Linear Quadratic Game of N Persons as the Analog of Antagonistic Game
Authors: Zhukovskiy V. I. , Samsonov S. P. , Romanova V. E. , Zhukovskaya L. V. , Mukhina Yu. S. 
Published: October 2021

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 <<Introduction in the theory of differential games of n-persons and its application>> M.: Sovetskoye Radio, 1980, and in monography of Zhukovskiy <<Equilibrium of objections and counterobjections>>, 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 <<the reigning position>> 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) <<for fundamental analysis of equilibria in noncooperative game theory>>. 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 <<And in the sun there are the spots>>, 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 $f_i(x^{(1)})=0<f_i(x^{(2)})=1~(i=1,2)$ and therefore the set $X^e$ is internally unstable, secondly, $f_i(x^{(1)})=0<f_i(\frac{1}{4},\frac{1}{3})~(i=1,2)$ and therefore the set $X^e$ is externally unstable. The external just as the internal instability of set of Nash equilibrium is negative for its practical use. In the first case there exists situation which dominates NE (for all players), in the second case this situation is Nash equilibrium. Pareto maximality of Nash equilibrium situation would allow to avoid consequences of external and internal instability. However such coincidence is an exotic phenomenon. Thus to avoid trouble connected with external and internal instability then we add the requirement of Pareto maximality to the notion of equilibrium of objections and counterobjections offered below. However we first of all reduce generally accepted solution concepts "--- NE and BE for the game $G_N$.
It is proved in the article that in mathematical model both NE and BE are absent but there exist equilibria of objections and conterobjections as well as sanctions and countersanctions and simultaneously Pareto maximality.