Guaranteed escaping strategies

To generate evasion strategies and evaluate corresponding guaranteed miss distances from $E$ to $\mathcal P_{j_1,\ldots,j_n} = \{P_{j_1},\ldots, P_{j_n}\}$, $ n \geq 3$, we set up two basic problems for the players with simple motions. In the first one, $E$ maximizes the miss distance to $P_a\in \mathcal P_{j_1,\ldots,j_n}$ when she moves along a given straight-line. In the second one, $E$ seeks to cross the intercept $P_b P_c$ just once and to maximize the miss distance to either of $P_b$ and $P_c$ during the infinite period of manoeuvring. In the game with a group of three or more pursuers, for a given history, we evaluate the minimum of the guaranteed miss distances when $E$ passing between $P_b$ and $ P_c$, $\forall b,c \in \{j_1,\ldots,j_n\}, b\not = c,$ and the guaranteed miss distance to $P_a$, $\forall a \in \{j_1,\ldots,j_n\}\backslash\{b,c\}$. After that, we are able to choose the best alternative for assigning $b$ and $c$.