Linear Quadratic Game of N Persons as the Analog of Antagonistic Game

Abstract

Publications on mathematical game theory with many (not less than 2) players one can conditionally distribute in four directions: noncooperative, hierarchical, cooperative and coalition games. The two last in its turn are divided in the games with side and nonside payments and respectively in the games with transferable and nontransferable payoffs. If the first ones are being actively investigated (St. Petersburg State Faculty of Applied Mathematics and Control Processes, St. Petersburg Economics and Mathematics Institute, Institute of Applied Mathematical Research of Karelian Research Centre RAS), then the games with nontransferable payoffs are not covered. Here we suggest to base on conception of objections and counterobjections. The initial investigations were published in two monographies of E.\;I.\;Vilkas, the Lithuanian mathematician (the pupil of N.\;N.\;Vorobjev, the professor of St. Petersburg University). For the differential games this conception was first applied by E.\;M.\;Waisbord in 1974, then it was continued by the first author of the present article combined with E.\;M.\;Waisbord in the book <> M.: Sovetskoye Radio, 1980, and in monography of Zhukovskiy <>, M.: KRASAND, 2010. However before formulating tasks of coalition games, to which this article is devoted, we return to noncoalition variant of many persons game. Namely we consider noncooperation game in normal form, defined by ordered triple: $$G_N=\langle\mathbb{N}, \{X_i\}_{i \in \mathbb{N}}, \{f_i (x)\}_{i \in \mathbb{N}}\rangle.$$ Here $\mathbb{N}=\{1,2,\ldots , N\ge2\}$ "--- set of ordinal numbers of players, each of them (see later) chooses its strategy $x_i\in X_i\subseteq \mathbb{R}^{n_i}$ (where by the symbol $\R^k$, $k\ge 1$, as usual, is denoted $k$-dimensional real Euclidean space, its elements are ordered sets of $k$-dimensional numbers, as well Euclidean norm $\parallel \cdot \parallel$ is used); as a result situation $x=(x_1,x_2,\ldots,x_N)\in X=\prod \limits_{i\in \mathbb{N}}X_i\subseteq \mathbb{R}^{\sum \limits_{i\in \mathbb{N}}n_i}$ form in the game. Payoff functions $f_i (x)$ are defined on set $X$ of situations $x$ for each players: \begin{gather*} f_1(x)=\sum_{j=1}^{N}x_{j}^{'} D_{1j}x_{j}+2d_{11}^{'} x_1,\\ f_2(x)=\sum_{j=1}^{N}x_{j}^{'} D_{2j}x_{j}+2d_{22}^{'} x_2,\\ \ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\ f_N(x)=\sum_{j=1}^{N}x_{j}^{'} D_{Nj}x_{j}+2d_{NN}^{'} x_N. \end{gather*} In the opinion of luminaries of game theory to equilibrium as acceptable solution of differential game has to be inherent the property of stability: the deviation from it of individual player cannot increase the payoff of deviated player. The solution suggested in 1949 (at that time by the 21 years old postgraduate student of Prinston University John Forbs Nash (jun.) and later named Nash equilibrium "--- NE) meets entirely this requirement. The NE gained certainly <> in economics, sociology, military sciences. In 1994 J.\;F.\;Nash was awarded Nobel Prize in economics (in a common effort with John Harsanyi and R.\;Selten) <>. Actually Nash developed the foundation of scientific method that played the great role in the development of world economy. If we open any scientific journal in economics, operation research, system analysis or game theory we certainly find publications concerned NE. However <>, situations sets of Nash equilibrium must be internally and externally unstable. Thus, in the simplest noncoalition game of 2 persons in the normal form $$ \langle\{1,2\},~\{X_i=[-1,1]\}_{i=1,2},~ \{f_i(x_1,x_2)=2x_1x_2-x_i^2\}_{i=1,2}\rangle$$ set os Nash equilibrium situations will be $$X^{e}=\{x^{e}=(x_1^{e},~x_2^{e})=(\alpha, \alpha)~|~\forall \alpha=const\in [-1,1]\},~f_i(x^e)=\alpha^2~(i\in 1,2).$$ For elements of this set (the segment of bisectrix of the first and the third quarter of coordinate angle), firstly, for $x^{(1)}=(0,0)\in X^{(e)}$ and $x^{(2)}=(1,1)\in X^{e}$ we have $f_i(x^{(1)})=0

分享

查看原文 本刊更多论文

作为对抗博弈模拟的N人线性二次博弈

有许多(不少于2)个参与者的数学博弈论出版物可以有条件地分布在四个方向上:非合作、分层、合作和联合博弈。在有一方支付和非一方支付的博弈中，以及在有可转移支付和不可转移支付的博弈中，最后两者分别被划分。如果第一个游戏正在积极调查(圣彼得堡国立应用数学和控制过程学院，圣彼得堡经济和数学研究所，卡累利阿研究中心RAS应用数学研究所)，那么具有不可转移收益的游戏就不包括在内。在此，我们建议以异议和反异议的概念为基础。最初的研究发表在立陶宛数学家E.\;I.\;Vilkas(圣彼得堡大学教授N.\;N.\;Vorobjev的学生)的两本专著中。对于微分对策，这一概念首先由e \;M \;Waisbord于1974年应用，然后由本文的第一作者与e \;M \;Waisbord在《M: Sovetskoye Radio》(1980)一书中以及Zhukovskiy的专著《M: KRASAND》(2010)中继续应用。然而，在确定本文所讨论的联合对策的任务之前，我们先回到多人对策的非联合变体。即我们考虑非合作博弈的标准形式，定义为有序三元组: $$G_N=\langle\mathbb{N}, \{X_i\}_{i \in \mathbb{N}}, \{f_i (x)\}_{i \in \mathbb{N}}\rangle.$$ 这里 $\mathbb{N}=\{1,2,\ldots , N\ge2\}$ ——一组序数玩家，他们每个人(见后文)选择自己的策略 $x_i\in X_i\subseteq \mathbb{R}^{n_i}$ (这里用符号表示 $\R^k$， $k\ge 1$，如往常一样，表示 $k$-维实欧几里德空间，它的元素是的有序集合 $k$维数，以及欧几里得范数 $\parallel \cdot \parallel$ 是用的);结果就是 $x=(x_1,x_2,\ldots,x_N)\in X=\prod \limits_{i\in \mathbb{N}}X_i\subseteq \mathbb{R}^{\sum \limits_{i\in \mathbb{N}}n_i}$ 游戏中的形式。收益函数 $f_i (x)$ 是在集合上定义的 $X$ 各种情况 $x$ 对于每个玩家:\begin{gather*}f_1(x)=\sum_{j=1}^{N}x_{j}^{'} D_{1j}x_{j}+2d_{11}^{'} x_1,\\f_2(x)=\sum_{j=1}^{N}x_{j}^{'} D_{2j}x_{j}+2d_{22}^{'} x_2,\\\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\ldots\\f_N(x)=\sum_{j=1}^{N}x_{j}^{'} D_{Nj}x_{j}+2d_{NN}^{'} x_N.\end{gather*}博弈论学者认为均衡作为微分博弈的可接受解必须具有固有的稳定性，即个体的偏离不能增加偏离者的收益。1949年提出的解决方案(当时21岁的普林斯顿大学研究生约翰·福布斯·纳什(John forbes Nash, 6月)，后来被命名为“纳什均衡”——NE)完全符合这一要求。东北大学在经济学、社会学和军事科学方面取得了很大的进步。1994年，J.\;F.\;纳什(与约翰·海萨尼和R.\;塞尔滕共同获得诺贝尔经济学奖)。实际上，纳什发展了科学方法的基础，对世界经济的发展起了巨大的作用。如果我们打开任何一本经济学、运筹学、系统分析或博弈论的科学杂志，我们肯定会发现有关新能源的出版物。然而，纳什均衡的情况集必须是内外不稳定的。因此，在最简单的二人非联盟博弈中 $$ \langle\{1,2\},~\{X_i=[-1,1]\}_{i=1,2},~ \{f_i(x_1,x_2)=2x_1x_2-x_i^2\}_{i=1,2}\rangle$$ 一组纳什均衡情况是 $$X^{e}=\{x^{e}=(x_1^{e},~x_2^{e})=(\alpha, \alpha)~|~\forall \alpha=const\in [-1,1]\},~f_i(x^e)=\alpha^2~(i\in 1,2).$$ 对于该集合的元素(坐标角的1 / 4和3 / 4的二等分线段)，首先，为 $x^{(1)}=(0,0)\in X^{(e)}$ 和 $x^{(2)}=(1,1)\in X^{e}$ 我们有 $f_i(x^{(1)})=0<f_i(x^{(2)})=1~(i=1,2)$ 因此这个集合 $X^e$ 内部不稳定，其次， $f_i(x^{(1)})=0<f_i(\frac{1}{4},\frac{1}{3})~(i=1,2)$ 因此这个集合 $X^e$ 外部不稳定。在实际应用中，纳什均衡集的外部不稳定性如同内部不稳定性一样是负的。在第一种情况下，存在支配NE(所有参与者)的情况，在第二种情况下，这种情况是纳什均衡。纳什均衡条件下的帕累托极大值可以避免外部和内部不稳定的后果。然而，这种巧合是一种奇特的现象。因此，为了避免与外部和内部不稳定相关的麻烦，我们在下面提出的反对和反反对的平衡概念中增加了帕累托最大值的要求。然而，我们首先要减少普遍接受的解决方案概念”——游戏的NE和BE $G_N$．

本文章由计算机程序翻译，如有差异，请以英文原文为准。