Game Theory in MMOs

Game theory is a field of mathematics that “attempts to mathematically capture behavior in strategic situations, in which an individual’s success in making choices depends on the choices of others” (Wikipedia). It’s a relatively simple model for making optimal decisions and predicting the behavior of others. A lot of times while playing, I often wonder why players behave the way they do and whether they make the best decisions. For example, I hear all the time in Arathi Basin that if we currently control 3 nodes, we should just defend them, as we only need 3 to win. However, in Warsong, once we cap the flag once, I don’t hear any suggestions that we should just defend until time runs out. These seem like very similar strategies, and I’m curious if they are truly similar and good strategies. As such, I’ve decided to explore how game theory can be applied to World of Warcraft, beginning with battlegrounds.

First, lets get some game theory basics out of the way. In game theory, the principle object we are dealing with is games. The “game” we are talking about here is not World of Warcraft, but rather a meta-game, such as deciding which node to attack in Eye of the Storm. Anything that we consider a “game” needs three components:

  • Players“, which is an decision-making entity. It could be all 80 players in a full Alterac Valley, or it could be the 2 teams in Warsong Gulch.
  • Strategies“, which are choices for each player to make. Each player must have the ability to make decisions, even if they are bad decisions. Decisions could be sending 11 players to the flag in Eye of the Storm if our player is a team, or walking into the enemy’s flag room is our player is a single person.
  • Payoffs“, or the ultimate rewards (or repercussions) for your strategies subject to all other player’s strategies. For example, heading to the center of Eye of the Storm and attempting to capture the flag could have several payoffs: earning a flag capture worth perhaps 100 points if enough players decide to help you, having no net effect if enough players from each faction are in the center, or wasting your time and possibly losing a node if the enemy attacks your bases.

That last description is a little vague, so lets jump into how we actually analyse something and get into examples.

One of the most common forms for a game theory simulation is “normal form“. It is a simply a matrix that shows decisions and playoff for both players. Here is one of the most simple models:

Player 1 ⇓/Player 2 ⇒ Player 2’s First Option Player 2’s Second Option
Player 1’s First Option Payoff to Player 1/
Payoff to Player 2
Payoff to Player 1/
Payoff to Player 2
Player 1’s Second Decision Payoff to Player 1/
Payoff to Player 2
Payoff to Player 1/
Payoff to Player 2

Imagine that there is a certain rare item, Awesomesauce, that only two players have obtained, and both are going to sell this item on the auction house. Imagine also that there is a person who will buy the cheapest Awesomesauce on the auction house, no matter what the price, and any unsold items are destroyed. Player 1 is considering selling it for 3,000 or 4,000 gold. Player 2 is considering 2,500 or 3,500 gold. Here are the possible outcomes:

Player 1 ⇓/Player 2 ⇒ Player 2 lists for 2,500 gold Player 2 lists for 3,500 gold
Player 1 lists for 3,000 gold Player 1 receives 0 gold/
Player 2 receives 2,500 gold
Player 1 receives 3,000 gold/
Player 2 receives 0 gold
Player 1 lists for 4,000 gold Player 1 receives 0 gold/
Player 2 receives 2,500 gold
Player 1 receives 0 gold/
Player 2 receives 3,500 gold

This is obviously not a very realistic or useful example, but an easy to understand illustration of a game. In reality, these players could sell for a lot of different prices, including any prices between 2,500 and 4,000 gold, and players have the ability to relist items. However, it is very clear why each of the outcomes happens.

Lets look at a more realistic example with equally as made up numbers:

Gollum is accused by his guild mate of ninjaing (stealing) a ring worth 500g from another guild mate, Frodo. Gollum equipped the ring and is unable to give it to Frodo or sell it. Frodo is upset and Gollum denies the incident ever happened. Their guild leader, Gandalf, tells players that they must decide in 1 hour to privately tell him what they will do. Gollum has the following options:

  • Tell Gandalf that he ninja’d the ring and give Frodo to buy his own ring.
  • Deny that he ninja’d the ring.

Frodo has the following options:

  • Insist that Gollum stole the ring.
  • Allow the incident to pass without interference from Gadalf.

These are the outcomes, and all players know all of these consequences:

  • If Gollum admits to the incident and Frodo insists that Gollum stole it, Gollum’s gives 500g to Frodo so that he can buy his own ring.
  • If Gollum denies the incident and Frodo insists that Gollum stole it, Frodo will ask a GM to look into the incident. The GM is known to be lazy and will just delete the ring and fine Gollum 200g.
  • If Gollum admits to the incident and Frodo drops the matter, Gandalf gives both Frodo and Gollum 100g each “for being good sports.”
  • If Gollum denies the incident and Frodo drops the incident, then nothing happens.

Additionally, Gollum decides that if he loses the ring in any manner, he likes it so much that he will buy another one for himself. If Frodo doesn’t get the ring, he also decides to buy one for himself. As such, both players will have a ring after the incident; we are only concerned in their change in gold held, ignoring interpersonal feelings and the “moral choice”.

Here is the normal form of the game:

Gollum⇓/Frodo Pursue matter Drop matter
Admit to theft -500g/0g 100g/-400g
Deny theft -700g/-500g 0g/-500g

In this situation, let’s consider what Frodo is thinking. If he drops the matter, he loses 400g or 500g regardless (mostly for buying a new ring). However, if he pursues the matter, he has a chance to “break even” at 0g. As such, the logical choice for Frodo is to pursue the matter, as no matter Gollum’s strategy, the pursuit strategy nets him equal or more gold. This is called a winning strategy – a strategy that is equal to or superior to all other of the player’s strategies in all situations.

Gollum is in a similar situation. If he admits to stealing the ring, no matter what Frodo does, he will receive more gold than he would by denying the theft! This is a significant result; no matter whether he actually stole the ring, he will still be better off saying that he did in fact steal the ring. As such, if both players are rational (ignoring all other factors), Gollum will admit to stealing the ring and Frodo will pursue the matter, and Gollum will turn over 500g to Frodo.

In a simple game theory model, this would be the end of the situation. If you only understand up to here, you have the gist of game theory. However, there is a twist in the story. If Frodo and Gollum discuss the situation, can they make a decision that gains them more net wealth? In other words, which outcome has the highest total payoff? If we sum the payoff in each cell, we find that Gollum admitting to theft and Frodo dropping the matter nets them the greatest profit:

Gollum⇓/Frodo Pursue matter Drop matter
Admit to theft -500g -300g
Deny theft -1200g -500g

The extra 200g from the guild bank to settle the matter accounts for the extra profit. Opposing players working together to achieve a higher total payoff is called cooperation. This doesn’t really happen very much in the type of strictly competitive/no communication situations we are looking at, like PvP, so we will look at this further in a future, extra-credit post.

Next time we will look at Game Theory in Arathi Basin!


  1. Interesting, and enlightening. I’ve yet to learn much about Game Development / Programming, and I’m sure this will be an beginning course.

    Thanks for all this, Heartbourne!

    Question is, what degree are you currently finishing up? I’ve always wondered and at first thought something Accounting from your Goblinism 101 post, but now I’m not 100% sure.

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