## Wednesday, February 24, 2010

### Gaming Math - Problem 14

Okay, as promised, here's the one about eigenvectors:

Problem 14: Community Events
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The "World of Warcraft" tabletop roleplaying game is based in part on the computer game of the same name, but also includes other features, including rules for in-game towns and communities. In the game, communities have a "community behavior map" consisting of a list of attributes, including "wealth", "greed", "happiness", "disaster", and so on, each with a numerical value. Different events can affect different attributes. Also, there are "links" going from one attribute to another, and each link has a different "intensity," which can be positive or negative. If a link goes from A to B with intensity I, then whenever an event affects attribute A, attribute B also changes by an amount equal to the amount attribute A changed, times the intensity of the link. For example maybe "wealth" and "happiness" have a link with an intensity of 0.5. Then if a group of adventurers brings back lots of treasure, increasing the "wealth" by 10, the "happiness" will increase by 10 x 0.5 = 5. Of course the intensity can be negative - an increase in the "disaster" attribute will probably reduce other attributes.

The rules specifically state that impact is only felt one link away from a factor. For example, suppose attribute A is linked to B and attribute B is linked to C. If an event affects A, then B will be affected by the link, but that change does not then continue on to affect C through its link. Suppose, however, that this rule were removed; you calculate the changes based on the initial event; then the changes based on the links; then the changes caused by the links from those links, and so on.

What conditions on the links and their intensities are necesary to ensure that this process always converges?