Game Theory in Starcraft II
http://www.teamliquid.net/forum/starcraft-2/241679-game-theory-and-starcraft-2
First released on July 27, 2010, Starcraft II: Wings of Liberty was the second game released by Blizzard modeled in the Starcraft universe. As a competitive real-time strategy game, Starcraft II had players fight each other on a map with the goal of producing units to destroy the opponents base. Once players lose their last structure, they lose the game. To build advantages that enable an opponent to defeat the other, most players will adopt play styles to open the game that generally fall into 3 categories: Aggressive (All-in), Standard, and Greedy (Scaling). Aggressive openers will generally grant an advantage over greedy openers but grant a disadvantage over a standard opener. Greedy openers will generally grant an advantage over a standard opener. (Aggressive << Standard << Greedy << Aggressive). The article makes a general analysis of the game openers, places them into categories, and concludes that all 3 openers are valid and important parts of developing any strategy for the game.
However, when we look at the strategies adopted by players on the competitive field, there is a definite preference toward different openers that seem to correspond with the general rank of the player. Although we don’t have data on the specific play style used by players in a game, we can make generalizations on the play style used and the length of the game. In general, games won with an aggressive opener will be short (<10 minutes), games won with a standard opener will be of medium length (10-20 minutes), and games won with a greedy opener will be of long length(20+ minutes). If we assume game length is strongly dependent on the opener used, we can see from the average game length of various ranking ladders that players ranked low (bottom 20%) tend to have on average medium-length games. Players in the middle 40% will on average have longer games, with the game length significantly more precise (15-35 minutes). Players in the top 40% will have games shorter than the middle 40%, with players in the top 5% averaging 8-11 minutes a game. From game lengths, and knowing how long it takes players to win with different openers, we can generally conclude that players ranked low on average will play a standard play style, players in the middle will play a standard or greedy style, and players in the upper tiers are more willing to utilize aggressive play styles. However with all 3 play styles valid and leading to generally predictable advantages over other styles, why do people in different ranks have different strategies?
In setting up this problem to look for a nash equilibrium, we know we’re dealing with a zero-sum game. Starcraft 2 has one clear winner and one clear loser. Any advantage gained by one opponent is a disadvantage to the other. An important thing to note is that a standard opener will net a strong advantage over an aggressive one, but only a slight disadvantage over a greedy one. An aggressive play-style will net a strong advantage over a greedy one. (Aggressive << Standard < Greedy << Aggressive). In addition, player skill strongly affects how effectively the player can play each opener, which affects how much a player actually wins in a winning opening move. With this in mind, we can construct games that could explain the differences in the trends for strategies at different ranks.
In general, standard openers are easier for beginners to learn and use than greedy or aggressive openers. The game may reflect something as like this:
| Aggressive | Standard | Greedy | |
| Aggressive | 0, 0 | -4,4 | 6,-6 |
| Standard | 4,-4 | 0,0 | -0.5,0.5 |
| Greedy | -6,6 | 0.5,-0.5 | 0,0 |
While games played by advanced players may reflect something like this:
| Aggressive | Standard | Greedy | |
| Aggressive | 0, 0 | -2,2 | 2,-2 |
| Standard | 2,-2 | 0,0 | -1,1 |
| Greedy | -2,2 | 1,-1 | 0,0 |
And games played by experienced players may reflect something like this:
| Aggressive | Standard | Greedy | |
| Aggressive | 0, 0 | -1,1 | 1,-1 |
| Standard | 1,-1 | 0,0 | -1,1 |
| Greedy | -1,1 | 1,-1 | 0,0 |
How then do players in the top 5% average games of 8-11 minutes? Averaging games of that length may suggest that players would have to play aggressive all the time, but players in the top 5% can accomplish a lot with 8-11 minutes, pushing an advantage relentlessly the moment it is presented, which is often distinguishable by the first 5 minutes. Players in the top 5% are really really good. Really really good.