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This text delves into the complexities of debate structures concerning game versions and their restrictions. It introduces various operational constraints like win, fault, and control for player 'x', alongside complementary operations. The analysis explores union and set theories relative to debates, analyzing how restrictions shape outcomes in various scenarios. By focusing on nuanced relationships—like win conditions and fault occurrences—the piece offers insights into strategic gameplay and competitive formats, setting a foundation for understanding varied game dynamics and interactions.
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Restricted Versions of Games • Win restriction: [w x]: only games that x wins • Fault restriction: [fl x]: only games in which x makes a fault • Control restriction: [c x]: only games that x controls • [c x] = [w x] union [fl x] • Also add a complement restriction operation for each of the restriction operations: [!w x],[!fl x],[!c x] • [w x] union [!w x] = all debates • x!=y: [fl y] [fl x] = empty set of debates =[w y][w x]
Focus on 2, 3, 4, 5 All debates 1 [!fl y] [w x] = [!c y] [w x] 2 [!w y] [fl x] = [!c y] [fl x] 3 [fl y] [w x] = [c y] [w x] 4 [w y] [fl x] = [c y] [fl x] 5 [!w x] [fl y] = [!c x] [fl y] 6 [!fl x] [w y] = [!c x] [w y] 7 [!c x] [!c y] c x c y x wins y at fault 1 5 3 4 2 6 y wins x at fault 7
