 Research Article
 Open Access
Budget Allocation in a Competitive Communication Spectrum Economy
 MingHua Lin^{1},
 JungFa Tsai^{2}Email author and
 Yinyu Ye^{3}
https://doi.org/10.1155/2009/963717
© MingHua Lin et al. 2009
 Received: 15 August 2008
 Accepted: 4 February 2009
 Published: 4 March 2009
Abstract
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 payoff function that includes the effect of interferences. A market equilibrium is a budget allocation, price spectrum, and tone power distribution that independently and simultaneously maximizes each user's utility. The equilibrium conditions of the market are formulated and analyzed, and the existence of an equilibrium is proved. Computational results and comparisons between the competitive equilibrium and Nash equilibrium solutions are also presented, which show that the competitive market equilibrium solution often provides more efficient power distribution.
Keywords
 Nash Equilibrium
 Power Allocation
 Competitive Equilibrium
 Power Demand
 Budget Allocation
1. 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 his/her utility, subject to his/her budget constraints, and the limited power bandwidth resource is efficiently utilized. Although the study of the competitive equilibrium can date back to Walras [1] 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 preassigning them to the users on a nonoverlapping 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 [2–5]; or as a cooperative utility maximization problem [6, 7]. Several algorithms were proposed to compute a Nash equilibrium solution (Iterative Waterfilling Algorithm (IWFA) [2, 4]; Linear Complementarity Problem (LCP) [3]) or globally optimal power allocations (Dual decomposition method, [8–10]) 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 (i.e., the sum of all users' utilities) 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 weakinterference 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.
 (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.
 (2)
A competitive equilibrium where all users have identical utility value always exists for the communication spectrum market with Shannon utilities.
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 demonstrates 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.
2. Mathematical Notations
First, a few mathematical notations. Let denote the dimensional Euclidean space; denote the subset of where each coordinate is nonnegative. and denote the set of real numbers and the set of nonnegative real numbers, respectively.
Let denote the set of ordered tuples and let denote the set of ordered tuples , where for . For each , suppose there is a real utility function , defined over . Let be a subset of defining for each point , then the sequence will be termed an abstract economy. Here represents the feasible action set of agent 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 for agent depends on his or her actions , as well as actions made by all other agents. Also, denote for a given .
A function is said to be concave if for any and any , we have ; and it is strictly concave if for . It is monotone increasing if for any , implies that .
3. Competitive Communication Spectrum Market
where and are power units purchased by all other users, and is the unit price for tone in the market.
where parameter denotes the normalized background noise power for user at tone , and parameter is the normalized crosstalk ratio from user to user at tone . Due to normalization we have for all . Clearly, is a continuous concave and monotone increasing function in for every .
There are four types of agents in this market. The firsttype agents are users. Each user aims to maximize its own utility under its budget constraint and the decisions by all other users. The secondtype agent, "Producer or Provider," who installs power capacity supply to the market from a convex and compact set to maximize his or her utility. We assume that they are fixed as in this paper, and , 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" , which can be interpreted as a "preference or ranking" of tones . For example, and simply mean that users may use one unit of to trade for two units of .
The fourth agent, "Budgeting," allocates "monetary" budget to user from a bounded total budget, say .
4. Budget Allocation in Competitive Communication Spectrum Market
In this section, we discuss how to adjust "monetary" budget to satisfy each user's prespecified physical power demand or to balance all individual utilities in a competitive spectrum market.
4.1. 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 that may be composed from any tone combination. We give an affirmative answer in this section.
 (i)
 (ii)
(market efficiency) , , for all . This condition says that if tone power capacity is greater than or equal to the total power consumption for tone , , then its equilibrium price ;
 (iii)
This condition says that if user 's power demand is not met, that is, , then one should allocate more or all "money budget" to user . Any budget allocation is optimal if for all , 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.
Corollary 4.1.
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.
where denotes any subgradient vector of with respect to .
that is, every inequality in the sequence must be tight, which implies for all .
On the other hand, the 4–6th conditions in (6) are optimality conditions of budget allocator's linear program, where is the dual variable. Then, we have a characterization theorem of a competitive equilibrium that satisfies power demands.
Theorem 4.2.
 (1)
 (2)
 (3)
 (4)
 (5)
Proof.
so that the first inequality of (6) implies that .
The second property is from , and .
The third is from for all and .
which is a contradiction to the assumption .
The last one is from the complementarity condition of user optimality.
Note that the constraint is merely a normalizing constraint and it can be replaced by another type of normalizing constraint such as . Moreover, multiple competitive equilibria may exist due to the nonconvexity of the optimality conditions of the spectrum management problem with minimal user power demands.
4.2. 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.
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.
Corollary 4.3.
The communication spectrum market with Shannon utilities has a competitive equilibrium that balances each user's utility value.
Note that the conditions for all are implied by the conditions in (13). On the other hand, the 4–6th conditions in (13) are optimality conditions of budget allocator's linear program for balancing utilities, where is the dual variable.
Again, we have a characterization theorem of a competitive equilibrium that balances individual utilities.
Theorem 4.4.
Every equilibrium of the discretized communication spectrum market with the Shannon utility that balances individual utilities has the following properties:
(1) (every tone power has a price);
(2) (all powers are allocated);
(3) (all user budgets are spent);
Proof.
The proof of properties 1, 2, 3, and 5 are the same as Theorem 4.2. The fourth property is from the 5th condition of (13). If , then the user cannot participate the game. Therefore, and by the 5th condition of (13), which implies all user utilities are identical.
5. An Illustration Example
and let the aggregate social utility be the sum of the two individual user utilities.
where the utility of user is 0.3522, the utility of user is 0.2139, and the social utility has value 0.5661.
where the utility of user is 0.4771, the utility of user 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 and user as 2 and get a Nash equilibrium , , , , where the utility of user is 0.3010, the utility of user is 0.1938, and the social utility has value . Since the power resource supply of each channel is assumed to be unconstrained in the Nash model, we see that Channel supplies units power and Channel supplies . 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.
where the utilities of user and user are both , and the social utility is .
If the power constraints of user and user are set as and , respectively, then the Nash equilibrium will be , , , , where the utility of user is 0.1761, the utility of user is 0.2730, and the social utility has value . Comparing the competitive equilibrium and Nash equilibrium solutions again, one can see that the competitive equilibrium provides a power distribution that not only makes both users with an identical utility value but also achieves a higher social utility than the Nash equilibrium does.
6. Numerical Simulations
where represent the average of normalized crosstalk ratios for . Furthermore, we assume , that is, the average crossinterference ratio is not above or it is less than the selfinterference ratio (always normalized to ). In all simulated cases, the channel background noise levels are chosen randomly from the interval , and the normalized crosstalk ratios are chosen randomly from the interval [0, 1]. The power supply of each channel is . The total budget is . All simulations are run on a Genuine Intel CPU 1.66 GHz Notebook.
6.1. Budget Allocation on Satisfying Individual Power Demands
In this section, we compute the budget allocation where the competitive equilibrium meets power demands or 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 (11) by optimization solver LINGO; the other is iteratively adjusting total budget among different users based on whether their power demands are satisfied or not. In the iterative algorithm, all user budgets 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 is summarized in Algorithm 1.
Algorithm 1

Step 3: Loop:
 (i)
Compute competitive economy equilibrium under ,
according to the model in [11].
 (ii)
 (iii)
 (iv)
 (i)
No. of channels  No. of users  

2  4  6  8  10  
M1  M2  M1  M2  M1  M2  M1  M2  
2  0.033  1.085  0.049  1.228  0.069  1.358  0.088  1.624  0.136  1.882 
4  0.022  1.164  0.080  1.479  0.267  2.255  0.463  3.465  1.011  6.450 
6  0.028  1.270  0.106  2.207  0.312  5.129  0.639  10.545  1.947  19.406 
8  0.025  1.516  0.103  3.788  0.510  10.305  0.875  25.697  2.592  51.210 
10  0.035  1.889  0.130  7.222  0.525  27.027  0.938  44.909  2.455  111.270 
12  0.028  2.482  0.158  12.558  0.603  41.747  1.816  93.028  3.164  190.489 
14  0.028  3.231  0.161  20.454  0.528  66.719  2.464  150.099  2.708  322.979 
16  0.039  4.793  0.184  33.251  0.684  102.846  1.260  263.820  6.006  519.137 
18  0.041  6.529  0.250  46.043  0.627  150.047  2.181  385.401  5.781  773.646 
20  0.042  9.322  0.247  66.839  0.703  215.038  2.645  553.401  4.689  1179.129 
No. of channels  No. of users  

2  4  6  8  10  
Social*  Social  Indiv  Social  Indiv  Social  Indiv  Social  Indiv  
2  9.20%  83%  8.51%  58%  7.85%  53%  8.42%  51%  9.38%  51% 
4  6.78%  87%  6.21%  70%  6.21%  62%  6.40%  58%  6.11%  57% 
6  5.91%  88%  6.20%  81%  5.68%  71%  5.59%  64%  5.57%  62% 
8  6.83%  92%  5.26%  78%  5.65%  71%  5.46%  69%  5.19%  67% 
10  6.14%  94%  5.82%  80%  5.31%  74%  5.26%  70%  5.05%  67% 
12  6.18%  94%  5.76%  84%  5.50%  77%  5.50%  74%  5.24%  71% 
14  5.73%  95%  5.49%  84%  5.55%  79%  5.26%  74%  5.04%  70% 
16  6.24%  97%  5.35%  83%  5.27%  81%  5.03%  75%  5.02%  74% 
18  5.62%  96%  5.64%  86%  5.33%  82%  5.22%  77%  5.28%  76% 
20  5.83%  97%  5.26%  88%  5.34%  85%  5.25%  81%  5.02%  74% 
6.2. Budget Allocation on Balancing Individual Utilities
To consider fairness, we adjust each user's endowed monetary budget to reach a competitive equilibrium where the individual utilities are balanced. Herein we also adopt two approaches to find out the budget allocation: one is solving the entire optimality conditions in (14) by optimization solver LINGO; the other is iteratively adjusting total budget among different users based on their individual utilities.The iterative algorithm that shifts some budget from highutility users to lowutility users and keeps the total budget as is summarized in Algorithm 2.
Algorithm 2

Step 3: Loop:
 (i)
Compute competitive economy equilibrium under ,
according to the model in [11].
 (ii)
 (iii)
 (iv)
 (i)
Number of iterations required to achieve the budget allocation where the competitive equilibrium has balanced individual utilities by the iterative algorithm, difference tolerance = 0.01 and average of 10 simulation runs.
No. of channels  No. of users  

2  4  6  8  10  
Iter*  Diff ^{ + }  Iter  Diff  Iter  Diff  Iter  Diff  Iter  Diff  
2  5  0.0076  10  0.0078  14  0.0079  18  0.0080  97  0.0080 
4  4  0.0075  20  0.0079  27  0.0080  21  0.0080  74  0.0080 
6  4  0.0077  9  0.0079  18  0.0080  22  0.0081  46  0.0081 
8  4  0.0080  8  0.0078  40  0.0079  149  0.0081  33  0.0080 
10  4  0.0080  13  0.0081  17  0.0079  57  0.0081  24  0.0080 
12  4  0.0078  16  0.0080  35  0.0079  31  0.0080  29  0.0080 
14  5  0.0078  8  0.0080  13  0.0079  21  0.0080  67  0.0080 
16  5  0.0078  9  0.0080  12  0.0079  27  0.0080  48  0.0080 
18  4  0.0076  6  0.0079  10  0.0078  18  0.0079  26  0.0080 
20  4  0.0077  7  0.0078  8  0.0079  11  0.0079  20  0.0079 
Comparisons of CPU time (seconds) required to achieve the budget allocation where competitive equilibrium has balanced individual utilities between two approaches, difference tolerance = 0.01 and average of 10 simulation runs.
No. of channels  No. of users  

2  4  6  8  10  
M1  M2  M1  M2  M1  M2  M1  M2  
2  0.048  0.330  0.056  0.364  0.061  0.447  0.060  0.575  0.239  0.837 
4  0.046  0.377  0.088  0.647  0.127  1.100  0.148  1.892  0.738  3.606 
6  0.048  0.467  0.075  1.005  0.116  2.469  0.353  5.425  1.550  11.273 
8  0.041  0.641  0.069  2.052  0.319  5.555  1.663  13.305  1.422  26.173 
10  0.070  0.872  0.113  3.366  0.214  10.264  1.759  27.294  1.056  54.902 
12  0.063  1.247  0.139  6.345  0.397  19.048  0.919  47.069  1.428  101.013 
14  0.064  1.822  0.095  9.692  0.217  32.551  0.577  81.780  2.633  168.536 
16  0.056  2.542  0.119  14.928  0.216  52.972  0.953  123.817  3.320  274.966 
18  0.058  3.328  0.103  22.686  0.261  74.310  1.117  191.992  1.733  401.128 
20  0.057  4.333  0.098  31.805  0.192  102.436  0.506  272.994  1.674  557.339 
Number of iterations and CPU time (seconds) required to achieve the budget allocation where the competitive equilibrium has balanced individual utilities in largescale problems by the iterative method, difference tolerance = 0.05 and average of 10 simulation runs.
No. of channels  2 users  10 users  50 users  100 users  

Iterations  Time  Iterations  Time  Iterations  Time  Iterations  Time  
256  1  0.119  4  6.775  8  646.620  16  4005.344 
512  1  0.211  3  14.309  6  964.164  8  4750.842 
1024  1  0.452  3  35.631  5  1663.111  5  6326.120 
Comparisons of social utility and individual utility between competitive equilibrium (CE) with balanced individual utilities and Nash equilibrium (NE), difference tolerance = 0.01 and average of 100 simulation runs.
No. of channels  No. of users  

2  4  6  8  10  
Social*  Social  Indiv  Social  Indiv  Social  Indiv  Social  Indiv  
2  0.96%  46%  45%  47%  48%  48%  
4  1.05%  55%  0.83%  53%  0.42%  50%  0.01%  48%  48%  
6  1.14%  56%  0.08%  50%  48%  48%  0.01%  49%  
8  1.27%  58%  0.66%  57%  0.17%  51%  0.00%  49%  0.19%  52% 
10  1.15%  61%  0.88%  58%  0.52%  54%  0.22%  52%  53%  
12  1.35%  66%  0.78%  57%  0.50%  56%  0.39%  54%  0.16%  52% 
14  1.43%  67%  0.85%  60%  0.28%  55%  0.10%  52%  0.16%  54% 
16  1.60%  75%  0.88%  60%  0.42%  55%  0.30%  54%  0.29%  55% 
18  1.63%  72%  0.71%  59%  0.34%  54%  0.14%  55%  0.16%  52% 
20  1.50%  73%  0.80%  59%  0.30%  56%  0.11%  54%  0.17%  54% 
Comparisons of social utility and individual utility between competitive equilibrium(CE) with balanced individual utilities and Nash equilibrium(NE) under two tiers of channels, difference tolerance = 0.01 and average of 100 simulation runs.
No. of channels  No. of users  

2  4  6  8  10  
Social*  Social  Indiv  Social  Indiv  Social  Indiv  Social  Indiv  
2  9.02%  81%  9.86%  73%  10.01%  69%  8.85%  67%  9.62%  67% 
4  7.68%  80%  7.67%  78%  8.30%  77%  8.54%  71%  8.23%  71% 
6  6.06%  87%  6.55%  81%  6.87%  77%  7.43%  76%  7.16%  75% 
8  5.56%  88%  6.24%  81%  6.41%  78%  6.80%  76%  6.75%  75% 
10  5.46%  87%  6.12%  84%  6.18%  80%  6.47%  77%  6.38%  75% 
12  5.66%  88%  5.75%  84%  5.88%  80%  5.94%  79%  6.38%  77% 
14  5.65%  91%  6.01%  85%  5.65%  82%  5.80%  81%  5.86%  77% 
16  5.63%  92%  5.84%  89%  5.75%  83%  5.86%  82%  5.70%  79% 
18  5.78%  94%  5.84%  88%  5.80%  83%  5.81%  83%  5.54%  80% 
20  5.66%  94%  5.58%  88%  5.77%  86%  5.72%  81%  5.63%  80% 
7. Conclusions
This study proposes two competitive equilibrium models: (1) to satisfy each user's physical power demand and (2) to balance all individual utilities in a competitive communication spectrum economy. Theoretically, we prove that a competitive equilibrium with physical power demand requirements always exists for the communication spectrum market with Shannon utility if the total power demand is less than or equal to the available total power supply. A competitive equilibrium with identical individual utilities also exists for the communication spectrum market with Shannon utility. Computationally, we use two approaches to find out the budget allocation where the competitive equilibrium satisfies power demand or balances individual utilities: one solves the characteristic equilibrium conditions and the other employs an iterative tatonamenttype method by adjusting budget to each user. The iterative method performs significantly faster and can efficiently solve largescale problems, which makes the competitive economy equilibrium model applicable in realtime spectrum management.
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. 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 (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.
Declarations
Acknowledgments
This research is supported in part by Taiwan NSC Grants NSC095SAFI564635TMS, NSC 962416H158003MY3, and the Fulbright Scholar Program. The research also is supported in part by Taiwan NSC Grants NSC095SAFI564640TMS, NSC 962416H027004MY3, and the Fulbright Scholar Program, and supported in part by NSF DMS0604513.
Authors’ Affiliations
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