The Value of Information with and without Control

Gordon Hazen, Northwestern University

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Collaborators Detlof von Winterfeldt

International Institute for Applied Systems Analysis

Robert KavetElectric Power Research Institute

Mayank MohanLoyola Law School

Stephen PeckElectric Power Research Institute (emeritus)

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Motivation An environmental policy decision

e = environmental impact without policy change D = policy (Strict or No change) eD = impact under policy D V = overall value

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Motivation The option of gathering more information /

doing research Ie = information from research

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Motivation: How valuable is commissioning research when another agent acts on it?

Choose I

Choose NoI

Stakeholder considers commissioning research (Choose I vs. Choose NoI) on environmental impacts e. Industry or industry

consortium Environmental group

Federal agency implements policy D based on research results Ie.

From stakeholder point of view, Federal policy D is an uncertainty, not a decision.

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Motivation: How valuable is recommending a medical test?

Choose I

Choose NoI

Governing body considers whether (Choose I vs. Choose NoI) to include a medical test in practice guidelines for potential disease e

Practicing physician implements treatment D based on test results Ie

From governing body’s point of view, treatment D implemented by physician is an uncertainty, not a decision.

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In general, how valuable is information when another agent acts on it to produce value? Approaches to the question

1. Treating the agent’s actions as uncertain Advantage: Can make qualitative statements about

information value with few assumptions on value Disadvantage: Need a model of uncertain agent

choice

2. Stackelberg leader-follower game Disadvantage: Need to account for value differences

between information-commissioning agent and policy-making agent

Advantage: Need only assume utility maximizing agents.

This talk will focus on the first approach.

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Information and ControlNo Information Information

Control

No Contro

l

VOIC

VOINo

C

VO

CI

VO

CN

oI

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Relationships

VOINoC can be positive or negative

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Relationships Theorem 1: The three quantities VOIC, VOCI, VOCNoI are

all nonnegative. Moreover, the incremental value of information (control vs. no control) is equal to the incremental value of control (information vs. no information).

VOIC VOINoC = VOCI VOCNoI

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Additivity Theorem 2: Suppose V = V1 + V2. Then

VOINoC,V =

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Nonnegativity of VOINoC

VOINoC can be positive or negative. When is it nonnegative?

Preliminary assumption: Suppose that uncertainty e is independent of whether or

not research concerning e is conducted, that is, e is independent of the events Choose I vs. Choose NoI.

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Nonnegativity of VOINoC

Theorem 3: A sufficient condition for the nonnegativity (nonpositivity) of the value VOINoC of information on e without control is that for all values e and all alternatives d:

that is, for all e and all d: Given e, the probability of a high-valued decision (i.e.

one exceeding d in value) increases (decreases) in expectation when one chooses to obtain information.

Note: The conditioning on e is probabilistic, not informative – there is no assumption one learns e before deciding.

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Nonnegativity of VOINoC

Case: D can take on only two possible values E.g., D {Act, Don’t act}

The key condition in Theorem 3 is equivalent to: The higher-value decision under e is more likely

when one chooses to acquire information than when one chooses not to.

(Again, this language is not meant to imply one observes e before deciding.)

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A Completely Binary Model Uncertainty quantity e can be High or Low Decisions D can be Act or Don’t act Research information Ie can indicate High e or

Low e.

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Parameters

𝑝=𝑃 (𝑒=𝐻 )

𝑡=𝑃 (𝐷=𝐴𝑐𝑡∨𝑁𝑜𝐼 )

Type-1 error:

Power:

tH

tL

𝑝=𝑃 (𝑒=𝐻 )

DtLDtH

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Results: Completely binary case Assumptions

D = Act has higher value V when e = High. D = Don’t act has higher value V when e = Low

The key condition for VOINoC 0: The higher-value decision under e is more likely when one

chooses to acquire information than when one chooses not to.

Translates to Under e = Low, the policy D = Don’t act is more likely when

one chooses to acquire information than when one chooses not to.

Under e = High, the policy D = Act is more likely when one chooses to acquire information than when one chooses not to.

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Results: Completely binary case

Under e = Low, the policy D = Don’t act is more likely when one chooses to acquire information than when one chooses not to.

𝛼≤∆ 𝑡𝐿

∆ 𝑡𝐿+∆ 𝑡𝑈≤ 𝛽

Under e = High, the policy D = Act is more likely when one chooses to acquire information than when one chooses not to.

tH = the increase in probability of acting if research indicates High e.

tL = the increase in probability of not acting if research indicates Low e.

Type-1 error Power

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Generality of results One can reach these conclusions knowing

almost nothing about the value structure . The only assumptions used:

D = Act has higher value when e = High. D = Don’t act has higher value when e = Low

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Relax assumption of binary research outcome Ie.

Parameters: P(D = Act | ie, Choose I) = t(ie) P(D = Act | Choose NoI) = t0

Same assumptions D = Act has higher value V when e = High. D = Don’t act has higher value V when e = Low

Result: A sufficient condition for VOINoC 0 is that obtaining information increases the probability of acting when e = High,

and decreases the probability of acting when e = Low

that is,

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VOI Application

D. von Winterfeldt, R.Kavet, S. Peck, M. Mohan, G. Hazen, (2011) “The Value of Environmental Information When Subsequent Decisions are Uncertain”.

What is the value of commissioning research on the health effects of overhead transmission lines? Stakeholders potentially commissioning research:

Research institutes (EPRI), medical foundations, energy facility investors.

Policy makers: Federal agencies. Potential policy mandates: Undergrounding

through residential areas, compaction or split phasing elsewhere.

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Application parameters

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Partial results Value ($millions) of information with and without control

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Conclusion VOI without control:

has potentially important applications in policy venues with multiple stakeholders;

has convenient mathematical properties. Alternate approach not considered here:

Stackleberg leader/follower game. Questions?