Timeline for Can the thief escape (from a smooth, simple closed curve)?
Current License: CC BY-SA 4.0
26 events
| when toggle format | what | by | license | comment | |
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| Oct 10, 2021 at 11:42 | vote | accept | Eric | ||
| Oct 7, 2021 at 22:32 | answer | added | Yaakov Baruch | timeline score: 7 | |
| Oct 6, 2021 at 15:37 | comment | added | Alessandro Della Corte | I believe that this is quite closely related: arxiv.org/pdf/1903.00688.pdf. | |
| Oct 6, 2021 at 8:08 | history | edited | Wlod AA | CC BY-SA 4.0 |
still simpler
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| Oct 6, 2021 at 8:01 | comment | added | Ville Salo | @Eric Two other possibilities which I think can both happen when you consider very general games: 1) the winner of a game is not even determined for some strategies, in the sense that the behavior is a differential equation with no solution, 2) there is no winning strategy for either player, i.e. any strategy of one can be beaten by the other, by changing the strategy (non-determinacy). For 2) I don't know "this type" of example, but a standard example is the "pick the bigger number" game. | |
| Oct 6, 2021 at 7:26 | comment | added | Eric | @VilleSalo What could be a fourth possibility? | |
| Oct 6, 2021 at 7:02 | answer | added | Yaakov Baruch | timeline score: 0 | |
| Oct 6, 2021 at 5:53 | comment | added | Ville Salo | I think this was sort of said already, but I am pretty sure that the "There're 3 possibilities" trichotomy makes a determinacy type assumption that typically is simply not true for this type of games | |
| Oct 6, 2021 at 5:30 | history | edited | Eric | CC BY-SA 4.0 |
added 11 characters in body
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| Oct 6, 2021 at 5:20 | history | edited | Eric | CC BY-SA 4.0 |
response to comments; tags
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| Oct 6, 2021 at 4:39 | comment | added | Anton Petrunin | @YaakovBaruch seems right --- make an answer from your comment. | |
| Oct 6, 2021 at 3:54 | comment | added | Eric | @MichaelRenardy They have perfect information about each other's current position. | |
| Oct 6, 2021 at 0:29 | comment | added | Christian Remling | This also has a slightly paradoxical feel to it because no matter what strategy the police chose, it clearly was stupid since they could have deployed an additional countable squad at no cost. | |
| Oct 6, 2021 at 0:27 | comment | added | Christian Remling | @DavidESpeyer: I'm not optimistic about this, as an attempt to give rigorous definitions. One would have to solve an infinite system which is about the same thing as a (non-linear) PDE, so we can't expect any existence or uniqueness results to be within reach. | |
| Oct 5, 2021 at 20:55 | comment | added | Yaakov Baruch | What about 1 cop (for short) trying to catch the thief crossing a segment? If the cop moves on the segment at top speed towards the thief (provided the thief starts far enough and the cop's reaction time is 0) then they should meet. Likewise a dense set of cops on a convex curve: if they run towards the thief only when the latter is much closer to them than, say, the minimum radius of curvature, while remaining dense all the time. On a non convex curve, the thief may try to trap them in a cul de sac, but that should be thwarted by the cops remaining dense always. So my guess is 2. | |
| Oct 5, 2021 at 20:42 | comment | added | Alessandro Della Corte | "Countably infinite officers on đ¶" means that they can be placed densely on $C$. I guess a dense initial placement can't be worse than any placement whose closure does not include $C$. And if they are dense I fail to see how the shape of $C$ can play a role, provided it is smooth. The thief should head for a point of $C$ where there is no officer and the game becomes a local approximation problem. In the survey mentioned by David E Speyer the pursuers are always finite, if I'm not mistaken. | |
| Oct 5, 2021 at 18:54 | comment | added | David E Speyer | There is a whole study of games of this sort; here is a survey arxiv.org/abs/2003.05013 . I couldn't find a quick source to link to for a precise definition, but the definition I would try is that the robber specifies a function giving their velocity as a function of the positions of themself and the police; the police do likewise, and then the trajectories of each person are obtained by solving the resulting ODE. Of course, one has to make sure that the function is nice enough that the ODE is solvable, but maybe we can make progress without digging into the details of this? | |
| Oct 5, 2021 at 18:42 | comment | added | Christian Remling | I suppose both thief and police may adjust their movements, depending on the current configuration. But this seems hard to formalize. It would perhaps work better in a discrete model. | |
| Oct 5, 2021 at 18:08 | comment | added | Michael Renardy | What knowledge do the thief and officers have of the other's position? | |
| Oct 5, 2021 at 17:27 | comment | added | dhy | @BD107 No, because the police bureau can place the officers wherever they wish. If the thief could escape no matter the officer placements from some point $p$, they can escape from any point by first heading to $p$. | |
| Oct 5, 2021 at 17:23 | comment | added | Eric | @PietroMajer I've edited the question to address that. | |
| Oct 5, 2021 at 17:22 | history | edited | Eric | CC BY-SA 4.0 |
maximum speed is 1 for everyone.
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| Oct 5, 2021 at 16:51 | comment | added | BD107 | couldn't it also depend on where the thief and officers start? | |
| Oct 5, 2021 at 16:49 | comment | added | Pietro Majer | âThey move with equal speedâ: what is exactly the regularity of their motion? | |
| Oct 5, 2021 at 16:38 | history | edited | Eric | CC BY-SA 4.0 |
added 13 characters in body; edited title
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| Oct 5, 2021 at 15:58 | history | asked | Eric | CC BY-SA 4.0 |