Corrigible Assistance in One Round: Pragmatic-Pedagogic Best Response

Elle Lazarski and Jaime Fernández Fisac study a class of human–robot assistance games in which a person’s action can resolve the robot’s uncertainty in one time step. This entry reviews the July 29, 2026 arXiv v1 manuscript. It combines a formal result with a simple collaborative block-building example, rather than a deployed robot or human-participant study.^[1]

How one round works

The method nests four decision models. H0 is a human completing the task alone. R1 is a robot that infers a goal assuming H0, then chooses an action using that belief. H2 is a teaching human who anticipates R1’s interpretation and response. Finally, R3 infers the goal assuming H2 and acts on that inference. The construction stops at this level, rather than iterating indefinitely. Its short-horizon planning approximation assumes that the goal becomes known after the current step; the theorem identifies conditions under which that optimistic assumption is actually fulfilled (§4).^[1]

The inference ceiling

A robot may assume that a person acts only to complete their own task. But an action can be optimal under several possible goals, so seeing even perfectly task-optimal behavior need not reveal which goal the person intends. The paper derives a ceiling on the robot’s posterior confidence under this model (§5.1). In its two-goal example, the ceiling is 2b/(1+b), where b is the robot’s prior probability of one goal; it stays below certainty for an interior prior. With b<1/3 it cannot even cross the robot’s 1/2 decision threshold in the example, regardless of how perfectly task-rational the human is assumed to be.^[1]

The proposed alternative models the person as considering how an action will influence the robot’s response. The robot then interprets the action as a purposeful cue. An action that looks equivalent for task execution alone can become informative through this interaction. This extends the teaching-and-learning structure of Cooperative Inverse Reinforcement Learning.^[1][2]

The block-building task makes this ambiguity concrete. The partners take turns adding blocks to complete one of two upright target shapes. Translations and rotations on the ground plane are treated as equivalent, leaving side-versus-up placements as the relevant choices. Actions such as placing a block far from the structure are excluded from the analysis. At the illustrated intermediate state, a side placement is optimal for both goals under the task-only model; anticipating the robot’s response can give it a distinct teaching meaning (§§3, 5).^[1]

A conditional one-round result

The authors define an action-separable game at a particular state and prior. Each possible goal must have a distinct set of signaling actions, and the signals must preserve the value achievable by a fully informed team: communication has zero task cost. Under these conditions and the limiting rational human and robot models, one round of best responses reaches an optimal equilibrium for the full horizon (Definitions 3–4 and Theorem 1, pp.12–14).^[1]

Pragmatic-Pedagogic Best Response (PPBR) checks these conditions and computes the signaling map. Its stated work scales with the product of the goal and action-set sizes given the oracle task values. Those values come from solving the fully informed task; the method does not eliminate that planning cost (§6, Algorithm 1).^[1]

Figures 2–3 show how the block-building model’s posterior varies with prior beliefs and rationality parameters. Finite rationality can materially affect inference; the exact equilibrium claim concerns the stated limit. The relative scaling matters: when the underlying R1 response becomes almost deterministic, the teaching advantage can become exponentially small, so human rationality must grow fast enough to exploit it. Sending both rationality parameters upward without attending to that relationship is insufficient (§5.3). The curves illustrate the model and do not measure real users’ ability to understand or produce its signals.^[1]

Relation to assistance learning

The paper builds on CIRL and pragmatic-pedagogic assistance research. Its contribution narrows the computational problem to a class whose communication structure permits an exact short-horizon solution. It also engages earlier concerns about misspecifying whether humans intend to teach; its proposed responsiveness is a theoretical perspective within the modeled setting (§2).^[1]

What corrigibility means here

The proposed robot gives the human a way to convey and achieve each modeled goal despite an initially unfavorable belief that still gives the goal positive probability. This is a useful form of responsiveness within the game. The broader 2015 correction problem also includes correcting the objective-learning rule and maintaining intervention mechanisms; those properties are not established by this theorem.^[1][3]

The result depends on the modeled goals, signaling opportunities, task dynamics and common payoff. Section 7 leaves shared signals that identify more than one goal and a multi-step extension outside scope. The paper’s favorable interpretation of pragmatic inference therefore needs to be read alongside its special game class, rather than as proof that richer human models always improve real-world alignment.^[1]