text: Enabling Radiative Transfer on AMR grids in CRASH 7 1 1 0.1 0.1 x , no AMR HII 0.01 xHII, CRASH3 0.01 xHI, no AMR xHI, CRASH3 xHII, AMR (1 r.l.) 0.001 xHII, CRASH-AMR (no AMR) 0.001 xHI, AMR (1 r.l.) x , CRASH-AMR (no AMR) HI 0.0001 0.0001 xxHHIII 10 xxHHIII 0.01 1 0.1 0.001 0.01 4 4 T, CRASH3 T, no AMR T, CRASH-AMR (no AMR) T, AMR (1 r.l.) 2 2 1 1 1 T 1 T 0.1 0.1 0.01 0.01 0.001 0.001 1 1 0.1 0.1 0.01 xHeII, CRASH3 0.01 xHeII, no AMR xHeIII, CRASH3 xHeIII, no AMR 0.001 xHeII, CRASH-AMR (no AMR) 0.001 xHeII, AMR (1 r.l.) xHeIII, CRASH-AMR (no AMR) xHeIII, AMR (1 r.l.) 0.0001 0.0001 0.1 xxHHeeIIIII 1 xxHHeeIIIII 0.01 0.1 0.001 0.01 0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 d [kpc] d [kpc] Figure 3. Spherically-averaged proﬁles at time t = 500 Myr Figure 4. Spherically-averaged proﬁles at time t = 500 Myr for for Test 1a. The colours refer to CRASH3 (green) and CRASH-AMR Test 1b. The colours refer to CRASH-AMR (AMR disabled; red) and (AMR disabled; red). The bottom sub-panels show ∆ between CRASH-AMR with one reﬁnement level (1 r.l.; blue). The bottom the CRASH3 and CRASH-AMR (AMR disabled) results. Top: Proﬁles sub-panels show ∆ between the CRASH-AMR (AMR disabled) and of xHII (dashed lines) and xHI (dotted lines). Middle: Proﬁle of CRASH-AMR (1 r.l.) results. Top: Proﬁles of xHII (dashed lines) and T (dash-dot lines). Bottom: Proﬁles of xHeII (solid lines) and xHI (dashed lines). Middle: Proﬁle of T (dash-dot lines). Bot- xHeIII (dot-dash lines). tom: Proﬁles of xHeII (solid lines) and xHeIII (dot-dash lines). put the results at intermediate times t = 10, 50, 100 and and CRASH-AMR, respectively. From the Figure it is clear that 200 Myr as in the original set-up. It should be noted that there is no signiﬁcant diﬀerence between the two codes, while ∼ simpler tests (e.g., with a gas composed by H only or with the spikes that we see in ∆ (with a maximum of 1%, but the temperature kept constant) have been run as well, and mostly below 0.1%) are due to numerical artefacts caused by give results similar to those discussed in the following. optimisation of CRASH-AMR involving rearranging of double- 5.1.1 Test 1a: AMR disabled precision ﬂoating point arithmetic expressions, and are not due to the changes associated with CHOMBO. This shows that To verify that the changes done to enable RT on AMR the AMR feature in CRASH-AMR is isolated from the rest of the grids do not introduce any numerical noise, we have run code and can be disabled without introducing any numerical Test 1 with AMR disabled in CRASH-AMR and compared the noise into the results. results to those from CRASH3. 5.1.2 Test 1b: AMR enabled The outcome is given in Figure 3, where each panel shows spherically-averaged physical quantities as a function Our next step has been to run the previous test with of d, together with the percentage diﬀerence (∆) between the AMR enabled, reﬁning 1003 cells around the source by a − CRAS· H3 and the CRASH-AMR results. We deﬁne ∆ = (Rref factor of 2, with the ICs set up on the base grid as well Ri) 100 / Rref, where Rref and Ri refer to results of CRASH3 as on the reﬁned grid. 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text: Enabling

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text: al.,

text: preprint


text: Interpolating between matching and hedonic ∗ pricing models † Brendan Pass May 29, 2020 Abstract We consider the theoretical properties of a model which encompasses bi-partite matching under transferable utility on the one hand, and hedo- nic pricing on the other. This framework is intimately connected to tri- partite matching problems (known as multi-marginal optimal transport problems in the mathematical literature). We exploit this relationship in two ways; ﬁrst, we show that a known structural result from multi- marginal optimal transport can be used to establish an upper bound on the dimension of the support of stable matchings. Next, assuming the distribution of agents on one side of the market is continuous, we iden- tify a condition on their preferences that ensures purity and uniqueness of the stable matching; this condition is a variant of a known condition in the mathematical literature, which guarantees analogous properties in the multi-marginal optimal transport problem. We exhibit several examples of surplus functions for which our condition is satisﬁed, as well as some for which it fails. 1 Introduction This paper considers the theoretical properties of a general model, which encompasses both the transferable utility matching model of Shapley and Shubik [25] and Becker [1], extended to continuous type spaces by Gretsky, Ostroy and Zame [12], and the hedonic model of Rosen [22], whose theoretical properties (in a continuous, multi-dimensional setting) were studied by Ekeland [7,8] and Chiappori-McCann-Nesheim [5]. In the hedonic model, agents on two sides of a market (eg, buyers and sellers) are matched together according to their preferences to exchange certain goods, ∗ The author is pleased to acknowledge the support of a University of Alberta start-up grant and National Sciences and Engineering Research Council of Canada Discovery Grant number 412779-2012. † Department of Mathematical and Statistical Sciences, 632 CAB, University of Alberta, Edmonton, Alberta, Canada, T6G 2G1 pass@ualberta.ca. 1 ##LTLine##

text: Interpolating

label: title

text: between

text: matching

text: and

text: hedonic

text: pricing

text: models

label: author

text: Brendan

text: Pass

text: May

text: Abstract

text: We

label: abstract

text: consider

text: the

text: theoretical

text: properties

text: of

text: a

text: model

text: which

text: encompasses

text: bi-partite

text: matching

text: under

text: transferable

text: utility

text: on

text: the

text: one

text: hand,

text: and

text: hedo-

text: nic

text: pricing

text: on

text: the

text: other.

text: This

text: framework

text: is

text: intimately

text: connected

text: to

text: tri-

text: partite

text: matching

text: problems

text: (known

text: as

text: multi-marginal

text: optimal

text: transport

text: problems

text: in

text: the

text: mathematical

text: literature).

text: We

text: exploit

text: this

text: relationship

text: in

text: two

text: ways;

text: ﬁrst,

text: we

text: show

text: that

text: a

text: known

text: structural

text: result

text: from

text: multi-

text: marginal

text: optimal

text: transport

text: can

text: be

text: used

text: to

text: establish

text: an

text: upper

text: bound

text: on

text: the

text: dimension

text: of

text: the

text: support

text: of

text: stable

text: matchings.

text: Next,

text: assuming

text: the

text: distribution

text: of

text: agents

text: on

text: one

text: side

text: of

text: the

text: market

text: is

text: continuous,

text: we

text: iden-

text: tify

text: a

text: condition

text: on

text: their

text: preferences

text: that

text: ensures

text: purity

text: and

text: uniqueness

text: of

text: the

text: stable

text: matching;

text: this

text: condition

text: is

text: a

text: variant

text: of

text: a

text: known

text: condition

text: in

text: the

text: mathematical

text: literature,

text: which

text: guarantees

text: analogous

text: properties

text: in

text: the

text: multi-marginal

text: optimal

text: transport

text: problem.

text: We

text: exhibit

text: several

text: examples

text: of

text: surplus

text: functions

text: for

text: which

text: our

text: condition

text: is

text: satisﬁed,

text: as

text: well

text: as

text: some

text: for

text: which

text: it

text: fails.

text: Introduction

text: This

text: paper

text: considers

text: the

text: theoretical

text: properties

text: of

text: a

text: general

text: model,

text: which

text: encompasses

text: both

text: the

text: transferable

text: utility

text: matching

text: model

text: of

text: Shapley

text: and

text: Shubik

text: and

text: Becker

text: extended

text: to

text: continuous

text: type

text: spaces

text: by

text: Gretsky,

text: Ostroy

text: and

text: Zame

text: and

text: the

text: hedonic

text: model

text: of

text: Rosen

text: whose

text: theoretical

text: properties

text: (in

text: a

text: continuous,

text: multi-dimensional

text: setting)

text: were

text: studied

text: by

text: Ekeland

text: and

text: Chiappori-McCann-Nesheim

text: In

text: the

text: hedonic

text: model,

text: agents

text: on

text: two

text: sides

text: of

text: a

text: market

text: (eg,

text: buyers

text: and

text: sellers)

text: are

text: matched

text: together

text: according

text: to

text: their

text: preferences

text: to

text: exchange

text: certain

text: goods,

text: The

text: author

text: is

text: pleased

text: to

text: acknowledge

text: the

text: support

text: of

text: a

text: University

text: of

text: Alberta

text: start-up

text: grant

text: and

text: National

text: Sciences

text: and

text: Engineering

text: Research

text: Council

text: of

text: Canada

text: Discovery

text: Grant

text: number

text: Department

text: of

text: Mathematical

text: and

text: Statistical

text: Sciences,

text: CAB,

text: University

text: of

text: Alberta,

text: Edmonton,

text: Alberta,

text: Canada,

text: pass@ualberta.ca.

text: ##LTLine##


text: assuming they are indiﬀerent to which partner they do business with. Letting X and Y be spaces of buyers and sellers, respectively, and Z the set of goods that can feasibly be produced, we assume that agents’ preferences are encoded respectively by functions u(x, z) and v(y, z), expressing the utilities of buyer ∈ ∈ ∈ x X and seller y Y if they purchase or produce a product of type z Z, respectively. The main problem is then to determine which buyers match with which sellers, which goods they exchange and the prices they exchange for them, in equilibirum. In the matching model, agent x (respectively y) has a preference u(x, y) (respectively v(x, y)) to match with agent y (respectively x). Together, a match between x and y generates a joint utility, or surplus, s(x, y) = u(x, y) + v(x, y). Utility can then be transferred in the form of a payment from one agent to the other; by this mechanism, the total surplus s(x, y) can be divided in any way between the two agents. Here the good z to be exchanged (or the non monetary terms of the contract) does not aﬀect agents’ preferences. Recently, Dupuy, Galichon and Zhao [6] formulated a hybrid model in which agents have preferences which depend on both their partners and the product under exchange. In that paper, x and y represent agents on diﬀerent sides of a marriage market, and z the location where they would agree to settle; however, as was noted by the authors, the model has much wider potential applicabil- ity. In family economics, for instance, when a couple match together, there are many conditions of the match which can aﬀect the surplus generated. In addition to location, a couple may choose to marry or to live together without marrying, and to have one or more children, for example. These decisions aﬀect the surplus diﬀerently depending on the couple; see, for example, [17]. When considering buyers and sellers, it also seems natural in many scenarios to allow consumers’ preferences to depend on both the producer he does business with and the good he receives. For example, consumers often exhibit brand loyalty; they may be willing to pay more (for the same good) when dealing with one company rather than another. On the other hand, producers’ preferences can also depend on the consumers they sell to; for example, mortgage lenders of- ten oﬀer better rates to clients with higher credit scores, reﬂecting the fact that they prefer to do business with more credit worthy borrowers. Another example occurs in insurance models like the one of Rothschild and Stiglitz [23]; insur- ance companies oﬀer the same policy at diﬀerent prices to diﬀerent consumers depending on how their characteristics inﬂuence the risk of a claim. The theortical properties of these hybrid models do not seem to have received much attention. In this paper, we study this model when the type spaces X and Y and the space of feasible contracts Z are all continuous. In both the classical matching problem, and the hedonic problem, conditions are well known which ensure that the equilibrium, or stable matching, is unique and pure. In the matching problem, this condition is known as the twist, or generalized Spence- Mirrlees condition. One natural question in the present setting is to identify conditions on the joint surplus function (which is now a function of x, y and z, without any speciﬁc form) in the more general model which ensure purity and uniqueness of the stable match. 2

text: assuming

text: they

text: are

text: indiﬀerent

text: to

text: which

text: partner

text: they

text: do

text: business

text: with.

text: Letting

text: X

text: and

text: Y

text: be

text: spaces

text: of

text: buyers

text: and

text: sellers,

text: respectively,

text: and

text: Z

text: the

text: set

text: of

text: goods

text: that

text: can

text: feasibly

text: be

text: produced,

text: we

text: assume

text: that

text: agents’

text: preferences

text: are

text: encoded

text: respectively

text: by

text: functions

text: u(x,

text: z)

text: and

text: v(y,

text: z),

text: expressing

text: the

text: utilities

text: of

text: buyer

text: x

text: X

text: and

text: seller

text: y

text: Y

text: if

text: they

text: purchase

text: or

text: produce

text: a

text: product

text: of

text: type

text: z

text: Z,

text: respectively.

text: The

text: main

text: problem

text: is

text: then

text: to

text: determine

text: which

text: buyers

text: match

text: with

text: which

text: sellers,

text: which

text: goods

text: they

text: exchange

text: and

text: the

text: prices

text: they

text: exchange

text: for

text: them,

text: in

text: equilibirum.

text: In

text: the

text: matching

text: model,

text: agent

text: x

text: (respectively

text: y)

text: has

text: a

text: preference

text: u(x,

text: y)

text: (respectively

text: v(x,

text: y))

text: to

text: match

text: with

text: agent

text: y

text: (respectively

text: x).

text: Together,

text: a

text: match

text: between

text: x

text: and

text: y

text: generates

text: a

text: joint

text: utility,

text: or

text: surplus,

text: s(x,

text: y)

text: u(x,

text: y)

text: v(x,

text: y).

text: Utility

text: can

text: then

text: be

text: transferred

text: in

text: the

text: form

text: of

text: a

text: payment

text: from

text: one

text: agent

text: to

text: the

text: other;

text: by

text: this

text: mechanism,

text: the

text: total

text: surplus

text: s(x,

text: y)

text: can

text: be

text: divided

text: in

text: any

text: way

text: between

text: the

text: two

text: agents.

text: Here

text: the

text: good

text: z

text: to

text: be

text: exchanged

text: (or

text: the

text: non

text: monetary

text: terms

text: of

text: the

text: contract)

text: does

text: not

text: aﬀect

text: agents’

text: preferences.

text: Recently,

text: Dupuy,

text: Galichon

text: and

text: Zhao

text: formulated

text: a

text: hybrid

text: model

text: in

text: which

text: agents

text: have

text: preferences

text: which

text: depend

text: on

text: both

text: their

text: partners

text: and

text: the

text: product

text: under

text: exchange.

text: In

text: that

text: paper,

text: x

text: and

text: y

text: represent

text: agents

text: on

text: diﬀerent

text: sides

text: of

text: a

text: marriage

text: market,

text: and

text: z

text: the

text: location

text: where

text: they

text: would

text: agree

text: to

text: settle;

text: however,

text: as

text: was

text: noted

text: by

text: the

text: authors,

text: the

text: model

text: has

text: much

text: wider

text: potential

text: applicabil-

text: ity.

text: In

text: family

text: economics,

text: for

text: instance,

text: when

text: a

text: couple

text: match

text: together,

text: there

text: are

text: many

text: conditions

text: of

text: the

text: match

text: which

text: can

text: aﬀect

text: the

text: surplus

text: generated.

text: In

text: addition

text: to

text: location,

text: a

text: couple

text: may

text: choose

text: to

text: marry

text: or

text: to

text: live

text: together

text: without

text: marrying,

text: and

text: to

text: have

text: one

text: or

text: more

text: children,

text: for

text: example.

text: These

text: decisions

text: aﬀect

text: the

text: surplus

text: diﬀerently

text: depending

text: on

text: the

text: couple;

text: see,

text: for

text: example,

text: When

text: considering

text: buyers

text: and

text: sellers,

text: it

text: also

text: seems

text: natural

text: in

text: many

text: scenarios

text: to

text: allow

text: consumers’

text: preferences

text: to

text: depend

text: on

text: both

text: the

text: producer

text: he

text: does

text: business

text: with

text: and

text: the

text: good

text: he

text: receives.

text: For

text: example,

text: consumers

text: often

text: exhibit

text: brand

text: loyalty;

text: they

text: may

text: be

text: willing

text: to

text: pay

text: more

text: (for

text: the

text: same

text: good)

text: when

text: dealing

text: with

text: one

text: company

text: rather

text: than

text: another.

text: On

text: the

text: other

text: hand,

text: producers’

text: preferences

text: can

text: also

text: depend

text: on

text: the

text: consumers

text: they

text: sell

text: to;

text: for

text: example,

text: mortgage

text: lenders

text: of-

text: ten

text: oﬀer

text: better

text: rates

text: to

text: clients

text: with

text: higher

text: credit

text: scores,

text: reﬂecting

text: the

text: fact

text: that

text: they

text: prefer

text: to

text: do

text: business

text: with

text: more

text: credit

text: worthy

text: borrowers.

text: Another

text: example

text: occurs

text: in

text: insurance

text: models

text: like

text: the

text: one

text: of

text: Rothschild

text: and

text: Stiglitz

text: insur-

text: ance

text: companies

text: oﬀer

text: the

text: same

text: policy

text: at

text: diﬀerent

text: prices

text: to

text: diﬀerent

text: consumers

text: depending

text: on

text: how

text: their

text: characteristics

text: inﬂuence

text: the

text: risk

text: of

text: a

text: claim.

text: The

text: theortical

text: properties

text: of

text: these

text: hybrid

text: models

text: do

text: not

text: seem

text: to

text: have

text: received

text: much

text: attention.

text: In

text: this

text: paper,

text: we

text: study

text: this

text: model

text: when

text: the

text: type

text: spaces

text: X

text: and

text: Y

text: and

text: the

text: space

text: of

text: feasible

text: contracts

text: Z

text: are

text: all

text: continuous.

text: In

text: both

text: the

text: classical

text: matching

text: problem,

text: and

text: the

text: hedonic

text: problem,

text: conditions

text: are

text: well

text: known

text: which

text: ensure

text: that

text: the

text: equilibrium,

text: or

text: stable

text: matching,

text: is

text: unique

text: and

text: pure.

text: In

text: the

text: matching

text: problem,

text: this

text: condition

text: is

text: known

text: as

text: the

text: twist,

text: or

text: generalized

text: Spence-

text: Mirrlees

text: condition.

text: One

text: natural

text: question

text: in

text: the

text: present

text: setting

text: is

text: to

text: identify

text: conditions

text: on

text: the

text: joint

text: surplus

text: function

text: (which

text: is

text: now

text: a

text: function

text: of

text: x,

text: y

text: and

text: z,

text: without

text: any

text: speciﬁc

text: form)

text: in

text: the

text: more

text: general

text: model

text: which

text: ensure

text: purity

text: and

text: uniqueness

text: of

text: the

text: stable

text: match.


text: of xy may be interpreted as consumers with more disposable income having stronger preferences for ethically produced goods than their lower income coun- terparts). Producers’ preferences are independent of consumers, but their costs yz + z2/2 include a quadratic term in good quality and also a supermodular term yz, meaning more ethical ﬁrms have higher marginal production costs (for instance, producing a higher quality good may take more hours of labour than a lower quality good – the resulting diﬀerence in cost will be higher for a ﬁrm paying higher wages). Now note that for U(x) = x2 and V (y) = y2 , we have 2 2 | − − | − − − x y z 2 ≤ s(x, y, z) U(x) V (y) = 0 (8) 2 with equality when x = y + z. Then taking γ to be uniform measure on the × × ∩ { } set X Y Z x = y + z , we immediately get that γ is a stable matching measure for its marginals µ = (P ) γ and ν = (P ) γ, with payoﬀ funtions X # Y # U and V . This matching is certainly not pure; each consumer x is indiﬀerent among a continuum of choices of producers y. We note that when consumer x and producer y match together, they ex- − − − change product z = x y, for price p = u(x, y, z) U(x) = x(y+z) x2/2 = x,y,z x2/2 . A y varies, increasing favourability of the ﬁrm y to the consumer x is − exactly oﬀset by the decreasing quality of the good z = x y they exchange, and the price remains constant. − − Remark 6.8. By example (6.6), the surplus function s(x, y, z) = xy+xz yz az2 is twisted on z-trivial splitting sets for any constant a other than a = 1, 2 indicating that the previous example is highly non generic. 7 Conclusion This paper studies a general hybrid matching-hedonic model where agents match according to their preferences for both their partners and the good or con- tract they exchange. In contrast to strict matching and strict hedonic problems, these mixed models do not seem to have received much theoretical attention yet, but are quite natural in a variety of settings. The hybrid problem has a natural connection with tripartite matching, or multi-marginal optimal transport; speciﬁcally, every stable matching in the hy- brid model solves a corresponding optimal transport problem. This observa- tion, together with known results on multi-marginal optimal transport, can be exploited to reveal information on the structure of matching patters. In par- ticular, locally, the dimension of the support of a stable matching measure is controlled in terms of the mixed second order partial derivatives of the surplus; this result holds without any conditions on the distributions µ and ν of agents. In addition, if µ is absolutely continuous, the twist on splitting sets condition is known to imply purity and uniqueness of stable matchings in multi-marginal optimal transport and therefore immediately implies the same for the hybrid 18 ##LTLine## ##LTLine## ##LTLine## ##LTLine##

text: of

text: xy

text: may

text: be

text: interpreted

text: as

text: consumers

text: with

text: more

text: disposable

text: income

text: having

text: stronger

text: preferences

text: for

text: ethically

text: produced

text: goods

text: than

text: their

text: lower

text: income

text: coun-

text: terparts).

text: Producers’

text: preferences

text: are

text: independent

text: of

text: consumers,

text: but

text: their

text: costs

text: yz

text: include

text: a

text: quadratic

text: term

text: in

text: good

text: quality

text: and

text: also

text: a

text: supermodular

text: term

text: yz,

text: meaning

text: more

text: ethical

text: ﬁrms

text: have

text: higher

text: marginal

text: production

text: costs

text: (for

text: instance,

text: producing

text: a

text: higher

text: quality

text: good

text: may

text: take

text: more

text: hours

text: of

text: labour

text: than

text: a

text: lower

text: quality

text: good

text: the

text: resulting

text: diﬀerence

text: in

text: cost

text: will

text: be

text: higher

text: for

text: a

text: ﬁrm

text: paying

text: higher

text: wages).

text: Now

text: note

text: that

text: for

text: U(x)

text: and

text: V

text: (y)

text: we

text: have

text: x

text: y

text: z

text: s(x,

text: y,

text: z)

text: U(x)

text: V

text: (y)

text: with

text: equality

text: when

text: x

text: y

text: z.

text: Then

text: taking

text: γ

text: to

text: be

text: uniform

text: measure

text: on

text: the

text: set

text: X

text: Y

text: Z

text: x

text: y

text: z

text: we

text: immediately

text: get

text: that

text: γ

text: is

text: a

text: stable

text: matching

text: measure

text: for

text: its

text: marginals

text: µ

text: (P

text: γ

text: and

text: ν

text: (P

text: γ,

text: with

text: payoﬀ

text: funtions

text: X

text: Y

text: U

text: and

text: V

text: This

text: matching

text: is

text: certainly

text: not

text: pure;

text: each

text: consumer

text: x

text: is

text: indiﬀerent

text: among

text: a

text: continuum

text: of

text: choices

text: of

text: producers

text: y.

text: We

text: note

text: that

text: when

text: consumer

text: x

text: and

text: producer

text: y

text: match

text: together,

text: they

text: ex-

text: change

text: product

text: z

text: x

text: y,

text: for

text: price

text: p

text: u(x,

text: y,

text: z)

text: U(x)

text: x(y+z)

text: x,y,z

text: A

text: y

text: varies,

text: increasing

text: favourability

text: of

text: the

text: ﬁrm

text: y

text: to

text: the

text: consumer

text: x

text: is

text: exactly

text: oﬀset

text: by

text: the

text: decreasing

text: quality

text: of

text: the

text: good

text: z

text: x

text: y

text: they

text: exchange,

text: and

text: the

text: price

text: remains

text: constant.

text: Remark

text: By

text: example

text: the

text: surplus

text: function

text: s(x,

text: y,

text: z)

text: xy+xz

text: yz

text: is

text: twisted

text: on

text: z-trivial

text: splitting

text: sets

text: for

text: any

text: constant

text: a

text: other

text: than

text: a

text: indicating

text: that

text: the

text: previous

text: example

text: is

text: highly

text: non

text: generic.

text: Conclusion

text: This

text: paper

text: studies

text: a

text: general

text: hybrid

text: matching-hedonic

text: model

text: where

text: agents

text: match

text: according

text: to

text: their

text: preferences

text: for

text: both

text: their

text: partners

text: and

text: the

text: good

text: or

text: con-

text: tract

text: they

text: exchange.

text: In

text: contrast

text: to

text: strict

text: matching

text: and

text: strict

text: hedonic

text: problems,

text: these

text: mixed

text: models

text: do

text: not

text: seem

text: to

text: have

text: received

text: much

text: theoretical

text: attention

text: yet,

text: but

text: are

text: quite

text: natural

text: in

text: a

text: variety

text: of

text: settings.

text: The

text: hybrid

text: problem

text: has

text: a

text: natural

text: connection

text: with

text: tripartite

text: matching,

text: or

text: multi-marginal

text: optimal

text: transport;

text: speciﬁcally,

text: every

text: stable

text: matching

text: in

text: the

text: hy-

text: brid

text: model

text: solves

text: a

text: corresponding

text: optimal

text: transport

text: problem.

text: This

text: observa-

text: tion,

text: together

text: with

text: known

text: results

text: on

text: multi-marginal

text: optimal

text: transport,

text: can

text: be

text: exploited

text: to

text: reveal

text: information

text: on

text: the

text: structure

text: of

text: matching

text: patters.

text: In

text: par-

text: ticular,

text: locally,

text: the

text: dimension

text: of

text: the

text: support

text: of

text: a

text: stable

text: matching

text: measure

text: is

text: controlled

text: in

text: terms

text: of

text: the

text: mixed

text: second

text: order

text: partial

text: derivatives

text: of

text: the

text: surplus;

text: this

text: result

text: holds

text: without

text: any

text: conditions

text: on

text: the

text: distributions

text: µ

text: and

text: ν

text: of

text: agents.

text: In

text: addition,

text: if

text: µ

text: is

text: absolutely

text: continuous,

text: the

text: twist

text: on

text: splitting

text: sets

text: condition

text: is

text: known

text: to

text: imply

text: purity

text: and

text: uniqueness

text: of

text: stable

text: matchings

text: in

text: multi-marginal

text: optimal

text: transport

text: and

text: therefore

text: immediately

text: implies

text: the

text: same

text: for

text: the

text: hybrid

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##

text: ##LTLine##


text: { ∈ } S = (y, z) : z argmax(s(x, y, z)) x Then S is a z-trivial splitting set for s at x, with splitting function V , as x for any y, z we have by deﬁnition ≥ V (y) = s¯(x, y) s(x, y, z) ∈ with equality whenever (y, z) S . x Now, choose y , y satisfying 0 1 D s¯(x, y ) = D s¯(x, y ); (13) x 0 x 1 ∈ ∈ we want to show y = y . We can choose z argmax (s(x, y , z)) and z 0 1 ∈0 z 0 1 argmax (s(x, y , z)), so that (y , z ), (y , z ) S . By the envelope condition, z 1 0 0 1 1 x we have D s¯(x, y ) = D s(x, y , z ) x 0 x 0 0 and D s¯(x, y ) = D s(x, y , z ) x 1 x 1 1 Combined with (13), this implies that D s(x, y , z ) = D s(x, y , z ). As x 0 0 x 1 1 (y , z ) and (y , z ) both belong to the z- trivial splitting set S, the twist on 0 0 1 1 z-trivial splitting sets hypothesis now implies (y , z ) = (y , z ); in particular, 0 0 1 1 y = y as desired. 0 1 ∈ Conversely, assume s¯ is twisted, and suppose that (y , z ), (y , z ) S , 0 0 1 1 x where S is a z-trivial splitting set at x, such that x D s(x, y , z ) = D s(x, y , z ); (14) x 0 0 x 1 1 we need to show (y , z ) = (y , z ). Let V be the splitting function for S ; 0 0 1 1 x then ≥ V (y ) s(x, y , z) 0 0 for all z, with equality for z = z . As the left hand side is independent of z, this ∈ 0 tells us that z argmax s(x, y , z) and so s¯(x, y ) = s(x, y , z ). The envelope 0 z 0 0 0 0 theorem then yields D s¯(x, y ) = D s(x, y , z ). x 0 x 0 0 An identical argument implies D s¯(x, y ) = D s(x, y , z ) x 1 x 1 1 24

text: S

text: (y,

text: z)

text: z

text: argmax(s(x,

text: y,

text: z))

text: x

text: Then

text: S

text: is

text: a

text: z-trivial

text: splitting

text: set

text: for

text: s

text: at

text: x,

text: with

text: splitting

text: function

text: V

text: as

text: x

text: for

text: any

text: y,

text: z

text: we

text: have

text: by

text: deﬁnition

text: V

text: (y)

text: s¯(x,

text: y)

text: s(x,

text: y,

text: z)

text: with

text: equality

text: whenever

text: (y,

text: z)

text: S

text: x

text: Now,

text: choose

text: y

text: y

text: satisfying

text: D

label: equation

text: s¯(x,

text: y

text: D

text: s¯(x,

text: y

text: x

text: x

text: we

text: want

text: to

text: show

text: y

text: y

text: We

text: can

text: choose

text: z

text: argmax

text: (s(x,

text: y

text: z))

text: and

text: z

text: ∈0

text: z

text: argmax

text: (s(x,

text: y

text: z)),

text: so

text: that

text: (y

text: z

text: (y

text: z

text: S

text: By

text: the

text: envelope

text: condition,

text: z

text: x

text: we

text: have

text: D

text: s¯(x,

text: y

text: D

text: s(x,

text: y

text: z

text: x

text: x

text: and

text: D

text: s¯(x,

text: y

text: D

text: s(x,

text: y

text: z

text: x

text: x

text: Combined

text: with

text: this

text: implies

text: that

text: D

text: s(x,

text: y

text: z

text: D

text: s(x,

text: y

text: z

text: As

text: x

text: x

text: (y

text: z

text: and

text: (y

text: z

text: both

text: belong

text: to

text: the

text: z-

text: trivial

text: splitting

text: set

text: S,

text: the

text: twist

text: on

text: z-trivial

text: splitting

text: sets

text: hypothesis

text: now

text: implies

text: (y

text: z

text: (y

text: z

text: in

text: particular,

text: y

text: y

text: as

text: desired.

text: Conversely,

text: assume

text: s¯

text: is

text: twisted,

text: and

text: suppose

text: that

text: (y

text: z

text: (y

text: z

text: S

text: x

text: where

text: S

text: is

text: a

text: z-trivial

text: splitting

text: set

text: at

text: x,

text: such

text: that

text: x

text: D

text: s(x,

text: y

text: z

text: D

text: s(x,

text: y

text: z

text: x

text: x

text: we

text: need

text: to

text: show

text: (y

text: z

text: (y

text: z

text: Let

text: V

text: be

text: the

text: splitting

text: function

text: for

text: S

text: x

text: then

text: V

text: (y

text: s(x,

text: y

text: z)

text: for

text: all

text: z,

text: with

text: equality

text: for

text: z

text: z

text: As

text: the

text: left

text: hand

text: side

text: is

text: independent

text: of

text: z,

text: this

text: tells

text: us

text: that

text: z

text: argmax

text: s(x,

text: y

text: z)

text: and

text: so

text: s¯(x,

text: y

text: s(x,

text: y

text: z

text: The

text: envelope

text: z

text: theorem

text: then

text: yields

text: D

text: s¯(x,

text: y

text: D

text: s(x,

text: y

text: z

text: x

text: x

text: An

text: identical

text: argument

text: implies

text: D

text: s¯(x,

text: y

text: D

text: s(x,

text: y

text: z

text: x

text: x


text: × × For a given measure γ on X Y Z, we denote by γ its projection × × XY × onto X Y , a measure on X Y deﬁned by γ (C) = γ(C Z) for any ⊂ × XY C X Y . The measures γ and γ are deﬁned analogously. Note that XZ × Y×Z → × γ = (P ) γ, where P : X Y Z X Y is the projection map, XY XY # XY P (x, y, z) := (x, y). It is also worth noting that if γ is a matching of µ and XY ν, then γ has marginals µ and ν. XY Given a matching γ, functions U(x) and V (y) are called payoﬀ functions for γ if U(x) + V (y) = s(x, y, z) for γ almost every (x, y, z). For any matching, points in the support2 spt(γ) of { } γ (and hence in the equality set U(x)+V (y) = u(x, y, z)+v(x, y, z) for payoﬀ functions U and V ) represent buyer-seller pairs who are matched together by γ, together with the good they exchange. Payoﬀ functions then represent a division ∈ of the surplus between matched pairs. Given such a triple (x, y, z) spt(γ) and payoﬀs U(x) and V (y), the price3 that y charges x in exchange for the good z is given − − p := u(x, y, z) U(x) = V (y) v(x, y, z). (1) x,y,z The matching is called stable if there exist payoﬀ functions U(x) and V (y) such that ≥ U(x) + V (y) u(x, y, z) + v(x, y, z) (2) for all (x, y, z). The condition (2) ensures stability of the matching in the sense that no pair of unmatched agents would both prefer to leave their current partners and match together. If (2) failed, so that U(x) + V (y) < u(x, y, z) + v(x, y, z) for ∈ some unmatched buyer-seller-good triple (x, y, z) (that is, (x, y, z) / spt(γ)), then buyer x and seller y would be incentivized to exchange good z for a price p such that − − V (y) v(x, y, z) < p < u(x, y, z) U(x), − resulting in increased payoﬀs U¯(x) := u(x, y, z) p > U(x) and V¯ (y) := v(x, y, z) + p > V (y) for both x and y. Finally, we turn our attention to purity of matchings. There are several relevant notions of purity here, corresponding to various relationships between buyers, sellers, and goods. Deﬁnition 2.1. A matching γ is called buyer-seller pure if its projection γ × XY onto X Y is concentrated on a graph over X; that is, if there exists a function ⊂ × × 2The support of γ is the smallest closed set spt(γ) X Y Z with full mass, γ(spt(γ)) = 1. 3In contrast to the strict hedonic problem, one cannot hope for a market clearing pricing function p(z) which is independent of x and y here; it is possible that the same good may be exchanged between diﬀerent pairs of buyers and seller for diﬀerent prices. 5 ##LTLine##

text: For

text: a

text: given

text: measure

text: γ

text: on

text: X

text: Y

text: Z,

text: we

text: denote

text: by

text: γ

text: its

text: projection

text: XY

text: onto

text: X

text: Y

text: a

text: measure

text: on

text: X

text: Y

text: deﬁned

text: by

text: γ

text: (C)

text: γ(C

text: Z)

text: for

text: any

text: XY

text: C

text: X

text: Y

text: The

text: measures

text: γ

text: and

text: γ

text: are

text: deﬁned

text: analogously.

text: Note

text: that

text: XZ

text: Y×Z

text: γ

text: (P

text: γ,

text: where

text: P

text: X

text: Y

text: Z

text: X

text: Y

text: is

text: the

text: projection

text: map,

text: XY

text: XY

text: XY

text: P

text: (x,

text: y,

text: z)

text: (x,

text: y).

text: It

text: is

text: also

text: worth

text: noting

text: that

text: if

text: γ

text: is

text: a

text: matching

text: of

text: µ

text: and

text: XY

text: ν,

text: then

text: γ

text: has

text: marginals

text: µ

text: and

text: ν.

text: XY

text: Given

text: a

text: matching

text: γ,

text: functions

text: U(x)

text: and

text: V

text: (y)

text: are

text: called

text: payoﬀ

text: functions

text: for

text: γ

text: if

text: U(x)

text: V

text: (y)

text: s(x,

text: y,

text: z)

text: for

text: γ

text: almost

text: every

text: (x,

text: y,

text: z).

text: For

text: any

text: matching,

text: points

text: in

text: the

text: spt(γ)

text: of

text: γ

text: (and

text: hence

text: in

text: the

text: equality

text: set

text: U(x)+V

text: (y)

text: u(x,

text: y,

text: z)+v(x,

text: y,

text: z)

text: for

text: payoﬀ

text: functions

text: U

text: and

text: V

text: represent

text: buyer-seller

text: pairs

text: who

text: are

text: matched

text: together

text: by

text: γ,

text: together

text: with

text: the

text: good

text: they

text: exchange.

text: Payoﬀ

text: functions

text: then

text: represent

text: a

text: division

text: of

text: the

text: surplus

text: between

text: matched

text: pairs.

text: Given

text: such

text: a

text: triple

text: (x,

text: y,

text: z)

text: spt(γ)

text: and

text: payoﬀs

text: U(x)

text: and

text: V

text: (y),

text: the

text: that

text: y

text: charges

text: x

text: in

text: exchange

text: for

text: the

text: good

text: z

text: is

text: given

text: p

text: u(x,

text: y,

text: z)

text: U(x)

text: V

text: (y)

text: v(x,

text: y,

text: z).

text: x,y,z

text: The

text: matching

text: is

text: called

text: stable

text: if

text: there

text: exist

text: payoﬀ

text: functions

text: U(x)

text: and

text: V

text: (y)

text: such

text: that

text: U(x)

text: V

text: (y)

text: u(x,

text: y,

text: z)

text: v(x,

text: y,

text: z)

text: for

text: all

text: (x,

text: y,

text: z).

text: The

text: condition

text: ensures

text: stability

text: of

text: the

text: matching

text: in

text: the

text: sense

text: that

text: no

text: pair

text: of

text: unmatched

text: agents

text: would

text: both

text: prefer

text: to

text: leave

text: their

text: current

text: partners

text: and

text: match

text: together.

text: If

text: failed,

text: so

text: that

text: U(x)

text: V

text: (y)

text: u(x,

text: y,

text: z)

text: v(x,

text: y,

text: z)

text: for

text: some

text: unmatched

text: buyer-seller-good

text: triple

text: (x,

text: y,

text: z)

text: (that

text: is,

text: (x,

text: y,

text: z)

text: spt(γ)),

text: then

text: buyer

text: x

text: and

text: seller

text: y

text: would

text: be

text: incentivized

text: to

text: exchange

text: good

text: z

text: for

text: a

text: price

text: p

text: such

text: that

text: V

text: (y)

text: v(x,

text: y,

text: z)

text: p

text: u(x,

text: y,

text: z)

text: U(x),

text: resulting

text: in

text: increased

text: payoﬀs

text: U¯(x)

text: u(x,

text: y,

text: z)

text: p

text: U(x)

text: and

text: V¯

text: (y)

text: v(x,

text: y,

text: z)

text: p

text: V

text: (y)

text: for

text: both

text: x

text: and

text: y.

text: Finally,

text: we

text: turn

text: our

text: attention

text: to

text: purity

text: of

text: matchings.

text: There

text: are

text: several

text: relevant

text: notions

text: of

text: purity

text: here,

text: corresponding

text: to

text: various

text: relationships

text: between

text: buyers,

text: sellers,

text: and

text: goods.

text: Deﬁnition

text: A

text: matching

text: γ

text: is

text: called

text: buyer-seller

text: pure

text: if

text: its

text: projection

text: γ

text: XY

text: onto

text: X

text: Y

text: is

text: concentrated

text: on

text: a

text: graph

text: over

text: X;

text: that

text: is,

text: if

text: there

text: exists

text: a

text: function

text: support

text: of

text: γ

text: is

text: the

text: smallest

text: closed

text: set

text: spt(γ)

text: X

text: Y

text: Z

text: with

text: full

text: mass,

text: γ(spt(γ))

text: contrast

text: to

text: the

text: strict

text: hedonic

text: problem,

text: one

text: cannot

text: hope

text: for

text: a

text: market

text: clearing

text: pricing

text: function

text: p(z)

text: which

text: is

text: independent

text: of

text: x

text: and

text: y

text: here;

text: it

text: is

text: possible

text: that

text: the

text: same

text: good

text: may

text: be

text: exchanged

text: between

text: diﬀerent

text: pairs

text: of

text: buyers

text: and

text: seller

text: for

text: diﬀerent

text: prices.

text: ##LTLine##


text: 4.1 Classical matching and hedonic problems We ﬁrst review a purity result in the straight matching case, s = s(x, y). Theorem 4.2. (Matching problems) Assume that u = u(x, y) and v = v(x, y) depend only on x and y. Assume that µ is absolutely continuous with − respect to Lebesgue measure and s(x, y) = u(x, y)+v(x, y) is x y twisted. Then × any stable matching is buyer-seller pure and its projection γ onto X Y is XY uniquely determined; that is, if γ and γ¯ are stable matchings, γ = γ¯ . XY XY This result is well known; a proof can be found in [5]. Indeed, in the math- ematics literature, comparable results regarding the equivalent optimal trans- port problem were established, in various levels of generality, by Brenier [2], Gangbo [10], Levin [14], Gangbo-McCann [11] Caﬀarelli [3]. × × Note that in our terminology, stable matchings are measures on X Y Z, whereas in the literature the strict matching problem is usually formulated × instead in terms of measures on X Y , as the good z plays no role in the surplus function. In our formulation, we would not have full uniqueness; any measure × × × on X Y Z, whose projection onto X Y is γ is a stable matching, as XY both agents x and y are indiﬀerent to the superﬂuous good z. We now turn to the fully hedonic case, where agents’ preferences u = u(x, z) and v = v(y, z) depend on goods but not on their partners. Theorem 4.3. (Hedonic problems) Assume that agents’ preferences u = u(x, z) and v = v(y, z) depend only on (x, z) and (y, z), respectively, and that µ is absolutely continuous with respect to Lebesgue measure. Then: − 1. If u is x z twisted, the stable matching is buyer-good pure and it’s pro- × jection γ onto X Z is uniquely determined. XZ − 2. If in addition, v is z y twisted, and, for each ﬁxed x, y, every maximum (cid:55)→ of the mapping z u(x, z) + v(y, z) over Z occurs on the interior of Z, the stable matching measure is buyer-(seller,good) pure and unique. The proof of part 1 can be found in [7], while the proof of the second assertion requires a minor additional argument. Proof. (of assertion 2) → Using part 1), we have the existence of a unique map F : X Z such that, Z for γ almost every (x, y, z), z = F (x). Now, by a result in [5], we also have for Z (cid:48) (cid:55)→ (cid:48) (cid:48) γ almost every (x, y, z) that z maximizes z u(x, z ) + v(y, z ), so that − D u(x, z) = D v(y, z) z z or − D u(x, F (x)) = D v(y, F (x)). (5) z Z z Z 10

label: section

text: Classical

text: matching

text: and

text: hedonic

text: problems

text: We

text: ﬁrst

text: review

text: a

text: purity

text: result

text: in

text: the

text: straight

text: matching

text: case,

text: s

text: s(x,

text: y).

text: Theorem

text: (Matching

text: problems)

text: Assume

text: that

text: u

text: u(x,

text: y)

text: and

text: v

text: v(x,

text: y)

text: depend

text: only

text: on

text: x

text: and

text: y.

text: Assume

text: that

text: µ

text: is

text: absolutely

text: continuous

text: with

text: respect

text: to

text: Lebesgue

text: measure

text: and

text: s(x,

text: y)

text: u(x,

text: y)+v(x,

text: y)

text: is

text: x

text: y

text: twisted.

text: Then

text: any

text: stable

text: matching

text: is

text: buyer-seller

text: pure

text: and

text: its

text: projection

text: γ

text: onto

text: X

text: Y

text: is

text: XY

text: uniquely

text: determined;

text: that

text: is,

text: if

text: γ

text: and

text: γ¯

text: are

text: stable

text: matchings,

text: γ

text: γ¯

text: XY

text: XY

text: This

text: result

text: is

text: well

text: known;

text: a

text: proof

text: can

text: be

text: found

text: in

text: Indeed,

text: in

text: the

text: math-

text: ematics

text: literature,

text: comparable

text: results

text: regarding

text: the

text: equivalent

text: optimal

text: trans-

text: port

text: problem

text: were

text: established,

text: in

text: various

text: levels

text: of

text: generality,

text: by

text: Brenier

text: Gangbo

text: Levin

text: Gangbo-McCann

text: Caﬀarelli

text: Note

text: that

text: in

text: our

text: terminology,

text: stable

text: matchings

text: are

text: measures

text: on

text: X

text: Y

text: Z,

text: whereas

text: in

text: the

text: literature

text: the

text: strict

text: matching

text: problem

text: is

text: usually

text: formulated

text: instead

text: in

text: terms

text: of

text: measures

text: on

text: X

text: Y

text: as

text: the

text: good

text: z

text: plays

text: no

text: role

text: in

text: the

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text: Theorem

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text: preferences

text: u

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text: z

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