Trading

Networks

with

Price-Setting

Agents

Larry

Blume

Dept.

of

Economics

Cornell

University,

Ithaca

NY

David

Easley

Dept.

of

Economics

Cornell

University,

Ithaca

NY

Jon

Kleinberg

Dept.

of

Computer

Science

Cornell

University,

Ithaca

NY

kleinber@cs.cornell.edu

´Eva

Tardos

Dept.

of

Computer

Science

Cornell

University,

Ithaca

NY

eva@cs.cornell.edu

ABSTRACT

In

a

wide

range

of

markets,

individual

buyers

and

sellers

often

trade

through

intermediaries,

who

determine

prices

via

strategic

considerations.

Typically,

not

all

buyers

and

sellers

have

access

to

the

same

intermediaries,

and

they

trade

at

correspondingly

different

prices

that

reflect

their

relative

amounts

of

power

in

the

market.

We

model

this

phenomenon

using

a

game

in

which

buyers,

sellers,

and

traders

engage

in

trade

on

a

graph

that

represents

the

access

each

buyer

and

seller

has

to

the

traders.

In

this

model,

traders

set

prices

strategically,

and

then

buyers

and

sellers

react

to

the

prices

they

are

offered.

We

show

that

the

resulting

game

always

has

a

subgame

perfect

Nash

equilibrium,

and

that

all

equilibria

lead

to

an

efficient

(i.e.

socially

optimal)

allocation

of

goods.

We

extend

these

results

to

a

more

general

type

of

matching

market,

such

as

one

finds

in

the

matching

of

job

applicants

and

employers.

Finally,

we

consider

how

the

profits

obtained

by

the

traders

depend

on

the

underlying

graph

roughly,

a

trader

can

command

a

positive

profit

if

and

only

if

it

has

an

essential

connection

in

the

network

structure,

thus

providing

a

graph-theoretic

basis

for

quantifying

the

amount

of

competition

among

traders.

Our

work

differs

from

recent

studies

of

how

price

is

affected

by

network

structure

through

our

modeling

of

price-setting

as

a

strategic

activity

carried

out

by

a

subset

of

agents

in

the

system,

rather

than

studying

prices

set

via

competitive

equilibrium

or

by

a

truthful

mechanism.

Categories

and

Subject

Descriptors

[Social

and

Behavioral

Sciences]:

Economics

General

Terms

Economics,

Theory

INTRODUCTION

In

a

range

of

settings

where

markets

mediate

the

interactions

of

buyers

and

sellers,

one

observes

several

recurring

properties:

Individual

buyers

and

sellers

often

trade

through

intermediaries,

not

all

buyers

and

sellers

have

access

to

the

same

intermediaries,

and

not

all

buyers

and

sellers

trade

at

the

same

price.

One

example

of

this

setting

is

the

trade

of

agricultural

goods

in

developing

countries.

Given

inadequate

transportation

networks,

and

poor

farmers"

limited

access

to

capital,

many

farmers

have

no

alternative

to

trading

with

middlemen

in

inefficient

local

markets.

A

developing

country

may

have

many

such

partially

overlapping

markets

existing

alongside

modern

efficient

markets

Financial

markets

provide

a

different

example

of

a

setting

with

these

general

characteristics.

In

these

markets

much

of

the

trade

between

buyers

and

sellers

is

intermediated

by

a

variety

of

agents

ranging

from

brokers

to

market

makers

to

electronic

trading

systems.

For

many

assets

there

is

no

one

market;

trade

in

a

single

asset

may

occur

simultaneously

on

the

floor

of

an

exchange,

on

crossing

networks,

on

electronic

exchanges,

and

in

markets

in

other

countries.

Some

buyers

and

sellers

have

access

to

many

or

all

of

these

trading

venues;

others

have

access

to

only

one

or

a

few

of

them.

The

price

at

which

the

asset

trades

may

differ

across

these

trading

venues.

In

fact,

there

is

no

price

as

different

traders

pay

or

receive

different

prices.

In

many

settings

there

is

also

a

gap

between

the

price

a

buyer

pays

for

an

asset,

the

ask

price,

and

the

price

a

seller

receives

for

the

asset,

the

bid

price.

One

of

the

most

striking

examples

of

this

phenomenon

occurs

in

the

market

for

foreign

exchange,

where

there

is

an

interbank

market

with

restricted

access

and

a

retail

market

with

much

more

open

access.

Spreads,

defined

as

the

difference

between

bid

and

ask

prices,

differ

significantly

across

these

markets,

even

though

the

same

asset

is

being

traded

in

the

two

markets.

In

this

paper,

we

develop

a

framework

in

which

such

phenomena

emerge

from

a

game-theoretic

model

of

trade,

with

buyers,

sellers,

and

traders

interacting

on

a

network.

The

edges

of

the

network

connect

traders

to

buyers

and

sellers,

and

thus

represent

the

access

that

different

market

participants

have

to

one

another.

The

traders

serve

as

intermediaries

in

a

two-stage

trading

game:

they

strategically

choose

bid

and

ask

prices

to

offer

to

the

sellers

and

buyers

they

are

connected

to;

the

sellers

and

buyers

then

react

to

the

prices

they

face.

Thus,

the

network

encodes

the

relative

power

in

the

structural

positions

of

the

market

participants,

including

the

implicit

levels

of

competition

among

traders.

We

show

that

this

game

always

has

a

subgame

perfect

Nash

equilibrium,

and

that

all

equilibria

lead

to

an

efficient

(i.e.

socially

optimal)

allocation

of

goods.

We

also

analyze

how

trader

profits

depend

on

the

network

structure,

essentially

characterizing

in

graph-theoretic

terms

how

a

trader"s

payoff

is

determined

by

the

amount

of

competition

it

experiences

with

other

traders.

Our

work

here

is

connected

to

several

lines

of

research

in

economics,

finance,

and

algorithmic

game

theory,

and

we

discuss

these

connections

in

more

detail

later

in

the

introduction.

At

a

general

level,

our

approach

can

be

viewed

as

synthesizing

two

important

strands

of

work:

one

that

treats

buyer-seller

interaction

using

network

structures,

but

without

attempting

to

model

the

processses

by

which

prices

are

actually

formed

and

another

strand

in

the

literature

on

market

microstructure

that

incorporates

price-setting

intermediaries,

but

without

network-type

constraints

on

who

can

trade

with

whom

By

developing

a

network

model

that

explicitly

includes

traders

as

price-setting

agents,

in

a

system

together

with

buyers

and

sellers,

we

are

able

to

capture

price

formation

in

a

network

setting

as

a

strategic

process

carried

out

by

intermediaries,

rather

than

as

the

result

of

a

centrally

controlled

or

exogenous

mechanism.

The

Basic

Model:

Indistinguishable

Goods.

Our

goal

in

formulating

the

model

is

to

express

the

process

of

price-setting

in

markets

such

as

those

discussed

above,

where

the

participants

do

not

all

have

uniform

access

to

one

another.

We

are

given

a

set

B

of

buyers,

a

set

S

of

sellers,

and

a

set

T

of

traders.

There

is

an

undirected

graph

G

that

indicates

who

is

able

to

trade

with

whom.

All

edges

have

one

end

in

B

S

and

the

other

in

T;

that

is,

each

edge

has

the

form

(i,

t)

for

i

S

and

t

T,

or

(j,

t)

for

j

B

and

t

T.

This

reflects

the

constraints

that

all

buyer-seller

transactions

go

through

traders

as

intermediaries.

In

the

most

basic

version

of

the

model,

we

consider

identical

goods,

one

copy

of

which

is

initially

held

by

each

seller.

Buyers

and

sellers

each

have

a

value

for

one

copy

of

the

good,

and

we

assume

that

these

values

are

common

knowledge.

We

will

subsequently

generalize

this

to

a

setting

in

which

goods

are

distinguishable,

buyers

can

value

different

goods

differently,

and

potentially

sellers

can

value

transactions

with

different

buyers

differently

as

well.

Having

different

buyer

valuations

captures

settings

like

house

purchases;

adding

different

seller

valuations

as

well

captures

matching

markets

for

example,

sellers

as

job

applicants

and

buyers

as

employers,

with

both

caring

about

who

ends

up

with

which

good

(and

with

traders

acting

as

services

that

broker

the

job

search).

Thus,

to

start

with

the

basic

model,

there

is

a

single

type

of

good;

the

good

comes

in

individisible

units;

and

each

seller

initially

holds

one

unit

of

the

good.

All

three

types

of

agents

value

money

at

the

same

rate;

and

each

i

B

S

additionally

values

one

copy

of

the

good

at

θi

units

of

money.

No

agent

wants

more

than

one

copy

of

the

good,

so

additional

copies

are

valued

at

Each

agent

has

an

initial

endowment

of

money

that

is

larger

than

any

individual

valuation

θi;

the

effect

of

this

is

to

guarantee

that

any

buyer

who

ends

up

without

a

copy

of

the

good

has

been

priced

out

of

the

market

due

to

its

valuation

and

network

position,

not

a

lack

of

funds.

We

picture

each

good

that

is

sold

flowing

along

a

sequence

of

two

edges:

from

a

seller

to

a

trader,

and

then

from

the

trader

to

a

buyer.

The

particular

way

in

which

goods

flow

is

determined

by

the

following

game.

First,

each

trader

offers

a

bid

price

to

each

seller

it

is

connected

to,

and

an

ask

price

to

each

buyer

it

is

connected

to.

Sellers

and

buyers

then

choose

from

among

the

offers

presented

to

them

by

traders.

If

multiple

traders

propose

the

same

price

to

a

seller

or

buyer,

then

there

is

no

strict

best

response

for

the

seller

or

buyer.

In

this

case

a

selection

must

be

made,

and,

as

is

standard

(see

for

example

we

(the

modelers)

choose

among

the

best

offers.

Finally,

each

trader

buys

a

copy

of

the

good

from

each

seller

that

accepts

its

offer,

and

it

sells

a

copy

of

the

good

to

each

buyer

that

accepts

its

offer.

If

a

particular

trader

t

finds

that

more

buyers

than

sellers

accept

its

offers,

then

it

has

committed

to

provide

more

copies

of

the

good

than

it

has

received,

and

we

will

say

that

this

results

in

a

large

penalty

to

the

trader

for

defaulting;

the

effect

of

this

is

that

in

equilibrium,

no

trader

will

choose

bid

and

ask

prices

that

result

in

a

default.

More

precisely,

a

strategy

for

each

trader

t

is

a

specification

of

a

bid

price

βti

for

each

seller

i

to

which

t

is

connected,

and

an

ask

price

αtj

for

each

buyer

j

to

which

t

is

connected.

(We

can

also

handle

a

model

in

which

a

trader

may

choose

not

to

make

an

offer

to

certain

of

its

adjacent

sellers

or

buyers.)

Each

seller

or

buyer

then

chooses

at

most

one

incident

edge,

indicating

the

trader

with

whom

they

will

transact,

at

the

indicated

price.

(The

choice

of

a

single

edge

reflects

the

facts

that

(a)

sellers

each

initially

have

only

one

copy

of

the

good,

and

(b)

buyers

each

only

want

one

copy

of

the

good.)

The

payoffs

are

as

follows:

For

each

seller

i,

the

payoff

from

selecting

trader

t

is

βti,

while

the

payoff

from

selecting

no

trader

is

θi.

(In

the

former

case,

the

seller

receives

βti

units

of

money,

while

in

the

latter

it

keeps

its

copy

of

the

good,

which

it

values

at

θi.)

For

each

buyer

j,

the

payoff

from

selecting

trader

t

is

θj

−αtj,

whle

the

payoff

from

selecting

no

trader

is

(In

the

former

case,

the

buyer

receives

the

good

but

gives

up

αtj

units

of

money.)

For

each

trader

t,

with

accepted

offers

from

sellers

is

and

buyers

jb,

the

payoff

is

P

r

αtjr

P

r

βtir

minus

a

penalty

π

if

b

s.

The

penalty

is

chosen

to

be

large

enough

that

a

trader

will

never

incur

it

in

equilibrium,

and

hence

we

will

generally

not

be

concerned

with

the

penalty.

This

defines

the

basic

elements

of

the

game.

The

equilibrium

concept

we

use

is

subgame

perfect

Nash

equilibrium.

Some

Examples.

To

help

with

thinking

about

the

model,

we

now

describe

three

illustrative

examples,

depicted

in

Figure

To

keep

the

figures

from

getting

too

cluttered,

we

adopt

the

following

conventions:

sellers

are

drawn

as

circles

in

the

leftmost

column

and

will

be

named

from

top

to

bottom;

traders

are

drawn

as

squares

in

the

middle

column

and

will

be

named

from

top

to

bottom;

and

buyers

are

drawn

as

circles

in

the

rightmost

column

and

will

be

named

from

top

to

bottom.

All

sellers

in

the

examples

will

have

valuations

for

the

good

equal

to

the

valuation

of

each

buyer

is

drawn

inside

its

circle;

and

the

bid

or

ask

price

on

each

edge

is

drawn

on

top

of

the

edge.

In

Figure

we

show

how

a

standard

second-price

auction

arises

naturally

from

our

model.

Suppose

the

buyer

valuations

from

top

to

bottom

are

w

x

y

z.

The

bid

and

ask

prices

shown

are

consistent

with

an

equilibrium

in

which

and

accept

the

offers

of

trader

and

no

other

buyer

accepts

the

offer

of

its

adjacent

trader:

thus,

trader

receives

the

good

with

a

bid

price

of

x,

and

makes

w

x

by

selling

the

good

to

buyer

for

w.

In

this

way,

we

can

consider

this

particular

instance

as

an

auction

for

a

single

good

in

which

the

traders

act

as

proxies

for

their

adjacent

buyers.

The

buyer

with

the

highest

valuation

for

the

good

ends

up

with

it,

and

the

surplus

is

divided

between

the

seller

and

the

associated

trader.

Note

that

one

can

construct

a

k-unit

auction

with

k

buyers

just

as

easily,

by

building

a

complete

bipartite

graph

on

k

sellers

and

traders,

and

then

attaching

each

trader

to

a

single

distinct

buyer.

In

Figure

we

show

how

nodes

with

different

positions

in

the

network

topology

can

achieve

different

payoffs,

even

when

all

w

x

y

z

x

w

x

x

y

y

z

z

(a)

Auction

x

x

x

x

(b)

Heterogeneous

outcomes

x

x

x

x

(c)

Implicit

perfect

competition

Figure

(a)

An

auction,

mediated

by

traders,

in

which

the

buyer

with

the

highest

valuation

for

the

good

ends

up

with

it.

(b)

A

network

in

which

the

middle

seller

and

buyer

benefit

from

perfect

competition

between

the

traders,

while

the

other

sellers

and

buyers

have

no

power

due

to

their

position

in

the

network.

(c)

A

form

of

implicit

perfect

competition:

all

bid/ask

spreads

will

be

zero

in

equilibrium,

even

though

no

trader

directly

competes

with

any

other

trader

for

the

same

buyer-seller

pair.

buyer

valuations

are

the

same

numerically.

Specifically,

seller

and

buyer

occupy

powerful

positions,

because

the

two

traders

are

competing

for

their

business;

on

the

other

hand,

the

other

sellers

and

buyers

are

in

weak

positions,

because

they

each

have

only

one

option.

And

indeed,

in

every

equilibrium,

there

is

a

real

number

x

such

that

both

traders

offer

bid

and

ask

prices

of

x

to

and

respectively,

while

they

offer

bids

of

and

asks

of

to

the

other

sellers

and

buyers.

Thus,

this

example

illustrates

a

few

crucial

ingredients

that

we

will

identify

at

a

more

general

level

shortly.

Specifically,

and

experience

the

benefits

of

perfect

competition,

in

that

the

two

traders

drive

the

bid-ask

spreads

to

in

competing

for

their

business.

On

the

other

hand,

the

other

sellers

and

buyers

experience

the

downsides

of

monopoly

they

receive

payoff

since

they

have

only

a

single

option

for

trade,

and

the

corresponding

trader

makes

all

the

profit.

Note

further

how

this

natural

behavior

emerges

from

the

fact

that

traders

are

able

to

offer

different

prices

to

different

agents

capturing

the

fact

that

there

is

no

one

fixed

price

in

the

kinds

of

markets

that

motivate

the

model,

but

rather

different

prices

reflecting

the

relative

power

of

the

different

agents

involved.

The

previous

example

shows

perhaps

the

most

natural

way

in

which

a

trader"s

profit

on

a

particular

transaction

can

drop

to

when

there

is

another

trader

who

can

replicate

its

function

precisely.

(In

that

example,

two

traders

each

had

the

ability

to

move

a

copy

of

the

good

from

to

But

as

our

subsequent

results

will

show,

traders

make

zero

profit

more

generally

due

to

global,

graph-theoretic

reasons.

The

example

in

Figure

gives

an

initial

indication

of

this:

one

can

show

that

for

every

equilibrium,

there

is

a

y

such

that

every

bid

and

every

ask

price

is

equal

to

y.

In

other

words,

all

traders

make

zero

profit,

whether

or

not

a

copy

of

the

good

passes

through

them

and

yet,

no

two

traders

have

any

seller-buyer

paths

in

common.

The

price

spreads

have

been

driven

to

zero

by

a

global

constraint

imposed

by

the

long

cycle

through

all

the

agents;

this

is

an

example

of

implicit

perfect

competition

determined

by

the

network

topology.

Extending

the

Model

to

Distinguishable

Goods.

We

extend

the

basic

model

to

a

setting

with

distinguishable

goods,

as

follows.

Instead

of

having

each

agent

i

B

S

have

a

single

numerical

valuation

θi,

we

index

valuations

by

pairs

of

buyers

and

sellers:

if

buyer

j

obtains

the

good

initially

held

by

seller

i,

it

gets

a

utility

of

θji,

and

if

seller

i

sells

its

good

to

buyer

j,

it

experiences

a

loss

of

utility

of

θij

This

generalizes

the

case

of

indistinguishable

goods,

since

we

can

always

have

these

pairwise

valuations

depend

only

on

one

of

the

indices.

A

strategy

for

a

trader

now

consists

of

offering

a

bid

to

each

seller

that

specifies

both

a

price

and

a

buyer,

and

offering

an

ask

to

each

buyer

that

specifies

both

a

price

and

a

seller.

(We

can

also

handle

a

model

in

which

a

trader

offers

bids

(respectively,

asks)

in

the

form

of

vectors,

essentially

specifying

a

menu

with

a

price

attached

to

each

buyer

(resp.

seller).)

Each

buyer

and

seller

selects

an

offer

from

an

adjacent

trader,

and

the

payoffs

to

all

agents

are

determined

as

before.

This

general

framework

captures

matching

markets

for

example,

a

job

market

that

is

mediated

by

agents

or

employment

search

services

(as

in

hiring

for

corporate

executives,

or

sports

or

entertainment

figures).

Here

the

sellers

are

job

applicants,

buyers

are

employers,

and

traders

are

the

agents

that

mediate

the

job

market.

Of

course,

if

one

specifies

pairwise

valuations

on

buyers

but

just

single

valuations

for

sellers,

we

model

a

setting

where

buyers

can

distinguish

among

the

goods,

but

sellers

don"t

care

whom

they

sell

to

this

(roughly)

captures

settings

like

housing

markets.

Our

Results.

Our

results

will

identify

general

forms

of

some

of

the

principles

noted

in

the

examples

discussed

above

including

the

question

of

which

buyers

end

up

with

the

good;

the

question

of

how

payoffs

are

differently

realized

by

sellers,

traders,

and

buyers;

and

the

question

of

what

structural

properties

of

the

network

determine

whether

the

traders

will

make

positive

profits.

To

make

these

precise,

we

introduce

the

following

notation.

Any

outcome

of

the

game

determines

a

final

allocation

of

goods

to

some

of

the

agents;

this

can

be

specified

by

a

collection

M

of

triples

(ie,

te,

je),

where

ie

S,

te

T,

and

je

B;

moreover,

each

seller

and

each

buyer

appears

in

at

most

one

triple.

The

meaning

is

for

each

e

M,

the

good

initially

held

by

ie

moves

to

je

through

te.

(Sellers

appearing

in

no

triple

keep

their

copy

of

the

good.)

We

say

that

the

value

of

the

allocation

is

equal

to

P

e∈M

θjeie

θieje

Let

θ∗

denote

the

maximum

value

of

any

allocation

M

that

is

feasible

given

the

network.

We

show

that

every

instance

of

our

game

has

an

equilibrium,

and

that

in

every

such

equilibrium,

the

allocation

has

value

θ∗

in

other

words,

it

achieves

the

best

value

possible.

Thus,

equilibria

in

this

model

are

always

efficient,

in

that

the

market

enables

the

right

set

of

people

to

get

the

good,

subject

to

the

network

constraints.

We

establish

the

existence

and

efficiency

of

equilibria

by

constructing

a

linear

program

to

capture

the

flow

of

goods

through

the

network;

the

dual

of

this

linear

program

contains

enough

information

to

extract

equilibrium

prices.

By

the

definition

of

the

game,

the

value

of

the

equilibrium

allocation

is

divided

up

as

payoffs

to

the

agents,

and

it

is

interesting

to

ask

how

this

value

is

distributed

in

particular

how

much

profit

a

trader

is

able

to

make

based

on

its

position

in

the

network.

We

find

that,

although

all

equilibria

have

the

same

value,

a

given

trader"s

payoff

can

vary

across

different

equilibria.

However,

we

are

able

to

characterize

the

maximum

and

minimum

amounts

that

a

given

trader

is

able

to

make,

where

these

maxima

and

minima

are

taken

over

all

equilibria,

and

we

give

an

efficient

algorithm

to

compute

this.

In

particular,

our

results

here

imply

a

clean

combinatorial

characterization

of

when

a

given

trader

t

can

achieve

non-zero

payoff:

this

occurs

if

and

only

there

is

some

edge

e

incident

to

t

that

is

essential,

in

the

sense

that

deleting

e

reduces

the

value

of

the

optimal

allocation

θ∗

We

also

obtain

results

for

the

sum

of

all

trader

profits.

Related

Work.

The

standard

baseline

approach

for

analyzing

the

interaction

of

buyers

and

sellers

is

the

Walrasian

model

in

which

anonymous

buyers

and

sellers

trade

a

good

at

a

single

market

clearing

price.

This

reduced

form

of

trade,

built

on

the

idealization

of

a

market

price,

is

a

powerful

model

which

has

led

to

many

insights.

But

it

is

not

a

good

model

to

use

to

examine

where

prices

come

from

or

exactly

how

buyers

and

sellers

and

trade

with

each

other.

The

difficulty

is

that

in

the

Walrasian

model

there

is

no

agent

who

sets

the

price,

and

agents

don"t

actually

trade

with

each

other.

In

fact

there

is

no

market,

in

the

everyday

sense

of

that

word,

in

the

Walrasian

model.

That

is,

there

is

no

physical

or

virtual

place

where

buyers

and

sellers

interact

to

trade

and

set

prices.

Thus

in

this

simple

model,

all

buyers

and

sellers

are

uniform

and

trade

at

the

same

price,

and

there

is

also

no

role

for

intermediaries.

There

are

several

literatures

in

economics

and

finance

which

examine

how

prices

are

set

rather

than

just

determining

equilibrium

prices.

The

literature

on

imperfect

competition

is

perhaps

the

oldest

of

these.

Here

a

monopolist,

or

a

group

of

oliogopolists,

choose

prices

in

order

to

maximize

their

profits

(see

for

the

standard

textbook

treatment

of

these

markets).

A

monopolist

uses

its

knowledge

of

market

demand

to

choose

a

price,

or

a

collection

of

prices

if

it

discriminates.

Oliogopolists

play

a

game

in

which

their

payoffs

depend

on

market

demand

and

the

actions

of

their

competitors.

In

this

literature

there

are

agents

who

set

prices,

but

the

fiction

of

a

single

market

is

maintained.

In

the

equilibrium

search

literature,

firms

set

prices

and

consumers

search

over

them

(see

Consumers

do

end

up

paying

different

prices,

but

all

consumers

have

access

to

all

firms

and

there

are

no

intermediaries.

In

the

general

equilibrium

literature

there

have

been

various

attempts

to

introduce

price

determination.

A

standard

proof

technique

for

the

existence

of

competitive

equilibrium

involves

a

price

adjustment

mechanism

in

which

prices

respond

to

excess

demand.

The

Walrasian

auctioneer

is

often

introduced

as

a

device

to

explain

how

this

process

works,

but

this

is

a

fundamentally

a

metaphor

for

an

iterative

priceupdating

algorithm,

not

for

the

internals

of

an

actual

market.

More

sophisticated

processes

have

been

introduced

to

study

the

stability

of

equilibrium

prices

or

the

information

necessary

to

compute

them.

But

again

there

are

no

price-setting

agents

here.

In

the

finance

literature

the

work

on

market

microstructure

does

have

price-setting

agents

(specialists),

parts

of

it

do

determine

separate

bid

and

ask

prices,

and

different

agents

receive

different

prices

for

the

same

asset

(see

for

a

treatment

of

microstructure

theory).

Work

in

information

economics

has

identified

similar

phenomena

(see

e.g.

But

there

is

little

research

in

these

literatures

examining

the

effect

of

restrictions

on

who

can

trade

with

whom.

There

have

been

several

approaches

to

studying

how

network

structure

determines

prices.

These

have

posited

price

determination

through

definitions

based

on

competitive

equilibrium

or

the

core,

or

through

the

use

of

truthful

mechanisms.

In

briefly

reviewing

this

work,

we

will

note

the

contrast

with

our

approach,

in

that

we

model

prices

as

arising

from

the

strategic

behavior

of

agents

in

the

system.

In

recent

work,

Kakade

et

al.

have

studied

the

distribution

of

prices

at

competitive

equilibrium

in

a

bipartite

graph

on

buyers

and

sellers,

generated

using

a

probabilistic

model

capable

of

producing

heavy-tailed

degree

distributions

Even-Dar

et

al.

build

on

this

to

consider

the

strategic

aspects

of

network

formation

when

prices

arise

from

competitive

equilibrium.

Leonard

Babaioff

et

al.

and

Chu

and

Shen

consider

an

approach

based

on

mechanism

design:

buyers

and

sellers

reside

at

different

nodes

in

a

graph,

and

they

incur

a

given

transportation

cost

to

trade

with

one

another.

Leonard

studies

VCG

prices

in

this

setting;

Babaioff

et

al.

and

Chu

and

Shen

additionally

provide

a

a

budget-balanced

mechanism.

Since

the

concern

here

is

with

truthful

mechanisms

that

operate

on

private

valuations,

there

is

an

inherent

trade-off

between

the

efficiency

of

the

allocation

and

the

budget-balance

condition.

In

contrast,

our

model

has

known

valuations

and

prices

arising

from

the

strategic

behavior

of

traders.

Thus,

the

assumptions

behind

our

model

are

in

a

sense

not

directly

comparable

to

those

underlying

the

mechanism

design

approach:

while

we

assume

known

valuations,

we

do

not

require

a

centralized

authority

to

impose

a

mechanism.

Rather,

price-setting

is

part

of

the

strategic

outcome,

as

in

the

real

markets

that

motivate

our

work,

and

our

equilibria

are

simultaneously

budget-balanced

and

efficient

something

not

possible

in

the

mechanism

design

frameworks

that

have

been

used.

Demange,

Gale,

and

Sotomayor

and

Kranton

and

Minehart

analyze

the

prices

at

which

trade

occurs

in

a

network,

working

within

the

framework

of

mechanism

design.

Kranton

and

Minehart

use

a

bipartite

graph

with

direct

links

between

buyers

and

sellers,

and

then

use

an

ascending

auction

mechanism,

rather

than

strategic

intermediaries,

to

determine

the

prices.

Their

auction

has

desirable

equilibrium

properties

but

as

Kranton

and

Minehart

note

it

is

an

abstraction

of

how

goods

are

allocated

and

prices

are

determined

that

is

similar

in

spirit

to

the

Walrasian

auctioneer

abstraction.

In

fact,

we

can

show

how

the

basic

model

of

Kranton

and

Minehart

can

be

encoded

as

an

instance

of

our

game,

with

traders

producing

prices

at

equilibrium

matching

the

prices

produced

by

their

auction

Finally,

the

classic

results

of

Shapley

and

Shubik

on

the

assignment

game

can

be

viewed

as

studying

the

result

of

trade

on

a

bipartite

graph

in

terms

of

the

core.

They

study

the

dual

of

a

linear

program

based

on

the

matching

problem,

similar

to

what

we

use

for

a

reduced

version

of

our

model

in

the

next

section,

but

their

focus

is

different

as

they

do

not

consider

agents

that

seek

to

set

prices.

MARKETS

WITH

PAIR-TRADERS

For

understanding

the

ideas

behind

the

analysis

of

the

general

model,

it

is

very

useful

to

first

consider

a

special

case

with

a

Kranton

and

Minehart,

however,

can

also

analyze

a

more

general

setting

in

which

buyers

values

are

private

and

thus

buyers

and

sellers

play

a

game

of

incomplete

information.

We

deal

only

with

complete

information.

stricted

form

of

traders

that

we

refer

to

as

pair-traders.

In

this

case,

each

trader

is

connected

to

just

one

buyer

and

one

seller.

(Thus,

it

essentially

serves

as

a

trade

route

between

the

two.)

The

techniques

we

develop

to

handle

this

case

will

form

a

useful

basis

for

reasoning

about

the

case

of

traders

that

may

be

connected

arbitrarily

to

the

sellers

and

buyers.

We

will

relate

profits

in

a

subgame

perfect

Nash

equilibrium

to

optimal

solutions

of

a

certain

linear

program,

use

this

relation

to

show

that

all

equilibria

result

in

efficient

allocation

of

the

goods,

and

show

that

a

pure

equilibrium

always

exists.

First,

we

consider

the

simplest

model

where

sellers

have

indistinguishable

items,

and

each

buyer

is

interested

in

getting

one

item.

Then

we

extend

the

results

to

the

more

general

case

of

a

matching

market,

as

discussed

in

the

previous

section,

where

valuations

depend

on

the

identity

of

the

seller

and

buyer.

We

then

characterize

the

minimum

and

maximum

profits

traders

can

make.

In

the

next

section,

we

extend

the

results

to

traders

that

may

be

connected

to

any

subset

of

sellers

and

buyers.

Given

that

we

are

working

with

pair-traders

in

this

section,

we

can

represent

the

problem

using

a

bipartite

graph

G

whose

node

set

is

B

S,

and

where

each

trader

t,

connecting

seller

i

and

buyer

j,

appears

as

an

edge

t

(i,

j)

in

G.

Note,

however,

that

we

allow

multiple

traders

to

connect

the

same

pair

of

agents.

For

each

buyer

and

seller

i,

we

will

use

adj(i)

to

denote

the

set

of

traders

who

can

trade

with

i.

Indistinguishable

Goods

The

socially

optimal

trade

for

the

case

of

indistinguishable

goods

is

the

solution

of

the

transportation

problem:

sending

goods

along

the

edges

representing

the

traders.

The

edges

along

which

trade

occurs

correspond

to

a

matching

in

this

bipartite

graph,

and

the

optimal

trade

is

described

by

the

following

linear

program.

max

SV

(x)

X

t∈T

:t=(i,j)

xt(θj

θi)

xt

∀t

T

X

t∈adj(i)

xt

∀i

S

X

t∈adj(j)

xt

∀j

B

Next

we

consider

an

equilibrium.

Each

trader

t

(i,

j)

must

offer

a

bid

βt

and

an

ask

αt.

(We

omit

the

subscript

denoting

the

seller

and

buyer

here

since

we

are

dealing

with

pair-traders.)

Given

the

bid

and

ask

price,

the

agents

react

to

these

prices,

as

described

earlier.

Instead

of

focusing

on

prices,

we

will

focus

on

profits.

If

a

seller

i

sells

to

a

trader

t

adj(i)

with

bid

βt

then

his

profit

is

pi

βt

θi.

Similarly,

if

a

buyer

j

buys

from

a

trader

t

adj(j)

with

ask

αt,

then

his

profit

is

pj

θj

αt.

Finally,

if

a

trader

t

trades

with

ask

αt

and

bid

βt

then

his

profit

is

yt

αt

βt.

All

agents

not

involved

in

trade

make

profit.

We

will

show

that

the

profits

at

equilibrium

are

an

optimal

solution

to

the

following

linear

program.

min

sum(p,

y)

X

i∈B∪S

pi

X

t∈T

yt

yt

∀t

T

pi

∀i

S

B

yt

(θj

pj)

(θi

pi)

∀t

(i,

j)

T

LEMMA

At

equilibrium

the

profits

must

satisfy

the

above

inequalities.

Proof.

Clearly

all

profits

are

nonnegative,

as

trading

is

optional

for

all

agents.

To

see

why

the

last

set

of

inequalities

holds,

consider

two

cases

separately.

For

a

trader

t

who

conducted

trade,

we

get

equality

by

definition.

For

other

traders

t

(i,

j),

the

value

pi

+θi

is

the

price

that

seller

i

sold

for

(or

θi

if

seller

i

decided

to

keep

the

good).

Offering

a

bid

βt

pi

θi

would

get

the

seller

to

sell

to

trader

t.

Similarly,

θj

pj

is

the

price

that

buyer

j

bought

for

(or

θj

if

he

didn"t

buy),

and

for

any

ask

αt

θj

pj,

the

buyer

will

buy

from

trader

t.

So

unless

θj

pj

θi

pi

the

trader

has

a

profitable

deviation.

Now

we

are

ready

to

prove

our

first

theorem:

THEOREM

In

any

equilibrium

the

trade

is

efficient.

Proof.

Let

x

be

a

flow

of

goods

resulting

in

an

equilibrium,

and

let

variables

p

and

y

be

the

profits.

Consider

the

linear

program

describing

the

socially

optimal

trade.

We

will

also

add

a

set

of

additional

constraints

xt

for

all

traders

t

T;

this

can

be

added

to

the

description,

as

it

is

implied

by

the

other

constraints.

Now

we

claim

that

the

two

linear

programs

are

duals

of

each

other.

The

variables

pi

for

agents

B

S

correspond

to

the

equations

P

t∈adj(i)

xt

The

additional

dual

variable

yt

corresponds

to

an

additional

inequality

xt

The

optimality

of

the

social

value

of

the

trade

will

follow

from

the

claim

that

the

solution

of

these

two

linear

programs

derived

from

an

equilibrium

satisfy

the

complementary

slackness

conditions

for

this

pair

of

linear

programs,

and

hence

both

x

and

(p,

y)

are

optimal

solutions

to

the

corresponding

linear

programs.

There

are

three

different

complementary

slackness

conditions

we

need

to

consider,

corresponding

to

the

three

sets

of

variables

x,

y

and

p.

Any

agent

can

only

make

profit

if

he

transacts,

so

pi

implies

P

t∈adj(i)

xt

and

similarly,

yt

implies

that

xt

also.

Finally,

consider

a

trader

t

with

xt

that

trades

between

seller

i

and

buyer

j,

and

recall

that

we

have

seen

above

that

the

inequality

yt

(θj

pj)

(θi

pi)

is

satisfied

with

equality

for

those

who

trade.

Next

we

argue

that

equilibria

always

exist.

THEOREM

For

any

efficient

trade

between

buyers

and

sellers

there

is

a

pure

equilibrium

of

bid-ask

values

that

supports

this

trade.

Proof.

Consider

an

efficient

trade;

let

xt

if

t

trades

and

otherwise;

and

consider

an

optimal

solution

(p,

y)

to

the

dual

linear

program.

We

would

like

to

claim

that

all

dual

solutions

correspond

to

equilibrium

prices,

but

unfortunately

this

is

not

exactly

true.

Before

we

can

convert

a

dual

solution

to

equilibrium

prices,

we

may

need

to

modify

the

solution

slightly

as

follows.

Consider

any

agent

i

that

is

only

connected

to

a

single

trader

t.

Because

the

agent

is

only

connected

to

a

single

trader,

the

variables

yt

and

pi

are

dual

variables

corresponding

to

the

same

primal

inequality

xt

and

they

always

appear

together

as

yt

pi

in

all

inequalities,

and

also

in

the

objective

function.

Thus

there

is

an

optimal

solution

in

which

pi

for

all

agents

i

connected

only

to

a

single

trader.

Assume

(p,

y)

is

a

dual

solution

where

agents

connected

only

to

one

trader

have

pi

For

a

seller

i,

let

βt

θi

pi

be

the

bid

for

all

traders

t

adjacent

to

i.

Similarly,

for

each

buyer

j,

let

αt

θj

pj

be

the

ask

for

all

traders

t

adjacent

to

j.

We

claim

that

this

set

of

bids

and

asks,

together

with

the

trade

x,

are

an

equilibrium.

To

see

why,

note

that

all

traders

t

adjacent

to

a

seller

or

buyer

i

offer

the

same

ask

or

bid,

and

so

trading

with

any

trader

is

equally

good

for

agent

i.

Also,

if

i

is

not

trading

in

the

solution

x

then

by

complementary

slackness

pi

and

hence

not

trading

is

also

equally

good

for

i.

This

shows

that

sellers

and

buyers

don"t

have

an

incentive

to

deviate.

We

need

to

show

that

traders

have

no

incentive

to

deviate

either.

When

a

trader

t

is

trading

with

seller

i

and

buyer

j,

then

profitable

deviations

would

involve

increasing

αt

or

decreasing

βt.

But

by

our

construction

(and

assumption

about

monopolized

agents)

all

sellers

and

buyers

have

multiple

identical

ask/bid

offers,

or

trade

is

occurring

at

valuation.

In

either

case

such

a

deviation

cannot

be

successful.

Finally,

consider

a

trader

t

(i,

j)

who

doesn"t

trade.

A

deviation

for

t

would

involve

offering

a

lower

ask

to

seller

i

and

a

higher

bid

to

seller

j

than

their

current

trade.

However,

yt

by

complementary

slackness,

and

hence

pi

θi

θj

pj,

so

i

sells

for

a

price

at

least

as

high

as

the

price

at

which

j

buys,

so

trader

t

cannot

create

profitable

trade.

Note

that

a

seller

or

buyer

i

connected

to

a

single

trader

t

cannot

have

profit

at

equilibrium,

so

possible

equilibrium

profits

are

in

one-to-one

correspondence

with

dual

solutions

for

which

pi

whenever

i

is

monopolized

by

one

trader.

A

disappointing

feature

of

the

equilibrium

created

by

this

proof

is

that

some

agents

t

may

have

to

create

ask-bid

pairs

where

βt

αt,

offering

to

buy

for

more

than

the

price

at

which

they

are

willing

to

sell.

Agents

that

make

such

crossing

bid-ask

pairs

never

actually

perform

a

trade,

so

it

does

not

result

in

negative

profit

for

the

agent,

but

such

pairs

are

unnatural.

Crossing

bid-ask

pairs

are

weakly

dominated

by

the

strategy

of

offering

a

low

bid

β

and

an

extremely

high

ask

to

guarantee

that

neither

is

accepted.

To

formulate

a

way

of

avoiding

such

crossing

pairs,

we

say

an

equilibrium

is

cross-free

if

αt

βt

for

all

traders

t.

We

now

show

there

is

always

a

cross-free

equilibrium.

THEOREM

For

any

efficient

trade

between

buyers

and

sellers

there

is

a

pure

cross-free

equilibrium.

Proof.

Consider

an

optimal

solution

to

the

dual

linear

program.

To

get

an

equilibrium

without

crossing

bids,

we

need

to

do

a

more

general

modification

than

just

assuming

that

pi

for

all

sellers

and

buyers

connected

to

only

a

single

trader.

Let

the

set

E

be

the

set

of

edges

t

(i,

j)

that

are

tight,

in

the

sense

that

we

have

the

equality

yt

(θj

pj)

(θi

pi).

This

set

E

contain

all

the

edges

where

trade

occurs,

and

some

more

edges.

We

want

to

make

sure

that

pi

for

all

sellers

and

buyers

that

have

degree

at

most

in

E.

Consider

a

seller

i

that

has

pi

We

must

have

i

involved

in

a

trade,

and

the

edge

t

(i,

j)

along

which

the

trade

occurs

must

be

tight.

Suppose

this

is

the

only

tight

edge

adjacent

to

agent

i;

then

we

can

decrease

pi

and

increase

yt

till

one

of

the

following

happens:

either

pi

or

the

constraint

of

some

other

agent

t

adj(i)

becomes

tight.

This

change

only

increases

the

set

of

tight

edges

E,

keeps

the

solution

feasible,

and

does

not

change

the

objective

function

value.

So

after

doing

this

for

all

sellers,

and

analogously

changing

yt

and

pj

for

all

buyers,

we

get

an

optimal

solution

where

all

sellers

and

buyers

i

either

have

pi

or

have

at

least

two

adjacent

tight

edges.

Now

we

can

set

asks

and

bids

to

form

a

cross-free

equilibrium.

For

all

traders

t

(i,

j)

associated

with

an

edge

t

E

we

set

αt

and

βt

as

before:

we

set

the

bid

βt

pi

θi

and

the

ask

αt

θj

−pj.

For

a

trader

t

(i,

j)

E

we

have

that

pi

+θi

θj

−pj

and

we

set

αt

βt

to

be

any

value

in

the

range

[θj

pj,

pi

θi].

This

guarantees

that

for

each

seller

or

buyer

the

best

sell

or

buy

offer

is

along

the

edge

where

trade

occurs

in

the

solution.

The

askbid

values

along

the

tight

edges

guarantee

that

traders

who

trade

cannot

increase

their

spread.

Traders

t

(i,

j)

who

do

not

trade

cannot

make

profit

due

to

the

constraint

pi

θi

θj

pj

(a)

No

trader

profit

x

x

x

x

x

x

x

(b)

Trader

profit

Figure

Left:

an

equilibrium

with

crossing

bids

where

traders

make

no

money.

Right:

an

equilibrium

without

crossing

bids

for

any

value

x

Total

trader

profit

ranges

between

and

Distinguishable

Goods

We

now

consider

the

case

of

distinguishable

goods.

As

in

the

previous

section,

we

can

write

a

transshipment

linear

program

for

the

socially

optimal

trade,

with

the

only

change

being

in

the

objective

function.

max

SV

(x)

X

t∈T

:t=(i,j)

xt(θji

θij)

We

can

show

that

the

dual

of

this

linear

program

corresponds

to

trader

profits.

Recall

that

we

needed

to

add

the

constraints

xt

for

all

traders.

The

dual

is

then:

min

sum(p,

y)

X

i∈B∪S

pi

X

t∈T

yt

yt

∀t

T

pi

∀i

S

B

yt

(θji

pj)

(θij

pi)

∀t

(i,

j)

T

It

is

not

hard

to

extend

the

proofs

of

Theorems

to

this

case.

Profits

in

an

equilibrium

satisfy

the

dual

constraints,

and

profits

and

trade

satisfy

complementary

slackness.

This

shows

that

trade

is

socially

optimal.

Taking

an

optimal

dual

solution

where

pi

for

all

agents

that

are

monopolized,

we

can

convert

it

to

an

equilibrium,

and

with

a

bit

more

care,

we

can

also

create

an

equilibrium

with

no

crossing

bid-ask

pairs.

THEOREM

All

equilibria

for

the

case

of

pair-traders

with

distinguishable

goods

result

in

socially

optimal

trade.

Pure

noncrossing

equilibria

exist.

Trader

Profits

We

have

seen

that

all

equilibria

are

efficient.

However,

it

turns

out

that

equilibria

may

differ

in

how

the

value

of

the

allocation

is

spread

between

the

sellers,

buyers

and

traders.

Figure

depicts

a

simple

example

of

this

phenomenon.

Our

goal

is

to

understand

how

a

trader"s

profit

is

affected

by

its

position

in

the

network;

we

will

use

the

characterization

we

obtained

to

work

out

the

range

of

profits

a

trader

can

make.

To

maximize

the

profit

of

a

trader

t

(or

a

subset

of

traders

T

all

we

need

to

do

is

to

find

an

optimal

solution

to

the

dual

linear

program

maximizing

the

value

of

yt

(or

the

sum

P

t∈T

yt).

Such

dual

solutions

will

then

correspond

to

equilibria

with

non-crossing

prices.

THEOREM

For

any

trader

t

or

subset

of

traders

T

the

maximum

total

profit

they

can

make

in

any

equilibrium

can

be

computed

in

polynomial

time.

This

maximum

profit

can

be

obtained

by

a

non-crossing

equilibrium.

One

way

to

think

about

the

profit

of

a

trader

t

(i,

j)

is

as

a

subtraction

from

the

value

of

the

corresponding

edge

(i,

j).

The

value

of

the

edge

is

the

social

value

θji

θij

if

the

trader

makes

no

profit,

and

decreases

to

θji

θij

yt

if

the

trader

t

insists

on

making

yt

profit.

Trader

t

gets

yt

profit

in

equilibrium,

if

after

this

decrease

in

the

value

of

the

edge,

the

edge

is

still

included

in

the

optimal

transshipment.

THEOREM

A

trader

t

can

make

profit

in

an

equilibrium

if

and

only

if

t

is

essential

for

the

social

welfare,

that

is,

if

deleting

agent

t

decreases

social

welfare.

The

maximum

profit

he

can

make

is

exactly

his

value

to

society,

that

is,

the

increase

his

presence

causes

in

the

social

welfare.

If

we

allow

crossing

equilibria,

then

we

can

also

find

the

minimum

possible

profit.

Recall

that

in

the

proof

of

Theorem

traders

only

made

money

off

of

sellers

or

buyers

that

they

have

a

monopoly

over.

Allowing

such

equilibria

with

crossing

bids

we

can

find

the

minimum

profit

a

trader

or

set

of

traders

can

make,

by

minimizing

the

value

yt

(or

sum

P

t∈T

yt)

over

all

optimal

solutions

that

satisfy

pi

whenever

i

is

connected

to

only

a

single

trader.

THEOREM

For

any

trader

t

or

subset

of

traders

T

the

minimum

total

profit

they

can

make

in

any

equilibrium

can

be

computed

in

polynomial

time.

GENERAL

TRADERS

Next

we

extend

the

results

to

a

model

where

traders

may

be

connected

to

an

arbitrary

number

of

sellers

and

buyers.

For

a

trader

t

T

we

will

use

S(t)

and

B(t)

to

denote

the

set

of

buyers

and

sellers

connected

to

trader

t.

In

this

section

we

focus

on

the

general

case

when

goods

are

distinguishable

(i.e.

both

buyers

and

sellers

have

valuations

that

are

sensitive

to

the

identity

of

the

agent

they

are

paired

with

in

the

allocation).

In

the

full

version

of

the

paper

we

also

discuss

the

special

case

of

indistinguishable

goods

in

more

detail.

To

get

the

optimal

trade,

we

consider

the

bipartite

graph

G

(S

B,

E)

connecting

sellers

and

buyers

where

an

edge

e

(i,

j)

connects

a

seller

i

and

a

buyer

j

if

there

is

a

trader

adjacent

to

both:

E

{(i,

j)

adj(i)

adj(j)

On

this

graph,

we

then

solve

the

instance

of

the

assignment

problem

that

was

also

used

in

Section

with

the

value

of

edge

(i,

j)

equal

to

θji

θij

(since

the

value

of

trading

between

i

and

j

is

independent

of

which

trader

conducted

the

trade).

We

will

also

use

the

dual

of

this

linear

program:

min

val(z)

X

i∈B∪S

zi

zi

∀i

S

B.

zi

zj

θji

θij

∀i

S,

j

B

adj(i)

adj(j)

Bids

and

Asks

and

Trader

Optimization

First

we

need

to

understand

what

bidding

model

we

will

use.

Even

when

goods

are

indistinguishable,

a

trader

may

want

to

pricediscriminate,

and

offer

different

bid

and

ask

values

to

different

sellers

and

buyers.

In

the

case

of

distinguishable

goods,

we

have

to

deal

with

a

further

complication:

the

trader

has

to

name

the

good

she

is

proposing

to

sell

or

buy,

and

can

possibly

offer

multiple

different

products.

There

are

two

variants

of

our

model

depending

whether

a

trader

makes

a

single

bid

or

ask

to

a

seller

or

buyer,

or

she

offers

a

menu

of

options.

(i)

A

trader

t

can

offer

a

buyer

j

a

menu

of

asks

αtji,

a

vector

of

values

for

all

the

products

that

she

is

connected

to,

where

αtji

is

the

ask

for

the

product

of

seller

i.

Symmetrically,

a

trader

t

can

offer

to

each

seller

i

a

menu

of

bids

βtij

for

selling

to

different

buyers

j.

(ii)

Alternatively,

we

can

require

that

each

trader

t

can

make

at

most

one

ask

to

each

seller

and

one

bid

for

each

buyer,

and

an

ask

has

to

include

the

product

sold,

and

a

bid

has

to

offer

a

particular

buyer

to

sell

to.

Our

results

hold

in

either

model.

For

notational

simplicity

we

will

use

the

menu

option

here.

Next

we

need

to

understand

the

optimization

problem

of

a

trader

t.

Suppose

we

have

bid

and

ask

values

for

all

other

traders

t

T,

t

t.

What

are

the

best

bid

and

ask

offers

trader

t

can

make

as

a

best

response

to

the

current

set

of

bids

and

asks?

For

each

seller

i

let

pi

be

the

maximum

profit

seller

i

can

make

using

bids

by

other

traders,

and

symmetrically

assume

pj

is

the

maximum

profit

buyer

j

can

make

using

asks

by

other

traders

(let

pi

for

any

seller

or

buyer

i

who

cannot

make

profit).

Now

consider

a

seller-buyer

pair

(i,

j)

that

trader

t

can

connect.

Trader

t

will

have

to

make

a

bid

of

at

least

βtij

θij

+pi

to

seller

i

and

an

ask

of

at

most

αtji

θji

−pj

to

buyer

j

to

get

this

trade,

so

the

maximum

profit

she

can

make

on

this

trade

is

vtij

αtji

βtij

θji

pj

(θij

pi).

The

optimal

trade

for

trader

t

is

obtained

by

solving

a

matching

problem

to

find

the

matching

between

the

sellers

S(t)

and

buyers

B(t)

that

maximizes

the

total

value

vtij

for

trader

t.

We

will

need

the

dual

of

the

linear

program

of

finding

the

trade

of

maximum

profit

for

the

trader

t.

We

will

use

qti

as

the

dual

variable

associated

with

the

constraint

of

seller

or

buyer

i.

The

dual

is

then

the

following

problem.

min

val(qt)

X

i∈B(t)∪S(t)

qti

qti

∀i

S(t)

B(t).

qti

qtj

vtij

∀i

S(t),

j

B(t).

We

view

qti

as

the

profit

made

by

t

from

trading

with

seller

or

buyer

i.

Theorem

summarizes

the

above

discussion.

THEOREM

For

a

trader

t,

given

the

lowest

bids

βtij

and

highest

asks

αtji

that

can

be

accepted

for

sellers

i

S(t)

and

buyers

j

B(t),

the

best

trade

t

can

make

is

the

maximum

value

matching

between

S(t)

and

B(t)

with

value

vtij

αtji

βtij

for

the

edge

(i,

j).

This

maximum

value

is

equal

to

the

minimum

of

the

dual

linear

program

above.

Efficient

Trade

and

Equilibrium

Now

we

can

prove

trade

at

equilibrium

is

always

efficient.

THEOREM

Every

equilibrium

results

in

an

efficient

allocation

of

the

goods.

Proof.

Consider

an

equilibrium,

with

xe

if

and

only

if

trade

occurs

along

edge

e

(i,

j).

Trade

is

a

solution

to

the

transshipment

linear

program

used

in

Section

Let

pi

denote

the

profit

of

seller

or

buyer

i.

Each

trader

t

currently

has

the

best

solution

to

his

own

optimization

problem.

A

trader

t

finds

his

optimal

trade

(given

bids

and

asks

by

all

other

traders)

by

solving

a

matching

problem.

Let

qti

for

i

B(t)∪S(t)

denote

the

optimal

dual

solution

to

this

matching

problem

as

described

by

Theorem

When

setting

up

the

optimization

problem

for

a

trader

t

above,

we

used

pi

to

denote

the

maximum

profit

i

can

make

without

the

offer

of

trader

t.

Note

that

this

pi

is

exactly

the

same

pi

we

use

here,

the

profit

of

agent

i.

This

is

clearly

true

for

all

traders

t

that

are

not

trading

with

i

in

the

equilibrium.

To

see

why

it

is

true

for

the

trader

t

that

i

is

trading

with

we

use

that

the

current

set

of

bid-ask

values

is

an

equilibrium.

If

for

any

agent

i

the

bid

or

ask

of

trader

t

were

the

unique

best

option,

then

t

could

extract

more

profit

by

offering

a

bit

larger

ask

or

a

bit

smaller

bid,

a

contradiction.

We

show

the

trade

x

is

optimal

by

considering

the

dual

solution

zi

pi

P

t

qti

for

all

agents

i

B

S.

We

claim

z

is

a

dual

solution,

and

it

satisfies

complementary

slackness

with

trade

x.

To

see

this

we

need

to

show

a

few

facts.

We

need

that

zi

implies

that

i

trades.

If

zi

then

either

pi

or

qti

for

some

trader

t.

Agent

i

can

only

make

profit

pi

if

he

is

involved

in

a

trade.

If

qti

for

some

t,

then

trader

t

must

trade

with

i,

as

his

solution

is

optimal,

and

by

complementary

slackness

for

the

dual

solution,

qti

implies

that

t

trades

with

i.

For

an

edge

(i,

j)

associated

with

a

trader

t

we

need

to

show

the

dual

solution

is

feasible,

that

is

zi

zj

θji

θij

Recall

vtij

θji

−pj

−(θij

+pi),

and

the

dual

constraint

of

the

trader"s

optimization

problem

requires

qti

qtj

vtij.

Putting

these

together,

we

have

zi

zj

pi

qti

pj

qtj

vtij

pi

pj

θji

θij

Finally,

we

need

to

show

that

the

trade

variables

x

also

satisfy

the

complementary

slackness

constraint:

when

xe

for

an

edge

e

(i,

j)

then

the

corresponding

dual

constraint

is

tight.

Let

t

be

the

trader

involved

in

the

trade.

By

complementary

slackness

of

t"s

optimization

problem

we

have

qti

qtj

vtij.

To

see

that

z

satisfies

complementary

slackness

we

need

to

argue

that

for

all

other

traders

t

t

we

have

both

qt

i

and

qt

j

This

is

true

as

qt

i

implies

by

complementary

slackness

of

t

"s

optimization

problem

that

t

must

trade

with

i

at

optimum,

and

t

t

is

trading.

Next

we

want

to

show

that

a

non-crossing

equilibrium

always

exists.

We

call

an

equilibrium

non-crossing

if

the

bid-ask

offers

a

trader

t

makes

for

a

seller-buyer

pair

(i,

j)

never

cross,

that

is

βtij

αtji

for

all

t,

i,

j.

THEOREM

There

exists

a

non-crossing

equilibrium

supporting

any

socially

optimal

trade.

Proof.

Consider

an

optimal

trade

x

and

a

dual

solution

z

as

before.

To

find

a

non-crossing

equilibrium

we

need

to

divide

the

profit

zi

between

i

and

the

trader

t

trading

with

i.

We

will

use

qti

as

the

trader

t"s

profit

associated

with

agent

i

for

any

i

S(t)

B(t).

We

will

need

to

guarantee

the

following

properties:

Trader

t

trades

with

agent

i

whenever

qti

This

is

one

of

the

complementary

slackness

conditions

to

make

sure

the

current

trade

is

optimal

for

trader

t.

For

all

seller-buyer

pairs

(i,

j)

that

a

trader

t

can

trade

with,

we

have

pi

qti

pj

qtj

θji

θij

which

will

make

sure

that

qt

is

a

feasible

dual

solution

for

the

optimization

problem

faced

by

trader

t.

We

need

to

have

equality

in

when

trader

t

is

trading

between

i

and

j.

This

is

one

of

the

complementary

slackness

conditions

for

trader

t,

and

will

ensure

that

the

trade

of

t

is

optimal

for

the

trader.

Finally,

we

want

to

arrange

that

each

agent

i

with

pi

has

multiple

offers

for

making

profit

pi,

and

the

trade

occurs

at

one

of

his

best

offers.

To

guarantee

this

in

the

corresponding

bids

and

asks

we

need

to

make

sure

that

whenever

pi

there

are

multiple

t

adj(i)

that

have

equation

in

the

above

constraint

We

start

by

setting

pi

zi

for

all

i

S

B

and

qti

for

all

i

S

B

and

traders

t

adj(i).

This

guarantees

all

invariants

except

the

last

property

about

multiple

t

adj(t)

having

equality

in

We

will

modify

p

and

q

to

gradually

enforce

the

last

condition,

while

maintaining

the

others.

Consider

a

seller

with

pi

By

optimality

of

the

trade

and

dual

solution

z,

seller

i

must

trade

with

some

trader

t,

and

that

trader

will

have

equality

in

for

the

buyer

j

that

he

matches

with

i.

If

this

is

the

only

trader

t

that

has

a

tight

constraint

in

involving

seller

i

then

we

increase

qti

and

decrease

pi

till

either

pi

or

another

trader

t

t

will

be

achieve

equality

in

for

some

buyer

edge

adjacent

to

i

(possibly

a

different

buyer

j

This

change

maintains

all

invariants,

and

increases

the

set

of

sellers

that

also

satisfy

the

last

constraint.

We

can

do

a

similar

change

for

a

buyer

j

that

has

pj

and

has

only

one

trader

t

with

a

tight

constraint

adjacent

to

j.

After

possibly

repeating

this

for

all

sellers

and

buyers,

we

get

profits

satisfying

all

constraints.

Now

we

get

equilibrium

bid

and

ask

values

as

follows.

For

a

trader

t

that

has

equality

for

the

seller-buyer

pair

(i,

j)

in

we

offer

αtji

θji

pj

and

βtij

θij

pi.

For

all

other

traders

t

and

seller-buyer

pairs

(i,

j)

we

have

the

invariant

and

using

this

we

know

we

can

pick

a

value

γ

in

the

range

θij

+pi+qti

γ

θji

(pj

qtj

We

offer

bid

and

ask

values

βtij

αtji

γ.

Neither

the

bid

nor

the

ask

will

be

the

unique

best

offer

for

the

buyer,

and

hence

the

trade

x

remains

an

equilibrium.

Trader

Profits

Finally

we

turn

to

the

goal

of

understanding,

in

the

case

of

general

traders,

how

a

trader"s

profit

is

affected

by

its

position

in

the

network.

First,

we

show

how

to

maximize

the

total

profit

of

a

set

of

traders.

The

profit

of

trader

t

in

an

equilibrium

is

P

i

qti.

To

find

the

maximum

possible

profit

for

a

trader

t

or

a

set

of

traders

T

we

need

to

do

the

following:

Find

profits

pi

and

qti

so

that

zi

pi

P

t∈adj(i)

qti

is

an

optimal

dual

solution,

and

also

satisfies

the

constraints

for

any

seller

i

and

buyer

j

connected

through

a

trader

t

T.

Now,

subject

to

all

these

conditions,

we

maximize

the

sum

P

t∈T

P

i∈S(t)∪B(t)

qti.

Note

that

this

maximization

is

a

secondary

objective

function

to

the

primary

objective

that

z

is

an

optimal

dual

solution.

Then

we

use

the

proof

of

Theorem

shows

how

to

turn

this

into

an

equilibrium.

THEOREM

The

maximum

value

for

P

t∈T

P

i

qti

above

is

the

maximum

profit

the

set

T

of

traders

can

make.

Proof.

By

the

proof

of

Theorem

the

profits

of

trader

t

can

be

written

in

this

form,

so

the

set

of

traders

T

cannot

make

more

profit

than

claimed

in

this

theorem.

To

see

that

T

can

indeed

make

this

much

profit,

we

use

the

proof

of

Theorem

We

modify

that

proof

to

start

with

profit

vectors

p

and

qt

for

t

T

and

set

qt

for

all

traders

t

T

We

verify

that

this

starting

solution

satisfies

the

first

three

of

the

four

required

properties,

and

then

we

can

follow

the

proof

to

make

the

fourth

property

true.

We

omit

the

details

of

this

in

the

present

version.

In

Section

we

showed

that

in

the

case

of

pair

traders,

a

trader

t

can

make

money

if

he

is

essential

for

efficient

trade.

This

is

not

Figure

The

top

trader

is

essential

for

social

welfare.

Yet

the

only

equilibrium

is

to

have

bid

and

ask

values

equal

to

and

the

trader

makes

no

profit.

true

for

the

type

of

more

general

traders

we

consider

here,

as

shown

by

the

example

in

Figure

However,

we

still

get

a

characterization

for

when

a

trader

t

can

make

a

positive

profit.

THEOREM

A

trader

t

can

make

profit

in

an

equilibrium

if

and

only

if

there

is

a

seller

or

buyer

i

adjacent

to

t

such

that

the

connection

of

trader

t

to

agent

i

is

essential

for

social

welfarethat

is,

if

deleting

agent

t

from

adj(i)

decreases

the

value

of

the

optimal

allocation.

Proof.

First

we

show

the

direction

that

if

a

trader

t

can

make

money

there

must

be

an

agent

i

so

that

t"s

connection

to

i

is

essential

to

social

welfare.

Let

p,

q

be

the

profits

in

an

equilibrium

where

t

makes

money,

as

described

by

Theorem

with

P

i∈S(t)∪B(t)

qti

So

we

have

some

agent

i

with

qti

We

claim

that

the

connection

between

agent

i

and

trader

t

must

be

essential,

in

particular,

we

claim

that

social

welfare

must

decrease

by

at

least

qti

if

we

delete

t

from

adj(t).

To

see

why

note

that

decreasing

the

value

of

all

edges

of

the

form

(i,

j)

associated

with

trader

t

by

qti

keeps

the

same

trade

optimum,

as

we

get

a

matching

dual

solution

by

simply

resetting

qti

to

zero.

To

see

the

opposite,

assume

deleting

t

from

adj(t)

decreases

social

welfare

by

some

value

γ.

Assume

i

is

a

seller

(the

case

of

buyers

is

symmetric),

and

decrease

by

γ

the

social

value

of

each

edge

(i,

j)

for

any

buyer

j

such

that

t

is

the

only

agent

connecting

i

and

j.

By

assumption

the

trade

is

still

optimal,

and

we

let

z

be

the

dual

solution

for

this

matching.

Now

we

use

the

same

process

as

in

the

proof

of

Theorem

to

create

a

non-crossing

equilibrium

starting

with

pi

zi

for

all

i

S

∪B,

and

qti

γ,

and

all

other

q

values

This

creates

an

equilibrium

with

non-crossing

bids

where

t

makes

at

least

γ

profit

(due

to

trade

with

seller

i).

Finally,

if

we

allow

crossing

equilibria,

then

we

can

find

the

minimum

possible

profit

by

simply

finding

a

dual

solution

minimizing

the

dual

variables

associated

with

agents

monopolized

by

some

trader.

THEOREM

For

any

trader

t

or

subset

of

traders

T

the

minimum

total

profit

they

can

make

in

any

equilibrium

can

be

computed

in

polynomial

time.

REFERENCES

M.

Babaioff,

N.

Nisan,

E.

Pavlov.

Mechanisms

for

a

Spatially

Distributed

Market.

ACM

EC

Conference,

C.

Barrett,

E.

Mutambatsere.

Agricultural

markets

in

developing

countries.

The

New

Palgrave

Dictionary

of

Economics,

edition,

forthcoming.

Kenneth

Burdett

and

Kenneth

Judd.

Equilibrium

Price

Disperison.

Econometrica,

July

L.

Chu,

Z.-J.

Shen.

Agent

Competition

Double

Auction

Mechanism.

Management

Science,

G.

Demange,

D.

Gale,

M.

Sotomayor.

Multi-item

auctions.

J.

Political

Econ.

E.

Even-Dar,

M.

Kearns,

S.

Suri.

A

Network

Formation

Game

for

Bipartite

Exchange

Economies.

ACM-SIAM

Symp.

on

Discrete

Algorithms

(SODA),

J.

Kephart,

J.

Hanson,

A.

Greenwald.

Dynamic

Pricing

by

Software

Agents.

Computer

Networks,

S.

Kakade,

M.

Kearns,

L.

Ortiz,

R.

Pemantle,

S.

Suri.

Economic

Properties

of

Social

Networks.

NIPS

R.

Kranton,

D.

Minehart.

A

Theory

of

Buyer-Seller

Networks.

American

Economic

Review

June

H.

Leonard.

Elicitation

of

Honest

Preferences

for

the

Assignment

of

Individuals

to

Positions.

J.

Pol.

Econ,

M.

E.

J.

Newman.

The

structure

and

function

of

complex

networks.

SIAM

Review,

M.

O"Hara.

Market

Microstructure

Theory.

Blackwell

Publishers,

Cambridge,

MA,

L.

Shapley

M.

Shubik,

The

Assignment

Game

I:

The

Core.

Intl.

J.

Game

Theory

Jean

Tirole.

The

Theory

of

Industrial

Organization.

The

MIT

Press,

Cambridge,

MA,

trading network
interaction of buyer and seller
buyer and seller interaction
money initial endowment
maximum and minimum amount
strategic behavior of trader
trader strategic behavior
trade network

algorithmic game theory
market
initial endowment of money
bid price
perfect competition
benefit
economics and finance
complementary slackness
monopoly