THEORY-527: If agents are the systems that would adapt their policy were their actions to affect the world differently, then in mechanised causal games meeting Kenton et al.'s five assumptions (one agent per connected component of the decision–utility graph, best response to every mechanism intervention, a fixed tie-breaking order, terminal edges only into decision mechanisms, and mechanism interventions able to set any deterministic function) the game graph, with its decisions, utilities and agents, is recovered exactly from interventions on mechanisms

Source

Kenton, Kumar, Farquhar, Richens, MacDermott & Everitt (2022), “Discovering Agents”, LIT-909, read in full in NOTE-707 from the arXiv PDF (v2).

What was actually shown

• The definition (§1.2). Agents are moved by reasons: a system “would act differently if [it] ‘knew’ that the world worked differently”. So “agents are systems that would adapt their policy if their actions influenced the world in a different way”.
• The models. Mechanised causal models add mechanism variables, which “parameterise how the object-level variables depend on their object-level parents”. A mechanism intervention is a soft intervention held across time steps, so a system can adapt to it.
• The theorem. Under five assumptions (each connected component of the decision–utility graph is one agent; agents play a subgame-perfect best response; ties break in a fixed order; only decision mechanisms have incoming terminal edges; mechanism interventions can instantiate any deterministic function), Algorithm 1 followed by Algorithm 2 recovers the true game graph (Theorem 1). Theorems 2 and 3 show Algorithms 2 and 3 are left inverses. In the authors’ words, this is “a precise experimental process that can, in principle and under some assumptions, be done to assess whether something is an agent”.
• The verdicts. “An existing thermostat will not adapt to this change, and is therefore not an agent”, but “the thermostat with its creation process is an agent”. “We consider the system of the RL training process to be an agent, but the learnt RL policy itself, in general, won’t be”. An actor-critic is two agents. A chain X → Y → Z has none.
• Failures. Assistance games, where human and robot share a utility, merge into one agent. A utility not downstream of its decision is mislabelled.

The theorem could have failed and did not. The verdicts follow from the definition; they test it only against intuition.

What this does not say

• That the definition is the right one. It is offered, not derived. Its verdicts on thermostats and trained policies are argued from it (NOTE-707, C3, C4). That it “reliably classifies humans as agents” is asserted, not demonstrated.
• That the verdicts are frame-free. “What is a decision or a utility node is frame-dependent” (§5.1), relative to the chosen variables, and the choice of coarse-graining can change whether agents are found (§5.2, asserted).
• That the procedure can be run on a given system. It needs interventional data on mechanisms. For most systems of interest, including deployed models, those interventions are not available as the theorem assumes.
• That a frozen policy has no goals. It says such a policy “derive[s] much of [its] goal-directedness from the process that creates [it]”, which locates the agent elsewhere, not that the policy’s behaviour is aimless.
• That shared goals make one agent in fact. The assistance-game merger is listed by the authors as a failure.

Connections

• Rival. Orseau, McGill & Legg (THEORY-624) define agency by which description, goal-optimizing or input-output, compresses the behaviour better for an observer. Kenton et al. argue that “the compression-based approach only considers what the system actually does”, and so may count as agents “systems with accidentally optimal input-output mappings”. The two give different verdicts there.
• THEORY-249 argues that agency is frame-dependent, and cites this method giving opposite verdicts under two choices of variables. The paper’s own §5.1 concedes the dependence.
• THEORY-208 counts a smart thermostat as an artefactual agent. This account does not, unless its design process is included.
• CLAIM-174 takes this definition as the adaptation half of its admission test, and CLAIM-170 reads the location of the agent in the governing process as an agent individuated by the reach of what governs it.
• On QUESTION-032, the level- and variable-relative answer (a factor is an agent if its decision mechanism responds to interventions on how its actions affect its utility) is the definition, which THEORY-801 carries since the narrowing. This entry adds only that, under the definition and the five assumptions, agents are recoverable from interventions.

Note of 2026-10-20: Theorem 1, re-derived

• Lemmas 1–2. Leave-one-out discovery declares W → V when some intervention on all other variables and two values of W change P(V), which is the definition of a direct edge, cyclic or not. Algorithm 1 then tests terminality by Definition 3, given structural interventions on every node.
• Decisions. For a decision D of agent A, Assumption 1 gives a utility U of A reached by a path from D through no other utility of A. Assumption 5 sets the path to copy D, makes U = 1 for one value of D, and makes every other mechanism ignore D. By Assumption 2 A picks that value; inverting U’s mechanism makes another uniquely optimal, so the mechanism edge from U to D is found. With D’s effects cut, every rule ties and Assumption 3 forbids a switch, so the edge is kept as terminal. Terminal edges into a non-decision’s mechanism do not exist (Assumption 4), so none is kept.
• Utilities. The same construction gives the edge for every utility. For a non-utility W with its children cut and every other mechanism fixed, W’s mechanism can reach expected utility only through W, whose effects are cut, so no rule changes and W gets no terminal edge.
• Agents. The terminal edges are the reversed edges of the agent subgraphs, whose weakly connected components are the agents by Assumption 1, and Algorithm 2 colours components.

The theorem does not test the definition in its antecedent, and the paper’s verdicts on particular systems are not consequences of it.

Note of 2026-10-20: the wider claim, filed Proposed

Under ADR-047 v4 the narrowing-loss sweep (batch 069) filed THEORY-801 as Proposed, extending this entry. It carries Kenton et al.’s definition of agency and the verdicts they argue from it (thermostat and design process, RL training and learnt policy, humans), which this entry’s title carried before batch 055 narrowed it to the recovery theorem. This entry is its evidence and is not changed by the filing.