When I just
started to play team tourney I heard that “champions in a team tourneys make it
unable to calculate the whole pairings process throughout”. But how? I decided
to investigate that with a minimax strategy taken from the game theory of the
mathematical statistics. Let’s look at the pairings evaluations table, like I
posted in the previous aticle (values range from 0 to 20 according to the
expectations of points taken from the game):
Enemy team
|
|||||||
GK
|
CSM
|
Eld
|
Tyr
|
BA
|
DE
|
||
Our team
|
Eld
|
5
|
13
|
10
|
15
|
11
|
6
|
IG
|
13
|
13
|
16
|
2
|
4
|
16
|
|
Dae
|
15
|
5
|
3
|
2
|
14
|
3
|
|
GK
|
9
|
14
|
13
|
14
|
8
|
13
|
|
SM
|
6
|
5
|
6
|
9
|
16
|
4
|
|
SW
|
6
|
14
|
17
|
16
|
10
|
14
|
I did this
example in 5th edition, so don’t try to think on the values itself:)
If you want
you can count that the average of this table is 10. So. It must be the game
with equal composition strength.
Now let us
consider the game of pairings without champion game! If both team’s evaluation
tables are the same, then one can calculate all 518400 combinations of pairins
for this table. On each step each captain chooses the best variant for his
team. I did this calculations for the table and the results are:
1. If we put the first defender – we’ll
get 53 points from this game minimum.
2. If opponent puts the first defender –
we’ll get at least 66 points.
So,
actually, if there is no champion’s game – we can roll the dice and see which
team is the winner:)
I’ll show
how minimax optimal strategy for the game of pairings differ from the simple
strategy of “best defender – best attacker”
Let’s
imagine that we are playing the game of pairings with the evaluations table
given above. For example, we’ve lost the roll-off and have to put the first
defender. Minimax evaluation is 53 points for our team. But we’ll follow the
best defender – best attacker procedure, as well as opponent do.
Our best
first defender is GK. Opponent can’t get more then 12 points out of him. Opponent
attacks with BA = 12 points (best attacker).
Opponent
defends with CSM. We attack CSM with SW to take 14 points and defend with
Eldars. In the end we’ll get the following game table:
1
|
GK
|
8
|
BA
|
2
|
4
|
SW
|
14
|
CSM
|
3
|
5
|
Eld
|
5
|
GK
|
6
|
8
|
SM
|
9
|
Tyr
|
7
|
9
|
IG
|
16
|
DE
|
10
|
12
|
Dae
|
3
|
Eld
|
11
|
=55
|
Left and
right columns are positions at which players get into the game. Left side is
our team, while right side is opponents team. In the center – the resulting
score we get from each pair and the total sum.
The sum
total is 55 points. It looks like we get more then it was expected by minimax
strategy! It means that opponent did a mistake in his strategy. Let’s see how
we can perform, using the minimax strategy. Our opponent will continue to play “best
defender – best attacker game”
I will not
deep into calculations, but show you the result:
1. According to the calculations, our
first defender is GK (same as above).
2. Opponent attacks with BA (as above)
3. Opponent defends with CSM (as above)
– it’s the mistake for our opponent. It was better to put DE or Eldar in
defend.
4. We attack with Eldars (pure math, no
magic – later we’ll get more profit with SW)
5. We defend with SM – optimal minimax
choice.
6. Opponent attack with DE for our SM.
7. Now opponent defends with GK
8. And we attack with Daemons. The last pairs are not of any interest – the result
is obvious:
The game
table will look like this:
GK
|
8
|
BA
|
2
|
|
4
|
Eld
|
13
|
CSM
|
3
|
5
|
SM
|
4
|
DE
|
6
|
8
|
Dae
|
15
|
GK
|
7
|
9
|
IG
|
2
|
Tyr
|
10
|
12
|
SW
|
17
|
Eld
|
11
|
=59
|
So, the
total sum is 59 points – it’s much better, then 53 or even 55 points, don’t it?
Opponent could win with 67 points if he play using the optimal strategy!
The same
will be with the won roll-off. But the game of pairings were to decide between
draw and victory of our team in that case.
As you can
see, team tourney without champion’s game is a game with full information and
all possible outcomes can be precalculated as strictly as strict is you
pairings evaluation table. To make the game of pairings unpredictable, the
champion’s game is used. The selection of champions before the roll-off makes
it subjective and too complex to precalculate. So, you can use some strategies
to minimize your risk or maximize outcome, but cannot build the optimal
strategy. Later I’ll try to find some applications for minimax strategy
evaluations in order to maximize the outcome or stabilize the result (minimize
risks).
For
discussion:
Do you use
pairings evaluations tables? Do you use math statistics to calculate the game
outcome?
No comments:
Post a Comment