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math bork 2nd try

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Date Editor Before After
5/3/2016 1:30:25 AMPLrankAdminSprung before revert after revert
5/3/2016 1:15:02 AMPLrankAdminSprung before revert after revert
5/3/2016 1:13:00 AMPLrankAdminSprung before revert after revert
Before After
1 [img]http://zero-k.info/img/Awards/trophy_rezz.png[/img] 1 [img]http://zero-k.info/img/Awards/trophy_rezz.png[/img]
2 TL;DR: teamstrength as a fundamental. 2 TL;DR: teamstrength as a fundamental.
3 \n 3 \n
4 I spent like 5 minutes thinking about this so this is probably full of logic holes but: 4 I spent like 5 minutes thinking about this so this is probably full of logic holes but:
5 \n 5 \n
6 @Brackman 's [url=http://zero-k.info/Forum/Thread/20712]system of calculating FFA win chance of any given team is[/url] 6 @Brackman 's [url=http://zero-k.info/Forum/Thread/20712]system of calculating FFA win chance of any given team is[/url]
7 {{{(sum of probabilities of win against each enemy team) * 2 /(N * (N-1))}}} 7 {{{(sum of probabilities of win against each enemy team) * 2 /(N * (N-1))}}}
8 \n 8 \n
9 I claim this is wrong if we use the pairwise values Elo gives us. FFAs do not work that way - everyone interacts with everyone else simultaneously, it's not pairwise. You can construct an counterexample, too: check out the chance of a guy with X elo as X approaches infinity, against two people both with some constant C elo. The pairwise winchance vectors are 9 I claim this is wrong if we use the pairwise values Elo gives us. FFAs do not work that way - everyone interacts with everyone else simultaneously, it's not pairwise. You can construct an counterexample, too: check out the chance of a guy with X elo as X approaches infinity, against two people both with some constant C elo. The pairwise winchance vectors are
10 {{{ 10 {{{
11 {-> 1, -> 1} 11 {-> 1, -> 1}
12 {0.5, -> 0} 12 {0.5, -> 0}
13 {0.5, -> 0} 13 {0.5, -> 0}
14 }}} 14 }}}
15 \n 15 \n
16 so (sum * 2 / (N * (N-1)) approaches {1/6, 1/6, 2/3} respectively. This literally gives you a 1/3 chance to lose against two Null AIs! 16 so (sum * 2 / (N * (N-1)) approaches {1/6, 1/6, 2/3} respectively. This literally gives you a 1/3 chance to lose against two Null AIs!
17 \n 17 \n
18 Teamstrength - used as a fundamental, not just as a way of averaging Elo - doesn't have this problem. If teams' strengths are 1, 2 and 4 then their chances to win are in proportion 1:2:4 (ie. 1/7, 2/7 and 4/7). 18 Teamstrength - used as a fundamental, not just as a way of averaging Elo - doesn't have this problem. If teams' strengths are 1, 2 and 4 then their chances to win are in proportion 1:2:4 (ie. 1/7, 2/7 and 4/7).
19 \n 19 \n
20 Now, the basic Elo requirements. The Elo change is (1-winchance)*K for a win and winchance*K for a lose (K is the Elo K-factor, ie. some scaling constant). 20 Now, the basic Elo requirements. The Elo change is (1-winchance)*K for a win and winchance*K for a lose (K is the Elo K-factor, ie. some scaling constant).
21 Here's an example of Elo gains teams of winstrength 1, 2 and 4 (sans K-factor): 21 Here's an example of Elo gains teams of winstrength 1, 2 and 4 (sans K-factor):
22 \n 22 \n
23 ||team||win||lose|| 23 ||team||win||lose||
24 ||1||+6||-1|| 24 ||1||+6||-1||
25 ||2||+5||-2|| 25 ||2||+5||-2||
26 ||4||+3||-4|| 26 ||4||+3||-4||
27 \n 27 \n
28 Win chances are 1/7, 2/7 and 4/7 -> sum = 1. 28 Win chances are 1/7, 2/7 and 4/7 -> sum = 1.
29 Expected change for team 1 is (1/7 * +6) - (6/7 * -1) == 0. 29 Expected change for team 1 is (1/7 * +6) - (6/7 * -1) == 0.
30 Expected change for team 2 is (2/7 * +5) - (5/7 * -2) == 0. 30 Expected change for team 2 is (2/7 * +5) - (5/7 * -2) == 0.
31 Expected change for team 4 is (4/7 * +3) - (3/7 * -4) == 0. 31 Expected change for team 4 is (4/7 * +3) - (3/7 * -4) == 0.
32 If team 1 wins, it gets +6 while teams 2/4 get -2/-4. Sum = 0. 32 If team 1 wins, it gets +6 while teams 2/4 get -2/-4. Sum = 0.
33 If team 2 wins, it gets +5 while teams 1/4 get -1/-4. Sum = 0. 33 If team 2 wins, it gets +5 while teams 1/4 get -1/-4. Sum = 0.
34 If team 3 wins, it gets +3 while teams 2/4 get -1/-2. Sum = 0. 34 If team 4 wins, it gets +3 while teams 1/2 get -1/-2. Sum = 0.
35 \n 35 \n
36 It's probably not hard to generalize but hopefully this illustrates the idea. 36 It's probably not hard to generalize but hopefully this illustrates the idea.
37 \n 37 \n
38 Effectively this reduces Elo to a wrapper around teamstrength meant to calculate change after game, and to show people some number that doesn't make them feel like crap (if you're a newb, Elo 1200 vs 2000 sounds okayish, but teamstrength 1 vs 100 doesn't -- of course in a teamstrength-based system the Elo values would become less extreme but e^x always looks more brutal than just x). 38 Effectively this reduces Elo to a wrapper around teamstrength meant to calculate change after game, and to show people some number that doesn't make them feel like crap (if you're a newb, Elo 1200 vs 2000 sounds okayish, but teamstrength 1 vs 100 doesn't -- of course in a teamstrength-based system the Elo values would become less extreme but e^x always looks more brutal than just x).