新的 VP Scale 計算公式已經貼在 Facebook 的 BTU 社團中. 以下是公式設計人 Henry Bethe 和我的信件往返. 他已經把設計原理解釋得差不多, 只剩下最後提到的一些細節沒講, 包括 * 有身價 game (+10/-6 IMP) 對函數值的影響, 指數函數與其他有何明顯不同? * 為何指數函數的二階微分比其他函數好很多 (much better)? 雖然無法體會這兩件事, 但我不打算再繼續追問. Henry Bethe 的父親是聲名顯赫的 Hans Bethe, 1967 諾貝爾物理獎得主. 對 Henry 提到牛頓冷卻定律簡直是班門弄斧, 但我還是提了, 因為這樣才能完整表達原先想法. 有興趣者可以試著翻譯並解釋 Henry 在信中所解釋的公式設計原理. 我寫的部分只是 為了拋磚引玉, 大家就不用管磚頭了. :) ---------- [From jmchen to Henry Bethe] ---------- I am curious about the rationale behind your formula, so I try to find it by myself. For simplicity, let's consider the case of 16-board only here. The complete formula for an arbitrary number of boards can be derived in a similar way easily. Let's denote y as winner's VP and x as IMP difference. With some assumptions, the function y = y(x) satisfies a differential equation of the form y' = k (h-y) where h and k are constants to be determined. I solved this ODE (separable variables) and obtained a solution which is exactly the same as your formula. Here are the conditions: 1. The graph of the function y = y(x) passes through 3 points -- (x,y) = (0, 10), (20, 15), and (60, 20). 2. Assume a horizontal line L with the equation y = h > 20. The rate of change of y = y(x) at the point P(t, y(t)) is proportional to the distance of P to L. According to the second condition, an ODE is set up: y'(x) / (h - y(x)) = k where k is a constant to be determined. Combining the first condition to solve the ODE, we get h = 2.5 (7+ 5^(0.5)) whose approximation is 23.09, k = -ln(tau)/20, and y = y(x) is identical to your formula. ---------- [From Henry Bethe to jmchen] ---------- We produced the formula quite differently: 1. The initial formula for the USBF scales was produced by a small group: me, Bart Bramley, Chip Martel and Jeff Rubens. 2. The impetus was dissatisfaction with scales that lumped imp results into one VP result. It was our sense that each imp should be worth something, with, perhaps, some limit. That is, a unique imp margin should map to a unique VP award. 3. We felt that the imps that determine whether you win or lose a match should be more important than imps that add to the margin of victory. Thus the conversion of imps to VP should not be linear. (VP scales in North America and England were never linear. The old WBF scale was essentially linear.) 4. We looked at imp margins in a number of settings: USBF Round Robins; World Championship Round Robins; North American Championship Swiss events (to the extent that we could get them); Quarterly and total results in a number of KO team events; finally a couple of lesser team events with seven and nine board matches. 5. We found that the median margin in all of these was approximately five times the square root of the number of boards. The average margin is a little more. We found that over 90% of the results had margins of three times the median or less. In the context of 16 board matches this means the median margin is 20 imps; 60 imps covers over 90% of matches. 6. We concluded that half the "winner's VP" should be awarded to a median margin win. There are 10 winner's VP. (The scale runs from 0 to 20 rather than -10 to +10 because there is a psychological effect from "losing" VP.) So winning a 16 board match by 20 VP should get 5 winner's VP. Winning by 60 should get all 10 winner's VP. 7. We experimented with a number of different formulations that satisfied the criteria that the formula adopted pass through the three defined points [(0,0), (20,5) and (60,10)] and that the first derivative of the value of successive imps be negative throughout the range. Among the types of formula we looked at were polynomial, logarithmic, a formula using the square root of the margin, and exponential. We concluded that the exponential was most satisfactory. 8. Once you do that, tau sort of falls out. This process was repeated by a committee of the WBF, that included Eugenio D'Orsi (Brazil), Max Bevan (UK - chief TD for the WBF), Maurizzio di Sacco (Italy - head of IP at WBF tournaments), me, Bart Bramley and Peter Buchen (Australia, a professor of applied mathematics). The group reached the same conclusion and recommended adoption of the USBF scales by the WBF. So, yes you can, as you should be able to, derive the formula from the scales. But what we did was to create a formula that fit our predilections of what a formula should do and that was consistent with match data. ---------- [From jmchen to Henry Bethe] ---------- As far as I know, the old WBF VP scale has been proportional to the square root of the number of boards already. It is not a surprise that this property is kept in your VP scale. When I first saw your formula, I believed that the point (20, 5) must have some significant meaning in 16-board matches based on an analysis of a large amount of data. Very glad to learn from you that 20 is the median of winners' margin. Also I was pretty sure that your formula was based on (0,0), (20,5), and (60,10). What puzzled me most was: Why an exponential function? It is equally easy to produce a polynomial, logarithmic, square root, or even trigonometric function passing through these three points with negative second derivative on the whole interval (0,60). In my humble opinion, the exponential is indeed the most natural and scientific choice. Just like population growth, radioactive decay, and Newton's law of cooling: When the rate of change (first derivative) is proportional to a linear function of current value, the solution to the constructed differential equation must be an exponential. That is why I tried to derive your formula in a way similar to Newton's law of cooling. http://www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html (Probably I should not mention physics. Your family is too good, especially your father Hans.) It is remarkable that you had tried several types of functions and concluded that the exponential was most satisfactory. All roads lead to Rome. All approaches lead to the exponential. ---------- [From Henry Bethe to jmchen] ---------- One of the things we did was to look at the effect of the choice of formula on the VP impact of bidding a vul game, e.g. the +10 vs -6 effect. The choice of formula has significantly different outcomes. Another issue for us was the second derivative, which was much better for the exponential function. And we decided that we did not want a function that, like a quadratic, would have a maximum value outside the range and then turn down. We were not so much being scientific as being aesthetic and pragmatic. ---------- [From jmchen to Henry Bethe] ---------- In the old WBF scale, after 25:5, there are 25:4 ~ 25:0 to penalize the loser of losing too much. We believed that you had discussed the possibility of 20:-1 and so on before determine your scale. The reason of not taking negative VP was also from psychological effect of "losing" VP? ---------- [From Henry Bethe to jmchen] ---------- No: in the scheme of things the value of additional imps beyond the 20-0 blitz becomes vanishingly small. The next 20 imps (in a 16 board match) would be worth about 1.2VP. It quite simply did not seem worth it. The old WBF scale was roughly linear over the 10 winner's VP available (e.g. from 15-15 to 25-5); about 4 imps needed to gain a VP. The new scale reaches 20-0 at about the same point that the old one reached 25-2. We unanimously felt that giving value to further imps was not necessary. --

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