A

Dynamic

Pari-Mutuel

Market

for

Hedging,

Wagering,

and

Information

Aggregation

David

M.

Pennock

Yahoo!

Research

Labs

N.

Pasadena

Ave,

Floor

Pasadena,

CA

USA

pennockd@yahoo-inc.com

ABSTRACT

I

develop

a

new

mechanism

for

risk

allocation

and

information

speculation

called

a

dynamic

pari-mutuel

market

(DPM).

A

DPM

acts

as

hybrid

between

a

pari-mutuel

market

and

a

continuous

double

auction

(CDA),

inheriting

some

of

the

advantages

of

both.

Like

a

pari-mutuel

market,

a

DPM

offers

infinite

buy-in

liquidity

and

zero

risk

for

the

market

institution;

like

a

CDA,

a

DPM

can

continuously

react

to

new

information,

dynamically

incorporate

information

into

prices,

and

allow

traders

to

lock

in

gains

or

limit

losses

by

selling

prior

to

event

resolution.

The

trader

interface

can

be

designed

to

mimic

the

familiar

double

auction

format

with

bid-ask

queues,

though

with

an

addition

variable

called

the

payoff

per

share.

The

DPM

price

function

can

be

viewed

as

an

automated

market

maker

always

offering

to

sell

at

some

price,

and

moving

the

price

appropriately

according

to

demand.

Since

the

mechanism

is

pari-mutuel

(i.e.,

redistributive),

it

is

guaranteed

to

pay

out

exactly

the

amount

of

money

taken

in.

I

explore

a

number

of

variations

on

the

basic

DPM,

analyzing

the

properties

of

each,

and

solving

in

closed

form

for

their

respective

price

functions.

Categories

and

Subject

Descriptors

[Computer

Applications]:

Social

and

Behavioral

Sciences-Economics

General

Terms

Algorithms,

Design,

Economics,

Theory.

INTRODUCTION

A

wide

variety

of

financial

and

wagering

mechanisms

have

been

developed

to

support

hedging

(i.e.,

insuring)

against

exposure

to

uncertain

events

and/or

speculative

trading

on

uncertain

events.

The

dominant

mechanism

used

in

financial

circles

is

the

continuous

double

auction

(CDA),

or

in

some

cases

the

CDA

with

market

maker

(CDAwMM).

The

primary

mechanism

used

for

sports

wagering

is

a

bookie

or

bookmaker,

who

essentially

acts

exactly

as

a

market

maker.

Horse

racing

and

jai

alai

wagering

traditionally

employ

the

pari-mutuel

mechanism.

Though

there

is

no

formal

or

logical

separation

between

financial

trading

and

wagering,

the

two

endeavors

are

socially

considered

distinct.

Recently,

there

has

been

a

move

to

employ

CDAs

or

CDAwMMs

for

all

types

of

wagering,

including

on

sports,

horse

racing,

political

events,

world

news,

and

many

other

uncertain

events,

and

a

simultaneous

and

opposite

trend

to

use

bookie

systems

for

betting

on

financial

markets.

These

trends

highlight

the

interchangeable

nature

of

the

mechanisms

and

further

blur

the

line

between

investing

and

betting.

Some

companies

at

the

forefront

of

these

movements

are

growing

exponentially,

with

some

industry

observers

declaring

the

onset

of

a

revolution

in

the

wagering

Each

mechanism

has

pros

and

cons

for

the

market

institution

and

the

participating

traders.

A

CDA

only

matches

willing

traders,

and

so

poses

no

risk

whatsoever

for

the

market

institution.

But

a

CDA

can

suffer

from

illiquidity

in

the

form

huge

bid-ask

spreads

or

even

empty

bid-ask

queues

if

trading

is

light

and

thus

markets

are

thin.

A

successful

CDA

must

overcome

a

chicken-and-egg

problem:

traders

are

attracted

to

liquid

markets,

but

liquid

markets

require

a

large

number

of

traders.

A

CDAwMM

and

the

similar

bookie

mechanism

have

built-in

liquidity,

but

at

a

cost:

the

market

maker

itself,

usually

affiliated

with

the

market

institution,

is

exposed

to

significant

risk

of

large

monetary

losses.

Both

the

CDA

and

CDAwMM

offer

incentives

for

traders

to

leverage

information

continuously

as

soon

as

that

information

becomes

available.

As

a

result,

prices

are

known

to

capture

the

current

state

of

information

exceptionally

well.

Pari-mutuel

markets

effectively

have

infinite

liquidity:

anyone

can

place

a

bet

on

any

outcome

at

any

time,

without

the

need

for

a

matching

offer

from

another

bettor

or

a

market

maker.

Pari-mutuel

markets

also

involve

no

risk

for

the

market

institution,

since

they

only

redistribute

money

from

losing

wagers

to

winning

wagers.

However,

pari-mutuel

kets

are

not

suitable

for

situations

where

information

arrives

over

time,

since

there

is

a

strong

disincentive

for

placing

bets

until

either

all

information

is

revealed,

or

the

market

is

about

to

close.

For

this

reason,

pari-mutuel

prices

prior

to

the

market"s

close

cannot

be

considered

a

reflection

of

current

information.

Pari-mutuel

market

participants

cannot

buy

low

and

sell

high:

they

cannot

cash

out

gains

(or

limit

losses)

before

the

event

outcome

is

revealed.

Because

the

process

whereby

information

arrives

continuously

over

time

is

the

rule

rather

than

the

exception,

the

applicability

of

the

standard

pari-mutuel

mechanism

is

questionable

in

a

large

number

of

settings.

In

this

paper,

I

develop

a

new

mechanism

suitable

for

hedging,

speculating,

and

wagering,

called

a

dynamic

parimutuel

market

(DPM).

A

DPM

can

be

thought

of

as

a

hybrid

between

a

pari-mutuel

market

and

a

CDA.

A

DPM

is

indeed

pari-mutuel

in

nature,

meaning

that

it

acts

only

to

redistribute

money

from

some

traders

to

others,

and

so

exposes

the

market

institution

to

no

volatility

(no

risk).

A

constant,

pre-determined

subsidy

is

required

to

start

the

market.

The

subsidy

can

in

principle

be

arbitrarily

small

and

might

conceivably

come

from

traders

(via

antes

or

transaction

fees)

rather

than

the

market

institution,

though

a

nontrivial

outside

subsidy

may

actually

encourage

trading

and

information

aggregation.

A

DPM

has

the

infinite

liquidity

of

a

pari-mutuel

market:

traders

can

always

purchase

shares

in

any

outcome

at

any

time,

at

some

price

automatically

set

by

the

market

institution.

A

DPM

is

also

able

to

react

to

and

incorporate

information

arriving

over

time,

like

a

CDA.

The

market

institution

changes

the

price

for

particular

outcomes

based

on

the

current

state

of

wagering.

If

a

particular

outcome

receives

a

relatively

large

number

of

wagers,

its

price

increases;

if

an

outcome

receives

relatively

few

wagers,

its

price

decreases.

Prices

are

computed

automatically

using

a

price

function,

which

can

differ

depending

on

what

properties

are

desired.

The

price

function

determines

the

instantaneous

price

per

share

for

an

infinitesimal

quantity

of

shares;

the

total

cost

for

purchasing

n

shares

is

computed

as

the

integral

of

the

price

function

from

to

n.

The

complexity

of

the

price

function

can

be

hidden

from

traders

by

communicating

only

the

ask

prices

for

various

lots

of

shares

(e.g.,

lots

of

shares),

as

is

common

practice

in

CDAs

and

CDAwMMs.

DPM

prices

do

reflect

current

information,

and

traders

can

cash

out

in

an

aftermarket

to

lock

in

gains

or

limit

losses

before

the

event

outcome

is

revealed.

While

there

is

always

a

market

maker

willing

to

accept

buy

orders,

there

is

not

a

market

maker

accepting

sell

orders,

and

thus

no

guaranteed

liquidity

for

selling:

instead,

selling

is

accomplished

via

a

standard

CDA

mechanism.

Traders

can

always

hedge-sell

by

purchasing

the

opposite

outcome

than

they

already

own.

BACKGROUND

AND

RELATED

WORK

Pari-mutuel

markets

Pari-mutuel

markets

are

common

at

horse

races

dog

races,

and

jai

alai

games.

In

a

pari-mutuel

market

people

place

wagers

on

which

of

two

or

more

mutually

exclusive

and

exhaustive

outcomes

will

occur

at

some

time

in

the

future.

After

the

true

outcome

becomes

known,

all

of

the

money

that

is

lost

by

those

who

bet

on

the

incorrect

outcome

is

redistributed

to

those

who

bet

on

the

correct

outcome,

in

direct

proportion

to

the

amount

they

wagered.

More

formally,

if

there

are

k

mutually

exclusive

and

exhaustive

outcomes

(e.g.,

k

horses,

exactly

one

of

which

will

win),

and

Mk

dollars

are

bet

on

each

outcome,

and

outcome

i

occurs,

then

everyone

who

bet

on

an

outcome

j

i

loses

their

wager,

while

everyone

who

bet

on

outcome

i

receives

Pk

Mj/Mi

dollars

for

every

dollar

they

wagered.

That

is,

every

dollar

wagered

on

i

receives

an

equal

share

of

all

money

wagered.

An

equivalent

way

to

think

about

the

redistribution

rule

is

that

every

dollar

wagered

on

i

is

refunded,

then

receives

an

equal

share

of

all

remaining

money

bet

on

the

losing

outcomes,

or

P

j=i

Mj/Mi

dollars.

In

practice,

the

market

institution

(e.g.,

the

racetrack)

first

takes

a

certain

percent

of

the

total

amount

wagered,

usually

about

in

the

United

States,

then

redistributes

whatever

money

remains

to

the

winners

in

proportion

to

their

amount

bet.

Consider

a

simple

example

with

two

outcomes,

A

and

B.

The

outcomes

are

mutually

exclusive

and

exhaustive,

meaning

that

Pr(A

B)

and

Pr(A)

Pr(B)

Suppose

is

bet

on

A

and

on

B.

Now

suppose

that

A

occurs

(e.g.,

horse

A

wins

the

race).

People

who

wagered

on

B

lose

their

money,

or

in

total.

People

who

wagered

on

A

win

and

each

receives

a

proportional

share

of

the

total

wagered

(ignoring

fees).

Specifically,

each

wager

on

A

entitles

its

owner

a

share

of

the

or

Every

dollar

bet

in

a

pari-mutuel

market

has

an

equal

payoff,

regardless

of

when

the

wager

was

placed

or

how

much

money

was

invested

in

the

various

outcomes

at

the

time

the

wager

was

placed.

The

only

state

that

matters

is

the

final

state:

the

final

amounts

wagered

on

all

the

outcomes

when

the

market

closes,

and

the

identity

of

the

correct

outcome.

As

a

result,

there

is

a

disincentive

to

place

a

wager

early

if

there

is

any

chance

that

new

information

might

become

available.

Moreover,

there

are

no

guarantees

about

the

payoff

rate

of

a

particular

bet,

except

that

it

will

be

nonnegative

if

the

correct

outcome

is

chosen.

Payoff

rates

can

fluctuate

arbitrarily

until

the

market

closes.

So

a

second

reason

not

to

bet

early

is

to

wait

to

get

a

better

sense

of

the

final

payout

rates.

This

is

in

contrast

to

CDAs

and

CDAwMMs,

like

the

stock

market,

where

incentives

exist

to

invest

as

soon

as

new

information

is

revealed.

Pari-mutuel

bettors

may

be

allowed

to

switch

their

chosen

outcome,

or

even

cancel

their

bet,

prior

to

the

market"s

close.

However,

they

cannot

cash

out

of

the

market

early,

to

either

lock

in

gains

or

limit

losses,

if

new

information

favors

one

outcome

over

another,

as

is

possible

in

a

CDA

or

a

CDAwMM.

If

bettors

can

cancel

or

change

their

bets,

then

an

aftermarket

to

sell

existing

wagers

is

not

sensible:

every

dollar

wagered

is

worth

exactly

up

until

the

market"s

close-no

one

would

buy

at

greater

than

and

no

one

would

sell

at

less

than

Pari-mutuel

bettors

must

wait

until

the

outcome

is

revealed

to

realize

any

profit

or

loss.

Unlike

a

CDA,

in

a

pari-mutuel

market,

anyone

can

place

a

wager

of

any

amount

at

any

time-there

is

in

a

sense

infinite

liquidity

for

buying.

A

CDAwMM

also

has

built-in

liquidity,

but

at

the

cost

of

significant

risk

for

the

market

maker.

In

a

pari-mutuel

market,

since

money

is

only

redistributed

among

bettors,

the

market

institution

itself

has

no

risk.

The

main

drawback

of

a

pari-mutuel

market

is

that

it

is

useful

only

for

capturing

the

value

of

an

uncertain

asset

at

some

instant

in

time.

It

is

ill-suited

for

situations

where

information

arrives

over

time,

continuously

updating

the

estimated

value

of

the

asset-situations

common

in

most

all

trading

and

wagering

scenarios.

There

is

no

notion

of

buying

low

and

selling

high,

as

occurs

in

a

CDA,

where

buying

when

few

others

are

buying

(and

the

price

is

low)

is

rewarded

more

than

buying

when

many

others

are

buying

(and

the

price

is

high).

Perhaps

for

this

reason,

in

most

dynamic

environments,

financial

mechanisms

like

the

CDA

that

can

react

in

real-time

to

changing

information

are

more

typically

employed

to

facilitate

speculating

and

hedging.

Since

a

pari-mutuel

market

can

estimate

the

value

of

an

asset

at

a

single

instant

in

time,

a

repeated

pari-mutuel

market,

where

distinct

pari-mutuel

markets

are

run

at

consecutive

intervals,

could

in

principle

capture

changing

information

dynamics.

But

running

multiple

consecutive

markets

would

likely

thin

out

trading

in

each

individual

market.

Also,

in

each

individual

pari-mutuel

market,

the

incentives

would

still

be

to

wait

to

bet

until

just

before

the

ending

time

of

that

particular

market.

This

last

problem

might

be

mitigated

by

instituting

a

random

stopping

rule

for

each

individual

pari-mutuel

market.

In

laboratory

experiments,

pari-mutuel

markets

have

shown

a

remarkable

ability

to

aggregate

and

disseminate

information

dispersed

among

traders,

at

least

for

a

single

snapshot

in

time

A

similar

ability

has

been

recognized

at

real

racetracks

Financial

markets

In

the

financial

world,

wagering

on

the

outcomes

of

uncertain

future

propositions

is

also

common.

The

typical

market

mechanism

used

is

the

continuous

double

auction

(CDA).

The

term

securities

market

in

economics

and

finance

generically

encompasses

a

number

of

markets

where

speculating

on

uncertain

events

is

possible.

Examples

include

stock

markets

like

NASDAQ,

options

markets

like

the

CBOE

futures

markets

like

the

CME

other

derivatives

markets,

insurance

markets,

political

stock

markets

idea

futures

markets

decision

markets

and

even

market

games

Securities

markets

generally

have

an

economic

and

social

value

beyond

facilitating

speculation

or

wagering:

they

allow

traders

to

hedge

risk,

or

to

insure

against

undesirable

outcomes.

So

if

a

particular

outcome

has

disutility

for

a

trader,

he

or

she

can

mitigate

the

risk

by

wagering

for

the

outcome,

to

arrange

for

compensation

in

case

the

outcome

occurs.

In

this

sense,

buying

automobile

insurance

is

effectively

a

bet

that

an

accident

or

other

covered

event

will

occur.

Similarly,

buying

a

put

option,

which

is

useful

as

a

hedge

for

a

stockholder,

is

a

bet

that

the

underlying

stock

will

go

down.

In

practice,

agents

engage

in

a

mixture

of

hedging

and

speculating,

and

there

is

no

clear

dividing

line

between

the

two

Like

pari-mutuel

markets,

often

prices

in

financial

markets

are

excellent

information

aggregators,

yielding

very

accurate

forecasts

of

future

events

A

CDA

constantly

matches

orders

to

buy

an

asset

with

orders

to

sell.

If

at

any

time

one

party

is

willing

to

buy

one

unit

of

the

asset

at

a

bid

price

of

pbid,

while

another

party

is

willing

to

sell

one

unit

of

the

asset

at

an

ask

price

of

pask,

and

pbid

is

greater

than

or

equal

to

pask,

then

the

two

parties

transact

(at

some

price

between

pbid

and

pask).

If

the

highest

bid

price

is

less

than

the

lowest

ask

price,

then

no

transactions

occur.

In

a

CDA,

the

bid

and

ask

prices

rapidly

change

as

new

information

arrives

and

traders

reassess

the

value

of

the

asset.

Since

the

auctioneer

only

matches

willing

bidders,

the

auctioneer

takes

on

no

risk.

However,

buyers

can

only

buy

as

many

shares

as

sellers

are

willing

to

sell;

for

any

transaction

to

occur,

there

must

be

a

counterparty

on

the

other

side

willing

to

accept

the

trade.

As

a

result,

when

few

traders

participate

in

a

CDA,

it

may

become

illiquid,

meaning

that

not

much

trading

activity

occurs.

The

spread

between

the

highest

bid

price

and

the

lowest

ask

price

may

be

very

large,

or

one

or

both

queues

may

be

completely

empty,

discouraging

One

way

to

induce

liquidity

is

to

provide

a

market

maker

who

is

willing

to

accept

a

large

number

of

buy

and

sell

orders

at

particular

prices.

We

call

this

mechanism

a

CDA

with

market

maker

Conceptually,

the

market

maker

is

just

like

any

other

trader,

but

typically

is

willing

to

accept

a

much

larger

volume

of

trades.

The

market

maker

may

be

a

person,

or

may

be

an

automated

algorithm.

Adding

a

market

maker

to

the

system

increases

liquidity,

but

exposes

the

market

maker

to

risk.

Now,

instead

of

only

matching

trades,

the

system

actually

takes

on

risk

of

its

own,

and

depending

on

what

happens

in

the

future,

may

lose

considerable

amounts

of

money.

Wagering

markets

The

typical

Las

Vegas

bookmaker

or

oddsmaker

functions

much

like

a

market

maker

in

a

CDA.

In

this

case,

the

market

institution

(the

book

or

house)

sets

the

initially

according

to

expert

opinion,

and

later

in

response

to

the

relative

level

of

betting

on

the

various

outcomes.

Unlike

in

a

pari-mutuel

environment,

whenever

a

wager

is

placed

with

a

bookmaker,

the

odds

or

terms

for

that

bet

are

fixed

at

the

time

of

the

bet.

The

bookmaker

profits

by

offering

different

odds

for

the

two

sides

of

the

bet,

essentially

defining

a

bidask

spread.

While

odds

may

change

in

response

to

changing

information,

any

bets

made

at

previously

set

odds

remain

in

effect

according

to

the

odds

at

the

time

of

the

bet;

this

is

precisely

in

analogy

to

a

CDAwMM.

One

difference

between

a

bookmaker

and

a

market

maker

is

that

the

former

usually

operates

in

a

take

it

or

leave

it

mode:

bettors

cannot

place

their

own

limit

orders

on

a

common

queue,

they

can

in

effect

only

place

market

orders

at

prices

defined

by

the

bookmaker.

Still,

the

bookmaker

certainly

reacts

to

bettor

demand.

Like

a

market

maker,

the

bookmaker

exposes

itself

to

significant

risk.

Sports

betting

markets

have

also

been

shown

to

provide

high

quality

aggregate

forecasts

Market

scoring

rule

Hanson"s

market

scoring

rule

(MSR)

is

a

new

mechanism

for

hedging

and

speculating

that

shares

some

properties

in

common

with

a

DPM.

Like

a

DPM,

an

MSR

can

be

conceptualized

as

an

automated

market

maker

always

willing

to

accept

a

trade

on

any

event

at

some

price.

An

MSR

requires

a

patron

to

subsidize

the

market.

The

patron"s

final

loss

is

variable,

and

thus

technically

implies

a

degree

of

risk,

though

the

maximum

loss

is

bounded.

An

MSR

maintains

a

probability

distribution

over

all

events.

At

any

time

any

Thin

markets

do

occur

often

in

practice,

and

can

be

seen

in

a

variety

of

the

less

popular

markets

available

on

or

in

some

financial

options

markets,

for

example.

A

very

clear

example

of

a

CDAwMM

is

the

interactive

betting

market

on

Or,

alternatively,

the

bookmaker

sets

the

game

line

in

order

to

provide

even-money

odds.

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.

An

MSR

is

not

pari-mutuel

in

nature,

as

the

patron

in

general

injects

a

variable

amount

of

money

into

the

system.

An

MSR

provides

a

two-sided

automated

market

maker,

while

a

DPM

provides

a

one-sided

automated

market

maker.

In

an

MSR,

the

vector

of

payoffs

across

outcomes

is

fixed

at

the

time

of

the

trade,

while

in

a

DPM,

the

vector

of

payoffs

across

outcomes

depends

both

on

the

state

of

wagering

at

the

time

of

the

trade

and

the

state

of

wagering

at

the

market"s

close.

While

the

mechanisms

are

quite

different-and

so

trader

acceptance

and

incentives

may

strongly

differ-the

properties

and

motivations

of

DPMs

and

MSRs

are

quite

similar.

Hanson

shows

how

MSRs

are

especially

well

suited

for

allowing

bets

on

a

combinatorial

number

of

outcomes.

The

patron"s

payment

for

subsidizing

trading

on

all

possible

combinations

of

n

events

is

no

larger

than

the

sum

of

subsidizing

the

n

event

marginals

independently.

The

mechanism

was

planned

for

use

in

the

Policy

Analysis

Market

(PAM),

a

futures

market

in

Middle

East

related

outcomes

and

funded

by

DARPA

until

a

media

firestorm

killed

the

As

of

this

writing,

the

founders

of

PAM

were

considering

reopening

under

private

A

DYNAMIC

PARI-MUTUEL

MARKET

High-level

description

In

contrast

to

a

standard

pari-mutuel

market,

where

each

dollar

always

buys

an

equal

share

of

the

payoff,

in

a

DPM

each

dollar

buys

a

variable

share

in

the

payoff

depending

on

the

state

of

wagering

at

the

time

of

purchase.

So

a

wager

on

A

at

a

time

when

most

others

are

wagering

on

B

offers

a

greater

possible

profit

than

a

wager

on

A

when

most

others

are

also

wagering

on

A.

A

natural

way

to

communicate

the

changing

payoff

of

a

bet

is

to

say

that,

at

any

given

time,

a

certain

amount

of

money

will

buy

a

certain

number

of

shares

in

one

outcome

the

other.

Purchasing

a

share

entitles

its

owner

to

an

equal

stake

in

the

winning

pot

should

the

chosen

outcome

occur.

The

payoff

is

variable,

because

when

few

people

are

betting

on

an

outcome,

shares

will

generally

be

cheaper

than

at

a

time

when

many

people

are

betting

that

outcome.

There

is

no

pre-determined

limit

on

the

number

of

shares:

new

shares

can

be

continually

generated

as

trading

proceeds.

For

simplicity,

all

analyses

in

this

paper

consider

the

binary

outcome

case;

generalizing

to

multiple

discrete

outcomes

should

be

straightforward.

Denote

the

two

outcomes

A

and

B.

The

outcomes

are

mutually

exclusive

and

See

for

more

information,

or

for

commentary.

haustive.

Denote

the

instantaneous

price

per

share

of

A

as

and

the

price

per

share

of

B

as

Denote

the

payoffs

per

share

as

and

respectively.

These

four

numbers,

are

the

key

numbers

that

traders

must

track

and

understand.

Note

that

the

price

is

set

at

the

time

of

the

wager;

the

payoff

per

share

is

finalized

only

after

the

event

outcome

is

revealed.

At

any

time,

a

trader

can

purchase

an

infinitesimal

quantity

of

shares

of

A

at

price

(and

similarly

for

B).

However,

since

the

price

changes

continuously

as

shares

are

purchased,

the

cost

of

buying

n

shares

is

computed

as

the

integral

of

a

price

function

from

to

n.

The

use

of

continuous

functions

and

integrals

can

be

hidden

from

traders

by

aggregating

the

automated

market

maker"s

sell

orders

into

discrete

lots

of,

say,

shares

each.

These

ask

orders

can

be

automatically

entered

into

the

system

by

the

market

institution,

so

that

traders

interact

with

what

looks

like

a

more

familiar

CDA;

we

examine

this

interface

issue

in

more

detail

below

in

Section

For

our

analysis,

we

introduce

the

following

additional

notation.

Denote

as

the

total

amount

of

money

wagered

on

A,

as

the

total

amount

of

money

wagered

on

B,

T

as

the

total

amount

of

money

wagered

on

both

sides,

as

the

total

number

of

shares

purchased

of

A,

and

as

the

total

number

of

shares

purchased

of

B.

There

are

many

ways

to

formulate

the

price

function.

Several

natural

price

functions

are

outlined

below;

each

is

motivated

as

the

unique

solution

to

a

particular

constraint

on

price

dynamics.

Advantages

and

disadvantages

To

my

knowledge,

a

DPM

is

the

only

known

mechanism

for

hedging

and

speculating

that

exhibits

all

three

of

the

following

properties:

guaranteed

liquidity,

no

risk

for

the

market

institution,

and

continuous

incorporation

of

information.

A

standard

pari-mutuel

fails

A

CDA

fails

A

CDAwMM,

the

bookmaker

mechanism,

and

an

MSR

all

fail

Even

though

technically

an

MSR

exposes

its

patron

to

risk

(i.e.,

a

variable

future

payoff),

the

patron"s

maximum

loss

is

bounded,

so

the

distinction

between

a

DPM

and

an

MSR

in

terms

of

these

three

properties

is

more

technical

than

practical.

DPM

traders

can

cash

out

of

the

market

early,

just

like

stock

market

traders,

to

lock

in

a

profit

or

limit

a

loss,

an

action

that

is

simply

not

possible

in

a

standard

pari-mutuel.

A

DPM

also

has

some

drawbacks.

The

payoff

for

a

wager

depends

both

on

the

price

at

the

time

of

the

trade,

and

on

the

final

payoff

per

share

at

the

market"s

close.

This

contrasts

with

the

CDA

variants,

where

the

payoff

vector

across

possible

future

outcomes

is

fixed

at

the

time

of

the

trade.

So

a

trader"s

strategic

optimization

problem

is

complicated

by

the

need

to

predict

the

final

values

of

and

If

P

changes

according

to

a

random

walk,

then

traders

can

take

the

current

P

as

an

unbiased

estimate

of

the

final

P,

greatly

decreasing

the

complexity

of

their

optimization.

If

P

does

not

change

according

to

a

random

walk,

the

mechanism

still

has

utility

as

a

mechanism

for

hedging

and

speculating,

though

optimization

may

be

difficult,

and

determining

a

measure

of

the

market"s

aggregate

opinion

of

the

probabilities

of

A

and

B

may

be

difficult.

We

discuss

the

implications

of

random

walk

behavior

further

below

in

Section

in

the

discussion

surrounding

Assumption

A

second

drawback

of

a

DPM

is

its

one-sided

nature.

While

an

automated

market

maker

always

stands

ready

to

accept

buy

orders,

there

is

no

corresponding

market

maker

to

accept

sell

orders.

Traders

must

sell

to

each

other

using

a

standard

CDA

mechanism,

for

example

by

posting

an

ask

order

at

a

price

at

or

below

the

market

maker"s

current

ask

price.

Traders

can

also

always

hedge-sell

by

purchasing

shares

in

the

opposite

outcome

from

the

market

maker,

thereby

hedging

their

bet

if

not

fully

liquidating

it.

Redistribution

rule

In

a

standard

pari-mutuel

market,

payoffs

can

be

computed

in

either

of

two

equivalent

ways:

each

winning

wager

receives

a

refund

of

the

initial

paid,

plus

an

equal

share

of

all

losing

wagers,

or

each

winning

wager

receives

an

equal

share

of

all

wagers,

winning

or

losing.

Because

each

dollar

always

earns

an

equal

share

of

the

payoff,

the

two

formulations

are

precisely

the

same:

Mlose

Mwin

Mwin

Mlose

Mwin

In

a

dynamic

pari-mutuel

market,

because

each

dollar

is

not

equally

weighted,

the

two

formulations

are

distinct,

and

lead

to

significantly

different

price

functions

and

mechanisms,

each

with

different

potentially

desirable

properties.

We

consider

each

case

in

turn.

The

next

section

analyzes

case

where

only

losing

money

is

redistributed.

Section

examines

case

where

all

money

is

redistributed.

DPM

I:

LOSING

MONEY

REDISTRIBUTED

For

the

case

where

the

initial

payments

on

winning

bets

are

refunded,

and

only

losing

money

is

redistributed,

the

respective

payoffs

per

share

are

simply:

So,

if

A

occurs,

shareholders

of

A

receive

all

of

their

initial

payment

back,

plus

dollars

per

share

owned,

while

shareholders

of

B

lose

all

money

wagered.

Similarly,

if

B

occurs,

shareholders

of

B

receive

all

of

their

initial

payment

back,

plus

dollars

per

share

owned,

while

shareholders

of

A

lose

all

money

wagered.

Without

loss

of

generality,

I

will

analyze

the

market

from

the

perspective

of

A,

deriving

prices

and

payoffs

for

A

only.

The

equations

for

B

are

symmetric.

The

trader"s

per-share

expected

value

for

purchasing

an

infinitesimal

quantity

of

shares

of

A

is

E[

shares]

Pr(A)

E

Pr(A))

E[

shares]

Pr(A)

E

A

Pr(A))

where

is

an

infinitesimal

quantity

of

shares

of

A,

Pr(A)

is

the

trader"s

belief

in

the

probability

of

A,

and

is

the

instantaneous

price

per

share

of

A

for

an

infinitesimal

quantity

of

shares.

is

the

trader"s

expectation

of

the

payoff

per

share

of

A

after

the

market

closes

and

given

that

A

occurs.

This

is

a

subtle

point.

The

value

of

does

not

matter

if

B

occurs,

since

in

this

case

shares

of

A

are

worthless,

and

the

current

value

of

does

not

necessarily

matter

as

this

may

change

as

trading

continues.

So,

in

order

to

determine

the

expected

value

of

shares

of

A,

the

trader

must

estimate

what

he

or

she

expects

the

payoff

per

share

to

be

in

the

end

(after

the

market

closes)

if

A

occurs.

If

E[

shares]/

a

risk-neutral

trader

should

purchase

shares

of

A.

How

many

shares?

This

depends

on

the

price

function

determining

In

general,

increases

as

more

shares

are

purchased.

The

risk-neutral

trader

should

continue

purchasing

shares

until

E[

shares]/

(A

riskaverse

trader

will

generally

stop

purchasing

shares

before

driving

E[

shares]/

all

the

way

to

zero.)

Assuming

riskneutrality,

the

trader"s

optimization

problem

is

to

choose

a

number

of

shares

n

of

A

to

purchase,

in

order

to

maximize

E[n

shares]

Pr(A)·n·E

Z

n

It"s

easy

to

see

that

the

same

value

of

n

can

be

solved

for

by

finding

the

number

of

shares

required

to

drive

E[

shares]/

to

zero.

That

is,

find

n

satisfying

Pr(A)

E

Pr(A))

if

such

a

n

exists,

otherwise

n

Market

probability

As

traders

who

believe

that

E[

shares

of

A]/

purchase

shares

of

A

and

traders

who

believe

that

E[

shares

of

B]/

purchase

shares

of

B,

the

prices

and

change

according

to

a

price

function,

as

prescribed

below.

The

current

prices

in

a

sense

reflect

the

market"s

opinion

as

a

whole

of

the

relative

probabilities

of

A

and

B.

Assuming

an

efficient

marketplace,

the

market

as

a

whole

considers

E[

shares]/

since

the

mechanisms

is

a

zero

sum

game.

For

example,

if

market

participants

in

aggregate

felt

that

E[

shares]/

then

there

would

be

net

demand

for

A,

driving

up

the

price

of

A

until

E[

shares]/

Define

MPr(A)

to

be

the

market

probability

of

A,

or

the

probability

of

A

inferred

by

assuming

that

E[

shares]/

We

can

consider

MPr(A)

to

be

the

aggregate

probability

of

A

as

judged

by

the

market

as

a

whole.

MPr(A)

is

the

solution

to

MPr(A)

MPr(A))

Solving

we

get

MPr(A)

At

this

point

we

make

a

critical

assumption

in

order

to

greatly

simplify

the

analysis;

we

assume

that

That

is,

we

assume

that

the

current

value

for

the

payoff

per

share

of

A

is

the

same

as

the

expected

final

value

of

the

payoff

per

share

of

A

given

that

A

occurs.

This

is

certainly

true

for

the

last

(infinitesimal)

wager

before

the

market

closes.

It"s

not

obvious,

however,

that

the

assumption

is

true

well

before

the

market"s

close.

Basically,

we

are

assuming

that

the

value

of

moves

according

to

an

unbiased

random

walk:

the

current

value

of

is

the

best

expectation

of

its

future

value.

I

conjecture

that

there

are

reasonable

market

efficiency

conditions

under

which

assumption

is

true,

though

I

have

not

been

able

to

prove

that

it

arises

naturally

from

rational

trading.

We

examine

scenarios

below

in

which

assumption

seems

especially

plausible.

Nonetheless,

the

assumption

effects

our

analysis

only.

Regardless

of

whether

is

true,

each

price

function

derived

below

implies

a

welldefined

zero-sum

game

in

which

traders

can

play.

If

traders

can

assume

that

is

true,

then

their

optimization

problem

is

greatly

simplified;

however,

optimizing

does

not

depend

on

the

assumption,

and

traders

can

still

optimize

by

strategically

projecting

the

final

expected

payoff

in

whatever

complicated

way

they

desire.

So,

the

utility

of

DPM

for

hedging

and

speculating

does

not

necessarily

hinge

on

the

truth

of

assumption

On

the

other

hand,

the

ability

to

easily

infer

an

aggregate

market

consensus

probability

from

market

prices

does

depend

on

Price

functions

A

variety

of

price

functions

seem

reasonable,

each

exhibiting

various

properties,

and

implying

differing

market

probabilities.

Price

function

I:

Price

of

A

equals

payoff

of

B

One

natural

price

function

to

consider

is

to

set

the

price

per

share

of

A

equal

to

the

payoff

per

share

of

B,

and

set

the

price

per

share

of

B

equal

to

the

payoff

per

share

of

A.

That

is,

Enforcing

this

relationship

reduces

the

dimensionality

of

the

system

from

four

to

two,

simplifying

the

interface:

traders

need

only

track

two

numbers

instead

of

four.

The

relationship

makes

sense,

since

new

information

supporting

A

should

encourage

purchasing

of

shares

A,

driving

up

both

the

price

of

A

and

the

payoff

of

B,

and

driving

down

the

price

of

B

and

the

payoff

of

A.

In

this

setting,

assumption

seems

especially

reasonable,

since

if

an

efficient

market

hypothesis

leads

prices

to

follow

a

random

walk,

than

payoffs

must

also

follow

a

random

walk.

The

constraints

lead

to

the

following

derivation

of

the

market

probability:

MPr(A)

MPr(B)

MPr(A)

MPr(B)

MPr(A)

MPr(B)

MPr(A)

The

constraints

specify

the

instantaneous

relationship

between

payoff

and

price.

From

this,

we

can

derive

how

prices

change

when

(non-infinitesimal)

shares

are

purchased.

Let

n

be

the

number

of

shares

purchased

and

let

m

be

the

amount

of

money

spent

purchasing

n

shares.

Note

that

dm/dn,

the

instantaneous

price

per

share,

and

m

R

n

Substituting

into

equation

we

get:

dm

dn

m

dm

m

dn

Z

dm

m

Z

dn

m)

n

C

m

h

e

n

i

Equation

gives

the

cost

of

purchasing

n

shares.

The

instantaneous

price

per

share

as

a

function

of

n

is

dm

dn

e

n

Note

that

as

required.

The

derivation

of

the

price

function

for

B

is

analogous

and

the

results

are

symmetric.

The

notion

of

buying

infinitesimal

shares,

or

integrating

costs

over

a

continuous

function,

are

probably

foreign

to

most

traders.

A

more

standard

interface

can

be

implemented

by

discretizing

the

costs

into

round

lots

of

shares,

for

example

lots

of

shares.

Then

ask

orders

of

shares

each

at

the

appropriate

price

can

be

automatically

placed

by

the

market

institution.

For

example,

the

market

institution

can

place

an

ask

order

for

shares

at

price

another

ask

order

for

shares

at

price

a

third

ask

for

shares

at

etc.

In

this

way,

the

market

looks

more

familiar

to

traders,

like

a

typical

CDA

with

a

number

of

ask

orders

at

various

prices

automatically

available.

A

trader

buying

less

than

shares

would

pay

a

bit

more

than

if

the

true

cost

were

computed

using

but

the

discretized

interface

would

probably

be

more

intuitive

and

transparent

to

the

majority

of

traders.

The

above

equations

assume

that

all

money

that

comes

in

is

eventually

returned

or

redistributed.

In

other

words,

the

mechanism

is

a

zero

sum

game,

and

the

market

institution

takes

no

portion

of

the

money.

This

could

be

generalized

so

that

the

market

institution

always

takes

a

certain

amount,

or

a

certain

percent,

or

a

certain

amount

per

transaction,

or

a

certain

percent

per

transaction,

before

money

in

returned

or

redistributed.

Finally,

note

that

the

above

price

function

is

undefined

when

the

amount

bet

or

the

number

of

shares

are

zero.

So

the

system

must

begin

with

some

positive

amount

on

both

sides,

and

some

positive

number

of

shares

outstanding

on

both

sides.

These

initial

amounts

can

be

arbitrarily

small

in

principle,

but

the

size

of

the

initial

subsidy

may

affect

the

incentives

of

traders

to

participate.

Also,

the

smaller

the

initial

amounts,

the

more

each

new

dollar

effects

the

prices.

The

initialization

amounts

could

be

funded

as

a

subsidy

from

the

market

institution

or

a

patron,

which

I"ll

call

a

seed

wager,

or

from

a

portion

of

the

fees

charged,

which

I"ll

call

an

ante

wager.

Price

function

II:

Price

of

A

proportional

to

money

on

A

A

second

price

function

can

be

derived

by

requiring

the

ratio

of

prices

to

be

equal

to

the

ratio

of

money

wagered.

That

is,

In

other

words,

the

price

of

A

is

proportional

to

the

amount

of

money

wagered

on

A,

and

similarly

for

B.

This

seems

like

a

particularly

natural

way

to

set

the

price,

since

the

more

money

that

is

wagered

on

one

side,

the

cheaper

becomes

a

share

on

the

other

side,

in

exactly

the

same

proportion.

Using

Equation

along

with

and

we

can

derive

the

implied

market

probability:

MPr(A)

MPr(B)

MPr(B)

MPr(A)

MPr(A)

MPr(B)

MPr(A)

We

can

solve

for

the

instantaneous

price

as

follows:

MPr(A)

MPr(B)

Working

from

the

above

instantaneous

price,

we

can

derive

the

implied

cost

function

m

as

a

function

of

the

number

n

of

shares

purchased

as

follows:

dm

dn

m

n

Z

dm

m

Z

dn

n

m)

C

m

e

r

−2

r

From

this

we

get

the

price

function:

dm

dn

p

e

r

−2

r

Note

that,

as

required,

and

If

one

uses

the

above

price

function,

then

the

market

dynamics

will

be

such

that

the

ratio

of

the

(instantaneous)

prices

of

A

and

B

always

equals

the

ratio

of

the

amounts

wagered

on

A

and

B,

which

seems

fairly

natural.

Note

that,

as

before,

the

mechanism

can

be

modified

to

collect

transaction

fees

of

some

kind.

Also

note

that

seed

or

ante

wagers

are

required

to

initialize

the

system.

DPM

II:

ALL

MONEY

REDISTRIBUTED

Above

we

examined

the

policy

of

refunding

winning

wagers

and

redistributing

only

losing

wagers.

In

this

section

we

consider

the

second

policy

mentioned

in

Section

all

money

from

all

wagers

are

collected

and

redistributed

to

winning

wagers.

For

the

case

where

all

money

is

redistributed,

the

respective

payoffs

per

share

are:

T

T

where

T

is

the

total

amount

of

money

wagered

on

both

sides.

So,

if

A

occurs,

shareholders

of

A

lose

their

initial

price

paid,

but

receive

dollars

per

share

owned;

shareholders

of

B

simply

lose

all

money

wagered.

Similarly,

if

B

occurs,

shareholders

of

B

lose

their

initial

price

paid,

but

receive

dollars

per

share

owned;

shareholders

of

A

lose

all

money

wagered.

In

this

case,

the

trader"s

per-share

expected

value

for

purchasing

an

infinitesimal

quantity

of

shares

of

A

is

E[

shares]

Pr(A)

E

A

risk-neutral

trader

optimizes

by

choosing

a

number

of

shares

n

of

A

to

purchase,

in

order

to

maximize

E[n

shares]

Pr(A)

n

E

Z

n

Pr(A)

n

E

m

The

same

value

of

n

can

be

solved

for

by

finding

the

number

of

shares

required

to

drive

E[

shares]/

to

zero.

That

is,

find

n

satisfying

Pr(A)

E

if

such

a

n

exists,

otherwise

n

Market

probability

In

this

case

MPr(A),

the

aggregate

probability

of

A

as

judged

by

the

market

as

a

whole,

is

the

solution

to

MPr(A)

Solving

we

get

MPr(A)

As

before,

we

make

the

simplifying

assumption

that

the

expected

final

payoff

per

share

equals

the

current

payoff

per

share.

The

assumption

is

critical

for

our

analysis,

but

may

not

be

required

for

a

practical

implementation.

Price

functions

For

the

case

where

all

money

is

distributed,

the

constraints

that

keep

the

price

of

A

equal

to

the

payoff

of

B,

and

vice

versa,

do

not

lead

to

the

derivation

of

a

coherent

price

function.

A

reasonable

price

function

can

be

derived

from

the

constraint

employed

in

Section

where

we

require

that

the

ratio

of

prices

to

be

equal

to

the

ratio

of

money

wagered.

That

is,

In

other

words,

the

price

of

A

is

proportional

to

the

amount

of

money

wagered

on

A,

and

similarly

for

B.

Using

Equations

and

we

can

derive

the

implied

market

probability:

MPr(A)

MPr(B)

T

T

MPr(A)

MPr(B)

MPr(A)

MPr(B)

MPr(A)

Interestingly,

this

is

the

same

market

probability

derived

in

Section

for

the

case

of

losing-money

redistribution

with

the

constraints

that

the

price

of

A

equal

the

payoff

of

B

and

vice

versa.

The

instantaneous

price

per

share

for

an

infinitesimal

quantity

of

shares

is:

Working

from

the

above

instantaneous

price,

we

can

derive

the

number

of

shares

n

that

can

be

purchased

for

m

dollars,

as

follows:

dm

dn

m

n

dn

dm

n

m

n

T

m)

ln

m)

m)

Note

that

we

solved

for

n(m)

rather

than

m(n).

I

could

not

find

a

closed-form

solution

for

m(n),

as

was

derived

for

the

two

other

cases

above.

Still,

n(m)

can

be

used

to

determine

how

many

shares

can

be

purchased

for

m

dollars,

and

the

inverse

function

can

be

approximated

to

any

degree

numerically.

From

n(m)

we

can

also

compute

the

price

function:

dm

dn

denom

where

denom

ln

m)

m)

Note

that,

as

required,

If

one

uses

the

above

price

function,

then

the

market

dynamics

will

be

such

that

the

ratio

of

the

(instantaneous)

prices

of

A

and

B

always

equals

the

ratio

of

the

amounts

wagered

on

A

and

B.

This

price

function

has

another

desirable

property:

it

acts

such

that

the

expected

value

of

wagering

on

A

and

simultaneously

wagering

on

B

equals

zero,

assuming

That

is,

of

A

of

B]

The

derivation

is

omitted.

Comparing

DPM

I

and

II

The

main

advantage

of

refunding

winning

wagers

(DPM

I)

is

that

every

bet

on

the

winning

outcome

is

guaranteed

to

at

least

break

even.

The

main

disadvantage

of

refunding

winning

wagers

is

that

shares

are

not

homogenous:

each

share

of

A,

for

example,

is

actually

composed

of

two

distinct

parts:

the

refund,

or

a

lottery

ticket

that

pays

$p

if

A

occurs,

where

p

is

the

price

paid

per

share,

and

one

share

of

the

final

payoff

if

A

occurs.

This

complicates

the

implementation

of

an

aftermarket

to

cash

out

of

the

market

early,

which

we

will

examine

below

in

Section

When

all

money

is

redistributed

(DPM

II),

shares

are

homogeneous:

each

share

entitles

its

owner

to

an

equal

slice

of

the

final

payoff.

Because

shares

are

homogenous,

the

implementation

of

an

aftermarket

is

straightforward,

as

we

shall

see

in

Section

On

the

other

hand,

because

initial

prices

paid

are

not

refunded

for

winning

bets,

there

is

a

chance

that,

if

prices

swing

wildly

enough,

a

wager

on

the

correct

outcome

might

actually

lose

money.

Traders

must

be

aware

that

if

they

buy

in

at

an

excessively

high

price

that

later

tumbles

allowing

many

others

to

get

in

at

a

much

lower

price,

they

may

lose

money

in

the

end

regardless

of

the

outcome.

From

informal

experiments,

I

don"t

believe

this

eventuality

would

be

common,

but

nonetheless

it

requires

care

in

communicating

to

traders

the

possible

risks.

One

potential

fix

would

be

for

the

market

maker

to

keep

track

of

when

the

price

is

going

too

low,

endangering

an

investor

on

the

correct

outcome.

At

this

point,

the

market

maker

could

artificially

stop

lowering

the

price.

Sell

orders

in

the

aftermarket

might

still

come

in

below

the

market

maker"s

price,

but

in

this

way

the

system

could

ensure

that

every

wager

on

the

correct

outcome

at

least

breaks

even.

OTHER

VARIATIONS

A

simple

ascending

price

function

would

set

and

where

α

In

this

case,

prices

would

only

go

up.

For

the

case

of

all

money

being

redistributed,

this

would

eliminate

the

possibility

of

losing

money

on

a

wager

on

the

correct

outcome.

Even

though

the

market

maker"s

price

only

rises,

the

going

price

may

fall

well

below

the

market

maker"s

price,

as

ask

orders

are

placed

in

the

aftermarket.

I

have

derived

price

functions

for

several

other

cases,

using

the

same

methodology

above.

Each

price

function

may

have

its

own

desirable

properties,

but

it"s

not

clear

which

is

best,

or

even

that

a

single

best

method

exists.

Further

analyses

and,

more

importantly,

empirical

investigations

are

required

to

answer

these

questions.

AFTERMARKETS

A

key

advantage

of

DPM

over

a

standard

pari-mutuel

market

is

the

ability

to

cash

out

of

the

market

before

it

closes,

in

order

to

take

a

profit

or

limit

a

loss.

This

is

accomplished

by

allowing

traders

to

place

ask

orders

on

the

same

queue

as

the

market

maker.

So

traders

can

sell

the

shares

that

they

purchased

at

or

below

the

price

set

by

the

market

maker.

Or

traders

can

place

a

limit

sell

order

at

any

price.

Buyers

will

purchase

any

existing

shares

for

sale

at

the

lower

prices

first,

before

purchasing

new

shares

from

the

market

maker.

Aftermarket

for

DPM

II

For

the

second

main

case

explored

above,

where

all

money

is

redistributed,

allowing

an

aftermarket

is

simple.

In

fact,

aftermarket

may

be

a

poor

descriptor:

buying

and

selling

are

both

fully

integrated

into

the

same

mechanism.

Every

share

is

worth

precisely

the

same

amount,

so

traders

can

simply

place

ask

orders

on

the

same

queue

as

the

market

maker

in

order

to

sell

their

shares.

New

buyers

will

accept

the

lowest

ask

price,

whether

it

comes

from

the

market

maker

or

another

trader.

In

this

way,

traders

can

cash

out

early

and

walk

away

with

their

current

profit

or

loss,

assuming

they

can

find

a

willing

buyer.

Aftermarket

for

DPM

I

When

winning

wagers

are

refunded

and

only

losing

wagers

are

redistributed,

each

share

is

potentially

worth

a

different

amount,

depending

on

how

much

was

paid

for

it,

so

it

is

not

as

simple

a

matter

to

set

up

an

aftermarket.

However,

an

aftermarket

is

still

possible.

In

fact,

much

of

the

complexity

can

be

hidden

from

traders,

so

it

looks

nearly

as

simple

as

placing

a

sell

order

on

the

queue.

In

this

case

shares

are

not

homogenous:

each

share

of

A

is

actually

composed

of

two

distinct

parts:

the

refund

of

p

dollars,

and

the

payoff

of

dollars,

where

p

is

the

per-share

price

paid

and

is

the

indicator

function

equalling

if

A

occurs,

and

otherwise.

One

can

imagine

running

two

separate

aftermarkets

where

people

can

sell

these

two

respective

components.

However,

it

is

possible

to

automate

the

two

aftermarkets,

by

automatically

bundling

them

together

in

the

correct

ratio

and

selling

them

in

the

central

DPM.

In

this

way,

traders

can

cash

out

by

placing

sell

orders

on

the

same

queue

as

the

DPM

market

maker,

effectively

hiding

the

complexity

of

explicitly

having

two

separate

aftermarkets.

The

bundling

mechanism

works

as

follows.

Suppose

the

current

price

for

share

of

A

is

A

buyer

agrees

to

purchase

the

share

at

The

buyer

pays

dollars

and

receives

dollars.

If

there

is

enough

inventory

in

the

aftermarkets,

the

buyer"s

share

is

constructed

by

bundling

together

from

the

first

aftermarket,

and

from

the

second

aftermarket.

The

seller

in

the

first

aftermarket

receives

dollars,

and

the

seller

in

the

second

aftermarket

receives

dollars.

Pseudo

aftermarket

for

DPM

I

There

is

an

alternative

pseudo

aftermarket

that"s

possible

for

the

case

of

DPM

I

that

does

not

require

bundling.

Consider

a

share

of

A

purchased

for

The

share

is

composed

of

and

Now

suppose

the

current

price

has

moved

from

to

per

share

and

the

trader

wants

to

cash

out

at

a

profit.

The

trader

can

sell

share

at

market

price

share

for

receiving

all

of

the

initial

investment

back,

and

retaining

share

of

A.

The

share

is

worth

either

some

positive

amount,

or

nothing,

depending

on

the

outcome

and

the

final

payoff.

So

the

trader

is

left

with

shares

worth

a

positive

expected

value

and

all

of

his

or

her

initial

investment.

The

trader

has

essentially

cashed

out

and

locked

in

his

or

her

gains.

Now

suppose

instead

that

the

price

moves

downward,

from

to

per

share.

The

trader

decides

to

limit

his

or

her

loss

by

selling

the

share

for

The

buyer

gets

the

share

plus

(the

buyer"s

price

refunded).

The

trader

(seller)

gets

the

plus

what

remains

of

the

original

price

refunded,

or

The

trader"s

loss

is

now

limited

to

at

most

instead

of

If

A

occurs,

the

trader

breaks

even;

if

B

occurs,

the

trader

loses

Also

note

that-in

either

DPM

formulation-traders

can

always

hedge

sell

by

buying

the

opposite

outcome

without

the

need

for

any

type

of

aftermarket.

CONCLUSIONS

I

have

presented

a

new

market

mechanism

for

wagering

on,

or

hedging

against,

a

future

uncertain

event,

called

a

dynamic

pari-mutuel

market

(DPM).

The

mechanism

combines

the

infinite

liquidity

and

risk-free

nature

of

a

parimutuel

market

with

the

dynamic

nature

of

a

CDA,

making

it

suitable

for

continuous

information

aggregation.

To

my

knowledge,

all

existing

mechanisms-including

the

standard

pari-mutuel

market,

the

CDA,

the

CDAwMM,

the

bookie

mechanism,

and

the

MSR-exhibit

at

most

two

of

the

three

properties.

An

MSR

is

the

closest

to

a

DPM

in

terms

of

these

properties,

if

not

in

terms

of

mechanics.

Given

some

natural

constraints

on

price

dynamics,

I

have

derived

in

closed

form

the

implied

price

functions,

which

encode

how

prices

change

continuously

as

shares

are

purchased.

The

interface

for

traders

looks

much

like

the

familiar

CDA,

with

the

system

acting

as

an

automated

market

maker

willing

to

accept

an

infinite

number

of

buy

orders

at

some

price.

I

have

explored

two

main

variations

of

a

DPM:

one

where

only

losing

money

is

redistributed,

and

one

where

all

money

is

redistributed.

Each

has

its

own

pros

and

cons,

and

each

supports

several

reasonable

price

functions.

I

have

described

the

workings

of

an

aftermarket,

so

that

traders

can

cash

out

of

the

market

early,

like

in

a

CDA,

to

lock

in

their

gains

or

limit

their

losses,

an

operation

that

is

not

possible

in

a

standard

pari-mutuel

setting.

FUTURE

WORK

This

paper

reports

the

results

of

an

initial

investigation

of

the

concept

of

a

dynamic

pari-mutuel

market.

Many

avenues

for

future

work

present

themselves,

including

the

following:

Random

walk

conjecture.

The

most

important

question

mark

in

my

mind

is

whether

the

random

walk

assumption

can

be

proven

under

reasonable

market

efficiency

conditions

and,

if

not,

how

severely

it

effects

the

practicality

of

the

system.

Incentive

analysis.

Formally,

what

are

the

incentives

for

traders

to

act

on

new

information

and

when?

How

does

the

level

of

initial

subsidy

effect

trader

incentives?

Laboratory

experiments

and

field

tests.

This

paper

concentrated

on

the

mathematics

and

algorithmics

of

the

mechanism.

However,

the

true

test

of

the

mechanism"s

ability

to

serve

as

an

instrument

for

hedging,

wagering,

or

information

aggregation

is

to

test

it

with

real

traders

in

a

realistic

environment.

In

reality,

how

do

people

behave

when

faced

with

a

DPM

mechanism?

DPM

call

market.

I

have

derived

the

price

functions

to

react

to

wagers

on

one

outcome

at

a

time.

The

mechanism

could

be

generalized

to

accept

orders

on

both

sides,

then

update

the

prices

wholistically,

rather

than

by

assuming

a

particular

sequence

on

the

wagers.

Real-valued

variables.

I

believe

the

mechanisms

in

this

paper

can

easily

be

generalized

to

multiple

discrete

outcomes,

and

multiple

real-valued

outcomes

that

always

sum

to

some

constant

value

(e.g.,

multiple

percentage

values

that

must

sum

to

However,

the

generalization

to

real-valued

variables

with

arbitrary

range

is

less

clear,

and

open

for

future

development.

Compound/combinatorial

betting.

I

believe

that

DPM

may

be

well

suited

for

compound

or

combinatorial

betting,

for

many

of

the

same

reasons

that

market

scoring

rules

are

well

suited

for

the

task.

DPM

may

also

have

some

computational

advantages

over

MSR,

though

this

remains

to

be

seen.

Acknowledgments

I

thank

Dan

Fain,

Gary

Flake,

Lance

Fortnow,

and

Robin

Hanson.

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O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B I O B I O O O O O B O B O O B O O O O O O B I I B O O O O O O O O O O O O B O O O O O B I O O B I O O B O B O O O O O O O O O O O O O O O O O O O O O O B O O B I O B I O O O O O O O B I I O O O O O O O O O O B I I O B B I O O O O O B I I O O O O O O B O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O B O O O O O O B O O O O O O B O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O B O O O O O O O O O O O O O O O O O O O O O B O O O O B I O B O O O O O O O O O O O O O O B O B I O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O B I O O O O O O O O O O B I O O O B O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O 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O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O O B I I O O O O O B O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O B O O O B I I O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O B I I O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O B O O O O B O O O O O B O O O O B I I O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I I O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O B I O B O O O O O O O O B I O O O O O O B O O O O O O O B I I O O O O O B O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O B I O O O B I O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O B O B I O B O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O B I I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O B O O O O O O O B O O O O O O O O O O O B O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O B O O O O O O O O O O O O O O B O O O O O O O O O O O O O B O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B O O O O O B I O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O B O O O O O O B I I O O O O O O O O O O O B O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B O O B O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

dynamic pari-mutuel market
pari-mutuel market
infinite buy-in liquidity
gain
bid-ask queue
automate market maker
compound security market
combinatorial bet
speculate
gamble

risk allocation
information speculation
dpm
hybrid
continuous double auction
cda
zero risk
market institution
event resolution
selling
loss
trader interface
double auction format
payoff per share
price function
automated market maker
demand
price
information aggregation
trade
hedge
bet
wager