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A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.

• In a fugitive move, the fugitive can travel no more than a distance of $\delta$
• In an officer move, the sum of distances travelled by all officers can be no more than $\delta$

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the officers' initial distribution?

If distance between the fugitive and an officer is $0$ in finite moves, the fugitive is caught, otherwise they escape. I strongly suspect the fugitive can escape if $\delta$ is small enough, but am unable to give a proof. I created this problem myself and know no other existing sources.

Addressing issues in the comments:

• A natural strategy for the officers is to stay on a circle centered round the fugitive, and somehow try to gradually shrink it while preventing the fugitive from escaping out of it, as suggested by Will. But it seems this strategy wouldn't work as expected, as pointed out by usul.

• TimothyChow mentioned the angel problem and fox games, both of which are games played on 2D lattice. I'm not sure ideas in lattice games would help, but let's see if any progress could be made if I "reformulate" the game in that fashion: On an infinite chessboard there's a single white king and $N$ black kings. The nearest black king must be $D$ moves away from the white king. Given $N$, white dictates the value of $D$, then black places their kings. Can white always force a draw without capturing any black kings? Of course, we all understand those kings are just the fugitive (white) and officers (black) dressing and moving like chess kings, so ignore niceties in chess rules such as stalemate, etc. For that purpose you can also replace the kings with power-one rooks (which can move only one square in four cardinal directions). Hopefully an answer for the game could shed some light on the original game, though I have no ready answer in mind so far.

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.

• In a fugitive move, the fugitive can travel no more than a distance of $\delta$
• In an officer move, the sum of distances travelled by all officers can be no more than $\delta$

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the officers' initial distribution?

If distance between the fugitive and an officer is $0$ in finite moves, the fugitive is caught, otherwise they escape. I strongly suspect the fugitive can escape if $\delta$ is small enough, but am unable to give a proof. I created this problem myself and know no other existing sources.

Addressing issues in the comments:

• A natural strategy for the officers is to stay on a circle centered round the fugitive, and somehow try to gradually shrink it while preventing the fugitive from escaping out of it, as suggested by Will. But it seems this strategy wouldn't work as expected, as pointed out by usul.

• TimothyChow mentioned the angel problem and fox games, both of which are games played on 2D lattice. I'm not sure ideas in lattice games would help, but let's see if any progress could be made if I "reformulate" the game in that fashion: On an infinite chessboard there's a single white king and $N$ black kings. The nearest black king must be $D$ moves away from the white king. Given $N$, white dictates the value of $D$, then black places their kings. Can white always force a draw without capturing any black kings?（Let's ignore niceties in chess rules such as stalemate, etc.） For that purpose you can also replace the kings with power-one rooks (which can move only one square in four cardinal directions). Hopefully an answer for the game could shed some light on the original game, though I have no ready answer in mind so far.

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.

• In a fugitive move, the fugitive can travel no more than a distance of $\delta$
• In an officer move, the sum of distances travelled by all officers can be no more than $\delta$

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the officers' initial distribution?

If distance between the fugitive and an officer is $0$ in finite moves, the fugitive is caught, otherwise they escape. I strongly suspect the fugitive can escape if $\delta$ is small enough, but am unable to give a proof. I created this problem myself and know no other existing sources.

Addressing issues in the comments:

• TimothyChow mentioned the angel problem and fox games, both of which are games played on 2D lattice. I'm not sure ideas in lattice games would help, but let's see if any progress could be made if I "reformulate" the game in that fashion: On an infinite chessboard there's a single white king and $N$ black kings. The nearest black king must be $D$ moves away from the white king. Given $N$, white dictates the value of $D$, then black places their kings. Can white always force a draw without capturing any black kings? Of course, we all understand those kings are just the fugitive (white) and officers (black) dressing and moving like chess kings, so ignore niceties in chess rules such as stalemate, etc. For that purpose you can also replace the kings with power-one rooks (which can move only one square in four cardinal directions). Hopefully an answer for the game could shed some light on the original game, though I have no ready answer in mind so far.
• A natural strategy for the officers is to stay on a circle centered round the fugitive, and somehow try to gradually shrink it while preventing the fugitive from escaping out of it, as suggested by Will. But it seems this strategy wouldn't work as expected, as pointed out by usul.

A fugitive is surrounded by $N$ police officers, with the nearest one at distance $1$ away. The fugitive and the officers move alternatively.

• In a fugitive move, the fugitive can travel no more than a distance of $\delta$
• In an officer move, the sum of distances travelled by all officers can be no more than $\delta$

Is it true that for $\forall N$, $\exists \delta$ such that the fugitive can escape regardless of the officers' initial distribution?

If distance between the fugitive and an officer is $0$ in finite moves, the fugitive is caught, otherwise they escape. I strongly suspect the fugitive can escape if $\delta$ is small enough, but am unable to give a proof. I created this problem myself and know no other existing sources.

Addressing issues in the comments:

• A natural strategy for the officers is to stay on a circle centered round the fugitive, and somehow try to gradually shrink it while preventing the fugitive from escaping out of it, as suggested by Will. But it seems this strategy wouldn't work as expected, as pointed out by usul.

• TimothyChow mentioned the angel problem and fox games, both of which are games played on 2D lattice. I'm not sure ideas in lattice games would help, but let's see if any progress could be made if I "reformulate" the game in that fashion: On an infinite chessboard there's a single white king and $N$ black kings. The nearest black king must be $D$ moves away from the white king. Given $N$, white dictates the value of $D$, then black places their kings. Can white always force a draw without capturing any black kings? Of course, we all understand those kings are just the fugitive (white) and officers (black) dressing and moving like chess kings, so ignore niceties in chess rules such as stalemate, etc. For that purpose you can also replace the kings with power-one rooks (which can move only one square in four cardinal directions). Hopefully an answer for the game could shed some light on the original game, though I have no ready answer in mind so far.