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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?