Betting

Boolean-Style:

A

Framework

for

Trading

in

Securities

Based

on

Logical

Formulas

Lance

Fortnow

Joe

Kilian

NEC

Laboratories

America

Independence

Way

Princeton,

NJ

David

M.

Pennock

Overture

Services,

Inc.,

floor

N.

Pasadena

Ave

Pasadena,

CA

Michael

P.

Wellman

University

of

Michigan

AI

Laboratory

Beal

Avenue

Ann

Arbor,

MI

ABSTRACT

We

develop

a

framework

for

trading

in

compound

securities:

financial

instruments

that

pay

off

contingent

on

the

outcomes

of

arbitrary

statements

in

propositional

logic.

Buying

or

selling

securities-which

can

be

thought

of

as

betting

on

or

against

a

particular

future

outcome-allows

agents

both

to

hedge

risk

and

to

profit

(in

expectation)

on

subjective

predictions.

A

compound

securities

market

allows

agents

to

place

bets

on

arbitrary

boolean

combinations

of

events,

enabling

them

to

more

closely

achieve

their

optimal

risk

exposure,

and

enabling

the

market

as

a

whole

to

more

closely

achieve

the

social

optimum.

The

tradeoff

for

allowing

such

expressivity

is

in

the

complexity

of

the

agents"

and

auctioneer"s

optimization

problems.

We

develop

and

motivate

the

concept

of

a

compound

securities

market,

presenting

the

framework

through

a

series

of

formal

definitions

and

examples.

We

then

analyze

in

detail

the

auctioneer"s

matching

problem.

We

show

that,

with

n

events,

the

matching

problem

is

co-NP-complete

in

the

divisible

case

and

Σp

in

the

indivisible

case.

We

show

that

the

latter

hardness

result

holds

even

under

severe

language

restrictions

on

bids.

With

log

n

events,

the

problem

is

polynomial

in

the

divisible

case

and

NP-complete

in

the

indivisible

case.

We

briefly

discuss

matching

algorithms

and

tractable

special

cases.

Categories

and

Subject

Descriptors

[Theory

of

Computation]:

Analysis

of

Algorithms

and

Problem

Complexity-Nonnumerical

Algorithms

and

Problems;

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms

Algorithms,

Economics,

Theory.

INTRODUCTION

Securities

markets

effectively

allow

traders

to

place

bets

on

the

outcomes

of

uncertain

future

propositions.

Examples

include

stock

markets

like

NASDAQ,

options

markets

like

CBOE

futures

markets

like

CME

other

derivatives

markets,

insurance

markets,

political

stock

markets

sports

betting

markets

horse

racing

markets

idea

futures

markets

decision

markets

and

even

market

games

The

economic

value

of

securities

markets

is

two-fold.

First,

they

allow

traders

to

hedge

risk,

or

to

insure

against

undesirable

outcomes.

For

example,

the

owner

of

a

stock

might

buy

a

put

option

(the

right

to

sell

the

stock

at

a

particular

price)

in

order

to

insure

against

a

stock

downturn.

Or

the

owner

of

a

house

may

purchase

an

insurance

contract

to

hedge

against

unforeseen

damage

to

the

house.

Second,

securities

markets

allow

traders

to

speculate,

or

to

obtain

a

subjective

expected

profit

when

market

prices

do

not

reflect

their

assessment

of

the

likelihood

of

future

outcomes.

For

example,

a

trader

might

buy

a

call

option

if

he

believes

that

the

likelihood

is

high

that

the

price

of

the

underlying

stock

will

go

up,

regardless

of

risk

exposure

to

changes

in

the

stock

price.

Because

traders

stand

to

earn

a

profit

if

they

can

make

effective

probability

assessments,

often

prices

in

financial

markets

yield

very

accurate

aggregate

forecasts

of

future

events

Real

securities

markets

have

complex

payoff

structures

with

various

triggers.

However,

these

can

all

be

modeled

as

collections

of

more

basic

or

atomic

Arrow-Debreu

securities

One

unit

of

one

Arrow-Debreu

security

pays

off

one

dollar

if

and

only

if

(iff)

a

corresponding

binary

event

occurs;

it

pays

nothing

if

the

event

does

not

occur.

So,

for

example,

one

unit

of

a

security

denoted

might

pay

iff

Acme"s

stock

is

above

on

January

An

Acme

stock

option

as

it

would

be

defined

on

a

cial

exchange

can

be

though

of

as

a

portfolio

of

such

atomic

In

this

paper,

we

develop

and

analyze

a

framework

for

trading

in

compound

securities

markets

with

payoffs

contingent

on

arbitrary

logical

combinations

of

events,

including

conditionals.

For

example,

given

binary

events

A,

B,

and

C,

one

trader

might

bid

to

buy

three

units

of

a

security

denoted

A

¯B

C

that

pays

off

iff

the

compound

event

A

¯B

C

occurs

for

thirty

cents

each.

Another

trader

may

bid

to

sell

six

units

of

a

security

A|C

that

pays

off

iff

A

occurs

for

fifty-five

cents

each,

conditional

on

event

C

occurring,

meaning

that

the

transaction

is

revoked

if

C

does

not

occur

(i.e.,

no

payoff

is

given

and

the

price

of

the

security

is

refunded)

Bids

may

also

be

divisible,

meaning

that

bidders

are

willing

to

accept

less

than

the

requested

quantity,

or

indivisible,

meaning

that

bids

must

be

fulfilled

either

completely

or

not

at

all.

Given

a

set

of

such

bids,

the

auctioneer

faces

a

complex

matching

problem

to

decide

which

bids

are

accepted

for

how

many

units

at

what

price.

Typically,

the

auctioneer

seeks

to

take

on

no

risk

of

its

own,

only

matching

up

agreeable

trades

among

the

bidders,

but

we

also

consider

alternative

formulations

where

the

auctioneer

acts

as

a

market

maker

willing

to

accept

some

risk.

We

examine

the

computational

complexity

of

the

auctioneer"s

matching

problem.

Let

the

length

of

the

description

of

all

the

available

securities

be

O(n).

With

n

events,

the

matching

problem

is

co-NP-complete

in

the

divisible

case

and

Σp

in

the

indivisible

case.

This

Σp

hardness

holds

even

when

the

bidding

language

is

significantly

restricted.

With

log

n

events,

the

problem

is

polynomial

in

the

divisible

case

and

NP-complete

in

the

indivisible

case.

Section

presents

some

necessary

background

information,

motivation,

and

related

work.

Section

formally

describes

our

framework

for

compound

securities,

and

defines

the

auctioneer"s

matching

problem.

Section

briefly

discusses

natural

algorithms

for

solving

the

matching

problem.

Section

proves

our

central

computational

complexity

results.

Section

discusses

the

possibility

of

tractable

special

cases.

Section

concludes

with

a

summary

and

some

ideas

of

future

directions.

PRELIMINARIES

Background

and

notation

Imagine

a

world

where

there

are

only

two

future

uncertain

events

of

any

consequence:

the

event

that

one"s

house

is

struck

by

lightning

by

December

denoted

struck,

and

the

event

that

Acme"s

stock

price

goes

above

by

January

denoted

In

this

simple

world

there

are

four

possible

future

states-all

possible

combinations

of

the

binary

events"

outcomes:

struck

struck

struck

struck

Hedging

risk

can

be

thought

of

as

an

action

of

moving

money

between

various

possible

future

states.

For

example,

Technically,

an

option

is

a

portfolio

of

infinitely

many

atomic

securities,

though

it

can

be

approximately

modeled

with

a

finite

number.

ing

one"s

house

transfers

money

from

future

states

where

struck

is

not

true

to

states

where

it

is.

Selling

a

security

denoted

-that

pays

off

iff

the

event

occurs-transfers

money

from

future

states

where

Acme"s

price

is

above

on

January

to

states

where

it"s

not.

Speculating

is

also

an

act

of

transferring

money

between

future

states,

though

usually

associated

with

maximizing

expected

return

rather

than

reducing

risk.

For

example,

betting

on

a

football

team

moves

money

from

the

team

loses

state

to

the

team

wins

state.

In

practice,

agents

engage

in

a

mixture

of

hedging

and

speculating,

and

there

is

no

clear

dividing

line

between

the

two

All

possible

future

outcomes

form

a

state

space

Ω,

consisting

of

mutually

exclusive

and

exhaustive

states

ω

Ω.

Often

a

more

natural

way

to

think

of

possible

future

outcomes

is

as

an

event

space

A

of

linearly

independent

events

A

A

that

may

overlap

arbitrarily.

So

in

our

toy

example

struck

is

one

of

the

four

disjoint

states,

while

struck

is

one

of

the

two

events.

Note

that

a

set

of

n

linearly

independent

events

defines

a

state

space

Ω

of

size

consisting

of

all

possible

combinations

of

event

outcomes.

Conversely,

any

state

space

Ω

can

be

factored

into

log

|Ω|

events.

Suppose

that

A

exhaustively

covers

all

meaningful

future

outcomes

(i.e.,

covers

all

eventualities

that

agents

may

wish

to

hedge

against

and/or

speculate

upon).

Then

the

existence

of

linearly

independent

securities-called

a

complete

market-allows

agents

to

distribute

their

wealth

arbitrarily

across

future

An

agent

may

create

any

hedge

or

speculation

it

desires.

Under

classical

conditions,

agents

trading

in

a

complete

market

form

an

equilibrium

where

risk

is

allocated

Pareto

optimally.

If

the

market

is

incomplete,

meaning

it

consists

of

fewer

than

linearly

independent

securities,

then

in

general

agents

cannot

construct

arbitrary

hedges

and

equilibrium

allocations

may

be

nonoptimal

In

real-world

settings,

the

number

of

meaningful

events

n

is

large

and

thus

the

number

of

securities

required

for

completeness

is

intractable.

No

truly

complete

market

exists

or

will

ever

exist.

One

motivation

behind

compound

securities

markets

is

to

provide

a

mechanism

that

supports

the

most

transfer

of

risk

using

the

least

number

of

transactions

possible.

Compound

securities

allow

a

high

degree

of

expressivity

in

constructing

bids.

The

tradeoff

for

increased

expressivity

is

increased

computational

complexity,

from

both

the

bidder"s

and

auctioneer"s

point

of

view.

Related

work

The

quest

to

reduce

the

number

of

financial

instruments

required

to

support

an

optimal

allocation

of

risk

dates

to

Arrow"s

original

work

The

requirement

stated

above

of

only

linearly-independent

securities

is

itself

a

reduction

from

the

most

straightforward

formulation.

In

an

economy

with

k

standard

goods,

the

most

straightforward

complete

market

contains

securities,

each

paying

off

in

one

good

under

one

state

realization.

Arrow

showed

that

a

market

where

securities

and

goods

are

essentially

separated,

with

securities

paying

off

in

a

single

numeraire

good

plus

k

spot

markets

in

the

standard

goods,

is

also

complete.

For

our

purposes,

we

need

consider

only

the

securities

market.

By

linearly

independent

securities,

we

mean

that

the

vectors

of

payoffs

in

all

future

states

of

these

securities

are

linearly

independent.

Varian

shows

that

a

complete

market

can

be

constructed

using

fewer

than

securities,

replacing

the

missing

securities

with

options.

Still,

the

number

of

linearly

independent

financial

instruments-securities

plus

optionsmust

be

to

guarantee

completeness.

Though

the

requirement

of

financial

instruments

cannot

be

relaxed

if

one

wants

to

guarantee

completeness

in

all

circumstances,

Pennock

and

Wellman

explore

conditions

under

which

a

smaller

securities

market

may

be

operationally

complete,

meaning

that

its

equilibrium

is

Pareto

optimal

with

respect

to

the

agents

involved,

even

if

the

market

contains

less

than

securities.

The

authors

show

that

in

some

cases

the

market

can

be

structured

and

compacted

in

analogy

to

Bayesian

network

representations

of

joint

probability

distributions

They

show

that,

if

all

agents"

risk-neutral

independencies

agree

with

the

independencies

encoded

in

the

market

structure,

then

the

market

is

operationally

complete.

For

collections

of

agents

all

with

constant

absolute

risk

aversion,

agreement

on

Markov

independencies

is

sufficient.

Bossaerts,

Fine,

and

Ledyard

develop

a

mechanism

they

call

combined-value

trading

(CVT)

that

allows

traders

to

order

an

arbitrary

portfolio

of

securities

in

one

bid,

rather

than

breaking

up

the

order

into

a

sequence

of

bids

on

individual

securities.

If

the

portfolio

order

is

accepted,

all

of

the

implied

trades

on

individual

securities

are

executed

simultaneously,

thus

eliminating

so-called

execution

risk

that

prices

will

change

in

the

middle

of

a

planned

sequence

of

orders.

The

authors

conduct

laboratory

experiments

showing

that,

even

in

thin

markets

where

ordinary

sequential

trading

breaks

down,

CVT

supports

efficient

pricing

and

allocation.

Note

that

CVT

differs

significantly

from

compound

securities

trading.

CVT

allows

instantaneous

trading

of

any

linear

combination

of

securities,

while

compound

securities

allow

more

expressive

securities

that

can

encode

nonlinear

boolean

combinations

of

events.

For

example,

CVT

may

allow

an

agent

to

order

securities

A

and

B

in

a

bundle

that

pays

off

as

a

linear

combination

of

A

and

but

CVT

won"t

allow

the

construction

of

a

compound

security

A

B

that

pays

off

iff

both

A

and

B

occur,

or

a

compound

security

A|B

Related

to

CVT

are

combinatorial

auctions

and

exchanges

mechanisms

that

have

recently

received

quite

a

bit

of

attention

in

the

economics

and

computer

science

literatures.

Combinatorial

auctions

allow

bidders

to

place

distinct

values

on

all

possible

bundles

of

goods

rather

than

just

on

individual

goods.

In

this

way

bidders

can

express

substitutability

and

complementarity

relationships

among

goods

that

cannot

be

expressed

in

standard

parallel

or

sequential

auctions.

Compound

securities

differ

from

combinatorial

auctions

in

concept

and

complexity.

Compound

securities

allow

bidders

to

construct

an

arbitrary

bet

on

any

of

the

possible

compound

events

expressible

as

logical

functions

of

the

n

base

events,

conditional

on

any

other

of

the

compound

events.

Agents

optimize

based

on

their

own

subjective

probabilities

and

risk

attitude

(and

in

general,

their

beliefs

about

other

agents"

beliefs

and

utilities,

ad

infinitum).

The

central

auctioneer

problem

is

identifying

arbitrage

opportunities:

that

is,

to

match

bets

together

without

taking

on

any

risk.

Combinatorial

auctions,

on

the

other

hand,

allow

bids

on

any

of

the

bundles

of

n

goods.

Typically,

Specifically,

one

unit

of

each

pays

off

iff

both

A

and

B

occur,

iff

A

or

B

occurs

(but

not

both),

and

otherwise.

uncertainty-and

thus

risk-is

not

considered.

The

central

auctioneer

problem

is

to

maximize

social

welfare.

Also

note

that

the

problems

lie

in

different

complexity

classes.

While

clearing

a

combinatorial

auction

is

polynomial

in

the

divisible

case

and

NP-complete

in

the

indivisible

case,

matching

in

a

compound

securities

market

is

NP-complete

in

the

divisible

case

and

Σp

in

the

indivisible

case.

In

fact,

even

the

problem

of

deciding

whether

two

bids

on

compound

securities

match,

even

in

the

divisible

case,

is

NP-complete

(see

Section

There

is

a

sense

in

which

it

is

possible

to

translate

our

matching

problem

for

compound

securities

into

an

analogous

problem

for

clearing

two-sided

combinatorial

exchanges

of

exponential

size.

Specifically,

if

we

regard

payoff

in

a

particular

state

as

a

good,

then

compound

securities

can

be

viewed

as

bundles

of

(fractional

quantities

of)

such

goods.

The

material

balance

constraint

facing

the

combinatorial

auctioneer

corresponds

to

a

restriction

that

the

compoundsecurity

auctioneer

be

disallowed

from

assuming

any

risk.

Note

that

this

translation

is

not

at

all

useful

for

addressing

the

compound-security

matching

problem,

as

the

resulting

combinatorial

exchange

has

an

exponential

number

of

goods.

Hanson

develops

a

market

mechanism

called

a

market

scoring

rule

that

is

especially

well

suited

for

allowing

bets

on

a

combinatorial

number

of

outcomes.

The

mechanism

maintains

a

joint

probability

distribution

over

all

states,

either

explicitly

or

implicitly

using

a

Bayesian

network

or

other

compact

representation.

At

any

time

any

trader

who

believes

the

probabilities

are

wrong

can

change

any

part

of

the

distribution

by

accepting

a

lottery

ticket

that

pays

off

according

to

a

scoring

rule

(e.g.,

the

logarithmic

scoring

rule)

as

long

as

that

trader

also

agrees

to

pay

off

the

most

recent

person

to

change

the

distribution.

In

the

limit

of

a

single

trader,

the

mechanism

behaves

like

a

scoring

rule,

suitable

for

polling

a

single

agent

for

its

probability

distribution.

In

the

limit

of

many

traders,

it

produces

a

combined

estimate.

Since

the

market

essentially

always

has

a

complete

set

of

posted

prices

for

all

possible

outcomes,

the

mechanism

avoids

the

problem

of

thin

markets,

or

illiquidity,

that

necessarily

plagues

any

market

containing

an

exponential

number

of

alternative

investments.

The

mechanism

requires

a

patron

to

pay

off

the

final

person

to

change

the

distribution,

though

the

patron"s

payment

is

bounded.

Though

Hanson

offers

some

initial

suggestions,

several

open

problems

remain,

including

efficient

methods

for

representing

and

updating

the

joint

distribution

and

recording

traders

positions

and

portfolios,

without

resorting

to

exponential

time

and

space

algorithms.

Fagin,

Halpern,

and

Megiddo

give

a

sound

and

complete

axiomatization

for

deciding

whether

sets

of

probabilistic

inequalities

are

consistent.

Bids

for

compound

securities

can

be

thought

of

as

expressions

of

probabilistic

inequalities:

for

example,

a

bid

to

buy

A

B

at

price

is

a

statement

that

the

probability

of

A

B

is

greater

than

If

a

set

of

single-unit

bids

correspond

to

a

set

of

inconsistent

probabilistic

inequalities,

then

there

is

a

match.

However,

because

they

are

interested

in

a

much

different

framework,

Fagin

et

al.

do

not

consider

several

complicating

factors

specific

to

the

securities

market

framework:

namely,

handling

multi-unit

or

fractional

bid

quantities,

identifying

matches,

choosing

among

multiple

matches,

and

optimizing

based

on

probabilities

and

risk

attitudes.

We

address

these

issues

below.

FRAMEWORK

FOR

TRADING

IN

COMPOUND

SECURITIES

High-level

description

Common

knowledge

among

agents

is

the

set

of

events

A.

There

are

no

predefined

securities.

Instead,

agents

offer

to

buy

or

sell

securities

of

their

own

design

that

pay

off

contingent

on

logical

combinations

of

events

and

event

negations.

Combination

operators

may

include

conjunctions,

disjunctions,

and

conditionals.

For

all

practical

purposes,

it

is

impossible

for

agents

to

trade

in

enough

securities

to

form

a

complete

market,

so

agents

must

devise

their

best

tradeoff

between

the

number

and

complexity

of

their

bids,

and

the

extent

to

which

their

risks

are

hedged

and

desirable

bets

are

placed.

In

its

most

general

form,

the

problem

is

game-theoretic

in

nature,

since

what

an

agent

should

offer

depends

on

what

it

believes

other

agents

will

accept.

At

the

other

end

of

the

spectrum,

a

simplified

version

of

the

problem

is

to

optimize

bids

only

on

currently

available

securities

at

current

prices.

In

between

these

two

formulations

are

other

possible

interesting

optimization

problems.

Approximation

algorithms

might

also

be

pursued.

The

auctioneer

faces

a

nontrivial

problem

of

matching

buy

and

sell

orders

to

maximize

surplus

(the

cash

and

securities

left

over

after

accepted

bids

are

fulfilled).

For

example,

offers

to

sell

at

and

at

can

match

with

an

offer

to

buy

at

with

surplus

Or

an

offer

to

sell

at

can

match

with

an

offer

to

buy

at

with

surplus

in

cash

and

in

securities.

In

general,

a

single

security

might

qualify

for

multiple

matches,

but

only

one

can

be

transacted.

So

the

auctioneer

must

find

the

optimal

set

of

matches

that

maximizes

surplus,

which

could

be

measured

in

a

number

of

ways.

Again,

approximation

algorithms

might

be

considered.

In

another

formulation,

the

auctioneer

functions

as

a

market

maker

willing

to

take

on

a

certain

amount

of

risk.

Informally,

our

motivation

is

to

provide

a

mechanism

that

allows

a

very

high

degree

of

expressivity

in

placing

hedges

and

bets,

and

is

also

capable

of

approximating

the

optimal

(complete-market)

allocation

of

risk,

trading

off

the

number

and

complexity

of

securities

and

transactions

needed.

Formal

description

Securities

We

use

φ

and

ψ

to

denote

arbitrary

boolean

formulas,

or

logical

combinations

of

events

in

A.

We

denote

securities

φ|ψ

Securities

pay

off

if

and

only

if

(iff)

φ

and

ψ

are

true,

pay

off

iff

φ

is

false

and

ψ

is

true,

and

are

canceled

(i.e.,

any

price

paid

is

refunded)

iff

ψ

is

false.

We

define

T

Ω

to

be

the

event

true

and

F

to

be

the

event

false.

We

abbreviate

φ|T

as

φ

Orders

Agents

place

orders,

denoted

o,

of

the

form

q

units

of

φ|ψ

at

price

p

per

unit,

where

q

implies

a

buy

order

and

q

implies

a

sell

order.

We

assume

agents

submitting

buy

(sell)

orders

will

accept

any

price

p∗

p

(p∗

p).

We

distinguish

between

divisible

and

indivisible

orders.

Agents

submitting

divisible

orders

will

accept

any

quantity

αq

where

α

Agents

submitting

indivisible

orders

will

accept

only

exactly

q

units,

or

at

all.

We

believe

that,

given

the

nature

of

what

is

being

traded

(state-contingent

dollars),

most

agents

will

be

content

to

trade

using

divisible

orders.

Every

order

o

can

be

translated

into

a

payoff

vector

Υ

across

all

states

ω

Ω.

The

payoff

Υ

ω

in

state

ω

is

q

1ω∈ψ(1ω∈φ

p),

where

1ω∈E

equals

iff

ω

E

and

zero

otherwise.

Recall

that

the

states

correspond

to

the

possible

combinations

of

event

outcomes.

We

index

multiple

orders

with

subscripts

(e.g.,

oi

and

Υi).

Let

the

set

of

all

orders

be

O

and

the

set

of

all

corresponding

payoff

vectors

be

P.

Example

(Translating

orders

into

payoff

vectors)

Suppose

that

|A|

Consider

an

order

to

buy

two

units

of

at

price

The

corresponding

payoff

vector

is:

Υ

Υ

Υ

Υ

Υ

The

matching

problem

The

auctioneer"s

task,

called

the

matching

problem,

is

to

determine

which

orders

to

accept

among

all

orders

o

O.

Let

αi

be

the

fraction

of

order

oi

accepted

by

the

auctioneer

(in

the

indivisible

case,

αi

must

be

either

or

in

the

divisible

case,

αi

can

range

from

to

If

αi

then

order

oi

is

considered

rejected

and

no

transactions

take

place

concerning

this

order.

For

accepted

orders

(αi

the

auctioneer

receives

the

money

lost

by

bidders

and

pays

out

the

money

won

by

bidders,

so

the

auctioneer"s

payoff

vector

is:

Υauc

X

Υi∈P

−αiΥi.

We

also

call

the

auctioneer"s

payoff

vector

the

surplus

vector,

since

it

is

the

(possibly

state-contingent)

money

left

over

after

all

accepted

orders

are

filled.

Assume

that

the

auctioneer

wants

to

choose

a

set

of

orders

so

that

it

is

guaranteed

not

to

lose

any

money

in

any

future

state,

but

that

the

auctioneer

does

not

necessarily

insist

on

obtaining

a

positive

benefit

from

the

transaction

(i.e.,

the

auctioneer

is

content

to

break

even).

Definition

(Matching

problem,

indivisible

case)

Given

a

set

of

orders

O,

does

there

exist

αi

with

at

least

one

αi

such

that

∀ω,

Υ

ω

auc

In

other

words,

does

there

exist

a

nonempty

subset

of

orders

that

the

auctioneer

can

accept

without

risk?

If

∀ω,

Υ

ω

auc

c

where

c

is

nonnegative,

then

the

surplus

leftover

after

processing

this

match

is

c

dollars.

Let

m

minω[Υ

ω

auc].

In

general,

processing

a

match

leaves

m

dollars

in

cash

and

Υ

ω

auc

m

in

state-contingent

dollars,

which

can

then

be

translated

into

securities.

Example

(Indivisible

order

matching)

Suppose

|A|

Consider

an

order

to

buy

one

unit

of

at

price

and

an

order

to

sell

one

unit

of

at

price

The

corresponding

payoff

vectors

are:

Υ1

Υ

,Υ

,Υ

,Υ

Υ2

The

auctioneer"s

payoff

vector

(the

negative

of

the

componentwise

sum

of

the

above

two

vectors)

is:

Υauc

−Υ1

Υ2

Since

all

components

are

nonnegative,

the

two

orders

match.

The

auctioneer

can

process

both

orders,

leaving

a

surplus

of

in

cash

and

one

unit

of

in

securities.

Now

consider

the

divisible

case,

where

order

can

be

partially

filled.

Definition

(Matching

problem,

divisible

case)

Given

a

set

of

orders

O,

does

there

exist

αi

with

at

least

one

αi

such

that

∀ω,

Υ

ω

auc

Example

(Divisible

order

matching)

Suppose

|A|

Consider

an

order

to

sell

one

unit

of

at

price

an

order

to

buy

one

unit

of

at

price

and

an

order

to

buy

one

unit

of

at

price

The

corresponding

payoff

vectors

are:

Υ1

Υ

,Υ

,Υ

,Υ

Υ2

Υ3

It

is

clear

by

inspection

that

no

non-empty

subset

of

whole

orders

constitutes

a

match:

in

all

cases

where

αi

(other

than

all

αi

at

least

one

state

sums

to

a

positive

amount

(negative

for

the

auctioneer).

However,

if

α1

α2

and

α3

then

the

auctioneer"s

payoff

vector

is:

Υauc

Υ1

Υ2

Υ3

constituting

a

match.

The

auctioneer

can

process

of

the

first

and

second

orders,

and

all

of

the

third

order,

leaving

a

surplus

of

units

of

In

this

example,

a

divisible

match

exists

even

though

an

indivisible

match

is

not

possible;

we

examine

the

distinction

in

detail

in

Section

where

we

separate

the

two

matching

problems

into

distinct

complexity

classes.

The

matching

problems

defined

above

are

decision

problems:

the

task

is

only

to

show

the

existence

or

nonexistence

of

a

match.

However,

there

may

be

multiple

matches

from

which

the

auctioneer

can

choose.

Sometimes

the

choices

are

equivalent

from

the

auctioneer"s

perspective;

alternatively,

an

objective

function

can

be

used

to

find

an

optimal

match

according

to

that

objective.

Example

(Auctioneer

alternatives

I)

Suppose

|A|

Consider

an

order

to

sell

one

unit

of

at

price

an

order

to

sell

one

unit

of

at

price

an

order

to

buy

one

unit

of

at

price

an

order

to

buy

one

unit

of

at

price

and

an

order

to

buy

one

unit

of

at

price

The

corresponding

payoff

vectors

are:

Υ1

Υ2

Υ3

Υ4

Υ5

Consider

the

indivisible

case.

The

auctioneer

could

choose

to

accept

bids

and

together,

or

the

auctioneer

could

choose

to

accept

bids

and

together.

Both

constitute

matches,

and

in

fact

both

yield

identical

payoffs

(Υauc

or

in

cash)

for

the

auctioneer.

Example

(Auctioneer

alternatives

II)

Suppose

|A|

Consider

an

order

to

sell

two

units

of

at

price

an

order

to

buy

one

unit

of

at

price

and

an

order

to

buy

one

unit

of

at

price

The

corresponding

payoff

vectors

are:

Υ1

Υ2

Υ3

Consider

the

divisible

case.

The

auctioneer

could

choose

to

accept

one

unit

each

of

all

three

bids,

yielding

a

payoff

to

the

auctioneer

of

in

cash

(Υauc

Alternatively,

the

auctioneer

could

choose

to

accept

units

of

bid

and

one

unit

each

of

bids

and

yielding

a

payoff

to

the

auctioneer

of

units

of

security

Both

choices

constitute

matches

(in

fact,

accepting

any

number

of

units

of

bid

between

and

can

be

part

of

a

match),

though

depending

on

the

auctioneer"s

objective,

one

choice

might

be

preferred

over

another.

For

example,

if

the

auctioneer

believes

that

is

very

likely

to

occur,

it

may

prefer

to

accept

units

of

bid

There

are

many

possible

criteria

for

the

auctioneer

to

decide

among

matches,

all

of

which

seem

reasonable

in

some

circumstances.

One

natural

quantity

to

maximize

is

the

volume

of

trade

among

bidders;

another

is

the

auctioneer"s

utility,

either

with

or

without

the

arbitrage

constraint.

Definition

(Trade

maximization

problem)

Given

a

set

of

indivisible

(divisible)

orders

O,

choose

αi

(αi

to

maximize

X

i

αiqi,

under

the

constraint

that

∀ω,

Υ

ω

auc

Another

reasonable

variation

is

to

maximize

the

total

percent

of

orders

filled,

or

P

i

αi,

under

the

same

(risk-free)

constraint

that

∀ω,

Υ

ω

auc

Definition

(Auctioneer

risk-free

utility-maximization

problem)

Let

the

auctioneer"s

subjective

probability

for

each

state

ω

be

Pr(ω),

and

let

the

auctioneer"s

utility

for

y

dollars

be

u(y).

Given

a

set

of

indivisible

(divisible)

orders

O,

choose

αi

(αi

to

maximize

X

ω∈Ω

Pr(ω)u(Υ

ω

auc),

under

the

constraint

that

∀ω,

Υ

ω

auc

Definition

(Auctioneer

standard

utility-maximization

problem)

Let

the

auctioneer"s

subjective

probability

for

each

state

ω

be

Pr(ω),

and

let

the

auctioneer"s

utility

for

y

dollars

be

u(y).

Given

a

set

of

indivisible

(divisible)

orders

O,

choose

αi

(αi

to

maximize

X

ω∈Ω

Pr(ω)u

Υ

ω

auc

This

last

objective

function

drops

the

risk-free

(arbitrage)

constraint.

In

this

case,

the

auctioneer

is

a

market

maker

with

beliefs

about

the

likelihood

of

outcomes,

and

the

auctioneer

may

actually

lose

money

is

some

outcomes.

Still

other

variations

and

other

optimization

criteria

seem

reasonable,

including

social

welfare,

etc.

It

also

seems

reasonable

to

suppose

that

the

surplus

be

shared

among

bidders

and

the

auctioneer,

rather

than

retained

solely

by

the

auctioneer.

This

is

analogous

to

choosing

a

common

transaction

price

in

a

double

auction

(e.g.,

the

midpoint

between

the

bid

and

ask

prices),

rather

than

the

buyer

paying

the

bid

price

and

the

seller

receiving

the

ask

price,

with

the

difference

going

to

the

auctioneer.

The

problem

becomes

more

complicated

when

dividing

surplus

securities,

in

part

because

they

are

valued

differently

by

different

agents.

Formulating

reasonable

sharing

rules

and

examining

the

resulting

incentive

properties

seems

a

rich

and

promising

avenue

for

further

investigation.

MATCHING

ALGORITHMS

The

straightforward

algorithm

for

solving

the

divisible

matching

problem

is

linear

programming;

we

set

up

an

appropriate

linear

program

in

Section

The

straightforward

algorithm

for

solving

the

indivisible

matching

problem

is

integer

programming.

With

n

events,

to

set

up

the

appropriate

linear

or

integer

programs,

simply

writing

out

the

payoff

vectors

in

the

straightforward

way

requires

space.

There

is

some

hope

that

specialized

algorithms

that

exploit

structure

among

bids

can

perform

better

in

terms

of

average-case

time

and

space

complexity.

For

example,

in

some

cases

matches

can

be

identified

using

logical

reduction

techniques,

without

writing

down

the

full

payoff

vectors.

So

a

match

between

the

following

bids:

sell

of

at

sell

of

at

buy

of

at

can

be

identified

by

reducing

the

first

two

bids

to

an

equivalent

offer

to

sell

at

that

clearly

matches

with

the

third

bid.

Formalizing

a

logical-reduction

algorithm

for

matching,

or

other

algorithms

that

can

exploit

special

structure

among

the

bids,

is

a

promising

avenue

for

future

work.

THE

COMPUTATIONAL

COMPLEXITY

OF

MATCHING

In

this

section

we

examine

the

computational

complexity

of

the

auctioneer"s

matching

problem.

Here

n

refers

to

the

problem"s

input

size

that

includes

descriptions

of

all

of

the

buy

and

sell

orders.

We

also

assume

that

n

bounds

the

number

of

base

securities.

We

consider

four

cases

based

on

two

parameters:

Whether

to

allow

divisible

or

indivisible

orders.

The

number

of

securities.

We

consider

two

possibilities:

(a)

O(log

n)

base

securities

yielding

a

polynomial

number

of

states.

(b)

An

unlimited

number

of

base

securities

yielding

an

exponential

number

of

states.

We

show

the

following

results.

Theorem

The

matching

problem

is

computable

in

polynomial-time

for

O(log

n)

base

securities

with

divisible

orders.

co-NP-complete

for

unlimited

securities

with

divisible

orders.

NP-complete

for

O(log

n)

base

securities

with

indivisible

orders.

Σp

for

unlimited

securities

with

indivisible

orders.

Small

number

of

securities

with

divisible

orders

We

can

build

a

linear

program

based

on

Definition

We

have

variables

αi.

For

each

i,

we

have

αi

and

for

each

state

ω

in

Ω

we

have

the

constraint

Υ

ω

auc

X

i

−αiΥ

ω

i

Given

these

constraints

we

maximize

X

i

αi.

A

set

of

orders

has

a

matching

exactly

when

P

i

αi

With

O(log

n)

base

securities,

we

have

|Ω|

bounded

by

a

polynomial

so

we

can

solve

this

linear

program

in

polynomial

time.

Note

that

one

might

argue

that

one

should

maximize

some

linear

combination

of

the

−Υ

ω

i

s

to

maximize

the

surplus.

However

this

approach

will

not

find

matchings

that

have

zero

surplus.

Large

number

of

securities

with

divisible

orders

With

unlimited

base

securities,

the

linear

program

given

in

Section

has

an

exponential

number

of

constraint

equations.

Each

constraint

is

short

to

describe

and

easily

computable

given

ω.

Let

m

n

be

the

total

number

of

buy

and

sell

orders.

By

the

theory

of

linear

programming,

an

upper

bound

on

the

objective

function

can

be

forced

by

a

collection

of

m

constraints.

So

if

no

matching

exists

there

must

exist

m

constraints

that

force

all

the

αi

to

zero.

In

nondeterministic

polynomial-time

we

can

guess

these

constraints

and

solve

the

reduced

linear

program.

This

shows

that

matching

is

in

co-NP.

To

show

co-NP-completeness

we

reduce

the

NP-complete

problem

of

Boolean

formula

satisfiability

to

the

nonexistence

of

a

matching.

Fix

a

formula

φ.

Let

the

base

securities

be

the

variables

of

φ

and

consider

the

single

security

φ

with

a

buy

order

of

If

the

formula

φ

is

satisfiable

then

there

is

some

state

with

payoff

and

no

fractional

unit

of

the

security

φ

is

a

matching.

If

the

formula

φ

is

not

satisfiable

then

every

state

has

an

auctioneer"s

payoff

of

and

a

single

unit

of

the

security

φ

is

a

matching.

One

could

argue

that

if

the

formula

φ

is

not

satisfiable

then

no

fully

rational

buyer

would

want

to

buy

φ

for

a

cost

of

We

can

get

around

this

problem

by

adding

auxiliary

base

securities,

A

and

B,

and

defining

two

securities

τ

(φ

A)

(A

B)

τ

(φ

A)

(A

B)

with

separate

buy

orders

of

on

each.

If

φ

were

satisfiable

then

in

the

state

corresponding

to

the

satisfying

assignment

and

both

A

and

B

to

be

true,

τ

and

τ

both

have

an

auctioneer"s

payoff

of

so

even

no

divisible

matching

can

exist.

If

φ

were

not

satisfiable

then

one

unit

of

each

would

be

a

matching

since

in

every

state

at

least

one

of

τ

or

τ

are

false.

Small

number

of

securities

with

indivisible

orders

This

case

is

easily

seen

to

be

in

NP:

Just

nondeterministically

guess

a

nonempty

subset

S

of

orders

and

check

for

each

state

ω

in

Ω

that

Υ

ω

auc

X

i∈S

−Υ

ω

i

Since

|Ω|

and

|S|

are

bounded

by

a

polynomial

in

n,

the

verification

can

be

done

in

polynomial

time.

To

show

that

matching

is

NP-complete

we

reduce

the

NPcomplete

problem

EXACT

COVER

BY

to

a

matching

of

securities.

The

input

to

consists

of

a

set

X

and

a

collection

C

of

subsets

of

X.

The

input

(X,

C)

is

in

if

C

contains

an

exact

cover

of

X,

i.e.,

there

is

a

subcollection

C

of

C

such

that

every

element

of

X

occurs

in

exactly

one

member

of

C

Karp

showed

that

is

NP-complete.

Suppose

we

have

an

instance

(X,

C)

with

the

vector

X

and

C

cm}.

We

set

Ω

r,

s}

and

define

securities

labelled

φ1

φm

ψ1

ψq

and

τ

as

follows:

Security

φi

is

true

in

state

r,

and

is

true

in

state

ek

if

k

is

not

in

ci.

Security

ψj

is

true

only

in

state

s.

Security

τ

is

true

in

each

state

ek

but

not

r

or

s.

We

have

buy

orders

on

each

φi

and

ψj

security

for

and

a

buy

order

on

τ

for

We

claim

that

a

matching

exists

if

and

only

if

(X,

C)

is

in

If

(X,

C)

is

in

let

C

be

the

subcollection

that

covers

each

element

of

X

exactly

once.

Note

that

|C

q.

We

claim

the

collection

consisting

of

φi

for

each

ci

in

C

every

ψj

and

τ

has

a

matching.

In

each

state

ek

we

have

an

auctioneer"s

payoff

of

(q

In

states

r

and

s

the

auctioneer"s

payoffs

are

−5

Suppose

now

that

(X,

C)

is

not

in

but

there

is

a

matching.

Consider

the

number

q

of

the

φi

in

that

matching

and

q

the

number

of

ψj

in

the

matching.

Since

a

matching

requires

a

nonempty

subset

of

the

orders

and

τ

by

itself

is

not

a

matching

we

have

q

q

We

have

three

cases.

q

q:

In

state

r,

the

auctioneer"s

payoff

is

−q

−(q

q)

q

q

In

state

s,

the

auctioneer"s

payoff

is

−q

q

−(q

q

q

q

q:

Consider

the

set

C

consisting

of

the

ci

where

φi

is

in

the

matching.

There

must

be

some

state

ek

not

in

any

of

the

ci

or

C

would

be

an

exact

cover.

The

auctioneer"s

payoff

in

ek

is

at

most

−q

q

−(q

q

Large

Number

of

Securities

with

Indivisible

Orders

The

class

Σp

is

the

second

level

of

the

polynomial-time

hierarchy.

A

language

L

is

in

Σp

if

there

exists

a

polynomial

p

and

a

set

A

in

P

such

that

x

is

in

L

if

and

only

if

there

is

a

y

with

|y|

p(|x|)

such

that

for

all

z,

with

|z|

p(|x|),

(x,

y,

z)

is

in

A.

The

class

Σp

contains

both

NP

and

coNP.

Unless

the

polynomial-time

hierarchy

collapses

(which

is

considered

unlikely),

a

problem

that

is

complete

for

Σp

is

not

contained

in

NP

or

co-NP.

We

will

show

that

computing

a

matching

is

Σp

and

remains

so

even

for

quite

restricted

types

of

securities,

and

hence

is

(likely)

neither

in

NP

or

co-NP.

While

it

may

seem

that

being

NP-complete

or

co-NP-complete

is

hard

enough,

there

are

further

practical

consequences

of

being

outside

of

NP

and

co-NP.

If

the

matching

problem

were

in

NP,

one

could

use

heuristics

to

search

for

and

verify

a

match

if

it

exists;

even

if

such

heuristics

fail

in

the

worst

case,

they

may

succeed

for

most

examples

in

practice.

Similarly,

if

the

matching

problem

were

in

co-NP,

one

might

hope

to

at

least

heuristically

rule

out

the

possibility

of

matching.

But

for

problems

outside

of

NP

or

co-NP,

there

is

no

framework

for

verifying

that

a

heuristically

derived

answer

is

correct.

Less

formally,

for

NP

(or

co-NP)-complete

problems,

you

have

to

be

lucky;

for

Σp

problems,

you

can"t

even

tell

if

you"ve

been

lucky.

We

note

that

the

existence

of

a

matching

is

in

Σp

We

use

y

to

choose

a

subset

of

the

orders

and

z

to

represent

a

state

ω

with

(x,

y,

z)

in

A

if

the

set

of

orders

has

a

total

nonpositive

auctioneer"s

payoff

in

state

ω.

We

prove

a

stronger

theorem

which

implies

that

matching

is

in

Σp

Let

Sn

be

a

set

of

securities,

where

each

security

Si

has

cost

ci

and

pays

off

pi

whenever

formula

Ci

is

satisfied.

The

problem

asks

whether

one

can,

by

accepting

either

or

of

each

security,

guarantee

a

worst-case

payoff

strictly

larger

than

the

total

cost.

Theorem

The

problem

is

Σp

Furthermore,

the

problem

remains

Σp

under

the

following

two

special

cases:

For

all

i,

Ci

is

a

conjunction

of

base

events

(or

their

negations),

pi

and

ci

cj

for

all

i

and

j.

For

all

i,

Ci

is

a

conjunction

of

at

most

base

securities

(or

their

negations).

These

hardness

results

hold

even

if

there

is

a

promise

that

no

subset

of

the

securities

guarantees

a

worst-case

payoff

identical

to

their

cost.

To

prove

Theorem

we

reduce

from

the

standard

Σp

problem

that

we

call

T∃∀BF.

Given

a

boolean

formula

φ

with

variables

xn

and

yn

is

the

following

fullyquantified

formula

true

∀yn

xn,

yn)?

The

problem

remains

Σp

when

xn,

yn)

is

restricted

to

being

a

disjunction

of

conjunctions

of

at

most

variables

(or

their

negations),

e.g.,

xn,

yn)

This

form,

without

the

bound

on

the

conjunction

size,

is

known

as

disjunctive

normal

form

(DNF);

the

restriction

to

conjunctions

of

variables

is

We

reduce

T∃∀BF

to

finding

a

matching.

For

the

simplest

reduction,

we

consider

the

matching

problem

where

one

has

a

set

of

Arrow-Debreu

securities

whose

payoff

events

are

conjunctions

of

the

base

securities,

or

their

negations.

The

auctioneer

has

the

option

of

accepting

either

or

of

each

of

the

given

securities.

We

first

reduce

to

the

case

where

the

payoff

events

are

conjunctions

of

arbitrarily

many

base

events

(or

their

negations).

By

a

standard

trick

we

can

reduce

the

number

of

base

events

in

each

conjunction

to

and

with

a

slight

twist

we

can

even

ensure

that

all

securities

have

the

same

price

as

well

as

the

same

payoff.

Finally,

we

show

that

the

problem

remains

hard

even

if

only

conjunctions

of

variables

are

allowed,

though

with

securities

that

deviate

slightly

from

Arrow-Debreu

securities

in

that

they

may

have

varying,

non

unit

payoffs.

The

basic

reduction

Before

describing

the

securities,

we

give

some

intuition.

The

T∃∀BFproblem

may

be

viewed

as

a

game

between

a

selector

and

an

adversary.

The

selector

sets

the

xi

variables,

and

then

the

adversary

sets

the

yi

variables

so

as

to

falsify

the

formula

φ.

We

can

view

the

problem

as

one

in

which

the

auctioneer

is

a

buyer

who

buys

securities

corresponding

to

disjunctions

of

the

base

events,

and

then

the

adversary

sets

the

values

of

the

base

events

to

minimize

the

payoff

from

the

securities.

We

construct

our

securities

so

that

the

optimal

buying

strategy

is

to

buy

n

expensive

securities

along

with

a

set

of

cheap

securities,

of

negligible

cost

(for

some

cases

we

can

modify

the

construction

so

that

all

securities

have

the

same

cost).

The

total

cost

of

the

securities

will

be

just

under

and

each

security

pays

off

so

the

adversary

must

ensure

that

of

the

securities

pays

off.

Each

expensive

security

forces

the

adversary

to

set

some

variable,

xi

to

a

particular

value

to

prevent

the

security

from

paying

off;

this

corresponds

to

setting

the

xi

variables

in

the

original

game.

The

cheap

securities

are

such

that

preventing

every

one

of

of

these

securities

from

paying

off

is

equivalent

to

falsifying

φ

in

the

original

game.

Among

the

technical

difficulties

we

face

is

how

to

prevent

the

buyer

from

buying

conflicting

securities,

e.g.,

one

that

forces

xi

and

the

other

that

forces

xi

allowing

for

a

trivial

arbitrage.

Secondly,

for

our

analysis

we

need

to

ensure

that

a

trader

cannot

spend

more

to

get

more,

say

by

spending

for

a

set

of

securities

with

the

property

that

at

least

securities

pay

off

under

all

possible

events.

For

each

of

the

variables

{xi},

{yi}

in

φ,

we

add

a

corresponding

base

security

(with

the

same

labels).

For

each

existential

variable

xi

we

add

additional

base

securities,

ni

and

zi.

We

also

include

a

base

security

Q.

In

our

basic

construction,

each

expensive

security

costs

C

and

each

cheap

security

costs

all

securities

pay

off

We

require

that

Cn+

(|cheap

securities|)

and

That

is,

one

can

buy

n

expensive

securities

and

all

of

the

cheap

securities

for

less

than

but

one

cannot

buy

n

expensive

securities

for

less

than

We

at

times

refer

to

a

security

by

its

payoff

clause.

Remark:

We

may

loosely

think

of

as

However,

this

would

allow

one

to

buy

a

security

for

nothing

that

pays

(in

the

worst

case)

nothing.

By

making

we

can

show

it

hard

to

distinguish

portfolios

that

guarantee

a

positive

profit

from

those

that

risk

a

positive

loss.

Setting

will

also

allow

us

to

show

hardness

results

for

the

case

where

all

securities

have

the

same

cost.

For

i

n,

we

have

two

expensive

securities

with

payoff

clauses

(¯xi

∧Q)

and

(¯ni

∧Q)

and

two

cheap

securities

with

payoff

clauses

(xi

¯zi)

and

(ni

¯zi).

For

each

clause

C

φ,

we

convert

every

negated

variable

¯xi

into

ni

and

add

the

conjunction

zn.

Thus,

for

a

clause

C

we

construct

a

cheap

security

SC,

with

payoff

clause

zn

Finally,

we

have

a

cheap

security

with

payoff

clause

¯Q).

We

now

argue

that

a

matching

exists

iff

∀yn

xn,

yn).

We

do

this

by

successively

constraining

the

buyer

and

the

adversary,

eliminating

behaviors

that

would

cause

the

other

player

to

win.

The

resulting

reasonable

strategies

correspond

exactly

to

the

game

version

of

T∃∀BF.

First,

observe

that

if

the

adversary

sets

all

of

the

base

securities

to

false

then

only

the

¯Q)

security

will

pay

off.

Thus,

no

buyer

can

buy

more

than

n

expensive

securities

and

guarantee

a

profit.

The

problem

is

thus

whether

one

can

buy

n

expensive

securities

and

all

the

cheap

securities,

so

that

at

for

any

setting

of

the

base

events

at

least

one

security

will

pay

off.

Clearly,

the

adversary

must

make

Q

hold,

or

the

¯Q)

security

will

pay

off.

Next,

we

claim

that

for

each

i,

i

i,

the

auctioneer

must

buy

at

least

one

of

the

(¯xi

Q)

and

(¯ni

Q)

securities.

This

follow

from

the

fact

that

if

the

adversary

sets

xi,

ni

and

zi

to

be

false,

and

every

other

base

event

to

be

true,

then

only

the

(¯xi

Q)

and

(¯ni

Q)

securities

will

pay

off.

As

no

auctioneer

can

buy

more

than

n

expensive

securities,

it

must

therefore

buy

exactly

one

of

(¯xi

Q)

or

(¯ni

Q),

for

each

i,

i

n.

For

the

rest

of

the

analysis,

we

assume

that

the

auctioneer

follows

this

constraint.

Suppose

that

the

buyer

buys

(¯xi

∧Q).

Then

the

adversary

must

set

xi

to

be

true

(since

it

must

set

Q

to

be

true),

or

the

security

will

pay

off.

It

must

then

set

zi

to

be

true

or

(xi∧¯zi)

will

pay

off.

Since

the

buyer

doesn"t

buy

(¯ni

Q)

(by

the

above

constraint),

and

all

the

other

securities

pay

the

same

or

less

when

ni

is

made

false,

we

can

assume

without

loss

of

generality

that

the

adversary

sets

ni

to

be

false.

Similarly,

if

the

buyer

buys

(¯ni

Q),

then

the

adversary

must

set

ni

and

zi

to

be

true,

and

we

can

assume

without

loss

of

generality

that

the

adversary

sets

xi

to

be

false.

Note

that

the

adversary

must

in

all

cases

set

each

zi

event

to

be

true.

Summarizing

the

preceding

argument,

there

is

an

exact

correspondence

between

the

rational

strategies

of

the

buyer

and

settings

for

the

xi

variables

forced

on

the

adversary.

Furthermore,

the

adversary

is

also

constrained

to

set

the

variables

Q,

zn

to

be

true,

and

without

loss

of

generality

may

be

assumed

to

set

ni

¯xi.

Under

these

constraints,

those

securities

not

corresponding

to

clauses

in

φ

are

guaranteed

to

not

pay

off.

The

adversary

also

decides

the

value

of

the

ym

base

events.

Recall

that

for

each

clause

C

φ

there

is

a

corresponding

security

SC.

Given

that

zi

is

true

and

ni

¯xi

(without

loss

of

generality),

it

follows

from

the

construction

of

SC

that

the

setting

of

the

yis

causes

SC

to

pay

off

iff

it

satisfies

C.

This

establishes

the

reduction

from

T∃∀BF

to

the

matching

problem,

when

the

securities

are

constrained

to

be

a

conjunction

of

polynomially

many

base

events

or

their

negations.

Reducing

to

conjunctions

There

are

standard

methods

for

reducing

DNF

formulae

to

formulae,

which

are

trivially

modifiable

to

our

securities

framework;

we

include

the

reduction

for

completeness.

Given

a

security

S

whose

payoff

clause

is

C

vk)

(variable

negations

are

irrelevant

to

this

discussion),

cost

c

and

payoff

p,

introduce

a

new

auxiliary

variable,

w,

and

replace

the

security

with

two

securities,

and

with

payoff

clauses,

w)

and

¯w

vk).

The

securities

both

have

payoff

p,

and

their

costs

can

be

any

positive

values

that

sum

to

c.

Note

that

at

most

one

of

the

securities

can

pay

off

at

a

time.

If

only

one

security

is

bought,

then

the

adversary

can

always

set

w

so

that

it

won"t

pay

off;

hence

the

auctioneer

will

buy

either

both

or

neither,

for

a

total

cost

of

c

(here

we

use

the

fact

that

one

is

only

allowed

to

buy

either

or

shares

of

each

security).

Then,

it

may

be

verified

that,

given

the

ability

to

set

w

arbitrarily,

the

adversary

can

cause

C

to

be

unsatisfied

iff

it

can

cause

both

and

to

be

unsatisfied.

Hence,

owning

one

share

each

of

and

is

equivalent

to

owning

one

share

of

S.

Note

that

has

three

variables

and

has

variables.

By

applying

the

transformation

successively,

one

obtains

an

equivalent

set

of

securities,

of

polynomial

size,

whose

payoff

clauses

have

at

most

variables.

We

note

that

in

the

basic

construction,

all

of

the

clauses

with

more

than

variables

are

associated

with

cheap

securities

(cost

Instead

of

subdividing

costs,

we

can

simply

make

all

of

the

resulting

securities

have

cost

the

constraints

on

C

and

must

reflect

the

new,

larger

number

of

cheap

securities.

One

can

ensure

that

all

of

the

payoff

clauses

have

exactly

variables,

with

a

similar

construction.

A

security

S

with

cost

c,

payoff

p

and

defining

clause

(x

y)

can

be

replaced

by

securities

and

with

cost

payoff

p

and

defining

clauses

(x∧y∧w)

and

(x∧y∧

¯w),

where

w

is

a

new

auxiliary

variable.

Essentially

the

same

analysis

as

given

above

applies

to

this

case.

The

case

of

single-variable

payoff

clauses

is

handled

by

two

applications

of

this

technique.

Reducing

to

equi-cost

securities

By

setting

C

and

appropriately,

one

can

ensure

that

in

the

basic

reduction

every

security

costs

a

polynomially

bounded

integer

multiple

of

call

this

ratio

r.

We

now

show

how

to

reduce

this

case

to

the

case

where

every

security

costs

Recall

that

the

expensive

securities

have

payoff

clauses

(¯xi

Q)

or

(¯ni

Q).

Assume

that

security

S

has

payoff

clause

(¯xi

Q)

(the

other

case

is

handled

identically).

Replace

S

with

security

S

with

payoff

clause

(¯xi

Q

are

auxiliary

variables;

fresh

variables

are

chosen

for

each

clause),

and

also

with

payoff

clauses

and(

Clearly,

buying

of

the

new

securities

is

equivalent

to

not

buying

the

original

security.

We

show

that

buying

all

of

the

new

securities

is

equivalent

to

buying

the

original

security,

and

that

buying

a

proper,

nonempty

subset

of

the

securities

is

irrational.

We

first

note

that

if

the

buyer

buys

securities

then

the

adversary

must

set

to

be

true,

or

one

of

the

securities

will

pay

off.

To

see

this,

note

that

if

wi

is

set

to

false,

then

¯wi

will

be

true

unless

is

set

to

false;

thus,

setting

to

false

forces

the

adversary

to

set

to

false,

causing

the

final

clause

to

be

true.

Having

set

true,

the

adversary

can

set

to

false,

ensuring

that

of

the

securities

pays

out.

If

wi

is

true,

then

(¯xi

Q

is

equivalent

to

(¯xi

Q).

So

buying

all

of

the

replacement

securities

for

each

is

equivalent

to

buying

the

original

security

for

r.

It

remains

to

show

that

buying

a

proper,

nonempty

subset

of

the

securities

is

irrational.

If

one

doesn"t

buy

S

then

the

adversary

can

set

the

w

variables

so

that

of

will

pay

off;

any

money

spent

on

these

securities

is

wasted.

If

one

doesn"t

buy

the

adversary

can

set

all

w

to

false,

in

which

case

of

the

new

clauses

will

pay

off,

regardless

of

the

value

of

xi

and

Q.

Similarly,

if

one

doesn"t

buy

Si,

for

i

r

the

adversary

can

set

to

be

true,

all

the

other

w

variables

to

be

false,

and

again

there

is

no

payoff,

regardless

of

the

value

of

xi

and

Q.

Thus,

buying

a

proper

subset

of

these

securities

will

not

increase

ones

payoff.

We

note

that

this

reduction

can

be

combined

trivially

with

the

reduction

that

ensures

that

all

of

the

defining

clauses

have

or

fewer

variables.

With

a

slightly

messier

argument,

all

of

the

defining

clauses

can

be

set

up

to

have

exactly

variables.

Reducing

to

clauses

of

at

most

variables

If

we

allow

securities

to

have

variable

payoffs

and

prices,

we

can

reduce

to

the

case

where

each

security"s

payoff

clause

is

a

conjunction

of

at

most

variables

or

their

negations.

Given

a

security

s

with

payoff

clause

(X

Y

Z),

cost

c

and

payoff

we

introduce

fresh

auxiliary

variables,

and

(new

variables

are

used

for

each

clause)

and

replace

S

with

the

following

securities:

Securities

and

each

with

cost

and

payoff

with

respective

payoff

clauses

(X

(Y

and

(Z

Securities

each

with

cost

and

payoff

with

payoff

clauses,

Here,

is

a

tiny

positive

quantity,

described

later.

By

a

simple

case

analysis,

we

have

the

following.

Observations:

For

any

i,

there

exists

a

setting

of

an

such

that

of

the

S

securities

only

Si

pays

off.

For

any

setting

of

and

at

least

one

of

the

S

securities

will

pay

off.

If

and

are

all

false,

all

of

the

S

securities

will

pay

off.

Setting

one

of

or

to

be

true,

and

the

others

to

be

will

cause

exactly

one

of

the

S

securities

to

pay

off.

By

Observation

there

is

no

point

in

buying

a

nonempty

proper

subset

of

the

S

securities:

The

adversary

can

ensure

that

of

the

bought

securities

will

pay

off,

and

even

if

all

the

S

securities

pay

off,

it

will

not

be

sufficient

to

recoup

the

cost

of

buying

a

single

S

security.

By

Observation

if

one

buys

all

the

S

securities,

one

is

guaranteed

to

almost

make

back

ones

investment

(except

for

in

which

case

by

Observations

and

the

adversaries

optimal

strategy

is

to

make

exactly

one

of

or

true.

We

set

C,

and

so

that

Cn

(|cheap

securities|)

Thus,

the

accumulated

losses

of

can

never

spell

the

difference

between

making

a

guaranteed

profit

and

making

no

profit

at

all.

Note

also

that

by

making

positive

we

prevent

the

existence

of

break-even

buying

strategies

in

which

the

buyer

only

purchases

S

securities.

Summarizing

the

previous

argument,

we

may

assume

without

loss

of

generality

that

the

buyer

buys

all

of

the

S

securities

(for

all

clauses),

and

that

for

each

clause

the

adversary

sets

exactly

one

of

that

clause"s

auxiliary

variables

or

to

be

true.

For

the

rest

of

the

discussion,

we

assume

that

the

players

follow

these

constraints.

We

next

claim

that

a

rational

buyer

will

either

buy

all

of

or

or

of

them.

If

the

buyer

doesn"t

buy

then

if

the

adversary

makes

true

and

and

false,

neither

nor

will

pay

off,

regardless

of

how

the

adversary

sets

X,

Y

and

Z.

Hence,

there

is

no

point

in

buying

either

or

if

one

doesn"t

buy

Applying

the

same

argument

to

and

establishes

the

claim.

Clearly,

buying

of

and

has,

up

to

negligible

factors,

the

same

price/payoff

behavior

as

not

buying

S.

We

next

argue

that,

subject

to

the

established

constraints

put

on

the

players"

behaviors,

buying

all

of

and

has

the

same

price/payoff

behavior

(again

ignoring

factors)

as

buying

S,

regardless

of

how

the

adversary

sets

X,

Y

and

Z.

First,

in

both

cases,

the

cost

is

c.

If

the

adversary

makes

X,

Y

and

Z

true,

then

S

pays

off

and

(assuming

that

exactly

one

of

and

is

true),

exactly

one

of

or

will

pay

off

If

X

is

false,

then

S

doesn"t

pay

off,

and

the

adversary

can

set

true

(and

and

false),

ensuring

that

of

and

pays

off.

The

same

argument

holds

if

Y

or

Z

are

false.

TRACTABLE

CASES

The

logical

question

to

ask

in

light

of

these

complexity

results

is

whether

further,

more

severe

restrictions

on

the

space

of

securities

can

enable

tractable

matching

algorithms.

Although

we

have

not

systematically

explored

the

possibilities,

the

potential

for

useful

tractable

cases

certainly

exists.

Suppose,

for

example,

that

bids

are

limited

to

unit

quantities

of

securities

of

the

following

two

forms:

Disjunctions

of

positive

events:

Ak

Single

negative

events:

¯Ai

Let

p

be

the

price

offered

for

a

disjunction

Ak

and

qi

the

maximal

price

offered

for

the

respective

negated

disjuncts.

This

disjunction

bid

is

part

of

a

match

iff

p

+P

i

qi

k.

Evaluating

whether

this

condition

is

satisfied

by

a

subset

of

bids

is

quite

straightforward.

Although

this

example

is

contrived,

its

application

is

not

entirely

implausible.

For

example,

the

disjunctions

may

correspond

to

insurance

customers,

who

want

an

insurance

contract

to

cover

all

the

potential

causes

of

their

asset

loss.

The

atomic

securities

are

sold

by

insurers,

each

of

whom

specialize

in

a

different

form

of

disaster

cause.

CONCLUSIONS

AND

FUTURE

DIRECTIONS

We

have

analyzed

the

computational

complexity

of

matching

for

securities

based

on

logical

formulas.

Many

possible

avenues

for

future

work

exist,

including

Analyzing

the

agents"

optimization

problem:

How

to

choose

quantities

and

bid/ask

prices

for

a

collection

of

securities

to

maximizes

one"s

expected

utility,

both

for

linear

and

nonlinear

utility

functions.

How

to

choose

securities;

that

is,

deciding

on

what

collection

of

boolean

formulas

to

offer

to

trade,

subject

to

constraints

or

penalties

on

the

number

or

complexity

of

bids.

How

do

make

the

above

choices

in

a

game

theoretically

sound

way,

taking

into

account

the

choices

of

other

traders,

their

reasoning

about

other

traders,

etc.

Although

matching

is

likely

intractable,

are

there

good

heuristics

that

achieve

matches

in

many

cases

or

approximate

a

matching?

Exploring

sharing

rules

for

dividing

the

surplus,

and

incentive

properties

of

the

resulting

mechanisms.

We

may

consider

a

market

to

be

in

computational

equilibrium

if

no

computationally-bounded

player

can

find

a

strategy

that

increases

utility.

With

few

exceptions

little

is

known

about

computational

equilibriums.

A

natural

question

is

to

determine

whether

a

market

can

achieve

a

computational

equilibrium

that

is

not

a

true

equilibrium,

and

under

what

circumstances

this

may

occur.

Acknowledgments

We

thank

Rahul

Sami

for

his

help

with

Section

We

thank

Rahul,

Joan

Feigenbaum

and

Robin

Hanson

for

useful

discussions.

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