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[img]http://zero-k.info/img/Awards/trophy_rezz.png[/img]
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[img]http://zero-k.info/img/Awards/trophy_rezz.png[/img]
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TL;DR: teamstrength as a fundamental.
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TL;DR: teamstrength as a fundamental.
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I spent like 5 minutes thinking about this so this is probably full of logic holes but:
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I spent like 5 minutes thinking about this so this is probably full of logic holes but:
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@Brackman 's [url=http://zero-k.info/Forum/Thread/20712]system of calculating FFA win chance of any given team is[/url]
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@Brackman 's [url=http://zero-k.info/Forum/Thread/20712]system of calculating FFA win chance of any given team is[/url]
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{{{(sum of probabilities of win against each enemy team) * 2 /(N * (N-1))}}}
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{{{(sum of probabilities of win against each enemy team) * 2 /(N * (N-1))}}}
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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
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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
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{{{
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{{{
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{-> 1, -> 1}
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{-> 1, -> 1}
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{0.5, -> 0}
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{0.5, -> 0}
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{0.5, -> 0}
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{0.5, -> 0}
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}}}
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}}}
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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!
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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!
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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).
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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).
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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).
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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).
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Here's an example of Elo gains teams of winstrength 1, 2 and 4 (sans K-factor):
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Here's an example of Elo gains teams of winstrength 1, 2 and 4 (sans K-factor):
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||team||win||lose||
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||team||win||lose||
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||1||+6||-1||
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||1||+6||-1||
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||2||+5||-2||
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||2||+5||-2||
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||4||+3||-4||
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||4||+3||-4||
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Win chances are 1/7, 2/7 and 4/7 -> sum = 1.
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Win chances are 1/7, 2/7 and 4/7 -> sum = 1.
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Expected change for team 1 is (1/7 * +6) - (6/7 * -1) == 0.
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Expected change for team 1 is (1/7 * +6) - (6/7 * -1) == 0.
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Expected change for team 2 is (2/7 * +5) - (5/7 * -2) == 0.
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Expected change for team 2 is (2/7 * +5) - (5/7 * -2) == 0.
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Expected change for team 4 is (4/7 * +3) - (3/7 * -4) == 0.
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Expected change for team 4 is (4/7 * +3) - (3/7 * -4) == 0.
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If team 1 wins, it gets +6 while teams 2/4 get -2/-4. Sum = 0.
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If team 1 wins, it gets +6 while teams 2/4 get -2/-4. Sum = 0.
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If team 2 wins, it gets +5 while teams 1/4 get -1/-4. Sum = 0.
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If team 2 wins, it gets +5 while teams 1/4 get -1/-4. Sum = 0.
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If
team
3
wins,
it
gets
+3
while
teams
2/4
get
-1/-2.
Sum
=
0.
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If
team
4
wins,
it
gets
+3
while
teams
1/2
get
-1/-2.
Sum
=
0.
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It's probably not hard to generalize but hopefully this illustrates the idea.
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It's probably not hard to generalize but hopefully this illustrates the idea.
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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).
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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).
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