博弈论与控制理论结合：Pareto-Nash-Stackelberg策略

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"Pareto-Nash-Stackelberg 游戏与控制理论" 本书由 Valeriu Ungureanu 编著，重点关注博弈论在控制理论中的应用，特别是在解决系统优化问题方面。这本书将博弈论的理论与控制理论相结合，为智能创新、系统和技术提供了新的研究视角。 " Nash 均衡"是博弈论中的一个核心概念，由 John Nash 提出，描述了在一个多人博弈中，每个参与者都无法通过单方面改变策略来提高自己的利益，前提是其他参与者的策略保持不变。这种状态被称为纳什均衡，是理解许多现实世界竞争情况的基础，如经济学、政治学和生物学等领域。 " Pareto 最优"是多目标优化中的一个关键概念，它指的是一个解决方案，在不使任何一方的福利变得更差的情况下，无法进一步改善至少一方的福利。在博弈论中，Pareto 最优可以帮助识别对所有参与者都有利的结果。 " Stackelberg 游戏"或称为 Stackelberg 领导者-跟随者博弈，是一种顺序博弈模型，其中一个领导者（通常具有更多的信息或决策优势）首先行动，然后跟随者根据领导者的选择作出反应。这在控制理论中特别有用，因为它模拟了如市场领导者与竞争对手之间的动态交互，或者在安全和军事策略中指挥官与敌方的决策过程。书中涉及的模型适用于扩展形式游戏，这种形式的游戏允许更复杂的时间结构和信息结构。作者深入研究了战略形式游戏，这是非合作博弈的主要研究领域，其中每个参与者独立地选择策略，而不与其他参与者协商。此外，书中的内容需要读者具备一定的数学背景，包括博弈论、优化方法和多准则决策分析等领域的基础知识。对于应用数学系的研究生来说，这是一个挑战性的读物，但它提供了一个深入理解博弈论与控制理论如何协同工作的平台，这对于解决现实世界的复杂问题，如网络安全性、交通管理和能源分配等具有重要意义。 "Pareto-Nash-Stackelberg 游戏与控制理论"是一部综合了博弈论理论与控制策略的专著，适合于那些希望在系统优化问题中应用博弈论分析的专业人士，以及对博弈论和控制理论有深厚兴趣的学者和研究人员。通过深入学习，读者将能够掌握如何利用这些理论工具解决实际问题，尤其是在那些涉及多个决策者和复杂互动的领域。

2 1 Introduction

There is an entire descriptive science dedicated to conﬂicts and their resolutions:

conﬂict theory and conﬂict resolution/reconciliation (see, e.g., [1, 2]) mostly applied

in sociology, social psychology and politics [3–5].

But, what is the role of game theory in such case?

1.1.2 What is Game Theory?

Game theory is characterized by its dominant features as

a theory of mathematical

models of decision making in situations and processes of conﬂict and cooperation

between intelligent rational players.

By intelligent players we mean generally the persons who have exhaustive knowl-

edge. By rational players we assume the persons that maximize their pay-off.

According to Robert Aumann, “Interactive Decision Theory” would be a more

appropriate name/title/description for a discipline traditionally called “Game The-

ory” [6]. Sure, an ideal name for a discipline is hard to ﬁnd. The last one has its

shortcomings, e.g. there are some situations [7] that may be called interactive deci-

sions, but they do not fulﬁl the main characteristics of the situations of conﬂict and

cooperation traditionally considered in classical game theory. Other deﬁnitions of

game theory may be presented, for example Shoham and Leyton-Brown in [8] deﬁnes

game theory as a mathematical study of interaction among self-interested decision

makers, Osborne in [9] deﬁnes game theory as a bag of analytical tools which are

created with the aim to support humans to understand phenomena which appear in

the process of decision makers interaction, Tadelis in [10] describes game theory as a

framework created on the bases of mathematical models of conﬂict and cooperation

phenomena among rational decision makers. Sure, there are other game theory deﬁ-

nitions, but all of them have in common the notion of conﬂict, cooperation, decision

makers, interactivity, and mathematical study. So, we propose to limit ourselves to

our deﬁnition because it includes all highlighted components, reﬂects the content of

this book and lets us get to the notion of decision making.

1.1.3 Decision Making

Decision making may be interpreted as a choice problem that is usually solved

by every being at different moments of his life. Intuitively, the simplest decision

making problem consists in choosing a decision from a set of admissible decisions.

If the beings are rational, they will choose admissible decisions, optimal according

to some criteria. Formally, such decision making problems can be formulated as

single-objective optimization problems:

f (x) → max, x ∈ X, (1.1)

1.1 Basic Preliminary Terminology 3

where X is the set of admissible decisions, f : X → R is an objective function

that corresponds to a decision making criterion. Evidently, the objective may be to

maximize or minimize the value of the objective function. In this context, it must be

mention that we do not concern in this book any theoretical aspects both of Zermelo’s

axiom of choice [11], in particular, and axiomatic set theory [12], in general.

An essential feature of Problem (1.1) consists of its deterministic characteristic

— all the data about Problem (1.1) are known and the effect of the decision variable

x on the objective function may be computed with certainty. So, Problem (1.1)is

considered and solved in conditions of certainty. Traditional optimization or mathe-

matical programming problems are typical examples of decision making problems

in conditions of certainty.

Generally, Problem (1.1) is not a simple problem. There are no general methods

for its solving. But for different special classes of Problem (1.1) (e.g., linear pro-

gramming [13, 14] and convex programming [15–18]), efﬁcient solving algorithms

exist.

1.1.4 Multi-Objective Optimization

If in (1.1) a vector criterion and, consequently, a vector objective f unction is con-

sidered, then the decision making problem becomes a multi-objective optimization

problem:

f(x) → max, x ∈ X, (1.2)

where X remains the set of admissible decisions, but f : X → R

becomes an objec-

tive vector-function that corresponds to m criteria of decision making. Problem (1.2)

requires a deﬁnition of optimality [19] because the traditional notions of maximum

and minimum are not sufﬁcient for Problem (1.2) solving. From this point of view,

Problem (1.2) is more difﬁcult than Problem (1.1).

The notion of efﬁciency or Pareto optimality is the main in studying multi-objective

optimization problems. Generally, in such problems the solution consists of a set of

equivalent efﬁcient points. From the theoretical point of view, such a solution is good

enough. But, from the practical point of view only an instance of the efﬁcient points

is needed. So the decision makers have to decide which one from the set of efﬁcient

points ﬁts better their expectations.

Problem (1.2) may be generalized by considering partial order relations or prefer-

ences on the set of admissible decisions, i.e. this means that criteria are not obligatory

of numeric types:

f(x) → max

, x ∈ X, (1.3)

where the notation max

means maximization according to partial order relation R ⊆

X×X. Evidently, the partial order relation may be not only of binary type, but also of

multiple type. Dominant and non-dominated decisions are traditionally considered

4 1 Introduction

optimal. But there is a problem with such solutions. Often, there are no dominant

decisions according to considered partial order relation. Nevertheless, from practical

point of view such an optimal solution must be found. In such cases, on the bases of

existent partial order relation another one is constructed on which the decisions can be

ordered. Evidently, such a relation is very subjective and depends mainly of decision

maker and implied experts. Problem (1.3) is referred to generalized mathematical

programming [20, 21]. The decision ordering may be achieved by special methods

such as: ORESTE, PROMETHEE and ELECTRE [22].

1.1.5 Multi-Agent Decision Problems

In Problems (1.1)–(1.3) decisions are made by a single person. But there are problems

in which the values of objective function depends not only of a single person decision

x ∈ X, but depends of decisions y ∈ Y of a series of persons which have generally

different interests:

f(x, y) → max, x ∈ X. (1.4)

This kind of problems are studied by game theory. It is simply to observe that problem

(1.4) may be seen as a parametric programming/optimization problem, where x is

the optimization variable, and y ∈ Y are the parameters. Let us observe, also, that

for every particular person an appropriate “parametric” problem must be solved.

A new type of “optimization” problems appear (see bellow normal form of the

noncooperative game) and a new type of solution concepts are required such as, e.g.,

Nash equilibrium concept.

A special case of Problem (1.4)is:

f(x,ξ)→ max, x ∈ X, (1.5)

where ξ ∈ Y is a random variable. Problem (1.5) can be treated as a zero-sum two-

player game theory problem in which the ﬁrst player is the decision maker, and the

second is nature.

Evidently, a decision making problem can possess the properties of all Problems

(1.1)–(1.5). Such a problem implies at least three players, and one of them is the

nature.

Among decision making problems (1.5) we can distinguish problems in conditions

of risk and problems in conditions of uncertainty.

1.1.6 Decision Making Problems in Situations of Risk

The random parameter in decision making problems in situations of risk trans-

forms the objective function into a random factor. This means that the traditional

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