| 1 |
We have a team FFA with N teams. We can describe it with 3 N-dimensional vectors:
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1 |
We have a team FFA with N teams. We can describe it with 3 N-dimensional vectors:
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| 2 |
- a vector [b]p[/b] of the teams' win probabilities calculated by my [url=http://zero-k.info/Forum/Thread/20712]generalization of the elo system for team FFA[/url] (or [url=http://zero-k.info/Forum/Thread/21102]even more general systems[/url]); The sum of its components must be 1.
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2 |
- a vector [b]p[/b] of the teams' win probabilities calculated by my [url=http://zero-k.info/Forum/Thread/20712]generalization of the elo system for team FFA[/url] (or [url=http://zero-k.info/Forum/Thread/21102]even more general systems[/url]); The sum of its components must be 1.
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| 3 |
- a vector [b]res[/b] of the teams' ressource factors (has been 100% for any team in earlier systems, but can now be changed arbitrarily as a handycap)
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3 |
- a vector [b]res[/b] of the teams' ressource factors (has been 100% for any team in earlier systems, but can now be changed arbitrarily as a handycap)
|
| 4 |
-
a
vector
[b]R[/b]
of
result
ratings
that
describes
the
game
outcome,
usually
1
for
the
winner
team,
0
for
all
other
teams;
The
sum
of
its
components
must
be
1.
For
exampe
if
there
is
a
draw
between
two
teams,
their
ratings
are
1/2.
If
3
teams
die
in
a
FFA
and
there
is
a
draw
between
the
remainung
teams,
their
ratings
are
1/(
N-3)
and
so
on.
|
4 |
-
a
vector
[b]R[/b]
of
result
ratings
that
describes
the
game
outcome,
usually
1
for
the
winner
team,
0
for
all
other
teams;
The
sum
of
its
components
must
be
1.
For
exampe
if
there
is
a
draw
between
two
teams,
their
ratings
are
1/2.
If
3
teams
die
in
a
FFA
and
there
is
a
draw
between
the
remaining
teams,
their
ratings
are
1/(
N-3)
and
so
on.
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| 5 |
\n
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5 |
\n
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| 6 |
Then the probability vector has to be changed to account for different res: {{{ p'_k = res_k * p_k / (scalar product of res and p). }}} The elo change of team number k must then be c(R_k - p'_k), where c is the elo change constant (usually around 30; becomes higher if players have low weightings, but is the same for all teams). I have proved that this solution makes the sum of win probabilities=1, the sum of elo changes=0 and the expectation value of elo change for any team=0.
|
6 |
Then the probability vector has to be changed to account for different res: {{{ p'_k = res_k * p_k / (scalar product of res and p). }}} The elo change of team number k must then be c(R_k - p'_k), where c is the elo change constant (usually around 30; becomes higher if players have low weightings, but is the same for all teams). I have proved that this solution makes the sum of win probabilities=1, the sum of elo changes=0 and the expectation value of elo change for any team=0.
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| 7 |
\n
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7 |
\n
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| 8 |
Any res would be possible, but in order to give every team a win chance of p'_k=1/N, we have to use {{{ res_k = 1/((N-1)p_k) * sum from j=1 to N with j not= k (res_j p_j). }}} It can still be freely chosen if the better team should get less ressources, the worse more or a bit of both, which is the same as multiplying the vector res with a factor.
|
8 |
Any res would be possible, but in order to give every team a win chance of p'_k=1/N, we have to use {{{ res_k = 1/((N-1)p_k) * sum from j=1 to N with j not= k (res_j p_j). }}} It can still be freely chosen if the better team should get less ressources, the worse more or a bit of both, which is the same as multiplying the vector res with a factor.
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| 9 |
\n
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9 |
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| 10 |
This "res" is only a mathematical value. It is not defined how it is realized ingame. There are many possibilities. The best approach is probably as a ressource income factor, maybe with ressource income factor of team k = sqrt(res_k) because win probability might be proportional to the square of resource income factor because of Lanchester's square law.
|
10 |
This "res" is only a mathematical value. It is not defined how it is realized ingame. There are many possibilities. The best approach is probably as a ressource income factor, maybe with ressource income factor of team k = sqrt(res_k) because win probability might be proportional to the square of resource income factor because of Lanchester's square law.
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| 11 |
\n
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11 |
\n
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| 12 |
This kind of ressource handycap is not meant to be the normal way to play, but maybe a good alternative for a 1v1 with high skill difference. The worse player wouldn't have to fear higher loss probability and the better player wouldn't have to fear unjustified elo loss when playing with handycap. Maybe make handycap an option in lobby and a room "1v1 skill dependent ressources -> equal win chances"?
|
12 |
This kind of ressource handycap is not meant to be the normal way to play, but maybe a good alternative for a 1v1 with high skill difference. The worse player wouldn't have to fear higher loss probability and the better player wouldn't have to fear unjustified elo loss when playing with handycap. Maybe make handycap an option in lobby and a room "1v1 skill dependent ressources -> equal win chances"?
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How to make win chances equal and account for draws in any team FFA