Multi-Value Responsibility Anticipation

Context

Parker2024 - Responsibility in a Multi-Value Strategic Setting studies agents choosing strategies when several possibly conflicting values are represented as finite-trace temporal formulas. The formal problem is to define responsibility sets that are not just pointwise blame labels but can guide strategy choice before a history is realized.

Formal Statement

A moral action system is nabla = (G, s_0, k, Omega), where G = (S, tau) is a multiagent transition system, s_0 is the start state, k is a finite horizon, and Omega = (Omega_1, ..., Omega_m) is a prioritized value base of LTLf formulas.

For value sets $X,Y\subseteq {\Omega }^{+}$, define the priority-level score:

$$scor{e}_n(X)=|X\cap {\Omega }_n|-|X\cap {\Omega }_n^{-}|.$$

The comparison $X⪯Y$ is lexicographic over priority levels: at the first priority level where the scores differ, the lower score is worse; if all scores tie, the sets are equivalent for comparison.

Given a strategy profile sigma, an agent i, and a unilateral alternative sigma_i', responsibility via strategy is:

$$X=Sat(Play(\sigma ,\nabla ),{\Omega }^{+})\setminus Sat(Play(({\sigma }_i^{\prime },{\sigma }_{-i}),\nabla ),{\Omega }^{+}).$$

Passive responsibility holds when such an alternative exists. Inexcusable passive responsibility further requires that no weak excuse exists: for negative responsibility $X⪯\varnothing$, there is no possible strategy of the other agents under which the original sigma_i strictly outperforms the accusing alternative sigma_i'.

The paper’s main anticipation results are:

• passive responsibility-minimizing strategies are exactly regret-minimizing strategies;
• inexcusable passive responsibility-minimizing strategies are exactly non-dominated strategies;
• in the finite setting, there exists a strategy that is both regret-minimizing and non-dominated.

Derivation / Construction

1. Evaluate each possible finite history with Sat(H, Omega^+), where Omega^+ includes both the positive value formulas and their negations.
2. Compare value sets lexicographically by priority level, counting positive satisfied values against violated values.
3. For an actual strategy profile, compare the actual play to the play induced by a unilateral alternative for agent i while holding the other agents’ strategy fixed.
4. Use the set difference between actual and counterfactual satisfactions as the responsibility set.
5. For anticipation, range over possible strategies of the other agents and take the worst single-history responsibility set rather than a union over histories.
6. Identify passive-responsibility anticipation with the same worst-case relative-difference object used for symbolic regret.
7. Identify inexcusable-responsibility anticipation with non-dominance because any asymmetrically dominated strategy lacks a weak excuse against the dominating alternative.

Implications

• The construction explains why multi-value responsibility needs set-level counterfactuals: avoidability must be assessed against one alternative history, not value by value across incompatible alternatives.
• Symbolic regret can produce explanation-friendly responsibility sets such as “the avoidable violation of value omega_1” rather than only scalar regret.
• The equivalence to non-dominance makes “having an excuse” a strategic property: a strategy is defensible when it is not asymmetrically weakly dominated.
• The framework is intentionally pessimistic because it assumes no probabilities over other agents’ strategies.
• The current formalism is finite and fully observable; compact strategy representations, heterogeneous values, and probabilistic beliefs are open extensions.

Links

• Responsibility Anticipation
• Parker2024 - Responsibility in a Multi-Value Strategic Setting
• Strategic Reasoning
• Decision Theory
• Linear Temporal Logic