Budget Allocation in a Competitive Communication Spectrum Economy

This study discusses how to adjust “monetary budget” to meet each user’s physical power demand or balance all individual utilities in a competitive “spectrum market” of a communication system. In the market, multiple users share a common frequency or tone band and each of them uses the budget to purchase its own transmit power spectra (taking others as given) in maximizing its Shannon utility or pay-oﬀ function that includes the eﬀect of interferences. A market equilibrium is a budget allocation, price spectrum


Introduction
The competitive economy equilibrium problem of a communication system consists of finding a set of prices and distributions of frequency or tone power spectra to users such that each user maximizes her utility, subject to her budget constraints, and the limited power bandwidth resource is efficiently utilized. Although the study of the competitive equilibrium can date back to Walras [8] work in 1874, the concepts applied to a communication system just emerged few years ago because of the great advances in communication technology recently. In a modern communication system such as cognitive radio or digital subscriber lines (DSL), users share the same frequency band and how to mitigate interference is a major design and management concern. The Frequency Division Multiple Access (FDMA) mechanism is a standard approach to eliminate interference by dividing the spectrum into multiple tones and pre-assigning them to the users on a non-overlapping basis.
However, this approach may lead to high system overhead and low bandwidth utilization. Therefore, how to optimize users' utilities without sacrificing the bandwidth utilization through spectrum management becomes an important issue. That is why the spectrum management problem has recently become a topic of intensive research in the signal processing and digital communication community.
From the optimization perspective, the problem can be formulated either as a noncooperative Nash game ( [4], [7], [12], [9]); or as a cooperative utility maximization problem ( [2], [14]). Several algorithms were proposed to compute a Nash equilibrium solution (Iterative Waterfilling Algorithm (IWFA) [4], [12]; Linear Complementarity Problem (LCP) [7]) or globally optimal power allocations (Dual decomposition method, [3], [6], [13]) for the cooperative game. Due to the problem's nonconvex nature, these algorithms either lack global convergence or may converge to a poor spectrum sharing strategy. Moreover, the Nash equilibrium solution may not achieve social communication economic efficiency; and, on the other hand, an aggregate social utility maximization model may not simultaneously optimize each user's individual utility.
Recently, Ye [11] proposed a competitive economy equilibrium solution that may achieve both social economic efficiency and individual optimality in dynamic spectrum management. He proved that a competitive equilibrium always exists for the communication spectrum market with Shannon utility for spectrum users, and under a weak-interference condition the equilibrium can be computed in a polynomial time. In [11], Ye assumes that the budget is fixed, but this paper deals how adjusting the budget can further improve the social utility and/or meet each individual physical demand.
This adds another level of resource control to improve spectrum utilization.
This study investigates how to allocate budget between users to meet each user's physical power demand or balance all individual utilities in the competitive communication spectrum economy. We prove that 1. A competitive equilibrium that satisfies each user's physical power demand always exists for the communication spectrum market with Shannon utilities if the total power demand is less than or equal to the available total power supply. Computational results and comparisons between the competitive equilibrium and Nash equilibrium solutions are also presented. The simulation results indicate that the competitive economy equilibrium solution provides more efficient power distribution to achieve a higher social utility in most cases. Besides, the competitive economy equilibrium solution can make more users to obtain higher individual utilities than the Nash equilibrium solution does in most cases. Moreover, the competitive economy equilibrium takes the power supply capacity of each channel into account, while the Nash equilibrium model assumes the supply unlimited where each user just needs to satisfy its power demand.
The remainder of this paper is organized as follows. The mathematical notations are illustrated in Section 2. Section 3 describes the competitive communication spectrum market considered in this study. Section 4 formulates two competitive equilibrium models that address budget allocation on satisfying power demands and budget allocation on balancing individual utilities. Section 5 demonstrations a toy example of two users and two channels. Section 6 describes how to solve the market equilibrium and presents the computational results. Finally, conclusions are made in the last section.

Mathematical Notations
First, a few mathematical notations. Let R n denote the n-dimensional Euclidean space; R n + denote the subset of R n where each coordinate is non-negative. R and R + denote the set of real numbers and the set of non-negative real numbers, respectively. Let X ∈ R mn + denote the set of ordered m-tuples X = (x 1 , ..., x m ) and letX i ∈ R (m−1)n + denote the set of ordered (m − 1)-tuplesX = (x 1 , ..., For each i, suppose there is a real utility function u i , defined over X. Let A i (x i ) be a subset ofX i defining for each pointx i ∈X i , then the sequence [X 1 , ..., X m , u 1 , ..., u m , A 1 (x 1 ), ..., A m (x m )] will be termed an abstract economy. Here A i (x i ) represents the feasible action set of agent i that is possibly restricted by the actions of others, such as the budget restraint that the cost of the goods chosen at current prices dose not exceed his income, and the prices and possibly some or all of the components of his income are determined by choices made by other agents. Similarly, utility function u i (x i ,x i ) for agent i depends on his or her actions x i , as well as actionsx i made by all other agents. Also, denote x j = (x 1j , ..., x mj ) ∈ R m for a given x ∈ X.

Competitive Communication Spectrum Market
Let the multi-user communication system consist of m transmitter-receiver pairs sharing a common frequency band discretized by n tones. For simplicity, we will call each of such transmitter-receiver pair a "User". Each user i will be endowed a "monetary" budget w i > 0 and use it to "purchase" powers, x ij , across tones j = 1, ..., n, from an open market so as to maximize its own utility where x i = (x i1 , ..., x in ) ∈ R n + andx i ∈ R (m−1)n + are power units purchased by all other users, and p j is the unit price for tone j in the market.
A commonly recognized utility for user i, i = 1, ..., m, in communication is the Shannon utility [5]: where parameter σ ij denotes the normalized background noise power for user i at tone j, and parameter a i kj is the normalized crosstalk ratio from user k to user i at tone j. Due to normalization we have a i ij = 1 for all i, j. Clearly, u i (x i ,x i ) is a continuous concave and monotone increasing function in There are four types of agents in this market. The first-type agents are users. Each user aims to maximize its own utility under its budget constraint and the decisions by all other users.
The second-type agent, "Producer or Provider", who installs power capacity supply s j ≥ 0 to the market from a convex and compact set S to maximize his or her utility. We assume that they are fixed ass in this paper, and i d i ≤ js j , that is, the total power demand is less than or equal to the available total power supply.
The third agent, "Market", sets tone power unit "price" p j ≥ 0, which can be interpreted as a "preference or ranking" of tones j = 1, ..., n. For example, p 1 = 1 and p 2 = 2 simply mean that users may use one unit ofs 2 to trade for two units ofs 1 .
The fourth agent, "Budgeting", allocates "monetary" budget w i > 0 to user i from a bounded total budget, say i w i = m.

Market
In this section, we discuss how to adjust "monetary" budget to satisfy each user's physical power demand or to balance all individual utilities in a competitive spectrum market.

Budget Allocation on Satisfying Individual Power Demands
The first question is whether or not the "Budgeting" agent can adjust "monetary budget" for each user to meet each user's desired total physical power demand d i that may be composed from any tone combination. We give an affirmative answer in this section.
A competitive market equilibrium is a power distribution [x * 1 , ..., x * m , p * , w * ] such that • (User optimality) x * i is a maximizer of (1) givenx * i , p * and w * i for every i.
This condition says that if tone power capacitys j is greater than or equal to the total power consumption for tone j, m i=1 x * ij , then its equilibrium price p * j = 0.
• (Budgeting according to demands) Given x * , w * is a maximizer of This condition says that if user i's power demand is not met, that is, d i − j x ij > 0, then one should allocate more or all "money budget" to user i. And any budget allocation is optimal if d i − j x ij ≤ 0 for all i, that is, if every user's physical power demand is met.
Since the "Budgeting" agent's problem is a bounded linear maximization, and all other agents' problems are identical to those in Ye [11], we have the following corollary: The communication spectrum market with Shannon utilities has a competitive equilibrium that satisfies each user's tone power demand, if the total power demand is less than or equal to the available total power supply.
Now consider the KKT conditions of (1): where The complete necessary and sufficient conditions for a competitive equilibrium with satisfied power demands can be summarized as: Note that the conditions (p * ) T x * i = w i for all i are implied by the conditions in (4): multiplying x * i ≥ 0 to both sides of the first inequality, we have (p * ) T x * i ≥ w i for all i, which, together with other inequality conditions in (4), imply that is, every inequality in the sequence must be tight, which implies (p * ) T x * i = w i for all i.
On the other hand, the 4-6th conditions in (4) are optimality conditions of budget allocator's linear program, where λ is the dual variable. Then, we have a characterization theorem of a competitive equilibrium that satisfies power demands.
Theorem 1. Every equilibrium of the discretized communication spectrum market with the Shannon utility that satisfies power demands has the following properties 1. p * > 0 (every tone power has a price); for all i, j (every user only purchases most valuable tone power).
Proof. Note that Since w i cannot be zero for all i, there is at least one i such that so that the first inequality of (4) implies that p * > 0.
The second property is from ( The third is from (p * ) T x * i = w i for all i and i x * i =s.
We prove the fourth property by contradiction. Suppose, d i − j x * ij > 0 for i ∈Ī for a non-empty index setĪ. Then, w i = 0 for all i ∈Ī so that x * i = 0 for all i ∈Ī. Then, which is a contradiction to the assumption i d i ≤ js j .
The last one is from the complementarity condition of user optimality.
The fourth property of Theorem 1 implies that equilibrium conditions (4) can be simplified to Note that the constraint i w * i = m is merely a normalizing constraint and it can be replaced by other type of normalizing constraint such as i w * i ≥ 1. Moreover, multiple competitive equilibria may exist due to the non-convexity of the optimality conditions of the spectrum management problem with minimal user power demands.

Budget Allocation on Balancing Individual Utilities
The second question is whether or not the "Budgeting" agent can adjust "monetary budget" for each user such that a certain fairness is achieved in the spectrum market; for example, every user obtains the same utility value, which is also a critical issue in spectrum management. We again give an affirmative answer in this section.
Here, a competitive market equilibrium is a density point [ • (Budgeting according to individual utilities) Given x * , w * is a minimizer of This condition says that if user i's utility is higher than any others', that is, , then one should shift "money budget" from user i to user j. And any budget allocation is optimal if u i (x * i ,x * i ) are identical for all i, that is, if every user has the same utility value.
Since the "Budgeting" agent's problem is again a bounded linear maximization, and all other agents' problems are identical to those in Ye [11], we have the following corollary: The communication spectrum market with Shannon utilities has a competitive equilibrium that balances each user's utility value.
The complete necessary and sufficient conditions for a competitive equilibrium with balanced utilities can be summarized as: Note that the conditions (p * ) T x * i = w i for all i are implied by the conditions in (6). On the other hand, the 4-6th conditions in (6) are optimality conditions of budget allocator's linear program for balancing utilities, where λ is the dual variable.
Again, we have a characterization theorem of a competitive equilibrium that balances individual utilities.
Theorem 2. Every equilibrium of the discretized communication spectrum market with the Shannon utility that balances individual utilities has the following properties 1. p * > 0 (every tone power has a price); are identical for all i (all user utilities are the same); for all i, j (every user only purchases most valuable tone power).
Proof. The proof of properties 1,2,3 and 5 are the same as Theorem 1. The fourth property is from the 5th condition of (6). If w i = 0, then the user can not participate the game. Therefore, w i > 0 ∀i by the 5th condition of (6), which implies all user utilities are identical.
The fourth property of Theorem 2 implies that equilibrium conditions (6) can be simplified to

An Illustration Example
Consider two channels f 1 and f 2 , and two users x and y. Let the Shannon utility function for user and one for user y be and let the aggregate social utility be the sum of the two individual user utilities.
Assume a competitive spectrum market with power supply for two channels is s 1 = s 2 = 2 and the initial endowments for two users is w x = w y = 1. Then the competitive solution is p 1 =3/5 and p 2 =2/5, proposed method, we can adjust the initial budget endowments to w x = 6/5 and w y = 4/5, then the equilibrium price will remain the same and the equilibrium allocation will be x 1 =2 and x 2 =0, where the utility of user x is 0.4771, the utility of user y is 0.1761, and the social utility has value 0.6532.
Since the Nash equilibrium model only considers each user's power demand, we set the power constraints of user x and user y as 2 and get a Nash equilibrium x 1 = 2, x 2 = 0, y 1 = 1, y 2 = 1, where the utility of user x is 0.3010, the utility of user y is 0.1938, and the social utility has value 0.4948. Since the power resource supply of each channel is assumed to be unconstrained in the Nash model, we see that Channel 1 supplies 3 units power and Channel 2 supplies 1. Even though, comparing the competitive equilibrium and Nash equilibrium solutions, one can see that the competitive equilibrium provides a power distribution that not only meets physical power demand and supply constraints but also achieves a much higher social utility than the Nash equilibrium does.
Now consider user x and user y need to have more balanced individual utilities. By the proposed method, we can adjust the initial endowments to w x = 4/5 and w y = 6/5, then the equilibrium price will remain the same and the equilibrium power distribution will be where a i j represent the average of normalized crosstalk ratios for k = i. Furthermore, we assume 0 ≤ a i j ≤ 1, that is, the average cross-interference ratio is not above 1 or it is less than the selfinterference ratio (always normalized to 1). In all simulated cases, the channel background noise level σ ij are chosen randomly from the interval (0,m], and the normalized crosstalk ratio a i j are chosen randomly from the interval [0,1]. The power supply of each channel j iss j = m, j = 1, ...n.
The total budget is i w i = m. All simulations are run on a Genuine Intel CPU 1.66GHz Notebook.

Budget Allocation on Satisfying Individual Power Demands
In this section, we compute the budget allocation where the competitive equilibrium meets power demands d i = 0.5 js j /m or d i = js j /m for all users under various number of channels and number of users. Two approaches are adopted to find out the budget allocation strategy: one is solving the entire optimality conditions in (5) by optimization solver LINGO; the other is iteratively adjusting total budget m among different users based on whether their power demands are satisfied or not. In the iterative algorithm, all user budget w i are set as 1 initially, then the competitive equilibrium can be derived from given channel capacity and user budget. If some user's power demand is not satisfied in the resulting competitive equilibrium, the budgeting agent reallocates budget to users and computes a new competitive equilibrium. The procedure reiterates until a desired competitive equilibrium is reached for satisfying power demands. The iterative algorithm that allocates more budget to the users with more power shortage and keeps the total budget as m is summarized in the following: Iterative algorithm for budget allocation on satisfying power demands Step 1: Set power supply of each channels j = m, j = 1, ..., n.
Step 3: Loop: i) Compute competitive economy equilibrium [x * 1 , ..., x * m , p * ] unders j , w i according to the model in [11]. ii) Obtain total allocated power for each user i, j x * ij . iii) Calculate average power shortage, avg short = i (di− j x * ij ) m , and minimal user budget, min w = min In each iteration, given channel capacitys j and user budget w i , the competitive equilibrium is derived by an iterative water-filling method [10]. Since the competitive equilibrium in each iteration satisfies i x * ij =s j = m and i w * i = m, and each user optimizes his own utility under his budget constraint and the equilibrium prices, relatively increasing one user's budget makes him obtain more powers and others obtain fewer powers. In the above algorithm, the user budget is  Table 1 lists the number of iterations required to find out the budget allocation with d i = 0.5 js j /m and d i = js j /m by the above iterative algorithm. The cases of d i = js j /m need more iterations since the total power demand i d i is equal to the total channel capacity js j . This requirement is tight and the budget allocation makes each user get the same physical power in the competitive equilibrium, that is, j x * ij = n, ∀i. Table 2 compares the CPU time used by two different approaches under power demands d i = js j /m. The iterative algorithm spends much less time than the method of solving entire optimal conditions on finding out the budget allocation and the competitive equilibrium. We can also use the iterative method to solve large scale problems. The number of iterations and the CPU time required to solve large scale problems are listed in Table 3. We observe that more iterations and CPU time spending for 100 users and 256 channels than those spending for 100 users and 1024 channels because the stop condition of the iterative algorithm is " ≤ error tolerance". In our simulations in Table 3 In comparing competitive equilibrium with Nash equilibrium, the total power allocated to user i, j x * ij , in competitive equilibrium is used as the power constraint for user i in Nash equilibrium model to derive a Nash equilibrium. The simulation results averaged over 100 independent runs indicates that the average social utility of competitive equilibrium is higher than that of Nash equilibrium in all cases with d i = 0.5 js j /m and in most cases with d i = js j /m, even though the difference is not significant. However, in certain type of problems, for instance, the channels being divided into two categories: high-quality and low-quality, the competitive equilibrium solution performs much better than the Nash equilibrium solution does. Table 4 compares social utility and individual utility between the competitive equilibrium and the Nash equilibrium when one half of channels with σ ij , j = 1, ..., n/2, chosen randomly from the interval (0, 0.1] and the other half of channels with σ ij , j = n/2 + 1, ..., n, chosen randomly from the interval [1, m]. One can see that the competitive equilibrium significantly outperforms the Nash equilibrium in the social utility value and a much higher portion of users obtain higher individual utilities in the competitive equilibrium than those in the Nash equilibrium.
Step 3: Loop: i) Compute competitive economy equilibrium [x * 1 , ..., x * m , p * ] unders j , w i according to the model in [11]. ii) Obtain individual utility of each user u i .
iii) Calculate average reciprocal of individual utility, avg rec u = This algorithm is similar to the algorithm for budget allocation on satisfying power demands.
For balancing individual utilities, herein the user budget is adjusted based on the individual utility in the equilibrium solution. The idea of using the reciprocal of individual utility makes some budget be transferred from the high-utility users to low-utility users. Since relatively increasing one user's budget makes him obtain more powers and others obtain fewer powers, this will decrease the difference between highest individual utility and lowest individual utility. The term min w aims to keep new w i not less than 0. The difference tolerance significantly affects the number of iterations required to converge to the budget allocation. Figure 3 indicates the convergence behavior of the iterative algorithm for balancing individual utilities for the case of 2 users and 2 channels illustrated in Section 5. The difference tolerance is set as 0.01. As the figure shows, at first, the difference is higher than 0.6, then the algorithm converges after eighteen iterations and the difference is below difference tolerance 0.01. Table 5 lists the number of iterations required to converge to the budget allocation for balancing individual utilities by the iterative algorithm. Table 6  method of solving the entire optimal conditions. Treating the budget allocation problem by solving the entire optimal conditions can obtain a budget allocation where the competitive equilibrium has exactly identical individual utility value for each user. Table 7 lists the number of iterations and the CPU time required to solve large scale problems for balancing utilities by the iterative method. We observe that more iterations are required for 100 users and 256 channels than those required for 100 users and 1024 channels because the stop condition of the proposed algorithm is " difference tolerance". In our simulations in Table 7, the balanced individual utilities for 100 users and 1024 channels are higher than those for 100 users and 256 channels, therefore the case of 100 users and 1024 channels requires fewer iterations to reach the difference tolerance 0.05 than the case of 100 users and 256 channels does. However the CPU time spending for one iteration in the case of 100 users and 256 channels is less than that in the case of 100 users and 1024 channels.
In comparing competitive equilibrium with Nash equilibrium, the total power allocated to each user in competitive equilibrium is also used as the power constraint to derive a Nash equilibrium.
The simulation results averaged over 100 independent runs are displayed in Table 8. We find that, in most cases, more users get higher individual utilities in competitive equilibrium than those in Nash equilibrium and the social utility of competitive equilibrium remains higher than that of Nash equilibrium. Table 9 lists the comparisons in the communication environment involving two tiers of channels, one half of channels with σ ij , j = 1, ..., n/2, chosen randomly from the interval (0, 0.1] and the other half of channels with σ ij , j = n/2 + 1, ..., n, chosen randomly from the interval [1, m]. We can observe that the competitive equilibrium not only makes more users obtain higher individual utilities but also significantly enhances the social utility. In other words, using budget allocation we can derive a competitive equilibrium that provides a power allocation strategy to balance individual utilities without sacrificing the social utility. Moreover, in the competitive equilibrium model with balanced individual utilities, all users have identical utility value. However, in the Nash equilibrium model the average difference between maximal individual utility and minimal individual utility is over 15%.

Conclusions
This study proposes two competitive equilibrium models 1) to satisfy each user's physical power demand, 2) to balance all individual utilities in a competitive communication spectrum economy.
In comparing with the Nash equilibrium solution under the identical power usage of each user obtained from the competitive equilibrium model, our computational results show that the social utility of the competitive equilibrium solution is better than that of the Nash equilibrium solution in most cases. And under the equilibrium condition with balanced individual utilities, the competitive economy equilibrium solution makes more users obtain higher individual utilities than Nash equilibrium solution does without sacrificing the social utility.
In this study, we propose a centralized algorithm to reach a desired competitive equilibrium for satisfying power demands or balancing individual utilities. In the future, a distributed algorithm should be developed especially when a centralized controller is not available in the network. Besides, although the iterative method works well in our computational experiments, its convergence is unproven. We plan to do so in future work. We would also consider further study in how to adjust another exogenous factor s (power supply) to achieve a better social solution while maintaining individual satisfaction. That is, how to set the power supply capacity for each channel to make spectrum power allocation more efficient under the competitive equilibrium market model.