Access Network Synthesis Game in Next
Generation Networks
Josephina Antoniou a,∗, Ioannis Koukoutsidis b, Eva Jaho b,
Andreas Pitsillides a, and Ioannis Stavrakakis b
a University
of Cyprus
Dept. Computer Science
75 Kallipoleos Street, P.O. Box 20537, CY-1678 Nicosia, CYPRUS
b National
& Kapodistrian University of Athens
Dept. Informatics & Telecommunications
Ilissia, 157 84, Athens, GREECE
Abstract
In next generation communication networks, multiple access networks will coexist
on a common service platform. In cases where network resource planning indicates
that individual access network resources are insufficient to meet service demands,
these networks can cooperate and combine their resources to form a unified network
that satisfies these demands. We introduce and study the Network Synthesis game,
in which individual access networks with insufficient resources form coalitions in
order to satisfy service demands. The formation of stable coalitions in the core
of the game is investigated, in both cases where payoffs are transferable or are
attributed in proportion to the contribution of each member of the coalition. We
also consider an alternative payoff allocation approach, according to the value of
the well-known Shapley-Shubik, Banzhaf and Holler-Packel power indices, which
represent the relative power each player has in the formation of coalitions. Using the
knowledge attained from the coalition game analysis, we propose a new power index,
called Popularity Power Index, which is based on the number of stable coalitions
an access network would participates in if payoffs were assigned in a fair manner.
Key words: next generation networks, access network synthesis, game theory,
coalitions, power indices
∗ Corresponding author
Email addresses: [email protected] (Josephina Antoniou),
[email protected] (Ioannis Koukoutsidis), [email protected] (Eva
Jaho), [email protected] (Andreas Pitsillides),
[email protected] (Ioannis Stavrakakis).
Preprint submitted to Elsevier
18 March 2009
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Introduction
Future communication networks, also called Next Generation Networks (NGNs),
are envisioned to be based upon a common, flexible and scalable convergence
platform, where different access networks, terminals and services can coexist
[1,2]. Access networks may employ different wireless and mobile technologies,
such as WLAN, WiMax, Ad-Hoc/Sensor as well as GSM and CDMA/WCDMA technologies and will be interconnected by an IP packet-switched core
network [3]. This IP core network deals with all network functionality and coordinates the participating access networks, in order for the system to behave
as a unified platform.
This interconnection allows individual access network components to easily
cooperate and share resources. Access networks in the NGN may need to cooperate in order to jointly meet service requirements 1 . The need for cooperation
arises when traffic forecast indicates that no single network alone can handle
the anticipated demand over a certain period (days, months, etc.). Potentially,
many different combinations of access networks can jointly provide sufficient
resources to meet service demands. The access networks participating in the
selected combination are expected to receive a payment by the NGN platform
administrator.
Let’s consider a specific example of access network cooperation – as part of
network resource planning – required to accommodate a service. Assume that
a number of collocated NGN users (e.g. participating in a conference) have
subscribed for the same multimedia multicast service, its starting time and
duration known to the NGN. Each of the subscribed users have the same
interfaces to the access networks participating in the NGN in their specific
location (e.g. WiFi, WiMax and WCDMA). Considering that none of the
participating access networks can alone support all the users subscribed to
the multicast service, network cooperation can ensure a more efficient network
resource planning and service support to all users. A user or application will
be indifferent to this cooperation, as long as its Quality of Service (QoS)
requirements are satisfied.
In this paper, we consider that different access networks are under different
administration authority or ownership. Consequently the decision of whether
to participate in a certain resource combination (or network coalition) or not,
would be shaped by the access network’s goal to maximize its benefits from the
contributed resources. As a result, coalition games arise in this environment,
with each access network aiming at participating in a “prevailing” coalition
(the one that will win) and at yielding the largest possible benefit to itself. The
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Non-cooperative solutions for meeting service requirements in NGN environments
exist [4]
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formation of coalitions depends on the available resources of each access network, as well as on the way payoffs are allocated to the participating networks.
The formation of composite networking structures, through the contributions
of resources from multiple independent networks, will be the outcome of a
game referred to as the “Network Synthesis” (NS) game.
As it is the case in games among rational players, the payoffs to the players will
shape the outcome of the NS game. In this paper, we consider both transferable
and non-transferable payoffs. In the transferable payoff case, individual access
networks can transfer any portion of their payoff to other members of the
coalition, as long as their final payoff remains greater than zero. In the nontransferable case, such side payments are not allowed and we will consider
that access networks attain a payoff that is proportional to their resource
contribution.
We study the stability of coalitions according to the core and inner core
concepts [5]. When payoffs are transferable, this leads to trivial solutions of
minimal-sized coalitions. When payoffs are non-transferable, the proportional
payoff allocation case is more interesting and requires the notion of a “by-least
winning” coalition. In all cases, when access networks have more than one preferences for coalitions, the game results in a coordination game with conflicting
preferences. We show that there exists at least one coalition which satisfies service requirements 2 and in which all its members attain their highest possible
payoff. When there are multiple stable coalitions, we introduce the concept
of stability under uncertainty of formation, to single out coalitions which are
more likely to be formed.
We also consider an alternative payoff allocation approach according to values of (normalized) power indices. In game theory, power indices are used to
measure the influence of a player on the formation of coalitions and thus on
the game itself. Such a payoff allocation is appropriate when we have a common pool of resources and access networks share their resources (i.e., as if
the core administrator would select them from a common pool). Payoffs are
thus determined based on the power of each access network in the game, i.e.
its index. We consider well-known indices, such the Shapley-Shubik index [6],
Banzhaf index [7] and the Holler-Packel index [8]. We compute values of these
power indices – and hence payoff allocations – for different numbers of access
networks and different distributions of available resources of the networks.
Using the knowledge attained from the coalition game analysis we propose a
new index, called the Popularity Power Index, which associates the popularity
of each access network to the number of stable coalitions it participates in. This
new index aims to achieve fairness, in the sense that it only considers the stable
2
We consider the case where the total available resources of the participating access
networks are greater than the service resource requirements.
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coalitions that would be formed if payoffs were assigned proportionally to the
players’ contributions, i.e. in a fair manner. Summarizing, we consider either
that access networks independently form coalitions and accordingly receive
payoffs, or that they share their resources (as if the core administrator would
select them from a common pool) and payoffs are determined based on the
power of each access network in the game.
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The paper is organized as follows. Section 2 discusses in more detail the motivation for this work. Section 3 describes the theoretical framework, while
Section 4 elaborates on the payoff allocation for the network synthesis game.
We discuss stable coalitions in Section 5. Section 6 goes through some resolutions for the coordination game that results when there are multiple stable
coalitions, and offers a number of illustrative examples. Section 7 considers
solutions based on power indices, elaborated through numerical results for the
well-known power indices and the proposed power index in Section 8. Finally,
Section 9 offers some conclusions.
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Motivation of the game
In a NGN system, the communication between two end-users, or a user and a
service provider (in case of multicast communication), is likely to involve more
than one wireless access network. This can be because of lack of resources
(bandwidth, channels or coverage) of each network (e.g., a call originating in
a certain network may end up being supported by multiple access networks, if
the first for example is congested and experiences lack of resources to support
the call).
The goal is for diverse access network technologies to be integrated seamlessly and inter-operate to provide global connectivity, fully broadband access
and adequate QoS. The integration of different technologies faces several challenges, resulting from the different bit rates, protocols, bandwidth allocation,
channel characteristics, and handoff mechanisms [3].
A major role will be played by the integration architecture. It was common in early integration architectures to require multi-mode user devices,
equipped with multiple interfaces to access different wireless networks. The
devices would incorporate most of the functionality without requiring major
interworking components between the different access networks. However, this
approach is not consistent with the vision of a unified network where different access networks would cooperate and share resources, transparently to the
end-user. It also has to deal with latencies during the handoff process, reconfiguration problems at the user devices and incoherent billing systems between
different network operators.
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In another architecture proposed in [9], a user accesses universal gateways, or
Universal Access Points (UAPs) interconnected by an overlay network on the
IP core. The UAPs, managed by the core administrator, are responsible for
selecting an access network, or a combination of access networks, based on
availability, QoS specifications, and user-defined choices. They must incorporate all different technologies and manage all functionalities for the integration
of these technologies. They should also handle all QoS negotiations between a
user and a network, as well as store user, network, and device capabilities and
preferences. This architecture has the advantage of removing the integration
burden from the mobile devices and can easier support single billing and subscription. It is also more in line with the ultimate goal of cooperation between
different access networks.
A similar role of a mediator between a user and multiple access networks is
also played by virtual network operators in current cellular systems. A virtual
network operator is a company that does not own any frequency spectrum or
network infrastructure, but resells wireless services under its own brand name,
using the network of one or more others. Given the opportunities presented,
more virtual network operators are likely to sprout in a NGN environment,
having as a task not only to interface the customer, but also to integrate the
different wireless access technologies [10].
In such a unifying architecture, the way that different access networks will
inter-operate (and co-operate) depends not only on technical, but also on economic factors. Different networks, especially if owned and operated by different
companies, must acquire a revenue from such cooperation. Therefore, interoperation must be preceded by a process of negotiation or coalition formation
between access network operators, in which the “form” of cooperation (i.e.,
the resources provided by each partner) and revenue allocation is determined.
Our network synthesis game aims to examine dynamic coalition formation
and payoff allocation in a NGN system with such an architecture. A practical
application of such a synthesis would be in the case that a forecast of resource
demands by users is available, based on which coalitions and payoffs will be
determined. We consider either that access networks independently form coalitions and accordingly receive payoffs, or that they share their resources (as if
the core administrator would select them from a common pool) and payoffs
are determined based on the power of each access network in the game.
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3.1 Network Synthesis Game: Definitions
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Theoretical Framework
The network synthesis game is described as follows. Let N = {1, 2, . . . , N} denote the set of players (access networks) and S the set of all possible coalitions,
i.e., the set of all non-empty subsets of N . Let B denote the least amount of
resources needed for accommodating service demands, and let bi denote the
amount of available resources of the ith member of a coalition (members can
be ordered arbitrarily). It will be assumed that the available resources of each
member are known to all members of a coalition 3 .
The characteristic function of the game is

1,
P|S|
if i=1 bi ≥ B
v(S) =
0, otherwise .
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(1)
That is, a coalition has positive value only if the sum of available resources
of its members is greater or equal to the resource threshold B. This definition corresponds to a simple (or 0-1) game; the game is also monotonic since
v(S1 ) ≤ v(S2 ) for all S1 ⊆ S2 . A coalition S is said to be winning if v(S) = 1,
otherwise it is said to be losing. A player i ∈ S is said to be a null player for
coalition S if v(S) = v(S \ {i}). It is generally called a null player if this holds
for every coalition S to which it may belong.
To avoid trivialities, we will generally assume that bi < B ∀i ∈ N , and that
PN
i=1 bi ≥ B.
As will be seen in Sect. 5, the so-called minimal winning coalitions are candidate solutions to the game. A winning coalition is said to be minimal if it
becomes a losing one upon departure of any member. A related notion is that
P|S|
of a by-least winning coalition. 4 Denoting by W (S) = i=1 bi (the sum of the
available resources of the members of S), a coalition S is said to be by-least
3
In any wireless environment there are certain constraints such as signal interference and user mobility that make the estimation of available resources a difficult
task and furthermore, the competing nature of the participating access networks
may urge them to withhold or distort this information. It will be assumed that
appropriate mechanisms and policies are in place that make the available resources
vector (b1 , . . . , bN ) known to all operators. A formal study of this is not part of this
paper.
4 In [11], the authors define a coalition S to be “least-winning” if it is minimal
winning and for any other minimal winning coalition S 0 it holds that W (S) ≤ W (S 0 ).
We introduce a related definition here from the point of view of each player i ∈ S.
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winning with (or for) player i, if it is a minimal winning coalition, it contains
i, and for any other minimal winning coalition S 0 containing i it holds that
W (S) ≤ W (S 0 ).
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3.2 Equivalence to the Weighted Voting Game
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This section demonstrates the equivalence of the network synthesis game to
the weighted voting game, a well-studied paradigm from which many useful
conclusions can directly apply.
Definition 1 A weighted voting game consists of N players and a weight
vector w = (w1 , w2 , . . . , wN ), where wi reflects the “voting weight” of player i.
P
Let W = N
i=1 wi . For a coalition S, the characteristic function of the game is

1,
P|S|
if i=1 wi >
v(S) = 
0, otherwise .
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W
2
(2)
We assume wi ≤ W/2 ∀i ∈ N .
To prove the equivalence, it suffices to show that there exists a one-to-one
P
mapping between vectors w and b = (b1 , . . . , bN ) so that i∈S wi > W/2 iff
P
i∈S bi ≥ B ∀S ⊆ N . A mapping satisfying these requirements is readily
obtained by setting wi = bi /2 and W to a number B ∗ arbitrarily close to B
P
P
such that i∈S bi ≥ B iff i∈S bi > B ∗ . (It is straightforward that such a
number exists, since we have discrete bi values.)
4
Payoff Allocation
Networks, i.e. players, participate in a coalition and offer their resources in return for some benefit (payoff). For example, the NGN platform administrator
may have a fixed amount of money to distribute to access networks in a coalition, as a reward for reserving their resources to handle the specific service(s).
It is considered that all players are independent and rational, and that the
objective of each player is to maximize its payoff. Clearly, which coalition(s)
will finally be formed depends on how payoffs are allocated.
Let the total payoff allocated to the set of players be P and the payoff allocation vector be p = (p1 , p2 , . . . , pN ), such that pi ≥ 0 ∀i = 1, . . . , N and
PN
i=1 pi = P ; an allocation satisfying the above conditions is said to be feasible.
We consider two cases: a) non-transferable payoffs, and b) transferable payoffs
between the access networks. In the first case the access networks get a fixed
7
payoff, which is based on their resource contribution relative to the other members of the coalition. More specifically, we consider in this case that payoffs
are allocated to members of a winning coalition proportionally to their contributions in the coalition. If this winning coalition is K consisting of K ≤ N
access networks with available resources b1 , b2 , . . . , bK , then


 PKbi
pi = 

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(It holds that
PK
j=1 bj
b
j=1 j
P, if i ∈ K
0,
otherwise .
(3)
≥ B.)
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In the second case, we are not interested in how exactly the payoff is allocated to the members of the winning coalition; this allocation can be made in
any way, as long as all members get a payoff greater than zero. Considering
transferable payoffs accounts for cases where the members of a coalition make
side-payments, transferring any part of their payoff to other members of their
coalition. Side-payments can be used as a means to “attract” other players
in a specific coalition. They can also be made in situations where there are
dynamic changes in the resources contributed by the members. As a realistic
example of the latter situation, consider the case that the NS game is performed for an online service, e.g. multicast. A network that participates in
fulfilling the service may experience a sudden increase in workload, due e.g.
to mobility and subsequent re-distribution of users. Such a change might force
the network to “subcontract”, i.e. to transfer part of its payoff to another network in the same coalition during the service session, in order to handle part
of the workload.
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Stable coalitions
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We will use the well-known concept of the core (see, e.g. [5]) to examine the
stability of coalitions formed according to the payoff allocations. Descriptively,
a payoff allocation to a set of N players is in the core of a coalitional game
if there is no coalition wherein each member can get a strictly higher payoff
than dictated by the allocation. Such an allocation, as well as the coalitions
that it induces, can be called stable, since there would not be a consensus to
break these coalitions and form other ones.
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The following definitions have been taken from [5] and adapted to our game.
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In order to apply the core concept, we slightly modify (1) in the transferable
8
payoff case to the following:
v(S) =
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
P,
P|S|
if i=1 bi ≥ B
otherwise .
0,
(4)
That is, when the minimum necessary resources are available, the value of the
characteristic function equals the total payoff.
For an allocation to be in the core of the transferable-payoff game – since
we are not interested in how the payoff is divided among the members of the
coalition – we only require that the worth of the coalition does not exceed the
sum of the payoff allocations.
Definition 2 An allocation p = (p1 , p2 , . . . , pN ) is said to be in the core of
the access network synthesis game with transferable payoffs iff
X
i∈N
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pi = P and
X
pi ≥ v(S), ∀S ⊆ N .
i∈S
Since v(S) takes either the value 0 or P in our game, this trivially reduces to
the requirement that for every winning coalition that could be formed by the
networks, the sum of payoff allocations should always equal P .
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This implies that there is no coalition S such that the players in S could all
do strictly better than as dictated by p by dividing the payoff v(S) among
themselves. (In transferable payoff games we are not interested in how the
payoff is divided among the members of the coalition.)
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For the non-transferable payoff case, we have the following definition:
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Definition 3 An allocation p = (p1 , p2 , . . . , pN ) is said to be in the core of the
P
access network synthesis game with non-transferable payoffs iff i∈N pi = P
and there exists no other payoff allocation y = (y1 , y2 , . . . , yN ) derived according to (3) for which yi > pi , ∀i ∈ S ⊆ N , for any S ⊆ N .
That is, the allocation must be feasible and there should exist no other allocation which gives strictly higher payoff to all members of a coalition.
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A single winning coalition can be mapped to an allocation in the core. In
the non-transferable payoff case, this is defined to be the coalition K, based
on which the payoff vector is derived. In the transferable payoff case, this is
defined as {i ∈ N : pi > 0}, the set of players with positive payoff. We can
then speak about “coalitions in the core”, as the set of winning coalitions for
which their corresponding allocations are in the core.
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Not all winning coalitions are in the core. In fact, we have the following:
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Theorem 4 In both the transferable and non-transferable payoff cases defined
above, only minimal winning coalitions are in the core.
PROOF. In the transferable payoff case, notice that for any non-minimal
winning coalition, a corresponding minimal one can be formed by the players
that are non-null (in the coalition). Then for any payoff allocation to players in
the non-minimal winning coalition, the players in the minimal one can divide
the excess payoff in such a way that they all get strictly higher payoff. In
the non-transferable payoff case, the statement of the theorem follows directly
from (3). 2
Remark 5 This theorem is an adaptation of Riker’s size principle [12] to this
game, which was also shown for weighted voting games in [13].
Hence it is reasonable to direct our attention to minimal winning coalitions.
Let M(N, v) denote the set of all minimal winning coalitions. 5 One could
argue that every minimal winning coalition could potentially be a solution of
the game. However, simple arguments show that the set of possible solutions
can be further reduced.
In the case of transferable payoffs, only minimal winning coalitions which are
also minimal in size for at least one of their members will be in the core. A
coalition S is said to be minimal in size for i ∈ S, iff |S| ≤ |S 0 | ∀S 0 3 i, where
S, S 0 ∈ M(N, v). This is evident because since we insist that no member of a
coalition in the core gets zero payoff, no player would want to transfer part of
its payoff to an additional member. For later usage, we shall let Zi (N, v) be
the set of coalitions which are minimal in size for player i.
In non-transferable payoff games, only coalitions that are by-least winning
for at least one player are solutions in the core of the game. This is a consequence of the proportional payoff allocation rule: if a player i belongs to two
P
P
minimal winning coalitions S and S 0 , then (bi / j∈S bj )P > (bi / j∈S 0 bj )P
P
P
if j∈S bj < j∈S 0 bj , and hence it would get a higher payoff in the by-least
winning coalition. The set of all by-least winning coalitions for player i will
be denoted by Li (N, v), or simply by Li .
We further proceed to study which coalitions are in the inner core of the game.
The inner core [5] is a subset of the core that contains coalitions that are “more
stable”, in the sense that there exists no randomized plan that could prevent
their formation.
5
Although the characteristic function v is defined only for transferable payoff
games, it is also used in this paper in notations concerning the non-transferable
payoff game such as M (N, v) and Li (N, v), since, along with (3), it can be used to
define the latter game.
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Definition 6 A randomized plan is any pair (η(S), y(S)), S ⊆ N , where
η is a probability distribution on the set of coalitions, and y is the vector
of payoff allocations for the members of coalition S, y(S) = (yi(S))i∈S . For
non-transferable payoff games, the inner core is composed of all allocations p
P
P
(or corresponding coalitions) for which S⊇{i} η(S)yi(S) < S⊇{i} η(S)pi, for
some i ∈ N , in all randomized plans (η(S), y(S)).
The inner core concept is used in cases where there is a mediator that invites
individual players to form a coalition which is not known deterministically,
but only with a certain probability distribution. Consider for example that
in the NS game, the platform administrator (mediator) informs the networks
that, in case they don’t arrive to an agreement by themselves, then a coalition
of its choice will be selected, which will be S1 with probability p1 , or S2 with
probability p2 = 1 − p1 . Then, in order for a coalition to be stable, the payoff
given to each network should not be smaller than the mean payoff anticipated
in the coalition of the platform administrator’s choice.
In the NS game, if an access network participates in several coalitions that
are by-least winning with it, then in the non-transferable payoff case it would
get the highest possible reward in every one of these coalitions. This reward
would further be the same in every randomized plan among these coalitions,
and lower for randomized plans containing coalitions other than the by-least
winning.
Since, for a player i, the allocation in a by-least winning coalition is the maximum it can get, there exists no randomized plan that could block these coalitions from forming and hence the latter are in the inner core of the game. We
summarize this in the following theorem.
Theorem 7 In the access network synthesis game with non-transferable payoffs proportional to the available resources of the participating networks, all
coalitions which are by-least winning for at least one of their members are in
the inner core of the game.
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Of all by-least winning coalitions which are in the inner core, we may informally state that those that are by-least winning for all their members are
“most stable”. This will be made formal by defining a new concept of “stability
under uncertainty of formation” in the next section.
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A nice property of the game is formulated in the following
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Theorem 8 In the access network synthesis game with non-transferable payoffs proportional to the available resources of the participating networks, there
exists at least one coalition which is by-least winning for all its members. Further, regardless of the payoff allocation, there exists at least one coalition that
is minimal in size for all its members.
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PROOF. We demonstrate the theorem only for by-least winning coalitions.
The proof for minimal in size coalitions – which of course, shouldn’t depend
on payoff allocations – is similar and is omitted here.
The proof is straightforward when there exists a coalition S, such that i∈S bi =
P
B, since then S is by-least winning for all its members. When i∈S bi > B for
some S ∈ ∪N
/ Lj , then necessarily
i=1 Li , and there exists j ∈ S such that S ∈
P
P
another coalition S1 6= S exists such that S1 ∈ Lj and i∈S1 bi < i∈S bi .
Similarly now, if there exists k ∈ S1 , k 6= j, such that S1 ∈
/ Lk , then there exP
P
ists another coalition S2 ∈
/ {S1 , S} such that S2 ∈ Lk and i∈S2 bi < i∈S1 bi .
Continuing this procedure, since we have a finite number of players, a fiP
nite sequence of coalitions S, S1 , S2 , . . . , Sm is produced, for which i∈S bi >
P
P
P
i∈Sm bi > B and Sm is by-least winning for all
i∈S2 bi > · · · >
i∈S1 bi >
its members. 2
P
338
Hence, in both transferable and non-transferable payoff cases studied, interests
of at least some players coincide.
339
6
337
340
341
342
343
344
345
The Coordination Game
We have established that each access network i would maximize its payoff
and hence prefer one of the coalitions in Zi (N, v) (transferable payoff case) or
T
Li (N, v) (proportional payoff case) to eventually be formed. Unless N
i=1 Zi 6= ∅
TN
in the former, or i=1 Li 6= ∅ in the latter case, there is no mutually preferred
coalition and we are led to a coordination game where at least one player has
conflicting preferences with one or more of the others.
351
In the next subsection, we shall present a possible resolution of such a coordination game, based on the idea of calculating the probability that these
coalitions would randomly form. The analysis applies equally to the transferable and non-transferable payoff cases, and in what follows we shall use
Gi (N, v) (or simply Gi ) to denote either Zi (N, v) or Li (N, v), depending on
which case we study.
352
6.1 Resolution of the game
346
347
348
349
350
(i)
For i = 1, . . . , N, we define the probability of formation Pf (S) to be the
probability that player i would anticipate coalition S to be formed, if player
i participated in it and all other players j ∈ S, j 6= i would independently
choose to participate in one of their preferred coalitions with equal probability.
12
That is,
(i)
Pf (S)
353
354
355
356
357
358
359
Q
 j∈S,
=
j6=i
1
,
|Gj |
if S ∈
0,
j∈S,j6=i Gj
T
otherwise .
(5)
Then it is reasonable that a player would ultimately prefer to participate in
the coalition which has the highest probability of formation, and hence, under
such uncertainty, would offer it the greatest expected payoff.
To appoint a name to this concept of stability, we define a coalition S to be
stable under uncertainty of formation if and only if there exists no other coalition S 0 with a common member with S that anticipates a higher probability
of formation for S 0 .
Definition 9 In the access network synthesis game with transferable or nontransferable payoffs, a coalition S is stable under uncertainty of formation iff
(i)
Pf (S) > 0 ∀i ∈ S and 6
(j)
(j)
@S 0 6= S s.t Pf (S 0 ) > Pf (S) for j ∈ S ∩ S 0 .
360
361
362
363
364
365
This notion would help to refine solutions of the coordination game. It also
creates a formal ground to confine solutions to coalitions which are minimal in
size (transferable payoff case) or by-least winning (proportional payoff case)
for all their members: by definition, only such coalitions are stable under
uncertainty of formation (in view of (5), other coalitions have zero probability
of formation for at least one member).
367
In the following we present two examples of games with non-transferable payoff, to illustrate these ideas.
368
6.1.1 Examples
366
371
These examples consider 6 players (Pi, i = 1, . . . , 6) aiming to form coalitions
in order to satisfy a demand request of 1 unit. Their normalized available
resources are indicated in the tables below.
372
Example 1.
369
370
373
374
375
player
P1
P2
P3
P4
P5
P6
bi /B
0.65
0.53
0.48
0.39
0.24
0.18
We have 9 minimal winning coalitions: {P1,P2}, {P1,P3}, {P1,P4}, {P1,P5,P6},
{P2,P3}, {P2,P4,P5}, {P2,P4,P6}, {P3,P4,P5}, and {P3,P4,P6}. Of these,
{P1,P4} is by-least winning for P1, P4; {P2,P3} for P2, P3; {P1,P5,P6} for
6
s.t means “such that”.
13
381
P5 and {P3,P4,P6} for P6. The two coalitions {P1,P4} and {P2,P3} are byleast winning for all their members and both have probability of formation 1
for all their members, since there are no alternative by-least winning coalitions
for them. Hence they are both stable under uncertainty of formation, and constitute candidate solutions of the game. Based on (5) and Definition 9, it is
easy to calculate that the other coalitions are not.
382
Example 2.
376
377
378
379
380
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
player
P1
P2
P3
P4
P5
P6
bi /B
0.6
0.5
0.4
0.3
0.2
0.1
We have 9 minimal winning coalitions: {P1,P2}, {P1,P3}, {P1,P4,P5}, {P1,P4,
P6}, {P2,P3,P4}, {P2,P3,P5}, {P2,P3,P6}, {P2,P4,P5}, and {P3,P4,P5,P6}.
Of these, {P1,P3}, {P1,P4,P6}, {P2,P3,P6}, {P2,P4,P5}, and {P3,P4,P5,P6}
are by-least winning for all their members. However, only coalitions {P1,P3}
and {P2,P4,P5} are stable under uncertainty of formation. This can be easily shown by calculating all probabilities according to (5). For example, to
(P2)
see that the remaining coalitions are not: Pf ({P2,P3,P6}) = 1/9 < 1/6 =
(P2)
(P1)
(P1)
Pf ({P2,P4,P5}), Pf ({P1,P4,P6}) = 1/9 < 1/3 = Pf ({P1,P3}), and
(P3)
(P3)
Pf ({P3,P4,P5,P6}) = 1/18 < 1/2 = Pf ({P1,P3}).
7
Solutions with Power Indices
So far we have studied payoff allocation when coalitions are explicitly formed
by access networks. For simple (0-1) games, a popular alternative method is
by using power indices. A power index (or value) is commonly used to measure
the influence of a player on the formation of coalitions and most importantly
on the outcome of the game. The notion of power indices can be used as a
naive solution concept for the game itself. One can postulate that, if no specific
payoff allocation rule (e.g., proportional payoff allocation) is specified a priori,
then normalized power indices can be used to allocate payoffs. Alternatively, it
can be used for payoff allocation when there is a common pool in which access
networks share their resources and there is no coalition formation process by
the networks themselves.
Widely used power indices are the Shapley-Shubik value φ (abbreviated here
as SSPI) [6], and the Banzhaf value β (abbreviated here as BPI) [7]. These are
generally sums of the marginal contributions (v(S) − v(S \ {i})) of a player
i to each coalition, weighted by different probability distributions over the
set of coalitions. We say that player i is critical to coalition S if its marginal
contribution is 1, otherwise it is non-critical.
14
The SSPI assumes all permutations of the order that members form a coalition
are equally likely, and is defined by:
φi (N, v) =
X
S⊆N
(S3i)
(|S| − 1)!(N − |S|)!
(v(S) − v(S \ {i})) .
N!
(6)
The BPI assumes, on the other hand, that all possible coalitions that contain
i are equally likely and is defined by:
βi (N, v) =
410
1
2N −1
X
(v(S) − v(S \ {i})) ,
(7)
S⊆N
(S3i)
for i = 1, . . . , N.
Another popular power index based on minimal winning coalitions is the
Holler-Packel index η (HPI) [8]. For simple games, the HPI is defined as:
ηi (N, v) =
X
(v(S) − v(S \ {i}))
S∈M (N,v)
(8)
= |{S ∈ M(N, v) : i ∈ S}| ,
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
for i = 1, . . . , N, where M(N, v) is the set of all minimal winning coalitions.
P
We define also the normalized Holler-Packel value η̄i = ηi / N
i=1 ηi , which represents the proportion of minimal winning coalitions player i is in.
An important property of both the SSPI and the BPI indices in monotonic
simple games is that players with greater weight (contribution) also get a
greater index. This is evident here since if a player i is critical to a coalition
S ∪ {i}, then a player i0 with bi0 > bi is also critical to coalition S ∪ {i0 }. This is
also referred to as “monotonicity of the players’ power indices to the weights”.
Also, based on the equivalence of the game to the weighted voting one, two
useful observations made in [13,11] about the behaviour of the power indices
can be transferred here: First, that restricting our attention to minimal winning coalitions as with the HPI results in weaker players (in our case, players
with relatively smaller available resources) getting higher power, compared to
the measurement with the SSPI and the BPI. Secondly, that with the HPI
the monotonicity of the players’ power indices to their weights may not be
preserved: a player with smaller weight may get a higher HPI ranking than a
player with greater weight.
15
428
429
430
431
432
433
434
435
436
7.1 A New Power Index
For cases where payoffs are attributed to players proportionally to their contributions in the coalition, the analysis in Section 5 has established that all
minimal winning coalitions are not equally likely. Rather, each player has specific preferences to be in one or more coalitions, which are by-least winning
with it.
In this section we use the stability analysis in Section 5 to motivate the introduction of a new index, based on the popularity of all coalitions which are in
∪N
i=1 Li (N, v) (a subset of M(N, v)), and hence in the inner core of the game.
For each minimal winning coalition S ∈ M(N, v), we define as its preference
index ω(S) the total number of preferences it gathers by all players:
ω(S) = |{i ∈ N : S ∈ Li (N, v)}| .
(9)
We define a new index, ζ , which we will call the Popularity Power Index
(PPI), as
X
ω(S)
IiS ,
(10)
ζi (N, v) =
P
k∈M (N,v) ω(k)
S∈M (N,v)
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
where IiS equals 1 if i ∈ S and 0 otherwise. In plain words, the index ζi equals
the probability that, if we were to pick a coalition by asking one player in
N randomly (and further, if when this player had multiple equal preferences,
he would select one of them with equal probability), then a winning coalition
would be selected that contains player i. Hence, this index relates the popularity of minimal winning coalitions a player belongs in, to this player’s power.
As with the other indices, we can also define a normalized form of this index:
P
ζ̄i = ζi / N
i=1 ζi .
8
Behaviour of power indices
This section examines the numerical behaviour of all power indices described
in the previous section.
Power index values are examined for different numbers of players and different
distributions of available resources. Even though the theory extends to many
players, we have selected game instances of a few players only (three, four
and five players) for illustrative purposes. Individual resources for each test
instance presented sum up to the same total resource amount; this is done
in order to better compare results between the different cases for the same
number of players.
16
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
For each set of players we consider different distributions Di (i = 0, . . . , 5)
of available resources, from the case i = 0 where resources are uniformly
distributed between access networks, i.e. each network has equal available
resources, to non-uniform cases (i = 1, . . . , 5), carefully selected to exhibit
varying values of the power indices when available resources are about the
same, or resources are concentrated in only a few of the networks. Specifically,
the following distributions have been selected:
- D0 is a uniform distribution of resources, i.e. where all networks have an
equal amount of resources.
- D1 represents a distribution of resources where the largest amount is concentrated in one “big” network, and all other networks have small and equal
resources. It is worth noting that no coalition satisfies the resource demand
without the contribution of the player with the highest resources.
- D2 to D5 represent distributions in which all players have unequal resources.
In distributions D2 and D3 , there exists again one network that has the
largest portion of required resources, while all the other networks are “small”
players that have small amounts of resources. In the situations considered
in D4 and D5 , the differences in amounts of resources are smaller, and the
distinction between “big” and “small” players is less explicit. In the cases
of 4 and 5 players, it can be seen that there exist winning coalitions that
do not include the player with the highest amount of resources.
To avoid taking absolute values, we have considered available resources of each
player i normalized with respect to the minimum resource requirement B, i.e.,
bi /B. Then, a winning coalition is formed only if the sum of their resources
exceeds one. We have only considered examples where the sum of available
resources of all members is greater than one. For each of the power indices
BPI, SSPI, HPI and PPI, we have examined both the values of the indices as
well as their rankings.
The values of normalized available resources are shown in Tables 1, 3, and 5,
for the cases of three, four, and five players respectively. For each one of these
cases the three power indices are generated also in a normalized form so that
they add up to one. This is done so that we may intuitively relate the power
index of each network to its payoff allocation. The values of indices for these
cases are shown in Tables 2, 4, and 6.
Differences between the indices’ values become more pronounced as the number of players increases (the increased number of possible coalitions allows such
differences to show). In general, the SSPI and BPI give similar values, favoring
the players with greater available resources. (A closer inspection reveals that
SSPI systematically does that to a slightly greater extent than BPI). On the
other hand, the HPI and PPI give a higher power to relatively weaker players.
This is because weaker players have smaller contributions and hence are more
17
Table 1
Instance 1: 3 players
Distribution
b1
B
b2
B
b3
B
D0
0.4
0.4
0.4
D1
0.8
0.2
0.2
D2
0.8
0.3
0.1
D3
0.9
0.2
0.1
D4
0.6
0.35
0.25
D5
0.55
0.45
0.25
Table 2
Instance 1 Indices
Distribution
Index
D0
D1
D2
D3
D4
D5
Player 1
Player 2
Player 3
BPI
0.33
0.33
0.33
SSPI
0.33
0.33
0.33
HPI
0.33
0.33
0.33
PPI
0.33
0.33
0.33
BPI
0.6
0.2
0.2
SSPI
0.67
0.17
0.17
HPI
0.5
0.25
0.25
PPI
0.5
0.25
0.25
BPI
0.5
0.5
0
SSPI
0.67
0.33
0
HPI
0.5
0.5
0
PPI
0.5
0.5
0
BPI
0.6
0.2
0.2
SSPI
0.67
0.17
0.17
HPI
0.5
0.25
0.25
PPI
0.5
0
0.5
BPI
0.33
0.33
0.33
SSPI
0.33
0.33
0.33
HPI
0.33
0.33
0.33
PPI
0.33
0.33
0.33
BPI
0.5
0.5
0
SSPI
0.67
0.33
0
HPI
0.5
0.5
0
PPI
0.5
0.5
0
18
Table 3
Instance 2: 4 players
Distribution
b1
B
b2
B
b3
B
b4
B
D0
0.4
0.4
0.4
0.4
D1
0.85
0.25
0.25
0.25
D2
0.8
0.55
0.15
0.1
D3
0.95
0.45
0.1
0.1
D4
0.6
0.4
0.35
0.25
D5
0.55
0.5
0.3
0.25
Table 4
Instance 2 Indices
Distribution
Index
D0
D1
D2
D3
D4
D5
Player 1
Player 2
Player 3
Player 4
BPI
0.25
0.25
0.25
0.25
SSPI
0.25
0.25
0.25
0.25
HPI
0.25
0.25
0.25
0.25
PPI
0.25
0.25
0.25
0.25
BPI
0.7
0.1
0.1
0.1
SSPI
0.75
0.083
0.083
0.083
HPI
0.5
0.17
0.17
0.17
PPI
0.5
0.17
0.17
0.17
BPI
0.5
0.3
0.1
0.1
SSPI
0.75
0.17
0.042
0.042
HPI
0.4
0.2
0.2
0.2
PPI
0.33
0
0.33
0.33
BPI
0.7
0.1
0.1
0.1
SSPI
0.75
0.083
0.083
0.083
HPI
0.5
0.17
0.17
0.17
PPI
0.5
0
0.25
0.25
BPI
0.33
0.33
0.17
0.17
SSPI
0.33
0.33
0.17
0.17
HPI
0.25
0.25
0.25
0.25
PPI
0.2
0.4
0.2
0.2
BPI
0.33
0.33
0.17
0.17
SSPI
0.33
0.33
0.17
0.17
HPI
0.25
0.25
0.25
0.25
PPI
0.2
0.4
0.2
0.2
19
Table 5
Instance 3: 5 players
Distribution
b1
B
b2
B
b3
B
b4
B
b5
B
D0
0.3
0.3
0.3
0.3
0.3
D1
0.7
0.2
0.2
0.2
0.2
D2
0.85
0.2
0.15
0.15
0.15
D3
0.9
0.25
0.15
0.15
0.05
D4
0.5
0.3
0.3
0.2
0.2
D5
0.45
0.35
0.3
0.25
0.15
Table 6
Instance 3 Indices
Distribution
Index
D0
D1
D2
D3
D4
D5
Player 1
Player 2
Player 3
Player 4
Player 5
BPI
0.2
0.2
0.2
0.2
0.2
SSPI
0.2
0.2
0.2
0.2
0.2
HPI
0.2
0.2
0.2
0.2
0.2
PPI
0.2
0.2
0.2
0.2
0.2
BPI
0.48
0.13
0.13
0.13
0.13
SSPI
0.8
0.05
0.05
0.05
0.05
HPI
0.33
0.17
0.17
0.17
0.17
PPI
0.33
0.17
0.17
0.17
0.17
BPI
0.79
0.05
0.05
0.05
0.05
SSPI
0.8
0.05
0.05
0.05
0.05
HPI
0.5
0.125
0.125
0.125
0.125
PPI
0.5
0
0.17
0.17
0.17
BPI
0.54
0.15
0.15
0.15
0
SSPI
0.8
0.067
0.067
0.067
0
HPI
0.5
0.17
0.17
0.17
0
PPI
0.5
0
0.25
0.25
0
BPI
0.31
0.22
0.22
0.125
0.125
SSPI
0.33
0.18
0.18
0.18
0.15
HPI
0.33
0.2
0.2
0.13
0.13
PPI
0.33
0.17
0.17
0.17
0.17
BPI
0.3
0.22
0.22
0.22
0.04
SSPI
0.33
0.18
0.18
0.18
0.15
HPI
0.23
0.23
0.23
0.23
0.08
PPI
0.33
0
0.33
0.33
0
20
496
497
often found in coalitions which are minimal winning, or by-least winning for
some players.
511
The HPI and PPI are more appropriate indices for the game, since they exclude
a number of coalitions which are not stable and hence would not appear if
access networks formed coalitions independently. Comparing these two indices,
we can argue that the PPI is more fair, in the sense that it only considers stable
coalitions in the inner core that would be formed if payoffs were allocated
proportionally to players’ contributions, i.e. in a fair manner. In Tables 2,
4, 6, we may note that often a player is allocated zero payoff with the PPI,
whereas non-zero with the HPI. These are cases where this player participates
in minimal winning coalitions, but not in any of the by-least winning coalitions.
For example, in Table 2, case D3 , player 2 is allocated 0.25 value with the HPI,
whereas 0 with the PPI, since it does not participate in the by-least winning
coalition. The monotonicity of index values to players’ contributions also does
not hold for the PPI in this example. Table ?? summarizes the performance
of different indices.
512
9
498
499
500
501
502
503
504
505
506
507
508
509
510
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
Conclusions
In this paper, we have introduced and studied the Network Synthesis game
which may arise in a NGN system, when individual access networks have insufficient resources and must form coalitions in order to satisfy service demands.
We have considered both transferable and non-transferable payoff games. The
stability analysis has shown that, in transferable payoff games, only winning
coalitions that are minimal in size for at least one player are in the core. In the
case of non-transferable payoffs allocated proportionally to the players’ contributions, we have shown that all coalitions which are by-least winning for
at least one player are in the inner core. Using the new concept of formation
probability, we finally argued that out of these coalitions, those that are minimal in size or by-least winning for all their members are most likely to form
in the transferable and non-transferable payoff cases respectively.
Furthermore, we argued that payoffs could be allocated according to the normalized values of power indices of the players in the game. The stability analysis has led to the proposal of a new index, called Popularity Power Index,
which associates a player’s value to the number of stable coalitions it participates in; the higher this number, the greater the index value. Compared to
the well-known Shapley-Shubik, Banzhaf and Holler-Packel indices, this index
is more fair in the sense that it only considers stable coalitions that would be
formed if payoffs were assigned in a proportional manner to players’ contributions. Although we have not associated such fairness with any specific metric,
21
534
535
stability of the underlying coalitions under proportional payoff allocation is a
natural qualitative property that a fair index should satisfy.
546
In conclusion, despite the fact that the integration architecture of a NGN environment is yet unknown, our paper constitutes a first step in studying the
economics and cooperation relationships of such an environment with multiple
access networks. This work can be extended by studying factors which may influence the coalition-formation process. Such are the order in which members
are invited in a coalition (e.g., in the proportional payoff case, it is reasonable
to anticipate that a player would participate in the by-least winning coalition
in which it is first invited) or explicit user preferences for operators or QoS
parameters. Finally, a more complete model should take into account negotiations between all interested parties, i.e. between different network operators,
or different network operators and users.
547
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536
537
538
539
540
541
542
543
544
545
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
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