The Generalized Symmetric Tequila Problem: Influence and Independence in N-Player Games

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This paper extends results from Mike Steel and Amelia Taylor's paper The Structure of Symmetric N-Player Games when Influence and Independence Collide. These games include n causes, which are dichotomous random variables whose values determine the probabilities of the values of n dichotomous effects. We denote the probability spaces that exhibit independence and influence among n players as Indₙ and Infₙ respectively. We define the solution space of the "generalized symmetric tequila problem," GSTₙ, as the set of probabilities for a set of given effects such that the causes and effects are independent and each cause influences the effects, that is GSTₙ is the intersection of Indₙ and Infₙ. Steel and Taylor showed that GSTₙ is connected for n greater than or equal to 8 and disconnected for n = 3, 4. We prove that for n = 5, 6, 7, GSTₙ is connected and determine the number of connected components of GST₄.