EPTIn Starcraft II, the meta-game (often shortened to simply “meta”) refers to strategic decisions made made on limited information. While the meta has many components, one of the most important decisions is which units and buildings to build first. This is known as the “opening”, one of many Starcraft terms borrowed from chess terminology. Interestingly, there is no universal “best” opening for any of the three races (Protoss, Terran, or Zerg). In other words, if a player, A, knows ahead of time what opening their opponent, B, will choose, then A always has the opportunity to choose an opening which will give A a significant advantage. This is called “countering” B's strategy. In extreme cases where it is unreasonable to hope that B can overcome the disadvantage of being countered, B's eventual defeat is known as a “build order loss”. Conversely, A's victory is a “build order win”. While there is a large number of opening strategies, we can begin understand how they interact by simplification: we divide them into three broad categories.
* Aggressive openings – aggressive openings emphasize fielding military units early, and quickly attacking the opponent. In general, aggressive openings will sacrifice long-term economic development in exchange for short term military capability.
* Standard openings – standard openings are balanced strategies which both field enough troops to keep a player safe from enemy attacks, and a developed economy which will be able to support expensive armies later in the game.
* Greedy openings – greedy openings neglect safety from enemy attacks in order to expand the economy as fast as possible. If the opponent under-reacts, these openings are the ideal set up for a deadly late game army.
In the simplest conceptual model, these there categories of strategy interact just like a game of rock-paper-scissors:
![[image loading]](http://i.imgur.com/lFMHyFV.png)
However, oftentimes the magnitude of a player's build order advantage is not always the same across differing strategic interactions. To put it in terms of our simple example, a player with a standard strategy is only at a slight disadvantage to a greedy one compared to the build order loss of either standard play holding off cheese, or of cheese breaking down a greedy style.
![[image loading]](http://i.imgur.com/dSDKwsC.png)
Keeping this asymmetry in mind, we can build a basic theoretical model to help get a rough understanding of how the Sc2 meta-game works. Here, (x,y) represents (player 1 payoff, player 2 playoff). If you want to make this more concrete, you can pretend each payoff point represents a 25% advantage in winning: a “-2” as a 0% chance of winning, “-1” a 25% chance of winning, etc. However, the magnitude of the correlation is arbitrary. A “payoff point” could represent, say, a 5% advantage without affecting the mathematical results. The numbers here are just an example; the relative strengths of the openings are less important than understanding how they interact.
![[image loading]](http://i.imgur.com/NOAm5nU.png)
There's several things of note here. First, the two numbers in parenthesis always add up to 0. This is no accident. Starcraft II is a zero-sum game: if one player has an advantage, the other is at a disadvantage. If one player wins, the other must lose. Second, the payoffs are mirror images across the diagonal of the above table.(a symmetric game) Unlike the zero-sum condition, this is an artifact of our simple model. In reality, differences in races, skill, knowledge, map balance, spawning positions, etc. will almost always disrupt this symmetry to some extent.
There is an important distinction to make between a build order and an overall strategy. A very simple strategy may well be strict coherence to a single build order, but strategies can be a whole lot more. You can take account of the strengths and weaknesses of the individual opponent/map/race etc. or employ a mix of different builds. In game theory, a simple one-build strategy is a “pure strategy” whereas a “mixed strategy” picks from two or more builds at some given set of probabilities.
Let's take a look at the heart of meta-game strategy, countering your opponent. Pure strategies are simple to counter.(e.g. if you know your opponent always cheeses, your optimal response is to always play defensive) Mixed strategies are slightly more complicated to counter. However, because our model is straightforward, even the general case is not too complex:
For our strategy (x1,x2,x3) where x1,x2,x3 >= 0 and x1+x2+x3=1, and a payoff matrix Pij, and a constant opponent's strategy of (y1,y2,y3) the average payoff is
![[image loading]](http://i.imgur.com/aG6q0lD.png)
For our strategy to be optimal, the payoff must be at a maximum. This comes in several forms: a mix of one, two, or three strategies. Given that the one and two strategy cases are trivial, we can focus on the mix of three case. We can solve the three strategy mix this explicitly by differentiating with respect to x1,x2, and x3. Treating payoff as a multivariable equation, all the critical points must follow:
![[image loading]](http://i.imgur.com/yxZQuzn.png)
So, y2=y3, 2y1=y3 and 2y1=y2. Thus, our extrema is when our opponent plays (20% aggressive, 40% standard, 40% greedy).
What is the nature of this point? Without going deeper into mathematics, the answer is a Nash equilibrium. In a Nash equilibrium, neither player can improve their average payoffs by switching strategies. Specifically, because of the zero-sum condition, our model fits the conditions of Von Neuman's mini-max theorem, and this point is the mini-max point. For a general mini-max point, all non-dominated pure strategies fare equally well against the (mixed) mini-max strategy. In a symmetrical game like our model,(where Pij=Pji) this implies that players can adopt “safe” strategy which averages to even games, even if the opponent knows the exact percentages of that safe strategy.
Note that the equilibrium strategy is not necessarily an even mix of viable builds. Determining the Nash equilibrium is dependent not only on having a range of builds to draw from, but also understanding the strengths and weaknesses in each of them.
So why would players ever deviate from the Nash equilibrium? Does that only happen when they miscalculate the equilibrium solution? Not at all. The equilibrium in our model is not stable. That is, even though neither player gains anything from switching strategies in the long run, neither player loses anything from switching strategies either. An individual equilibrium player vs. unbalanced player game is as dead even as an equilibrium vs. equilibrium one. Therefore, the other viable option is “exploitative” strategies, which can pull a consistent advantage in the long term by taking advantage of imbalance in the opponent's or opponents' meta-game strategy, and do no worse than breaking even if there are no meta-game imbalances. However, unlike a safe strategy, an exploitative strategy can also result in a long term disadvantage if the opponent is able to consistently anticipate how you play.
The safeness of a meta-game strategy is not binary; it is possible to hedge bets between a completely “safe” strategy and purely “exploitative” one. A slightly imbalanced strategy will yield much less advantage(against a non-equilibrium opponent) than a full imbalance, though it comes with much less risk.(this risk/reward dynamic is in the nature of a zero-sum game)
Despite the fact that exploitative player behavior can remain outside the Nash equilibrium indefinitely,(i.e. you can choose to play greedy every game) interestingly, there is a tendency for the aggregate strategy to remain close to the Nash equilibrium as long as the players are part of what you might call an “open system” in analogy to thermodynamics. This is true even in the extreme case where there are no equilibrium strategy players. The mechanism that “enforces” the meta on a broad scale is of course, a player's hidden MMR. If the aggregate meta-game strays from the Nash equilibrium at a certain skill level, weaker players playing a counter-strategy will be brought into the system because of their comparative meta-game advantage.(e.g. if all the masters players are playing greedy 100% of the time, aggressive diamond players will gain an advantage against the masters players, helping them toward promotion, whereas safe/greedy diamond players' chances will be unaffected or even worse) If there is an upper end of the league, counter-strategy players will remain near the top of the system, creating even stronger demand for counter players at the lower end. Note that this is not true for the bottom of the skill range, where there is both no way to either demote bad players out of play or bring in additional worse players to balance the meta.
Now for some prediction: we should expect that the frequencies of a set of particular openers to be predictable from the win percentages of each build order match-up. If we know how good each strategy is, we can predict how often it is played.
As a test case, take the meta-game of ZvZ at the masters level. In order to test the three build model, divide builds into three (nearly arbitrary) categories: pool before 75 seconds (aggressive), pool between 75 and before 130 seconds (standard) and pool 130 seconds and after (greedy). The main concern is that none of the strategies are dominated by the others, because of the trivial results.(each of our pure strategies should be the optimal response to another of our pure strategies) Replays were worldwide ZvZ replays with an average league rating of "masters" during a roughly two week period. After 101 replays, the results were tallied by the winner's strategy and loser's strategy:
![[image loading]](http://i.imgur.com/RrOxhAA.png)
![[image loading]](http://i.imgur.com/r0bVrcW.png)
Big thanks to both the sc2reader project for providing the tools for automated replay analysis, and ggtracker.com's collection of replays for providing the large number of replays needed for a reasonably representative sample.
As you can see, the resulting frequencies of each build are relatively well-determined by our theoretical model. This is not generally true if the values of Pij were arbitrary, because adding in additional “standard” replays at the same win percentage would cause predicted frequencies to be at odds with our sample. So the model both works, and makes non-trivial predictions.
In summary, players can average out the effects of the meta-game in the long term by playing at the Nash equilibrium, a probabilistic mix of build openers. This is the “safe” meta-game strategy. Alternatively, if you believe the opponent is playing outside the Nash equilibrium, you can exploit this by taking the risk of purposeful imbalance in your own strategy. The higher the imbalance, the greater the advantage/disadvantage in payoff.
Overall, this is not intended to be a complete guide for how to open. The meta-game shifts all the time, and many meta-game skills are very difficult/impossible to emulate mathematically, like extrapolation from limited data, and creating better “boxes” to group strategies. This is just a basic framework for understanding what meta-game choices are available to a player, and how these affect their chance of winning. This is also just the tip of the iceberg. There are obviously more than three openers, more meta-game decisions than the opening, asymmetry between the players, additional considerations for formats other than a best of 1, and much, much more. The meta-game is a big place. I'd love to hear your thoughts.
I like the computations, and I agree with them (in the abstract sense). 
