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We show that it is possible to understand and identify a decision maker’s subjective causal judgements by observing her preferences over interventions. Following Pearl [2000, DOI: doi.org/10.1017/S0266466603004109 ], we represent causality using causal models (also called structural equations models), where the world is described by a collection of variables, related by equations. We show that if a preference relation over interventions satisfies certain axioms (related to standard axioms regarding counterfactuals), then we can define (i) a causal model, (ii) a probability capturing the decision-maker’s uncertainty regarding the external factors in the world and (iii) a utility on outcomes such that each intervention is associated with an expected utility and such that intervention A is preferred to B iff the expected utility of A is greater than that of B. In addition, we characterize when the causal model is unique. Thus, our results allow a modeler to test the hypothesis that a decision maker’s preferences are consistent with some causal model and to identify causal judgements from observed behavior. 

We focus on explaining image classifiers, taking the work ofMothilal et al. 2021 (MMTS) as our point of departure. We observe that, although MMTS claim to be using the definition of explanation proposed by Halpern 2016, they do not quite do so. Roughly speaking, Halpern’s definition has a necessity clause and a sufficiency clause. MMTS replace the necessity clause by a requirement that, as we show, implies it. Halpern’s definition also allows agents to restrict the set of options considered.While these difference may seem minor, as we show, they can have a nontrivial impact on explanations.We also show that, essentially without change, Halpern’s definition can handle two issues that have proved difficultfor other approaches: explanations of absence (when, for example, an image classifier for tumors outputs “no tumor”) and explanations of rare events (such as tumors). 

Envy-free cake-cutting protocols procedurally divide an infinitely divisible good among a set of agents so that no agent prefers another's allocation to their own. These protocols are highly complex and difficult to prove correct. Recently, Bertram, Levinson, and Hsu introduced a language called Slice for describing and verifying cake-cutting protocols. Slice programs can be translated to formulas encoding envy-freeness, which are solved by SMT. While Slice works well on smaller protocols, it has difficulty scaling to more complex cake-cutting protocols. We improve Slice in two ways. First, we show any protocol execution in Slice can be replicated using piecewise uniform valuations. We then reduce Slice's constraint formulas to formulas within the theory of linear real arithmetic, showing that verifying envy-freeness is efficiently decidable. Second, we design and implement a linear type system which enforces that no two agents receive the same part of the good. We implement our methods and verify a range of challenging examples, including the first nontrivial four-agent protocol.