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.

Would a rational actor ever build a weapon that would destroy both himself and his opponent?

If we take it at face value, it would seem that it’s never a good idea to make something that would harm you and your opponent. Afterall, despite man’s impeccable’s record for finding new ways of killing, he still values self preservation fairly highly. Even in scenarios such as suicide bombing, the bomber is usually convinced that he’s dying for a good cause or that there is an afterlife which is worth blowing himself up for.

So here’s a basic sequential game. The actor, row, goes first and the actor, column, goes second. Row’s nuclear weapon does not harm himself while column’s nuclear weapon is self destructive. A utility of 1 means survival. A utility of 0 means destruction.

The Nash Equilibrium for this game is 1, 1. If row decides to Nuke, column can punish row by nuking as well. To column, it makes no difference. Once row has launched his missiles, he is as good as dead. Because row knows that if he nukes, column will punish, his best strategy is to not nuke. Column’s best decision is then to not nuke.

So the above scenario is still fairly abstract. There are a few things to note. The game does not predict what will happen in the future. It may well be that both parties will decide to develop even more destructive and effective weapons, or it could be that both parties will try to negotiate disarmament. I cannot forecast how the game will change, but I can say that under the circumstances in the above game, it is entirely rational to have a weapon that’s self destructive.

The key is that column does not build the self destructive weapon because he wants everybody to die. In fact, everybody dying is equally as bad as having row nuke for column. He builds the weapon because row will react differently if he knows that he cannot nuke with impunity. Having the weapon is better than having no weapon. Having no weapon is effectively declaring to row that you will always play a no nuke strategy in which case column’s fate depends on whether or not row feels like destroying column. If we modify row’s utility value from deciding to nuke to be 1.1 so that it’s higher than 1, row still does not have an incentive deviate if it knows that column will nuke if it does so. We can interpret the 1.1 utility as row’s increased utility from not having an opponent. Now if column is forced to play a not nuke strategy because it does not have the weapon available, then row’s best decision would be to nuke. Column has a clear incentive to have the weapon.