We continue the investigation of analytic spaces from the perspective of computable structure theory. We show that if p ≥ 1 is a computable real, and if Ω is a nonzero, non-atomic, and separable measure space, then every computable presentation of L p (Ω) is computably linearly isometric to the standard computable presentation of L p [0, 1]; in particular, L p [0, 1] is computably categorical. We also show that there is a measure space Ω that does not have a computable presentation even though L p (Ω) does for every computable real p ≥ 1.Laboratoire lorrain de recherche en informatique et ses applications,

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In this work, we provide a structural characterization of the possible Nash equilibria in the well-studied class of security games with additive utility. Our analysis yields a classification of possible equilibria into seven types and we provide closed-form feasibility conditions for each type as well as closed-form expressions for the expected outcomes to the players at equilibrium. We provide uniqueness and multiplicity results for each type and utilize our structural approach to propose a novel algorithm to compute equilibria of each type when they exist. We then consider the special cases of security games with fully protective resources and zero-sum games. Under the assumption that the defender can perturb the payoffs to the attacker, we study the problem of optimizing the defender expected outcome at equilibrium. We show that this problem is weakly NPhard in the case of Stackelberg equilibria and multiple attacker resources and present a pseudopolynomial time procedure to solve this problem for the case of Nash equilibria under mild assumptions. Finally, to address non-additive security games, we propose a notion of nearest additive game and demonstrate the existence and uniqueness of a such a nearest additive game for any non-additive game.