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APPENDIX
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| Equations (Eq): | |||||
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Eq
1:
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| We formally define the number of wins required after round r for team i as Par winsi (r); and the total number of wins for team i at the completion of round r as TWi(r). Using the 8th ranked team at any round r as the ideal Par proportion in determining the wins required to make the finals. | |||||
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Eq
2:
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Pri(F8|r)=1{Par Winsi(r)=}+1{(Par Winsi(r)>0)∩(Par Winsi(r)<22-r)}[1-B(Par Winsi(r)-1;22-r,Pi)] | ||||
| where is 1{a} the indicator function taking value 1 if condition a is true and 0 if false. | |||||
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Eq
3:
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Ii(r)=Pri(Make F8|Win Match r + 1)-Pri(Make F|Lose Match r + 1) | ||||
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Eq
4:
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Pri(Make F8|Win Match r + 1) = 1{Par Winsi(r)=}+1{(Par Winsi(r)>0)∩(Par Winsi(r)<22-r)}[1-B(Par Winsi(r)-2;22-(r+1),p)] | ||||
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Eq
5:
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Pri(Make F8|Lost Match r + 1) = 1{Par Winsi(r)=}+1{(Par Winsi(r)>0)∩(Par Winsi(r)<22-r)}[1-B(Par Winsi(r)-1;22-(r+1),p)] | ||||
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Eq
6:
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Ui(r)=1-Ii(r) | ||||
| =1+Pri(Make F8|Win Match r + 1)-Pri(Make F|Lose Match r + 1) | |||||
| =1+[1-B(x;n,p)]-[1-B(x-1;n,p)] | |||||
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| =1+[1-(b(0;n,p)+b(1;n,p)+...+b(x;n,p))]-[1-(b(0;n,p)+b(1;n,p)+...+b(x-1;n,p))] | |||||
| =1-b(x;n,p) | |||||
| So, | |||||
| Ui(r)=1-Par Winsi;22-(r+1),p) | |||||
| Noting b(x;n,p) (the discrete binomial distribution function with x = number of successes, n = number of trials and p = probability of success) in the final result, Unimportance is simple to evaluate, relying on a discrete rather than continuous result, and given the values of Par Wins, can be easily computed using a scientific calculator. | |||||
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Eq
7:
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DPRi(r)=1{i wins matcher}·1{r>6}·Ui(r)·(1-Pri(F8|r)) | ||||
| where is 1{a} the indicator function taking value 1 if condition a is true and 0 if false. The Draft Score, or DScore, for team i at round r is simply the sum of the DPR: | |||||
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Eq
8:
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