Open Access

Budget Allocation in a Competitive Communication Spectrum Economy

EURASIP Journal on Advances in Signal Processing20092009:963717

DOI: 10.1155/2009/963717

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 pay-off 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.

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 [25]; 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, [810]) 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 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 what follows.
  1. (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. (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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq1_HTML.gif denote the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq2_HTML.gif -dimensional Euclidean space; https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq3_HTML.gif denote the subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq4_HTML.gif where each coordinate is nonnegative. https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq6_HTML.gif denote the set of real numbers and the set of nonnegative real numbers, respectively.

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq7_HTML.gif denote the set of ordered https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq8_HTML.gif -tuples https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq9_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq10_HTML.gif denote the set of ordered https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq11_HTML.gif -tuples https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq12_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq13_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq14_HTML.gif . For each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq15_HTML.gif , suppose there is a real utility function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq16_HTML.gif , defined over https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq17_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq18_HTML.gif be a subset of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq19_HTML.gif defining for each point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq20_HTML.gif , then the sequence https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq21_HTML.gif will be termed an abstract economy. Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq22_HTML.gif represents the feasible action set of agent https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq23_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq24_HTML.gif for agent https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq25_HTML.gif depends on his or her actions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq26_HTML.gif , as well as actions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq27_HTML.gif made by all other agents. Also, denote https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq28_HTML.gif for a given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq29_HTML.gif .

A function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq30_HTML.gif is said to be concave if for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq31_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq32_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq33_HTML.gif ; and it is strictly concave if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq34_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq35_HTML.gif . It is monotone increasing if for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq37_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq38_HTML.gif .

3. Competitive Communication Spectrum Market

Let the multiuser communication system consist of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq39_HTML.gif transmitter-receiver pairs sharing a common frequency band discretized by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq40_HTML.gif tones. For simplicity, we will call each of such transmitter-receiver pair a "User." Each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq41_HTML.gif will be endowed a "monetary" budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq42_HTML.gif and use it to "purchase" powers, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq43_HTML.gif , across tones https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq44_HTML.gif , from an open market so as to maximize its own utility https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq45_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ1_HTML.gif
(1)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq47_HTML.gif are power units purchased by all other users, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq48_HTML.gif is the unit price for tone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq49_HTML.gif in the market.

A commonly recognized utility for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq51_HTML.gif , in communication is the Shannon utility [12]:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ2_HTML.gif
(2)

where parameter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq52_HTML.gif denotes the normalized background noise power for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq53_HTML.gif at tone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq54_HTML.gif , and parameter https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq55_HTML.gif is the normalized crosstalk ratio from user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq56_HTML.gif to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq57_HTML.gif at tone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq58_HTML.gif . Due to normalization we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq59_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq60_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq61_HTML.gif is a continuous concave and monotone increasing function in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq62_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq63_HTML.gif .

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq64_HTML.gif to the market from a convex and compact set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq65_HTML.gif to maximize his or her utility. We assume that they are fixed as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq66_HTML.gif in this paper, and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq67_HTML.gif , 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" https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq68_HTML.gif , which can be interpreted as a "preference or ranking" of tones https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq69_HTML.gif . For example, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq71_HTML.gif simply mean that users may use one unit of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq72_HTML.gif to trade for two units of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq73_HTML.gif .

The fourth agent, "Budgeting," allocates "monetary" budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq74_HTML.gif to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq75_HTML.gif from a bounded total budget, say https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq76_HTML.gif .

Figure 1 illustrates the interaction among four types of agents in the proposed competitive spectrum market. Each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq77_HTML.gif determines its power allocation https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq78_HTML.gif under its budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq79_HTML.gif , power spectra unit price density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq80_HTML.gif and the decisions by all other users https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq81_HTML.gif . The producer installs power capacity spectra density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq82_HTML.gif based on power spectra unit price density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq83_HTML.gif to maximize his or her utility. The market sets power spectra unit price density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq84_HTML.gif based on tone power distribution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq85_HTML.gif and power capacity spectra density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq86_HTML.gif to make market clear. The budgeting agent allocates budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq87_HTML.gif to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq88_HTML.gif from a bounded total budget according to tone power distribution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq89_HTML.gif for satisfying power demands or balancing individual utilities. In this study, we assume power capacity spectra density https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq90_HTML.gif is fixed. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq91_HTML.gif is determined first in the system.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Fig1_HTML.jpg
Figure 1

Interaction among four types of agents in the proposed competitive spectrum market.

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq92_HTML.gif that may be composed from any tone combination. We give an affirmative answer in this section.

A competitive market equilibrium is a power distribution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq93_HTML.gif such that
  1. (i)

    (user optimality) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq94_HTML.gif is a maximizer of (1) given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq95_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq97_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq98_HTML.gif ;

     
  2. (ii)

    (market efficiency) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq99_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq100_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq101_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq102_HTML.gif . This condition says that if tone power capacity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq103_HTML.gif is greater than or equal to the total power consumption for tone https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq104_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq105_HTML.gif , then its equilibrium price https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq106_HTML.gif ;

     
  3. (iii)
    (budgeting according to demands) given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq107_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq108_HTML.gif is a maximizer of
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ3_HTML.gif
    (3)

    This condition says that if user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq109_HTML.gif 's power demand is not met, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq110_HTML.gif , then one should allocate more or all "money budget" to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq111_HTML.gif . Any budget allocation is optimal if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq112_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq113_HTML.gif , 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.

Now consider the KKT conditions of (1):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ4_HTML.gif
(4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq114_HTML.gif denotes any subgradient vector of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq115_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq116_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq118_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq119_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq120_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq121_HTML.gif . The optimality conditions in (4) can be simplified to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ5_HTML.gif
(5)
The complete necessary and sufficient conditions for a competitive equilibrium with satisfied power demands can be summarized as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ6_HTML.gif
(6)
Note that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq122_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq123_HTML.gif are implied by the conditions in (6): multiplying https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq124_HTML.gif to both sides of the first inequality, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq125_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq126_HTML.gif , which, together with other inequality conditions in (6), imply
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ7_HTML.gif
(7)

that is, every inequality in the sequence must be tight, which implies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq127_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq128_HTML.gif .

On the other hand, the 4–6th conditions in (6) are optimality conditions of budget allocator's linear program, where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq129_HTML.gif is the dual variable. Then, we have a characterization theorem of a competitive equilibrium that satisfies power demands.

Theorem 4.2.

Every equilibrium of the discretized communication spectrum market with the Shannon utility that satisfies power demands has the following properties:
  1. (1)

    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq130_HTML.gif (every tone power has a price);

     
  2. (2)

    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq131_HTML.gif (all powers are allocated);

     
  3. (3)

    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq132_HTML.gif (all user budgets are spent);

     
  4. (4)

    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq133_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq134_HTML.gif (all user demands are met);

     
  5. (5)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq135_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq136_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq137_HTML.gif (every user only purchases most valuable tone power).

     

Proof.

Note that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ8_HTML.gif
(8)
Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq138_HTML.gif cannot be zero for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq139_HTML.gif , there is at least one https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq140_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ9_HTML.gif
(9)

so that the first inequality of (6) implies that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq141_HTML.gif .

The second property is from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq142_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq143_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq144_HTML.gif .

The third is from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq145_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq147_HTML.gif .

We prove the fourth property by contradiction. Suppose, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq148_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq149_HTML.gif for a nonempty index set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq150_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq151_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq152_HTML.gif so that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq153_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq154_HTML.gif . Then,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ10_HTML.gif
(10)

which is a contradiction to the assumption https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq155_HTML.gif .

The last one is from the complementarity condition of user optimality.

The fourth property of Theorem 4.2 implies that equilibrium conditions (6) can be simplified to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ11_HTML.gif
(11)

Note that the constraint https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq156_HTML.gif is merely a normalizing constraint and it can be replaced by another type of normalizing constraint such as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq157_HTML.gif . 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.

Here, a competitive market equilibrium is a density point https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq158_HTML.gif such that
  1. (i)

    (user optimality) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq159_HTML.gif is a maximizer of (1) given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq162_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq163_HTML.gif ;

     
  2. (ii)

    (market efficiency) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq166_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq167_HTML.gif ;

     
  3. (iii)
    (budgeting according to individual utilities) Given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq169_HTML.gif is a minimizer of
    https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ12_HTML.gif
    (12)

    This condition says that if user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq170_HTML.gif 's utility is higher than any others', that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq171_HTML.gif , then one should shift "money budget" from user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq172_HTML.gif to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq173_HTML.gif . Any budget allocation is optimal if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq174_HTML.gif are identical for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq175_HTML.gif , 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.

Corollary 4.3.

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
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ13_HTML.gif
(13)

Note that the conditions https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq176_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq177_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq178_HTML.gif 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) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq179_HTML.gif (every tone power has a price);

(2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq180_HTML.gif (all powers are allocated);

(3) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq181_HTML.gif (all user budgets are spent);

(4) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq182_HTML.gif are identical for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq183_HTML.gif (all user utilities are the same);
  1. (5)

    If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq184_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq185_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq186_HTML.gif (every user only purchases most valuable tone power).

     

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq187_HTML.gif , then the user cannot participate the game. Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq188_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq189_HTML.gif by the 5th condition of (13), which implies all user utilities are identical.

The fourth property of Theorem 4.4 implies that equilibrium conditions (13) can be simplified to
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ14_HTML.gif
(14)

5. An Illustration Example

Consider two channels https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq190_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq191_HTML.gif , and two users https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq192_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq193_HTML.gif . Let the Shannon utility function for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq194_HTML.gif be
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ15_HTML.gif
(15)
and one for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq195_HTML.gif be
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ16_HTML.gif
(16)

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq196_HTML.gif and the initial endowments for two users is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq197_HTML.gif . Then the competitive solution is
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ17_HTML.gif
(17)

where the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq198_HTML.gif is 0.3522, the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq199_HTML.gif is 0.2139, and the social utility has value 0.5661.

Now consider each of them has a physical power demand https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq200_HTML.gif . From above example we find https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq201_HTML.gif can not satisfy user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq202_HTML.gif 's power demand https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq203_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq204_HTML.gif . By the proposed method, we can adjust the initial budget endowments to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq205_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq206_HTML.gif , then the equilibrium price will remain the same and the equilibrium allocation will be
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ18_HTML.gif
(18)

where the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq207_HTML.gif is 0.4771, the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq208_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq209_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq210_HTML.gif as 2 and get a Nash equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq211_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq212_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq214_HTML.gif , where the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq215_HTML.gif is 0.3010, the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq216_HTML.gif is 0.1938, and the social utility has value https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq217_HTML.gif . Since the power resource supply of each channel is assumed to be unconstrained in the Nash model, we see that Channel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq218_HTML.gif supplies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq219_HTML.gif units power and Channel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq220_HTML.gif supplies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq221_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq222_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq223_HTML.gif need to have more balanced individual utilities. By the proposed method, we can adjust the initial endowments to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq224_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq225_HTML.gif , then the equilibrium price will remain the same and the equilibrium power distribution will be
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ19_HTML.gif
(19)

where the utilities of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq226_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq227_HTML.gif are both https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq228_HTML.gif , and the social utility is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq229_HTML.gif .

If the power constraints of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq230_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq231_HTML.gif are set as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq233_HTML.gif , respectively, then the Nash equilibrium will be https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq234_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq237_HTML.gif , where the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq238_HTML.gif is 0.1761, the utility of user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq239_HTML.gif is 0.2730, and the social utility has value https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq240_HTML.gif . 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

This section presents some computer simulation results on using two different approaches to achieve budget allocation for satisfying each user's power demand or balancing individual utilities. We compare the competitive equilibrium solution with Nash equilibrium solution in social utility and individual utilities under various number of channels and number of users in a weak-interference communication environment. In a weak-interference communication channel, the Shannon utility function is approximated by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Equ20_HTML.gif
(20)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq241_HTML.gif represent the average of normalized crosstalk ratios for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq242_HTML.gif . Furthermore, we assume https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq243_HTML.gif , that is, the average cross-interference ratio is not above https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq244_HTML.gif or it is less than the self-interference ratio (always normalized to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq245_HTML.gif ). In all simulated cases, the channel background noise levels https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq246_HTML.gif are chosen randomly from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq247_HTML.gif , and the normalized crosstalk ratios https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq248_HTML.gif are chosen randomly from the interval [0, 1]. The power supply of each channel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq249_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq250_HTML.gif . The total budget is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq251_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq252_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq253_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq254_HTML.gif among different users based on whether their power demands are satisfied or not. In the iterative algorithm, all user budgets https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq255_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq256_HTML.gif is summarized in Algorithm 1.

Algorithm 1

Iterative algorithm for budget allocation on satisfying power demands
  • Step 1:  Set power supply of each channel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq257_HTML.gif .

  • Step 2:  Initialize budget assigned to each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq258_HTML.gif .

  • Step 3:  Loop:
    1. (i)

      Compute competitive economy equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq259_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq260_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq261_HTML.gif

      according to the model in [11].

       
    2. (ii)

      Obtain total allocated power for each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq262_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq263_HTML.gif .

       
    3. (iii)

      Calculate average power shortage, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq264_HTML.gif ,

      and minimal user budget, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq265_HTML.gif .

       
    4. (iv)

      Update https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq266_HTML.gif .

      Until https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq267_HTML.gif error tolerance, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq268_HTML.gif .

       
In each iteration, given channel capacity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq269_HTML.gif and user budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq270_HTML.gif , the competitive equilibrium is derived by an iterative waterfilling method [13]. Since the competitive equilibrium in each iteration satisfies https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq272_HTML.gif , 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 Algorithm 1, the user budget is reassigned according to the power shortage of each user in the equilibrium solution. The idea of comparing the user's power shortage with average shortage makes more budget be allocated to the users with higher power shortage and the total budget remains https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq273_HTML.gif . The term https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq274_HTML.gif aims to keep new https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq275_HTML.gif not less than 0. The power demand value and the error tolerance have a significant impact on the number of iterations required to converge to the budget allocation where the competitive equilibrium meets the power demands. Figure 2 indicates the convergence behavior of the iterative algorithm for satisfying power demands for the case of 2 users and 2 channels illustrated in Section 5. Each user has a physical power demand https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq276_HTML.gif . The error tolerance is set as 0.01. As the figure shows, at first, user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq277_HTML.gif has power shortage and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq278_HTML.gif has power surplus, then the algorithm converges after eight iterations and the errors https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq279_HTML.gif for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq280_HTML.gif and user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq281_HTML.gif are both below error tolerance 0.01.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Fig2_HTML.jpg
Figure 2

Convergence of iterative algorithm for satisfying power demands.

Table 1 lists the number of iterations required to find out the budget allocation with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq283_HTML.gif by the above iterative algorithm. The cases of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq284_HTML.gif need more iterations since the total power demand https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq285_HTML.gif is equal to the total channel capacity https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq286_HTML.gif . This requirement is tight and the budget allocation makes each user get the same physical power in the competitive equilibrium, that is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq287_HTML.gif . Table 2 compares the CPU time used by two different approaches under power demands https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq288_HTML.gif . 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 " https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq289_HTML.gif error tolerance." In our simulations in Table 3, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq290_HTML.gif for 100 users and 256 channels and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq291_HTML.gif for 100 users and 1024 channels, therefore the case of 100 users and 1024 channels requires fewer iterations and less total CPU time to reach the error 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.
Table 1

Number of iterations required to achieve the budget allocation where the competitive equilibrium satisfies power demands https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq293_HTML.gif by the iterative algorithm, error tolerance = 0.01, average of 10 simulation runs.

No. of channels

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq294_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq295_HTML.gif

 

No. of users

No. of users

 

2

4

6

8

10

2

4

6

8

10

2

1

1

1

1

1

5

14

20

22

38

4

1

1

1

1

1

5

18

44

49

85

6

1

1

1

1

1

5

20

37

47

65

8

1

1

1

1

1

7

21

35

52

66

10

1

1

1

1

1

6

18

33

53

66

12

1

1

1

1

1

6

18

35

56

71

14

1

1

1

1

1

6

20

35

50

70

16

1

1

1

1

1

6

18

35

49

74

18

1

1

1

1

1

6

18

35

45

70

20

1

1

1

1

1

6

18

31

50

68

Table 2

Comparisons of CPU time (seconds) required to achieve the budget allocation where the competitive equilibrium satisfies power demands https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq296_HTML.gif between two approaches, error 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

0.033

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq299_HTML.gif : iterative algorithm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq300_HTML.gif : solving the entire optimal conditions.

Table 3

Number of iterations and CPU time (seconds) required to achieve the budget allocation where the competitive equilibrium satisfies power demands https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq301_HTML.gif in large-scale problems by the iterative method, error tolerance = 0.05 and average of 10 simulation runs.

No. of channels

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq302_HTML.gif

 

2 users

10 users

50 users

100 users

 

Iterations

Time

Iterations

Time

Iterations

Time

Iterations

Time

256

1

0.072

1

2.162

5

400.092

17

4751.578

512

1

0.255

1

5.656

1

148.097

3

2277.144

1024

1

0.388

1

15.978

1

365.290

1

1720.400

In comparing competitive equilibrium with Nash equilibrium, the total power allocated to user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq303_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq304_HTML.gif , in competitive equilibrium is used as the power constraint for user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq305_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq306_HTML.gif and in most cases with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq307_HTML.gif , 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq308_HTML.gif chosen randomly from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq309_HTML.gif and the other half of channels with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq310_HTML.gif chosen randomly from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq311_HTML.gif . 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.
Table 4

Comparisons of social utility and individual utility between competitive equilibrium(CE) with power demands https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq312_HTML.gif and Nash equilibrium(NE), error 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

9.20%

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%

*Social: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq314_HTML.gif

Indiv: average percentage in number of users obtaining higher individual utilities in CE than in NE.

6.2. Budget Allocation on Balancing Individual Utilities

To consider fairness, we adjust each user's endowed monetary budget https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq315_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq316_HTML.gif among different users based on their individual utilities.The iterative algorithm that shifts some budget from high-utility users to low-utility users and keeps the total budget as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq317_HTML.gif is summarized in Algorithm 2.

Algorithm 2

Iterative algorithm for budget allocation on balancing individual utilities
  • Step 1:  Set power supply of each channel https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq318_HTML.gif .

  • Step 2:  Initialize budget assigned to each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq319_HTML.gif .

  • Step 3:  Loop:
    1. (i)

      Compute competitive economy equilibrium https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq320_HTML.gif under https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq321_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq322_HTML.gif

      according to the model in [11].

       
    2. (ii)

      Obtain individual utility of each user https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq323_HTML.gif .

       
    3. (iii)

      Calculate average reciprocal of individual utility, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq324_HTML.gif ,

      and minimal user budget, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq325_HTML.gif .

       
    4. (iv)

      Update https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq326_HTML.gif

      Until https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq327_HTML.gif difference tolerance.

       
Algorithm 2 is similar to Algorithm 1 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq328_HTML.gif aims to keep new https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq329_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq330_HTML.gif is higher than 0.6, then the algorithm converges after eighteen iterations and the difference is below difference tolerance 0.01.
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_Fig3_HTML.jpg
Figure 3

Convergence of iterative algorithm for balancing individual utilities.

Table 5 lists the number of iterations required to converge to the budget allocation for balancing individual utilities by the iterative algorithm. Table 6 compares the CPU time used by two different approaches to achieve the budget allocation. The iterative algorithm spends less CPU time than the 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 " https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq331_HTML.gif 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.
Table 5

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

*Iter: number of iterations

+Diff: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq332_HTML.gif .

Table 6

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

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq335_HTML.gif : iterative algorithm

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq336_HTML.gif : solving entire optimal conditions.

Table 7

Number of iterations and CPU time (seconds) required to achieve the budget allocation where the competitive equilibrium has balanced individual utilities in large-scale 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

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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq337_HTML.gif chosen randomly from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq338_HTML.gif and the other half of channels with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq339_HTML.gif chosen randomly from the interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq340_HTML.gif . 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq341_HTML.gif .
Table 8

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%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq343_HTML.gif 0.59%

45%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq344_HTML.gif 0.41%

47%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq345_HTML.gif 0.34%

48%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq346_HTML.gif 0.44%

48%

4

1.05%

55%

0.83%

53%

0.42%

50%

0.01%

48%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq347_HTML.gif 0.42%

48%

6

1.14%

56%

0.08%

50%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq348_HTML.gif 0.15%

48%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq349_HTML.gif 0.08%

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%

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq350_HTML.gif 0.08%

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%

*Social: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq351_HTML.gif

Indiv: average percentage in number of users obtaining higher individual utilities in CE than in NE.

Table 9

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%

*Social: https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq353_HTML.gif

Indiv: average percentage in number of users obtaining higher individual utilities in CE than in NE.

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 tatonament-type method by adjusting budget to each user. The iterative method performs significantly faster and can efficiently solve large-scale problems, which makes the competitive economy equilibrium model applicable in real-time 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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F963717/MediaObjects/13634_2008_Article_2647_IEq354_HTML.gif (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 NSC-095-SAF-I-564-635-TMS, NSC 96-2416-H-158-003-MY3, and the Fulbright Scholar Program. The research also is supported in part by Taiwan NSC Grants NSC-095-SAF-I-564-640-TMS, NSC 96-2416-H-027-004-MY3, and the Fulbright Scholar Program, and supported in part by NSF DMS-0604513.

Authors’ Affiliations

(1)
Department of Information Technology and Management, Shih-Chien University
(2)
Department of Business Management, National Taipei University of Technology
(3)
Department of Management Science and Engineering, Stanford University

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© Ming-Hua Lin et al. 2009

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