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Agent Incentives: A Causal Perspective

Tom Everitt, Ryan Carey, Eric Langlois, Pedro A Ortega, Shane Legg

arXiv:2102.01685v2cs.AIcs.LG

TL;DR

The paper asks how to analyze whether an optimal agent policy depends on sensitive attributes or influences important variables, questions relevant to AI safety and fairness. It uses causal influence diagrams to define and test agent incentives, completing a graphical criterion for value of information and giving sound and complete criteria for new incentive concepts. The analysis is based on causal structure, with limits including the need to know causal relationships and difficulty representing dynamically changing ones.

  • Problem

    The paper asks whether an optimal policy depends on a sensitive attribute or seeks to influence an important variable, issues relevant to choosing objectives for safe and fair behavior.

    Source: Introduction · p. 1

  • Method

    The authors use causal influence diagrams and a structural causal influence model to define agent incentives and derive graphical criteria.

    Source: Introduction · p. 1, Structural Causal Influence Models · p. 3

  • Results

    The paper proves the value-of-information criterion complete and gives sound and complete graphical criteria for instrumental control and response incentives.

    Source: Introduction · p. 1, Value of Information · p. 4, Response Incentives · p. 5

  • Takeaways & Limitations

    The framework offers a unified perspective on agent incentives, with applications including value learning, interruptibility, conservatism, agent frameworks, and reward tampering.

    Source: Discussion and Conclusion · p. 8

    Because the results focus on causal structure, they cannot compare relative incentive strengths, require known causal relationships, and do not readily express dynamically changing causal relationships.

    Source: Discussion and Conclusion · p. 8

Abstract

from arXiv · show

We present a framework for analysing agent incentives using causal influence diagrams. We establish that a well-known criterion for value of information is complete. We propose a new graphical criterion for value of control, establishing its soundness and completeness. We also introduce two new concepts for incentive analysis: response incentives indicate which changes in the environment affect an optimal decision, while instrumental control incentives establish whether an agent can influence its utility via a variable X. For both new concepts, we provide sound and complete graphical criteria. We show by example how these results can help with evaluating the safety and fairness of an AI system.

Introduction

The paper uses causal influence diagrams to ask when an optimal policy depends on sensitive attributes or seeks to influence important variables. It develops graphical criteria and incentive concepts to help analyze safety and fairness questions.

Introduction

  • The paper uses influence diagrams and causal models to analyze and predict how machine-learning systems behave when trained to optimize an objective in an environment.The examples concern grade prediction and content recommendation.Source: Introduction · p. 1
  • The authors prove the first correct completeness result for the graphical criterion for value of information (VoI).They say the criterion is unique and cannot be further improved.Source: Introduction · p. 1
  • For value of control (VoC), the authors provide a complete graphical criterion with soundness and completeness proofs.The passage describes the earlier criterion as incomplete.Source: Introduction · p. 1
  • Instrumental control incentives refine VoC to variables an agent can influence, while response incentives identify environmental changes to which an optimal policy responds.The paper gives formal definitions and sound and complete graphical criteria for both concepts.Source: Introduction · p. 1
  • The analysis focuses on single-decision settings, which can model supervised learning, contextual bandits, and policy choice in an MDP.The passage also notes that prior work describes transforming multi-decision settings into single-decision ones.Source: Introduction · p. 1

Setup

The setup combines causal models with decision and utility nodes to represent how agents act in an environment. A policy specifies the decision rule and turns the model into a causal model on which interventions can be defined.

Setup

Structural Causal Models

  • A structural causal model (SCM) represents endogenous variables with acyclic structural functions and independent exogenous variables that encode uncertainty.Interventions are represented by replacing the structural functions for intervened variables; soft interventions replace a function with another function.Source: Structural Causal Models · p. 2

Causal Influence Diagrams

  • A causal influence diagram (CID) is a directed acyclic graph partitioned into structure, decision, and utility nodes, with utility nodes having no children.An edge into a decision represents information available when that decision is made.Source: Causal Influence Diagrams · p. 3

Structural Causal Influence Models

  • A structural causal influence model (SCIM) combines a CID with exogenous variables, structural functions for non-decision variables, and a distribution over the exogenous variables.In a single-decision SCIM, a policy maps observations and decision randomness to an action; selecting it turns the SCIM into an SCM, and an optimal policy maximizes expected utility.Source: Structural Causal Influence Models · p. 3

Materiality

Materiality asks whether an observation can improve optimal performance, while nonrequisiteness gives a graphical test for when an observation cannot provide useful information. The section notes that this test applies even when the diagram is not assumed to be causal.

Materiality

  • An observation is material when removing its information link to the decision lowers the maximum attainable utility.Materiality can also be assessed graphically through whether the observation is requisite.Source: Materiality · pp. 3–4
  • An observation cannot provide useful information when it is d-separated from utility conditional on the other observations; this is called nonrequisiteness.The materiality and value-of-information results in this section do not require causal influence diagrams.Source: Materiality · pp. 3–4
  • In the grade-prediction example, high school is a requisite observation, whereas gender is not.The example identifies which observations matter to the predictor in that model.Source: Materiality · p. 4

Value of Information

The paper extends materiality to value of information (VoI) by asking whether adding an observation can improve optimal utility. It proves a complete graphical criterion and illustrates its use in the grade-prediction example.

Value of Information

  • A variable has value of information when adding it as an observation makes it material, provided it is not a descendant of the decision.The added information link must not create a cycle.Source: Value of Information · p. 4
  • A single-decision CID admits VoI for a variable exactly when that variable is requisite after adding its information link to the decision.The paper establishes completeness as well as soundness for this graphical criterion.Source: Value of Information · p. 4
  • Graphical connectedness alone does not ensure that utility expectations or attainable utility change, so the completeness proof constructs a SCIM in which the variable is material.The authors distinguish this proof from an earlier attempt by Nielsen and Jensen (1999).Source: Value of Information · p. 4
  • In the grade-prediction example, the predictor has an incentive to use high school but not gender because gender provides no useful information for predicting university grade.Source: Value of Information · p. 4

Response Incentives

Response incentives identify environmental variables to which optimal decisions respond, with a complete graphical criterion based on paths in the minimal reduction. For sensitive attributes, they distinguish when counterfactual unfairness is merely possible from when all optimal policies exhibit it.

Response Incentive Criterion

  • A single-decision CID admits a response incentive on X if and only if its minimal reduction contains a directed path from X to D.The minimal reduction removes information links from nonrequisite observations.Source: Response Incentives · p. 4
  • The proof intuition is that an optimal decision responds only to effects that reach one of its requisite observations.For completeness, the authors construct a compatible SCIM in which a path in the minimal reduction supplies useful information that every optimal policy must use.Source: Response Incentives · p. 5

Fairness

  • A response incentive on a sensitive attribute A holds exactly when all optimal policies are counterfactually unfair with respect to A.This makes unfairness incentivised, not merely possible.Source: Response Incentives · p. 5
  • In the grade-prediction example, race but not gender has a response incentive, although both attributes can potentially support counterfactual unfairness.The minimal reduction retains the path from race through high school to predicted grade but excludes the gender-to-grade edge.Source: Response Incentives · p. 5
  • Removing the predictor’s access to high school removes the race path and leaves neither race nor gender with a response incentive.In the revised example, race can still have positive value of information without incentivised counterfactual unfairness.Source: Response Incentives · p. 5, Figure 3 · p. 5

Applications

  • The criterion applies to any fairness definition by checking whether the relevant unfairness occurs under all optimal policies.Response incentives can also be desirable, as when an AI system should respond to a shutdown button.Source: Response Incentives · p. 5

Value of Control

Value of control asks whether changing a variable could improve attainable utility. Its complete graphical criterion uses paths from that variable to utility nodes in the minimal reduction.

Value of Control

  • Value of control asks whether attainable utility could increase if the agent could set a variable’s value.The definition allows a soft intervention that replaces the variable’s structural function while respecting the graph.Source: Value of Control · pp. 5–6

Value of Control Criterion

  • A single-decision CID admits positive value of control for X if and only if the minimal reduction contains a directed path from X to a utility node.The minimal reduction rules out effects through observations that an optimal policy can ignore.Source: Value of Control · p. 6
  • A path can support value of control by directly affecting utility or by increasing useful information available at the decision.The completeness proof constructs a SCIM in which one of these effects occurs.Source: Value of Control · p. 6

Application

  • In the content-recommendation example, every non-decision node has positive value of control, including variables the decision cannot control.This motivates instrumental control incentives, which account for the agent’s limitations.Source: Value of Control · p. 6

Instrumental Control Incentive

Instrumental control incentives ask whether an agent can affect utility through a variable, separating useful control from control the decision can actually exert. A directed path from the decision through that variable to utility is the complete graphical criterion.

Definition

  • An instrumental control incentive asks whether choosing D to influence X independently of D’s other effects would matter for utility.It concerns utility achieved through control of X, not side-effects of an optimal policy; it corresponds to a natural indirect effect from D to U in the policy-completed model.Source: Instrumental Control Incentive · p. 6

Instrumental Control Incentive Criterion

  • A single-decision CID admits an instrumental control incentive on X if and only if a directed path runs from D through X to a utility node.The criterion is both sound and complete.Source: Instrumental Control Incentive · p. 6

Application

  • In the recommender example, influenced user opinions have an instrumental control incentive, so the agent may induce stronger emotions to make clicks more predictable.Clicks are also on a decision-to-utility path.Source: Instrumental Control Incentive · p. 7

Redesign

  • Rewarding predicted clicks from a separately trained preference model can remove the incentive to manipulate user opinions while preserving an incentive for predicted clicks.The prediction should estimate whether the original user would click each post if offered individually, excluding opinion changes from earlier posts.Source: Instrumental Control Incentive · p. 7, Figure 4 · p. 7

Related Work

The paper situates causal influence diagrams among related approaches to decision-making, fairness, and mechanism design. It emphasizes that causal relationships in agent environments are central to its incentive-analysis perspective.

Causal Influence Diagrams

  • Earlier causal influence-diagram work modelled rational decision-making but was less formal and focused on human decision-makers.The paper also notes an open-source Python implementation of CIDs.Source: Related Work · p. 7

Causal Models

  • Howard canonical-form influence diagrams permit counterfactual reasoning only downstream of decisions, which is inadequate for defining response incentives.The paper argues that a stronger causality property is necessary for most of its incentive concepts.Source: Related Work · p. 7

Value of Information and Control

  • For positive value of information, the paper presents a completeness result beyond earlier sound graphical criteria and discusses a distinct criterion proposed by Nielsen and Jensen (1999).The paper says that criterion, taken literally, is unsound for requisite nodes and positive value of information.Source: Related Work · p. 7

Fairness

  • Response incentives address counterfactual fairness for any CID, whereas the cited path-specific-effect incentive criterion is deferred to future work.The paper contrasts this with Nabi, Malinsky, and Shpitser’s (2019) approach, which assumes recall of all past events.Source: Related Work · p. 8

Mechanism Design

  • Mechanism design shares the broad aim of shaping rational-agent behaviour through objectives and environments, but this paper focuses on causal relationships in agent environments.The paper contrasts this with mechanism design’s core challenge of private information or preferences.Source: Related Work · p. 8

Discussion and Conclusion

The graphical criteria establish incentive conclusions from causal structure even when quantitative relationships are unknown, but cannot compare the relative strength of incentives. The discussion also notes that causal diagrams struggle with dynamically changing relationships and cites applications across several AI safety topics.

Discussion and Conclusion

  • The criteria reveal whether incentives are present from causal structure, even when quantitative relationships between variables are unknown.However, they do not support subtler comparisons such as the relative strength of different incentives.Source: Discussion and Conclusion · p. 8
  • Causal relationships between variables must be known, and causal diagrams struggle to express dynamically changing relationships.The paper notes that these constraints are common to causal models generally.Source: Discussion and Conclusion · p. 8
  • The framework has been applied to value learning, interruptibility, conservatism, agent-framework modeling, and reward tampering.The authors hope incentive analysis will ultimately contribute to safer and fairer AI systems.Source: Discussion and Conclusion · p. 8

A Causality Examples

The examples show why incentive analysis requires influence diagrams that represent the environment’s full causal structure: weaker representations can have the same policy-conditioned distributions yet indicate different incentives.

A Causality Examples

  • In Figure 5, only diagram 5a represents X affecting utility U, correctly identifying positive value of control and an instrumental control incentive for X.Diagram 5b omits the causal edge X → U.Source: A Causality Examples · p. 10
  • These examples establish that the stronger causal representation is necessary to correctly model instrumental control incentives, value of control, and response incentives.The Figure 5 example covers instrumental control and value of control; Figure 6 demonstrates the response-incentive case.Source: A Causality Examples · p. 10
  • In Figure 6, only the fully causal diagram 6a shows that Y can have a response incentive or positive value of control, although both diagrams have identical joint distributions given any policy.The alternative diagram reverses the edge Y → X.Source: A Causality Examples · p. 10

B Proof Preliminaries

The proof preliminaries collect causal-model results used in later proofs, including definitions and lemmas connecting graphical structure to counterfactual and probabilistic relationships.

B Proof Preliminaries

  • The preliminaries introduce causal irrelevance of X to Y given Z, with a condition quantified over disjoint sets W.They also state that Lemma 20 is proved by induction over variables, following Galles and Pearl (1997, Lemma 12).Source: B Proof Preliminaries · p. 11
  • A lemma equates conditional expectations of Y under intervention on X when Y is d-separated from Z in the graph with incoming edges to X removed.The stated result is attributed to Pearl (2009, Thm. 3.4.1, Rule 1).Source: B Proof Preliminaries · p. 11
  • The lemmas connect graph separation with independence across compatible probability functions and state when interventions on Y leave Pr(X|Z) unchanged.The intervention result requires X to be independent of Y given Z in the graph.Source: B Proof Preliminaries · p. 11

C.1 Value of Information Criterion

This section develops graphical criteria for value of information and incentive analysis, including response and control incentives, and connects response incentives to counterfactual fairness. It characterizes that connection in terms of whether an optimal policy can be chosen unchanged by interventions on A.

C.1 Value of Information Criterion

  • The VoI criterion is proved sound and complete; when X is d-connected to a utility descendant of D conditional on the decision and other parents, some compatible model gives X VoI.Every single-decision SCIM also has an optimal policy that depends only on requisite observations.Source: §C.1 · pp. 11–12

C.2 Response Incentive Criterion

  • If X reaches D in the minimal reduction of a single-decision CID, some compatible model has a response incentive on X.The completeness construction gives optimal decisions that depend on X, while intervening to set X to 0 reduces expected utility from 1 to 0.Source: §C.2 · pp. 12–13

C.3 Value of Control Criterion

  • If X reaches a utility node in the minimal reduction and X is not D, some compatible model gives X positive value of control.The construction covers paths both with and without D.Source: §C.3 · p. 14

C.4 Instrumental Control Incentive Criterion

  • If a single-decision CID contains a path D to X to a utility node, some compatible model has an instrumental control incentive on X.The construction makes the utility depend on D along the path, so intervening on X can change utility.Source: §C.4 · p. 14

C.5 Counterfactual Fairness

  • In a single-decision SCIM, all optimal policies are counterfactually unfair with respect to sensitive attribute A if and only if A has a response incentive.Source: §C.5 · p. 14
  • Choose an arbitrary ordering of dom(D) and let the policy select the minimal action in the support of the original optimal policy; the resulting policy is optimal.The passage states that this follows because the original policy is optimal.Source: §C.5 · p. 14
  • This policy makes the same decision with and without intervention in the specified decision contexts, so there is no response incentive on A.The passages express this as Da(ε) = D(ε) under the constructed optimal policy.Source: §C.5 · p. 14
  • Conversely, if there is no response incentive, an optimal policy whose decision is unchanged under every intervention on A is counterfactually fair.Because intervention cannot change D in any setting, its conditional probabilities remain the same under any condition and decision.Source: §C.5 · p. 14
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