Responsibility and blame: a structural-model approach

Responsibility

• The contingency-based definition handles both diffuse voting causes and Suzy’s rock causing the bottle to shatter despite Billy’s accurate throw.In the rock example, Suzy’s throw becomes causal when Billy’s throw is absent, while the model’s further subtleties exclude Billy as a cause.Source: §1 · p. 2

Motivation

• The paper extends Halpern and Pearl’s counterfactual account so responsibility reflects how many changes are needed before an action becomes critical.Under a contingency, A becomes critical for B when changing A changes B.Source: §1 · pp. 2–3

Responsibility

• In an 11–0 election, each winning voter has degree of responsibility 1/6, compared with 1 when every voter is critical in a 5–4 election.The measure is 1/(N + 1), where N is the minimal number of changes needed to create counterfactual dependence.Source: §1 · pp. 2–3

Blame

• The framework assumes complete knowledge of the causal model for responsibility but incorporates the agent’s actual or expected knowledge when assessing blame.The doctor example distinguishes causing a death from being blameworthy when adverse side effects were unknown or unknowable.Source: §1 · p. 3
• Degree of blame is the expected degree of responsibility over an agent’s epistemic state, excluding situations where the outcome or action already holds.With ten marksmen who do not know who has the live bullet, each has degree of blame 1/10 although only one has responsibility 1.Source: §1 · pp. 3–4

Scope

• The definitions are not intended to capture moral responsibility, because they omit intentions and possible alternative actions.The paper presents them as potentially useful for clarifying moral issues rather than directly resolving them.Source: §1 · p. 4

Causal Models: A Review

• A causal model consists of exogenous and endogenous variables, value ranges, and functions determining each endogenous variable from the others.A binary causal model restricts every variable to two possible values.Source: §2 · pp. 4–5
• Recursive models use an acyclic causal network, ensuring a unique solution for each context and restricting causal effects to descendants.Edges indicate which variables’ functions depend on one another.Source: §2 · p. 5
• Structural equations encode mechanisms and counterfactuals by specifying how variables respond when other variables are set externally.The intervention notation [Y1 ← y1, ..., Yk ← yk]ϕ describes ϕ in the counterfactual world produced by those settings.Source: §2 · pp. 5–6

Rock-Throwing Models

• The naive rock-throwing model cannot distinguish simultaneous impacts from Suzy’s rock arriving first.This modeling limitation makes both throws appear causally relevant under the basic representation.Source: §2 · p. 5, Figure 1 · p. 5
• The better model introduces hit variables so Billy’s rock hits only when Billy throws and Suzy’s rock has not already hit.Assuming Suzy throws first creates an edge from Suzy’s hit variable to Billy’s, but not the reverse.Source: §2 · p. 6, Figure 2 · p. 6

3.1 Causality

• Causality is defined through actual truth, counterfactual dependence under contingencies, and minimality, with AC2(b) preventing spurious causal attributions.The contingency may hold variables fixed while testing whether changing X changes ϕ, but X must suffice across subsets of the active causal process.Source: §3.1 · p. 7
• In the refined rock-throwing model, Suzy is a cause of the shattering but Billy is not because AC2(b) fails for every contingency.Allowable-setting restrictions can exclude contingencies such as Billy’s perfectly accurate throw missing the bottle.Source: §3.1 · p. 8

3.2 Responsibility

• Degree of responsibility is 1/(k + 1), where k is the minimum number of changes needed to make the outcome counterfactually depend on the candidate.Responsibility is 0 when no such contingency exists and 1 when changing the candidate alone falsifies the outcome.Source: §3.2 · p. 8
• In the voting example, each 11–0 winning vote has responsibility 1/6, whereas each 6–5 winning vote has responsibility 1.The first result requires five other votes to change before one vote becomes critical; the second requires no additional changes.Source: §3.2 · p. 9
• Responsibility also measures system criticality: a state uniquely satisfying ✸p has responsibility 1, while three such states each have responsibility 1/3.The measure can diagnose redundant states and indicate fault tolerance to simultaneous failures of at most two states.Source: §3.2 · p. 9

3.3 Blame

• Blame is the expected degree of responsibility of an action over the agent’s pre-action epistemic state, represented by possible model-context situations and probabilities.The framework separates uncertainty about the true situation from uncertainty about how the world works, encoded by contexts and structural equations.Source: §3.3 · p. 10
• The agent’s epistemic state changes blame: a doctor’s blame can be 0 before learning documented drug risks but high under the state he should have had.The framework’s definition uses the epistemic state before the action, whereas post-action knowledge may produce a higher blame assessment.Source: §3.3 · pp. 11–12

4.1 The complexity of computing responsibility

• Responsibility computation is FPΣP_2[log n]-complete for general recursive models and FPNP[log n]-complete for binary models.The general-model upper bound uses binary search with at most ⌈log n⌉ ΣP_2-oracle queries, while the binary-model result is attributed to [Chockler, Halpern, and Kupferman 2003].Source: §4.1 · pp. 13, 16
• The hardness proof reduces MAXQSAT2 and MINQSAT2 to responsibility, with the constructed model satisfying dr((M,⃗u), (X = 0), ψ) = 1/(MINQSAT2(Φ) + 2).MAXQSAT2 is solved using logarithmically many ΣP_2-oracle queries, and MINQSAT2 is polynomially interreducible with MAXQSAT2.Source: §4.1 · pp. 14, 16, 17, 18

4.2 The complexity of computing blame

• Computing blame in general models has an FPΣP_2[n] upper bound and an FPΣP_2[log n] lower bound, while binary models have FPNP[n] and FPNP[log n] bounds.The upper bound computes responsibility for each relevant augmented situation and then takes the expectation; the lower bound follows because responsibility is the one-situation special case of blame.Source: §4.2 · pp. 18–19
• Blame computation averages responsibility over the epistemic state, using at most the sum of ⌈log n_i⌉ oracle queries across its N augmented situations.The oracle is NP for binary models and ΣP_2 for general models.Source: §4.2 · p. 18
• Theorem 4.4 gives completeness classifications for computing blame in recursive causal models and in binary causal models.The theorem’s statement distinguishes recursive models from the binary-model case, with the detailed class notation split across the supplied passages.Source: §4.2 · pp. 19–20
• Parallelizing independent situations and replacing logarithmically many adaptive queries with polynomially many non-adaptive queries yields the stated upper bound for computing blame.The naive algorithm uses N log n* queries; queries across situations are independent, while Jenner and Toran’s transformation handles adaptivity.Source: §4.2 · p. 19
• A reduction from N closed quantified Boolean formulas encodes each formula’s truth into a separate bit group of the degree-of-blame value, establishing hardness for arbitrary recursive models.The constructed epistemic state has N+1 augmented situations, and the ith bit group is nonzero exactly when Φi belongs to TQBF.Source: §4.2 · pp. 19–20
• For binary models, hardness follows by reducing from determining which of N propositional formulas are satisfiable.The supplied passage states that the reduction establishes the corresponding hardness result and omits its details.Source: §4.2 · p. 20

• The framework defines responsibility and blame from Halpern and Pearl’s causality, while recognizing that usefulness rather than definitive correctness is the relevant standard.The authors emphasize examining whether the definitions capture intuitions.Source: §5 · p. 20
• The framework distinguishes causal responsibility from moral blame: an action can cause an outcome and have responsibility 1 while receiving blame 0 under a reasonable epistemic state.The authors note that their definitions do not model intentions or alternative actions sufficiently for moral responsibility.Source: §5 · pp. 20–21
• In the insecticide example, Joe’s blame depends on the epistemic state: after Glenn’s cold, Joe’s degree of blame is 1/3 and the suggested economic damage is $100.Before the outcome, expected loss differs by $30 depending on whether Joe sprays; after the cold, immunity is excluded from the epistemic state.Source: §5 · pp. 21–22
• Legal doctrine sometimes conflicts with the framework by rejecting reduced or differentiated responsibility when multiple wrongdoers contribute to an injury.Joint and several liability can require one defendant to pay the full loss even when another wrongdoer also contributed.Source: §5 · p. 22
• The framework makes legally relevant distinctions in cases where one action enables another, assigning causality, responsibility, and blame according to the causal model and epistemic state.For the cigarette-and-oil example, A may be a cause with responsibility 1/2 and relatively low blame, depending on whether B would have started the fire independently.Source: §5 · p. 23
• The current responsibility measure focuses on when an action becomes critical, so weighted or game-theoretic refinements may better represent diffuse or unequal contributions.The authors propose variable weights as one generalization and identify further clarification of responsibility and blame as important future work.Source: §5 · p. 23