Fair cost-sharing methods for the minimum spanning tree game

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Fair cost-sharing methods for the minimum spanning tree game

Eric Angel, Evripidis Bampis, Lélia Blin, Laurent Gourvès

To cite this version:

Eric Angel, Evripidis Bampis, Lélia Blin, Laurent Gourvès. Fair cost-sharing methods for the minimum spanning tree game. Information Processing Letters, Elsevier, 2006, 100, pp.29–35.

�10.1016/j.ipl.2006.05.007�. �hal-00341341�

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Eri Angel Evripidis Bampis LeliaBlin LaurentGourves

1. LaMI, CNRSUMR 8042, Universite d'

Evry,Tour

Evry 2, 523Plae desterrassesde l'agora,91000

Evry Cedex

fangel,bampis,lblin,lgourvesglami.univ-evry.f r

Abstrat

Westudytheproblemofsharinginafairmannertheostofaservieprovidedtoasetofplayers

intheontextofCooperativeGameTheory. Weintrodueanewfairnessmeasureapturingthe

dissatisfation(orhappiness)ofeahplayerandweproposetwoostsharingmethodsminimizing

themaximum oraveragedissatisfation of thelients forthelassialminimum spanning tree

game.

Keywords: ost sharing,fairness,minimumostspanningtree game, Birdrule

1 Introdution

Cooperative GameTheoryappliesinsituationswhere morethanone deisionmakersareinvolved

suhasintheasewhereagroupofdeisionmakersdeidetoundertakeaprojettogetherinorder

toinrease(resp. derease) theirtotal revenue(resp. ost). Theyhave thentosolvetwoproblems:

i)howto exeute theirprojetinan optimalwayand ii)howto alloatetherevenue/osts among

thepartiipants. Theseond problemis thesubjetof Cooperative GameTheory whihproposes

prot/ost alloations taking into aount the revenue/ost of all possible oalitions (subsets of

the set of partiipants). Indeed, if one or more players oneive of a proposed alloationasbeing

disadvantageous to them, they an deide to do not partiipate. Even when an alloation is

individually rational, there may be a problem if a group of players gure out that they an do

better without working with the others. This is not possible when the revenue ost alloation

belongsto theore ofthe game. Thus,one important issuein Cooperative GameTheory onsists

insearhingforarevenue/ostalloationbelongingtotheoreofthegame. But,ifforsomegames

the situation is problemati beause of the emptynness of the ore, for other gamesthe situation

beomes problematibeause of thehuge number of dierent alloations that belong to theore.

In this later ase, an individualplayer even ifhe has notinentive to do notpartiipate, he may

be unhappywith respet to the revenue/ost alloation when omparing with the best (for him)

alloation that belongs to the ore. Our goal, in this paper, is to introdue new tools allowing

to take into aount the dissatisfation (or happiness) of the players. We thus introdue a new

riterion formeasuring the dissatisfationof theplayers,the dissatisfation fator, and studythe

problemofprovidingasolutionintheoreandminimizingthedissatisfationfator. Weillustrate

ourapproah usingthe lassialminimum spanning tree game.

1.1 Denitions and Notations

More formally,a oalition gamewith transferable payohV;i onsistsof a niteset V of players,

and a funtion that assoiates with every nonempty subset S (a oalition) of V a real number

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intheoalitionS haveto payolletivelyinorderto haveaessto aservie. Letx

i

,fori2V,be

the ost that theplayer ihas to pay. The entralquestion is howto fairlyalloate the ost (V)

among thesetof playersV?

A solutionx =(x

i )

i2V

belongsto theore ifno oalition an obtainan outome better forall

its members than the urrent assignment (x

i )

i2V

[9 ℄. In other words, the ore C of the oalition

game with transferable payo hV;i is the set of ost vetors (x

i )

i2V

suh that P

i2V x

i

= (V)

and 8S V one has P

i2S x

i

(S). An equivalent denitionis to say that the ore is the set

of osts(x

i )

i2V

forwhih there isno oalitionS and ost vetor (y

i )

i2V

forwhih P

i2S y

i

=(S)

and y

i

<x

i

foralli2S. Therefore,given a solutionintheore, there isno inentive foranagent

to leavethe grandoalitionV.

However,eahagent mayompareitsurrentostwiththebestost(thesmallestone)itould

have payed inanother solutionintheore. Given a solution(x

i )

i2V

, we denethe dissatisfation

of agent ias:

i

(x;C)= x

i

min

y2C y

i :

Twooptimization problemsnaturallyarise:

mwd: minimize the worst dissatisfation, i.e. nd an alloation x = (x

i )

i2V

2 C whih

minimizesmax

i2V

i (x;C).

mad: minimize the average dissatisfation, i.e. nd an alloation x = (x

i )

i2V

2 C whih

minimizes P

i2V

i (x;C).

1.2 The spanning tree game and our ontribution

To illustrate our approah we onsider the problem of broadast routing. Suppose that a servie

has to be provided to a set of lients V over a network G(V r

;E). We onsider that G is an

undiretedonnetedgraph. AmongallnodesofG, wedistinguishtheroot(theprovider),denoted

by r. The set of nodes is denoted by V r

while V (the set of lients) denotes V r

nfrg. Eah

edge e 2 E has a non-negative integral ost

e

. The servie an be provided diretly to a lient

or via others. Thus, the minimum substruture that an be used is a tree spanning all lients.

This tree has a ost whih must be shared by the lients. This is a lassial ooperative game

problemsineooperationmay redueaggregate osts. Formally,one hasforanyoalitionS V,

(S)=minf P

e2T

S

e jT

S

isa spanningtree ofG[S[frg℄g,whereG[S[frg℄denotesthesubgraph

of Ginduedbythesetof vertiesS[frg.

CostsharingforthisproblemhasbeenrstaddressedbyClausandKleitman[2 ℄,whileBird[1 ℄

treated this problem with game theoreti methods and proposed a ost alloation rule known as

Bird's rule. It onsists in assigning to eah lient the ost of the edge inident upon him on the

uniquepathfrom himto thesoure/provider inaminimumost spanningtree. LetT

G

be theset

of all minimumost spanning treesof Gand let C

opt

be theost of anytree inthis set. Let T be

a treeinT

G

andv bea vertexinV. Thereis auniquepathbetweenr and v inT. This pathuses

exatlyone edge [x;v℄, wherex2V r

nfvg. Let(T;v)=

[x;v℄

bethe ostof thisedge. Bird'srule

alloates to v thequantity (T;v),i.e. thefee ofv isx

v

=(T;v)=

[x;v℄

.

Bird's ruleensures that no oalition hasinentive to be formed. In fat, theset of alloations

C

Bird

arising from this rule are always in the ore C of the minimum spanning tree game (see

GranotandHuberman[7 ℄,[3 ℄). Inaddition,thesetofBirdtreealloationsistheuniquenon-empty

solutionfor the minimumspanning tree game that satises three important properties eÆieny,

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sine in general there are more than one minimum ost spanning trees for a given network, this

wayofdividingtheostsdoesnotneessarilyleadtoauniqueostalloation. Evenworse,froman

individualpointof view,itmayleadto aostalloationthathargesaveryhighfee toalient(or

asubsetoflients)omparedto theonethathewouldpaywithadierentminimumostspanning

tree. As a onsequene, the dissatisfation of an agent grows as the prie he is harged deviates

from the best possible. We therefore dene the dissatisfation of v with respet to T and Bird's

ruleasfollows:

v (T;C

Bird )=

(T;v)

min

T 0

2T

G f(T

0

;v)g :

Thegoalofthisworkisto designalgorithmswhihprovideostalloationsbasedonBird'srule

inorder to ensurethat they always belong to theore of the game,whiletaking into aount the

happinessoftheusers. Todoso,asstatedbefore,twodiretionsanbetypiallyfollowed: Minimize

theworst dissatisfationover all agentsorminimizethe average dissatisfation. Therefore, we get

two ombinatorial optimization problemsthatwe all:

spanningtree-mwd: Finda minimumost spanningtree thatminimizes theworstdissat-

isfation.

spanning tree-mad: Finda minimum ost spanning tree that minimizes the average dis-

satisfation.

We providepolynomialtimealgorithmsforthese two problems.

Remark.

Notie that a dierent notion of fairness has been widely used in the ontext of bandwidth allo-

ation in network routing, namely the notion of max-min fairness (see e.g. [8 ℄). In our ontext

it orresponds to a min-max fairness ondition stating that in a fair alloationthe maximumfee

paidbyalientshouldbe assmallaspossible,thenone shouldmakesure thatthenextlargestfee

paid bya lient should be as smallas possible, and so on. In other words,an alloationx is fair

if there is no way of dereasing a fee x

i

without inreasing some other fee x

j

suh that x

j x

i .

Notiethatfortheproblemwe onsider,allalloationsfoundusingtheruleofBirdhave thesame

degree of fairnesswith respetto eah other, and therefore no disriminationispossiblewith this

notionof fairness. This is true sine themultiset ontaining fx

1

;:::;x

jVj

g isalways the same for

anyalloationx foundwiththeBird'srule.

Organization of the paper.

Setion2studythedissatisfationfator intheontext oftheBirdostalloation. Setions3 and

4 arerespetivelydevoted to thespanning tree-mwdand spanningtree-mad problemswhile

Setion5 givessome onludingremarks.

2 The dissatisfation fator with Bird's rule

Given a tree T 2 T

G

, the fee of a vertex in this tree using Bird's rule depends on the plae of

the root. This is why it is onvenient to deal with arboresenes instead of trees (we assume

that arboresenes are oriented from the root to the leaves). As a onsequene, we build from

G = (V r

;E;) a weighted digraph H = (V r

;A;) as follows: For eah ordered pair of nodes

x;y 2 V V r

suh that x 6= y, put two ars (x;y) and (y;x) in A ifthere is an edge [x;y℄ in E

and set

(x;y)

=

(y;x)

=

[x;y℄

. Thus, for eah minimum ost spanning tree of G, there exists a

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orrespondingminimumost spanningarboresenerootedinr (msa forshort),B,inH. Inthe

rest ofthepaper,weonlyworkwitharboreseneson thedigraphH,and wedenote byT

H

theset

of msa r

's ofH.

Inorder todeterminethedissatisfationofavertexv2V inan arboreseneB,itisneessary

to know what are the fees it an have to pay (this set is denoted by F(v)). It is lear by Bird's

rulethat (B;v) an onlytake valuesin f

(x;v)

j(x;v) 2Ag. This onditionis neessary butnot

suÆient. The followingproedure,denoted byFEE, provides theset of fees a vertex ould have

to pay. It onsists in removing some ars inident upon the vertex and enforing him to pay a

partiular prie. We note

jA

0 theset ofostson thesubset A 0

A.

FEE

Input: A digraphH =(V r

;A;), a vertexv2V

Step 1: Computea msa r

of H and let C

opt

be its total ost

Step 2: L:=f

(x;v)

j(x;v)2Ag

Step 3: F(v):=;

Step 4: For eahl of Ldo

A 0

:=Anf(x;v)j

(x;v) 6=lg

H 0

=(V r

;A 0

;

jA 0

)

Computea msa r

B 0

inH 0

IfB 0

existsand its total ost isC

opt

ThenF(v):=F(v)[flg

EndFor

Output: F(v)

Then, we are able to determine, for eah vertex v, the set F(v) of fees it ould pay in any

arboresene B 2 T

H

. Among these fees, we distinguish the lowest denoted by F

min

(v) and the

largest denoted by F

max

(v). Computing a msa r

an take O(mn) time (where m = jAj and

n=jV r

j) [4 ℄. Thus,FEEruns intimeO(mn 2

).

OneanremarkthatifwerunFEEoneahvertexthenoneangeneratethesamearboresenes

againandagain. Bykeepingtrakoftheostsinurredbyeahvertexinallarboresenesgenerated

bythealgorithm, we may save some omputation time. However, interms of worst ase analysis,

thetime omplexityofthealgorithm remainsunhanged.

In the next setion, we study the spanning tree-mwd problem for whih we propose an

optimalpolynomialtimealgorithm.

3 The minimum worst dissatisfation problem

Westudythefollowingproblem: Amongallspanningarboresenesofminimumost,ndonethat

minimizes theworst dissatisfation over all verties. Formally,the spanning tree-mwd problem

an bedesribed asfollows:

argmin

B2T

H max

v2V f

v (B;C

Bird )g

where

v (B;C

Bird )=

(B;v)

min

B 0

2T

H f(B

0

;v)g :

EventhoughtheproedureFEEdeterminesalltheoststhatBird'sruleanassigntothever-

ties,itisnotitselfsuÆienttondaminimumspanningtreeminimizingtheworstdissatisfation.

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PSfragreplaements

r

r r

r

a

a a

a

b

b b

b

d

d d

d

e

e e

e

f

f f

f

G T

1

T

2

T

3

T

4 1

1 1

1

1 1

1

2

2 2

2

2 2

2

3

3 3

3 3 3

3

3 3

3

Figure1: IfwerunFEEonGthenonlytreesT

2 ,T

3 andT

4

anbegeneratedwhileT

1

,anoptimum

forthespanningtree-mwd problem,is notbuilt.

To seeit, onsidertheinstanegiven inFigure 1forwhihtheoptimaltreedoesnotbelong tothe

set ofarboresenes generatedbyFEE.

The quantitymax

v2V fF

max (v)=F

min

(v)g is an upperboundon the worst dissatisfation. The

idea ofouralgorithm isto derease thisupperbounduntilitreahestheoptimalvalue. To do so,

we proposean algorithm whihiterativelydeletes somears of H untilanymsa r

of H isoptimal

withrespetto theworst dissatisfation.

ALGO 1

Input: A digraphH =(V r

;A;)

B ofH and letC

opt

beits ost

Step 2: Foreah vertexv2V,omputeF(v) withFEE

Step 3: A 0

:=A

Step 4: Selet v 0

2V suh that Fmax(v

0

)

Fmin(v 0

)

=max

v2V n

Fmax(v)

Fmin(v) o

Step 5: A 00

:=A 0

Step 6: A 0

:=A 0

nf(x;v 0

)j

(x;v 0

)

=F

max (v

0

)g

B 0

on H 0

=(V r

;A 0

;

jA 0)

If B 0

doesnot existorits ost isgreater thanC

opt

ThenGotoStep 8

Elseremove F

max (v

0

) from F(v 0

)

GotoStep 4

B 00

on H 00

=(V r

;A 00

;

jA 00)

Output: B 00

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PSfrag replaements

r r

r

x x

y y

z z

(a)

(b) ()

(d) (e) (f)

1 1 1 1

1 1 1

2 2 2

3

3 3 3

3

3 3 3

Figure2: The instane Gisgiven in(a). A minimumostspanning treeinG hasaost C

opt

=6.

The orrespondingdigraphH isgiven in(b). One hasF(x)=f2;3g,F(y)=f1;2;3g and F(z)=

f1;3g, therefore F

max (x)

F

min (x)

=3=2, F

max (y)

F

min (y)

=3 and F

max (z)

F

min (z)

=3. The worst dissatisfation ours on

nodes y and z. In (), the algorithm deletes (r;z). With this new instane, it is still possibleto

ompute a msa r

with ost 6. One has F

max (x)

F

min (x)

=3=2, F

max (y)

F

min (y)

= 3 and F

max (z)

F

min (z)

= 1. In (d), the

algorithm deletes (r;y). With thisnew instane, it is stillpossibleto omputea msa r

with ost

6. Onehas

Fmax(x)

Fmin(x)

=3=2,

Fmax(y)

Fmin(y)

=2and

Fmax(z)

Fmin(z)

=1. The worst dissatisfationourson vertex

y. In (e), the algorithm deletes (x;y) but there is no more a msa r

with ost 6. Therefore, the

algorithm omputes on (d) a msa r

and returns the orresponding tree (f). Finally, the worst

dissatisfationis 2.

Theorem 1 The algorithm ALGO 1 gives an optimal solution for the spanning tree-mwd

problem and runsin polynomial time.

Proof. Supposethattheminimumworstdissatisfationisequalto Æ

. TaketheoriginaldigraphH

and, foreah vertex v 6=r,remove every ar (x;v) suh that

(x;v)

>Æ

F

min

(v). This proessing

produesasubgraphH

forwhih,omputingamsa r

ispossibleandanyoneofthemisoptimalfor

thespanningtree-mwd problem. SupposeALGO1 returnsa msa r

withworst dissatisfation

Æ>Æ

. Thismeansonlyars(x;v)withoststritlylargerthanÆF

min

(v)wereremoved andthere

is at least one ar (x 0

;v 0

) suh that

(x 0

;v 0

)

= ÆF

min (v

0

). The algorithm stops if the removal of

(x 0

;v 0

)leadstoone ofthefollowingoutomes: No moremsa r

existsoranymsa r

hasatotal ost

stritlygreater thanC

opt

. However, suh adigraphis asubgraphof H

. We geta ontradition.

Step 2runs inO(mn 3

) timewhiletheloop betweenStep4 and 8runs inO(m 2

n) time. 2

A ompleteexampleisgiven inFigure 2.

Next setion addressestheproblemof minimizingtheaverage dissatisfationoverall lients.

4 The minimum average dissatisfation problem

Westudythefollowingproblem: Amongallspanningarboresenesofminimumost,ndonethat

minimizestheaverage dissatisfation. Notiethatsinethenumberof vertiesisxed,minimizing