Cloud platforms are critical infrastructure for digital economy. This study explains how large cloud providers use prices and service quality to allocate nonstorable computing time among customers with different willingness to pay and delay sensitivity. Pooling many uncorrelated workloads makes utilization predictable, reducing the need for idle backup capacity and improving operating efficiency. The predictability, however, allows providers to profitably throttle lower-tier customers through slower processing or higher interruption risk while keeping the whole market served. For cloud managers, the analysis shows how cross-region pooling, transparent interruption policies, and auction-based spot pricing can improve utilization, segment demand, and guide capacity planning. For customers and policymakers, the results identify why throttling persists: high switching costs and proprietary ecosystems weaken competitive pressure. Policies that improve workload portability, reduce data-egress frictions, and increase data interoperability can benefit the consumers and society without sacrificing the scale economies of large cloud networks. Structural remedies that reduce scale may curb market power but risk undermining operational efficiency.

Research on Nash equilibrium existence for infinite games has grown into a patchwork of technical preconditions and counterexamples. This paper presents a unified program in equilibrium theory by revising the predominant model of mixed strategies based on countable additivity. A game is specified by a nonempty set of players and, for each player, a nonempty action set and a bounded von Neumann-Morgenstern utility function. Every such game is shown to admit a Nash equilibrium in finitely additive mixed strategies. In addition, the equilibrium correspondence for any such game is shown to be nonempty, compact-valued, and upper hemicontinuous, and the same is true for equilibria obtained as limits of finite approximations. Techniques developed in this paper show that infinite games long treated as intractable become amenable to direct equilibrium analysis.

Measurement literacy is required for strong scientific reasoning, effective experimental design, conceptual and empirical validation of measurement quantities, and the intelligible interpretation of error in theory construction. This discourse examines how issues in measurement are posed and resolved and addresses potential misunderstandings. Examples drawn from across the sciences are used to show that measurement literacy promotes the goals of scientific discourse and provides the necessary foundation for carving out perspectives and carrying out interventions in science.

Patient and Pareto responsive (pPr) societal preferences were introduced and studied in Khan and Stinchcombe ( 2018 ). This paper develops a tractable subclass of the pPr preferences that satisfy a strong equity criterion formulated to match intuitions and results for large but finite models. In population models where the number and happiness of future people is stochastic, the only optimal policies require sustainability (resp. an abundance of effort) in the presence of irreversible (resp. difficult to reverse) negative externalities suffered by future generations. Partially ordering the preferences by increasing degrees of inequality aversion over generations, more inequality averse preferences give rise to choices that are counterintutive from population ethics viewpoint in smaller sets of problems.

A multiple-prior decision maker is open-minded if she can describe, as subjective uncertainty, all convex sets of distributions over payoff relevant consequences. Open-mindedness is equivalent to the ability to subjectively describe both the uniform distribution on an interval and the set of all distributions on an interval. Parameterized sets of i.i.d. distributions from classical statistics satisfy these conditions. The use of open-minded sets of priors to model decision makers allows the objective and the subjective approaches to uncertainty to inform each other and changes the implications of previously used axioms for multi-prior preferences. Subjective models with sets of priors that are not open-minded yield preferences only over those subjective sets of distributions that are describable. This preference incompleteness always implies the failure to rank elements in a dense class of set, and may rank so few elements that ambiguity attitudes do not affect choices between subjectively uncertain prospects.

Respect for first order distributional overtaking guarantees that social welfare functions for intergenerational problems treat present and future people equally and respect the Pareto criterion, modulo null sets. For weakly ergodic optimization problems, this class of social welfare functions yields solutions that respect welfare concerns, sharply contrasting with extant patient criteria. For problems in which the evolution of future paths hinges on early events and decisions, the curvature of our social welfare functions determines the risks that society is willing to undertake and leads to a variant of the precautionary principle.

A choice problem is risky (respectively ambiguous) if the decision maker is choosing between probability distributions (respectively sets of probability distributions) over utility relevant consequences. We provide an axiomatic foundation for and a representation of continuous linear preferences over sets of probabilities on consequences. The representation theory delivers: first and second order dominance for ambiguous problems; a utility interval based dominance relation that distinguishes between sources of uncertainty; a complete theory of updating convex sets of priors; a Bayesian theory of the value of ambiguous information structures; complete separations of attitudes toward risk and ambiguity; and new classes of preferences that allow decreasing relative ambiguity aversion and thereby rationalize recent challenges to many of the extant multiple prior models of ambiguity aversion. We also characterize a property of sets of priors, descriptive completeness, that resolves several open problems and allows multiple prior models to model as large a class of problems as the continuous linear preferences presented here.

Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Four Questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.A. Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.B. Expected Utility Maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.C. Specification Testing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.D. Games With Differential Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.E. Review Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2. Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.A. Norm Convergence in R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.B. The Dual Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.C. Semi-Norms for Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.D. Uniform Convergence on Subsets of Dual Spaces . . . . . . . . . . . . . . . . . . . . . 10 2.E. Finite Versus Infinite Dimensionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.F. Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3. Convex Cones, Partial Orders, and Pre-Orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.A. Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 3.B. Partial Orders and Pre-orders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.C. Dual Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.D. Reminders About Supermodularity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.E. The Monotone Likelihood Ratio Property (MLRP) . . . . . . . . . . . . . . . . . . . 16 3.F. Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4. Topological Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.A. Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.B. Topological Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.C. Vector Spaces with Compatible Topologies . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 4.D. Seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5. Duality and the Hahn-Banach Theorem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.A. Linear Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 5.B. The Hahn-Banach Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.C. Duality, Weak Topologies, and Weak∗ Topologies . . . . . . . . . . . . . . . . . . . . . 19 5.D. Polar Sets. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.E. Cones and Dual Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 5.F. Finite Dimensional Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 6. Dualities in Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Organizations face a competitive certification market for their statements, the statements do not convince third parties unless certified, the organizations are sometimes better served by a lie, and honest mistakes are possible. In our model of such a market: if certifiers are liable for mistakes, certifier contracts must be contingent; when certification is inelastically demanded, increases in certifier liability effectively reduce third party trust; organizational liability for mis-statements has a strong deterrent effect on mis-statements and increases third party trust; and after a strong negative shock to the financial system, loosening certification standards can only make it harder to raise third party trust levels.

A choice problem is risky (respectively ambiguous) if the decision maker is choosing between probability distributions (respectively sets of probability distributions) over utility relevant consequences. We provide an axiomatic foundation for and a representation of continuous linear preferences over sets of probabilities on consequences. The representation theory delivers: first and second order dominance for ambiguous problems; a utility interval based dominance relation that distinguishes between sources of uncertainty; a complete theory of updating convex sets of priors; a Bayesian theory of the value of ambiguous information structures; complete separations of attitudes toward risk and ambiguity; and new classes of preferences that allow decreasing relative ambiguity aversion and thereby rationalize recent challenges to many of the extant multiple prior models of ambiguity aversion. We also characterize a property of sets of priors, descriptive completeness, that resolves several open problems and allows multiple prior models to model as large a class of problems as the continuous linear preferences presented here. Roughly, risk refers to situations where the likelihood of relevant events can be represented by a probability measure, while ambiguity refers to situations where there is insufficient information available for the decision maker to assign probabilities to events. (Epstein and Zhang [22])

From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zero-one if there exists an E in F, a measurable, onto phi:Omega -> Pi such that for all p in Pi, p(phi^{-1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zero-one set of non-atomic probabilities on a standard measure space, that M is a complete, separable metric space, Delta_M is the set of Borel probabilities on M and Comp(Delta_M) is the class of non-empty, compact subsets of Delta_M with the Hausdorff metric. There exists a jointly measurable H: Comp(Delta_M) x Omega ->M such that for all K in Comp(Delta_M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) -->0, then for all p in Pi, p({omega: H(K_n,omega) -->H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's representation theorem.

If (X,X ) is a measure space and F◦ ⊂ X is a field generated either by a countable class of sets or by a Vapnik-Červonenkis class, then if μ is purely finitely additive, there exist uncountably many μ′ agreeing with μ on F◦ and having |μ(A) − μ′(A)| = 1 for uncountably many A. If μ is also non-atomic, then for any r ∈ (0, 1], |μ(Ar)− μ(Ar)| = r for uncountably many Ar. Al-Najjar’s [1] unlearnability result is: if (X,X ) belongs to a class of measure spaces not supporting countably additive nonatomic distributions and C is a Vapnik-Červonenkis class, then there exist purely finitely additive nonatomic probabilities, μ and μ′, agreeing on C and having |μ(Aα)− μ(Aα)| = α for any α ∈ (0, 12 ] for uncountably many Aα. Alice laughed. “There’s no use trying,” she said: “one ca’n’t believe impossible things.” “I daresay you haven’t had much practice,” said the Queen. “When I was your age, I always did it for halfan-hour a day. Why, sometimes I’ve believed as many as six impossible things before breakfast.” (Lewis Carroll, Through the Looking Glass)

Abstract A scoring rule is proper if it elicits an expert’s true beliefs as a probabilistic forecast, and it is strictly proper if it uniquely elicits an expert’s true beliefs. The value function associated with a (strictly) proper scoring rule is (strictly) convex on any convex set of beliefs. This paper gives conditions on compact sets of possible beliefs Θ that guarantee that every continuous value function on Θ is the value function associated with some strictly proper scoring rule. Compact subsets of many parametrized sets of distributions on Rk satisfy these conditions.

Under study are games in which players receive private signals and then simultaneously choose actions from compact sets. Payoffs are measurable in signals and jointly continuous in actions. Stinchcombe (2011) [19] proves the existence of correlated equilibria for this class of games. This paper is a study of the information structures for these games, the discontinuous expected utility functions they give rise to, and the notion of a balanced approximation to an infinite game with discontinuous payoffs.