The Duplicator-Spoiler Game for an Ordinal Number of Turns

In this paper, we investigate Duplicator-Spoiler Games, a method of comparing the properties of structures. These games are especially significant because they have deep connections to logical expressibility. Traditionally, Duplicator-Spoiler Games were played with a preset, finite number of turns. We do not examine such situations but instead consider games of arbitrary lengths, which we represent with ordinal notation. In our games, the game length is delayed to allow more freedom to distinguish structures, giving previously dead-end games new interest. We determine the number of turns required to win various ordinal Duplicator-Spoiler Games, including those based upon scattered orderings and operations. We then use several techniques to classify large groups of structures and find general game length bounds. These techniques include the inductive construction of large structures, a comparison of games to directed graph isomorphism, an adaption of cantor ordinal and an interpretation of turns as elements of free basis. Additionally, we investigate a relationship between our new ordinal game and logical expressibility, relating our research to the original motivation for the study of Duplicator-Spoiler Games.