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среда, 11 сентября 2013 г.

Team tourney theory – Pairings with full information


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?

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