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If this catches on, perhaps the rules will be referred to as the
Ingo rules ;-) Since this is based on a real world variant of Go, why not base epsilon on that? The fact that the limit of displacement from the intended position is limited to the immediately adjacent points, suggests that the thrower is pretty accurate. The distribution is narrow enough that the chance of going further afield is (effectively) zero. Therefore epsilon should be pretty tiny. It must be large enough that there is a chance of the frisbee being at least 50% over the line (i.e. epsilon > 0), but small enough that the chance of it going 70.7% over the line is vanishingly small (otherwise we would be allowing it to be displaced onto the diagonally adjacent positions). Assuming a Gaussian distribution (probably not true for frisbees but it will do) and assuming 3 standard deviations away is close enough to "vanishingly small", we have 3.sigma = 0.7071... (sqrt(0.5)), sigma = 0.2357 (sqrt(0.5)/3), tipping point for throw being >50% over the line t.sigma = 0.5, t = 0.5 / sigma = 3.sqrt(0.5) = 2.12 => epsilon = 0.017, approximately 1 in 60. Looking at this from a purely combinatorial point of view,t we need 1/epsilon > number-of-moves-in-a-game but 1/epsilon^2 << number-of-moves-in-a-game, which 0.017 seems to satisfy for all common board sizes. Hopefully such a small epsilon also avoids destroying the possibility of local tactical play but also introduces a new element to the game (75% chance of at least once displaced move in 81 move game, over 99% chance of at least one displaced move in a 361 move game). In fact to model the real world, epsilon ought to vary depending on the move. It should increase depending on distance from throwing position, and should not be equal for N,S,E and W displacement. Assuming standing south of the board, we expect epsilon N > S > E = W (range is normally harder to judge than direction and overthrows tend to be worse than under-throws). It seems to me this may bring in interesting elements to move choice - it may be better to play a weaker move which is closer and therefore more likely to be played successfully than a stronger move which is less likely to be played successfully. But perhaps this over complicates things - how does it work out with fixed epsilon around 0.017. Raffles On 11-Nov-15 15:29, Álvaro Begué wrote:
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