text: Value of information in noncooperative games ∗ † ‡ Nils Bertschinger 1, David H. Wolpert 2, Eckehard Olbrich 1, and Ju¨rgen § Jost 1,2 1Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, D-04103 Leipzig, Germany 2Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA May 26, 2020 Abstract In some games, additional information hurts a player, e.g., in games with ﬁrst-mover advantage, the second-mover is hurt by seeing the ﬁrst-mover’s move. What properties of a game determine whether it has such negative “value of information” for a particular player? Can a game have negative value of information for all players? To answer such questions, we general- ize the deﬁnition of marginal utility of a good to deﬁne the marginal utility of a parameter vector specifying a game. So rather than analyze the global structure of the relationship between a game’s parameter vector and player behavior, as in previous work, we focus on the local structure of that relation- ship. This allows us to prove that generically, every game can have negative marginal value of information, unless one imposes a priori constraints on allowed changes to the game’s parameter vector. We demonstrate these and related results numerically, and discuss their implications. 1 Introduction How a player in a noncooperative game behaves typically depends on what in- formation she has both about her physical environment and about the behavior of ∗ nils.bertschinger@mis.mpg.de † http://davidwolpert.weebly.com ‡ olbrich@mis.mpg.de § jjost@mis.mpg.de 1 ##LTLine## ##LTLine##

text: Value

label: title

text: of

text: information

text: in

text: noncooperative

text: games

label: author

font: MSZUCC+txsy

text: Nils

text: Bertschinger

label: paragraph

text: David

text: H.

text: Wolpert

text: Eckehard

text: Olbrich

text: and

text: Ju¨rgen

text: Jost

text: Planck

text: Institute

text: for

text: Mathematics

text: in

text: the

text: Sciences,

text: Inselstraße

text: Leipzig,

text: Germany

text: Fe

text: Institute,

text: Hyde

text: Park

text: Road,

text: Santa

text: Fe,

text: NM

text: USA

text: May

label: date

text: Abstract

text: In

label: abstract

text: some

text: games,

text: additional

text: information

text: hurts

text: a

text: player,

text: e.g.,

text: in

text: games

text: with

text: ﬁrst-mover

text: advantage,

text: the

text: second-mover

text: is

text: hurt

text: by

text: seeing

text: the

text: ﬁrst-mover’s

text: move.

text: What

text: properties

text: of

text: a

text: game

text: determine

text: whether

text: it

text: has

text: such

text: negative

text: “value

text: of

text: information”

text: for

text: a

text: particular

text: player?

text: Can

text: a

text: game

text: have

text: negative

text: value

text: of

text: information

text: for

text: all

text: players?

text: To

text: answer

text: such

text: questions,

text: we

text: general-

text: ize

text: the

text: deﬁnition

text: of

text: marginal

text: utility

text: of

text: a

text: good

text: to

text: deﬁne

text: the

text: marginal

text: utility

text: of

text: a

text: parameter

text: vector

text: specifying

text: a

text: game.

text: So

text: rather

text: than

text: analyze

text: the

text: global

text: structure

text: of

text: the

text: relationship

text: between

text: a

text: game’s

text: parameter

text: vector

text: and

text: player

text: behavior,

text: as

text: in

text: previous

text: work,

text: we

text: focus

text: on

text: the

text: local

text: structure

text: of

text: that

text: relation-

text: ship.

text: This

text: allows

text: us

text: to

text: prove

text: that

text: generically,

text: every

text: game

text: can

text: have

text: negative

text: marginal

text: value

text: of

text: information,

text: unless

text: one

text: imposes

text: a

text: priori

text: constraints

text: on

text: allowed

text: changes

text: to

text: the

text: game’s

text: parameter

text: vector.

text: We

text: demonstrate

text: these

text: and

text: related

text: results

text: numerically,

text: and

text: discuss

text: their

text: implications.

text: Introduction

text: How

text: a

text: player

text: in

text: a

text: noncooperative

text: game

text: behaves

text: typically

text: depends

text: on

text: what

text: in-

text: formation

text: she

text: has

text: both

text: about

text: her

text: physical

text: environment

text: and

text: about

text: the

text: behavior

text: of

text: nils.bertschinger@mis.mpg.de

text: olbrich@mis.mpg.de

text: jjost@mis.mpg.de

text: ##LTLine##

text: ##LTLine##


text: for the analysis of value of information in the Braess’ paradox and Cournot exam- ple of negative value of information, as discussed below in Sec. . It also means that even if for some particular f of interest one intuitively expects that increasing f should reduce expected utility, generically there will be inﬁnitesimal changes to θ that will increase f but increase expected utility. An illustration of this is also discussed in the “negative value of utility” example in Sec. . 4.3 Genericity of negative value of information Consider situations where f is a monotonically increasing function of V across ⊆ a compact S Θ. This means that f and V have the same level hypersurfaces across S (although, of course the values of V and f on any such common level hypersurface will in general be diﬀerent). The monotonicity implies that the linear order induced by values f (θ) relates the level hypersurfaces in the same way that the linear order induced by values V(θ) relates those level hypersurfaces. Say that in addition neither grad( f ) nor grad(V) equals 0 anywhere in S . So grad(V) and grad( f ) are proportional to one another throughout S (although the proportionality constant may change).12 So, by Prop. 1, for no θ in S is there a direction δθ such V that f,δθ(θ) < 0. In general though, level hypersurfaces will not match up throughout a region, as the condition that grad(V) and grad( f ) be proportional is very restrictive and spe- cial, and so is typically violated. When they do not, we have points in that region that have directions with negative value of f . We now derive a criterion involving both the gradients and the Hessians of V and f to identify such a mismatch. Proposition 3. Assume that both f and V are analytic at θ with nonzero gradients, and choose some (cid:15) > 0. Deﬁne ≡ ν gradV(θ) ≡ φ grad f (θ) ≡ { } B (θ) σ : dist(σ, θ) < (cid:15) (cid:15) Then ∈ 12Hypothesize that the gradients were not parallel at some θ S. Since neither gradient equals ∈ zero, this would mean that there is a direction δθ S that has zero inner product with gradf at θ but not with grad(V) there. δθ would lie in the level surface of f going through θ but not the level surface of V going through θ. This would contradict the supposition that those level surfaces are coincident throughout S. 25 ##LTLine##

text: for

text: the

text: analysis

text: of

text: value

text: of

text: information

text: in

text: the

text: Braess’

text: paradox

text: and

text: Cournot

text: exam-

text: ple

text: of

text: negative

text: value

text: of

text: information,

text: as

text: discussed

text: below

text: in

text: Sec.

text: It

text: also

text: means

text: that

text: even

text: if

text: for

text: some

text: particular

text: f

text: of

text: interest

text: one

text: intuitively

text: expects

text: that

text: increasing

text: f

text: should

text: reduce

text: expected

text: utility,

text: generically

text: there

text: will

text: be

text: inﬁnitesimal

text: changes

text: to

text: θ

font: AJCCEQ+rtxbmi

text: that

text: will

text: increase

text: f

text: but

text: increase

text: expected

text: utility.

text: An

text: illustration

text: of

text: this

text: is

text: also

text: discussed

text: in

text: the

text: “negative

text: value

text: of

text: utility”

text: example

text: in

text: Sec.

label: section

text: Genericity

text: of

text: negative

text: value

text: of

text: information

text: Consider

text: situations

text: where

text: f

text: is

text: a

text: monotonically

text: increasing

text: function

text: of

text: V

text: across

text: a

text: compact

text: S

text: Θ.

text: This

text: means

text: that

text: f

text: and

text: V

text: have

text: the

text: same

text: level

text: hypersurfaces

text: across

text: S

text: (although,

text: of

text: course

text: the

text: values

text: of

text: V

text: and

text: f

text: on

text: any

text: such

text: common

text: level

text: hypersurface

text: will

text: in

text: general

text: be

text: diﬀerent).

text: The

text: monotonicity

text: implies

text: that

text: the

text: linear

text: order

text: induced

text: by

text: values

text: f

text: (θ)

text: relates

text: the

text: level

text: hypersurfaces

text: in

text: the

text: same

text: way

text: that

text: the

text: linear

text: order

text: induced

text: by

text: values

text: V(θ)

text: relates

text: those

text: level

text: hypersurfaces.

text: Say

text: that

text: in

text: addition

text: neither

text: grad(

text: f

text: nor

text: grad(V)

text: equals

text: anywhere

text: in

text: S

text: So

text: grad(V)

text: and

text: grad(

text: f

text: are

text: proportional

text: to

text: one

text: another

text: throughout

text: S

text: (although

text: the

text: proportionality

text: constant

text: may

text: So,

text: by

text: Prop.

text: for

text: no

text: θ

text: in

text: S

text: is

text: there

text: a

text: direction

text: δθ

text: such

text: V

text: that

text: f,δθ(θ)

text: In

text: general

text: though,

text: level

text: hypersurfaces

text: will

text: not

text: match

text: up

text: throughout

text: a

text: region,

text: as

text: the

text: condition

text: that

text: grad(V)

text: and

text: grad(

text: f

text: be

text: proportional

text: is

text: very

text: restrictive

text: and

text: spe-

text: cial,

text: and

text: so

text: is

text: typically

text: violated.

text: When

text: they

text: do

text: not,

text: we

text: have

text: points

text: in

text: that

text: region

text: that

text: have

text: directions

text: with

text: negative

text: value

text: of

text: f

text: We

text: now

text: derive

text: a

text: criterion

text: involving

text: both

text: the

text: gradients

text: and

text: the

text: Hessians

text: of

text: V

text: and

text: f

text: to

text: identify

text: such

text: a

text: mismatch.

text: Proposition

text: Assume

text: that

text: both

text: f

text: and

text: V

text: are

text: analytic

text: at

text: θ

text: with

text: nonzero

text: gradients,

text: and

text: choose

text: some

text: Deﬁne

text: ν

text: gradV(θ)

text: φ

text: grad

text: f

text: (θ)

text: B

text: (θ)

text: σ

text: dist(σ,

text: θ)

text: Then

text: that

text: the

text: gradients

text: were

text: not

text: parallel

text: at

text: some

text: θ

text: S.

text: Since

text: neither

text: gradient

text: equals

text: zero,

text: this

text: would

text: mean

text: that

text: there

text: is

text: a

text: direction

text: δθ

text: S

text: that

text: has

text: zero

text: inner

text: product

text: with

text: gradf

text: at

text: θ

text: but

text: not

text: with

text: grad(V)

text: there.

text: δθ

text: would

text: lie

text: in

text: the

text: level

text: surface

text: of

text: f

text: going

text: through

text: θ

text: but

text: not

text: the

text: level

text: surface

text: of

text: V

text: going

text: through

text: θ.

text: This

text: would

text: contradict

text: the

text: supposition

text: that

text: those

text: level

text: surfaces

text: are

text: coincident

text: throughout

text: S.

text: ##LTLine##


text: (cid:48) independent of i. So evaluating at θ = θ and by the above proportionality, (18) simpliﬁes to | | ∇∇ − φ ∇∇ [ f ]i,j | | [ V]i,j = k[grad(V)]i[grad(V)]j (21) ν (cid:12) (cid:12) (cid:12) (cid:12) for some scalar k. Taking norms of(cid:12)(cid:12)both sides, yiel(cid:12)(cid:12)ds (cid:12) (cid:12) ∇∇ − | |∇∇ f |φ| V ν k = | | , (22) ν 2 whence (16). (cid:3) We will refer to the matrix with entries (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) | | | | ∇∇ f − |φ| ∇∇V ν|iν| j − [∇∇ f ]i + |φ| [∇∇ f ]i V ν ν 2 j ν j as the second-order mismatch. 13 Prop. 3 means that if there is nonzero second- order mismatch at some θ then C −(θ) cannot be empty throughout a d-dimensional + (cid:48) sphere centered at θ. In other words, it means that for any (cid:15) > 0, there is some θ (cid:48) (cid:48) within distance (cid:15) of θ and associated direction δθ such that at θ there are negative (cid:48) values of f in direction δθ .14 13 One might think that so long as the Hessian of V at θ is not a constant times the Hessian of f at θ, then the level curves of V and f will not match up throughout the vicinity of θ. Such mismatch in the level curves would in turn imply that even if the gradients of V and f were parallel at θ, there would be points inﬁnitesimally close to θ where the gradients of V and of f are not parallel. If this reasoning were valid, it would suggest that the condition in Prop. 3 that the second-order mismatch | | is non-zero could be tightened (by replacing that condition with the requirement that DijF (cid:44) |φ| DijV) ν and yet the implication given by that proposition — that there are points where the gradients are not parallel — would still follow. This is not the case though: Even if the Hessian of V is not a constant times the Hessian of f at θ, it is possible that the gradients of f and V are parallel throughout a neighborhood of θ. As a simple example, take d = 2,V(θ) = θ1, f(θ) = (1 + [θ1])2. Even though the Hessian of this V is not a constant times the Hessian of this f anywhere near the origin, the gradients of f and V are parallel everywhere near the origin. − 14 However it is still possible for C −(θ) to be empty throughout a (d 1)-dimensional hyperplane + T going through θ. As an example, choose d = 2,V(θ) = θ1, f(θ) = (1 + θ1)2(1 + [θ2]2). The second order mismatch is nonzero for this V and f at θ = (0,0). (The Hessian of f at the origin is twice the identity matrix, the Hessian of V there is th{e (cid:48)zero matrix, b∈ut }νiνj only has one non-zero entry, fo(cid:48)r i = j = 1.) However everywhere on the line θ = t(1,0) : t R going through the origin, gradf(θ ) (cid:48) and grad(V)(θ ) are parallel. 27 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: independent

text: of

text: i.

text: So

text: evaluating

text: at

text: θ

font: TCPZZG+rtxmi

font: VJTTCO+rtxr

text: θ

text: and

text: by

text: the

text: above

text: proportionality,

text: simpliﬁes

text: to

text: φ

text: f

text: ]i,j

text: V]i,j

text: k[grad(V)]i[grad(V)]j

text: ν

font: PXEDGA+txex

text: for

text: some

text: scalar

text: k.

text: Taking

text: norms

text: sides,

text: f

text: |φ|

text: V

text: ν

text: k

text: ν

text: whence

text: We

text: will

text: refer

text: to

text: the

text: matrix

text: with

text: entries

text: f

text: |φ|

text: ∇∇V

text: ν|iν|

text: j

text: f

text: ]i

text: |φ|

text: f

text: ]i

text: V

text: ν

text: ν

text: j

text: ν

text: j

text: as

text: the

text: second-order

text: mismatch.

text: Prop.

text: means

text: that

text: if

text: there

text: is

text: nonzero

text: second-

text: order

text: mismatch

text: at

text: some

text: θ

text: then

text: C

text: −(θ)

text: cannot

text: be

text: empty

text: throughout

text: a

text: d-dimensional

text: sphere

text: centered

text: at

text: θ.

text: In

text: other

text: words,

text: it

text: means

text: that

text: for

text: any

text: there

text: is

text: some

text: θ

text: within

text: distance

text: of

text: θ

text: and

text: associated

text: direction

text: δθ

text: such

text: that

text: at

text: θ

text: there

text: are

text: negative

text: values

text: of

text: f

text: in

text: direction

text: δθ

text: One

text: might

text: think

text: that

text: so

text: long

text: as

text: the

text: Hessian

text: of

text: V

text: at

text: θ

text: is

text: not

text: a

text: constant

text: times

text: the

text: Hessian

text: of

text: f

text: at

text: θ,

text: then

text: the

text: level

text: curves

text: of

text: V

text: and

text: f

text: will

text: not

text: match

text: up

text: throughout

text: the

text: vicinity

text: of

text: θ.

text: Such

text: mismatch

text: in

text: the

text: level

text: curves

text: would

text: in

text: turn

text: imply

text: that

text: even

text: if

text: the

text: gradients

text: of

text: V

text: and

text: f

text: were

text: parallel

text: at

text: θ,

text: there

text: would

text: be

text: points

text: inﬁnitesimally

text: close

text: to

text: θ

text: where

text: the

text: gradients

text: of

text: V

text: and

text: of

text: f

text: are

text: not

text: parallel.

text: If

text: this

text: reasoning

text: were

text: valid,

text: it

text: would

text: suggest

text: that

text: the

text: condition

text: in

text: Prop.

text: that

text: the

text: second-order

text: mismatch

text: is

text: non-zero

text: could

text: be

text: tightened

text: (by

text: replacing

text: that

text: condition

text: with

text: the

text: requirement

text: that

text: DijF

text: |φ|

text: DijV)

text: ν

text: and

text: yet

text: the

text: implication

text: given

text: by

text: that

text: proposition

text: that

text: there

text: are

text: points

text: where

text: the

text: gradients

text: are

text: not

text: parallel

text: would

text: still

text: follow.

text: This

text: is

text: not

text: the

text: case

text: though:

text: Even

text: if

text: the

text: Hessian

text: of

text: V

text: is

text: not

text: a

text: constant

text: times

text: the

text: Hessian

text: of

text: f

text: at

text: θ,

text: it

text: is

text: possible

text: that

text: the

text: gradients

text: of

text: f

text: and

text: V

text: are

text: parallel

text: throughout

text: a

text: neighborhood

text: of

text: θ.

text: As

text: a

text: simple

text: example,

text: take

text: d

text: f(θ)

text: Even

text: though

text: the

text: Hessian

text: of

text: this

text: V

text: is

text: not

text: a

text: constant

text: times

text: the

text: Hessian

text: of

text: this

text: f

text: anywhere

text: near

text: the

text: origin,

text: the

text: gradients

text: of

text: f

text: and

text: V

text: are

text: parallel

text: everywhere

text: near

text: the

text: origin.

text: However

text: it

text: is

text: still

text: possible

text: for

text: C

text: −(θ)

text: to

text: be

text: empty

text: throughout

text: a

text: (d

text: hyperplane

text: T

text: going

text: through

text: θ.

text: As

text: an

text: example,

text: choose

text: d

text: f(θ)

text: The

text: second

text: order

text: mismatch

text: is

text: nonzero

text: for

text: this

text: V

text: and

text: f

text: at

text: θ

text: (The

text: Hessian

text: of

text: f

text: at

text: the

text: origin

text: is

text: twice

text: the

text: identity

text: matrix,

text: the

text: Hessian

text: of

text: V

text: there

text: is

text: th{e

text: matrix,

text: b∈ut

text: }νiνj

text: only

text: has

text: one

text: non-zero

text: entry,

text: i

text: j

text: However

text: everywhere

text: on

text: the

text: line

text: θ

text: t

text: R

text: going

text: through

text: the

text: origin,

text: gradf(θ

text: and

text: grad(V)(θ

text: are

text: parallel.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: In the light of this, arguably the most important issues to analyze in any given game is not whether at some particular θ there is a direction in which some par- ticular type of information increases but expected utility decreases. That is almost always the case. Rather our results redirect attention to diﬀerent issues. One such issue is what fraction of all θ and all possible inﬁnitesimal changes to θ have neg- ative value of information. Another important issue arises whenever there are constraints on the allowed directions in Θ specifying how θ is changed. Such constraints are quite common in the real world. As an example, there might be inequality constraints on the information capacity of a channel, which would manifest themselves in constraints on the allowed change in parameter space for θ for which those constraints are tight. Another example of a constraint on the allowed change to the parameter vector is the stipulation that the change must be an insertion (or removal) of a garbling node, in the sense of Blackwell’s theorem [22]. With such constraints, in general the directions in the sets C −(θ) and C− (θ) arising in Prop. 1 may all + + violate the constraints, and so not be allowed. This is why, for example, the result of Blackwell (that there are no garbling nodes that one can insert just before a decision node in an ID that increase maximal expected utility for a rational player) is consistent with our results; the insertion of such a garbling node corresponds to a constrained change in θ, a change that lies in a cone that does not intersect C±(θ) (see Fig. 3). 4.4 Pareto negative values of information Recall that Prop. 1 established that there are directions in which the value of f is negative for player i iﬀ grad f and gradVi are not positively proportional. We can generalize this to give a condition for there to be directions in which the value of f is negative for all players, in which case we say that we have “Pareto negative V values of information”. To make this precise, deﬁne i (θ) as the value of f in f,δθ direction δθ to player i in a MAID. → ≡ { } Propositio⊥n 4. Given a set of functions Vi : Θ R, deﬁne C Con( grad(Vi) ). Assume C (cid:44) ∅. Then grad( f ) (cid:60) C ⇔ ∃ (cid:48) ∀ δθ : V i (θ ) < 0 i (23) f,δθ where all N + 1 gradients are evaluated at θ. 28

text: In

text: the

text: light

text: of

text: this,

text: arguably

text: the

text: most

text: important

text: issues

text: to

text: analyze

text: in

text: any

text: given

text: game

text: is

text: not

text: whether

text: at

text: some

text: particular

text: θ

text: there

text: is

text: a

text: direction

text: in

text: which

text: some

text: par-

text: ticular

text: type

text: of

text: information

text: increases

text: but

text: expected

text: utility

text: decreases.

text: That

text: is

text: almost

text: always

text: the

text: case.

text: Rather

text: our

text: results

text: redirect

text: attention

text: to

text: diﬀerent

text: issues.

text: One

text: such

text: issue

text: is

text: what

text: fraction

text: of

text: all

text: θ

text: and

text: all

text: possible

text: inﬁnitesimal

text: changes

text: to

text: θ

text: have

text: neg-

text: ative

text: value

text: of

text: information.

text: Another

text: important

text: issue

text: arises

text: whenever

text: there

text: are

text: constraints

text: on

text: the

text: allowed

text: directions

text: in

text: Θ

text: specifying

text: how

text: θ

text: is

text: changed.

text: Such

text: constraints

text: are

text: quite

text: common

text: in

text: the

text: real

text: world.

text: As

text: an

text: example,

text: there

text: might

text: be

text: inequality

text: constraints

text: on

text: the

text: information

text: capacity

text: of

text: a

text: channel,

text: which

text: would

text: manifest

text: themselves

text: in

text: constraints

text: on

text: the

text: allowed

text: change

text: in

text: parameter

text: space

text: for

text: θ

text: for

text: which

text: those

text: constraints

text: are

text: tight.

text: Another

text: example

text: of

text: a

text: constraint

text: on

text: the

text: allowed

text: change

text: to

text: the

text: parameter

text: vector

text: is

text: the

text: stipulation

text: that

text: the

text: change

text: must

text: be

text: an

text: insertion

text: (or

text: removal)

text: of

text: a

text: garbling

text: node,

text: in

text: the

text: sense

text: of

text: Blackwell’s

text: theorem

text: With

text: such

text: constraints,

text: in

text: general

text: the

text: directions

text: in

text: the

text: sets

text: C

text: −(θ)

text: and

text: C−

text: (θ)

text: arising

text: in

text: Prop.

text: may

text: all

text: violate

text: the

text: constraints,

text: and

text: so

text: not

text: be

text: allowed.

text: This

text: is

text: why,

text: for

text: example,

text: the

text: result

text: of

text: Blackwell

text: (that

text: there

text: are

text: no

text: garbling

text: nodes

text: that

text: one

text: can

text: insert

text: just

text: before

text: a

text: decision

text: node

text: in

text: an

text: ID

text: that

text: increase

text: maximal

text: expected

text: utility

text: for

text: a

text: rational

text: player)

text: is

text: consistent

text: with

text: our

text: results;

text: the

text: insertion

text: of

text: such

text: a

text: garbling

text: node

text: corresponds

text: to

text: a

text: constrained

text: change

text: in

text: θ,

text: a

text: change

text: that

text: lies

text: in

text: a

text: cone

text: that

text: does

text: not

text: intersect

text: C±(θ)

text: (see

text: Fig.

text: Pareto

text: negative

text: values

text: of

text: information

text: Recall

text: that

text: Prop.

text: established

text: that

text: there

text: are

text: directions

text: in

text: which

text: the

text: value

text: of

text: f

text: is

text: negative

text: for

text: player

text: i

text: iﬀ

text: grad

text: f

text: and

text: gradVi

text: are

text: not

text: positively

text: proportional.

text: We

text: can

text: generalize

text: this

text: to

text: give

text: a

text: condition

text: for

text: there

text: to

text: be

text: directions

text: in

text: which

text: the

text: value

text: of

text: f

text: is

text: negative

text: for

text: all

text: players,

text: in

text: which

text: case

text: we

text: say

text: that

text: we

text: have

text: “Pareto

text: negative

text: V

text: values

text: of

text: information”.

text: To

text: make

text: this

text: precise,

text: deﬁne

text: i

text: (θ)

text: as

text: the

text: value

text: of

text: f

text: in

text: f,δθ

text: direction

text: δθ

text: to

text: player

text: i

text: in

text: a

text: MAID.

text: Propositio⊥n

text: Given

text: a

text: set

text: of

text: functions

text: Vi

text: Θ

text: R,

text: deﬁne

text: C

text: Con(

text: grad(Vi)

text: Assume

text: C

text: Then

text: grad(

text: f

text: C

text: δθ

text: V

text: i

text: (θ

text: i

text: f,δθ

text: where

text: all

text: N

text: gradients

text: are

text: evaluated

text: at

text: θ.


text: This potential for an inconsistency between changes to information and util- ity is also illustrated by considering the gradient vectors gradI(X; S ) and grad(V) which are orthogonal to the isoclines (wrt/ the Fisher metric)18. By Prop. 1, only if those gradients are collinear is it the case that every change to the game parameter vector increasing the expected utility must necessarily increase the mutual infor- mation. However in Fig. 3 we clearly see that these gradients giving the directions of steepest ascent of I(X; S ) and V are diﬀerent. Thus, locally we can always ﬁnd directions with negative value of information. Note that this does not mean that the agent prefers less information globally, i.e. there is no path, along which the value of information is strictly negative, from no information, e.g. (cid:15)1 = (cid:15)2 = 12 to full information ((cid:15)1 = (cid:15)2 = 0). Conversely, Cor. 2 implies that we can ﬁnd inﬁnitesimal changes to the game parameters that cause both expected utility and mutual information to increase, in agreement with Blackwell’s theorem. In the present example, such changes arise if we simultaneously reduce both channel noises, e.g. by moving directly towards the origin. 5.2 Leader-follower example Next, we turn to a simple games involving two players, namely a leader A1 and a follower A2 (see Fig. 4). In contrast to the single-player game, now the distribution p(X) of the state of Nature is replaced by the equilibrium strategy σ(a1) of player 1, a strategy that will also depend on the parameters θ. Another diﬀerence is that there are now two utility functions, one for each player. As in the decision problem, we consider binary state spaces and an asymmetric binary channel with parameters θ = ((cid:15)1, (cid:15)2). We use the utility functions of the players analyzed in [4]19 follower u /u L R 1 2 r L (5, 2) (3, 1) e d a R (6, 3) (4, 4) e l 18Consider a (joint) distribution(cid:88)p(x;θ) that is parameterized by θ. Then, the Fisher information metric g is deﬁned as ij ∂log p(x;θ) ∂log p(x;θ) gij = p(x;θ) ∂θk ∂θl x The statistical origin of the Fisher metric lies in the task of estimating a probability distribution from a family parametrized by θ from observations of the variable x. The Fisher metric expresses the sensitivity of the dependence of the family on θ, that is, how well observations of x can discriminate among nearby values of θ. 19The game is a discretization of the Stackelberg duopoly game, with two moves for each player. 32 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: This

text: potential

text: for

text: an

text: inconsistency

text: between

text: changes

text: to

text: information

text: and

text: util-

text: ity

text: is

text: also

text: illustrated

text: by

text: considering

text: the

text: gradient

text: vectors

text: gradI(X;

font: XKVPOB+txtt

text: S

text: and

text: grad(V)

text: which

text: are

text: orthogonal

text: to

text: the

text: isoclines

text: (wrt/

text: the

text: Fisher

text: By

text: Prop.

text: only

text: if

text: those

text: gradients

text: are

text: collinear

text: is

text: it

text: the

text: case

text: that

text: every

text: change

text: to

text: the

text: game

text: parameter

text: vector

text: increasing

text: the

text: expected

text: utility

text: must

text: necessarily

text: increase

text: the

text: mutual

text: infor-

text: mation.

text: However

text: in

text: Fig.

text: we

text: clearly

text: see

text: that

text: these

text: gradients

text: giving

text: the

text: directions

text: of

text: steepest

text: ascent

text: of

text: I(X;

text: S

text: and

text: V

text: are

text: diﬀerent.

text: Thus,

text: locally

text: we

text: can

text: always

text: ﬁnd

text: directions

text: with

text: negative

text: value

text: of

text: information.

text: Note

text: that

text: this

text: does

text: not

text: mean

text: that

text: the

text: agent

text: prefers

text: less

text: information

text: globally,

text: i.e.

text: there

text: is

text: no

text: path,

text: along

text: which

text: the

text: value

text: of

text: information

text: is

text: strictly

text: negative,

text: from

text: no

text: information,

text: e.g.

text: to

text: full

text: information

text: Conversely,

text: Cor.

text: implies

text: that

text: we

text: can

text: ﬁnd

text: inﬁnitesimal

text: changes

text: to

text: the

text: game

text: parameters

text: that

text: cause

text: both

text: expected

text: utility

text: and

text: mutual

text: information

text: to

text: increase,

text: in

text: agreement

text: with

text: Blackwell’s

text: theorem.

text: In

text: the

text: present

text: example,

text: such

text: changes

text: arise

text: if

text: we

text: simultaneously

text: reduce

text: both

text: channel

text: noises,

text: e.g.

text: by

text: moving

text: directly

text: towards

text: the

text: origin.

text: Leader-follower

text: example

text: Next,

text: we

text: turn

text: to

text: a

text: simple

text: games

text: involving

text: two

text: players,

text: namely

text: a

text: leader

text: and

text: a

text: follower

text: (see

text: Fig.

text: In

text: contrast

text: to

text: the

text: single-player

text: game,

text: now

text: the

text: distribution

text: p(X)

text: of

text: the

text: state

text: of

text: Nature

text: is

text: replaced

text: by

text: the

text: equilibrium

text: strategy

text: of

text: player

text: a

text: strategy

text: that

text: will

text: also

text: depend

text: on

text: the

text: parameters

text: θ.

text: Another

text: diﬀerence

text: is

text: that

text: there

text: are

text: now

text: two

text: utility

text: functions,

text: one

text: for

text: each

text: player.

text: As

text: in

text: the

text: decision

text: problem,

text: we

text: consider

text: binary

text: state

text: spaces

text: and

text: an

text: asymmetric

text: binary

text: channel

text: with

text: parameters

text: θ

text: We

text: use

text: the

text: utility

text: functions

text: of

text: the

text: players

text: analyzed

text: in

text: follower

text: u

text: /u

text: L

text: R

text: r

text: L

text: e

text: d

text: a

text: R

text: e

text: l

text: a

text: (joint)

text: that

text: is

text: parameterized

text: by

text: θ.

text: Then,

text: the

text: Fisher

text: information

text: metric

text: g

text: is

text: deﬁned

text: as

text: ij

text: ∂log

text: p(x;θ)

text: ∂log

text: p(x;θ)

text: gij

text: p(x;θ)

text: ∂θk

text: ∂θl

text: x

text: The

text: statistical

text: origin

text: of

text: the

text: Fisher

text: metric

text: lies

text: in

text: the

text: task

text: of

text: estimating

text: a

text: probability

text: distribution

text: from

text: a

text: family

text: parametrized

text: by

text: θ

text: from

text: observations

text: of

text: the

text: variable

text: x.

text: The

text: Fisher

text: metric

text: expresses

text: the

text: sensitivity

text: of

text: the

text: dependence

text: of

text: the

text: family

text: on

text: θ,

text: that

text: is,

text: how

text: well

text: observations

text: of

text: x

text: can

text: discriminate

text: among

text: nearby

text: values

text: of

text: θ.

text: game

text: is

text: a

text: discretization

text: of

text: the

text: Stackelberg

text: duopoly

text: game,

text: with

text: two

text: moves

text: for

text: each

text: player.

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Isoclines: Beta = 5 0.5 0.6 V 0.45 MI grad V 0.5 0.4 grad MI 0.35 0.4 0.3 0.3 2 s 0.25 p e 0.2 0.2 0.15 0.1 0.1 0 0.05 0 -0.1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 eps1 Figure 3: Isoclines of expected utility V and mutual information I(X; S ) with corresponding gradient vectors showing the directions of steepest ascent (wrt/ the Fisher metric). The gray regions show which channels are more informative (dark gray) or less informative (light gray) than the channel with noise parameters ((cid:15)1 = 0.17, (cid:15)2 = 0.22), in the sense of the term arising in Blackwell’s analysis. Bagwell pointed out that in the pure strategy Nash equilibria the leader can only take advantage of moving ﬁrst (by playing L), when the follower can observe his move perfectly (Stackelberg solution). As soon as the slightest amount on noise is added to the channel only the equilibrium of the simultaneous move game (both playing R, the Cournot solution) remains20. Here we show that our diﬀerential analysis uncovers a much richer structure. In particular, we show there exist a QRE branch and parameters for the noise of the channel such that both players prefer more noise. In the decision case, we used I(X; S ) to quantify the amount of information that is available to the (single) player. In the multi-player game setting, the cor- responding quantity is I(A ; S ), and strongly depends on the move of the leader. 1 | As an illustration, consider a symmetric channel p(s a ) parametrized by the single ≡ 1 value (cid:15) (cid:15)1,2. Fig. 5 shows how the strategy of the leader depends on the channel noise and the rationality β = β1,2 of the players. This shows that for suﬃciently 20There are additional mixed equilibria, which change smoothly with the noise. These are men- 33 ##LTFigure## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Isoclines:

font: default

text: Beta

text: V

text: MI

text: grad

text: V

text: grad

text: MI

text: s

text: p

text: e

text: Figure

text: Isoclines

label: caption

text: of

text: expected

text: utility

text: V

text: and

text: mutual

text: information

text: I(X;

text: S

text: with

text: corresponding

text: gradient

text: vectors

text: showing

text: the

text: directions

text: of

text: steepest

text: ascent

text: (wrt/

text: the

text: Fisher

text: metric).

text: The

text: gray

text: regions

text: show

text: which

text: channels

text: are

text: more

text: informative

text: (dark

text: gray)

text: or

text: less

text: informative

text: (light

text: gray)

text: than

text: the

text: channel

text: with

text: noise

text: parameters

text: in

text: the

text: sense

text: of

text: the

text: term

text: arising

text: in

text: Blackwell’s

text: analysis.

text: Bagwell

text: pointed

text: out

text: that

text: in

text: the

text: pure

text: strategy

text: Nash

text: equilibria

text: the

text: leader

text: can

text: only

text: take

text: advantage

text: of

text: moving

text: ﬁrst

text: (by

text: playing

text: L),

text: when

text: the

text: follower

text: can

text: observe

text: his

text: move

text: perfectly

text: (Stackelberg

text: solution).

text: As

text: soon

text: as

text: the

text: slightest

text: amount

text: on

text: noise

text: is

text: added

text: to

text: the

text: channel

text: only

text: the

text: equilibrium

text: of

text: the

text: simultaneous

text: move

text: game

text: (both

text: playing

text: R,

text: the

text: Cournot

text: solution)

text: Here

text: we

text: show

text: that

text: our

text: diﬀerential

text: analysis

text: uncovers

text: a

text: much

text: richer

text: structure.

text: In

text: particular,

text: we

text: show

text: there

text: exist

text: a

text: QRE

text: branch

text: and

text: parameters

text: for

text: the

text: noise

text: of

text: the

text: channel

text: such

text: that

text: both

text: players

text: prefer

text: more

text: noise.

text: In

text: the

text: decision

text: case,

text: we

text: used

text: I(X;

text: S

text: to

text: quantify

text: the

text: amount

text: of

text: information

text: that

text: is

text: available

text: to

text: the

text: (single)

text: player.

text: In

text: the

text: multi-player

text: game

text: setting,

text: the

text: cor-

text: responding

text: quantity

text: is

text: I(A

text: S

text: and

text: strongly

text: depends

text: on

text: the

text: move

text: of

text: the

text: leader.

text: As

text: an

text: illustration,

text: consider

text: a

text: symmetric

text: channel

text: p(s

text: a

text: parametrized

text: by

text: the

text: single

text: value

text: Fig.

text: shows

text: how

text: the

text: strategy

text: of

text: the

text: leader

text: depends

text: on

text: the

text: channel

text: noise

text: and

text: the

text: rationality

text: β

text: of

text: the

text: players.

text: This

text: shows

text: that

text: for

text: suﬃciently

text: are

text: additional

text: mixed

text: equilibria,

text: which

text: change

text: smoothly

text: with

text: the

text: noise.

text: These

text: are

text: men-

text: ##LTFigure##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: Overall, this gives rise to a total of M equations for M unknown quantities | σi(av xpa(v)), Zi(xpa(v)). Using a vector valued function f we can abbreviate the above by the following equation: f(σ , Z , β, θ) = 0 (30) β,θ β,θ (cid:89) where σ is a vector of all strategies β,θ { | ∈ D ∈ X ∈ X } σi(av xv; βi, θ) : i = 1, . . . , n, v i, av v, xv u , ∈ u Pa(v) Z collects all normalization constants, and 0 is the vector of all 0’s. Note that β,θ in general, even once the distributions at all decision nodes have been ﬁxed, the | distributions at chance nodes aﬀect the value of E(ui av, xpa(v); β, θ). Therefore they aﬀect the value of the function f. This is why f can depend explicitly on θ, as well as depend directly on β. The (vector-valued) p(cid:34)artial (cid:35)deriva(cid:34)tive of the p(cid:35)osition of the QRE in (σθ, Zθ) with respect to θ is then given by implicit diﬀerentiation of eq. (30) : − ∂σθ − ∂ f ∂ f 1 ∂ f ∂θ = (31) ∂Zθ ∂σ (cid:104) ∂Z ∂(cid:105)θ ∂θ θ θ where the dependence on β is hidden for clarity, all partial derivatives are evaluated at the QRE, and we assume that the matrix ∂f ∂f is invertible at the point θ at which we are evaluating the partial derivati∂vσeθs. ∂Zθ These equations give the partial derivatives of the mixed strategy proﬁle. They apply to any MAID, and allow us to write the partial derivatives of other quantities of interest. In particular, the partial derivative of the expected utility of any player i is (cid:88) (cid:88) (cid:88) (cid:89) | ∂Vi = ui(x)∂p(x; θ) = ui(x) ∂p(xv xpa(v); θ) p(xv(cid:48) | xpa(v(cid:48)); θ) ∂θ x∈XV ∂θ x∈XV v∈V ∂θ v(cid:48)(cid:44)v (32) | where each term ∂p(xv xpa(v);θ) is given by the appropriate component of Eq. (31) ∂θ | if v is a decision node. (For the other, chance nodes, ∂p(xv xpa(v);θ) can be calculated ∂θ directly). Similarly, the partial derivatives of other functions of interest such as mutual informations between certain nodes of the MAID can be calculated from Eq. (31). Evaluating those derivatives and the additional ones needed for the Fisher met- ric by hand can be very tedious, even for small games. Here, we used automatic 48 ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: Overall,

text: this

text: gives

text: rise

text: to

text: a

text: total

text: of

text: M

text: equations

text: for

text: M

text: unknown

text: quantities

text: σi(av

text: xpa(v)),

text: Zi(xpa(v)).

text: Using

text: a

text: vector

text: valued

text: function

text: f

text: we

text: can

text: abbreviate

text: the

text: above

text: by

text: the

text: following

text: equation:

text: f(σ

text: Z

text: β,

text: θ)

text: β,θ

text: β,θ

text: where

text: σ

text: is

text: a

text: vector

text: of

text: all

text: strategies

text: β,θ

text: D

text: X

text: X

text: σi(av

text: xv;

text: βi,

text: θ)

text: i

text: n,

text: v

text: i,

text: av

text: v,

text: xv

text: u

text: u

text: Pa(v)

text: Z

text: collects

text: all

text: normalization

text: constants,

text: and

text: is

text: the

text: vector

text: of

text: all

text: Note

text: that

text: β,θ

text: in

text: general,

text: even

text: once

text: the

text: distributions

text: at

text: all

text: decision

text: nodes

text: have

text: been

text: ﬁxed,

text: the

text: distributions

text: at

text: chance

text: nodes

text: aﬀect

text: the

text: value

text: of

text: E(ui

text: av,

text: xpa(v);

text: β,

text: θ).

text: Therefore

text: they

text: aﬀect

text: the

text: value

text: of

text: the

text: function

text: f.

text: This

text: is

text: why

text: f

text: can

text: depend

text: explicitly

text: on

text: θ,

text: as

text: well

text: as

text: depend

text: directly

text: on

text: β.

text: The

text: (vector-valued)

text: of

text: the

text: of

text: the

text: QRE

text: in

text: (σθ,

text: Zθ)

text: with

text: respect

text: to

text: θ

text: is

text: then

text: given

text: by

text: implicit

text: diﬀerentiation

text: of

text: eq.

text: ∂σθ

text: f

text: f

text: f

text: ∂θ

text: ∂Zθ

text: ∂σ

text: ∂Z

text: ∂θ

text: θ

text: θ

text: where

text: the

text: dependence

text: on

text: β

text: is

text: hidden

text: for

text: clarity,

text: all

text: partial

text: derivatives

text: are

text: evaluated

text: at

text: the

text: QRE,

text: and

text: we

text: assume

text: that

text: the

text: matrix

text: ∂f

text: ∂f

text: is

text: invertible

text: at

text: the

text: point

text: θ

text: at

text: which

text: we

text: are

text: evaluating

text: the

text: partial

text: derivati∂vσeθs.

text: ∂Zθ

text: These

text: equations

text: give

text: the

text: partial

text: derivatives

text: of

text: the

text: mixed

text: strategy

text: proﬁle.

text: They

text: apply

text: to

text: any

text: MAID,

text: and

text: allow

text: us

text: to

text: write

text: the

text: partial

text: derivatives

text: of

text: other

text: quantities

text: of

text: interest.

text: In

text: particular,

text: the

text: partial

text: derivative

text: of

text: the

text: expected

text: utility

text: of

text: any

text: player

text: i

text: is

text: ∂Vi

text: ui(x)∂p(x;

text: θ)

text: ui(x)

text: ∂p(xv

text: xpa(v);

text: θ)

text: θ)

text: ∂θ

text: x∈XV

text: ∂θ

text: x∈XV

text: v∈V

text: ∂θ

text: where

text: each

text: term

text: ∂p(xv

text: xpa(v);θ)

text: is

text: given

text: by

text: the

text: appropriate

text: component

text: of

text: Eq.

text: ∂θ

text: if

text: v

text: is

text: a

text: decision

text: node.

text: (For

text: the

text: other,

text: chance

text: nodes,

text: ∂p(xv

text: xpa(v);θ)

text: can

text: be

text: calculated

text: ∂θ

text: directly).

text: Similarly,

text: the

text: partial

text: derivatives

text: of

text: other

text: functions

text: of

text: interest

text: such

text: as

text: mutual

text: informations

text: between

text: certain

text: nodes

text: of

text: the

text: MAID

text: can

text: be

text: calculated

text: from

text: Eq.

text: Evaluating

text: those

text: derivatives

text: and

text: the

text: additional

text: ones

text: needed

text: for

text: the

text: Fisher

text: met-

text: ric

text: by

text: hand

text: can

text: be

text: very

text: tedious,

text: even

text: for

text: small

text: games.

text: Here,

text: we

text: used

text: automatic

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: (cid:104) (cid:105) gradV, δθ (cid:104) (cid:105) (1) grad f, δθ Thus, a change to the bundle has a high diﬀerential value of f , e.g. of weight, when that change (in the direction δθ) results in a large gain in utility for a small change in f . In some situations one might wish to speak of the “value of f ” without speci- fying a speciﬁc direction of change to the vector-valued argument of f . Indeed, all previous quantiﬁcations of “value of information” that we know of, both in game theory and in other ﬁelds (e.g., analysis of inﬂuence diagrams [14, 18]), quantiﬁes value of information without specifying a direction. We can extend our approach to provide such a “direction-free” quantiﬁcation of value of information, by choosing the direction δθ = grad f which results in the largest change of f , i.e. steepest ascent. Thus, we can quantify the “diﬀerential value of f ” as (cid:104) (cid:105) grad(V), grad f || || (2) grad f 2 The measure in (2) says that if a small change in the value of f leads to a big change in expected utility, f is more valuable than if the same change in expected utility required a bigger change in the value of f . Furthermore, it expresses the value of information in units of value (of an inﬁnitesimal change to f ) per unit of f . Thus, we follow the conventional terminology where we would typically say that how much the consumer values a change to good j is given by the associated change in utility divided by the amount of change in good j. (After all, that tells us change in utility per unit change in the good.) All of the reasoning above can be carried over from the case of a single con- sumer to apply to multi-player scenarios. To see how, ﬁrst note that in the reasoning above, θ is simply the parameter vector determining the utility of the consumer. In other words, it is the parameter vector specifying the details of a game being played by a decision maker in a game against Nature. So it naturally generalizes to a multi-player game, as the parameter vector specifying the details of such a game. Next, replace the consumer player in the reasoning above by a particular player in the multi-player game. The key to the reasoning above is that specifying θ speciﬁes the expected utility of the consumer player. In the case of the consumer, that map from parameter vector to expected utility is direct. In a multi-player game, that direct map becomes an indirect map speciﬁed in two stages: First by the equilibrium concept, taking θ to the mixed strategy proﬁle of all the players, 6 ##LTLine## ##LTLine##

label: equation

text: gradV,

text: δθ

text: grad

text: f,

text: δθ

text: Thus,

text: a

text: change

text: to

text: the

text: bundle

text: has

text: a

text: high

text: diﬀerential

text: value

text: of

text: f

text: e.g.

text: of

text: weight,

text: when

text: that

text: change

text: (in

text: the

text: direction

text: δθ)

text: results

text: in

text: a

text: large

text: gain

text: in

text: utility

text: for

text: a

text: small

text: change

text: in

text: f

text: In

text: some

text: situations

text: one

text: might

text: wish

text: to

text: speak

text: of

text: the

text: “value

text: of

text: f

text: without

text: speci-

text: fying

text: a

text: speciﬁc

text: direction

text: of

text: change

text: to

text: the

text: vector-valued

text: argument

text: of

text: f

text: Indeed,

text: all

text: previous

text: quantiﬁcations

text: of

text: “value

text: of

text: information”

text: that

text: we

text: know

text: of,

text: both

text: in

text: game

text: theory

text: and

text: in

text: other

text: ﬁelds

text: (e.g.,

text: analysis

text: of

text: inﬂuence

text: diagrams

text: quantiﬁes

text: value

text: of

text: information

text: without

text: specifying

text: a

text: direction.

text: We

text: can

text: extend

text: our

text: approach

text: to

text: provide

text: such

text: a

text: “direction-free”

text: quantiﬁcation

text: of

text: value

text: of

text: information,

text: by

text: choosing

text: the

text: direction

text: δθ

text: grad

text: f

text: which

text: results

text: in

text: the

text: largest

text: change

text: of

text: f

text: i.e.

text: steepest

text: ascent.

text: Thus,

text: we

text: can

text: quantify

text: the

text: “diﬀerential

text: value

text: of

text: f

text: as

text: grad(V),

text: grad

text: f

text: grad

text: f

text: The

text: measure

text: in

text: says

text: that

text: if

text: a

text: small

text: change

text: in

text: the

text: value

text: of

text: f

text: leads

text: to

text: a

text: big

text: change

text: in

text: expected

text: utility,

text: f

text: is

text: more

text: valuable

text: than

text: if

text: the

text: same

text: change

text: in

text: expected

text: utility

text: required

text: a

text: bigger

text: change

text: in

text: the

text: value

text: of

text: f

text: Furthermore,

text: it

text: expresses

text: the

text: value

text: of

text: information

text: in

text: units

text: of

text: value

text: (of

text: an

text: inﬁnitesimal

text: change

text: to

text: f

text: per

text: unit

text: of

text: f

text: Thus,

text: we

text: follow

text: the

text: conventional

text: terminology

text: where

text: we

text: would

text: typically

text: say

text: that

text: how

text: much

text: the

text: consumer

text: values

text: a

text: change

text: to

text: good

text: j

text: is

text: given

text: by

text: the

text: associated

text: change

text: in

text: utility

text: divided

text: by

text: the

text: amount

text: of

text: change

text: in

text: good

text: j.

text: (After

text: all,

text: that

text: tells

text: us

text: change

text: in

text: utility

text: per

text: unit

text: change

text: in

text: the

text: good.)

text: All

text: of

text: the

text: reasoning

text: above

text: can

text: be

text: carried

text: over

text: from

text: the

text: case

text: of

text: a

text: single

text: con-

text: sumer

text: to

text: apply

text: to

text: multi-player

text: scenarios.

text: To

text: see

text: how,

text: ﬁrst

text: note

text: that

text: in

text: the

text: reasoning

text: above,

text: θ

text: is

text: simply

text: the

text: parameter

text: vector

text: determining

text: the

text: utility

text: of

text: the

text: consumer.

text: In

text: other

text: words,

text: it

text: is

text: the

text: parameter

text: vector

text: specifying

text: the

text: details

text: of

text: a

text: game

text: being

text: played

text: by

text: a

text: decision

text: maker

text: in

text: a

text: game

text: against

text: Nature.

text: So

text: it

text: naturally

text: generalizes

text: to

text: a

text: multi-player

text: game,

text: as

text: the

text: parameter

text: vector

text: specifying

text: the

text: details

text: of

text: such

text: a

text: game.

text: Next,

text: replace

text: the

text: consumer

text: player

text: in

text: the

text: reasoning

text: above

text: by

text: a

text: particular

text: player

text: in

text: the

text: multi-player

text: game.

text: The

text: key

text: to

text: the

text: reasoning

text: above

text: is

text: that

text: specifying

text: θ

text: speciﬁes

text: the

text: expected

text: utility

text: of

text: the

text: consumer

text: player.

text: In

text: the

text: case

text: of

text: the

text: consumer,

text: that

text: map

text: from

text: parameter

text: vector

text: to

text: expected

text: utility

text: is

text: direct.

text: In

text: a

text: multi-player

text: game,

text: that

text: direct

text: map

text: becomes

text: an

text: indirect

text: map

text: speciﬁed

text: in

text: two

text: stages:

text: First

text: by

text: the

text: equilibrium

text: concept,

text: taking

text: θ

text: to

text: the

text: mixed

text: strategy

text: proﬁle

text: of

text: all

text: the

text: players,

text: ##LTLine##

text: ##LTLine##


text: which we typically refer to as an information channel. If for some reason the player (cid:80) ignored x, then she would assign the a-priori likelihood p(y) to y, even though in | | fact its a-posteriori likelihood is p(y x) = (cid:48)p(py()yp(cid:48))(px(yx)|y(cid:48)). This change in the likeli- y hoods she would assign to y is a measure of the information that x provides about y. Arguably, this change of distribution is the core property of information that is of interest in economic scenarios. Averaging over observation x and state y’s, and working in log space, this change in the likelihood she would assign to the actual y if she ignored x (and | so used likelihoods p(y) (cid:88)rather than p(y x)(cid:20)) is (cid:21) | | p(y x) p(x)p(y x) ln . (3) p(y) x,y Thus, eq. (3) gives the (average) increase in information that player i has about y due to observing x. Note that this is true no matter how the variables X and Y arise in the strategic interaction. In particular, this interpretation of the quantity in Eq. (3) does not require that the value x arise directly through a pre-speciﬁed | distribution p(x y). x and y could instead be variables concerning the strategies of the players at a particular equilibrium. In this sense, we have shown that the quantity in Eq. (3) is a proper way to measure the information relating any pair of variables arising in a strategic sce- nario. None of the usual axiomatic arguments motivating Shannon’s information theory [24, 10] were used in doing this. Nevertheless, the quantity in Eq. (3) is just the mutual information between X and Y, as deﬁned by Shannon. Mutual informa- tion, together with the related quantity of entropy, forms the basis of information theory. It not only allows us to quantify information, but has many applications in diﬀerent areas ranging from coding theory to machine learning to evolutionary biology. In game theory, information is more commonly expressed in terms of an infor- {A A } mation partition. An information partition 1, . . . , ∈nX(on{ some m}easure space Ω) can be viewed as a random variable with values x = 1, . . . , n , i.e. the sig- nal x reveals which element of the partition was hit. Coarsening that partition can | ∈ X then be considered as a (deterministic) information channel p(y x) from x to ∈ Y a value y in the coarser partition. Now when we want to evaluate how much information the agent obtains from the coarser partition Y about some other ran- → → dom variable N, e.g. corresponding to a state of Nature, we see that N X Y is a Markov chain5. Thus, the data-processing inequality6 applies and the mutual in- → → | | 5X Y Z are said to form a Markov chain when p(x,y,z) = p(x)p(y x)p(z y). 6The data processing inequality is an important theorem in information theory which shows that 9 ##LTLine## ##LTLine## ##LTLine##

text: which

text: we

text: typically

text: refer

text: to

text: as

text: an

text: information

text: channel.

text: If

text: for

text: some

text: reason

text: the

text: player

text: ignored

text: x,

text: then

text: she

text: would

text: assign

text: the

text: a-priori

text: likelihood

text: p(y)

text: to

text: y,

text: even

text: though

text: in

text: fact

text: its

text: a-posteriori

text: likelihood

text: is

text: p(y

text: x)

text: This

text: change

text: in

text: the

text: likeli-

text: y

text: hoods

text: she

text: would

text: assign

text: to

text: y

text: is

text: a

text: measure

text: of

text: the

text: information

text: that

text: x

text: provides

text: about

text: y.

text: Arguably,

text: this

text: change

text: of

text: distribution

text: is

text: the

text: core

text: property

text: of

text: information

text: that

text: is

text: of

text: interest

text: in

text: economic

text: scenarios.

text: Averaging

text: over

text: observation

text: x

text: and

text: state

text: y’s,

text: and

text: working

text: in

text: log

text: space,

text: this

text: change

text: in

text: the

text: likelihood

text: she

text: would

text: assign

text: to

text: the

text: actual

text: y

text: if

text: she

text: ignored

text: x

text: (and

text: so

text: used

text: likelihoods

text: p(y)

text: than

text: p(y

text: is

text: p(y

text: x)

text: p(x)p(y

text: x)

text: ln

text: p(y)

text: x,y

text: Thus,

text: eq.

text: gives

text: the

text: (average)

text: increase

text: in

text: information

text: that

text: player

text: i

text: has

text: about

text: y

text: due

text: to

text: observing

text: x.

text: Note

text: that

text: this

text: is

text: true

text: no

text: matter

text: how

text: the

text: variables

text: X

text: and

text: Y

text: arise

text: in

text: the

text: strategic

text: interaction.

text: In

text: particular,

text: this

text: interpretation

text: of

text: the

text: quantity

text: in

text: Eq.

text: does

text: not

text: require

text: that

text: the

text: value

text: x

text: arise

text: directly

text: through

text: a

text: pre-speciﬁed

text: distribution

text: p(x

text: y).

text: x

text: and

text: y

text: could

text: instead

text: be

text: variables

text: concerning

text: the

text: strategies

text: of

text: the

text: players

text: at

text: a

text: particular

text: equilibrium.

text: In

text: this

text: sense,

text: we

text: have

text: shown

text: that

text: the

text: quantity

text: in

text: Eq.

text: is

text: a

text: proper

text: way

text: to

text: measure

text: the

text: information

text: relating

text: any

text: pair

text: of

text: variables

text: arising

text: in

text: a

text: strategic

text: sce-

text: nario.

text: of

text: the

text: usual

text: axiomatic

text: arguments

text: motivating

text: Shannon’s

text: information

text: theory

text: were

text: used

text: in

text: doing

text: this.

text: Nevertheless,

text: the

text: quantity

text: in

text: Eq.

text: is

text: just

text: the

text: mutual

text: information

text: between

text: X

text: and

text: Y,

text: as

text: deﬁned

text: by

text: Shannon.

text: Mutual

text: informa-

text: tion,

text: together

text: with

text: the

text: related

text: quantity

text: of

text: entropy,

text: forms

text: the

text: basis

text: of

text: information

text: theory.

text: It

text: not

text: only

text: allows

text: us

text: to

text: quantify

text: information,

text: but

text: has

text: many

text: applications

text: in

text: diﬀerent

text: areas

text: ranging

text: from

text: coding

text: theory

text: to

text: machine

text: learning

text: to

text: evolutionary

text: biology.

text: In

text: game

text: theory,

text: information

text: is

text: more

text: commonly

text: expressed

text: in

text: terms

text: of

text: an

text: infor-

text: {A

text: A

text: mation

text: partition.

text: An

text: information

text: partition

text: ∈nX(on{

text: some

text: m}easure

text: space

text: Ω)

text: can

text: be

text: viewed

text: as

text: a

text: random

text: variable

text: with

text: values

text: x

text: n

text: i.e.

text: the

text: sig-

text: nal

text: x

text: reveals

text: which

text: element

text: of

text: the

text: partition

text: was

text: hit.

text: Coarsening

text: that

text: partition

text: can

text: X

text: then

text: be

text: considered

text: as

text: a

text: (deterministic)

text: information

text: channel

text: p(y

text: x)

text: from

text: x

text: to

text: Y

text: a

text: value

text: y

text: in

text: the

text: coarser

text: partition.

text: Now

text: when

text: we

text: want

text: to

text: evaluate

text: how

text: much

text: information

text: the

text: agent

text: obtains

text: from

text: the

text: coarser

text: partition

text: Y

text: about

text: some

text: other

text: ran-

text: dom

text: variable

text: N,

text: e.g.

text: corresponding

text: to

text: a

text: state

text: of

text: Nature,

text: we

text: see

text: that

text: N

text: X

text: Y

text: is

text: a

text: Markov

text: Thus,

text: the

text: data-processing

text: applies

text: and

text: the

text: mutual

text: in-

text: Y

text: Z

text: are

text: said

text: to

text: form

text: a

text: Markov

text: chain

text: when

text: p(x,y,z)

text: p(x)p(y

text: x)p(z

text: y).

text: data

text: processing

text: inequality

text: is

text: an

text: important

text: theorem

text: in

text: information

text: theory

text: which

text: shows

text: that

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: of the process, both at leading order [22, 31], and at higher orders [16, 30]. Nevertheless, hh inclusive NLO [9, 32] and NNLO cross section calculations [20, 21] have been performed in the eﬀective theory approximation, giving an estimate of the size of the higher-order corrections in the full theory. Thus, in the absence of the full NLO calculation, the best one can do is merge samples of diﬀerent jet multiplicities in a consistent way, carefully avoiding any issues that may arise due to double-counting or phase space region mismatch. Such simulations have been shown to reliably describe the kinematical properties of experimental data (see for example the relevant experimental CMS [33] and ATLAS [34] analyses), modulo the correct normalisa- tion taken from higher-order cross section calculations. Here we perform such a merging of samples, including the full top and bottom mass dependence in the fermion loops of Fig. 1, as well as the higher-order real emission diagrams which we examine below. This paper is organised in the following way: in Section we brieﬂy describe the OpenLoops generator for one-loop matrix elements and provide cross sections for the various contributing exclusive channels. In Section we present results and examine the systematic uncertainties associated with the merging prescription, and in Section we investigate the phenomenological implications of including the merged higher-order matrix elements. We present our conclusions in Section . 2 OpenLoops and matrix elements 2.1 The OpenLoops matrix element generator The OpenLoops generator is based on the open-loops algorithm [35] for the eﬃcient evaluation of one-loop matrix elements. The algorithm employs a numerical recursion to construct the loop momentum dependence of the numerator of loop amplitudes combined with tensor integral reduction. The tensor integrals are computed by the Collier library, which implements the Denner-Dittmaier reduction procedure for the numerically stable evaluation of tensor integrals [36, 37] and the scalar integrals of Ref. [38]. Incidentally, using tensor integrals allows for a high degree of optimisation through caching, since the integrals can be shared across diﬀerent Feynman diagrams for all helicity and colour conﬁgurations. For on-shell reduction approaches this is only possible when the loop amplitude is interfered with a tree amplitude [35] and therefore not in calculations of loop induced processes, like the one presented here. 2.2 Higgs pair production matrix elements Like in the case of single Higgs boson production, the production cross section at hh hadron colliders is dominated by the gluon fusion channels. For a more detailed dissection of the leading order cross section, we refer the reader to Section 2 of Ref. [18]. The classes of higher-order real emission diagrams that we include in our calculation are shown in Fig. 2. The real emission process also contains diagrams with , and qg q¯g qq¯ initial states which are subdominant but non-negligible and must be included for consistent merging since the parton shower introduces splittings on the initial state gluons g → qq¯ of the 0-jet matrix elements [39]. It is important to note that the diagrams that involve – 3 –

text: of

text: the

text: process,

text: both

text: at

text: leading

text: order

text: and

text: at

text: higher

text: orders

text: Nevertheless,

text: hh

text: inclusive

text: NLO

text: and

text: NNLO

text: cross

text: section

text: calculations

text: have

text: been

text: performed

text: in

text: the

text: eﬀective

text: theory

text: approximation,

text: giving

text: an

text: estimate

text: of

text: the

text: size

text: of

text: the

text: higher-order

text: corrections

text: in

text: the

text: full

text: theory.

text: Thus,

text: in

text: the

text: absence

text: of

text: the

text: full

text: NLO

text: calculation,

text: the

text: best

text: one

text: can

text: do

text: is

text: merge

text: samples

text: of

text: diﬀerent

text: jet

text: multiplicities

text: in

text: a

text: consistent

text: way,

text: carefully

text: avoiding

text: any

text: issues

text: that

text: may

text: arise

text: due

text: to

text: double-counting

text: or

text: phase

text: space

text: region

text: mismatch.

text: Such

text: simulations

text: have

text: been

text: shown

text: to

text: reliably

text: describe

text: the

text: kinematical

text: properties

text: of

text: experimental

text: data

text: (see

text: for

text: example

text: the

text: relevant

text: experimental

text: CMS

text: and

text: ATLAS

text: analyses),

text: modulo

text: the

text: correct

text: normalisa-

text: tion

text: taken

text: from

text: higher-order

text: cross

text: section

text: calculations.

text: Here

text: we

text: perform

text: such

text: a

text: merging

text: of

text: samples,

text: including

text: the

text: full

text: top

text: and

text: bottom

text: mass

text: dependence

text: in

text: the

text: fermion

text: loops

text: of

text: Fig.

text: as

text: well

text: as

text: the

text: higher-order

text: real

text: emission

text: diagrams

text: which

text: we

text: examine

text: below.

text: This

text: paper

text: is

text: organised

text: in

text: the

text: following

text: way:

text: in

text: Section

text: we

text: brieﬂy

text: describe

text: the

text: OpenLoops

text: generator

text: for

text: one-loop

text: matrix

text: elements

text: and

text: provide

text: cross

text: sections

text: for

text: the

text: various

text: contributing

text: exclusive

text: channels.

text: In

text: Section

text: we

text: present

text: results

text: and

text: examine

text: the

text: systematic

text: uncertainties

text: associated

text: with

text: the

text: merging

text: prescription,

text: and

text: in

text: Section

text: we

text: investigate

text: the

text: phenomenological

text: implications

text: of

text: including

text: the

text: merged

text: higher-order

text: matrix

text: elements.

text: We

text: present

text: our

text: conclusions

text: in

text: Section

text: OpenLoops

text: and

text: matrix

text: elements

text: The

text: OpenLoops

text: matrix

text: element

text: generator

text: The

text: OpenLoops

text: generator

text: is

text: based

text: on

text: the

text: open-loops

text: algorithm

text: for

text: the

text: eﬃcient

text: evaluation

text: of

text: one-loop

text: matrix

text: elements.

text: The

text: algorithm

text: employs

text: a

text: numerical

text: recursion

text: to

text: construct

text: the

text: loop

text: momentum

text: dependence

text: of

text: the

text: numerator

text: of

text: loop

text: amplitudes

text: combined

text: with

text: tensor

text: integral

text: reduction.

text: The

text: tensor

text: integrals

text: are

text: computed

text: by

text: the

text: Collier

text: library,

text: which

text: implements

text: the

text: Denner-Dittmaier

text: reduction

text: procedure

text: for

text: the

text: numerically

text: stable

text: evaluation

text: of

text: tensor

text: integrals

text: and

text: the

text: scalar

text: integrals

text: of

text: Ref.

text: Incidentally,

text: using

text: tensor

text: integrals

text: allows

text: for

text: a

text: high

text: degree

text: of

text: optimisation

text: through

text: caching,

text: since

text: the

text: integrals

text: can

text: be

text: shared

text: across

text: diﬀerent

text: Feynman

text: diagrams

text: for

text: all

text: helicity

text: and

text: colour

text: conﬁgurations.

text: For

text: on-shell

text: reduction

text: approaches

text: this

text: is

text: only

text: possible

text: when

text: the

text: loop

text: amplitude

text: is

text: interfered

text: with

text: a

text: tree

text: amplitude

text: and

text: therefore

text: not

text: in

text: calculations

text: of

text: loop

text: induced

text: processes,

text: like

text: the

text: one

text: presented

text: here.

text: Higgs

text: pair

text: production

text: matrix

text: elements

text: Like

text: in

text: the

text: case

text: of

text: single

text: Higgs

text: boson

text: production,

text: the

text: production

text: cross

text: section

text: at

text: hh

text: hadron

text: colliders

text: is

text: dominated

text: by

text: the

text: gluon

text: fusion

text: channels.

text: For

text: a

text: more

text: detailed

text: dissection

text: of

text: the

text: leading

text: order

text: cross

text: section,

text: we

text: refer

text: the

text: reader

text: to

text: Section

text: of

text: Ref.

text: The

text: classes

text: of

text: higher-order

text: real

text: emission

text: diagrams

text: that

text: we

text: include

text: in

text: our

text: calculation

text: are

text: shown

text: in

text: Fig.

text: The

text: real

text: emission

text: process

text: also

text: contains

text: diagrams

text: with

text: and

text: qg

text: q¯g

text: qq¯

text: initial

text: states

text: which

text: are

text: subdominant

text: but

text: non-negligible

text: and

text: must

text: be

text: included

text: for

text: consistent

text: merging

text: since

text: the

text: parton

text: shower

text: introduces

text: splittings

text: on

text: the

text: initial

text: state

text: gluons

text: g

text: qq¯

text: of

text: the

text: matrix

text: elements

text: It

text: is

text: important

text: to

text: note

text: that

text: the

text: diagrams

text: that

text: involve


text: Correlation eﬀects in (111) bilayers of perovskite transition-metal oxides ∗ Satoshi Okamoto,1, Wenguang Zhu,2 Yusuke Nomura,3 Ryotaro Arita,3 Di Xiao,4 and Naoto Nagaosa3,5 1Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA 2ICQD/HFNL, University of Science and Technology of China, Hefei, Anhui 230026, China 3Department of Applied Physics, the University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113-8656, Japan 4Department of Physics, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA 5RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan We investigate the correlation-induced Mott, magnetic and topological phase transitions in artiﬁ- cial (111) bilayers of perovskite transition-metal oxides LaAuO and SrIrO for which the previous 3 3 density-functional theory calculations predicted topological insulating states. Using the dynamical- mean-ﬁeld theory with realistic band structures and Coulomb interactions, LaAuO bilayer is shown 3 to be far away from a Mott insulating regime, and a topological-insulating state is robust. On the other hand, SrIrO bilayer is on the verge of an orbital-selective topological Mott transition and turns 3 to a trivial insulator by an antiferromagnetic ordering. Oxide bilayers thus provide a novel class of topological materials for which the interplay between the spin-orbit coupling and electron-electron interactions is a fundamental ingredient. PACS numbers: 73.21.-b 73.43.-f, 71.27.+a I. INTRODUCTION (a) (b) B 2 B B Topological insulators (TIs) are novel quantum states B B’ B1 B1 of matter characterized by the nontrivial band topol- O ogy due to the relativistic spin-orbit coupling (SOC) [1– 7]. While the TI states are essentially single-particle B’ A B A B B 2 2 phenomena, the interplay between the SOC and strong B B Coulomb interactions in electronic systems has gained z z B1 considerable attention [8–10]. A variety of eﬀects be- x y x y tween these two couplings has been theoretically ex- plored, such as TI states induced by Coulomb interac- FIG. 1: (Color online) Buckled honeycomb lattice formed in tions [11, 12], competition and/or cooperation between (111) bilayer of perovskite TMO ABO . (a) (111) bilayer 3 the two in a Kane-Mele-Hubbard (KMH) model[13–15] of ABO sandwiched by insulating AB’O . (b) ABO forms 3 3 3 or a Bernevig-Hughes-Zhang (BHZ) model [16–18], and buckled honeycomb lattice with two sublattices B and B . 1 2 topological Kondo insulators [19–21]. Recently, based on the tight-binding modeling and sub- sequent density-functional theory (DFT) calculations, accurately using DFT [22, 28], including the SOC, the bilayers of perovskite transition-metal oxides (TMOs) eﬀect of electron-electron interactions has not been ad- ABO grown along the [111] crystallographic axis, i.e., dressed except for limiting cases for such TMO bilayers 3 (111) bilayers, were proposed as possible candidates for [22, 29–35]. two-dimensional TIs[22]. In such (111) bilayers, buckled In this paper, we provide concrete examples of the in- honeycomb lattice is formed by B-site transition-metal terplay between the SOC and correlation eﬀects in (111) ions (Fig. 1). TMOs are known to cover a vast extension TMO bilayers. As speciﬁc examples, we consider SrIrO 3 of unconventional phenomena due to the strong electron- (SIO) and LaAuO (LAO), for which the previous DFT 3 electron and electron-lattice interactions [23]. Because of calculations predicted TI states [22], and investigate the the active orbital degrees of freedom, the interplay be- correlation-induced phase transitions by means of the tween the SOC and correlation eﬀects could induce phe- dynamical-mean-ﬁeld theory (DMFT) [36]. It is shown nomena that are absent in abstract theoretical models or that LAO bilayer is far from Mott insulating and antifer- those for semiconductors, such as a spin-orbit Mott insu- romagnetic (AF) insulating regimes, and a TI state is ro- lator [24]. Furthermore, the recent development in syn- bust. On the contrary, the correlation eﬀect is signiﬁcant thesizing artiﬁcial TMO heterostructures provides great for SIO bilayer, and an AF trivial insulating state is re- tunability over fundamental physical parameters [25]. alized. This is induced by a relatively narrow bandwidth Once realized in TMOs, combining TIs with other novel of near-Fermi-level states with the dominant J = 1/2 eff states would allow us to study further novel phenomena character, which undergoes an orbital-selective topolog- utilizing proximity eﬀects involving TIs [26, 27]. How- ical Mott transition when the magnetic ordering is sup- ever, while single-electron properties are treated rather pressed. Our results indicate that the interplay between ##LTFigure## ##LTFigure##

text: Correlation

text: eﬀects

text: in

text: bilayers

text: of

text: perovskite

text: transition-metal

text: oxides

text: Satoshi

text: Wenguang

text: Yusuke

text: Ryotaro

text: Di

text: and

text: Naoto

text: Science

text: and

text: Technology

text: Division,

text: Oak

text: Ridge

text: National

text: Laboratory,

text: Oak

text: Ridge,

text: Tennessee

text: USA

text: University

text: of

text: Science

text: and

text: Technology

text: of

text: China,

text: Hefei,

text: Anhui

text: China

text: of

text: Applied

text: Physics,

text: the

text: University

text: of

text: Tokyo,

text: Hongo,

text: Bunkyo-ku,

text: Tokyo

text: Japan

text: of

text: Physics,

text: Carnegie

text: Mellon

text: University,

text: Pittsburgh,

text: Pennsylvania

text: USA

text: Center

text: for

text: Emergent

text: Matter

text: Science

text: (CEMS),

text: Wako,

text: Saitama

text: Japan

text: We

text: investigate

text: the

text: correlation-induced

text: Mott,

text: magnetic

text: and

text: topological

text: phase

text: transitions

text: in

text: artiﬁ-

text: cial

text: bilayers

text: of

text: perovskite

text: transition-metal

text: oxides

text: LaAuO

text: and

text: SrIrO

text: for

text: which

text: the

text: previous

text: density-functional

text: theory

text: calculations

text: predicted

text: topological

text: insulating

text: states.

text: Using

text: the

text: dynamical-

text: mean-ﬁeld

text: theory

text: with

text: realistic

text: band

text: structures

text: and

text: Coulomb

text: interactions,

text: LaAuO

text: bilayer

text: is

text: shown

text: to

text: be

text: far

text: away

text: from

text: a

text: Mott

text: insulating

text: regime,

text: and

text: a

text: topological-insulating

text: state

text: is

text: robust.

text: On

text: the

text: other

text: hand,

text: SrIrO

text: bilayer

text: is

text: on

text: the

text: verge

text: of

text: an

text: orbital-selective

text: topological

text: Mott

text: transition

text: and

text: turns

text: to

text: a

text: trivial

text: insulator

text: by

text: an

text: antiferromagnetic

text: ordering.

text: Oxide

text: bilayers

text: thus

text: provide

text: a

text: novel

text: class

text: of

text: topological

text: materials

text: for

text: which

text: the

text: interplay

text: between

text: the

text: spin-orbit

text: coupling

text: and

text: electron-electron

text: interactions

text: is

text: a

text: fundamental

text: ingredient.

text: PACS

text: numbers:

text: I.

text: INTRODUCTION

text: (a)

text: (b)

text: B

text: B

text: B

text: Topological

text: insulators

text: (TIs)

text: are

text: novel

text: quantum

text: states

text: B

text: B’

text: of

text: matter

text: characterized

text: by

text: the

text: nontrivial

text: band

text: topol-

text: O

text: ogy

text: due

text: to

text: the

text: relativistic

text: spin-orbit

text: coupling

text: (SOC)

text: While

text: the

text: TI

text: states

text: are

text: essentially

text: single-particle

text: B’

text: A

text: B

text: A

text: B

text: B

text: phenomena,

text: the

text: interplay

text: between

text: the

text: SOC

text: and

text: strong

text: B

text: B

text: Coulomb

text: interactions

text: in

text: electronic

text: systems

text: has

text: gained

text: z

text: z

text: considerable

text: attention

text: A

text: variety

text: of

text: eﬀects

text: be-

text: x

text: y

text: x

text: y

text: tween

text: these

text: two

text: couplings

text: has

text: been

text: theoretically

text: ex-

text: plored,

text: such

text: as

text: TI

text: states

text: induced

text: by

text: Coulomb

text: interac-

text: FIG.

text: (Color

text: online)

text: Buckled

text: honeycomb

text: lattice

text: formed

text: in

text: tions

text: competition

text: and/or

text: cooperation

text: between

text: bilayer

text: of

text: perovskite

text: TMO

text: ABO

text: (a)

text: bilayer

text: the

text: two

text: in

text: a

text: Kane-Mele-Hubbard

text: (KMH)

text: of

text: ABO

text: sandwiched

text: by

text: insulating

text: AB’O

text: (b)

text: ABO

text: forms

text: or

text: a

text: Bernevig-Hughes-Zhang

text: (BHZ)

text: model

text: and

text: buckled

text: honeycomb

text: lattice

text: with

text: two

text: sublattices

text: B

text: and

text: B

text: topological

text: Kondo

text: insulators

text: Recently,

text: based

text: on

text: the

text: tight-binding

text: modeling

text: and

text: sub-

text: sequent

text: density-functional

text: theory

text: (DFT)

text: calculations,

text: accurately

text: using

text: DFT

text: including

text: the

text: SOC,

text: the

text: bilayers

text: of

text: perovskite

text: transition-metal

text: oxides

text: (TMOs)

text: eﬀect

text: of

text: electron-electron

text: interactions

text: has

text: not

text: been

text: ad-

text: ABO

text: grown

text: along

text: the

text: crystallographic

text: axis,

text: i.e.,

text: dressed

text: except

text: for

text: limiting

text: cases

text: for

text: such

text: TMO

text: bilayers

text: bilayers,

text: were

text: proposed

text: as

text: possible

text: candidates

text: for

text: two-dimensional

text: In

text: such

text: bilayers,

text: buckled

text: In

text: this

text: paper,

text: we

text: provide

text: concrete

text: examples

text: of

text: the

text: in-

text: honeycomb

text: lattice

text: is

text: formed

text: by

text: B-site

text: transition-metal

text: terplay

text: between

text: the

text: SOC

text: and

text: correlation

text: eﬀects

text: in

text: ions

text: (Fig.

text: TMOs

text: are

text: known

text: to

text: cover

text: a

text: vast

text: extension

text: TMO

text: bilayers.

text: As

text: speciﬁc

text: examples,

text: we

text: consider

text: SrIrO

text: of

text: unconventional

text: phenomena

text: due

text: to

text: the

text: strong

text: electron-

text: (SIO)

text: and

text: LaAuO

text: (LAO),

text: for

text: which

text: the

text: previous

text: DFT

text: electron

text: and

text: electron-lattice

text: interactions

text: Because

text: of

text: calculations

text: predicted

text: TI

text: states

text: and

text: investigate

text: the

text: the

text: active

text: orbital

text: degrees

text: of

text: freedom,

text: the

text: interplay

text: be-

text: correlation-induced

text: phase

text: transitions

text: by

text: means

text: of

text: the

text: tween

text: the

text: SOC

text: and

text: correlation

text: eﬀects

text: could

text: induce

text: phe-

text: dynamical-mean-ﬁeld

text: theory

text: (DMFT)

text: It

text: is

text: shown

text: nomena

text: that

text: are

text: absent

text: in

text: abstract

text: theoretical

text: models

text: or

text: that

text: LAO

text: bilayer

text: is

text: far

text: from

text: Mott

text: insulating

text: and

text: antifer-

text: those

text: for

text: semiconductors,

text: such

text: as

text: a

text: spin-orbit

text: Mott

text: insu-

text: romagnetic

text: (AF)

text: insulating

text: regimes,

text: and

text: a

text: TI

text: state

text: is

text: ro-

text: lator

text: Furthermore,

text: the

text: recent

text: development

text: in

text: syn-

text: bust.

text: On

text: the

text: contrary,

text: the

text: correlation

text: eﬀect

text: is

text: signiﬁcant

text: thesizing

text: artiﬁcial

text: TMO

text: heterostructures

text: provides

text: great

text: for

text: SIO

text: bilayer,

text: and

text: an

text: AF

text: trivial

text: insulating

text: state

text: is

text: re-

text: tunability

text: over

text: fundamental

text: physical

text: parameters

text: alized.

text: This

text: is

text: induced

text: by

text: a

text: relatively

text: narrow

text: bandwidth

text: Once

text: realized

text: in

text: TMOs,

text: combining

text: TIs

text: with

text: other

text: novel

text: of

text: near-Fermi-level

text: states

text: with

text: the

text: dominant

text: J

text: eff

text: states

text: would

text: allow

text: us

text: to

text: study

text: further

text: novel

text: phenomena

text: character,

text: which

text: undergoes

text: an

text: orbital-selective

text: topolog-

text: utilizing

text: proximity

text: eﬀects

text: involving

text: TIs

text: How-

text: ical

text: Mott

text: transition

text: when

text: the

text: magnetic

text: ordering

text: is

text: sup-

text: ever,

text: while

text: single-electron

text: properties

text: are

text: treated

text: rather

text: pressed.

text: Our

text: results

text: indicate

text: that

text: the

text: interplay

text: between

text: ##LTFigure##

text: ##LTFigure##