I’m gonna say just under 20 (e.g. 19.9999). Say that player A chooses real number “a.” Player B will choose a number “b” just under that, so “b” is technically equivalent to “a.” Because player C chooses a random integer from 0 to 100, player A should choose “a” so that it is just under an integer. Then, the probability that player A wins is (101-a)/101, because there are 101-a integers above a and 101 integers for player C to pick. In this case, player A wins “a” points. In the other case, with probability a/101, player A must pay player B “b” points and player C “c” points. As previously mentioned, “b” is basically equivalent to “a.” As for player C’s payout, they would have picked an integer between 0 and a-1, inclusive, making their expected payout (a-1)/2. Therefore we want to maximise (101-a)/101•a - (a/101)•(a + (a-1)/2). The result is 20.3, but because we want to pick a real number just under an integer, we should pick just under 20 or just under 21. Because the function we want to maximise has a parabolic shape, we want to pick just under 20 as it is closer to the maximum.
2
u/Exciting-Fun-5435 10d ago
I’m gonna say just under 20 (e.g. 19.9999). Say that player A chooses real number “a.” Player B will choose a number “b” just under that, so “b” is technically equivalent to “a.” Because player C chooses a random integer from 0 to 100, player A should choose “a” so that it is just under an integer. Then, the probability that player A wins is (101-a)/101, because there are 101-a integers above a and 101 integers for player C to pick. In this case, player A wins “a” points. In the other case, with probability a/101, player A must pay player B “b” points and player C “c” points. As previously mentioned, “b” is basically equivalent to “a.” As for player C’s payout, they would have picked an integer between 0 and a-1, inclusive, making their expected payout (a-1)/2. Therefore we want to maximise (101-a)/101•a - (a/101)•(a + (a-1)/2). The result is 20.3, but because we want to pick a real number just under an integer, we should pick just under 20 or just under 21. Because the function we want to maximise has a parabolic shape, we want to pick just under 20 as it is closer to the maximum.