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пятница, 13 сентября 2013 г.

Team tourney theory - Advantage realization


In the previous article I discovered the situation, where you want to risk against more powerful, usually, opponent. But there can be different situation – when you have advantage in composition and don’t want to be outplayed in the game of pairing. So, this article is about “how to minimize risks versus less powerful teams”.

I showed that opponent can guess our champion to maximize points. So, it’s the worst case for our team, actually and let’s count that opponent successfully did it – he knows our champion! How to hold the ground if such a shit has happened?

How can we choose our champion?


If to say it simple, than just “choose defender as a champion”. If to be more correct, you should calculate minimax strategy for every champion variant for loosed rolloff. Let’s look at the pairings evaluations table with the advantage in composition:




Oponent's team
GK
CSM
Eld
Tyr
BA
DE
Our team
Eld
5
13
10
15
14
6
IG
15
16
18
2
7
16
Dae
15
5
3
2
14
3
GK
9
14
13
14
8
13
SM
6
5
8
9
16
4
SW
6
14
17
16
10
14


According to this table, we have quite strong team, but nevertheless we can lose the game of pairings. Let’s try to use our advantage! I’ve calculated minimax strategy for every player in our team set as a champion. Every time I allowed opponent to guees our champion correctly and play the best way he can! The resulting score for the team for different champion variants:

Eldar – 54
IG – 52
Daemons – 53
GK – 55
SM – 52
SW – 57

The main idea here: if we set SW as our champion – we CAN NOT GET LESS THEN 57 POINTS! So, WE CAN NOT LOSE AT ALL!!!

In practice the outcome can be much better because we can win the rolloff and opponent can make a mistake with high probability -> we’ll get more points!

Let’s go through the example:

We set SW as our champion and opponent know it (guessed it)! Then we lose the rolloff and have to set the defender first. The problem here is that WE DO NOT KNOW OPPONENTS CHAMPION. It means, that we have 5 players, but opponent has 6, so the opponent has more flexibility in taking decisions.

0 – If opponent can calculate minimax strategy correctly – he set BA as a champion!
1 – With the not square table the best first defender for our team is Daemons! Surprise!)
2 – Opponent, for sure, attacks with Tyranids.
3 – The best defender of the opponent, GK, defends.
4 – We attack GK with SM. Surprise again? Nope, pure math.
5 – We defend with GK, predictable.
6 – Opponent has not good pairing for our GK and has to attack with DE.
7 – The last pairings are obvious. And the resulting game table will look like this:

champ
SW
10
BA
champ
1
Dae
2
Tyr
2
4
SM
6
GK
3
5
GK
13
DE
6
8
IG
16
CSM
7
9
Eld
10
Eld
10
57


So, we played without mistakes, as well as our opponent and we have 57 points (a draw) as a result. It’s good for us, cause we lost the rolloff and opponent guessed our champion!

But one can say:
"Hey, stop! Why to put BA as a champion?! Let’s attack with GK and get more points on those SW!!!

Okay, let’s look at this example too:

0 – Champions are defined (SW against opponent’s GK)
1 – We do not know opponent’s champion and our strategy is the same as before! We lost the rolloff and defend with Daemons!
 2 – Opponent attacks with tyranids as before – it’s clear.
3 – Opponent defends with BA! It looks logical but it’s a mistake!
4 – We attack BA with SM (great match for our SM actually).
5 – We defend with GK, it’s clear too.
6 – Opponent has no good pair for GK again and attacks with DE again.
7 – The last pairs are predefined and bring us even more points!

The resulting game table:

champ
SW
6
GK
champ
1
Dae
2
Tyr
2
4
SM
16
BA
3
5
GK
13
DE
6
8
IG
16
CSM
7
9
Eld
10
Eld
10
63


So, opponent chose the best counter-champion but as a result got even less points. Actually, you can calculate it throughout and find that if we win the roll-off then we have almost no possibilities for a draw – only victory. With lost rolloff we get a draw with GK or BA as an opponent’s champion. Any other army as an opponent’s champion give us a victory!

The main goal of this strategy is not to lose the game if we have advantage in composition, even with lost rolloff and guessed champion. The probability of this is about 1/16 actually, so in most cases we’ll win anyway:)


Afterword.
I have finished my game of pairings theory translations, but not finished to discover the methods and strategies for the game of pairings.
I wonder if this can be interesting or useful for anybody, casue, actually a have almost no feedback. Maybe, it's because captains prefer to keep their strategies in secret, or because we are so lazy to use math statistics for the toy soldiers game.

4 комментария:

  1. I'm read your whole blog on those pairings. The outcome is a big surprise to me. Defending with Daemon seems odd for me.
    I will analyse your minimax theory because I don't understand it well enough for now.
    If the theory is good it must definitly be coded on a computer to use it.

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    Ответы
    1. I've used it already in a 5-player game. For 8-player game some code optimization must be done.

      But it was quite effective to set Tyranids as first defender in the last game of the Russian federation team cup - 2013:)

      I do not completely understand now, how to minimize the dispersion between pairings evaluation table and the real results (usually players can better do pairing estimation after a game is played, but it’s too late!!)

      And it’s not obvious how to use the theory. Actually, I look forward to use it as a HELPER to do some precalculations. I’m not sold on the idea to do pairings absolutely automatically :)

      So, now I’m very interested in the methods to get valid pairings evaluations table.
      1 – Who must fill the tables (players, captain, together)?
      2 – What range of scores should table include (from -1 to 1, from 0 to 20, percents)?
      3 – Is it necessary to do separate tables for each mission?
      4 – How to fill into the table the dispersion of the evaluations (as a range of numbers, 13-15; as a mean and dispersion, 14±1)?

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  2. Hello, thanks for translating and writing all this pairing game theory. As you've discovered, if it was a game with full information and there was no random factors (with the champion and on the tables) the pairing game would be pretty boring to play. However we don't have all the information.

    As "foreign consultant" to team Ukraine in 2010 and former captain in the French ETC team qualifiers my experience is that there are no perfect pairing tables.
    It's always a tradeoff between 2 factors:
    - time/work necessary to fill the tables
    - expertise of the fillers

    In a dream world, each list of the team would be pitted several(hundreds?) times against all opponents lists played by the opponent players for each scenario/table layout. This would allow a result distribution to be created and remove any guess work.

    Obviously, this is not feasible so we need to approximate that.

    We can suppose that each player should know his own army the best and has the best information to fill pairings against lists that they already faced (or similar ones).

    Thus I think that tables should be filled by players, with possibility to ask for help for certain lists that they don't know so well (either through games or through talks with someone familiar with the list).

    After that the captain (or possibly together) should make a pass to correct player bias (some players are conservative, other optimists, etc).

    As to the form of the table, it's also a tradeoff, if it's too precise (one per scenario, on a 0/20 scale, with alternatives (if choosing table, if winning roll-off, etc ),...) it will take a lot of time for the players and they will be less focused on each table cell. If it's too lax, you will loose potential information from your players. So it all depends on players/captain involvement.

    And don't forget that once you get the tables, the captain needs to choose the champion for each table, this might take a lot of time to do well.

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  3. 1- Players, always, they have to sign their objective, as in real life. Captain can correct some too optimistic evaluations or advice how to get more points to the player.
    2 - 0-20 : That's not acurate of course, but better have the best precision possible
    3 - In theory yes. depends on the time you and your team have.
    4 - I prefer 14 ±1 personnally. 14 is the most relevant figure, it must be clearly written.

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