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rmsgrey
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 Duck on a lake with unseen predator   « on: May 6th, 2023, 3:01am » Quote Modify

In the classic duck on a lake problem, a duck is near the middle of a circular lake, with a fox prowling the shore. The fox can run four times as fast as the duck can swim, but if the duck can make it to shore, it can then take off and escape (note: this is not ornithologically accurate - ducks can generally take off from open water).

With both duck and fox able to see each other at all times, the duck can always escape before the fox can intercept it (while the fox can cover more than half the circumference of the lake in the time it takes the duck to swim the radius, so long as the duck is within a quarter radius of the center, it can manage a higher angular velocity, so can stay opposite the fox until it gets that far out, where it takes less time to swim 3/4 of the radius than the fox does to run half the circumference)

What happens when the duck cannot see the fox, but the fox can see the duck? Does either have a strategy that's guaranteed to always win? (the duck escaping, or the fox preventing the escape) If not, what's the best each can manage?
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rmsgrey
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 Re: Duck on a lake with unseen predator   « Reply #1 on: May 15th, 2023, 3:10am » Quote Modify

Here's what I've come up with:

The fox cannot guarantee to capture the duck. If the duck swims in circles around the center of the lake just inside the 1/4 radius line, then, no matter what the fox does, the duck will have a greater angular velocity, and, over a long enough period, will sweep through every possible angle from the fox. If the duck circles for long enough, then swims radially to shore at a random angle, there's a chance that the fox is more than 3 radians from the duck and so unable to beat it to its landing point.

Against this strategy, the fox can attempt to minimise its time in the dead zone, by running with the duck whenever it's in range, and against it whenever it's out of range.

The size of the dead zone depends linearly on how close to the r/4 line the duck is circling (zero at 1 - [pi]/4), while the time the fox can stay out of the dead zone is inversely proportional to how close to the r/4 line the duck circles, so the optimum for the duck under this strategy would be to circle at (5-[pi])r/8.

On the other hand, if the fox uses the strategy of heading toward the duck's projected landing point whenever it can beat the duck there, and moving to exit the dead zone as soon as possible on the assumption the duck follows its most obvious projected motion (by whatever method you use to project it), the duck can counter that extremely effectively by hanging out somewhere off center long enough for the fox to take up position directly outside the duck, giving the fox a known starting position that the duck can then escape from provided the fox follows the obvious strategy.

To counter that sort of bait-and-escape strategy, the fox can ignore the duck whenever it's inside the r/4 line, picking a position around the circumference randomly and waiting for the duck to emerge and then pursuing it.

If that is the optimal strategy, then the fox has a flat 3/[pi] chance of catching the duck.

Of course, the duck also has the "flip the table" option of staying in the pond forever.
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