I'm a student at University of Milan of Computer Science.
I found a Nash Equilibrium Strategy (i.e. a strategy belonging to a Nash Equilibrium) for ZvZ openings.
If you don't know what is a NE, check out here:
https://en.wikipedia.org/wiki/Nash_equilibrium
Anyway, to make it simple, an NE strategy in a mirror game like ZvZ is a strategy that cannot lose and at least can win if the opponent make mistakes.
Being a mirror game, would mean that in a NE (i.è. if both opponent use NE strategies), they have both 50% of winning.
An NE strategy in this case is a strategy that has AT LEAST 50% of winning vs every possible opponent strategy.
I make this assumptions:
4P beats 12P 100% of the times.
9P beats 4P 100%.
12P vs 9P: in this case I assumed that the player with 12P take an advantage, I assumed he's probability of winning the rest of the game is now 60%. So from 50-50 the chances are now 60-40.
XPvsXP: of course if both player plays the same strategy they have both 50% of winning.
With a lot of math I found the NE of this game, and it is this mixed strategy (a mixed strategy is a strategy where we choose at random from pure strategies):
------------------------------------------------------------------------------------------------------------------------------------------
NE STRATEGY FOR ZVZ OPENING
1/11: 4P
5/11: 9P
5/11: 12P
------------------------------------------------------------------------------------------------------------------------------------------
So it means we go for 4Pool 1/11 of the times, 9P 5/11 of the times and 12P the other 5/11, choosing at random each game.
We now verify if it is really an equilibrium. We try every pure strategy vs our strategy and see what is his income.
4PvsNE: 1/11*0.5 + 5/11*0 + 5/11*1 = 0.5
So 4P has no advantage vs our strategy.
9PvsNE: 1/11*1 + 5/11*0.5 + 5/11*0.4 = 0.5
Same.
12PvsNE: 1/11*0 + 5/11*0.6 + 5/11*0.5 = 0.5
Same.
We saw that the opponent is indiffirent to play any of his strategy vs us.
Now let's take a look how other common strats performs vs our NE:
12H: I assumed that 12H has 0% winning vs 4P; 20% vs 9P, and 60% vs 12P.
12HvsNE: 1/11*0 + 5/11*0.2 + 5/11*0.6 = 36%
So if any opponent include the 12H strategy in his strategy and play with us, he will give us a LARGE edge, so overall we will have an advantage on him. Cool!
Let's see 9H: Now I assumed a 0% winning vs 4P, 70% vs 9P, and 30% vs 12P.
9HvsNE: 1/11*0 + 5/11*0.7 + 5/11*0.3 = 45%
We also take a SMALL edge with our NE strategy vs the 9H.
We can also question ourselves if that can be other equilibrium strategies in this subgame. We can easily proove there is not. Infact:
If an opponent uses any pure strategy (for example, 100% 9P), it is of course exploitable (i.e. exists a strategy that can win more than 50% vs him), so it cannot be an equilibrium strategy.
If an opponent uses a mixed strategy that has 9H or 12H with a probability >0%, he give up edge vs our strategy, so it has <50% of winning vs us, so it cannot be an equilibrium strategy.
If an opponent try to use a mixed strategy only between 9P and 12P (with both >0%), he is exploitable to 12P.
If an opponent try to use a mixed strategy only between 4P and 9P (with both >0%), he is exploitable to 9P.
If an opponent try to use a mixed strategy only between 4P and 12P (with both >0%), he is exploitable to 4P.
Finally, note that this is the only point of indifference for mixed strategies with 4P 9P and 12P, infact if you try any different frequencies you'll see they are exploitable. So, if you have 4P<1/11 you are exploitable with 12P. If you have 9P<5/11 you are exploitable with 4P. And if you have 12P<5/11, you are exploitable with 9P.
And now we saw that all the possible strategies cannot be equilibrium strategies, so the only equilibrium strategy is our super cool NE strategy:
1/11: 4P
5/11: 9P
5/11: 12P
EDIT:
------------------------------------------------------------------------------------------------------------------------------------------
CHANING A PARAMETER: (as suggested by Ty2) 4Pvs12P 80% instead of 100%
------------------------------------------------------------------------------------------------------------------------------------------
I also tried to calcolate the NE with:
4P beats 12P 80% (changing from 100% to 80%).
The Nash Equilibrium for this subgame becomes (instead of the 1/11 5/11 5/11):
1/9: 4P
3/9: 9P
5/9: 12P
I also checked 9H and 12H and they are still losing strategies vs this NE.
EDIT 2:
------------------------------------------------------------------------------------------------------------------------------------------
*******BEST STRATEGY*******
CHANING PARAMETERS WITH REAL DATA: (as suggested by Crunchums)
http://www.teamliquid.net/forum/bw-strategy/348493-zvz-build-order-statistics
AND AVOID PLAYING 4P
------------------------------------------------------------------------------------------------------------------------------------------
New parameters:
9P vs 12H: 74%
12H vs 12P: 62%
12P vs 9P: 54%
I found this AMAZING strategy, even WITHOUT using 4P (using 12H instead).
This strategy is also very very close to the equilibrium:
9P: 4/10
12P: 5/10
12H: 1/10
This strategy ensure a 49.6% vs 9P and 12P, a 50% vs 4P, and wins vs 12H and 9H. We had to give up that 0.4% edge to avoid playing 4P, but I think it's worth it.
Best strategy here!
------------------------------------------------------------------------------------------------------------------------------------------
Final remarks on strategies:
1) 9P and 12P: as we can see, it's always a good idea to have our 12P frequency >= our 9P frequency.
2) 9H: Consequently to the 1*, 9H is an overall bad strategy (only good for exploiting a leak), because it would be good only vs players who 9P more than anything else.
3) 12H: this is a nice build if we are not using 4P in our strategy, and only with very low frequency (like 4P).
Of course, when we are in game, while taking informations, we could dinamically change the strategy we are using. So if, for example, we are going to 12P and we scout the opponent who 12P too, we change to 12H to take an advantage.
Our frequencies must be used only at the start of the game, without any information, and can be changed during the game.
I hope you guys enjoyed my mathematical analysis.
Best regards!
Federico Distefano