text: 4 all degrees and is Σ A . 1P plays an important role in the The information needed for the calculation in Equatoin(12) i,j i,j i,j analysis of control power estimation and of control strategy, is the degree list of network G, which is far less than the as it represents the information about the matrix A. information of adjacency matrix A. In addition, degree lists follow the power-law distribution in most social networks [2], which means the degree list could be sampled from Theorem III.1. In a BA model, the probability 1P of two i,j the network. The computational cost of such calculation is nodes i and j being connected in network G with N nodes O(N), which is much more efﬁcient than the matrix inverse and a ﬁxed degree sequence is: calculation required by Equation (7). d d 1P = i j , (9) i,j ΣN d According to Equation (12), we can see that the control k=1 k strategy, as well as the degrees of the control nodes, have a where d is the degree corresponding to node k. direct impact on the control power. Without loss of generality, k we set the preferred sign of beliefs positive. If controlled The proof of Theorem III.1 is provided in Ap- ∗ ≤ beliefs B , i c, are maximized to be 1, and the private pendix IV-A2b. i belief w has zero mean, then, as shown in Theorem III.3, the i maximization of 1cp requires the selection of a control group In order to verify the correctness of Equation (9), we need C to include nodes with highest degrees in the network G. to generate sample social networks according to the basic assumption of the BA model. The network synthesis of a BA model is discussed in Baraba´a´si(2002) [2]. The input of Theorem III.3. Consider a social network G of N nodes, { } the synthesis process is the total number of nodes N and the with degree list d , i = 1, . . . , N, a private belief w with i ∗ ≤ average number of edges attached to each new incoming node zero mean and maximized control beliefs B = 1, i c. i m. The detailed process is shown in Appendix . Suppose further that the number of control nodes, c, is ﬁxed. { } ≥ The control set C = θ , θ , . . . , θ , where d d ≤ ≤ ≤o o1 o2 oc θoi θoj Theorem III.1 has revealed the statistical properties of ad- if i j, 1 i, j N, maximizes the control power 1cp: jacency matrix A. If we take A as a random matrix, combined with the formula to calculate a converged belief, as shown C = arg max 1cp(C) (13) o in Equation (5) and Equation (6), we are able to obtain the C expected value of converged beliefs for all the nodes in the network, which is shown in Theorem III.2. The proof of Theorem III.3 is provided in Appendix IV-A2b. Theorem III.2. In a BA model, the expected value of con- (cid:54)∈ verged belief 1B ∞ of a non-controlled node i, i C, is: B. Generalized Markov Graph Model i, Σc B∗ d + ΣN wθj d In [16], the BA model is shown to be a special case of a 1B ∞ = 1 di j=1 θj θj − j=c+1 1+dθj θj , Markov Graph model [10]. A Markov Graph model is based on i, ΣNk=1dk 1 + di 1 β1 (10) the dependence between pairs of nodes. The basic assumption of the BA model depends however on the degree of nodes, which is a speciﬁc description of pairwise relationship between where w is the private belief of node θ , m is the average θi i nodes. It is thus natural to extend the BA model to the GMG number of edges per node in a network G∗ with N nodes, di model to analyse the property of adjacency matrix A. is the degree corresponding to node i, B is the controlled belief of control node j, c is the numberθj of control nodes, In a GMG Model, the probabilistic dependence of an edge ∈ ≤ is extended from the other attached edges to attached triangles. θ C for i c, and β is a constant which is smaller than i 1 As degree is used to describe the dependence on attached 1: edges, a clustering coefﬁcient, which is related to both edges and triad relational structures, is added to the description d 2 β = ΣN θk /ΣN d . (11) of dependence in a GMG model. The assumption about the 1 k=c+1 1 + d k=1 k θk probability of a node i attached by a new edge in a GMG model then becomes: The proof of Theorem III.2 is in Appendix IV-A2b. Plugging Equation (10) into Equation (3), we obtain the ∂di ∼ d (1 + γ )α, (14) control power: ∂t i i 1cp = ΣN (di(Σcj=1Bθ∗jdθj + ΣNj=c+1−1+wdθjθj dθj) − w ) coewfﬁhecrieentdiofisnotdhee id,eganrede αo,fwnhoidceh iis, cγailleisd tthhee cclluusstteerriinngg i=1 NΣN d (1 + d )(1 β ) i weight, is determined by the property of the network G. k=1 k i 1 (12) The range of the clustering coefﬁcient lies in [0, 1]. If there is no triangle attached to a node i, the clustering coefﬁcient ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: all

text: degrees

text: and

text: is

text: Σ

text: A

text: plays

text: an

text: important

text: role

text: in

text: the

text: The

text: information

text: needed

text: for

text: the

text: calculation

text: in

text: i,j

text: i,j

text: i,j

text: analysis

text: of

text: control

text: power

text: estimation

text: and

text: of

text: control

text: strategy,

text: is

text: the

text: degree

text: list

text: of

text: network

text: G,

text: which

text: is

text: far

text: less

text: than

text: the

text: as

text: it

text: represents

text: the

text: information

text: about

text: the

text: matrix

text: A.

text: information

text: of

text: adjacency

text: matrix

text: A.

text: In

text: addition,

text: degree

text: lists

text: follow

text: the

text: power-law

text: distribution

text: in

text: most

text: social

text: networks

text: which

text: means

text: the

text: degree

text: list

text: could

text: be

text: sampled

text: from

text: Theorem

text: In

text: a

text: BA

text: model,

text: the

text: probability

text: of

text: two

text: i,j

text: the

text: network.

text: The

text: computational

text: cost

text: of

text: such

text: calculation

text: is

text: nodes

text: i

text: and

text: j

text: being

text: connected

text: in

text: network

text: G

text: with

text: N

text: nodes

text: O(N),

text: which

text: is

text: much

text: more

text: efﬁcient

text: than

text: the

text: matrix

text: inverse

text: and

text: a

text: ﬁxed

text: degree

text: sequence

text: is:

text: calculation

text: required

text: by

text: Equation

text: d

text: d

text: i

text: j

text: i,j

text: ΣN

text: d

text: According

text: to

text: Equation

text: we

text: can

text: see

text: that

text: the

text: control

text: k

text: strategy,

text: as

text: well

text: as

text: the

text: degrees

text: of

text: the

text: control

text: nodes,

text: have

text: a

text: where

text: d

text: is

text: the

text: degree

text: corresponding

text: to

text: node

text: k.

text: direct

text: impact

text: on

text: the

text: control

text: power.

text: Without

text: loss

text: of

text: generality,

text: k

text: we

text: set

text: the

text: preferred

text: sign

text: of

text: beliefs

text: positive.

text: If

text: controlled

text: The

text: proof

text: of

text: Theorem

text: is

text: provided

text: in

text: Ap-

text: beliefs

text: B

text: i

text: c,

text: are

text: maximized

text: to

text: be

text: and

text: the

text: private

text: pendix

text: i

text: belief

text: w

text: has

text: zero

text: mean,

text: then,

text: as

text: shown

text: in

text: Theorem

text: the

text: i

text: maximization

text: of

text: requires

text: the

text: selection

text: of

text: a

text: control

text: group

text: In

text: order

text: to

text: verify

text: the

text: correctness

text: of

text: Equation

text: we

text: need

text: C

text: to

text: include

text: nodes

text: with

text: highest

text: degrees

text: in

text: the

text: network

text: G.

text: to

text: generate

text: sample

text: social

text: networks

text: according

text: to

text: the

text: basic

text: assumption

text: of

text: the

text: BA

text: model.

text: The

text: network

text: synthesis

text: of

text: a

text: BA

text: model

text: is

text: discussed

text: in

text: The

text: input

text: of

text: Theorem

text: Consider

text: a

text: social

text: network

text: G

text: of

text: N

text: nodes,

text: the

text: synthesis

text: process

text: is

text: the

text: total

text: number

text: of

text: nodes

text: N

text: and

text: the

text: with

text: degree

text: list

text: d

text: i

text: N,

text: a

text: private

text: belief

text: w

text: with

text: i

text: average

text: number

text: of

text: edges

text: attached

text: to

text: each

text: new

text: incoming

text: node

text: zero

text: mean

text: and

text: maximized

text: control

text: beliefs

text: B

text: i

text: c.

text: i

text: m.

text: The

text: detailed

text: process

text: is

text: shown

text: in

text: Appendix

text: Suppose

text: further

text: that

text: the

text: number

text: of

text: control

text: nodes,

text: c,

text: is

text: ﬁxed.

text: The

text: control

text: set

text: C

text: θ

text: θ

text: θ

text: where

text: d

text: d

text: ≤o

text: oc

text: θoi

text: θoj

text: Theorem

text: has

text: revealed

text: the

text: statistical

text: properties

text: of

text: ad-

text: if

text: i

text: j,

text: i,

text: j

text: N,

text: maximizes

text: the

text: control

text: power

text: jacency

text: matrix

text: A.

text: If

text: we

text: take

text: A

text: as

text: a

text: random

text: matrix,

text: combined

text: with

text: the

text: formula

text: to

text: calculate

text: a

text: converged

text: belief,

text: as

text: shown

text: C

text: arg

text: max

text: o

text: in

text: Equation

text: and

text: Equation

text: we

text: are

text: able

text: to

text: obtain

text: the

text: C

text: expected

text: value

text: of

text: converged

text: beliefs

text: for

text: all

text: the

text: nodes

text: in

text: the

text: network,

text: which

text: is

text: shown

text: in

text: Theorem

text: The

text: proof

text: of

text: Theorem

text: is

text: provided

text: in

text: Appendix

text: Theorem

text: In

text: a

text: BA

text: model,

text: the

text: expected

text: value

text: of

text: con-

text: verged

text: belief

text: of

text: a

text: non-controlled

text: node

text: i,

text: i

text: C,

text: is:

text: B.

text: Generalized

text: Markov

text: Graph

text: Model

text: i,

text: Σc

text: B∗

text: d

text: ΣN

text: wθj

text: d

text: In

text: the

text: BA

text: model

text: is

text: shown

text: to

text: be

text: a

text: special

text: case

text: of

text: a

text: di

text: θj

text: θj

text: θj

text: Markov

text: Graph

text: model

text: A

text: Markov

text: Graph

text: model

text: is

text: based

text: on

text: i,

text: di

text: β1

text: the

text: dependence

text: between

text: pairs

text: of

text: nodes.

text: The

text: basic

text: assumption

text: of

text: the

text: BA

text: model

text: depends

text: however

text: on

text: the

text: degree

text: of

text: nodes,

text: which

text: is

text: a

text: speciﬁc

text: description

text: of

text: pairwise

text: relationship

text: between

text: where

text: w

text: is

text: the

text: private

text: belief

text: of

text: node

text: θ

text: m

text: is

text: the

text: average

text: θi

text: i

text: nodes.

text: It

text: is

text: thus

text: natural

text: to

text: extend

text: the

text: BA

text: model

text: to

text: the

text: GMG

text: number

text: of

text: edges

text: per

text: node

text: in

text: a

text: network

text: G∗

text: with

text: N

text: nodes,

text: di

text: model

text: to

text: analyse

text: the

text: property

text: of

text: adjacency

text: matrix

text: A.

text: is

text: the

text: degree

text: corresponding

text: to

text: node

text: i,

text: B

text: is

text: the

text: controlled

text: belief

text: of

text: control

text: node

text: j,

text: c

text: is

text: the

text: numberθj

text: of

text: control

text: nodes,

text: In

text: a

text: GMG

text: Model,

text: the

text: probabilistic

text: dependence

text: of

text: an

text: edge

text: is

text: extended

text: from

text: the

text: other

text: attached

text: edges

text: to

text: attached

text: triangles.

text: θ

text: C

text: for

text: i

text: c,

text: and

text: β

text: is

text: a

text: constant

text: which

text: is

text: smaller

text: than

text: i

text: As

text: degree

text: is

text: used

text: to

text: describe

text: the

text: dependence

text: on

text: attached

text: edges,

text: a

text: clustering

text: coefﬁcient,

text: which

text: is

text: related

text: to

text: both

text: edges

text: and

text: triad

text: relational

text: structures,

text: is

text: added

text: to

text: the

text: description

text: d

text: β

text: ΣN

text: θk

text: /ΣN

text: d

text: of

text: dependence

text: in

text: a

text: GMG

text: model.

text: The

text: assumption

text: about

text: the

text: d

text: k

text: θk

text: probability

text: of

text: a

text: node

text: i

text: attached

text: by

text: a

text: new

text: edge

text: in

text: a

text: GMG

text: model

text: then

text: becomes:

text: The

text: proof

text: of

text: Theorem

text: is

text: in

text: Appendix

text: Plugging

text: Equation

text: into

text: Equation

text: we

text: obtain

text: the

text: ∂di

text: d

text: γ

text: )α,

text: control

text: power:

text: ∂t

text: i

text: i

text: ΣN

text: dθj)

text: w

text: coewfﬁhecrieentdiofisnotdhee

text: id,eganrede

text: αo,fwnhoidceh

text: iis,

text: cγailleisd

text: tthhee

text: cclluusstteerriinngg

text: NΣN

text: d

text: d

text: β

text: i

text: weight,

text: is

text: determined

text: by

text: the

text: property

text: of

text: the

text: network

text: G.

text: k

text: i

text: The

text: range

text: of

text: the

text: clustering

text: coefﬁcient

text: lies

text: in

text: If

text: there

text: is

text: no

text: triangle

text: attached

text: to

text: a

text: node

text: i,

text: the

text: clustering

text: coefﬁcient

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: ##LTLine##

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text: ##LTLine##


text: Precision Asteroseismology (cid:13) Proceedings IAU Symposium No. 301, 2014 c 2014 International Astronomical Union J.A. Guzik, W.J. Chaplin, G. Handler & A. Pigulski, eds. DOI: 00.0000/X000000000000000X Pulsating variables from the OGLE and Araucaria projects G. Pietrzyn´ski1,2 1 Warsaw University Observatory, Al. Ujazdowskie 4, 00-478 Warszawa, Poland email: pietrzyn@astrouw.edu.pl 2Departamento de Astronom´ıa, Universidad de Concepcio´n, Casilla 160-C, Concepcio´n, Chile Abstract. We present some results of long term studies of pulsating stars conducted in the course of the OGLE and Araucaria projects. In particular very scarce eclipsing binaries con- taining pulsating stars are discussed. Such systems provide a unique opportunity to improve calibration of the cosmic distance scale and to better calibrate stellar evolutionary models. 1. OGLE and ARAUCARIA projects Optical Gravitational Lensing Experiment (OGLE) is one of the biggest astronomical surveys. This project has been in operation over the last 21 years. With the current instrumental setup, the OGLE team is capable to observe about one billion stars ev- ery night. The observations are mainly conducted in the Milky Way (bulge and disc) and Magellanic Clouds. Based on the collected data, an enormous amount of precise light curves for basically all kinds of pulsating stars were already published: Cepheids (Soszyn´ski et al. 2008, 2010a), RR Lyrae (Soszyn´ski et al. 2010b, 2011a), long period vari- ables (Soszyn´ski et al. 2011b, 2013), etc., and several new catalogs are in preparation. Thanks to an exquisite statistics and high quality of the data, many extremely scarce and very interesting objects have been also discovered, including very good candidates for pulsating stars in eclipsing binary systems (Soszyn´ski et al. 2008, 2011a) and a unique sample of eclipsing binary systems composed of clump giants (Graczyk et al. 2011). Araucaria project. The main goal of this project is to signiﬁcantly improve the calibra- tion of the cosmic distance scale based on observations of several distance indicators in nearby galaxies. As a ﬁrst step, we performed an optical survey of nine nearby galaxies discovering about 700 Cepheids (Pietrzyn´ski et al. 2002, 2004, 2006, 2007, 2010a). We also performed infrared observations of discovered Cepheids (Gieren et al. 2005b, 2006, 2009, 2013; Pietrzyn´ski et al. 2006), and RR Lyrae stars selected from the OGLE catalogs in the LMC, SMC and Sculptor galaxies (Pietrzyn´ski et al. 2008, Szewczyk et al. 2008, 2009). Recently, a very important part of the Araucaria project is related to detailed studies of eclipsing binary systems discovered by the OGLE project, which have very large potential for precise and accurate distance determination and also for improving our knowledge of basic physics of pulsating stars. During my talk I will focus on some results obtained for some of these systems. 2. Setting the zero point for P-L relation of pulsating stars The distance to the LMC is widely adopted as the zero point in the calibration of the cosmic distance scale. Therefore, a precise and accurate distance determination to this galaxy is paramount for astrophysics in general. Detached eclipsing double-lined spectroscopic binaries oﬀer a unique opportunity to measure distances directly (e.g. Kruszewski and Semeniuk 1999). Recently, we measured 1 ##LTLine## ##LTLine## ##LTLine##

text: Precision

label: paragraph

text: Asteroseismology

text: Proceedings

text: IAU

text: Symposium

text: No.

text: c

text: International

text: Astronomical

text: Union

text: J.A.

text: Guzik,

text: W.J.

text: Chaplin,

text: G.

text: Handler

text: A.

text: Pigulski,

text: eds.

text: DOI:

text: Pulsating

text: variables

text: from

text: the

text: OGLE

text: and

text: Araucaria

text: projects

text: G.

label: author

text: Warsaw

text: University

text: Observatory,

text: Al.

text: Ujazdowskie

text: Warszawa,

text: Poland

text: email:

text: pietrzyn@astrouw.edu.pl

text: de

text: Astronom´ıa,

text: Universidad

text: de

text: Concepcio´n,

text: Casilla

text: Concepcio´n,

text: Chile

text: Abstract.

text: We

label: abstract

text: present

text: some

text: results

text: of

text: long

text: term

text: studies

text: of

text: pulsating

text: stars

text: conducted

text: in

text: the

text: course

text: of

text: the

text: OGLE

text: and

text: Araucaria

text: projects.

text: In

text: particular

text: very

text: scarce

text: eclipsing

text: binaries

text: con-

text: taining

text: pulsating

text: stars

text: are

text: discussed.

text: Such

text: systems

text: provide

text: a

text: unique

text: opportunity

text: to

text: improve

text: calibration

text: of

text: the

text: cosmic

text: distance

text: scale

text: and

text: to

text: better

text: calibrate

text: stellar

text: evolutionary

text: models.

text: OGLE

text: and

text: ARAUCARIA

text: projects

text: Optical

text: Gravitational

text: Lensing

text: Experiment

text: (OGLE)

text: is

text: one

text: of

text: the

text: biggest

text: astronomical

text: surveys.

text: This

text: project

text: has

text: been

text: in

text: operation

text: over

text: the

text: last

text: years.

text: With

text: the

text: current

text: instrumental

text: setup,

text: the

text: OGLE

text: team

text: is

text: capable

text: to

text: observe

text: about

text: one

text: billion

text: stars

text: ev-

text: ery

text: night.

text: The

text: observations

text: are

text: mainly

text: conducted

text: in

text: the

text: Milky

text: Way

text: (bulge

text: and

text: disc)

text: and

text: Magellanic

text: Clouds.

text: Based

text: on

text: the

text: collected

text: data,

text: an

text: enormous

text: amount

text: of

text: precise

text: light

text: curves

text: for

text: basically

text: all

text: kinds

text: of

text: pulsating

text: stars

text: were

text: already

text: published:

text: Cepheids

text: (Soszyn´ski

text: et

text: al.

text: RR

text: Lyrae

text: (Soszyn´ski

text: et

text: al.

text: long

text: period

text: vari-

text: ables

text: (Soszyn´ski

text: et

text: al.

text: etc.,

text: and

text: several

text: new

text: catalogs

text: are

text: in

text: preparation.

text: Thanks

text: to

text: an

text: exquisite

text: statistics

text: and

text: high

text: quality

text: of

text: the

text: data,

text: many

text: extremely

text: scarce

text: and

text: very

text: interesting

text: objects

text: have

text: been

text: also

text: discovered,

text: including

text: very

text: good

text: candidates

text: for

text: pulsating

text: stars

text: in

text: eclipsing

text: binary

text: systems

text: (Soszyn´ski

text: et

text: al.

text: and

text: a

text: unique

text: sample

text: of

text: eclipsing

text: binary

text: systems

text: composed

text: of

text: clump

text: giants

text: (Graczyk

text: et

text: al.

text: Araucaria

text: project.

text: The

text: main

text: goal

text: of

text: this

text: project

text: is

text: to

text: signiﬁcantly

text: improve

text: the

text: calibra-

text: tion

text: of

text: the

text: cosmic

text: distance

text: scale

text: based

text: on

text: observations

text: of

text: several

text: distance

text: indicators

text: in

text: nearby

text: galaxies.

text: As

text: a

text: ﬁrst

text: step,

text: we

text: performed

text: an

text: optical

text: survey

text: of

text: nine

text: nearby

text: galaxies

text: discovering

text: about

text: Cepheids

text: (Pietrzyn´ski

text: et

text: al.

text: We

text: also

text: performed

text: infrared

text: observations

text: of

text: discovered

text: Cepheids

text: (Gieren

text: et

text: al.

text: Pietrzyn´ski

text: et

text: al.

text: and

text: RR

text: Lyrae

text: stars

text: selected

text: from

text: the

text: OGLE

text: catalogs

text: in

text: the

text: LMC,

text: SMC

text: and

text: Sculptor

text: galaxies

text: (Pietrzyn´ski

text: et

text: al.

text: Szewczyk

text: et

text: al.

text: Recently,

text: a

text: very

text: important

text: part

text: of

text: the

text: Araucaria

text: project

text: is

text: related

text: to

text: detailed

text: studies

text: of

text: eclipsing

text: binary

text: systems

text: discovered

text: by

text: the

text: OGLE

text: project,

text: which

text: have

text: very

text: large

text: potential

text: for

text: precise

text: and

text: accurate

text: distance

text: determination

text: and

text: also

text: for

text: improving

text: our

text: knowledge

text: of

text: basic

text: physics

text: of

text: pulsating

text: stars.

text: During

text: my

text: talk

text: I

text: will

text: focus

text: on

text: some

text: results

text: obtained

text: for

text: some

text: of

text: these

text: systems.

text: Setting

text: the

text: zero

text: point

text: for

text: P-L

text: relation

text: of

text: pulsating

text: stars

text: The

text: distance

text: to

text: the

text: LMC

text: is

text: widely

text: adopted

text: as

text: the

text: zero

text: point

text: in

text: the

text: calibration

text: of

text: the

text: cosmic

text: distance

text: scale.

text: Therefore,

text: a

text: precise

text: and

text: accurate

text: distance

text: determination

text: to

text: this

text: galaxy

text: is

text: paramount

text: for

text: astrophysics

text: in

text: general.

text: Detached

text: eclipsing

text: double-lined

text: spectroscopic

text: binaries

text: oﬀer

text: a

text: unique

text: opportunity

text: to

text: measure

text: distances

text: directly

text: (e.g.

text: Kruszewski

text: and

text: Semeniuk

text: Recently,

text: we

text: measured

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: ∗ The Empirical Implications of Privacy-Aware Choice † ‡ § Rachel Cummings Federico Echenique Adam Wierman May 26, 2020 Abstract This paper initiates the study of the testable implications of choice data in settings where agents have privacy preferences. We adapt the standard conceptualization of consumer choice theory to a situation where the consumer is aware of, and has preferences over, the information revealed by her choices. The main message of the paper is that little can be inferred about consumers’ preferences once we introduce the possibility that the consumer has concerns about privacy. This holds even when consumers’ privacy preferences are assumed to be monotonic and separable. This motivates the consideration of stronger assumptions and, to that end, we introduce an additive model for privacy preferences that does have testable implications. 1 Introduction The purpose of this paper is to study what an observer can learn about a consumer’s preferences and behavior when the consumer has concerns for her privacy and knows that she is being observed. The basic message of our results is that very little can be learned without strong assumptions on the form of the consumer’s privacy preferences. To motivate the problem under study, consider the following story. Alice makes choices on the internet. She chooses which websites to visit, what books to buy, which hotels to reserve, and which newspapers to read. She knows, however, that she is being watched. An external agent, “Big Brother” (BB), monitors her choices. BB could be a private ﬁrm like Google, or a government agency like the NSA. As a result of being watched, Alice is concerned for her privacy; and this concern aﬀects her behavior. Alice has deﬁnitive preferences over the things she chooses among. For example, given three political blogs a, b, and c, she may prefer to follow a. But, BB will observe such a choice, and infer that she prefers a over b and c. This is uncomfortable to Alice, because her preferences are shaped by her political views, and she does not like BB to know her ∗ We thank Jeﬀ Ely, and especially Katrina Ligett, for useful comments. This work was supported by NSF grants 1101470, 1216006, and CNS-1254169, as well as BFS grant 2012348 and the Charles Lee Powell Foundation. † Electrical Engineering and Computer Science, Northwestern University. rachelc@u.northwestern.edu ‡ Humanities and Social Sciences, California Institute of Technology. fede@hss.caltech.edu § Computing and Mathematical Sciences, California Institute of Technology. adamw@caltech.edu 1 ##LTLine##

label: title

text: The

text: Empirical

text: Implications

text: of

text: Privacy-Aware

text: Choice

text: Rachel

text: Cummings

text: Federico

text: Echenique

text: Adam

text: Wierman

text: May

text: Abstract

text: This

text: paper

text: initiates

text: the

text: study

text: of

text: the

text: testable

text: implications

text: of

text: choice

text: data

text: in

text: settings

text: where

text: agents

text: have

text: privacy

text: preferences.

text: We

text: adapt

text: the

text: standard

text: conceptualization

text: of

text: consumer

text: choice

text: theory

text: to

text: a

text: situation

text: where

text: the

text: consumer

text: is

text: aware

text: of,

text: and

text: has

text: preferences

text: over,

text: the

text: information

text: revealed

text: by

text: her

text: choices.

text: The

text: main

text: message

text: of

text: the

text: paper

text: is

text: that

text: little

text: can

text: be

text: inferred

text: about

text: consumers’

text: preferences

text: once

text: we

text: introduce

text: the

text: possibility

text: that

text: the

text: consumer

text: has

text: concerns

text: about

text: privacy.

text: This

text: holds

text: even

text: when

text: consumers’

text: privacy

text: preferences

text: are

text: assumed

text: to

text: be

text: monotonic

text: and

text: separable.

text: This

text: motivates

text: the

text: consideration

text: of

text: stronger

text: assumptions

text: and,

text: to

text: that

text: end,

text: we

text: introduce

text: an

text: additive

text: model

text: for

text: privacy

text: preferences

text: that

text: does

text: have

text: testable

text: implications.

text: Introduction

text: The

text: purpose

text: of

text: this

text: paper

text: is

text: to

text: study

text: what

text: an

text: observer

text: can

text: learn

text: about

text: a

text: consumer’s

text: preferences

text: and

text: behavior

text: when

text: the

text: consumer

text: has

text: concerns

text: for

text: her

text: privacy

text: and

text: knows

text: that

text: she

text: is

text: being

text: observed.

text: The

text: basic

text: message

text: of

text: our

text: results

text: is

text: that

text: very

text: little

text: can

text: be

text: learned

text: without

text: strong

text: assumptions

text: on

text: the

text: form

text: of

text: the

text: consumer’s

text: privacy

text: preferences.

text: To

text: motivate

text: the

text: problem

text: under

text: study,

text: consider

text: the

text: following

text: story.

text: Alice

text: makes

text: choices

text: on

text: the

text: internet.

text: She

text: chooses

text: which

text: websites

text: to

text: visit,

text: what

text: books

text: to

text: buy,

text: which

text: hotels

text: to

text: reserve,

text: and

text: which

text: newspapers

text: to

text: read.

text: She

text: knows,

text: however,

text: that

text: she

text: is

text: being

text: watched.

text: An

text: external

text: agent,

text: “Big

text: Brother”

text: (BB),

text: monitors

text: her

text: choices.

text: BB

text: could

text: be

text: a

text: private

text: ﬁrm

text: like

text: Google,

text: or

text: a

text: government

text: agency

text: like

text: the

text: NSA.

text: As

text: a

text: result

text: of

text: being

text: watched,

text: Alice

text: is

text: concerned

text: for

text: her

text: privacy;

text: and

text: this

text: concern

text: aﬀects

text: her

text: behavior.

text: Alice

text: has

text: deﬁnitive

text: preferences

text: over

text: the

text: things

text: she

text: chooses

text: among.

text: For

text: example,

text: given

text: three

text: political

text: blogs

text: a,

text: b,

text: and

text: c,

text: she

text: may

text: prefer

text: to

text: follow

text: a.

text: But,

text: BB

text: will

text: observe

text: such

text: a

text: choice,

text: and

text: infer

text: that

text: she

text: prefers

text: a

text: over

text: b

text: and

text: c.

text: This

text: is

text: uncomfortable

text: to

text: Alice,

text: because

text: her

text: preferences

text: are

text: shaped

text: by

text: her

text: political

text: views,

text: and

text: she

text: does

text: not

text: like

text: BB

text: to

text: know

text: her

text: We

text: thank

text: Jeﬀ

text: Ely,

text: and

text: especially

text: Katrina

text: Ligett,

text: for

text: useful

text: comments.

text: This

text: work

text: was

text: supported

text: by

text: NSF

text: grants

text: and

text: as

text: well

text: as

text: BFS

text: grant

text: and

text: the

text: Charles

text: Lee

text: Powell

text: Foundation.

text: Electrical

text: Engineering

text: and

text: Computer

text: Science,

text: Northwestern

text: University.

text: rachelc@u.northwestern.edu

text: Humanities

text: and

text: Social

text: Sciences,

text: California

text: Institute

text: of

text: Technology.

text: fede@hss.caltech.edu

text: Computing

text: and

text: Mathematical

text: Sciences,

text: California

text: Institute

text: of

text: Technology.

text: adamw@caltech.edu

text: ##LTLine##


text: views or her preferences. As a result, she may be reluctant to choose a. She may choose b instead because she is more comfortable with BB believing that she ranks b over a and c.1 Now, the question becomes, given observations of Alice’s behavior, what can we learn about her preferences? We might conjecture that her behavior must satisfy some kind of rationality axiom, or that one could back out, or reverse-engineer, her preferences from her behavior. After all, Alice is a fully rational consumer (agent), meaning that she maximizes a utility function (or a transitive preference relation). She has a well-deﬁned preference over the objects of choice, meaning that if she could ﬁx what BB learns about her—if what BB learns about her were independent from her choices—then she would choose her favorite object. Further, Alice’s preferences over privacy likely satisfy particular structural properties. For example, she has well-deﬁned preferences over the objects of choice, and she cares about the preference revealed by her choices: she always prefers revealing less to revealing more. In economics, preferences of this form are called separable and monotonic; and such preferences normally place strong restrictions on agents’ behavior. However, contrary to the above discussion, the results in this paper prove that nothing can be inferred about Alice’s preferences once we introduce the possibility that she has concerns about privacy. No matter what her behavior, it is compatible with some concerns over privacy, i.e., she always has an “alibi” that can explain her choices as a consequence of privacy concerns. The strongest version of this result is that all possible behaviors on the part of Alice are compatible with all possible preferences that Alice may have over objects: postulate some arbitrary behavior for Alice, and some arbitrary preference over objects, and the two will always be compatible. So BB’s objective is hopeless. He can never learn anything about Alice’s true pref- erences over political blogs, or over any other objects of choice. If BB tries to estimate preferences from some given choices by Alice, he ﬁnds that all preferences could be used to explain her choices. He cannot narrow down the set of preferences Alice might have, no matter what the observed behavior. The result continues to hold if BB adversarially sets up scenarios for Alice to choose from. That is, even if BB oﬀers Alice menus of choices so as to maximize what he can learn from her behavior, the result is still that nothing can be learned. The results in this paper have a variety of implications. First, they motivate the use of speciﬁc parametric models of preferences over privacy. Our main result makes strong qualitative assumptions about preferences (separability, monotonicity). Given that such assumptions lack empirical bite, one should arguably turn to stronger assumptions yet. The paper proposes an additive utility function that depends on the chosen object and on what is revealed by the consumer’s choices. If Alice chooses x then she obtains a utility u(x) and a “penalty” v(x, y) for not choosing y, for all non-chosen y, as she reveals to BB that she ranks x over y. This additive model does have restrictions for the consumer’s behavior, and could be estimated given data on Alice’s choices. The model is methodologically close to models used in economics to explain individual choices, and could be econometrically estimated using standard techniques. 1Like Alice, 85% of adult internet users have take steps to avoid surveillance by other people or orga- nizations, see [8]. 2 ##LTLine##

text: views

text: or

text: her

text: preferences.

text: As

text: a

text: result,

text: she

text: may

text: be

text: reluctant

text: to

text: choose

text: a.

text: She

text: may

text: choose

text: b

text: instead

text: because

text: she

text: is

text: more

text: comfortable

text: with

text: BB

text: believing

text: that

text: she

text: ranks

text: b

text: over

text: a

text: and

text: Now,

text: the

text: question

text: becomes,

text: given

text: observations

text: of

text: Alice’s

text: behavior,

text: what

text: can

text: we

text: learn

text: about

text: her

text: preferences?

text: We

text: might

text: conjecture

text: that

text: her

text: behavior

text: must

text: satisfy

text: some

text: kind

text: of

text: rationality

text: axiom,

text: or

text: that

text: one

text: could

text: back

text: out,

text: or

text: reverse-engineer,

text: her

text: preferences

text: from

text: her

text: behavior.

text: After

text: all,

text: Alice

text: is

text: a

text: fully

text: rational

text: consumer

text: (agent),

text: meaning

text: that

text: she

text: maximizes

text: a

text: utility

text: function

text: (or

text: a

text: transitive

text: preference

text: relation).

text: She

text: has

text: a

text: well-deﬁned

text: preference

text: over

text: the

text: objects

text: of

text: choice,

text: meaning

text: that

text: if

text: she

text: could

text: ﬁx

text: what

text: BB

text: learns

text: about

text: her—if

text: what

text: BB

text: learns

text: about

text: her

text: were

text: independent

text: from

text: her

text: choices—then

text: she

text: would

text: choose

text: her

text: favorite

text: object.

text: Further,

text: Alice’s

text: preferences

text: over

text: privacy

text: likely

text: satisfy

text: particular

text: structural

text: properties.

text: For

text: example,

text: she

text: has

text: well-deﬁned

text: preferences

text: over

text: the

text: objects

text: of

text: choice,

text: and

text: she

text: cares

text: about

text: the

text: preference

text: revealed

text: by

text: her

text: choices:

text: she

text: always

text: prefers

text: revealing

text: less

text: to

text: revealing

text: more.

text: In

text: economics,

text: preferences

text: of

text: this

text: form

text: are

text: called

text: separable

text: and

text: monotonic;

text: and

text: such

text: preferences

text: normally

text: place

text: strong

text: restrictions

text: on

text: agents’

text: behavior.

text: However,

text: contrary

text: to

text: the

text: above

text: discussion,

text: the

text: results

text: in

text: this

text: paper

text: prove

text: that

text: nothing

text: can

text: be

text: inferred

text: about

text: Alice’s

text: preferences

text: once

text: we

text: introduce

text: the

text: possibility

text: that

text: she

text: has

text: concerns

text: about

text: privacy.

text: No

text: matter

text: what

text: her

text: behavior,

text: it

text: is

text: compatible

text: with

text: some

text: concerns

text: over

text: privacy,

text: i.e.,

text: she

text: always

text: has

text: an

text: “alibi”

text: that

text: can

text: explain

text: her

text: choices

text: as

text: a

text: consequence

text: of

text: privacy

text: concerns.

text: The

text: strongest

text: version

text: of

text: this

text: result

text: is

text: that

text: all

text: possible

text: behaviors

text: on

text: the

text: part

text: of

text: Alice

text: are

text: compatible

text: with

text: all

text: possible

text: preferences

text: that

text: Alice

text: may

text: have

text: over

text: objects:

text: postulate

text: some

text: arbitrary

text: behavior

text: for

text: Alice,

text: and

text: some

text: arbitrary

text: preference

text: over

text: objects,

text: and

text: the

text: two

text: will

text: always

text: be

text: compatible.

text: So

text: BB’s

text: objective

text: is

text: hopeless.

text: He

text: can

text: never

text: learn

text: anything

text: about

text: Alice’s

text: true

text: pref-

text: erences

text: over

text: political

text: blogs,

text: or

text: over

text: any

text: other

text: objects

text: of

text: choice.

text: If

text: BB

text: tries

text: to

text: estimate

text: preferences

text: from

text: some

text: given

text: choices

text: by

text: Alice,

text: he

text: ﬁnds

text: that

text: all

text: preferences

text: could

text: be

text: used

text: to

text: explain

text: her

text: choices.

text: He

text: cannot

text: narrow

text: down

text: the

text: set

text: of

text: preferences

text: Alice

text: might

text: have,

text: no

text: matter

text: what

text: the

text: observed

text: behavior.

text: The

text: result

text: continues

text: to

text: hold

text: if

text: BB

text: adversarially

text: sets

text: up

text: scenarios

text: for

text: Alice

text: to

text: choose

text: from.

text: That

text: is,

text: even

text: if

text: BB

text: oﬀers

text: Alice

text: menus

text: of

text: choices

text: so

text: as

text: to

text: maximize

text: what

text: he

text: can

text: learn

text: from

text: her

text: behavior,

text: the

text: result

text: is

text: still

text: that

text: nothing

text: can

text: be

text: learned.

text: The

text: results

text: in

text: this

text: paper

text: have

text: a

text: variety

text: of

text: implications.

text: First,

text: they

text: motivate

text: the

text: use

text: of

text: speciﬁc

text: parametric

text: models

text: of

text: preferences

text: over

text: privacy.

text: Our

text: main

text: result

text: makes

text: strong

text: qualitative

text: assumptions

text: about

text: preferences

text: (separability,

text: monotonicity).

text: Given

text: that

text: such

text: assumptions

text: lack

text: empirical

text: bite,

text: one

text: should

text: arguably

text: turn

text: to

text: stronger

text: assumptions

text: yet.

text: The

text: paper

text: proposes

text: an

text: additive

text: utility

text: function

text: that

text: depends

text: on

text: the

text: chosen

text: object

text: and

text: on

text: what

text: is

text: revealed

text: by

text: the

text: consumer’s

text: choices.

text: If

text: Alice

text: chooses

text: x

text: then

text: she

text: obtains

text: a

text: utility

text: u(x)

text: and

text: a

text: “penalty”

text: v(x,

text: y)

text: for

text: not

text: choosing

text: y,

text: for

text: all

text: non-chosen

text: y,

text: as

text: she

text: reveals

text: to

text: BB

text: that

text: she

text: ranks

text: x

text: over

text: y.

text: This

text: additive

text: model

text: does

text: have

text: restrictions

text: for

text: the

text: consumer’s

text: behavior,

text: and

text: could

text: be

text: estimated

text: given

text: data

text: on

text: Alice’s

text: choices.

text: The

text: model

text: is

text: methodologically

text: close

text: to

text: models

text: used

text: in

text: economics

text: to

text: explain

text: individual

text: choices,

text: and

text: could

text: be

text: econometrically

text: estimated

text: using

text: standard

text: techniques.

text: Alice,

text: of

text: adult

text: internet

text: users

text: have

text: take

text: steps

text: to

text: avoid

text: surveillance

text: by

text: other

text: people

text: or

text: orga-

text: nizations,

text: see

text: ##LTLine##


text: A Deﬁnition 1. A choice problem (X, , c) is rationalizable (via privacy preferences) (cid:23) ∈ \ { } if there is a privacy preference such that if x = c(A) and y A x then { ∈ \ { }} (cid:31) { ∈ \ { }} (x, (x, z) : z A x ) (y, (y, z) : z A y ), ∈ A (cid:23) A for all A . In this case, we say that rationalizes (X, , c). Thus, a choice problem is rationalizable when there exists a privacy preference that “explains” the data, i.e., when there exists a privacy preference for which the observed choices are maximal. 3 The rationalizability of privacy-aware choice In this section, we present our main results, which characterize when choice data from privacy-aware consumers is rationalizable. Our results focus on the testable implications of structural assumptions about the form of the privacy preferences of the consumer. While a consumer’s preferences may, in general, be a complex combination of preferences over the choices and revealed preferences, there are some natural properties that one may expect to hold in many situations. In particular, we focus on three increasingly strong structural assumptions in the following three subsections: monotonicity, separability, and additivity. 3.1 Monotone privacy preferences A natural assumption on privacy preferences is “monotonicity”, i.e., the idea that re- vealing less information is always better. Monotonicity of privacy preferences is a common assumption in the privacy literature, e.g., see [13] and [6], but of course one can imagine situations where it may not hold, e.g., see [1] and [7]. In our context, we formalize monotone privacy preferences as follows. (cid:23) × × Deﬁnition 2. A binary relation over X 2X X is a monotone privacy preference when (cid:23) (i) is a linear order, and (cid:40) (cid:48) (cid:31) (cid:48) (ii) B B implies that (x, B) (x, B ). This deﬁnition formalizes the idea that revealing less information is better. In partic- (cid:40) (cid:48) (cid:48) ular, if B B , then fewer comparisons are being made in B than in B , so (x, B) reveals (cid:48) less information to the observer than (x, B ). Given the above deﬁnition, the question we address is “what are the empirical impli- cations of monotone privacy preferences?” That is, “Is monotonicity refutable via choice data?” The following proposition highlights that monotonicity is not refutable, so any choice data has a monotone privacy preference that explains it. Proposition 1. Any choice problem is rationalizable via monotone privacy preferences. Proof. We shall use the following notation: { ∈ \ { }} A = (x, y) : y A x . x 5

text: A

text: Deﬁnition

text: A

text: choice

text: problem

text: (X,

text: c)

text: is

text: rationalizable

text: (via

text: privacy

text: preferences)

text: if

text: there

text: is

text: a

text: privacy

text: preference

text: such

text: that

text: if

text: x

text: c(A)

text: and

text: y

text: A

text: x

text: then

text: (x,

text: (x,

text: z)

text: z

text: A

text: x

text: (y,

text: (y,

text: z)

text: z

text: A

text: y

text: A

text: A

text: for

text: all

text: A

text: In

text: this

text: case,

text: we

text: say

text: that

text: rationalizes

text: (X,

text: c).

text: Thus,

text: a

text: choice

text: problem

text: is

text: rationalizable

text: when

text: there

text: exists

text: a

text: privacy

text: preference

text: that

text: “explains”

text: the

text: data,

text: i.e.,

text: when

text: there

text: exists

text: a

text: privacy

text: preference

text: for

text: which

text: the

text: observed

text: choices

text: are

text: maximal.

text: The

text: rationalizability

text: of

text: privacy-aware

text: choice

text: In

text: this

text: section,

text: we

text: present

text: our

text: main

text: results,

text: which

text: characterize

text: when

text: choice

text: data

text: from

text: privacy-aware

text: consumers

text: is

text: rationalizable.

text: Our

text: results

text: focus

text: on

text: the

text: testable

text: implications

text: of

text: structural

text: assumptions

text: about

text: the

text: form

text: of

text: the

text: privacy

text: preferences

text: of

text: the

text: consumer.

text: While

text: a

text: consumer’s

text: preferences

text: may,

text: in

text: general,

text: be

text: a

text: complex

text: combination

text: of

text: preferences

text: over

text: the

text: choices

text: and

text: revealed

text: preferences,

text: there

text: are

text: some

text: natural

text: properties

text: that

text: one

text: may

text: expect

text: to

text: hold

text: in

text: many

text: situations.

text: In

text: particular,

text: we

text: focus

text: on

text: three

text: increasingly

text: strong

text: structural

text: assumptions

text: in

text: the

text: following

text: three

text: subsections:

text: monotonicity,

text: separability,

text: and

text: additivity.

text: Monotone

text: privacy

text: preferences

text: A

text: natural

text: assumption

text: on

text: privacy

text: preferences

text: is

text: “monotonicity”,

text: i.e.,

text: the

text: idea

text: that

text: re-

text: vealing

text: less

text: information

text: is

text: always

text: better.

text: Monotonicity

text: of

text: privacy

text: preferences

text: is

text: a

text: common

text: assumption

text: in

text: the

text: privacy

text: literature,

text: e.g.,

text: see

text: and

text: but

text: of

text: course

text: one

text: can

text: imagine

text: situations

text: where

text: it

text: may

text: not

text: hold,

text: e.g.,

text: see

text: and

text: In

text: our

text: context,

text: we

text: formalize

text: monotone

text: privacy

text: preferences

text: as

text: follows.

text: Deﬁnition

text: A

text: binary

text: relation

text: over

text: X

text: X

text: is

text: a

text: monotone

text: privacy

text: preference

text: when

text: (i)

text: is

text: a

text: linear

text: order,

text: and

text: (ii)

text: B

text: B

text: implies

text: that

text: (x,

text: B)

text: (x,

text: B

text: This

text: deﬁnition

text: formalizes

text: the

text: idea

text: that

text: revealing

text: less

text: information

text: is

text: better.

text: In

text: partic-

text: ular,

text: if

text: B

text: B

text: then

text: fewer

text: comparisons

text: are

text: being

text: made

text: in

text: B

text: than

text: in

text: B

text: so

text: (x,

text: B)

text: reveals

text: less

text: information

text: to

text: the

text: observer

text: than

text: (x,

text: B

text: Given

text: the

text: above

text: deﬁnition,

text: the

text: question

text: we

text: address

text: is

text: “what

text: are

text: the

text: empirical

text: impli-

text: cations

text: of

text: monotone

text: privacy

text: preferences?”

text: That

text: is,

text: “Is

text: monotonicity

text: refutable

text: via

text: choice

text: data?”

text: The

text: following

text: proposition

text: highlights

text: that

text: monotonicity

text: is

text: not

text: refutable,

text: so

text: any

text: choice

text: data

text: has

text: a

text: monotone

text: privacy

text: preference

text: that

text: explains

text: it.

text: Proposition

text: Any

text: choice

text: problem

text: is

text: rationalizable

text: via

text: monotone

text: privacy

text: preferences.

text: Proof.

text: We

text: shall

text: use

text: the

text: following

text: notation:

text: A

text: (x,

text: y)

text: y

text: A

text: x

text: x


text: × (cid:48) (cid:48) (cid:48) Deﬁne a binary relation E on X B(X) as follows: (x, B) E (x , B ) if either x = x (cid:40) (cid:48) (cid:54) (cid:48) ∈ A (cid:48) ∈ and B B , or x = x and there is A with x = c(A), x A and B = A while (cid:48) X B = A (cid:48). It will be useful for our proof to think of E as the edges of a directed graph x × (cid:48) G = (V, E), where V = X B(X). The edges where x = x result from the monotonicity (cid:54) (cid:48) requirement, and the edges where x = x result from the requirement that observed choices be rationalized. For shorthand, we will call these edges “monotone” and “rationalizing,” respectively. It should be clear that any linear order that extends B (i.e any linear order (cid:23) ⊆(cid:23) A with E ) is a monotone privacy preference that rationalizes (X, , c). By Lemma 1, we are done if we show that E is acyclic. To prove that E is acyclic, it is equivalent to show that the graph is acyclic. (cid:48) (cid:48) By the deﬁnition of E, for any pair (x, B) E (x , B ), the cardinality of B must be (cid:48) (cid:48) at most that of B , and if x = x then the cardinality must be strictly smaller due to monotonicity. Hence, there can be no cycles containing monotone edges. (cid:48) (cid:48) (cid:54) (cid:48) Thus any cycle must contain only rationalizing edges (x, B) E (x , B ) with x = x . ∈ A (cid:48) Each such edge arises from some A for which B = A while B = A (cid:48), and for ∈ X x each such A there is a unique x A with x = c(A). If the graph were to contain two consecutive rationalizing edges, it would contradict uniqueness of choice. Therefore there cannot be any cycles in E. Proposition 1 provides a contrast to the context of classical revealed preference theory, when consumers are privacy-oblivious. In particular, in the classical setting, choice be- havior that violates the strong axiom of revealed preferences (SARP) is not rationalizable, and thus refutes the consumer choice model. However, when privacy-aware consumers are considered, such a refutation of monotonic preferences is impossible. Interestingly, this means that while one may believe that preferences are non-monotonic, the form of data considered in this paper does not have the power to refute monotonicity.2 Note that the question addressed by Proposition 1 is only whether the consumer’s choice behavior is consistent with rational behavior, and is not about whether the con- sumer’s underlying preferences over outcomes in X can be learned. In particular, these underlying preferences may not even be well deﬁned for the general model considered to this point. We address this issue in the next section after imposing more structure on the privacy preferences. 3.2 Separable privacy preferences That all choice behavior is rationalizable via monotone privacy preferences can be attributed to the ﬂexibility provided by such preferences. Here we turn to a signiﬁcant restriction on the preferences one might use in rationalizing the consumer’s behavior. It is natural to postulate that the consumer would have some underlying, or intrinsic, preferences over possible options when her choices are not observed. Indeed, the observer is presumably trying to learn the agent’s preferences over objects. Such preferences should be well deﬁned: if outcome x is preferred to outcome y when both are paired with the same privacy set B, then it is natural that x will always be preferred to y when both are 2This phenomenon is common in the consumer choice formulation of the revealed preference problem, but it comes about for completely diﬀerent reasons. 6 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Deﬁne

text: a

text: binary

text: relation

text: E

text: on

text: X

text: B(X)

text: as

text: follows:

text: (x,

text: B)

text: E

text: (x

text: B

text: if

text: either

text: x

text: x

font: default

text: A

text: and

text: B

text: B

text: or

text: x

text: x

text: and

text: there

text: is

text: A

text: with

text: x

text: c(A),

text: x

text: A

text: and

text: B

text: A

text: while

text: X

text: B

text: A

text: It

text: will

text: be

text: useful

text: for

text: our

text: proof

text: to

text: think

text: of

text: E

text: as

text: the

text: edges

text: of

text: a

text: directed

text: graph

text: x

text: G

text: (V,

text: E),

text: where

text: V

text: X

text: B(X).

text: The

text: edges

text: where

text: x

text: x

text: result

text: from

text: the

text: monotonicity

text: requirement,

text: and

text: the

text: edges

text: where

text: x

text: x

text: result

text: from

text: the

text: requirement

text: that

text: observed

text: choices

text: be

text: rationalized.

text: For

text: shorthand,

text: we

text: will

text: call

text: these

text: edges

text: “monotone”

text: and

text: “rationalizing,”

text: respectively.

text: It

text: should

text: be

text: clear

text: that

text: any

text: linear

text: order

text: that

text: extends

text: B

text: (i.e

text: any

text: linear

text: order

text: A

text: with

text: E

text: is

text: a

text: monotone

text: privacy

text: preference

text: that

text: rationalizes

text: (X,

text: c).

text: By

text: Lemma

text: we

text: are

text: done

text: if

text: we

text: show

text: that

text: E

text: is

text: acyclic.

text: To

text: prove

text: that

text: E

text: is

text: acyclic,

text: it

text: is

text: equivalent

text: to

text: show

text: that

text: the

text: graph

text: is

text: acyclic.

text: By

text: the

text: deﬁnition

text: of

text: E,

text: for

text: any

text: pair

text: (x,

text: B)

text: E

text: (x

text: B

text: the

text: cardinality

text: of

text: B

text: must

text: be

text: at

text: most

text: that

text: of

text: B

text: and

text: if

text: x

text: x

text: then

text: the

text: cardinality

text: must

text: be

text: strictly

text: smaller

text: due

text: to

text: monotonicity.

text: Hence,

text: there

text: can

text: be

text: no

text: cycles

text: containing

text: monotone

text: edges.

text: Thus

text: any

text: cycle

text: must

text: contain

text: only

text: rationalizing

text: edges

text: (x,

text: B)

text: E

text: (x

text: B

text: with

text: x

text: x

text: A

text: Each

text: such

text: edge

text: arises

text: from

text: some

text: A

text: for

text: which

text: B

text: A

text: while

text: B

text: A

text: and

text: for

text: X

text: x

text: each

text: such

text: A

text: there

text: is

text: a

text: unique

text: x

text: A

text: with

text: x

text: c(A).

text: If

text: the

text: graph

text: were

text: to

text: contain

text: two

text: consecutive

text: rationalizing

text: edges,

text: it

text: would

text: contradict

text: uniqueness

text: of

text: choice.

text: Therefore

text: there

text: cannot

text: be

text: any

text: cycles

text: in

text: E.

text: Proposition

text: provides

text: a

text: contrast

text: to

text: the

text: context

text: of

text: classical

text: revealed

text: preference

text: theory,

text: when

text: consumers

text: are

text: privacy-oblivious.

text: In

text: particular,

text: in

text: the

text: classical

text: setting,

text: choice

text: be-

text: havior

text: that

text: violates

text: the

text: strong

text: axiom

text: of

text: revealed

text: preferences

text: (SARP)

text: is

text: not

text: rationalizable,

text: and

text: thus

text: refutes

text: the

text: consumer

text: choice

text: model.

text: However,

text: when

text: privacy-aware

text: consumers

text: are

text: considered,

text: such

text: a

text: refutation

text: of

text: monotonic

text: preferences

text: is

text: impossible.

text: Interestingly,

text: this

text: means

text: that

text: while

text: one

text: may

text: believe

text: that

text: preferences

text: are

text: non-monotonic,

text: the

text: form

text: of

text: data

text: considered

text: in

text: this

text: paper

text: does

text: not

text: have

text: the

text: power

text: to

text: refute

text: Note

text: that

text: the

text: question

text: addressed

text: by

text: Proposition

text: is

text: only

text: whether

text: the

text: consumer’s

text: choice

text: behavior

text: is

text: consistent

text: with

text: rational

text: behavior,

text: and

text: is

text: not

text: about

text: whether

text: the

text: con-

text: sumer’s

text: underlying

text: preferences

text: over

text: outcomes

text: in

text: X

text: can

text: be

text: learned.

text: In

text: particular,

text: these

text: underlying

text: preferences

text: may

text: not

text: even

text: be

text: well

text: deﬁned

text: for

text: the

text: general

text: model

text: considered

text: to

text: this

text: point.

text: We

text: address

text: this

text: issue

text: in

text: the

text: next

text: section

text: after

text: imposing

text: more

text: structure

text: on

text: the

text: privacy

text: preferences.

text: Separable

text: privacy

text: preferences

text: That

text: all

text: choice

text: behavior

text: is

text: rationalizable

text: via

text: monotone

text: privacy

text: preferences

text: can

text: be

text: attributed

text: to

text: the

text: ﬂexibility

text: provided

text: by

text: such

text: preferences.

text: Here

text: we

text: turn

text: to

text: a

text: signiﬁcant

text: restriction

text: on

text: the

text: preferences

text: one

text: might

text: use

text: in

text: rationalizing

text: the

text: consumer’s

text: behavior.

text: It

text: is

text: natural

text: to

text: postulate

text: that

text: the

text: consumer

text: would

text: have

text: some

text: underlying,

text: or

text: intrinsic,

text: preferences

text: over

text: possible

text: options

text: when

text: her

text: choices

text: are

text: not

text: observed.

text: Indeed,

text: the

text: observer

text: is

text: presumably

text: trying

text: to

text: learn

text: the

text: agent’s

text: preferences

text: over

text: objects.

text: Such

text: preferences

text: should

text: be

text: well

text: deﬁned:

text: if

text: outcome

text: x

text: is

text: preferred

text: to

text: outcome

text: y

text: when

text: both

text: are

text: paired

text: with

text: the

text: same

text: privacy

text: set

text: B,

text: then

text: it

text: is

text: natural

text: that

text: x

text: will

text: always

text: be

text: preferred

text: to

text: y

text: when

text: both

text: are

text: phenomenon

text: is

text: common

text: in

text: the

text: consumer

text: choice

text: formulation

text: of

text: the

text: revealed

text: preference

text: problem,

text: but

text: it

text: comes

text: about

text: for

text: completely

text: diﬀerent

text: reasons.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Proof of Proposition 3. To construct an example that is not rationalizable via additive { } privacy preferences, we begin by deﬁning the set of alternatives as X = x, y, z, w and { } { } the choice data as follows. It includes six observations: z = c( x, z ), x = c( x, y, z ), { } { } { } { } w = c( w, z ), z = c( w, y, z ), x = c( x, w ), w = c( x, y, w ). To see that this choice data is not rationalizable suppose, towards a contradiction, that { } the pair (u, v) rationalizes c. Then z = c( x, z ) implies that − − u(z) v(z, x) > u(x) v(x, z), { } while x = c( x, y, z ) implies that − − − − u(z) v(z, x) v(z, y) < u(x) v(x, z) v(x, y). Therefore v(z, y) > v(x, y). { } { } Similarly, we can argue that w = c( w, z ) and z = c( w, y, z ) together imply { } { } that v(w, y) > v(z, y), and x = c( x, w ) and w = c( x, y, w ) together imply that v(x, y) > v(w, y). This gives us a contradiction and so proves that the choice data is not rationalizable. Given that the structure imposed by additive privacy preferences is testable, the next task is to characterize data sets that are consistent with (or refute) the additive privacy preference model. The example given in the proof of Proposition 3 already suggests an important feature of choice data that must hold for it to be rationalizable. A ∈ Given a choice problem (X, , c) and an element y X, deﬁne the binary relation Ry ∈ A ∪ { } by x Ry z if there is A with z = c(A) and x = c(A y ). Our next result gives a test for additively rational preferences. It says that, if there are cycles in the binary relation Ry, then the choice data cannot be rationalized by additive privacy preferences. Proposition 4. A choice problem can be rationalized by additive privacy preferences only if Ry is acyclic, for all y. Proof. Let c be rationalizable by the additive privacy preferences characterized by (u, v). ∈ (cid:88) (cid:88) ∈ A For each x, z X such that x Ry z, then there is some A such that z = c(A) and ∈ x A, so − − u(z) v(z, t) > u(x) v(x, t). ∈ ∈ (cid:88) t A (cid:88)t A ∪ { } ∈ ∪ { } Similarly, x = c(A y ) and z A y , so − − − − u(z) v(z, t) v(z, y) > u(x) v(x, t) v(x, y). ∈ ∈ t A t A For both inequalities to be true simultaneously, we need v(z, y) > v(x, y). Thus, ⇒ x Ry z = v(z, y) > v(x, y). (1) · · · Now assume there exists a cycle in binary relation Ry: a Ry a Ry Ry a Ry a . ·1· · 2 k 1 Then by Equation (1), it must be that v(a , y) > v(a , y) > > v(a , y) > v(a , y). In 1 2 k 1 particular, v(a , y) > v(a , y) which is a contradiction. Then for choices to be rationalized, 1 ∈1 acyclicity of Ry for all y X is a necessary condition. 9 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Proof

text: of

text: Proposition

text: To

text: construct

text: an

text: example

text: that

text: is

text: not

text: rationalizable

text: via

text: additive

text: privacy

text: preferences,

text: we

text: begin

text: by

text: deﬁning

text: the

text: set

text: of

text: alternatives

text: as

text: X

text: x,

text: y,

text: z,

text: w

text: and

text: the

text: choice

text: data

text: as

text: follows.

text: It

text: includes

text: six

text: observations:

text: z

text: c(

text: x,

text: z

text: x

text: c(

text: x,

text: y,

text: z

text: w

text: c(

text: w,

text: z

text: z

text: c(

text: w,

text: y,

text: z

text: x

text: c(

text: x,

text: w

text: w

text: c(

text: x,

text: y,

text: w

text: To

text: see

text: that

text: this

text: choice

text: data

text: is

text: not

text: rationalizable

text: suppose,

text: towards

text: a

text: contradiction,

text: that

text: the

text: pair

text: (u,

text: v)

text: rationalizes

text: c.

text: Then

text: z

text: c(

text: x,

text: z

text: implies

text: that

text: u(z)

text: v(z,

text: x)

text: u(x)

text: v(x,

text: z),

text: while

text: x

text: c(

text: x,

text: y,

text: z

text: implies

text: that

text: u(z)

text: v(z,

text: x)

text: v(z,

text: y)

text: u(x)

text: v(x,

text: z)

text: v(x,

text: y).

text: Therefore

text: v(z,

text: y)

text: v(x,

text: y).

text: Similarly,

text: we

text: can

text: argue

text: that

text: w

text: c(

text: w,

text: z

text: and

text: z

text: c(

text: w,

text: y,

text: z

text: together

text: imply

text: that

text: v(w,

text: y)

text: v(z,

text: y),

text: and

text: x

text: c(

text: x,

text: w

text: and

text: w

text: c(

text: x,

text: y,

text: w

text: together

text: imply

text: that

text: v(x,

text: y)

text: v(w,

text: y).

text: This

text: gives

text: us

text: a

text: contradiction

text: and

text: so

text: proves

text: that

text: the

text: choice

text: data

text: is

text: not

text: rationalizable.

text: Given

text: that

text: the

text: structure

text: imposed

text: by

text: additive

text: privacy

text: preferences

text: is

text: testable,

text: the

text: next

text: task

text: is

text: to

text: characterize

text: data

text: sets

text: that

text: are

text: consistent

text: with

text: (or

text: refute)

text: the

text: additive

text: privacy

text: preference

text: model.

text: The

text: example

text: given

text: in

text: the

text: proof

text: of

text: Proposition

text: already

text: suggests

text: an

text: important

text: feature

text: of

text: choice

text: data

text: that

text: must

text: hold

text: for

text: it

text: to

text: be

text: rationalizable.

text: A

text: Given

text: a

text: choice

text: problem

text: (X,

text: c)

text: and

text: an

text: element

text: y

text: X,

text: deﬁne

text: the

text: binary

text: relation

text: Ry

text: A

text: by

text: x

text: Ry

text: z

text: if

text: there

text: is

text: A

text: with

text: z

text: c(A)

text: and

text: x

text: c(A

text: y

text: Our

text: next

text: result

text: gives

text: a

text: test

text: for

text: additively

text: rational

text: preferences.

text: It

text: says

text: that,

text: if

text: there

text: are

text: cycles

text: in

text: the

text: binary

text: relation

text: Ry,

text: then

text: the

text: choice

text: data

text: cannot

text: be

text: rationalized

text: by

text: additive

text: privacy

text: preferences.

text: Proposition

text: A

text: choice

text: problem

text: can

text: be

text: rationalized

text: by

text: additive

text: privacy

text: preferences

text: only

text: if

text: Ry

text: is

text: acyclic,

text: for

text: all

text: y.

text: Proof.

text: Let

text: c

text: be

text: rationalizable

text: by

text: the

text: additive

text: privacy

text: preferences

text: characterized

text: by

text: (u,

text: v).

text: A

text: For

text: each

text: x,

text: z

text: X

text: such

text: that

text: x

text: Ry

text: z,

text: then

text: there

text: is

text: some

text: A

text: such

text: that

text: z

text: c(A)

text: and

text: x

text: A,

text: so

text: u(z)

text: v(z,

text: t)

text: u(x)

text: v(x,

text: t).

text: t

text: A

text: A

text: Similarly,

text: x

text: c(A

text: y

text: and

text: z

text: A

text: y

text: so

text: u(z)

text: v(z,

text: t)

text: v(z,

text: y)

text: u(x)

text: v(x,

text: t)

text: v(x,

text: y).

text: t

text: A

text: t

text: A

text: For

text: both

text: inequalities

text: to

text: be

text: true

text: simultaneously,

text: we

text: need

text: v(z,

text: y)

text: v(x,

text: y).

text: Thus,

text: x

text: Ry

text: z

text: v(z,

text: y)

text: v(x,

text: y).

text: Now

text: assume

text: there

text: exists

text: a

text: cycle

text: in

text: binary

text: relation

text: Ry:

text: a

text: Ry

text: a

text: Ry

text: Ry

text: a

text: Ry

text: a

text: ·1·

text: k

text: Then

text: by

text: Equation

text: it

text: must

text: be

text: that

text: v(a

text: y)

text: v(a

text: y)

text: v(a

text: y)

text: v(a

text: y).

text: In

text: k

text: particular,

text: v(a

text: y)

text: v(a

text: y)

text: which

text: is

text: a

text: contradiction.

text: Then

text: for

text: choices

text: to

text: be

text: rationalized,

text: ∈1

text: acyclicity

text: of

text: Ry

text: for

text: all

text: y

text: X

text: is

text: a

text: necessary

text: condition.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: where t = min(r , s ), m = min(r , (cid:96) ), µ = min(p + 1, r , s ), λ = K K K K ||| − K |K|| (cid:107) K(cid:107) K K K K (cid:55)→ min(p + 1, r , (cid:96) ), and where R = o( u u ) z . Moreover, if the map v K R →KR K h,p + H2(Ω) R A(v)v: d d is twice continuously diﬀerentiable everywhere in d, then there exists a constant C such that (cid:32) (cid:33)   (cid:88)   − ≤ p3/2 h2µK 2 (cid:107) (cid:107) (cid:107) (cid:107) (25) R C max K K − u 2 z . K∈Th hdK/2 K∈Th p2KtK 3 HtK(K) H2(Ω) Before we embark on the proof of Theorem 4.5, we ﬁrst introduce an auxiliary result. (cid:55)→ R → R By our assumption that the map v A(v)v: d d is everywhere Fr´echet diﬀer- R (cid:55)→ → R entiable in d, we have that the map y N(y; v): V (h, p) is everywhere Fr´echet ∈ (cid:48) diﬀerentiable in V (h, p). Accordingly, for any v V (h, p), let N (q; w, v) denote the (cid:55)→ → R derivative of the map y N(y; v): V (h, p) at q in the direction w, given by − (cid:48) N(q + tw; v) N(q; v) ∈ N (q; w, v) = lim , q, w, v V (h, p). → t 0 t We introduce the following auxiliary result. ∈ ∩ Lemma 4.6. Let u Hs(Ω) C0(Ω)(cid:90), s > 3/2, denote the solution of (1), and suppose ∈ that the dual problem (22) has a strong solution z H2(Ω). Then, (cid:48) − (cid:48) ∇ · ∇ ∀ ∈ J (w) = N (u; w, z) (1 + θ) a ( u; [[w]]) z ds w V (h, p). ψ Γ h,0,D ∈ | ∈ F Proof. Since z H2(Ω), we have that [[z]] = 0 for all F . Accordingly, evaluat- (cid:48) ∈ F h,0 ing N (u; w, z) for any w V(cid:90)(h, p), we ﬁnd that (cid:90) (cid:88) (cid:48) (cid:48) ∇ ∇ · ∇ {{ (cid:48) ∇ · ∇ }} (26) N (u; w, z) = a ( u; w) z dx + θ a ( u; [[w]]) z ds. ∈T K Γ K h,0,D h Using the dual problem (22) and applying integration-by-parts, we also ﬁnd that, for ∈ all w V (h, p), (cid:90) (cid:88) − (cid:18)(cid:90) ∇ · ∗ ∇ ∇ (cid:90) (cid:19) J (w) = (cid:88) w (A ( u) z) dx ψ ∈T K K h (cid:90) ∗ ∇ ∇ · ∇ −(cid:90) ∗ ∇ ∇ · = (cid:88) A ( u) z w dx A ( u) z n w ds K ∈T K ∂K (27) K h (cid:90) ∗ ∇ ∇ · ∇ − ∗ ∇ ∇ {{ }} = A ( u) z w dx [[A ( u) z]] w ds ∈T K Γ K h,0 h − {{ ∗ ∇ ∇ }} · A ( u) z [[w]] ds. Γ h,0,D ∗ ∇ ∇ | ∈ F By [6, Lemma 1.24], it follows that [[A ( u) z]] = 0 weakly for all F . ∗F∇ ∇ · (cid:48) ∇ · ∇ h,0 Thence, comparing (26) and (27) while noting that A ( u) z w = a ( u; w) z for ∈ R all w d, we obtain the stated result. 14 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: where

text: t

text: min(r

text: s

text: m

text: min(r

text: µ

text: min(p

text: r

text: s

text: λ

text: K

text: K

text: K

text: K

text: K

text: |K||

text: K

text: K

text: K

text: K

text: min(p

text: r

text: and

text: where

text: R

text: o(

text: u

text: u

text: z

text: Moreover,

text: if

text: the

text: map

text: v

text: K

text: R

text: →KR

text: K

text: h,p

text: R

text: A(v)v:

text: d

text: d

text: is

text: twice

text: continuously

text: diﬀerentiable

text: everywhere

text: in

text: d,

text: then

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

label: equation

text: R

text: C

text: max

text: K

text: K

text: u

text: z

text: K∈Th

text: K∈Th

text: HtK(K)

text: Before

text: we

text: embark

text: on

text: the

text: proof

text: of

text: Theorem

text: we

text: ﬁrst

text: introduce

text: an

text: auxiliary

text: result.

text: R

text: R

text: By

text: our

text: assumption

text: that

text: the

text: map

text: v

text: A(v)v:

text: d

text: d

text: is

text: everywhere

text: Fr´echet

text: diﬀer-

text: R

text: R

text: entiable

text: in

text: d,

text: we

text: have

text: that

text: the

text: map

text: y

text: N(y;

text: v):

text: V

text: (h,

text: p)

text: is

text: everywhere

text: Fr´echet

text: diﬀerentiable

text: in

text: V

text: (h,

text: p).

text: Accordingly,

text: for

text: any

text: v

text: V

text: (h,

text: p),

text: let

text: N

text: (q;

text: w,

text: v)

text: denote

text: the

text: R

text: derivative

text: of

text: the

text: map

text: y

text: N(y;

text: v):

text: V

text: (h,

text: p)

text: at

text: q

text: in

text: the

text: direction

text: w,

text: given

text: by

text: N(q

text: tw;

text: v)

text: N(q;

text: v)

text: N

text: (q;

text: w,

text: v)

text: lim

text: q,

text: w,

text: v

text: V

text: (h,

text: p).

text: t

text: t

text: We

text: introduce

text: the

text: following

text: auxiliary

text: result.

text: Lemma

text: Let

text: u

text: Hs(Ω)

text: s

text: denote

text: the

text: solution

text: of

text: and

text: suppose

text: that

text: the

text: dual

text: problem

text: has

text: a

text: strong

text: solution

text: z

text: Then,

text: J

text: (w)

text: N

text: (u;

text: w,

text: z)

text: θ)

text: a

text: u;

text: [[w]])

text: z

text: ds

text: w

text: V

text: (h,

text: p).

text: ψ

text: Γ

text: F

text: Proof.

text: Since

text: z

text: we

text: have

text: that

text: [[z]]

text: for

text: all

text: F

text: Accordingly,

text: evaluat-

text: F

text: ing

text: N

text: (u;

text: w,

text: z)

text: for

text: any

text: w

text: p),

text: we

text: ﬁnd

text: that

text: N

text: (u;

text: w,

text: z)

text: a

text: u;

text: w)

text: z

text: dx

text: θ

text: a

text: u;

text: [[w]])

text: z

text: ds.

text: ∈T

text: K

text: Γ

text: K

text: h

text: Using

text: the

text: dual

text: problem

text: and

text: applying

text: integration-by-parts,

text: we

text: also

text: ﬁnd

text: that,

text: for

text: all

text: w

text: V

text: (h,

text: p),

text: J

text: (w)

text: w

text: (A

text: u)

text: z)

text: dx

text: ψ

text: ∈T

text: K

text: K

text: h

text: A

text: u)

text: z

text: w

text: dx

text: A

text: u)

text: z

text: n

text: w

text: ds

text: K

text: ∈T

text: K

text: ∂K

text: K

text: h

text: A

text: u)

text: z

text: w

text: dx

text: [[A

text: u)

text: z]]

text: w

text: ds

text: ∈T

text: K

text: Γ

text: K

text: h

text: A

text: u)

text: z

text: [[w]]

text: ds.

text: Γ

text: F

text: By

text: Lemma

text: it

text: follows

text: that

text: [[A

text: u)

text: z]]

text: weakly

text: for

text: all

text: F

text: ∗F∇

text: Thence,

text: comparing

text: and

text: while

text: noting

text: that

text: A

text: u)

text: z

text: w

text: a

text: u;

text: w)

text: z

text: for

text: R

text: all

text: w

text: d,

text: we

text: obtain

text: the

text: stated

text: result.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: deduce that (cid:90) (cid:88) (cid:48) − R = N (u; e, Π (z)) N(u; Π (z)) + N(u ; Π (z)) h,p h,p h,p h,p − (cid:90) ∇ ∇ · ∇ = r( u; e) (Π (z)) dx (cid:98) h,p ∈T K K h (cid:90) {{ ∇ ∇ · − ∇ }} + r( u, e) (θσ 1 (Π (z)) + [[Π (z)]]) ds σ h,p h,p Γ h,0,D − (cid:90) − {{ ∇ ∇ · ∇ }} θ(cid:88) σ 1 r( u, e) (Π (z)) ds h,p Γ h,0,D ≤ (cid:48) (cid:90) |∇ | |∇ | C e 2 (Π (cid:0) (z)) dx (cid:1) A (cid:98) h,p K K (cid:48) (cid:90) {{|∇ | | | − |∇ | | | }} + C e 2 θ σ 1 (Π (z)) + [[Π (z)]] ds A σ h,p h,p Γ h,0,D (cid:48) | | − {{|∇ | |∇ |}} + C θ σ 1 e 2 (Π (z)) ds. A h,p Γ h,0,D |{{ · }}| ≤ {{| | | |}} ≤ {{| |}} {{| |}} By Young’s inequality and the fact that q q q q 2 q q for ∈ T 1 2 1 2 1 2 all q , q [H1(Ω, )]d,(cid:90)we then obtain: 1 2 h(cid:88) ≤ (cid:48) |∇(cid:90) | |∇ | R C e 2 (cid:0)(Π (z)) dx (cid:1) A h,p K K | | (cid:90) (cid:48) − {{|∇ |}} | | {{|∇ |}} + 12 θ C (cid:0) σ 1 e 2 + σ(cid:1) [[e]] 2 (Π (z)) ds A h,p Γ h,0,D (cid:48) {{|∇ |}} | | | | + 4 C e 2 + σ2 [[e]] 2 [[Π (z)]] ds A h,p Γ ≤ | | h,0,(cid:48)D ||| ||| ||| ||| (32) (4 + 12 θ ) C e 2 Π (z) , A + h,p (cid:63) where (cid:13) (cid:13) (cid:13) (cid:13) ||| ||| (cid:107) (cid:13)(cid:107) (cid:13) {{|∇ |}} (cid:13) (cid:13) Πh,p(z) (cid:63) = Km∈aTx Πh,p(z) W∞1 (K) + F∈mFax (Πh,p(z)) L∞(F) h h,0,D | | (33) + F∈mFax σ [[Πh,p(z)]] L∞(F). h,0,D (cid:16) (cid:17) ||| ||| An upper bound for e is provided by Theorem 4.3. To prove (25), it thus remains + ||| ||| ≤ − (cid:107) (cid:107) to show that Π (z) C max ∈T p3/2 h d/2 z . To this end, let us note h,p (cid:63) K h K K H2(Ω) that, in view of Lemma 4.1 and the triangle inequality, there exists a constant C such that (cid:107) (cid:107) ≤ (cid:107) (cid:107) (cid:32) (cid:33) Π (z) C z . Thence, exploiting the inverse estimate (6), we have that h,p H2(K) H2(K) (cid:107) (cid:107) ≤ (cid:32) p (cid:33)(cid:107) (cid:107) K max Π (z) C max Π (z) K∈Th h,p W∞1 (K) K∈Th hd/2 h,p H1(K) K ≤ p (cid:107) (cid:107) K C max z . K∈Th hd/2 H2(Ω) K 17 ##LTLine## ##LTLine##

text: deduce

text: that

text: R

text: N

text: (u;

text: e,

text: Π

text: (z))

text: N(u;

text: Π

text: (z))

text: N(u

text: Π

text: (z))

text: h,p

text: h,p

text: h,p

text: h,p

text: r(

text: u;

text: e)

text: (Π

text: (z))

text: dx

text: h,p

text: ∈T

text: K

text: K

text: h

text: r(

text: u,

text: e)

text: (θσ

text: (Π

text: (z))

text: [[Π

text: (z)]])

text: ds

text: σ

text: h,p

text: h,p

text: Γ

text: σ

text: r(

text: u,

text: e)

text: (Π

text: (z))

text: ds

text: h,p

text: Γ

text: C

text: e

text: (Π

text: (z))

text: dx

text: A

text: h,p

text: K

text: K

text: C

text: e

text: θ

text: σ

text: (Π

text: (z))

text: [[Π

text: (z)]]

text: ds

text: A

text: σ

text: h,p

text: h,p

text: Γ

text: C

text: θ

text: σ

text: e

text: (Π

text: (z))

text: ds.

text: A

text: h,p

text: Γ

text: By

text: Young’s

text: inequality

text: and

text: the

text: fact

text: that

text: q

text: q

text: q

text: q

text: q

text: q

text: for

text: T

text: all

text: q

text: q

text: then

text: obtain:

text: R

text: C

text: e

text: (z))

text: dx

text: A

text: h,p

text: K

text: K

text: θ

text: C

text: σ

text: e

text: [[e]]

text: (Π

text: (z))

text: ds

text: A

text: h,p

text: Γ

text: C

text: e

text: σ2

text: [[e]]

text: [[Π

text: (z)]]

text: ds

text: A

text: h,p

text: Γ

text: θ

text: C

text: e

text: Π

text: (z)

text: A

text: h,p

text: where

text: Πh,p(z)

text: Km∈aTx

text: Πh,p(z)

text: (K)

text: F∈mFax

text: (Πh,p(z))

text: L∞(F)

text: h

text: F∈mFax

text: σ

text: [[Πh,p(z)]]

text: L∞(F).

text: An

text: upper

text: bound

text: for

text: e

text: is

text: provided

text: by

text: Theorem

text: To

text: prove

text: it

text: thus

text: remains

text: to

text: show

text: that

text: Π

text: (z)

text: C

text: max

text: ∈T

text: h

text: z

text: To

text: this

text: end,

text: let

text: us

text: note

text: h,p

text: K

text: h

text: K

text: K

text: that,

text: in

text: view

text: of

text: Lemma

text: and

text: the

text: triangle

text: inequality,

text: there

text: exists

text: a

text: constant

text: C

text: such

text: that

text: Π

text: (z)

text: C

text: z

text: Thence,

text: exploiting

text: the

text: inverse

text: estimate

text: we

text: have

text: that

text: h,p

text: p

text: K

text: max

text: Π

text: (z)

text: C

text: max

text: Π

text: (z)

text: K∈Th

text: h,p

text: (K)

text: K∈Th

text: h,p

text: K

text: p

text: K

text: C

text: max

text: z

text: K∈Th

text: K

text: ##LTLine##

text: ##LTLine##


text: ||| − ||| (cid:107) − (cid:107) ||| − ||| (cid:107) − (cid:107) p u u u u p u u u u h,p h,p L2(Ω) h,p h,p L2(Ω) 1 3.11E+00 — 4.46E-01 — 2 5.74E-01 — 8.60E-02 — 3 4.09E-02 (3.94) 3.30E-03 (4.47) 4 8.72E-03 (6.04) 6.74E-04 (7.00) 5 2.17E-03 (5.75) 1.51E-04 (6.03) 6 7.12E-04 (6.18) 3.71E-05 (7.15) 7 2.84E-04 (6.05) 1.35E-05 (7.18) 8 1.28E-04 (5.96) 5.00E-06 (6.97) 9 6.38E-05 (5.94) 1.97E-06 (7.66) 10 3.43E-05 (5.91) 9.28E-07 (7.55) 11 1.96E-05 (5.89) 4.51E-07 (7.34) 12 1.17E-05 (5.89) 2.37E-07 (7.49) 13 7.32E-06 (5.88) 1.31E-07 (7.43) 14 4.73E-06 (5.88) 7.52E-08 (7.45) 15 3.16E-06 (5.88) 4.50E-08 (7.44) 16 2.16E-06 (5.88) 2.78E-08 (7.46) 17 1.51E-06 (5.88) 1.76E-08 (7.50) 18 1.08E-06 (5.88) 1.15E-08 (7.49) 19 7.86E-07 (5.88) 7.64E-09 (7.51) 20 5.81E-07 (5.88) 5.19E-09 (7.54) 21 4.36E-07 (5.88) 3.59E-09 (7.55) 22 3.32E-07 (5.88) 2.53E-09 (7.56) 23 2.55E-07 (5.89) 1.80E-09 (7.57) 24 1.99E-07 (5.89) 1.30E-09 (7.59) ||| − ||| (cid:107) − (cid:107) Table 1: Example 2. Convergence of u u and u u with p-reﬁnement for mesh − h,p h,p L2(Ω) (a) and θ = 1. The results are grouped in odd and even values of p. The quantities in brackets indicate the p-convergence rates. ||| − ||| (cid:107) − (cid:107) ||| − ||| (cid:107) − (cid:107) p u u u u p u u u u h,p h,p L2(Ω) h,p h,p L2(Ω) 1 2.07E+00 — 2.31E-01 — 2 2.21E-01 — 2.12E-02 — 3 3.39E-02 (3.74) 2.22E-03 (4.23) 4 3.63E-03 (5.93) 4.62E-04 (5.52) 5 3.42E-03 (4.49) 1.67E-04 (5.07) 6 1.09E-03 (2.96) 1.50E-04 (2.78) 7 1.03E-03 (3.55) 4.39E-05 (3.96) 8 4.75E-04 (2.90) 6.64E-05 (2.83) 9 4.49E-04 (3.32) 1.81E-05 (3.53) 10 2.49E-04 (2.89) 3.50E-05 (2.86) 11 2.35E-04 (3.23) 9.29E-06 (3.32) 12 1.47E-04 (2.90) 2.07E-05 (2.89) 13 1.38E-04 (3.17) 5.44E-06 (3.20) 14 9.38E-05 (2.91) 1.32E-05 (2.90) 15 8.82E-05 (3.14) 3.48E-06 (3.13) 16 6.36E-05 (2.92) 8.97E-06 (2.91) 17 5.97E-05 (3.11) 2.36E-06 (3.09) 18 4.51E-05 (2.92) 6.36E-06 (2.92) 19 4.23E-05 (3.10) 1.68E-06 (3.06) 20 3.31E-05 (2.93) 4.67E-06 (2.93) 21 3.11E-05 (3.08) 1.24E-06 (3.04) 22 2.50E-05 (2.93) 3.53E-06 (2.94) 23 2.35E-05 (3.08) 9.42E-07 (3.02) 24 1.94E-05 (2.94) 2.73E-06 (2.94) ||| − ||| (cid:107) − (cid:107) Table 2: Example 2. Convergence of u u and u u with p-reﬁnement for mesh − h,p h,p L2(Ω) (b) and θ = 1. The results are grouped in odd and even values of p. The quantities in brackets indicate the p-convergence rates. 5.3 Example 3 In the third and ﬁnal example, we consider a case not fully covered by our theory. We ∇ |∇ | − consider the solution of the p(x)-Laplace equation with A(x, u) = u p(x) 2I, where − | | | | p(x) = 4 x 2. Note that A does not comply with Assumption 1.1 for x < 1. The − \ × − problem is posed on the L-shaped domain Ω = ( 1, 1)2 [0, 1) ( 1, 0] with Γ = − × { } ∪ {− } × − \ N [ 1, 1] 1 1 [ 1, 1] and Γ = ∂Ω Γ . The data f, g and g are chosen such D N D N that the solution is given by the smooth function u(x) = x1 ex1 x2. In Figure 4, we show the convergence of the DG-norm of the error with h-reﬁnement − ||| − ||| for p = 1, 2, 3, 4 and θ = 1, 1. As in Example 1, we observe that u u converges O → h,p to zero, for each ﬁxed value of p, at a rate (hp) as h 0. Note that this is in perfect agreement with the theoretical error estimate presented in Theorem 4.3, even though the underlying Assumption 1.1 is not met. Also note that the results are virtually distinguish- able between the two choices of the parameter θ. In Figure 5, we present the convergence − of the L2(Ω)-norm with h-reﬁnement for p = 1, 2, 3, 4 and θ = 1, 1. Here, as in Example − 1, signiﬁcant diﬀerences are observed between the two values of θ. For θ = 1, optimal 22 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

label: table

text: p

text: u

text: u

text: u

text: u

text: p

text: u

text: u

text: u

text: u

text: h,p

text: h,p

text: h,p

text: h,p

text: Table

text: Example

text: Convergence

text: of

text: u

text: u

text: and

text: u

text: u

text: with

text: p-reﬁnement

text: for

text: mesh

text: h,p

text: h,p

text: (a)

text: and

text: θ

text: The

text: results

text: are

text: grouped

text: in

text: odd

text: and

text: even

text: values

text: of

text: p.

text: The

text: quantities

text: in

text: brackets

text: indicate

text: the

text: p-convergence

text: rates.

text: p

text: u

text: u

text: u

text: u

text: p

text: u

text: u

text: u

text: u

text: h,p

text: h,p

text: h,p

text: h,p

text: Table

text: Example

text: Convergence

text: of

text: u

text: u

text: and

text: u

text: u

text: with

text: p-reﬁnement

text: for

text: mesh

text: h,p

text: h,p

text: (b)

text: and

text: θ

text: The

text: results

text: are

text: grouped

text: in

text: odd

text: and

text: even

text: values

text: of

text: p.

text: The

text: quantities

text: in

text: brackets

text: indicate

text: the

text: p-convergence

text: rates.

text: Example

text: In

text: the

text: third

text: and

text: ﬁnal

text: example,

text: we

text: consider

text: a

text: case

text: not

text: fully

text: covered

text: by

text: our

text: theory.

text: We

text: consider

text: the

text: solution

text: of

text: the

text: p(x)-Laplace

text: equation

text: with

text: A(x,

text: u)

text: u

text: p(x)

text: where

text: p(x)

text: x

text: Note

text: that

text: A

text: does

text: not

text: comply

text: with

text: Assumption

text: for

text: x

text: The

text: problem

text: is

text: posed

text: on

text: the

text: L-shaped

text: domain

text: Ω

text: with

text: Γ

text: N

text: and

text: Γ

text: ∂Ω

text: Γ

text: The

text: data

text: f,

text: g

text: and

text: g

text: are

text: chosen

text: such

text: D

text: N

text: D

text: N

text: that

text: the

text: solution

text: is

text: given

text: by

text: the

text: smooth

text: function

text: u(x)

text: In

text: Figure

text: we

text: show

text: the

text: convergence

text: of

text: the

text: DG-norm

text: of

text: the

text: error

text: with

text: h-reﬁnement

text: for

text: p

text: and

text: θ

text: As

text: in

text: Example

text: we

text: observe

text: that

text: u

text: u

text: converges

text: O

text: h,p

text: to

text: zero,

text: for

text: each

text: ﬁxed

text: value

text: of

text: p,

text: at

text: a

text: rate

text: (hp)

text: as

text: h

text: Note

text: that

text: this

text: is

text: in

text: perfect

text: agreement

text: with

text: the

text: theoretical

text: error

text: estimate

text: presented

text: in

text: Theorem

text: even

text: though

text: the

text: underlying

text: Assumption

text: is

text: not

text: met.

text: Also

text: note

text: that

text: the

text: results

text: are

text: virtually

text: distinguish-

text: able

text: between

text: the

text: two

text: choices

text: of

text: the

text: parameter

text: θ.

text: In

text: Figure

text: we

text: present

text: the

text: convergence

text: of

text: the

text: with

text: h-reﬁnement

text: for

text: p

text: and

text: θ

text: Here,

text: as

text: in

text: Example

text: signiﬁcant

text: diﬀerences

text: are

text: observed

text: between

text: the

text: two

text: values

text: of

text: θ.

text: For

text: θ

text: optimal

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text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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text: respective dual spaces, equiped with the norms (cid:104) (cid:105) (cid:104) (cid:105) (cid:107) (cid:107) f, v (cid:48) (cid:107) (cid:107) g, v (cid:48) f (cid:48) = sup (cid:107) (cid:107)U ,U , g (cid:48) = sup (cid:107) (cid:107)V ,V , U ∈ \{ } u V ∈ \{ } v u U 0 U v V 0 V (cid:104)· ·(cid:105) (cid:104)· ·(cid:105) (cid:48) (cid:48) where , (cid:48) and , (cid:48) are the duality pairings between U and U, and V and V , U ,U V ,V respectively. The ﬁrst result that we present constitutes a nonlinear extension of the classical well- posedness result of Banach, Neˇcas and Babuˇska; cf., for example, [6, Theorem 1.1]. The statement of the theorem and parts of its proof are adopted from [23, Appendix A], where the theorem is presented in a Hilbert space setting. We note that the theorem generalizes some other results from the literature; see, for example, [24, Theorem 25.B]. × → R Theorem A.1 (inf-sup conditions). Let a: U V be a semilinear form, such that − ≤ (cid:107) − (cid:107) (cid:107) (cid:107) ∀ ∈ ∀ ∈ (36) a(w ; v) a(w ; v) C w w v w .w U, v V 1 2 a 1 2 U V 1 2 for some constant C > 0. Then, the variational problem a ∈ ∀ ∈ (37) u U : a(u; v) = f(v) v V ∈ ∈ (cid:48) admits a unique solution u U for every f V if and only if − ∃ a(w ; v) a(w ; v) ≥ (38) M > 0 : inf sup (cid:107) 1 − (cid:107) (cid:107) 2(cid:107) M , a w1,w(cid:54) 2∈U v∈V \{0} w1 w2 U v V a w1=w2 ∀ ∈ \ { } (39) sup a(w; v) > 0 v V 0 . ∈ w U ∈ (cid:48) \ { } ∈ Moreover, for any g V f and corresponding u˜ U such that a(u˜; v) = g(v) for ∈ all v V , we have the following a priori estimate: (cid:107) − (cid:107) ≤ 1 (cid:107) − (cid:107) (40) u u˜ f g (cid:48). U V M a Proof. The proof proceeds in a similar manner as for the linear setting; cf., for example, ∈ → R (cid:55)→ [22]. For any ﬁxed w U, consider the linear functional φ : V of the form v ∈ w φ (v) := a(w; v) for all v V . By virtue of (36) with w = w and w = 0, we have that w 1 2 | | | | (cid:107) (cid:107) φ (v) a(w; v) ≤ (cid:107) (cid:107) φ (cid:48) = sup (cid:107)w(cid:107) = sup (cid:107) (cid:107) C w . w V ∈ \{ } v ∈ \{ } v a U v V 0 V v V 0 V ∈ (cid:48) → (cid:48) (cid:55)→ ∈ Hence, φ V . Now, let A: U V such that w A(w) := φ for all w U. The w ∈ w (cid:48) variational problem (37) is then equivalent to ﬁnding u U such that A(u) = f in V . 24 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: respective

text: dual

text: spaces,

text: equiped

text: with

text: the

text: norms

text: f,

text: v

text: g,

text: v

text: f

text: sup

text: ,U

text: g

text: sup

text: ,V

text: U

text: u

text: V

text: v

text: u

text: U

text: U

text: v

text: V

text: V

text: where

text: and

text: are

text: the

text: duality

text: pairings

text: between

text: U

text: and

text: U,

text: and

text: V

text: and

text: V

text: U

text: ,U

text: V

text: ,V

text: respectively.

text: The

text: ﬁrst

text: result

text: that

text: we

text: present

text: constitutes

text: a

text: nonlinear

text: extension

text: of

text: the

text: classical

text: well-

text: posedness

text: result

text: of

text: Banach,

text: Neˇcas

text: and

text: Babuˇska;

text: cf.,

text: for

text: example,

text: Theorem

text: The

text: statement

text: of

text: the

text: theorem

text: and

text: parts

text: of

text: its

text: proof

text: are

text: adopted

text: from

text: Appendix

text: A],

text: where

text: the

text: theorem

text: is

text: presented

text: in

text: a

text: Hilbert

text: space

text: setting.

text: We

text: note

text: that

text: the

text: theorem

text: generalizes

text: some

text: other

text: results

text: from

text: the

text: literature;

text: see,

text: for

text: example,

text: Theorem

text: R

text: Theorem

text: (inf-sup

text: conditions).

text: Let

text: a:

text: U

text: V

text: be

text: a

text: semilinear

text: form,

text: such

text: that

text: a(w

text: v)

text: a(w

text: v)

text: C

text: w

text: w

text: v

text: w

text: .w

text: U,

text: v

text: V

text: a

text: U

text: V

text: for

text: some

text: constant

text: C

text: Then,

text: the

text: variational

text: problem

text: a

text: u

text: U

text: a(u;

text: v)

text: f(v)

text: v

text: V

text: admits

text: a

text: unique

text: solution

text: u

text: U

text: for

text: every

text: f

text: V

text: if

text: and

text: only

text: if

text: a(w

text: v)

text: a(w

text: v)

text: M

text: inf

text: sup

text: M

text: a

text: v∈V

text: U

text: v

text: V

text: a

text: sup

text: a(w;

text: v)

text: v

text: V

text: w

text: U

text: Moreover,

text: for

text: any

text: g

text: V

text: f

text: and

text: corresponding

text: u˜

text: U

text: such

text: that

text: a(u˜;

text: v)

text: g(v)

text: for

text: all

text: v

text: V

text: we

text: have

text: the

text: following

text: a

text: priori

text: estimate:

text: u

text: u˜

text: f

text: g

text: U

text: V

text: M

text: a

text: Proof.

text: The

text: proof

text: proceeds

text: in

text: a

text: similar

text: manner

text: as

text: for

text: the

text: linear

text: setting;

text: cf.,

text: for

text: example,

text: R

text: For

text: any

text: ﬁxed

text: w

text: U,

text: consider

text: the

text: linear

text: functional

text: φ

text: V

text: of

text: the

text: form

text: v

text: w

text: φ

text: (v)

text: a(w;

text: v)

text: for

text: all

text: v

text: V

text: By

text: virtue

text: of

text: with

text: w

text: w

text: and

text: w

text: we

text: have

text: that

text: w

text: φ

text: (v)

text: a(w;

text: v)

text: φ

text: sup

text: sup

text: C

text: w

text: w

text: V

text: v

text: v

text: a

text: U

text: v

text: V

text: V

text: v

text: V

text: V

text: Hence,

text: φ

text: V

text: Now,

text: let

text: A:

text: U

text: V

text: such

text: that

text: w

text: A(w)

text: φ

text: for

text: all

text: w

text: U.

text: The

text: w

text: w

text: variational

text: problem

text: is

text: then

text: equivalent

text: to

text: ﬁnding

text: u

text: U

text: such

text: that

text: A(u)

text: f

text: in

text: V

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

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