A Mood Value for Fair Resource Allocations

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by  Francesca Fossati

A Mood Value for Fair Resource Allocations
Francesca Fossati∗, Stefano Moretti†, Stefano Secci∗
∗Sorbonne Universite´s, UPMC Univ Paris 06, LIP6, 75005 Paris, France
† CNRS UMR7243, PSL, Universite´ Paris-Dauphine, Paris, France.
Abstract—In networking and computing, resource allocation
is typically addressed using classical sharing protocols as, for
instance, the proportional division rule, the max-min fair allo-
cation, or other solutions inspired by cooperative game theory.
In this paper, we argue that, describing the resource allocation
problemasacooperativegame,suchclassicalresourceallocation
7
approaches, as well as associated notions of fairness, show
1
important limitations. We identify in the individual satisfaction Fig. 1: A critical resource allocation situation example
0
rate the key aspect of the challenge of defining a new notion
2
of fairness and, consequently, a resource allocation algorithm
is through the proportion of the demand that is satisfied by
n more appropriate for the cooperative context. We generalize the
an allocation. Large literature exists indeed in the networking
a conceptofusersatisfactionconsideringthesetofadmissiblesolu-
J tionsforbankruptcygames.WeadapttheJain’sfairnessindexto area on proportional resource allocations for many practical
7 include the new user satisfaction rate. Accordingly, we propose situations, from wireless networks to transport connection
2 a new allocation rule we call ‘Mood Value’. For each user it management [2], [3], [4].
equalizes our novel game-theoretic definition of user satisfaction In this paper, we are particularly interested instead in

] with respect to a distribution of the resource. We test the mood
I value and the new fairness index through extensive simulations cooperative networking contexts such that users can be aware
N showing how they better support the fairness analysis. of other users’ demands and the available amount. As such,
. rational users shall compute their satisfaction also based on
s
c I. INTRODUCTION the presence of other users. In fact, such networking contexts
[ In communication networks and computing systems, re- with demand and resource availability awareness are making

1 sourceallocation(insomecontextsalsoreferredtoasresource surface in wired and wireless network environments with
v scheduling, pooling, or sharing) is a phase, in a network an increasing level of programmability, i.e., using software-
1 protocol or system management stack, when a group of defined radio and network platforms that expose novel (north-
9
individual users or clients have to receive a portion of the bound)interfacestouserstodisseminateinformationandpilot
9
resource in order to operate a service. Resource allocation network resource allocations. Our main idea is defining a
7
0 becomesachallengingproblemwhentheavailableresourceis new notion of user satisfaction for such interactive resource
. limitedandnotenoughtofullysatisfyusers’demand.Insuch allocation situations with demand and resource awareness.
1
0 situations,resourceallocationalgorithmsneedtoensureaform Let us briefly clarify our motivation with the following
7 of fairness. Such situations emerge in a variety of contexts, allocation example. A user i asks a quantity of resource that
1 such as wireless access [1], [2], competitive routing [3], is bigger than the resource itself (as B in Fig. 1). Classical
v: transport control [4]. fairnessindices[6],[7],[8]tendtoqualifytheusersatisfaction
i The common methodology adopted in the literature is as maximum when i obtains exactly what it asks. In the case
X
to, on the one hand, determine allocation rules such that where i asks more than the available amount, it cannot reach
r
a they satisfy desirable properties [5], and, on the other hand, the maximum satisfaction due to the fact that its demand
analyse the fairness of a given allocation through indices, the exceeds the available resource. Instead, in demand and re-
most commonly used being the Jain’s index [6]. Allocation sourceawarenessconditions,itwouldbemorereasonablethat
rules and indices of fairness are commonly justified by some its satisfaction is maximum when it obtains all the available
fairness criteria. For instance, among two equivalent users resource. Furthermore, if all the other users together ask a
demandingthesameamountofresource,itmakessensenotto quantityofgoodinferiortotheresource,aminimumportionof
discriminate and to give to each of them the same portion of it,equaltothedifferencebetweentheresourceandthesumof
theresource.Insomecases,itcanbedesirabletoguaranteeat thedemandsofalltheothers,isguaranteedtoi.Underadual
leastaminimumamountoftheresourcesothatthemaximum reasoning, it also appears more acceptable that the minimum
number of users can be served. satisfactionofauserisreachedwhenitreceivestheminimum
Inthenetworkingliterature,theresourceallocationproblem portion of the available resource, instead of when it receives
is historically solved as a single-decision maker problem in zero.Ifusersareincompleteinformationcontexttheclassical
whichusersarepossiblynotawareoftheotherusers’demands approach can lead to not reasonable outcomes.
andofthetotalamountofavailableresource.Itfollowsthatthe In this perspective, in order to better describe the user
mostnaturalandintuitivewaytoquantifytheusersatisfaction satisfaction as a function of the available resource, and to
capture the interactions due to the networking context (e.g., in the following, we refer to the latter rule as ‘proportional’
networked users may be aware of respective demands, may instead of the previous (not weighted) one.
ally in the formulation of their demands, etc), we propose to The idea behind the max-min fairness (MMF) allocation
model the resource allocation problem as a coalitional game. is to maximize firstly the minimum allocation; secondly, the
Accordingly,wedefineanewsatisfactionrateforusers,ableto second lowest allocation, and so on [10], [11]. This solution
adapt to various configurations of the demands. Furthermore, coincides with the only feasible allocation such that, if the
we define a new resource allocation rule, called the ‘Mood allocation of some users is increased, the allocation of some
Value’,basedontheideathatthemostfairallocationistheone other users with smaller or equal amount is decreased.
thatequalizesthesatisfactionofeachplayer.Indeed,regardless Moregenerally,itispossibletoobtainafamilyofallocation
of the level of satisfaction, each player is not discriminated if rules maximizing a parametric utility function. The α-fair
its satisfaction is the same than the one of all the others. We utilityfunctionisdefinedas (cid:80)n x(i1−α) [8].Ifα→1thesolu-
also provide an interpretation of this approach positioning it i=1 1−α
tion of the optimization problem coincides with the weighted
with respect to classical traffic theory [9].
proportional allocation with w equal to 1, if α = 2 with
The paper is organized as follows. Section II presents the i
the minimum delay potential allocation, that is the allocation
state of the art on the topic. In Section III a new satisfaction
n
rate is proposed. In Section IV the mood value and a new obtainedminimizingthetotalpotentialdelay (cid:80)( 1 )[12],and
fairness index are described. In Section V we provide an i=1 xi
if α→∞ with the max-min fair allocation.
interpretation of the mood value with a traffic theory method-
ology. Section VI presents some numerical examples. Finally, B. Game theoretical allocation rules
Section VII concludes the paper. Recently game theory has been applied to communication
systems in order to model network interactions. For example,
II. BACKGROUND
in [13] a cooperative game model is proposed to select a fair
Aresourceallocationproblemcanbecharacterizedbyapair allocation of the transmission rate in multiple access channels
(c,E), in which c is the vector of demands (claims) from n and in [14] the authors studied,using coalitional game theory,
users (claimants) and E is the resource (estate) that should be the cooperation between rational users in wireless networks.
shared between them. The set of users is N ={1,...,n}. The Moreover, it is possible to analyze the allocation problem
resource allocation is a challenging problem when E is not as a Transferable Utility (TU) game [15], [16], [17], which
n
enough to satisfy all the demands ((cid:80)c ≥E). An allocation is defined as a pair (N,v), where N = {1,...,n} denotes
i
i=1 the set of players and v : 2N → R is the characteristic
x∈Rn isasolutionvectorthatsatisfiesthreebasicproperties:
function,(byconvention,v(∅)=0).Bankruptcygames[5],in
• Non-negativity: each user should receive at least zero. particular, deal with situations where the number of claimed
• Demands boundedness: each user cannot receive more resource exceeds that available. A Bankruptcy game is a TU-
than its demand. game (N,v) in which the value of the coalition is given by
• Efficiency: the sum of all allocations should be E. (cid:88)
v(S)=max{E− c ,0} (1)
An allocation rule is a function that associates a unique i
allocation vector x to each (c,E). i∈N\S
where E ≥0 represents the estate to be divided and c∈RN
A. Classical resource allocation rules (cid:80) +
is a vector of claims satisfying the condition c > E
i∈N i
Manyresourceallocationrulesareproposedintheliterature [18], [19]. The bankruptcy game is superadditive, that is:
and each of them is characterized by a set of properties
v(S∪T)≥v(S)+v(T), ∀S,T ⊆N|S∩T =∅ (2)
that justify the use of the given rule in order to find a
solution of the allocation problem [5]. In computer networks,
it is also supermodular (or, equivalently, convex), that is:
the most well-known rules are: the proportional rule and the
weighted proportional rule [9], the max-min fair allocation v(S∪T)+v(S∩T)≥v(S)+v(T) ∀S,T ⊆N (3)
(MMF)[10],[11],andtheα-fairallocation[8].Eachofthese
A classical set-value solution for a TU-game is the core
allocation rules, result of an optimization problem and/or an
C(v), which is is defined as the set of allocation vectors x∈
iterative algorithm, follows a fairness criterion.
RN for which no coalition has an incentive to leave the grand
The weighted proportional allocation rule is based on
coalition N, i.e.:
the idea that a logarithmic utility function captures well the
(cid:88) (cid:88)
individualevaluationoftheworthoftheresource[9].Oneway C(v)={x∈RN : x =v(N), x ≥v(S) ∀S ⊂N}.
i i
n
to compute it is via the maximization of (cid:80)w logx subject i∈N i∈S
i i (4)
i=1
to demand boundness and efficiency constraints. When w is A one-point solution (or simply a solution) for a class CN
i
equalto1theresultingallocationiscalledsimplyproportional of coalitional games is a function ψ :CN →RN that assigns
and when w is equal to c we obtain the allocation that actu- a payoff vector ψ(v) ∈ RN to every coalitional game in the
i i
ally produces allocations proportional to the demands; hence class. A well-known solution for TU-games is the Shapley
2
value [20] φ(v) of a game (N,v), defined as the weighted resources in a proportional way even when this allocation is
mean of the players’ marginal contributions over all possible notthemostsuitabletosolvetheproblem.Anotherwell-know
coalitions and computed as follows: index of fairness is the Atkinson’s index [7]; contrary to the
Jain’s index, it measures the degree of inequality of a given
(cid:88)
φ (v)= w (S)(v(S)−v(S\{i})), (5) allocation, taking value equal to 0 when the system is 100%
i i
fair in the MMF sense, and 1 when it is totally unfair.
S⊆N:i∈S
with w (S)= (s−1)!(n−s)! where s denotes the cardinality of Example 1. Let (c,E) be the situation of Fig. 1 with c =
i n!
S ⊆N. (3,13,2) and E =10. The discussed allocation rules provide
Another well studied solution for TU-games is the nucleo- values in Table I along with the Jain’s index and 1-Atkinson’s
lus, based on the idea of minimizing the maximum discontent index in order to have a measure of fairness.
[21]. Given a TU-game (N,v) and an allocation x∈RN, let
(cid:80)
e(S,x) = v(S)− x be the excess of coalition S over Userdemands Prop. MMF Shapley Nucleolus CEL
i∈S i
the allocation x, and let ≤ be the lexicographic order on
L A:3 1.67 3 1.5 1 0
R. Given an imputation x, θ(x) is the vector that arranges in
B:13 7.22 5 7.5 8 10
decreasingordertheexcessofthe2n−1non-emptycoalitions
C:2 1.11 2 1 1 0
over the imputation x. The nucleolus ν(v) is defined as the
(cid:80) Jain’sindex 1 0.882 0.995 0.946 0.333
imputationx(i.e., x =v(N)andx ≥v({i})foreach
i∈N i i Atkinson’sindex 0.844 0.965 0.821 0.777 0.333
i ∈ N) such that θ(x) ≤ θ(y) for all y imputations of the
L
TABLE I: Allocation rules: comparison (E =10, cf. Fig. 1).
game v.
Given a bankruptcy game, many other solutions can be
proposed [5]. As already introduced in the previous section, The axiomatic theory of fairness proposed in [22] shows
the proportional allocation assigns to player i an allocation that it exists an unique family of fairness measures, which
(cid:80)n includes the Jain’s and the Atkinson’s indices, satisfying a set
equaltoE·c / c .Forexample,itisworthmentioningthe
i i of reasonable axioms. In the rest of the paper, we consider
i=1
Constrained Equal Loss (CEL) allocation that divides equally only the Jain’s index because it is the one classically used
the difference between the sum of the demands and E, under in networking applications. MMF-driven inequality indices
the constraint that no player receives a negative amount. find their most appropriate use in socio-economical contexts,
because they are linked to the concept of welfare of an
C. Fairness indices
income distribution. Furthermore the Jain’s index is based
The evaluation of the fairness of the allocations, used as an on the idea of summarizing the information about the users’
important system performance metric especially in network- satisfaction, which is close to our methodology of redefining
ing,canbeusefultodiscriminateamongallocationrulesandto users’ satisfaction under demand and resource awareness, as
evaluatethelevelof‘justice’intherepartitionoftheresources. discussed in the following section.
Jain[6]introducesaformulaaimedatprovidingaquantitative
measure of the fairness of a resource sharing allocation. III. FROMDEMANDFRACTIONSATISFACTIONTOGAME
THEORETICALSATISFACTION
Definition 1 (Jain’s index). Given an allocation problem
In this section, we propose a game-theoretic approach to
(c,E) and an allocation x, the Jain’s fairness index is:
evaluate the satisfaction of a user for an allocation.
J =(cid:20)(cid:88)n (cid:0)xi(cid:1)(cid:21)2(cid:30)(cid:20)n(cid:88)n (cid:0)xi(cid:1)2(cid:21) (6) A. User satisfaction rate
c c
i i
i=1 i=1
A crucial issue in resource allocation is to jointly:
The Jain’s index is bounded between n1 and 1 [6]. The • find the best solution in terms of a certain goal;
maximum fairness is measured when all the users obtain • evaluate its fairness by referring to a fairness index.
the same fraction of demand and the minimum fairness is
With this purpose, it is important to evaluate the individual
measured when it exists only one user that receives all the
satisfaction rates and to summarize the information given by
resource. The Jain’s index has the following good properties:
each of them with a global fairness index.
• Population size independence: applicable to any user set, A natural way to quantify the satisfaction of a user, as
finite or infinite.
proposed by Jain, is through the proportion of the demand
• Scaleandmetricindependence:notaffectedbythescale. that is satisfied by an allocation [6].
• Boundedness: can be expressed as a percentage.
• Continuity: able to capture any change in the allocation. Definition 2 (Demand Fraction Satisfaction rate). Given the
user i with demand c and an allocation x , the Demand
The index considers the proportion of demand and it gives i i
Fraction Satisfaction (DFS) rate of i is:
the maximum fairness to the allocation for which all the
x
users receive the same proportion of the demand, regardless DFS = i. (7)
of the type of allocation problem, it suggests to allocate the i ci
3
This rate takes value between 0 and 1 since it represents In both cases the PS rate shows that player 2 is less satisfied
the percentage of the demand that is satisfied. than what expected with the DFS rate. This is due to the fact
Unavoidably, this way to quantify the user satisfaction that the game guarantees player 2 to get at least 5.
makes the weighted proportional allocation the fairest one
Thefollowingpropositionsshowsomeinterestingproperties
since it allocates proportionally to the demand. There are,
of the PS rate.
however, situations in which the common sense does not
suggest to allocate in a proportional way; e.g., if there is Proposition 1. If the allocation x belongs to the core of the
a big gap between the demands, in order to protect the bankruptcy game, PS ∈[0,1] ∀i∈N.
i
‘weaker’ users and guarantee them a minimum portion of the
Proof:Ifasolutionxbelongstoacoreitholds:x ≥v(i)
estate, the MMF allocation can be preferable. Furthermore, i
and x ≤ v(N)−v(N \i). Thus v(i) and v(N)−v(N \i)
as mentioned in the introduction, the presence of other users i
are the minimum and the maximum value that an allocation
should rationally be considered not to distort the satisfaction
in the core can take. If x = v(i) = min then PS = 0, if
of each user, in case of awareness about other users’ demand i i i
x =v(N)−v(N \i)=max then PS =1.
and the available demands. i i i
For these reasons, we aim at defining an alternative satis- Proposition 2. It is possible to summarize the bankruptcy
faction rate such that it satisfies the following two properties regimes of the PS rate in four possible cases as in Table II.
we name demand relativeness and relative null satisfaction:
• Demand relativeness: a user is fully satisfied when it re- ci<E ci≥E
ceives its maximal right, based on the available resource; PS case PS case
• Relative null satisfaction: a user has null satisfaction v(i)=0 xi GM xi GG
whenitreceivesexactlyitsminimalright,basedonother ci E
users’ demands and the available resource. v(i)(cid:54)=0 xi−v(i) MM xi−v(i) MG
ci−v(i) E−v(i)
The minimal right for a player is the difference between the
available amount and the sum of the demands of the other TABLE II: Value of PS in the four possible cases.
users (i.e., taking a worst case assumption that the others get
the totality of their demand), and the maximal right is equal Proof: Let us treat each possible cases of Table II:
to the maximum available resource, i.e., ci if ci < E, or it • Case GM: v(i)=0, ci <E
is equal to E otherwise. Remembering the definition of the Using the definition of bankruptcy game, it holds:
characteristic function of a bankruptcy game we have that: v(N)−v(N \i)=E−max{0,E−c }=E−E+c .
i i
• the minimal right for player i is v(i) It follows PSi =xi/ci.
• the maximal right for player i is v(N)−v(N \i) • Case GG: v(i)=0, ci ≥E
Using the definition of bankruptcy game, it holds:
Thusweintroducethe‘playersatisfaction(PS)rate’,which
v(N)−v(N\i)=E−max{0,E−c }=E. It follows
satisfies the above two properties by considering the value of i
PS =x /E
the bankruptcy game associated to the allocation problem. i i
• Case MM: v(i)(cid:54)=0, ci <E
Definition 3 (Player Satisfaction Rate). Given a bankruptcy AsincaseMG,v(N)−v(N\i)=E−max{0,E−ci}=
n
game such that (cid:80)ci > E and an allocation xi, the Player ci. It follows PSi =(xi−v(i))/(ci−v(i)).
i=1 • Case MG: v(i)(cid:54)=0, ci ≥E
Satisfaction (PS) rate for i is:
AsincaseGG,v(N)−v(N\i)=E−max{0,E−ci}=
PS = xi−mini , (8) E. It follows PSi =(xi−v(i))/(E−v(i)).
i max −min
i i Caseterminology: thePSratedifferentiates4possiblecases
where:mini =v(i),maxi =v(N)−v(N\i).If(cid:80)ni=1ci =E we name GM, GG, MM, MG. If a player asks less than E we
the player satisfaction rate is PSi =1, ∀i∈N. callitmoderateplayer(M)whileifitasksmorethanE itisa
greedyplayer(G).Insimilarway,ifthesumofthedemandof
PS ∈ [0,1] if the allocation belongs to the core (see
i a group of n−1 players exceeds E, that means v(i)=0, the
Proposition 1). Moreover it ‘corrects’ DFS since it replaces
i group is a group of greedy players (G) otherwise if v(i)(cid:54)=0
the interval of possible values [0,c ] for x with the interval
i i we have a group of moderate players (M).
[min ,max ].Consequently,iffortheDFSratethemaximum
i i Proposition 2 highlights that, not only there is a relation
satisfactionforiismeasuredwhenitgetsc andtheminimum
i between the DFS rate and the PS rate, the satisfaction of a
when it gets 0, with PS, i is measured to be totally satisfied
user should be modified when it is considered as a player
when it gets max and totally unsatisfied when it gets min .
i i inside a cooperative game. In particular, we can notice that
Example2. Consider(c,E)ofExample1(seeFig.1)andthe for case GM the PS rate coincides with the DFS one, i.e.,
corresponding bankruptcy game model. It holds: PSi = DFSi; for case GG, the user satisfaction measured
Proportional allocation: DFS =0.555 and PS =0.444 with the PS rate is higher than when it is measured with the
2 2
MMF allocation: DFS2 =0.3846 and PS2 =0. DFSrate,i.e.,PSi ≥DFSi;inthe MG case,wehaveinstead
4
that DFS ≥ PS . We can also notice that the denominator but v(N)−v(N\i)−v(i) isequalto 1 so d(x,i)= 1 −1.
i i xi−v(i) PSi PSi
of the PS rate is always different from zero. In cases GM and It is worth noting that if d(x,i) goes to infinity, then PSi
GG thisisobviouslytrue,incase MM thedenominatoriszero goes to 0 and if d(x,i)=0 then PSi =1. This gives another
when (cid:80)ni=1ci = E but in this case we set PSi = 1 and in interpretation of the PS rate. The higher the satisfaction is,
(cid:80)
case MG the denominator is zero when j∈N,j(cid:54)=icj =0 that the bigger the enthusiasm of i, for being in the coalition, is.
is impossible. Furthermore, from Proposition 2 it follows that On the contrary, the closer to zero the user satisfaction is, the
if an allocation, i.e. a solution of an allocation problem that higher the propensity of user i to leave the coalition is.
satisfiesefficiency,non-negativityanddemandboundedness,is
an imputation, then PS ∈ [0,1] for all the users. This holds IV. THEMOODVALUEANDTHEPLAYERFAIRNESSINDEX
i
due to the fact that for an allocation, in each of the 4 cases In this section, we define a new resource allocation rule
presented above, it is always verified that v(N)−v(N \i) is we call the Mood Value. The fairness idea behind this rule is
an upper bound for x . the same of the one behind the Jain’s index. A repartition
i
of a resource is fair when all the users have the same
B. Game-theoretical interpretation
satisfaction. Furthermore, we propose a novel fairness index
To support and justify the use of the new satisfaction rate, as a modification of the Jain’s index.
we show an interesting game-theoretic interpretation.
Gately [23] introduced the concept of propensity to disrupt A. The Mood Value
inordertoeliminatethelessfairimputationinsideofthecore. Using the defined PS rate, we can define the mood value.
The idea was to investigate the gain of the player from the
Definition 5 (Mood Value). Given an allocation problem
cooperationor,instead,itspropensitytoleavethecooperation,
characterizedby(c,E),theallocationxsuchthatPS =PS
and to eliminate the imputation for which the propensity to i j
∀i,j ∈N is called mood value.
leave the coalition for some players is excessively high. The
formal definition of the propensity to disrupt is given in [24]. Due to the relation between the propensity to disrupt and
the player satisfaction, the fairest solution corresponds to the
Definition4 (Propensitytodisrupt). Foranyallocationvector
one in which every player has the same propensity to leave
x,thepropensitytodisruptd(x,S)ofacoalitionS ∈N (S (cid:54)=
thecoalition.Equalizingthepropensitytodisruptoftheusers,
∅,N) is the ratio of the loss incurred by the complementary
thisallocationequalizesthemoodofeachplayer.Inparticular,
coalition N \S to the loss incurred by the coalition S itself
givenagame,itexistsauniquemoodsuchthatthesatisfaction
if the payoff vector is abandoned. In formula,
of each user is the same. The closer to zero the mood is, the
x(N \S)−v(N \S)
d(x,S)= . (9) more unsatisfied user i is; the closer to one the mood is, the
x(S)−v(S) more enthusiast the user i is.
An equivalent definition of d(x,S) is :
Proposition 4. Let (c,E) characterize an allocation problem.
d(x,S)= x(cid:101)(S)−v(S) −1 (10) It exists a unique mood m such that PSi =m ∀i∈N; it is:
x(S)−v(S)
E−min
m= (13)
where: x(cid:101)(S)=v(N)−v(N \S) [23]. max−min
The propensity to disrupt of a coalition S quantifies its de-
n n n
(cid:80) (cid:80) (cid:80)
sire to leave the coalition. When x(S)=v(S) the propensity where min = v(i) = min(i) and max = [E −
to disrupt of S is infinite and the desire of S to leave the i=1 i=1 i=1
n
(cid:80)
coalition is maximum; when x(S) > v(S) but x(S)−v(S) v(N \i)]= max(i). And the mood value is given by:
is small, the value of d(x,S) is very high and again S does i=1
not like the agreement; when x(S) = v(N)−v(N \S) the xm =v(i)+m(max(i)−min(i)). (14)
i
propensity to disrupt is zero and S has the propensity not to
Proof: Let PS =m ∀i∈N. It follows:
destroy the coalition; when x(S) > v(N) − v(N \ S) the i
indexisnegativeandthereisanhyper-enthusiasmforsuchan x =m(E−v(N \i))+(1−m)v(i). (15)
i
agreement.
Due to the efficiency property it holds:
It holds an interesting relationship between the propensity
to disrupt and the player satisfaction rate. n
(cid:88)
m(E−v(N \i))+(1−m)v(i)=E. (16)
Proposition 3. The relationship between the player satisfac-
i=1
tion rate and the propensity to disrupt is:
Thus (13). Since x is the mood value iff PS =m ∀i∈N:
i i
1
PSi = d(x,i)+1. (11) xi−v(i) =m (17)
E−v(N \i)−v(i)
Proof: Using the alternative definition of d(x,i) we have
∀i∈N and (14) remains proved.
v(N)−v(N \i)−v(i) From(13)wecannoticethatthemooddependsonlyonthe
d(x,i)= −1 (12)
x −v(i) game setting, thus, given a bankruptcy game, we can know a
i
5
priori the value of the mood that produces a fair allocation. Definition 6 (Players fairness index). Given a problem (c,E)
Knowing m, on can easily calculate the mood value xm. and an allocation x, the players fairness index is:
i
The formula (14) shows that each user receives the mini- (cid:20) n (cid:21)2(cid:30) n
(cid:88)(cid:0) (cid:1) (cid:88)(cid:0) (cid:1)2
mum possible allocation v(i) plus a portion m of the quantity J = PS n PS (18)
p i i
maxi−mini. The nearer to 1 is the mood m, the greater is i=1 i=1
thehappinessofeachuser,andtheclosertothemaximumthe Proposition6. Theplayersfairnessindextakesvaluein[1,1]
n
allocationis.Infact,whenmisequalto1,theplayerreceives when the allocation belongs to the core.
exactlyE−v(N\i),thatisthemaximumportionofresource
Proof: From Proposition 1 follows that PS belongs to
that it can get, being inside a bankruptcy game. i
n
(cid:80)
The mood value owns some interesting properties. It is an [0,1] and that PS is always not negative. The maximum
i
allocationthusitsatisfiesnon-negativity,demandboundedness i=1
fairnessismeasuredwhenalltheusershavethesamePSrate,
andefficiencyproperty;itisstable,thatmeansitbelongstothe
i.e.:
coreofthegame(prop.5)anditguaranteesmorethanminimal
right to each player (xmi > v(i)). Furthermore it satisfies the (cid:20)(cid:80)n (cid:0)PSi(cid:1)(cid:21)2 =(cid:0)nPSi(cid:1)2 ⇒n(cid:80)n (cid:0)PSi(cid:1)2 =nn(cid:0)PSi(cid:1)2.
followingproperty:ifv(i)=v(j)andv(N\i)=v(N\j)then i=1 i=1
xmi =xmj .Thisimpliestheequaltreatmentsofequals(ci =cj Thus Jp = 1. The minimum fairness is measured when ∃!k
⇒ xm =xm) and equal treatment of greedy claimants (given s.t. PS (cid:54)=0 and PS =0 ∀j (cid:54)=k. In this case:
i j k j
a bankruptcy game, let G be the set of greedy players, i.e. (cid:20)(cid:80)n (cid:0)PS (cid:1)(cid:21)2 =(cid:0)PS (cid:1)2 ⇒n(cid:80)n (cid:0)PS (cid:1)2 =n(cid:0)PS (cid:1)2 ⇒J = 1
such that c >E: if |G|≥2 then xm =xm ∀i,j ∈G). i k i k p n
i i j i=1 i=1
Proposition5. Themoodvaluebelongstothecoreof(N,v). For core allocations, Jp takes value in the same interval
of J making possible a comparison between the two indices.
Proof: We should prove that xmS ≥v(S), ∀S ⊆N. Furthermore, this index maintains all the good properties of
If v(S)=0 the condition holds due to the fact that xmi <0, the Jain’s index: the population size independence, the scale
∀i∈N.Nowconsiderthecasev(S)>0.SupposethatxmS < and metric independence, the boundedness and the continuity.
(cid:80)
v(S) = E − c . For the efficiency property it holds
E =xm+xm i∈,Nim\pSlyiing xm >(cid:80) c , which yields V. INTERPRETATIONWITHRESPECTTOTRAFFICTHEORY
S N\S N\S i∈N\S i
acontradictionwiththefactthat,accordingtothemoodvalue In the already cited seminal works about the definition of
solution, each user receives at most its demand. proportional and weighted proportional allocations in network
MoodValueComputationComplexity: Differentlyfromthe communications, network optimization models are defined
other allocation solutions inspired by game theory, in order to using as goal the maximization of an utility function. A
calculate this new allocation, only the value of 2n coalitions, typical application is the bandwidth sharing between elastic
i.e., the ones formed by the single players and the ones applications [9], i.e., protocols able to adapt the transmission
containing n−1 players, is needed. The time complexity of rateupondetectionofpacketloss.Inthiscontextweshowhow
mood value computation is dominated by the complexity of it is possible to revisit the mood value as a value resulting of
computing v(i) that is O(n). In dynamic situations, i.e. when thesumoftheminimumallocationandtheresultofaweighted
thevalueofeachofthencoalitionshastobeupdatedateach proportional allocation formulation where the weights are not
slot of time, the complexity is therefore O(n2), but it can the original demands, but new demands re-scaled accordingly
be reduced to O(n) where v(i) pre-computation is possible. to the maximum possible allocation knowing the available
This makes the mood value the best allocation rule in terms resource, and the minimum allocation under the awareness
of time complexity together with the proportional allocation: of other user’s demands and the available resource. More
the Shapley value has a time complexity of O(n!), while precisely, the mood value can be computed as the result of
iterative algorithms for the computation of MMF and CEL the following 4-step algorithm.
allocationshaveaO(n2logn)timecomplexity;theNucleolus
Step 1: We assign to each user the minimal right v(i).
computation is a NP-hard problem. Step 2: We set the new value of the estate E(cid:48) =E−min=
Intermsofspatialcomplexity,themoodvalue,proportional, n
MMF and CEL allocations can be considered as equivalent E− (cid:80)v(i) and the new demands c(cid:48)i =maxi−mini.
i=1
and in the order of O(n). Instead, the Shapley value and the
Nucleolus computations have a spatial complexity of O(2n). Step 3: We solve the following optimization problem
n
B. The Player Fairness Index maximize (cid:88)c(cid:48)logx
i i
Considering the observed good properties that make the x i=1
Jain’s index a strong fairness index, we propose its modifi- subject to x ≤c(cid:48), i=1,...,n
i i (19)
cation replacing the DFS rate of the Jain’s index with the PS
x ≥0, i=1,...,n
i
rate. The resulting new fairness index we propose takes value
n
1 when all the users have the same satisfaction, i.e., when the (cid:88)x =E(cid:48)
i
allocation is the mood value.
i=1
6
Step 4:Themoodvaluecoincideswiththesumoftheminimal
right and the allocation given by step 3: xm =v(i)+x .
i i
Proof:Weshouldprovethattheresultoftheoptimization
problem is x =mc(cid:48). The lagrangian of the problem is
i i
n n
(cid:88) (cid:88)
L(x,µ,λ)= c(cid:48)logx −µT(C−Ax)−λ(E(cid:48)− x )
i i i
Fig. 2: RB demand distribution and its Weibull fitting
i=1 i=1
where the vector µ and λ are the lagrangian multipliers (or
for demand generation is between 0 and 6000 units. It is
shadow prices), C is the vector of the demands [c(cid:48),...c(cid:48) ] and
∂1L nc(cid:48) worth noting that the Weibull distribution is quite close to the
A is the identity matrix of dimension n. Then, = i − Paretodistribution(bothareexponentialones),itsanddiscrete
∂y y
c(cid:48) i i variations (e.g., Zipf’s one), for example used in in-network
µi−λ. The optimum is given by yi = µ +i λ when µ ≥ 0, content caching resource allocation [17] .
i
n We run different instances with a ratio of E (available
Ay ≤ C, (cid:80)y = E(cid:48) and µT(C −Ay) = 0. This coincides
i resource) ranging from 5% to 95% of the global demand. We
i=1
with the case in which µT = 0 and λ (cid:54)= 0. In fact, we have firstsimulate300bankruptcygameswith3and5users.Fig.3
(cid:80)n c(cid:48)i = 1 (cid:80)n c(cid:48) = E(cid:48). It follows that λ = 1 (cid:80)n c(cid:48) is show the users configuration as a function of the available
λ λ i E(cid:48) i resource.With3users(Fig.3a,c),forlowvalueofEalmostall
i=1 i=1 i=1
greaterorequalto1andy = c(cid:48)i islessorequaltoc(cid:48),thatisan are greedy players (GG case) due to the fact that the resource
i λ i is small; increasing E the number of moderate players (GM)
1 n
admissible solution. We can now notice that λ= (cid:80)c(cid:48) = increases but also some users in configuration MG appear.
E(cid:48) i
i=1 In fact, increasing E some greedy players become moderate
max−min 1
= . It follows y =mc(cid:48). whiletheothersremaingreedyones;someofthemaregreedy
E−min m i i
insideagroupofgreedyusers(GG),whilesomeothersgreedy
Example 3. Let (c,E) be the allocation problem of Fig. 1. inside a group of moderate ones (MG). When the available
Following the algorithm we have: resource is higher than half of the global demand, greedy
Step 1: v(i)=[0,5,0]. Step 2: E(cid:48) =5, c(cid:48)i =[3,5,2]. players GG disappear and the number of moderate players
Step 3: x=[1.5,2.5,1] Step 4: xmi =[1.5,7.5,1]. increases.Inparticular,users MM appearandtheybecomethe
majority when the resource is large. With 5 users (Fig. 3b,d),
The algorithm shows that the mood value firstly assign the
we find a similar trend than with 3 users in the number of
minimal right (step 1) and secondly, considering the new
moderate players that increases when increasing E. However,
allocation problem resulting after the first assignment (step
2), it allocates in a proportional way the resources (step 3). MG usersarefew;infact,itholdsthatitcanexistatmostone
The proportion of resource to allocated is the mood. MG user in a game and, due to the higher number of users in
the system, it is very unlikely that there exists only a player
We provides two ways to compute the mood value: (14)
and the 4-step algorithm of Section V. It is clear that the MG in the system such that the sum of the demands of the
computation of the mood value throught the formula (14) is
less complex than the one using the 4 steps algorithm.
VI. NUMERICALEXAMPLES
We tested the mood value and the new fairness index in
a few significant configurations comparing them with the
classical allocations and the Jain’s index.
We considered two demands distributions: (i) a uniform
distribution, and (ii) a Weibull distribution. The former can
be considered as a baseline, while the latter a maybe more
(a)3users,uniform (b)5users,uniform
realistic one. To create the Weibull distribution, taking in-
spiration from cellular (OFDMA) resource allocation studies
we emulated an indoor scenario of femtocells using the
WINNERIIchannelmodel[25]:generatinginauniformway
10000 users around the cell station between 3 and 100 m,
we associate resource blocks (RBs) to each of them with a
transmit power between 1 and 100 dB; Fig. 2 is the resulting
RB histogram distribution, which is well fit by a Weibull
distribution f(x) = (a)(x)(a−1)e−(xb)a for x > 0 with scale
b b (c)3users,Weibull (d)5users,Weibull
parameter a=2400 and shape parameter b=1.4. The range
Fig. 3: User cases distribution
7
(a)Proportional (b)ShapleyValue (c)Nucleolus (a)Proportional (b)ShapleyValue (c)Nucleolus
(d)MoodValue (e)MMF (f)CEL (d)MoodValue (e)MMF (f)CEL
Fig. 4: Fairness as a function of E/demand (5 users, uniform) Fig. 6: Fairness as a function of E/demand (5 users, weibull)
is small (high congestion), i.e., when there are many greedy
users. In fact the MMF allocation and the mood value, are
close:insuchcases,bothhavethepropertyoftreatingequally
thegreedyclaimant,givingthemthesameportionofresource,
independently of their demands. Instead, when E increases,
(a)Proportional (b)ShapleyValue (c)Nucleolus the MMF one is not fair any longer because it satisfies more
the two users with less claim while it gives the minimal right
to the one with bigger claim; in fact, in such cases the mood
value becomes closer to the Proportional allocation, to the
ShapleyvalueandtotheNucleolus.Thesimilaritybetweenthe
Proportional allocation and the mood value is due to the fact
that the correct way to measure the satisfaction of moderate
players is through the DFS rate and increasing E the number
(d)MoodValue (e)MMF (f)CEL
of moderate players increases. It follows that the allocation
Fig. 5: Fairness as a function of E/demand (3 users, weibull) equalizingtheDFSrates,i.e.,theProportionalone,iscloseto
the one equalizing the PS rates of user, i.e., the Mood Value.
other n−1 players exceeds E. Thus, with a number of users With 5 users (Fig. 4 and Fig. 6), we can notice that, due to
higher than 5, one can practically reduce the number of user the fact that the group MG is little, when E reach the 40% of
cases from 4 to 3. For this reason, in order to capture all the theglobaldemandtheProportionalisequaltotheMoodvalue
possible scenarios, we choose a low number of user for the and it shows a PF index equal to 1. Furthermore the MMF
first round of simulations. allocation show again a high (PF) fairness when the resource
Fig. 4, 5 and 6 show the results of the first simulations. is little (5%), i.e., when there are many greedy players.
Weconsiderthesixallocationsdiscussedbefore:Proportional, With a second round of simulations we want to see how
Shapley, Nucleolus, Mood Value, MMF and CEL. We calcu- theDemandFraction(DFS)andPlayer(PS)Satisfactionrates
late the Jain’s fairness index and the players fairness index are distributed with an increasing number of users. Results
and we plot, for each ratio of E and each index, the mean are shown in Figure 7 as box-plots (minimum, quantiles and
value in between the first and third quantile lines. The Jain’s maximum,withoutoutliers),fortworegimes(highcongestion
index is depicted with the red color and square points while of a 5% E/demand ratio, and low congestion of a 95% ratio).
the players fairness index with the blue one and round points. We can notice that the value of the satisfaction is low when
Due to space limit, we do not show the case of 3 users with the resource is small (Fig.. 7a,b,c) while it is higher when
uniformdemanddistributionbecausesimilartothe3-userone the congestion is low (Fig. 7d,e,f). With 4 users, in terms
with the weibull distribution. of user satisfaction rate, the mood value is close to the MMF
In the 3-user scenario (Fig. 5) it is possible to notice allocationwhenthecongestionishigh,andtotheproportional
differences between the result obtained by the classical Jain’s allocation when the congestion is low. As the number of
index and our new players fairness (PF) index. It is worth players grows, the absolute difference between allocation in
recalling that the Jain’s index has value 1 when the allocation terms of distribution of the satisfaction decrease. In both the
is proportional while the PF index is 1 when the allocation congestionsituations(E equalto5%and95%ofthedemand)
is the mood value. We can notice that the PF index considers the Shapley value is the closest allocation to the mood value
the MMF allocation as a fair one when the available resource in terms of PS rate.
8
(a)4users,E/demand=5% (b)8users,E/demand=5% (c)16users,E/demand=5%
(d)4users,E/demand=95% (e)8users,E/demand=95% (f)16users,E/demand=95%
Fig. 7: Users satisfaction rates - E over global demand = 5% and 95% - uniform demand distribution
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