Gap-Free Decision-Making Mechanisms

• Gap-free decision-making mechanisms are systems where every failure to meet normative goals is counterfactually attributable to at least one agent.
• They are formalized using sequential decision trees, ensuring that each non-compliant outcome has a corresponding strategy to enforce compliance.
• These mechanisms often require centralized 'elected dictatorship' structures, posing challenges for design, fairness, and computational verification.

A gap-free decision-making mechanism is a system or protocol for individual or collective decisions in which, for every possible outcome that fails a normative requirement, at least one agent has or had a counterfactual strategy to avoid this failure. In other words, a mechanism is gap-free when responsibility for every non-compliant outcome is always attributable to at least one agent within the system, never leaving the situation where no agent could have prevented the undesirable result. This concept stands in contrast to mechanisms where a responsibility gap emerges—where some outcomes can occur for which no individual can be deemed responsible in the sense of having had the practical power to steer the outcome otherwise (2505.06312, 2506.07935, 2507.02582).

1. Formal Characterization of Gap-Free Mechanisms

In sequential decision-making models—typically represented as rooted trees with decision nodes assigned to agents and leaves labeled by outcomes—a mechanism is gap-free if every failure to achieve the normative goal is counterfactually attributable to an agent. More formally, let $\gamma(\vec{a})$ denote the deontic constraint on outcomes over action profiles $\vec{a}$, and ${\sf cf}_i(\vec{a})$ encode that agent $i$ has a counterfactual strategy to force $\gamma$. The gap-freeness property can be captured by the formula: $$\forall \vec{a}\; \Bigl[ \neg\gamma(\vec{a})\to\bigvee_{i}{\sf cf}_i(\vec{a}) \Bigr]$$ which asserts that in every non-compliant outcome, at least one agent, via a deviation or alternative strategy, could have ensured compliance (2507.02582).

In the perfect-information setting, gap-free mechanisms are exactly those where, along every path from the root to any leaf in the decision tree, there is a node from which some agent can unilaterally dictate the outcome. This property is formalized by the concept of an "elected dictatorship" (2505.06312). That is,

$$\forall \text{ root–leaf path, } \exists v, \exists a:\ v \in \text{win}_a(\text{Yes}) \cap \text{win}_a(\text{No})$$

where $\text{win}_a(o)$ is the set of nodes from which agent $a$ can guarantee outcome $o$ (yes or no).

2. Relationship to Responsibility and Diffusion of Responsibility

Gap-freeness is focused on the absence of situations where nobody is responsible. This differs from diffusion of responsibility, where more than one agent can simultaneously be deemed responsible for a particular outcome (i.e., a set of agents each with independent counterfactual power to change the result) (2506.07935, 2507.02582). Formally, a mechanism is diffusion-free if, for every non-compliant outcome, there is at most one agent with counterfactual responsibility.

Table: Key Responsibility Properties in Sequential Mechanisms

Property	Definition	Complexity
Gap-free	In every failure, some agent could have prevented it	$\Pi_3$-complete (2507.02582)
Diffusion-free	In every failure, at most one agent could have prevented it	$\Pi_2$-complete (2507.02582)
Both (unique responsibility)	In every failure, exactly one agent could have prevented it	$\Pi_2$-complete (2507.02582)

If both properties are enforced, responsibility attribution becomes unique and verification simplifies compared to gap-freeness alone.

3. Elected Dictatorships and Mechanism Structure

A fundamental result is that, in settings with perfect information, avoiding responsibility gaps requires mechanisms to be "elected dictatorships" (2505.06312, 2506.07935). In such mechanisms,

• Every outcome path reaches a stage where a single agent can enforce or prevent any possible outcome, regardless of other agents' actions.
• For two agents, gap-freeness equates to simple dictatorship by one agent; for more than two, every path must "elect" exactly one dictator at some stage.

In imperfect-information scenarios, the notion generalizes to epistemic dictatorships, where an agent not only controls the outcome but is aware of being in such a position. The class of gap-free mechanisms then sits between "elected epistemic dictatorship" and "semi-epistemic dictatorship," reflecting variations in agents' knowledge about their own power to control outcomes (2505.06312).

4. Computational Complexity of Verification

The verification of gap-freeness is computationally demanding. When encoded as logical formulas over symbolic mechanism representations, the gap-free property requires checking a formula with three quantifier alternations (a universal block over all outcome paths, an existential block for counterfactual strategies per agent, and a universal block for all possible reactions of others). This positions the problem as $\Pi_3$-complete in the polynomial hierarchy (2507.02582).

In comparison,

• Verifying diffusion-freeness alone is $\Pi_2$-complete.
• The intersection (i.e., verifying that every "failure" outcome is uniquely attributable to one agent) is also $\Pi_2$-complete, since requiring uniqueness reduces the complexity by one quantification layer.

This increased complexity implies practical challenges for systematic design and formal verification of large or complex decision systems seeking gap-freeness.

5. Mechanism Design Implications and Tradeoffs

To avoid gaps, mechanisms must ensure at all times that ultimate counterfactual responsibility is not left undetermined. This often leads to highly centralized structures, where at each decision branch a single agent is eventually empowered to make a binding choice.

Implications include:

• Decentralized or democratic mechanisms (such as majority voting) generally do not achieve gap-freeness, especially in the presence of ties or concurrent actions.
• Enforcing unique responsibility may conflict with other desiderata such as fairness, distribution of authority, or resilience to manipulation.
• Mechanism simplification via structure-preserving transformations (such as bisimulation) preserves or reveals the elected dictatorship property, enabling rigorous analysis and canonical representation (2506.07935).

Efforts to eliminate both diffusion and gap force the structure of the decision tree toward a unique-responsibility paradigm, with clear accountability at every possible outcome.

6. Broader Impact and Future Research

The formal analysis of gap-free mechanisms has practical implications across legal, ethical, and AI system domains where individual responsibility must be clearly assignable.

Open research directions include:

• Extending formal definitions and results to concurrent and imperfect-information mechanisms, stochastic or nondeterministic protocols, and settings with probabilistic outcomes or multiple normative constraints.
• Studying algorithms for efficient verification in restricted (but practically relevant) classes of mechanisms.
• Investigating tradeoffs between achieving unique responsibility and other governance, efficiency, or robustness constraints in multi-agent and collective AI systems.

In sum, gap-free decision-making mechanisms provide a rigorous framework for ensuring that every non-compliant outcome remains individually attributable, with direct consequences for the design, accountability, and regulation of complex decision systems (2505.06312, 2506.07935, 2507.02582).