4 ms·
why do you seek the fixed point and not the k value that maximizes s(k) = (1 - k)(1 + k) / 2 + k / 2 ?
by boromi 10y ago
why do you seek the fixed point and not the k value that maximizes s(k) = (1 - k)(1 + k) / 2 + k / 2 ?
- SonOfLilit 10y agos(k) is the function for "The number I should choose if my opponent chooses k" [1]. We arrived at the formula by asking "what is the best number for us to choose assuming our opponent chooses k?". Now, obviously a smart opponent would not choose a number k where k != s(k), because s(k) would be a better choice. So the only numbers it makes sense for him to choose would be fixed points of s(k), points where k = s(k). And the smartest thing for me to do in that case would be to choose s(k), which just so happens to be equal to k. [1] which may be more or less than s(k), e.g. if my opponent chooses to always redraw, i.e. k=1, it should be the same as if he chooses to never redraw, k=0, and we expect for both cases s(k)=0.5 < 1 (and it indeed is)
- murbard2 10y agoWhy do you bring up expected value in the calculation. You may end up with the right result, but there's no good justification for why expected value should come into play.