanced collection homogeneous if the (unique) balancing weights are

all equal.

Lemma 2 In a cyclic permutation game, if the sequence [(ih, jh) : h = 1, . . . , k] of o�-π pairs

generates a loop, the collection {[jh, ih] : h = 1, . . . , k} of intervals is a homogeneous minimal

balanced collection. Conversely, if a minimal balanced collection consists of intervals then it

9

is homogeneous, consequently, it is determined by a loop in the above way.

Proof Note that (ih, jh) 6∈ π implies the properness of [jh, ih] for all h = 1, . . . , k.

Since (i1, j1) and (i2, j2) are subsequent o�-π elements of a loop, we have π(i1) = j2, so

the related intervals [j1, i1] and [j2, i2] are neighbors. Clearly, the same holds all along the

sequence. Moreover, the last interval [jk, ik] and the �rst interval [j1, i1] are again neighbors,

since π(ik) = j1. Thus, if we list the intervals in the order given by the loop and the players

in the order given by π, we get a list that starts with j1; goes through i1, π(i1) = j2, . . ., i2,

π(i2) = j3, . . ., etc.; and ends in ik. Since this list contains every player the same number of

times, the intervals can be balanced with equal weights.

To see minimality, we observe that among the intervals only [j1, i1] ends in i1, and only

the interval [j2, i2] starts with j2, so any other interval either contains both i1 and j2 = π(i1)

or none of the two. Clearly, the same holds for any other in-π pairs of the loop. It follows

that no interval can be left out from the collection without loosing its balancedness.

The similar proof of the converse statement (that is not used in later proofs) is left to the

reader. ¤

As an example, let N = {1, . . . , 8} and [(3, 7), (7, 4), (5, 8), (6, 6)] be a sequence of

o�-π pairs that determines the loop [(3, 7), (3, 4), (7, 4), (7, 8), (5, 8), (5, 6), (6, 6), (6, 7)].

Then the collection {78123, 4567, 812345, 6} covers every player exactly twice, so it can be

balanced with equal 1/2 weights.

Since the nucleolus is an excess-based solution, the following result will play a key role in

our discussion.

Lemma 3 In a cyclic permutation game, for any S ∈ P there is a collection S ⊆ H∗ which

reproduces S (modulo N), and a constant tS ≥ 0 such that

e(S, x) =∑

T∈Se(T, x)− tS ∀x ∈ C. (4)

Proof Let S ∈ P and σS ∈ Π∗S . Since for any component [p, q] of S we have σS(q) ∈ S but

π(q) 6∈ S, the number of o�-π pairs in σS is at least the number of components of S. Thus,

|σS \ π| ≥ 1, moreover, |σS \ π| = 1 if and only if S = [j, i] and σS = σij for some (i, j) ∈ H.

The �rst step is to notice that on D = m−1(C),

e(S, u + v) =∑

(i,j)∈σS\πeij(u, v) ∀ (u, v) ∈ D, (5)

10

with at least two terms on the right side, unless S = [j, i] and σS = σij for some (i, j) ∈ H.

The second step is to see that for any (i, j) ∈ H,

eij(u, v) = e([j, i], u + v)− tij ∀ (u, v) ∈ D. (6)

Indeed, if we subtract u([j, i]) + v([j, i]) from both terms on the right side of (3) and notice

that∑

(p,q)∈σijepq(u, v) = eij(u, v) holds for all (u, v) ∈ D, we immediately get (6).

Combining (5) and (6) gives

e(S, x) =∑

(i,j)∈σS\πe([j, i], x)−

∑

(i,j)∈σS\πtij ∀x ∈ C. (7)

By Lemma 1, the collection of intervals associated with the pairs in σS \ π reproduces S

(modulo N), so we are done in case [j, i] ∈ H∗ for all (i, j) ∈ σS \ π.

In the other case, we have a collection S ⊆ H which reproduces S (modulo N), and a

constant tS ≥ 0 such that (4) holds, but there is an interval T ∈ S \ H∗. Then, by repeating

the above argument for T , there is a collection T ⊆ H which reproduces T (modulo N), and a

constant tT ≥ 0 such that e(T, x) =∑

R∈T e(R, x)− tT for all x ∈ C. Substituting this in (4)

and replacing T with T give a modi�ed collection S ′ = S\{T}∪T which reproduces S (modulo

N), and a constant tS′ = tS + tT ≥ 0 such that (4) holds. If S ′ ⊆ H∗, we are done, otherwise,

we repeat this kind of decomposition. Since for [q, p] 6∈ H∗ we have epq(u, v) < e([q, p], u + v)

for all (u, v) ∈ D, the decomposition process of [q, p] can not bring back [q, p] itself. Thus, in

a �nite number of steps we obtain a collection in H∗ (and a nonnegative constant) with which

the lemma holds. ¤

An important consequence of the lemma is that at each core allocation the excess of any

proper coalition is majorized by the excess of at least one interval coalition from the collection

H∗. Moreover,

maxS∈P

e(S, u + v) = maxT∈H∗

e(T, u + v) = max(p,q)∈H

epq(u, v) ∀ (u, v) ∈ D. (8)

5 The nucleolus in the cyclic case

In this section we show how the embedding of permutation games in (double-sized) assignment

games can be exploited in �nding the nucleolus in the cyclic case. Throughout the section we

11

assume that the underlying matrix is nonnegative. As we remarked earlier, this assumption

does not restrict the generality of our discussion.

One key point in Shapley and Shubik's (1972) characterization of the core of assignment

games is that, besides e�ciency, it su�ces to require rationality only for single-player and

mixed-pair coalitions. The reason is that this collection contains all essential coalitions in any

assignment game. It was observed by Huberman (1980) that for balanced games the nucleolus

is completely determined by the collection of essential coalitions. For assignment games with

player set N = I ∪J this means that the nucleolus is determined by procedure (1) even if it is

initiated with the much smaller collection Σ0 := { {k} : k ∈ N} ∪ { {(i, j) : i ∈ I, j ∈ J} (cf.

(Solymosi and Raghavan, 1994)). A similar simpli�cation is not possible for the whole class of

permutation games, since in Example 3 of the next section we present a permutation game in

which all coalitions are essential. In the cyclic case however, there is another small collection

of coalitions which completely determines the nucleolus: the collection H of (proper) intervals.

Next we show that the nucleolus is actually determined by the (sub)collection H∗. (The cyclicpermutation game in Example 3 is one in which H∗ = H.)

Lemma 4 The nucleolus of a cyclic permutation game is determined by the specialized version

(1)H∗ of procedure (1) in which Σ0 := H∗.

Proof It follows from (8) that in the �rst iteration of procedure (1) ε1 and X1 are actually

determined by the intervals in H∗.Let r ≥ 2 and S ∈ Σr−1 \ H∗. Then the excess of S is not constant on Xr−1. (Notice

that this need not be true for r = 1 in case ε1 = 0.) It follows from Lemma 3 that in any

decomposition of S there is an interval T ∈ H∗ on the right side of (4) whose excess is not

constant on Xr−1, so T ∈ Σr−1. Since e(S, x) ≤ e(T, x) for all x ∈ Xr−1, coalition S can be

ignored when εr and Xr are found. It follows that no coalition outside H∗ plays a role in the

optimization part of iteration r.

On the other hand, if iteration r is the �rst one when it becomes true that there is a

collection S ⊆ H∗ such that (4) holds and for all T ∈ S the excess of T is constant on Xr,

then S itself has a constant excess on Xr. Thus, S 6∈ Σr, so it has no chance to play a role in

the subsequent iterations of (1).

Therefore, if procedure (1) is initialized with Σ0 := H∗ (or with any collection containing

H∗), the generated sequences 0 ≥ ε1 > . . . > ε% and X0 ⊇ X1 ⊃ . . . ⊃ X% remain the same.

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¤

Based on the tight connection between the interval coalitions in a cyclic permutation game

and the pairs of the underlying matrix, next we show how the nucleolus can be obtained from

the �nal outcome of the following procedure. Its input is a nonnegative square matrix A with

row and column index set N = {1, . . . , n}, and a maximal N -permutation π in A. Let D+,

H, eij(., .) be de�ned as in section 3.

Let Y 0 := D+ and ∆0 := H. For t = 1, . . . , τ de�ne recursively

δt := min(u,v)∈Y t−1 max(i,j)∈∆t−1 eij(u, v) ,

Y t := {(u, v) ∈ Y t−1 : max(i,j)∈∆t−1 eij(u, v) = δt},∆t := {(i, j) ∈ ∆t−1 : min(u,v)∈Y t eij(u, v) = max(u,v)∈Y t eij(u, v)},∆t := ∆t−1 \∆t,

(9)

where τ is the �rst value of t for which ∆t = ∅.

We call the �nal set Y τ the pair-nucleolus of the matrix A. It is easily seen that the pair-

nucleolus is well de�ned, not empty, and it is typically a line segment (that shrinks to a

singleton if and only if aiπ(i) = 0 for some i ∈ N). Notice that for any (u, v) ∈ Y τ there is a

typically nondegenerate closed interval of values of s for which the allocation (u′, v′) given by

u′i = ui + s, v′i = vi − s, ∀ i ∈ N , also belongs to Y τ .

Our main result is the following

Theorem 1 If the cyclic permutation game (N, V ) is induced by the nonnegative matrix A,

then N (N, V ) = m(Y τ ), where Y τ is the pair-nucleolus of A.

Proof It follows from Lemma 4 that the nucleolus is the outcome of the specialized version

(1)H of procedure (1) in which Σ0 := H. We chose this (weaker) version to initially have the

one-to-one correspondence: [j, i] ∈ Σ0(= H) if and only if (i, j) ∈ ∆0(= H). We shall prove

that the relation X% = m(Y τ ) holds between the �nal sets in (1)H and (9).

Since X0 = C = m(D+) = m(Y 0), it follows from (8) that ε1 = δ1, and consequently,

that X1 = m(Y 1). Then (6) immediately implies that [j, i] ∈ Σ1 if and only if (i, j) ∈ ∆1.

Consequently, after the �rst iterations the one-to-one correspondence still holds between ∆1

and Σ1.

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For r ≥ 2, let us assume that Xr−1 = m(Y r−1) and [j, i] ∈ Σr−1 i� (i, j) ∈ ∆r−1 hold.

Then for any (i, j) ∈ ∆r−1, (6) implies eij(u, v) ≤ e([j, i], u + v) for all (u, v) ∈ Y r−1. Thus,

we have max(i,j)∈∆r−1 eij(u, v) ≤ max[j,i]∈Σr−1 e([j, i], u + v) for all (u, v) ∈ Y r−1.

Conversely, let S ∈ Σr−1 and σS ∈ Π∗S . Then e(S, .) is not constant on Xr−1, so by (5),

eij(., .) is not constant on Y r−1 for at least one (i, j) ∈ σS\π. This means that for any S ∈ Σr−1

there is at least one (i, j) ∈ ∆r−1 such that e(S, u + v) ≤ eij(u, v) for all (u, v) ∈ Y r−1. Thus,

we have maxS∈Σr−1 e(S, u + v) ≤ max(i,j)∈∆r−1 eij(u, v) for all (u, v) ∈ Y r−1.

Therefore, we have maxS∈Σr−1 e(S, u + v) = max(i,j)∈∆r−1 eij(u, v) for all (u, v) ∈ Y r−1,

that implies not just εr = δr, but also Xr = m(Y r). By (6) again, [j, i] ∈ Σr if and only if

(i, j) ∈ ∆r. Consequently, after the r-th iterations the one-to-one correspondence still holds

between ∆r and Σr.

It follows that the number of iterations, % in (1)H and τ in (9), are equal, and hence, that

the claimed relation X% = m(Y %) indeed holds. ¤

The pair-nucleolus of a nonnegative square matrix can be computed by the algorithm

developed by Solymosi and Raghavan (1994). Although it is presented there in a form best

suited for computing the nucleolus of an assignment game directly from the generating (not

necessarily square) matrix, it actually computes the �nal outcome of (9). Indeed, to get the

nucleolus of the assignment game induced by matrix A, one has to compute the pair-nucleolus

of the augmented matrix A′ = [a′ij ]i∈N0,j∈N0 , where N0 = N ∪ {0}, a′ij = aij ∀ i, j ∈ N , and

a′ij = 0 otherwise. (Notice that the pair-nucleolus of A′ is a singleton, since there must be a 0

element in any maximal permutation, furthermore, u0 = v0 = 0 throughout the procedure for

the two null-players.) The reason is that eij(., .) in (9) is exactly the excess in the assignment

game of the mixed-pair coalition {i, j} if i, j ∈ N , and of the single-player coalition {i} (resp.

{j}) if j (resp. i) is the null-player from the other side. Therefore, the assignment nucleolus

algorithm of Solymosi and Raghavan (1994) can be used to compute the nucleolus of a cyclic

permutation game, but the single-player coalitions (the pairs containing a null player) must

be ignored. From the complexity of that algorithm we get the following

Corollary 1 The nucleolus of an n-player cyclic permutation game can be computed directly

from the underlying n× n matrix in O(n4) time.

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6 Counter-examples

In Theorem 1 we showed that if the matrix A induces a cyclic permutation game VA, then

the nucleolus of VA is the image under the map m of the pair-nucleolus of A. The following

example demonstrates that this correspondence does not hold with the pair-nucleolus of the

augmented matrix A′, i.e., with the nucleolus of the related assignment game.

Example 1 The nucleolus of the cyclic permutation game induced by the matrix

A =

2 4 4

0 1 3

5 2 2

is not the merged nucleolus of the assignment game induced by A.

Since the (boxed) row maximums form a cyclic permutation of the players, the induced per-

mutation game is indeed cyclic. The reader may easily check that the nucleolus of the cyclic

permutation game is (9/2, 2, 11/2). It is indeed the image under the map m of the pair-

nucleolus (4, 2, 9/2 ; 1/2, 0, 1) of the matrix A. On the other hand, the nucleolus (3, 1, 3 ;

2, 1, 2) of the related assignment game is mapped by m to the core allocation (5, 2, 5).

Our second example demonstrates that, unlike in Theorem 1, for non-cyclic permutation

games the nucleolus need not be the merged pair-nucleolus of the underlying matrix.

Example 2 The nucleolus of the non-cyclic permutation game induced by the matrix

A =

0 8 0 9

2 0 0 0

0 1 0 4

0 0 2 0

is not the merged pair-nucleolus of A.

Clearly, the only maximal permutation in A consists of two cycles. The reader may easily

check that the nucleolus of this non-cyclic permutation game is (23/3, 7/3, 7/3, 11/3). On

the other hand, the image under the map m of the pair-nucleolus (8, 2, 2, 2 ; 0, 0, 0, 2) of the

matrix A is the core allocation t= (8, 2, 2, 4).

It remains an open problem to �nd an e�cient algorithm for the nucleolus of non-cyclic

permutation games. The above example also shows that computing the nucleoli of the cyclic

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subgames and putting them together might be far from the solution. Indeed, the nucleolus

(5, 5, . , . ) of the subgame on N1 = {1, 2} and the nucleolus ( . , . , 3, 3) of the subgame

on N2 = {3, 4} together gives the allocation (5, 5, 3, 3) that is not even in the core of the

permutation game, since 5 + 3 < 9 = V ({1, 4}).The following example is here to underline the importance of Lemma 4. It also shows a

cyclic permutation game in which H∗ = H.

Example 3 The upper-triangular matrix

A =

0 1 1 1

0 0 1 1

0 0 0 1

0 0 0 0

induces a 4-player (cyclic) permutation game in which all coalitions are essential. The example

clearly generalizes to any order n ≥ 2.

Since any principal submatrix is of the same type, we have v(S) = |S| − 1 for all S ⊆ N . It

follows that any S ⊆ N is essential, because v(S) = |S|−1 > |S1|−1+ |S2|−1 = v(S1)+v(S2)

for any partition S = S1 ∪ S2 of S.

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