Team tourney theory – Minimax strategy calculation

To clear out what is minimax strategy I’ll make an example with a 3x3 player game. We’ll have 3*3*2*2 = 36 pairing variants in this game, but not all of them are acceptable for us or our opponents! Let’s take a look at the evaluations table:

	BA	GK	SB
IG	3	10	11
DE	16	8	2
Tyr	6	4	15

We have IG, DE and Tyr in our team, while opponent has BA, GK and SB. We have to go first, for example. Let’s build a tree of variants for one branch:

Let’s follow the tree from leafs to roots.

 3rd attacker. The choice of third attacker is obvious – it’s always easy to choose 1 player out of 1 player;) In the very left branch such an attacker will bring us 15 points!

 2nd attacker. The coice of the second attacker is a bit more complex – we have to choose from 2 variants. Opponent defends with GK. We want to find the variant that give us more points total. If we choose DE we’ll get 15+8 = 23 points. If we choose tyranids then we’ll get 4+7=11 points. So, it’s easy to choose DE!

 2nd defender. Enemy think that something depends on his choice of second defender, but it’s easy to see that nothing depends on the choice. If opponent chooses GK, then we’ll follow the left branch to get 15+8 points. It’s easy to see that if opponent chooses SB, then we’ll get the same 15+8 points. Let’s assume that our opponent chose GK in this example.

 1st attacker. The choice of the first attacker in this game is the only point, where our opponent can choose something actually. But we now know that if he chooses BA, then we’ll get 15+8+3 points total. If to follow in the same manner the branches for SB and GK – we’ll get the numbers given in the picture. The decision, smart opponent will make, is to choose the minimum sum total variant – he chooses BA!

 1st defender. Now we now that if we choose IG, then opponent will set BA and the game result will be 15+8+3 = 26 points. We can calculate all the branches to see the number of points for the choice of DE and Tyr respectively. Our decision is based on the maximum sum total. In this example we’ll choose IG to get 15+8+3 = 26 points.

So, this example game according to the minimax strategy with ours first turn will bring us 26 points and has 2 pairings variants:

 1. IG-BA-GK-DE-TYR-SB

 2. IG-BA-SB-TYR-DE-GK

You can try to calculate this example for the opponent having the first turn and see that, actually, the outcome will be less than 26 points for us – that is the specific of a even-player game of pairings:)

Later I’ll try to find some applications for the minimax strategy for the games with a champion.

Maybe you can see how to use this strategy for ETC or other team tourneys?