I tried to find solutions for this one, actually found the correct answer for the first one and gave up on the second. Happy to see, that my assumptions were correct actually.
The weird solution has to be right where it is. There has to be a radius for player 1-3, when placing player 4 right next to player 1 would be better than on the other side of player 1 on the line, and the optimal solution has to balance these options.
Ohh that’s a very good point. It would seem that the solution occurs when those configurations yield the same maximum area for player 4. I’m realizing now that I probably forgot to account for the “outer” solution with the value I provided (I should have known better than to second guess Peter and Izumihara!) I think the value is cited might be a little too close to the center and opened up a better position for Player 4 in the “outer” configuration. I’ll come back to this and investigate further.
When the first three players form a triangle with a radius of 0.435, player 4 can secure a 22.9% chance of success by selecting a point very close to one of these players in the direction away from the origin.
I derived the solution I submitted previously by solving a series of equations---one of which enforces the condition that the probabilities of the "inner" and "outer" outcomes are equal.
Thus, as you may recall from the two figures, player 4 ends up with two solutions that are equally likely.
"If rPi is the distance Pi is away from the origin, then Player 4’s Voronoi cell will have a triangular shape (except that one of the sides is curvilinear) when rP4 > rP1 and will have a quadrilateral shape (again, with the exception of one side which is curvilinear) when rP4 > rP1."
good catch on the inequality, I've corrected it. it should read "...and will have a quadrilateral shape (again, with the exception of one side which is curvilinear) when rP4 < rP1."
Defining r_P_i was just so that I didn't have to define r_P_4 and r_P_1 separately. Not quite the right usage but I figured the reader would know what I meant.
Not exactly a fellow Badger, but I'm a fan!
I tried to find solutions for this one, actually found the correct answer for the first one and gave up on the second. Happy to see, that my assumptions were correct actually.
The weird solution has to be right where it is. There has to be a radius for player 1-3, when placing player 4 right next to player 1 would be better than on the other side of player 1 on the line, and the optimal solution has to balance these options.
Ohh that’s a very good point. It would seem that the solution occurs when those configurations yield the same maximum area for player 4. I’m realizing now that I probably forgot to account for the “outer” solution with the value I provided (I should have known better than to second guess Peter and Izumihara!) I think the value is cited might be a little too close to the center and opened up a better position for Player 4 in the “outer” configuration. I’ll come back to this and investigate further.
When the first three players form a triangle with a radius of 0.435, player 4 can secure a 22.9% chance of success by selecting a point very close to one of these players in the direction away from the origin.
I derived the solution I submitted previously by solving a series of equations---one of which enforces the condition that the probabilities of the "inner" and "outer" outcomes are equal.
Thus, as you may recall from the two figures, player 4 ends up with two solutions that are equally likely.
"If rPi is the distance Pi is away from the origin, then Player 4’s Voronoi cell will have a triangular shape (except that one of the sides is curvilinear) when rP4 > rP1 and will have a quadrilateral shape (again, with the exception of one side which is curvilinear) when rP4 > rP1."
1) This doesn't use rPi.
2) The 2 inequalities are the same.
good catch on the inequality, I've corrected it. it should read "...and will have a quadrilateral shape (again, with the exception of one side which is curvilinear) when rP4 < rP1."
Defining r_P_i was just so that I didn't have to define r_P_4 and r_P_1 separately. Not quite the right usage but I figured the reader would know what I meant.