Mechanism design and implementation theoretic perspective of interference coupled wireless systems

This paper investigates the properties of social choice functions, that represent resource allocation strategies in interference coupled wireless systems. The resources can be physical layer parameters such as power vectors or antenna weights. The paper investigates the permissible social choice functions, which can be implemented by a mechanism in either Nash equilibria or dominant strategy — for utility functions representing interference coupled wireless systems. Strategy proofness and efficiency properties of social choice functions are used to capture the properties of non-manipulability and Pareto optimality of solution outcomes of resource allocation strategies, respectively. The analysis indicates certain inherent limitations when designing strategy proof and efficient resource allocation strategies. These restrictions are investigated in an analytical mechanism design framework of interference coupled wireless systems. Furthermore, the paper characterizes the Pareto optimal boundary points (efficient) of utility sets of interference coupled wireless system. An axiomatic framework of interference functions is used to capture interference coupling. The Pareto optimal boundary points for the cases of individual power constraints and a total power constraint are described based on the properties of the underlying interference functions.


I. INTRODUCTION
The operator serving users, has to design and implement resource allocation strategies, extracting the true value of the resources from the users. We utilize the social choice function (SCF) to represent resource allocation strategies in interference coupled wireless systems. The goals of the designed resource allocation strategies can be viewed in terms of social choice, which is simply an aggregation of the preferences of the different users towards a single joint decision.
Each user, can choose their own utility function. For a user -announcing its true utilities to the operator might not be in its best interest, i.e. the users can choose to reveal a utility function, which differs from their true utility functions, so as to obtain more utility. Here the theory of mechanism design comes into play. Mechanism design can be thought as reverse game theory. This paper studies certain desirable properties of SCFs representing resource allocation strategies and has the following main contributions: 1) Let a mechanism implement a SCF (representing a resource allocation strategy) in Nash equilibrium. there exists a "point" in the set of physical layer resources, such that the SCF chooses this point for all possible utility functions in the family of signalto-interference (plus noise) ration (SINR) based utility functions. A similar result can be proved for dominant strategy implementation. We provide examples showing that, this has severe limitations on the designing of SCFs, in turn having limitations on designing resource allocation strategies, e.g. beamforming, minimum mean square error (MMSE) receiver design. 2) a) A resource allocation strategy is nonmanipulable (captured by strategy proofness), if and only if the signal-to-interference(-plus-noise) ratio (SINR) function γ k for a particular user k is a constant function. b) The constant from (2a) is independent of the utility of its own utility u k . c) The constant from (2a) depends on the utility functions u −k = [u 1 , . . . , u k−1 , u k+1 , . . . , u K ] of the other users (users other than user k).
3) The Pareto optimal points are the efficient points of the utility set. Characterization of the permitted resource allocations for efficient and strategy proof SCF for the case of 2 users. 4) Characterization of Pareto optimal boundary points of utility sets for the case of individual power constraints and total power constraints.
We briefly summarize certain previous work done in the framework of wireless or communication systems in relation to mechanism design. Our reference list is by no means comprehensive. [1], [2] calls nodes selfish if they are owned by independent users and their only objective is to maximize their individual goals. The paper presents a game theoretic framework for truthful broadcast protocol and strategy proof pricing mechanism. [3], [4] provide a tutorial on mechanism design and attempt to apply it to concepts in engineering. [5], [6] utilize SINR and power auctions to allocate resources in a wireless scenario and present an asynchronous distributed algorithm for updating power levels and prices to characterize convergence using supermodular game theory. We begin by presenting the the model in interference coupled wireless systems, in Section II below.

II. ANALYTICAL FRAMEWORK
In this paper we shall investigate the case of interference coupled wireless systems. Before we begin to describe our system model and present the relevant definitions, we provide certain notational conventions used in the paper in Section II-A below.

A. Preliminaries and Notation
Matrices and vectors are denoted by bold capital letters and bold lowercase letters, respectively. Let y be a vector, then y l = [y] l is the l th component. Let y −l denote the vector y without the l th component. Likewise G mn = [G] mn is a component of the matrix G. The notation y ≥ 0 implies that y l ≥ 0 for all components l. x y implies component-wise inequality with strict inequality for at least one component. Similar definitions hold for the reverse directions. Finally x = y implies that the vector differs in at least one component. The set of non-negative reals is denoted as R + . The set of positive reals is denoted as R ++ .

B. Interference Coupled Wireless Systems
In a wireless system, the users' utilities can strongly depend on the underlying physical layer. An important measure for the link performance is the SINR. Consider K users with transmit powers p = [p 1 , . . . , p K ] T and K := {1, . . . , K}. The noise power at each receiver is σ 2 . Hence the SINR at each receiver depends on the extended power vector where I k is the interference (plus noise) as a function of p. In order to model interference coupling, we shall follow the axiomatic approach proposed in [7], [8]. The general interference functions possess the properties of conditional positivity, scale invariance and monotonicity with respect to the power component and strict monotonicity with respect to the noise component. For further details, kindly refer to the Appendix VII. The structure of the SINR region depends on the interference coupling in the system. For axiomatic interference functions it is not obvious what would be an appropriate system to define interference coupling. We can define the system as coupled as follows. For given k, r we define such that the function f r (δ, p) = I k (p − δe(r)) is strictly monotone decreasing for 0 ≤ δ ≤ δ r (p) 0 otherwise. This condition can be further relaxed as follows. Instead of requiring the above property for a specific power vector p, we now define the system as "coupled" if there is some arbitrary power vector p such that the matrix [D I (p)] kr = 1. Thereby we obtain the global dependency matrix, which is independent of the choice of power vector p. Definition 1. Global dependency matrix: G I is the global dependency matrix, given by The non-zero entries in G I mark the transmitter/receiver pairs, which are coupled by interference. A zero entry implies that no interference is received, no matter how large the transmission power is. As an example, consider that users are assigned to different orthogonal resources separated by adaptive interference rejection techniques. Note that this coupling model includes the widely used concept of a "link gain matrix" as a special case. In the remainder of this paper we shall use G I in order to analyze the effects of interference coupling on the structure of the boundary. We denote the matrix G I =: G. We assume that G is an irreducible matrix [9], pp. 360-361 (see also the standard reference for nonnegative matrices [10]).
We now introduce certain important properties of interference functions, namely strict monotonicity, strict positivity and the dependency set which will be needed later in the analysis.
Definition 2. Dependency set: Based on the global dependency matrix as defined by (2), we say that L k is the dependency set of user k if This is the set of transmitters, which have an impact on user k. Such a framework is not limited to cellular wireless networks. In a general multi-point to multi-point system, all users interfere with every other users. Example 1. An example of the dependency set when all users interfere with each other is given by L k = {j ∈ K | j = k} for the case of no self interference. For the case with self interference an example of the dependency set is L k = K. Definition 3. Strict monotonicity: I k (p) is said to be strict monotone on their respective dependency set L k if p (1) ≥ p (2) with p (2) ).
In other words I k (p) is strict increasing in at least one component.
Definition 4. Strict positivity: I k (p) for all k ∈ K is said to be strict positive on the dependency set L k , if for p ≥ 0, p = [0, . . . , 0] T and p l > 0 for l ∈ L k , i.e. p is not the all-zero vector then we have that I(p) > 0. I 1 , . . . , I K are strict positive on their dependency sets L 1 , . . . , L K respectively. From a practical point of view strict positivity on the dependency set seems very natural in wireless systems. None the less, it is an important mathematical restriction whose impact shall be noticed in Section VI in proving the desired results. 2) I(p) = k p ω k k , ω ≥ 0, K k=1 ω k = 1 is not a strict positive interference function on the dependency set.
Until now we have focused on interference coupling aspects, where interference is a function of the powers of the various users and noise. We shall now analyze the resulting utility functions and utility sets in Section II-C.

C. Utility Sets based on SINR
We are particularly interested in analyzing the class of utility functions, which is function of SINR, given by (1). The utility functions, which shall be introduced are motivated based on two factors listed below.
• Users in a wireless system coupled by interference and can be competitive in nature. • Performance indicators in wireless systems are influenced by physical layer parameters.
Definition 5. SINR based utility (SBU) function: For user k, u k is said to be an SBU function, if there exists a strict monotone increasing continuous function q and an interference function I k such that Utilities: QoS region Let u = [u 1 , . . . , u K ] ∈ U K , where U K is the family of SBU functions for K users. In our paper, "utility" can represent certain arbitrary performance measures, which depend on the SINR by a strictly monotone and continuous functioñ u defined on R + . The utility of user k is An example of the above case if capacity:ũ(x) = log(1+x).
In the following performance indicators, we would like to minimize the objective function, e.g. MMSE:ũ(x) = 1/(1+ x), BER:ũ(x) = Q( √ x) and high-SNR approximation of We shall analyze the Pareto optimal boundary of certain sets for the following cases. 1) Utility sets with individual power constraints: In this case we have the following power vector p ∈ P, here 2) Utility sets with a total power constraint: In this case we again have the following power vector p ∈ P, here P = R K+1 + , with a total power constraint on the vector p ∈ R K + of k∈K p k ≤ P tot . Example 3. Consider a 2 user scenario, where the resources at the physical layer are only the powers of the users, i.e. R = P, with k∈K p k ≤ P total and the utility function is defined by (5). With this scenario, Figure 1 displays the concepts of the set of power vectors P and the corresponding quality of service (QoS) set of utilities Q 2 .

III. MECHANISM DESIGN
In this section we review certain mechanism design and implementation theoretic notation [11], in the context of interference coupled wireless systems. The set of users in the system is defined as K := {1, . . . , K}. We assume that the number of users K ≥ 2. Let R be an arbitrary set of outcomes at the physical layer, where R := × k∈K R k and r k ∈ R k . Resources at the physical layer are power, antenna weights etc. A combination of these could also be considered as resources and modeled by our framework.
Example 4. Consider a SIMO uplink scenario with a sum power constraint or a MISO downlink scenario with a sum power constraint P total and beamforming vectors for the users being ω k , for k ∈ K. The set of resources R can be represented in this scenario as follows.

A. Social Choice Functions and Mechanisms
Each user k has a preference relation defined over the set of outcomes R, which admits a numerical representation u k : R → R + . Different users in a wireless systems could have different preferences as to what they wish the solution outcomes of a resource allocation strategy should be. We shall utilize the SCF to characterize resource allocation strategies. If a particular property (axiom) is satisfied by the SCF, then the corresponding property is satisfied by the resource allocation strategy, i.e. we utilize certain properties (axioms) to emulate desirable and natural properties of resource allocation strategies.
A SCF aggregates the preferences of all the users into a social choice for the entire system, i.e. the solution outcome of a resource allocation strategy. Definition 6. Social choice function: A SCF is a function f : U K → R that associates with every u ∈ U K an unique outcomes f (u) in R. Figure 2 depicts an SCF and the corresponding domain and range. We shall now introduce the mechanism, whose properties shall be investigated throughout the paper. Definition 7. Mechanism: A mechanism is a function g : S K → R that assigns to every strategy K-tuple s ∈ S K a unique element r ∈ R.
A game form is a special case of a mechanism. A mechanism is a procedure for determining outcomes (see Figure 2). Who gets to choose the mechanism, i.e. who is mechanism designer depends on the scenario in question, e.g. base station, operator, regulator. A strategy is a complete contingent plan or decision rule that says what a user will do at each of its information sets. Let S k be the strategy set of a user k ∈ K and S K := × k∈K S k be the strategy set of the set of users K. We shall now present the implementation theoretic concepts in Section III-B below.

B. Implementation Concepts
Let g(S k , s −k ) be the attainable set of user k at s −k , i.e. the set of outcomes that user k can induce when the other users select s −k . For k ∈ K, u k ∈ U k and a power vector r ∈ R, let be the weak lower contour set of user k with u k at power vector r. We formalize certain desirable properties of resource allocation strategies by means of an axiomatic framework to capture these properties. Our interest being in exploring the interplay between the axiomatic framework and the possible implementable resource allocation strategies.
Definition 8. Nash equilibrium: Given a mechanism g : S K → R, the strategy profile s ∈ S K is a Nash equilibrium of g at u ∈ U K if for all k ∈ K, g(S k ), s −k ⊆ L(g(s), u k ) Let N g (u) be the set of Nash equilibria of g at u.
Definition 9. Nash equilibrium implementation: The mechanism g implements the SCF f in Nash equilibrium, if for each utility function K-tuple u ∈ U K , the following two conditions are fulfilled.
1) There exists a strategy K-tuple s ∈ N g (u) such that g(s) = f (u). 2) For any strategy K-tuple s ∈ N g (u), g(s) = f (u).
Remark 1. The SCF f is Nash implementable if there exists a mechanism that implements f in Nash equilibria.
The second condition in Definition 9 as it ensures, that irrespective of the choice of the strategy K-tuple in the set N g (u) -we always obtain the same outcome in the set of alternatives, namely f (u). Such a requirement is essential for implementation, since otherwise, we would not be in a position to characterize the properties of the SCF f .

Definition 10. Dominant strategy:
The strategy s k ∈ S k is a dominant strategy for user k ∈ K of g at u k ∈ U k if for Let DS g k (u k ) be the set of dominant strategies for user k of mechanism g at utility function u k .
Remark 2. The strategy K-tuple s ∈ S K is a dominant strategy equilibrium of g at utility K-tuple u ∈ U K if for all users k ∈ K, s k ∈ DS g k (u k ). Let DS g (u) be the set of dominant strategy equilibria of mechanism g at utility K-tuple u.
Definition 11. Dominant strategy implementation: A mechanism g implements a SCF f in dominant strategy equilibria if for each utility K-tuple u ∈ U K , 1) there exists a strategy K-tuple s ∈ DS g (u) such that g(s) = f (u) and 2) for any strategy K-tuple s ∈ DS g (u), g(s) = f (u).
The SCF f is dominant strategy implementable if there exists a mechanism that implements f in dominant strategy equilibria. The mechanism g is called the direct revelation Remark 3. We do not distinguish between the SCF f and the direct revelation mechanism associated with the SCF f . Throughout the rest of the paper, when we say the SCF f , we also mean the direct revelation mechanism g associated with the SCF f . We now present certain desired properties of SCF in Section III-C.

C. Desired Properties of Social Choice Functions
In this section we shall investigate certain desirable properties of SCFs. We begin by presenting the following example.
Example 5. Consider a wireless system of K users, where the strategy set of any user k ∈ K is given by 1 User k has an utility function u k , having an efficient solution outcome. The user is aware that, a better utility function will result in a better resource allocation (from its perspective) (since it is efficient), i.e. ifû k > u k andû = (û k , u −k ), u = (u k , u −k ), then γ k (f (û)) > γ k (f (u)). Therefore, we have u k (p) > u k (p). Then, of course a user has an incentive to misrepresenting its utility function.
By misrepresenting its utility function, a user can manipulate the outcome of a resource allocation strategy. Avoiding such behaviour is a desired property from the perspective of an operator or a regulator. The property, that a particular resource allocation strategy is non-manipulable is emulated by the SCF f satisfying the property strategy proofness.
Definition 12. Strategy Proof : A SCF f is said to be strategy-proof, if for all users k ∈ K and for all utility functions u k ,û k ∈ U, ∀û −k ∈ U K−1 , we have that A mechanism is said to be strategy-proof if it reveals the users true utilities.
, for all users k ∈ K and u k (r) > u k (f (u)) for some user k ∈ K.
Efficiency from the point of view of wireless communication (physical layer perspective) of the resource allocation strategies, implies choosing an operating point on the Pareto boundary of the feasible utility region [12].

Definition 14. Option set:
The option set of a user k ∈ K, given a utility function K-tuple u −k ∈ U K is the set {r ∈ R | ∃u k ∈ U, such thatf (u k , u −k ) = r}. (7) where r is a resource vector.
The option set O k , is the set of resources for all the users, which user k can influence with its utility function, given the utility function (K-1)-tuple u −k ∈ U K−1 . For certain wireless scenarios and corresponding domains of rate regions, we would like to characterize resource allocation strategies, which satisfy certain desirable properties from the axiomatic framework, which can be implemented using mechanisms (see Figure 3). Having equipped ourselves with

IV. NASH IMPLEMENTATION AND DOMINANT STRATEGY IMPLEMENTATION
In this section we present certain results pertaining to Nash implementation and dominant strategy implementation for the family of SINR based utility functions. We shall investigate the implementation aspects of resource allocation strategies in wireless networks. We begin by presenting Lemma 1, which characterizes the Nash equilibrium properties of a strategy K-tuple. Lemma 1. Let a particular strategy K-tuple s be in N g (u), where u ∈ U K . Then, the strategy K-tuple s is in N g (ũ), for allũ ∈ U K , i.e. the strategy K-tuple s is independent of the utility K-tuple u.
Proof: If a particular strategy K-tuple s ∈ N g (u), then we have that g(S k , s −k ) ⊆ L(g(s), u k ) for all users k ∈ K. This implies that, γ k (g(S k , s −k )) ≤ γ k (g(s)) for all k ∈ K. Hence, we have the desired result.
We shall now develop the connection between the Nash equilibrium and a SCF f , which can be implemented in Nash equilibrium. The Nash equilibrium implementation and the dominant strategy implementation results are presented in the Lemma below.
Based on the above lemmas we are now in position to present our main result pertaining to implementation of resource allocation strategies.
⇐=: The other direction can be easily verified. Hence we skip the proof.
The proof for the second statement of the theorem can be carried out in a similar manner. Example 6. Consider a multiuser multiple access channel (MAC), with a beamforming array at the basestation [13], [14]. For fixed channels, the optimal beamforming weight vectors ω opt k for the k th user, with respect to maximizing γ k (p, ω opt k ) can be calculated. The optimal SINR for the k th can be written as: γ k (p, ω opt k ) = p k h H k (σ 2 I + j =k p j h j h H j ) −1 h k where p k , h k and σ 2 k are the power, the channel vectors at the basestation array and the noise for the k th user respectively. The interference function for the k th user is, The structure of the feasible utility region depends on several factors, for instance, the receiver strategy. For one set of beamformers ω k , ∀k ∈ K corresponds to one particular utility region U(P total , ω) for fixed channels, where ω = [ω 1 , . . . , ω K ] and P total is the total power constraint.
Let a mechanism g implement f (γ This section investigates the property of strategy proofness along with efficiency and the limitations it imposes while obtaining resource allocation strategies. We begin by presenting a result, which states the following: a SCF f is strategy proof if and only if for all users k ∈ K, the outcome of the resource allocation for the k th user, i.e γ k (r) is a constant. The constant γ k (r) is independent of its own utility function u k ∈ U. However, it (γ k (r)) is dependent on the utility functions u −k ∈ U K−1 of the other users j ∈ K\k.
Theorem 2. For all users k ∈ K, for all utility function K-tuples u −k ∈ U K−1 , a SCF f is strategy proof, if and only if there exists a constant c k (u −k ) > 0, where for all resource vectors r ∈ O k (u −k ), γ k (r) = c k (u −k ), where γ k is the SINR function of the k th user.
Proof: The proof is available in [15]. We now present a result for 2 users, which shows the restriction of the available SCF f , if we want them to satisfy the properties of strategy proofness and efficient, i.e. the solution outcome of the resource allocation strategy is nonmanipulable and is Pareto optimal.
Theorem 3. Let the number of users K = 2. Then SCF f is efficient and strategy proof, if and only if there exists a resource vector r * ∈ R with γ(r * ) a Pareto optimal resource allocation and utility function K-tuple u ∈ U K with f (u) = r * .
Proof: The proof is available in [15]. To complement the results of efficient and strategy proof, we investigate the Pareto optimal points of utility sets in Section VI below.

VI. ANALYSIS OF THE PARETO OPTIMAL BOUNDARY OF UTILITY SETS
We shall investigate the property of boundary points, namely Pareto optimality of utility sets. Let S(I) be the SINR set depending on the interference function I. The characteristics of S(I) depends on the properties of interference balancing function C(γ, I), which in turn depends on the properties of the underlying interference functions I 1 , . . . , I K . Let γ k be the inverse function of φ k then γ k (q k ) is the minimum SINR level needed by the k th user to satisfy the QoS target q k . Let q ∈ Q be a vector of QoS values. QoS values q ∈ Q are feasible if and only if the interference balancing function C(γ(q), I) ≤ 1. The QoS feasible set is the sub-level set We are specifically interested in the boundary of Q, which is characterized by C(γ(q), I) = 1. The boundary is denoted as ∂Q. We shall now describe what we mean by weak Pareto optimal and Pareto optimal boundary points respectively.
From a practical point of view, this implies that it is not possible to collectively improve the performance of all the users in the system.
From a practical point of view, this implies that it is not possible to improve the performance of one user without simultaneously decreasing the performance of another user. Lemma 3. A boundary point q ∈ ∂Q is Pareto optimal if and only if γ(q) ∈ ∂S(I) is Pareto optimal.
With Lemma 3 we know that for any utility set according to the above definition we can analyze Pareto optimality by focusing on the underlying SINR set.

A. Analysis of Pareto Optimal Boundary of Utility Sets with Individual Power Constraints
In this section we shall analyze the properties of boundary points of utility sets with individual power constraints. We begin by presenting the structure of the QoS region for individual power constraints.
1) Structure of the QoS Region for Individual Power Constraints: Consider the SINR feasible region for users K := {1, . . . , K} with individual power constraints p ≤ p max , which is defined as the sub-level set where γ is a vector of SINR values, whose feasibility is determined by the min-max optimum (see e.g. [16]).
The structure of the SINR set S(I, p max ) depends on the properties of C(γ, I, p max ), which in turn depends on the properties of the underlying interference functions I 1 , . . . , I K as well as on the chosen power constraints p max . The corresponding QoS values q ∈ Q are feasible if and only if C γ(q), I, p max ≤ 1. The QoS feasible set is the sub-level set We are now interested in the boundary of Q, which is characterized by C(γ(q), I, p max ) = 1. The boundary is denoted as ∂Q. Since I 1 , . . . , I K are strict monotonic in p K+1 , there exists a γ > 0 such that p = p(γ), p k ≤ p max k , for K ≥ 2 with γ k I k (p) = C(γ, I, p max )p k , 1 ≤ k ≤ K. We now introduce the restricted global dependency matrix, which shall be used to obtain certain results in the following section.
Definition 17. Restricted global dependency matrix: G the global dependency matrix is a K × (K + 1) matrix and the dependency on noise is clear for every I 1 , . . . , I K . Then where G Res is the restricted global dependency matrix of dimension K × K and [1, . . . , 1] T is representing the dependency on noise.
G Res measures the "cross talk" in the system, i.e. the dependency between the users due to direct interference (as opposed to indirect dependency due to power constraints).
2) Pareto Optimal Boundary of Utility Sets with Individual Power Constraints: In the case of individual power constraints the structure of the SIR region and in turn of certain utility regions using (5) depends not only on the interference coupling but also on the noise. The following theorem describes the conditions for the boundary of utility sets with individual power constraints to be Pareto optimal.
Theorem 4. Let interference functions I 1 , . . . , I K be strict monotone on their respective dependency sets. Then each boundary point is Pareto optimal if and only if G Res is irreducible.
Proof: The proof is available in [17]. The above result says that for all SIR vectors γ, such that we can find a power vector p ≤ p max , we have that γ k ≤ p k /I k (p), 1 ≤ k ≤ K, p ≤ p max and there exists an index k 0 such that γ k0 < p k0 /I k0 (p), for all 1 ≤ k ≤ K. Then we can find a power vector 0 < p * < p such that γ k < p * k /I k (p * ) if and only if the matrix G Res is irreducible. The above result has investigated the Pareto boundary. Furthermore knowing the properties when the boundary points are Pareto optimal aid us in designing appropriate algorithms for resource allocation and utility maximization.
Remark 4. For downward comprehensive sets in the positive orthant, e.g. feasible QoS sets of feasible SINR -Pareto optimality of the boundary points is connected with the convexity of the set. Convexity helps in obtaining (at least) easy numerical solutions to the concerned problems.
Example 8. Consider a 2-user Gaussian multiple access channel (MAC) with successive interference cancellation, normalized noise σ 2 n = 1 and a given decoding order 1, 2. The SINR of the users are SINR 1 (p) = p1 p2+1 and SINR 2 (p) = p 2 . Assuming power constraints p 1 ≤ p max 1 = 1 and p 2 ≤ p max 2 = 1, we obtain an SINR region. Consider a boundary point γ, which is achieved by p * = [p max 1 /2, p max 2 /2] T . Hence p * ∈ P(γ, p max ). This vector achieves the SINR p with component-wise minimum power. However power vector p * is not the only element of P(γ, p max ). Due to interference cancellation we can increase the power, hence the SINR of user 1, without reducing the SINR of user 2. If both users transmit with maximum power p max then the corner pointγ is achieved. In addition, this power vector is contained in P(γ, p max ) sinceγ ≥ γ, such that the SIR targets γ are still fulfilled. For an arbitrary power vector p ∈ P(γ, p max ), consider the fixed point iteration p (n+1) k = γ k I k (p (n) ), p (0) k = p k , for all k ∈ K. The limit p * = lim n→∞ p (n) > 0 is special since it achieves the SIR vector γ with component-wise minimum power.

B. Analysis of the Pareto Optimal Boundary of Utility Sets with a Total Power Constraint
In this section we shall analyze the properties of boundary points of utility sets with a total power constraint. Here we consider interference functions from the Yates framework, which are strictly monotonic in the noise component. These results have direct impact on practical resource allocation strategies for utility sets of the specified type. We begin by presenting the structure of the QoS region for a total power constraint.
1) Structure of the QoS Region with a Total Power Constraint: The SINR region under a sum power constraint (total power constraint) is defined as S(I, P tot ) = {γ ∈ R K + | C(γ, I, P tot ) ≤ 1}, where C(γ, I, P tot ) = inf p>0, p 1≤Ptot max k∈K γ k I k (p) p k .
We shall see in the next section (Section VI-B) that the sum-power constrained region S(I, P tot ) is relatively easy to handle since the SINR tradeoffs are caused due to the sharing of a common power resource.
2) Pareto Optimal Boundary of Utility Sets with a Total Power Constraint: Let us assume that the sum of all transmission powers in limited by P tot . The next result shows that for arbitrary interference functions from the Yates framework, which are strictly monotonic with respect to noise, the boundary points of utility sets with a total power constraint and no self interference are Pareto optimal.
Theorem 5. Let I k for all k ∈ K be arbitrary interference functions. Then for all 0 < P tot < +∞, all boundary points of S(I, P tot ) are Pareto optimal.
Proof: The proof is available in [17].
Example 9. Consider a 2-user Gaussian broadcast channel (BC) with dirty paper coding normalized noise σ 2 n = 1. The SINR of the users γ = [γ 1 , γ 2 ] T are γ 1 = p1 p2+1 and γ 2 = p 2 . Then we have that the powers of the individual users are p 1 = γ 1 (p 2 +1) = γ 1 (γ 2 +1) and p 2 = γ 2 . Assuming a total power constraint p 1 + p 2 ≤ P tot , we obtain the SINR region. We have that the SINR of user 2 is given by γ 2 = Ptot+γ1 1+γ1 . The characterization of the boundary points of feasible utility sets for the special case of log-convex interference functions is discussed in [18]. Note that we require that I(p) is strictly monotone with respect to the last component p K+1 . An example is I(p) = v T p+σ 2 , where v ∈ R K + is a vector of interference coupling coefficients. The axiomatic framework A1-A4 is connected with the framework of standard interference functions [7]. The details about the relationship between the model A1-A4 and Yates' standard interference functions were discussed in [8] and further investigated in [19]. For the purpose of this paper it is sufficient to be aware that there exists a connection between these two models and the results of this paper are applicable to standard interference functions.