Equilibrium in Video Games

Most competitive games reach an equilibrium stage that’s often called the metagame. An important distinct to note is that while gamers can reach an equilibrium, it is not necessary the solution to the game, rather, the metagame is the solution to what everyone currently understands and know in the game’s player base.

There are also games that do not have an equilibrium. For instance, there is no equilibrium strategy to play in Rock, Paper, Scissors. By comparison, marines and medics is often considered an equilibrium strategy for playing Terran in Starcraft. The player can choose to build purely medics, purely marines, or a mixture of both. Their best decision tends to be to build a mixture. In game theory, this is called the dominant strategy because compared to the alternative strategies, the dominant strategy is always better.

Practically speaking, it is very difficult to design a game that has no equilibria. It is also very difficult to build complex games with an equilibrium in mind. Most games have evolved outside of their creator’s predictions; it’s why games tend to be imbalanced until substantial play testing has occurred. A game designer can try to influence the equilibrium, but it’s unlikely that he’ll predict what ultimately will be done in a competitive environment. The collective gaming community has proven to be far superior in finding the best strategies. Afterall, I doubt that the original designers of Starcraft even considered building supply depots in front of their bunkers.

From a game theory perspective, games that have no equilibrium have no solutions. To an extent, games, in particular, RTS, that do not have very strong equilibrium strategies tend to be casual. The lack of equilibrium strategies make it difficult to win, at least in terms of strategic thinking because there are no “solutions” that can be played by higher skilled players. In other words, there is no hierarchy; no ladder or ranking system because you cannot be better than another player. When strategic play becomes less important, the next most important factor is the player’s performance which is reflexes, mental state, micro, and other factors that can be improved by practice. It may not matter what you play in Rock, Paper, Scissors, but you can at least intimidate him with mind games which I suppose could give you the winning edge.

There is another form of competitive gaming that should be noted. There are games that are won by strategies which we can discuss using the tools of game theory, however, there are also games that rely on performance. Games like Soccer and Counter-Strike have some degrees of metagame, but compared to a RTS, they tend to be more dependent on player performance (ie. how fast you can headshot someone with the AK as opposed to choosing between building more troops versus expanding). Competitive gaming exists outside of game theory solutions, but those tend to be games of performance rather than strategy.

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Why massed marines beat massed zerglings

There is a very common phenomenon in RTS games that’s often forgotten: massed ranged units tend to beat massed melee units. Even more so, good massed ranged units will beat inferior massed ranged units. The benefit from massing ranged units is their ability to lower the firepower of opposing units before they can inflict harm. For instance, a pair of zerglings will beat a marine, but 24 zerglings will lose to 12 marines.

During my time as a volunteer balance developer for a mod, I had to understand and apply core game mechanics. Below is an instrument with a very basic model that I call the Effective Range Firepower (ERF). It measures how well a cluster of units perform. We can interpret X as simply the number of units. Alpha would be a measure of well the unit performs when massed. In Starcraft, this is the unit’s attack range. In a more complex game such as Company of Heroes, this is the unit’s attack range, accuracy, cooldown/reload modifiers, etc.

There are two assumptions that the model makes. 1) There is no limit to the number of units aggregated. This is generally true until your units start physically blocking one another from attacking. If that’s the case, their ERF simply comes to a plateau. 2) There are no stacking penalties. In certain games, units actually debuff each other.

In Company of Heroes Opposing Fronts, there is actually an upgrade for the Panzer Elite faction that makes aggregating units even better. Although it’s only speculations, the fact that the design team behind that title was inexperienced leads me to believe that whoever designed that ability didn’t have a good understanding of how massed ranged units function. Subsequently, there was a huge host of problems associated with the blobbing of Panzer Grenadiers who inherently already had pretty good ERF.

When balancing these effects, we have to take many other variables into account. Generally speaking, we ask ourselves how easy is it for players to pull off unit aggregations. We don’t want to make the unit so expensive that only the best players can make use of the unit. At the same time, if we price the unit so low so that even the weakest players can use the unit, we risk having skilled players maximizing the unit’s ERF. In the worst case scenarios, we employ hard caps which effectively stops the unit’s ERF at a certain value. My favorite solutions tend to be the introduction of abilities or other units that punish unit aggregations. These things can range from the ensnare ability on the queen from Starcraft, or something like the machine gun unit in Company of Heroes that gets more accurate as the enemy gets more units together. These creative solutions have their own dangers of introducing even more variables to a game, but the risk is often worth the freedom that it will ultimately grant players.

Anatomy of a game

A game is a system of constraints. Whether it’s Solitaire or Call of Duty, the player is restricted in what he can do. When we play any game, what we’re really doing is trying to beat the constraints. Sometimes beating the constraints can mean optimizing statistical probabilities in your favor. It could be reading your opponent and trying to figure out what sets of constraints you can impose on her, and conversely, how to deal with the constraints that she can impose on you. Whatever it is, constraints are central to any game.

Now let’s compare two similar games, Chess and Checkers. With tournaments across the globe and the prestigious title of Grandmaster, Chess is taken seriously. Checkers, on the other hand, is something you would play with your ten year old brother. The rules of Checkers is simple; your pieces can move forward in a diagonal pattern, and you remove opposing pieces from play by “leaping” over their pieces. If you get a piece to the opposite end of the board, it gets promoted and can now move backwards. Chess has a myriad of rules which are far too many to discuss.

An important distinction to note is that it’s not the number of rules that necessarily creates constraint. For instance, Go is very simple but is on parity with Chess in complexity. What ultimately matters is the ability to maneuver around constraints. In Starcraft, one of the major constraint is resources. Initially, the player has little choice in what he can do, but as the game progresses, he can choose to expand to acquire more resources, build less resource intensive units, or even cripple his opponent’s economy to give himself a relative advantage.

The key goal of a successful game is have enough maneuverability around constraints so that it grants the player enough freedom to experiment. When there’s too few constraints, the game becomes too easy, but with too many constraints, the game becomes static with little room for novel strategies.