博弈论在无线工程中的应用

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《无线工程师的博弈论》(Game Theory for Wireless Engineers)是由Allen B. MacKenzie和Luiz A. DaSilva合著的一本专著，版权属于Morgan & Claypool出版社，首次出版于2006年。本书旨在将博弈论这一数学工具应用于无线通信领域，但因为无线网络的复杂性——包括移动性和流量模型的动态特性、多用户交互等因素，使得传统的分析方法在解决实际问题上遇到了挑战。博弈论是研究决策者之间互动的一种理论，通常用于分析冲突或合作的情况。在无线工程中，这种理论的应用主要体现在优化网络资源分配、频谱管理、干扰控制、安全策略以及多接入技术等方面。例如，通过博弈论，可以设计出针对多个用户竞争无线信道的有效策略，如通过拍卖机制分配频谱使用权，或者通过纳什均衡来平衡网络中的功率控制和数据传输。书中可能讨论了以下关键知识点： 1. **非合作博弈与合作博弈**：介绍基本的博弈类型，如零和博弈（一方赢，另一方输）、囚徒困境等，并解释它们在无线网络中的应用，比如干扰管理和频谱效率最大化。 2. **动态博弈**：针对无线网络中的动态环境，如移动通信中的多用户调度，可能探讨了如何通过动态博弈模型预测用户行为，制定适应性强的策略。 3. **纳什均衡**：阐述如何在无线网络中寻找稳定的战略配置，即每个参与者在其他参与者策略不变的情况下选择最优行动的状态。 4. **信息不对称博弈**：无线环境中，信号强度、信道质量等信息可能不完全对所有用户公开，这可能导致信息不对称，作者可能会讨论如何处理这类情况下的决策问题。 5. **网络博弈论**：通过案例分析，展示如何将博弈论应用于大型无线网络架构，如蜂窝网络、无线传感器网络，甚至是5G和未来网络的多层设计中。 6. **算法设计**：书中可能包含针对特定问题的算法设计，如基于博弈论的分布式算法，以实现高效的无线网络协调和优化。 7. **应用实例与案例**：通过具体的无线通信系统或标准（如Wi-Fi、蜂窝通信）来展示博弈论的实际应用，让读者理解理论在实践中的作用。《无线工程师的博弈论》是一本深入浅出的教材，旨在帮助无线通信领域的专业人士理解和应用博弈论，以提升网络性能，优化资源配置，解决复杂的无线通信问题。随着网络技术的发展，博弈论在无线工程中的重要性只会日益凸显。

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4 GAME THEORY FOR WIRELESS ENGINEERS

Nonetheless, modern game theory usually traces its roots to the seminal Theory of Games

and Economic Behavior, by John von Neumann and Oskar Morgenstern, published in 1944.

Von Neumann and Morgenstern gave an axiomatic development of utility theory, which now

dominates economic thought and which we will explore in the next chapter, and also introduced

the formal notion of a cooperative game. Like much early work in game theory, though, their

work on noncooperative game theory (which now dominates the game theory literature) focused

on relatively simple two-person zero-sum games.

In papers between 1950 and 1953, John Nash contributed to the development of both

noncooperative and cooperative game theory [5–7]. His most important contribution, though,

was probably his existence proof of an equilibrium in noncooperative games, the Nash Equilib-

rium [5]; we will examine this proof in some detail in Chapter 3. For his contributions to game

theory, Nash shared in the 1994 Bank of Sweden Prize in Economic Sciences in Memory of

Alfred Nobel.

This early work was signiﬁcantly aided by work completed at the Rand Corporation,

which was formed after the World War II to connect military planning with research. With

this charter, the Rand Corporation sought to develop theories and tools for decision making

under uncertainty, leading to important contributions not only in game theory but also in

linear and dynamic programming and mathematical modeling and simulation. One of the early

applications of this theory was the study of the decision making of nations during the cold

war [8].

Since many of the early game theorists were economists, it is not surprising, perhaps, that

game theory’s greatestsuccess hasbeen in revolutionizingthe ﬁeldof economics. Ideasdeveloped

in the context of game theory now pervade almost every aspect of economic thought: utility

theory, strategic use of information, cooperative and team games, the problem of coordination

between independent players, auction theory, and implementation of incentive mechanisms.

Game theory has also made signiﬁcant contributions to other disciplines, including po-

litical science, sociology, biology, and military strategy. In the next section, we will discuss why

we believe that game theory has the potential to make signiﬁcant contributions to the study of

wireless communications and networking.

1.3 WHY IS GAME THEORY RELEVANT TO WIRELESS

COMMUNICATIONS AND NETWORKING?

Ad hoc networks have occupied a preeminent place in the wireless communications and net-

working literature for the last several years. An ad hoc network is a self-conﬁguring, multihop

network in which there is no central authority. Thus, every aspect of the conﬁguration and

operation of an ad hoc network must be completely distributed. Furthermore, nodes in an ad

hoc network are often severely energy and power constrained. In emerging wireless networks,

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INTRODUCTION TO GAME THEORY 5

such as sensor networks, mesh networks, and pervasive computing systems, many of these same

features—decentralized operation, self conﬁguration, and power/energy awareness—are often

desirable.

Game theory, as we have seen, is the study of the interaction of autonomous agents. At

this point, it should be clear how game theory may be of help when analyzing modern wireless

networks. In a modern wireless network, each node running a distributed protocol must make

its own decisions (possibly relying on information from other nodes). These decisions may

be constrained by the rules or algorithms of a protocol, but ultimately each node will have

some leeway in setting parameters or changing the mode of operation. These nodes, then, are

autonomous agents, making decisions about transmit power, packet forwarding, backoff time,

and so on.

In making these decisions, what does the node seek to optimize? In some cases, nodes

may seek the “greater good” of the network as a whole. In other cases, nodes may behave

selﬁshly, looking out for only their own user’s interests. In a ﬁnal case, nodes may behave

maliciously, seeking to ruin network performance for other users. In the second and third

cases, the application of game theory may be straightforward, as game theory traditionally

analyzes situations in which player objectives are in conﬂict. In the ﬁrst case, node objectives

may be aligned (as all players seek the “greater good” of the network), but game theory may

still offer useful insights. Even when nodes have shared objectives, they will each have a unique

perspective on the current network state,leading to possible conﬂicts regarding the best course of

action.

1.4 HOW CAN I USE GAME THEORY PROPERLY?

There are many potential pitfalls on the road to the application of game theory to wireless

communications and networking. One of these pitfalls has already been mentioned—mistaking

a simple optimization problem for a game. As we have already discussed, a problem is only a

game if there are multiple agents involved in making decisions. In some cases, it may be that

artiﬁcially creating multiple agents to play a game is worthwhile; in many cases, though, one

would be better served by using an optimization algorithm or technique. So, if it is not clear

who the agents are, one should carefully consider whether or not the use of game theory is

warranted.

A second common mistake in applications of game theory is confusion between the

theories of cooperative and noncooperative games. In this book, we focus on noncooperative

games, which have been primary in the study of game theory for the last 25 years or so.

Noncooperative game theory assumes agents with distinct interests that interact through a

predeﬁned mechanism; it encompasses concepts such as normal and extensive form games,

incomplete information, Nash equilibrium, and forward and backward induction. There is

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6 GAME THEORY FOR WIRELESS ENGINEERS

also a substantial body of literature on cooperative games, in which players form coalitions

and negotiate via an unspeciﬁed mechanism to allocate resources. Cooperative game theory

encompasses concepts such as the Nash bargaining solution and the Shapely value. Although

there are certainly proﬁtable applications of both theories and deep connections between the

two, the amateur game theorist would be well advised to mix the two theories with extreme

caution. Mixing the two theories can easily lead to results that are incomprehensible, at best,

and utter nonsense, at worst.

Perhaps the mostcommon mistake among legitimate applications ofgame theory, though,

is a failure to clearly deﬁne the game and the setting in which it is played. Who are the players of

the game? What actions are available to the players? What are the players’ objectives? Does an

equilibrium exist for the given game? Is it unique? Is there a dynamic process by which players

update their strategies? If so, what is it and is it guaranteed to converge?

Finally, there are two general “philosophies” on applying game theory, and it is extremely

important to consider which philosophy you are using. In many cases, researchers who have not

clearly thought through their reasoning will jump back and forth between the two philosophies.

This is bound to create holes in their arguments, as the philosophies are mutually exclusive.

The ﬁrst philosophy might be called a direct application of game theory. In this philoso-

phy, the users’ actual preferences are said to be modeled by the players’ utility functions. What

makes the application of this philosophy difﬁcult is that it is very difﬁcult to extract preferences

from users. Furthermore, nailing down a single explicit utility function that represents the

preferences of all users is not possible, since different users will have different preferences. It is

unlikely that even a parameterized utility function can represent the preferences of every user

except in very simple (perhaps contrived) situations. Even if one could postulate the existence

of such a function, it would take considerable effort to prove that such a formulation was valid

for the majority of users in the system.

Thus, if one wants to claim that the utility functions in a game represent actual users’

preferences, then one must operate with a bare minimum collection of assumptions about the

users’ preferences. Typical assumptions might include an assumption that users prefer higher

signal-to-interference-and-noise ratios (SINRs) to lower SINRs and that users prefer to spend

less power rather than more to obtain a given SINR. This approach can produce beautiful

results. It is, however, a very challenging task to execute well.

The second philosophy might be called an “engineering” application of game theory.

This philosophy assumes that the engineer is capable of programming the devices in the system

to behave as if they are maximizing a chosen utility function. Since the engineer chooses the

utility function, it can be whatever the engineer desires. The burden in this philosophy lies

in explaining why a particular utility function has been chosen and why game theory is being

employed at all. Though perhaps easier than demonstrating that a particular utility function

represents the preferences of all users, this may still be a difﬁcult challenge to meet.

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INTRODUCTION TO GAME THEORY 7

Note that there is a stark contrast between these two philosophies. Either you are claiming

thatyour utility functionsrepresentactual userpreferences—inwhichcase youhadbetter provide

some evidence or make the assumptions about those functions very light—or you are not—in

which case you had better provide some other compelling reason for using game theory.

The pointof thissection is notto discouragethe use ofgame theory. Obviously,the authors

think that game theory is a valuable tool for approaching a number of different problems; as

discussed before, game theory has revolutionized economics. The point, though, is that game

theory, like any tool, must be used carefully and on the right kind of problems. To quote a

familiar clich

e, “If the only tool you have is a hammer, then every problem looks like a nail.”

We must be vigilant not to fall into that trap.

1.5 INTRODUCTION TO EXAMPLES

In this work, we will discuss a variety of communications and networking examples to illustrate

the concepts that we introduce. These examples should provide a sense of how game theory

has been used by researchers and will hopefully inspire you to consider some of your own game

theory applications. The examples should also demonstrate (on a small scale) each of the two

philosophies discussed above.

This section brieﬂy introduces three examples to start your thinking along game theoretic

lines.

1.5.1 Power Control

The problem of uplink power control in a code division multiple access (CDMA) cellular system

has been considered by a number of authors [9–14]. For simplicity, we will discuss a single-cell

model, although multi-cell models (and, more recently, even ad hoc models) are also discussed

in the literature.

The players in a cellular power control game are the cellular telephones in the cell. This

is often assumed to be a ﬁxed, known number of players. Each player’s action is the power level

that she chooses. In many cases, this is limited to be a power level between 0 and p

max

, although

some games consider power levels between 0 and ∞.

The question of payoff functions, as usual, is the most difﬁcult question; we will consider

it further in the next chapter. For now, though, consider the following properties, which make

this problem appropriate for a game model.

•

The player’s payoff is a function of her own transmit power level and her SINR. The

player’s SINR is a function of her own transmit power and the transmit powers of the

other players in the cell.

•

When a player increases her power level, this will increase her own SINR, but will

decrease the SINRs of all other players.

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8 GAME THEORY FOR WIRELESS ENGINEERS

•

For a ﬁxed SINR, the players prefer lower power levels to higher ones. That is, players

wish to conserve power and extend their battery life when possible.

•

For a ﬁxed power level, players prefer higher SINR to lower SINR. That is, players

want the best possible channel conditions for a given expenditure of power.

Game theoretic analysis of this situation has yielded several conclusions. First, taken as

a simple game, the power control situation described here usually leads to extremely inefﬁcient

outcomes. What typically occurs is that each player increases her power to increase her SINR.

This power increase, though, decreases the SINRs of other users, who then increase their power

levels to compensate. By the time an equilibrium is reached, all users are blasting away at much

higher power levels than necessary—they could all reduce their powers and everyone would be

better off. This could perhaps be viewed as a negative result for game theory, but we believe

that it provides some insight into the reality of power control: left to their own devices, selﬁsh

users will behave inefﬁciently in the power control game.

Second, though, game theory provides some insight into ways to obtain better outcomes

in the power control scenario. For instance, it is possible to add external incentives against

power increases, such as charging users based on transmit power. Another option is to model

the scenario as a repeated game, where users can punish each other for using too much power.

We will look at some of these possibilities in a future chapter.

What makes this problem interesting to model as a game? There are multiple nodes

making individual power control decisions that impact the performance of every node in the

cell. Thus, we have an interactive decision problem of the type that is well modeled by game

theory.

1.5.2 Routing

Another problem that is well modeled by game theory is the problem of routing in a network.

Some examples of work in this area include [15–18]. For simplicity, we assume that a policy

of source routing is followed, where the complete route for each packet is determined by the

packet source.

In this problem, the players in the game could be viewed as the source nodes in the

network, but it is slightly more convenient to view a player as a source/destination pair.

(In reality, the decision making will be carried out by the source, but making the source/

destination pair the player allows for the existence of multiple ﬂows from a single source.) The

action set available to each player is the set of all possible paths from the source to the destina-

tion. (In many formulations of the routing game, nodes can actually choose multiple routes and

decide how much of their ﬂow to send on each route. We ignore this complication for now.)

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博弈论在无线工程中的应用

Game Theory in Wireless and Communication Networks

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