Betrayal at House on the Hill is a 3-6 player board game that lends itself very well to network analysis. The full rules of the game can be found here, but it is not particularly complicated: all players are in a house, the rooms of which need to be individually explored and revealed. At a certain poitrant in the game, the players will cause a “haunt”. Once this occurs, the game dynamics change in some interesting way,and it becomes clear to players what they must do to win. There are 50 unique haunt scenarios, so the strategy of the game is unknown until the haunt is revealed.
As this is a game, it seems obvious to first discuss the game theory behind the strategy in the game. The main complaint that true board game enthusiasts seem to have about Betrayal at House on the Hill is that there is no dominant strategy before the haunt begins. This is for two reasons: first, the game is quite complex, so strategies are hard to define; second, and more importantly, the “haunt” scenario determines the win conditions of each player. From a narrative perspective, this is fine (and really fun!), but from a strategic perspective, this makes the game somewhat non-competitive.
So far, I have discussed that “haunt” scenarios change the game, but I have not explained how. The particular way in which a haunt can change the game can be visualized and explained using networks. Assume there are six people playing the game; the graph below depicts the relationships between players when the game begins.
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
As you can see, all relationships are green, because no player is aware of his or her win condition. Once the haunt begins, there can be one of two scenarios. First, one player may turn traitor. The diagram would look like this:
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
This is an interesting configuration because players are split into two components (one containing the only traitor, and the other containing everyone else), and every member of one component is friends with everyone in their component, and enemies with everyone outside their component. In other words, this is a stable configuration. This means that “haunts” which cause a single player to turn traitor yield less chaotic, and more strategic games. Each player is on a team and has a goal. The other haunt configuration is:
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
In this type of haunt, all players are pitted against each other, and there are only a set number of winners. As a graph with all negative edges is unstable, these games tend to get out of hand quickly, and formal strategy goes out the window as players desperately try to do anything they can to win.
Ultimately, these analyses of Betrayal at House on the Hill are not meant to discredit the game as a competitive challenge, nor are they trying to oversimplify the player interaction that makes the game interesting. Instead, they are meant to provide a new perspecive on the game, and to show that some features of the game are predictable, and can be modeled with networks.
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