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

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 Ind_n and Inf_n respectively. We define the solution space of the "generalized symmetric tequila problem," GST_n, 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 GST_n is the intersection of Ind_n and Inf_n. Steel and Taylor showed that GST_n is connected for n greater than or equal to 8 and disconnected for n = 3, 4. We prove that for n = 5, 6, 7, GST_n is connected and determine the number of connected components of GST_4.