MATHEMATICS

L. A. PETROSYAN

DIFFERENTIAL SURVIVAL GAMES WITH MANY PARTICIPANTS

(Presented by Academician Yu. V. Linnik, 8 X 1964)

The games considered are models of pursuit in a certain convex set \(S\) in the plane with several participants: a group of pursuers \(P=\{P_1,\ldots,P_n\}\) and pursued objects \(E_1,\ldots,E_n\), which move in the set \(S\) with constant linear speeds and have the ability at each moment of time to change the direction of their motion. A pursued object \(E_j\) is considered caught as soon as there is an \(i\in\{1,2,\ldots,n\}\) such that the points \(P_i\) and \(E_j\) coincide. Each of the pursued objects is interested in reaching the boundary of the set \(S\) before being caught by one of the pursuers.

In what follows we shall use the definitions and notation of note \((^1)\).

1. The game \(\Gamma(n,1)\). The game \(\Gamma(n,1)\) is an antagonistic game of two players: a group of pursuers \(P=\{P_1,\ldots,P_n\}\) and a pursued object \(E\). In the plane there is given a certain closed convex set \(S\). Points \(\xi^1,\ldots,\xi^n,\eta\in S\), called the initial positions of the pursued \(P_1,\ldots,P_n\) and of the pursued object \(E\), and \(n+1\) numbers \(v_1,\ldots,v_n,u\), \(v_i>u\) for all \(i=1,2,\ldots,n\), are given.

The structure of the game is specified by the system of equations

\[ \dot{x}^{\,i}=\varphi^i[\psi(x^1,x^2,x^3,\ldots,x^n,t)], \]

\[ \dot{y}=\psi(x^1,x^2,\ldots,x^n,t),\quad i=1,2,\ldots,n, \]

with initial conditions \(x^i(0)=\xi^i,\ y(0)=\eta,\ i=1,2,\ldots,n\).

The set \(\mathfrak{E}\) of strategies of \(E\) and the sets \(\mathfrak{P}^i\) of strategies of \(P_i\) satisfy conditions 1—4 \((^1)\). From these conditions it follows directly that to each situation \((\varphi^1,\ldots,\varphi^n,\psi)\) and initial conditions \(\xi^1,\ldots,\xi^n,\eta\) there correspond uniquely \(2(n+1)\) continuous functions \(x_1^i(t), x_2^i(t), y_1(t), y_2(t)\), \(i=1,2,\ldots,n\).

The pair \(\{x_1^i(t),x_2^i(t)\}\) will be called the trajectory of \(P_i\). Let \(t_P=\min_i (t_P)^i\). The payoff function is defined in the same way as in the game \(\Gamma\).

Theorem 1. The game \(\Gamma(n,1)\) has an equilibrium situation in pure strategies. Moreover, the optimal strategy of the group \(P=\{P_1,\ldots,P_n\}\) consists in each \(P_i\) pursuing \(E\) according to the \(\Pi\)-strategy.

Theorem 2. Suppose that in the game \(\Gamma(n,1)\) there exists a circle \(B\), belonging to \(S\), of radius \(a\), such that: a) \(\xi^1,\ldots,\xi^n\in B\) at the initial moment; b) \(\eta\in B\) at the initial moment; c) we radially project the points corresponding to the positions of the pursuers at the initial moment onto the circumference of \(B\); denote these projections by \(\xi_B^1,\xi_B^2,\ldots,\xi_B^n\), where the indices \(1,2,\ldots,n\) increase in the clockwise direction; the distance between two neighboring projections must be less than \(a/3\).

If conditions a), b), c) are fulfilled, then \(E\) is caught in \(S\) for any \(v_i/u\ge 1,\ i=1,2,\ldots,n\).

1. The games \(\Gamma'\), \(\Gamma'_T\), \(\bar{\Gamma}'_T\), \(\Gamma'(1,2)\). The game \(\Gamma'\) differs from \(\Gamma\) only in the payoff function, which is defined as follows.

Let \(y(t_P)\) be the point of capture in the situation \((\varphi,\psi)\). Suppose that it belongs to \(S\). Denote by \(\rho[y(t_P),S]\) the distance from the point of capture to the boundary of \(S\). Let \(f(\rho)\) be a certain function having the following properties: a) \(f(\rho)\) is a strictly decreasing function of \(\rho\); b) let \(0<\varepsilon<1\), then \(f(0)=\varepsilon\); c) let \(\rho_0\) be the distance from \(\eta\) to \(S\); if capture occurs at the point \(\eta\), then \(f(\rho_0)=0\).

The payoff function in the game \(\Gamma'\), \(K'(\xi,\eta,\varphi,\psi)\), is defined through \(f(\rho)\) as follows.

\[ K'(\xi,\eta,\varphi,\psi)= \begin{cases} -1, & \text{if } t_P>t_{S_E},\\ 1-f(\rho), & \text{if } t_P\leqslant t_{S_E},\\ 0, & \text{if } t_P=t_{S_E}=\infty. \end{cases} \]

Theorem 3. In the game \(\Gamma'\) there exists an equilibrium situation in pure strategies. The player \(P\) has, moreover, a unique optimal strategy—the \(P\)-strategy.

Consider a certain generalization \(\Gamma'_T\) of the game \(\Gamma'\), in which \(P\) can begin pursuit only after some time \(T\) has elapsed from the start of the game. The game \(\Gamma'_T\) obtained in this way will be called the game with delay \(T\), or simply the game with delay.

Lemma 1. In the game \(\Gamma'_T\) the \(P\)-strategy is optimal for \(P\).

Suppose further that in the time interval \([0,T]\) the player \(E\) may use only some subclass \(\mathfrak{C}_T\) of his strategies. Denote this game by \(\bar{\Gamma}'_T\).

Lemma 2. In the game \(\bar{\Gamma}'_T\) the \(P\)-strategy is optimal for \(P\); for any \(\varepsilon>0\), \(E\) has an \(\varepsilon\)-optimal strategy.

We now turn to the solution of the game \(\Gamma'(1,2)\). The game \(\Gamma'(1,2)\) is a three-person game: the pursuer \(P\) and the pursued \(E_1\) and \(E_2\). We shall assume that the players \(E_1\) and \(E_2\) are forbidden to form a coalition. The strategy spaces are the same as in the game \(\Gamma'\). \(v\) is the linear velocity of the pursuer \(P\); \(u_1\) is the linear velocity of the pursued \(E_1\); \(u_2\) is the linear velocity of the pursued \(E_2\).

The inequality \(v>u_i,\ i=1,2\), holds. As in the game \(\Gamma'\), the strategies of the players are denoted by \(\varphi,\psi^1,\psi^2\).

The payoff of \(E_i,\ i=1,2\), is defined as the negative of the payoff of \(P\) in the game \(\Gamma'(P,E_i)\). Let \(K_i\) be the payoff of \(E_i,\ i=1,2\). Then the payoff of \(P\) is given by the expression \(K_P=-K_1-K_2\).

Thus the game \(\Gamma'(1,2)\) is a zero-sum game. In what follows, we shall denote by \(\psi^i(t)\) the optimal strategy of \(E_i\) in the game \(\Gamma'(P,E_i)\), beginning at time \(t\), and by \(\varphi^{i,3-i}\), \(i=1,2\), the strategy of the pursuer \(P\) consisting in the fact that \(P\) successively applies the \(P\)-strategy first to \(E_i\), and then to \(E_{3-i}\).

Consider two situations \((\vec{\psi}^{\,1},\vec{\psi}^{\,2},\varphi^{1,2})\) and \((\vec{\psi}^{\,1},\vec{\psi}^{\,2},\varphi^{2,1})\). Let \(K_P^{1,2}\) be the payoff in one situation, and \(K_P^{2,1}\) in the other.

Theorem 4. If \(K_P^{1,2}\geqslant K_P^{2,1}\), then an \(\varepsilon\)-Nash equilibrium situation in the game \(\Gamma'(1,2)\) is constructed as follows:

a) the optimal strategy of \(P\), \(\varphi^{1,2}\), until the moment while the inequality

\[ K_P(\vec{\psi}^{\,1},\vec{\psi}^{\,2},\varphi^{1,2}) \geqslant K(\vec{\psi}^{\,1},\psi^2,\varphi^{2,1}), \tag{1} \]

holds, and \(\varphi^{2,1}\) as soon as \(E_2\) begins to apply the strategy \(\psi^2\) that violates (1);

b) the optimal strategy of \(E_1\), \(\psi^1\);

c) an $\varepsilon$-optimal strategy $E_2$ is defined as an $\varepsilon$-optimal strategy $\psi^{\varepsilon 2}$ in the game $\Gamma'_p (E_2, P)$ under the restrictions on the class of admissible strategies

\[ K_P(\vec{\psi}^{\,1}, \psi^2, \varphi^{1,2}) \geqslant K_P(\vec{\psi}^{\,1}, \psi^2, \varphi^{2,1}), \tag{2} \]

\[ K_{E_2}(\psi^1, \psi^2, \varphi^{1,2}) \geqslant K_{E_2}(\vec{\psi}^{\,1}, \psi^2, \varphi^{2,1}). \]

The game $\Gamma(m,n)$. Under the preceding conditions there are $m$ pursuers $P_1,\ldots,P_m$ and $n$ evaders $E_1,\ldots,E_n$. We shall denote this game by $\Gamma(m,n)$. For solving this game, so far, using methods of integer linear programming, it has been possible only to estimate from below the value of the game $\operatorname{Val}\Gamma(m,n)$. The estimate of $\operatorname{Val}\Gamma(m,n)$ is obtained as follows. Suppose that the set of pursuers $P_1,\ldots,P_m$ is divided into $n$ pursuing squads $M_1,\ldots,M_n$ in such a way that, by the end of the game, each of the squads $M_1,\ldots,M_n$ can pursue only one of the evaders $E_i$ according to a $P$-strategy. For each fixed partition of the set of pursuers into squads $M_1,\ldots,M_n$, the greatest payoff of the set of pursuers $f(M_1,\ldots,M_n)$ is obtained as the solution of the following integer linear programming problem [2]:

\[ \max \sum_{i,j}^{m,n} \xi_{ij} a_{ij} \tag{3} \]

under the conditions

\[ \sum_{i=1}^{m} \xi_{ij} \leqslant 1,\quad \sum_{j=1}^{n} \xi_{ij} \leqslant 1 \]

for all $i,j$ (here $a_{ij}$ is the payoff of the $i$-th squad $M_i$ from pursuing $E_j$).

The evaders cannot enter into coalitions.
It is clear that

\[ \operatorname{Val}\Gamma(m,n) \geqslant \max_{\{M_1,\ldots,M_n\}} f(M_1,\ldots,M_n), \tag{4} \]

where $\{M_1,\ldots,M_n\}$ is the set of all possible partitions of $P_1,\ldots,P_m$ into $n$ groups.

Received
5 X 1964

CITED LITERATURE

$^{1}$ L. A. Petrosyan, DAN, 161, No. 1 (1965).
$^{2}$ D. Gale, The Theory of Linear Economic Models, N. Y., 1960.