Making Information More Valuable Mark Whitmeyer June 27 2024

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Making Information More Valuable

Mark Whitmeyer∗

June 27, 2024

Abstract

We study what changes to an agent’s decision problem increase her value for infor-

mation. We prove that information becomes more valuable if and only if the agent’s

reduced-form payoﬀin her belief becomes more convex. When the transformation

corresponds to the addition of an action, the requisite increase in convexity occurs if

and only if a simple geometric condition holds, which extends in a natural way to the

addition of multiple actions. We apply these ﬁndings to two scenarios: a monopolistic

screening problem in which the good is information and delegation with information

acquisition.

Keywords: Expected Utility, Selling Information, Information Acquisition, Delegation

JEL Classiﬁcations: D81; D82; D83

∗Arizona State University. Email: mark.whitmeyer@gmail.com. I am grateful to Simon Board, Costas

Cavounidis, Gregorio Curello, Dana Foarta, Rosemary Hopcroft, Vasudha Jain, Doron Ravid, Eddie Schlee,

Ludvig Sinander, Bruno Strulovici, Can Urgun, Tong Wang, Joseph Whitmeyer, Tom Wiseman, Renkun

Yang, Kun Zhang, and various seminar and conference audiences for their feedback. I also thank the editor,

Emir Kamenica, and four anonymous referees for their useful suggestions. This paper was formerly titled

“Flexibility and Information.”

arXiv:2210.04418v5 [econ.TH] 26 Jun 2024

When action grows unproﬁtable, gather information;

when information grows unproﬁtable, sleep.

Ursula Le Guin, The Left Hand of Darkness

1 Introduction

To a rational, expected-utility maximizing decision-maker, information is always valu-

able. However, some kinds of information are more valuable than others. Likewise, in-

formation is more valuable in certain decision problems than in others. The economics

literature, commencing with the seminal work of Blackwell (Blackwell (1951) and Black-

well (1953)), has largely focused on the ﬁrst comparison, between information structures.

In this paper, we study the second comparison, between decision problems. When can

we say that one agent values information more than another? Equivalently, suppose we

alter an agent’s decision problem. What sorts of modiﬁcations increase her value for in-

formation?

Blackwell’s way of comparing information structures is relatively detail-free. In his

ranking, information structure 1 is more valuable than information structure 2 if for any

prior held by the agent and any decision problem, the agent prefers 1 to 2. In this paper,

in specifying what it means for agent 1 to value information more than agent 2, we take

a similarly broad approach. Namely, we require that any information structure be more

valuable to agent 1 than agent 2, no matter the prior.

We begin by identifying that relative convexity distinguishes agents’ comparative love

of information. Agent 1 values information more than agent 2 if the diﬀerence in the

agents’ value functions1V1−V2is convex. Next, we turn our attention to the value func-

tions themselves. What modiﬁcations to an agent’s decision problem result in an increase

in convexity? One natural way to alter an agent’s decision problem is by allowing her an

additional action. How does increased ﬂexibility–a greater capacity to adapt her behavior

to new information–change an agent’s value for information?

1An agent’s value function, V(µ), is her maximal expected payoﬀat any belief µ∈∆(Θ), obtained by

plugging in an optimizing action. We deﬁne this object formally on page 6 in Expression 1.

In §4.2, we carry this analysis further by allowing the agent not just one but potentially

multiple additional actions. Next, in §4.3, we remove actions. We then leave the set of

actions unchanged (§4.4), but instead scale the agent’s utility. This allows us to speak to

the eﬀects of repetition and aggregate risk on the value of information. In §4.5, we reveal

that increased (or decreased) risk aversion has an ambiguous aﬀect on an agent’s value

for information.

Central to our study is the observation that a modiﬁcation to an agent’s decision prob-

lem alters her value for information through two channels. The ﬁrst is the agent’s sen-

sitivity to information–if her value for information increases (in the manner deﬁned in

this paper) it must be that she is more reactive to information. The second is the value

to the agent of distinguishing between actions. This must also increase if the agent’s

value for information is to increase. All in all, a transformation makes information more

valuable if and only if the agent becomes more sensitive to information and the value of

distinguishing between actions increases.

When the transformation to the decision problem is either the addition or subtraction

of actions, the ﬁrst channel is all-important. This is particularly stark when we modify

the agent’s decision problem by adding a single action (§4.1): the agent’s value for infor-

mation increases if and only if she becomes more sensitive to information. We uncover a

simple geometric condition necessary and suﬃcient for this to transpire and show that an

iterative version of this condition also guarantees an increase in the value of information

when multiple actions are added. Moreover, although this condition is not necessary to

make information more valuable when multiple actions are added, any failure of neces-

sity is not robust–perturbing the utilities from the new actions slightly will make it so

that the agent does not become more sensitive to information.

Perhaps unexpectedly, we discover that unless all of the remaining actions or all of

the removed actions were initially dominated, taking away actions can never lead to a

higher value for information. That is, it is only an elimination that results in a totally

new decision problem (in eﬀect) or the exact same decision problem, that can lead to an

increase in an agent’s value for information. Any removal other than these necessarily

makes the agent less sensitive to information.

1.1 Motivating Example

The question under study has signiﬁcant practical relevance. The job of a regulator is to

enact policies that modify the incentives of agents in some environment. This typically

entails the addition or subtraction of actions: there are contracts that an insurer may not

oﬀer, assets that an investment ﬁrm may not sell, and limits to how many ﬁsh a trawler

may catch.2Insurers themselves change agents’ payoﬀs by reducing their risk, ﬂattening

their payoﬀs. Firms do the opposite with their workers: bonus schemes tied to a worker’s

performance make her payoﬀsteeper and more sensitive to randomness.

Consider for instance an insurance provider dictating what treatments it will cover;

viz., what procedures a doctor may conduct. For simplicity, suppose there are three con-

ditions a patient with an injured hand may have–three states of the world. In one state,

state 0, the injury is just a sprain; in another, state 1, a bone is broken but not displaced;

and in state 2, the fracture is displaced.

Suppose ﬁrst the doctor may only oﬀer one treatment: place a cast on the hand (action

c). Accordingly, she has two possible actions, do nothing (action n), which is uniquely

optimal if the injury is just a sprain; or cast the hand, which is uniquely optimal if the

hand is broken. This decision problem is represented in Figure 1a: point (µ1,µ2)speciﬁes

the respective probabilities (beliefs) that the bone is broken but not displaced or broken

and displaced. Accordingly, the blue region is the region of probabilities in which nis

optimal; and the red region are those probabilities for which cis optimal.

Let us now consider two possible new treatments aﬀorded to the doctor. Suppose the

provider now covers surgery (action s). This is relatively high-risk and is only optimal if

the doctor is conﬁdent the bone is broken and displaced. Figure 1b represents this new

decision problem: sis optimal if and only if the doctor’s belief is in the purple region. On

the other hand, suppose the provider instead allows a conservative treatment consisting

of stretching and rehabilitating exercises (action r). This is better than nothing in the

case of a fracture, but is inferior to rest for sprains. This scenario is Figure 1c, where ris

2The verdicts that can be handed down in criminal cases are also legislated, so our results also speak to

what kinds of verdicts improve incentives for information acquisition (cf. Siegel and Strulovici (2020)).

optimal for beliefs in the black region.

Which of these new options, if either, does not dampen the doctor’s enthusiasm for

information, regardless of her prior or what that information may be? As we discover in

this paper, the answer is simple, only the former of the two potential new procedures,

surgery, makes information more valuable. Indeed, suppose that a sprain and a break

are equally likely. With only the initial two treatments to choose from, the doctor strictly

beneﬁts from any information. If we gave the doctor the conservative option, this would

clearly no longer be true: any information that doesn’t move her beliefs much is now

worthless, as the conservative treatment remains optimal at those beliefs. In contrast, the

surgery option makes information weakly more valuable.

The crucial diﬀerence between the two potential new actions is that the surgery option

is reﬁning: only the region of beliefs in which doing nothing is optimal shrinks. In

contrast, the conservative treatment partially replaces each pre-existing treatment.

2 The Model

There is a grand set of actions A. Our protagonist is a decision-maker, an agent who

initially possesses a compact set of actions A⊆A. There is an unknown state of the

world θ, which is drawn according to some full-support prior µ0from some ﬁnite set of

states Θ. Initially, the agent has some continuous utility function u:A×Θ→R.

DB(u,A,Θ)denotes the agent’s Initial Decision Problem. We are studying the eﬀect

of a transformation of the decision problem on the agent’s value for information.3To

that end, ˆ

DBˆ

u, ˆ

A,Θdenotes the agent’s Transformed Decision Problem. Here are a few

leading examples of such transformations:

Scenario 1. Becoming More Flexible. Ais ﬁnite. In ˆ

D, the agent’s utility function re-

mains unchanged, ˆ

u=u; and her new set of actions is ˆ

ABA∪Bfor some additional ﬁnite

set of actions B∈A\A.

Scenario 2. Becoming A Little More Flexible. This is the special case of the agent be-

3Equivalently, we are comparing two diﬀerent agents’ values for information.

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MakingInformationMoreValuableMarkWhitmeyer∗June27,2024AbstractWestudywhatchangestoanagent’sdecisionproblemincreasehervalueforinfor-mation.Weprovethatinformationbecomesmorevaluableifandonlyiftheagent’sreduced-formpayoffinherbeliefbecomesmoreconvex.Whenthetransformationcorrespondstotheadditionofanacti...

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