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Markets in UncertaintyRisk, Gambling, and

Information Aggregation

- a tutorial by
- David M. Pennock Michael P.

Wellman pennockd_at_yahoo-inc.com

wellman_at_umich.edu - dpennock.com

ai.eecs.umich.edu/people/wellman - presented at the 19th National Conference on

Artificial Intelligence, July 2004, San Jose, CA,

USA

MP1-1

Outline

- Overview tour 15 minWhat is a market in

uncertainty? - Background 30 min
- Single agent perspective
- Subjective probability
- Utility, risk, and risk aversion
- Decision making under uncertainty
- Multiagent perspective
- Trading/allocating risk
- Pareto optimality
- Securities Markets in uncertainty

Outline

- Mechanisms, examples andempirical studies

45 min - What how Instruments mechanisms
- Real-money markets Examples evaluations
- Iowa Electronic Market
- Options
- TradeSports Effects of war
- Horse racing, sports betting
- Play-money markets

Outline

- Lab experiments and theory 20 min
- Laboratory experiments, field tests
- Theoretical underpinnings
- Rational expectations
- Efficient markets hypothesis
- No-Trade Theorems
- Information aggregation

Outline

- Characterizing information 20 minaggregation
- Market as an opinion pool
- Market as a composite agent
- Market belief, utility
- Market Bayesian updates
- Market adaptation, dynamics
- Paradoxes, impossibilities
- Opinion pool impossibilities
- Composite agent non-existence

Outline

- Computational aspects 60 min
- Combinatorics
- Compact securities markets
- Combinatorial securities markets
- Compound securities markets
- Market scoring rules
- Dynamic pari-mutuel market
- Policy Analysis Market
- Distributed market computation
- Legal issues miscellaneous 5 min
- Discussion, QA 15 min

1. Overview tour

- What is a market in uncertainty ?

A market in uncertainty

- Take a random variable, e.g.
- Turn it into a financial instrument payoff

realized value of variable

US04Pres Bush?

2004 CAEarthquake?

Aside Terminology

- Key aspect payout is uncertain
- Called variously asset, security, contingent

claim, derivative (future, option), stock,

prediction market, information market, gamble,

bet, wager, lottery - Historically mixed reputation
- Esp. gambling aspect
- A time when options were frowned upon
- But when regulated serve important social roles...

Why? Reason 1

- Get information
- price ? expectation of random variable(in

theory, lab experiments, empirical studies,

...more later) - Do you have a random variable whose expectation

youd like to know?A market in uncertainty can

probably help

Why?Reason 1 Information

- Information market financial mechanism

designed to obtain estimates of expectations of

random variables - Easy as 1, 2, 3
- Take a random variable whose expectation youd

like to know - Turn it into a financial instrument (payoff

realized value of variable) - Open a market in the financial instrument
- ? price(t) ? EtX (in many cases, ... more later)

Getting information

- Non-market approach ask an expert
- How much would you pay for this?
- A 5/36 ? 0.1389
- caveat expert is knowledgeable
- caveat expert is truthful
- caveat expert is risk neutral, or RN for 1
- caveat expert has no significant outside stakes

Getting information

- Non-market approach pay an expert
- Ask the expert for his report r of the

probability P( ) - Offer to pay the expert
- 100 log r if
- 100 log (1-r) if
- It so happens that the expert maximizes expected

profit by reporting r truthfully - caveat expert is knowledgeable
- caveat expert is truthful
- caveat expert is risk neutral, or RN
- caveat expert has no significant outside stakes

logarithmic scoring rule, a proper scoring

rule

Getting information

- Market approach ask the publicexperts

non-experts alikeby opening a market - Let any person i submit a bid order an offer to

buy qi units at price pi - Let any person j submit an ask order an offer

to sell qj units at price pj(if you sell 1 unit,

you agree to pay 1 if ) - Match up agreeable trades (many poss. mechs...)

Getting information

- Market approach ask the publicexperts

non-experts alikeby opening a market - If, at any time, for any bidder i and ask-er j,

pi pj, then ij trade min(qi,qj) units at price

?pj,pi - In equilibrium (no trades)
- max bid pi
- bounds aggregate public opinion of expectation

Aside Mechanism alternatives

- This is the continuous double auction (CDA)
- Many other market auction mechanisms work
- call market
- pari-mutuel market
- market scoring rules
- CDA w/ market maker
- Vegas bookmaker, others
- Key Market price aggregate estimate of

expected value

Hanson 2002

(Real) Great expectations

- For dice example, no need for market Ex is

known no one should disagree - Real power comes for non-obvious expectations of

random variables, e.g.

I am entitled to

1 if

0 otherwise

I am entitled to

x if interest rate x on Jan 1, 2004

I am entitled to

max(0,x-k) if MSFT xon Jan 1, 2004

call option

I am entitled to

f(future weather)

weather derivative

I am entitled to

Bin Ladencaptured

1 if

0 otherwise

I am entitled to

1 if Kansas beats Marq.by 4.5 points 0

otherw.

http//tradesports.com

(No Transcript)

Play moneyReal expectations

http//www.hsx.com/

http//us.newsfutures.com/

http//www.ideosphere.com

Cancercuredby 2010

Machine Gochampionby 2020

Does it work?Yes...

- Evidence from real markets, laboratory

experiments, and theory indicate that markets are

good at gathering information from many sources

and combining it appropriately e.g. - Markets like the Iowa Electronic Market predict

election outcomes better than pollsForsythe

1992, 1999Oliven 1995Rietz 1998Berg

2001Pennock 2002 - Futures and options markets rapidly incorporate

information, providing accurate forecasts of

their underlying commodities/securitiesSherrick

1996Jackwerth 1996Figlewski 1979Roll

1984Hayek 1945 - Sports betting markets provide accurate forecasts

of game outcomes Gandar 1998Thaler

1988Debnath EC03Schmidt 2002

Does it work?Yes...

- E.g. (contd)
- Laboratory experiments confirm information

aggregationPlott 198219881997Forsythe

1990Chen, EC-2001 - And field tests Plott 2002
- Theoretical underpinnings rational

expectationsGrossman 1981Lucas 1972 - Procedural explanation agents learn from

pricesHanson 1998Mckelvey 1986Mckelvey

1990Nielsen 1990 - Proposals to use information markets to help

science Hanson 1995, policymakers, decision

makers Hanson 1999, government Hanson 2002,

military DARPA FutureMAP, PAM - Even market games work! Servan-Schreiber

2004Pennock 2001

Why? Reason 2

- Manage risk
- If is horribly terrible for

youBuy a bunch of and if

happens, you are

compensated

Why? Reason 2

- Manage risk
- If is horribly terrible for

youBuy a bunch of and if

happens, you are

compensated

I am entitled to

1 if

0 if

The flip-side of prediction Hedging (Reason 2)

- Allocate risk (hedge)
- insured transfers risk to insurer, for
- farmer transfers risk to futures speculators
- put option buyer hedges against stock drop

seller assumes risk

- Aggregate information
- price of insurance? prob of catastrophe
- OJ futures prices yield weather forecasts
- prices of options encode prob dists over stock

movements - market-driven lines are unbiased estimates of

outcomes - IEM political forecasts

Reason 2 Manage risk

- What is insurance?
- A bet that something bad will happen!
- E.g., Im betting my insurance co. that my house

will burn down theyre betting it wont. Note we

might agree on P(burn)! - Why? Because Ill be compensated if the bad thing

does happen - A risk-averse agent will seek to hedge (insure)

against undesirable outcomes

E.g. stocks, options, futures, insurance, ...,

sports bets, ...

- Allocate risk (hedge)
- insured transfers risk to insurer, for
- farmer transfers risk to futures speculators
- put option buyer hedges against stock drop

seller assumes risk - sports bet may hedge against other stakes in

outcome

- Aggregate information
- price of insurance? prob of catastrophe
- OJ futures prices yield weather forecasts
- prices of options encode prob dists over stock

movements - market-driven lines are unbiased estimates of

outcomes - IEM political forecasts

Examples

- I buy MSFT stock at s. Im afraid it will go

down. I buy a put option that pays Max0,k-s k

is strike price. If s goes down below k, my

stock investment goes down, but my option

investment goes up to compensate - Im a farmer. Im afraid corn prices will go too

low. I buy corn futures to lock in a price today.

Examples

- I own a house in CA. Im afraid of earthquakes. I

pay an insurance premium so that, if an

earthquake happens, I am compensated. - I am an Oscar-nominated actor. Im afraid Im

going to lose. I bet against myself on an

offshore gambling site. If I do lose, I am

compensated. (Except that the offshore site

disappears and refuses to pay?)

What am I buying?

- When you hedge/insure, you pay to reduce the

unpredictability of future wealth - Risk-aversion All else being equal, prefer

certainty to uncertainty in future wealth - Typically, a less risk-averse party (e.g., huge

insurance co, futures speculator) assumes the

uncertainty (risk) in return for an expected

profit

On hedging and speculating

- Hedging is an act to reduce uncertainty
- Speculating is an act to increase expected future

wealth - A given agent engages in a (largely inseparable)

mixture of the two - Both can be encoded together as a maximization of

expected utility, where utility is a function of

wealth, ... more later

On trading

- Why would two parties agree to trade in a market

in uncertainty? - They disagree on expected values (probs)
- They differ in their risk attitude or exposure

they trade to reallocate risk - Both (most likely)
- Aside legality is murky, though generally (2) is

legal in the US while (1) often is not. In

reality, it is nearly impossible to differentiate.

On computational issues

some

?

- Information aggregation is a form of distributed

computation - Agent level
- nontrivial optimization problem, even in 1

marketultimately a game-theoretic question - probability representation, updating algorithm

(Bayes net) - decision representation, algorithm (POMDP)
- agent problems computational complexity,

algorithms, approximations, incentives

On computational issues

some

?

- Mechanism level
- Single market
- What can a market compute?
- How fast (time complexity)?
- Do some mechanisms converge faster (e.g.,

subsidy) - Multiple markets
- How many securities to compute a given fn? How

many secs to support sufficient social

welfare?(expressivity and representational

compactness) - Nontrivial combinatorics (auctioneers

computational complexity algorithms

approximations incentives)

On computational issues

some

?

- Machine learning, data mining
- Beat the market (exploiting combinatorics?)
- Explain the market, information retrieval
- Detect fraud

2. Background

- Single agent perspective
- Subjective probability
- Utility, risk, and risk aversion
- Decision making under uncertainty
- Multiagent perspective
- Trading/allocating risk
- Pareto optimality
- Securities markets in uncertainty

Decision making under uncertainty

- How should agents behave (make decisions, choose

actions) when faced with uncertainty? - Decision theory Prescribes maximizing expected

utility

Why reason about uncertainty?

- Propositional logic No uncertaintyCould never

explain seatbelt use - Decisions D - drive car S - wear seatbelt
- Events A - accident occurs
- A ? ?D
- ?A ? ?S
- Cant explain DS
- Key A is uncertain

Why Bayesian uncertainty?

- E.g. You can buy skis for bOr you can rent for

b/k, k1 - Worst-case analysisRent for k days, then

buyYoull spend at most 2b - But what if you strongly believe youll skimore

than k times? ? Buy earlier - That k1st time is your last? ? Dont buy
- Expected (utility) case often more appropriate

Decision making under uncertainty, an example

ABC TVs Who Wants to be a Millionaire?

Decision making under uncertainty, an example

- v15 1,000,000 if correct 32,000 if

incorrect500,000 if walk away

Decision making under uncertainty, an example

- if you answer
- Ev15 1,000,000 Pr(correct)32,000

Pr(incorrect) - if you walk away
- 500,000

Decision making under uncertainty, an example

- if you answer
- Ev15 1,000,000 0.532,000 0.5
- 516,000
- if you walk away
- 500,000
- you should answer, right?

Decision making under uncertainty, an example

- Most people wont answer risk averse
- U(x) log(x)
- if you answer
- Eu15 log(1,000,000) 0.5log(32,000)

0.5 - 6/24.5/2 5.25
- if you walk away
- log(500,000) 5.7

Decision making under uncertainty, an example

- Maximizing Eui for i

Q7, L1,3

walk

answer

L1

L3

Q7, L3

? 0.4

X 0.6

log(2k)

walk

answer

L3

log(1k)

Q8, L1,3

? 0.8

X 0.2

log(2k)

log(1k)

Q8, L3

Decision making under uncertainty, in general

?set of all possible future states of the world

Decision making under uncertainty, in general

- ? are disjoint exhaustivestates of the world
- ?i rain tomorrow Bush elected Y! stock up

car not stolen ... - ?j rain tomorrow Bush elected Y! stock up

car stolen ...

?1

?2

?3

?i

?

?

??

Decision making under uncertainty, in general

- Equivalent, more natural
- Ei rain tomorrowEj Bush elected
- Ek Y! stock up
- El car stolen
- ?2n

E1

E2

?

Ei

En

Ej

Decision making under uncertainty, in general

- Preferences, utility
- ??i?j ? u(?i) u(?j)
- Expected utility
- Eu ?? Pr(?)u(?)
- Decisions (actions) can affect Pr(?)
- What you should dochoose actions to maximize

expected utility - Why? To avoid being a money pump de

Finetti74, among other reasons...

Preference under uncertainty

- Define a prospect, ? p, ?1 ?2
- Given the following axioms of ?
- orderability (?1 ? ?2) ? (?1 ? ?2) ? (?1 ?2)
- transitivity (?1 ? ?2) ? (?2 ? ?3) ? (?1 ? ?3)
- continuity ?1 ? ?2 ? ?3 ? ? p. ?2 p, ?1 ?3
- substitution ?1 ?2 ? p, ?1 ?3 p, ?2 ?3
- monotonicity ?1 ? ?2 ? pq ? p, ?1 ?2 ? q,

?? ?2 - decomposability
- p, ?1 q, ?2 ?3 q, p, ?1 ?2 p, ?1

?3 - Preference can be represented by a real-valued

expected utility function such that - u(p, ?1 ?2) p u(?1) (1p)u(?2)

Utility functions

- (??? a probability distribution over ?)
- Eu ????represents preferences,
- Eu(?) ? Eu(??) iff ??? ???
- Let ?(?) au(?) b, a0.
- Then E?(?) Eaub(?) a Eu(?) b.
- Since they represent the same preferences, ? and

u are strategically equivalent (? u).

Utility of money

- Outcomes are dollars
- Risk attitude
- risk neutral u(x) x
- risk averse (typical) u concave (u??(x) all x)
- risk prone u convex
- Risk aversion function
- r(x) u??(x) / u?(x)

Risk aversion hedging

- Eu.01 (4).99 (4.3) 4.2980
- Action buy 10,000 of insurance for 125
- Eu4.2983
- Even better, buy 5974.68 of insurance for 74.68
- Eu 4.2984 ? Optimal

?1 car stolenu(?1) log(10,000)

?2 car not stolenu(?2) log(20,000)

u(?1) log(19,875)

u(?2) log(19,875)

u(?1) log(15,900)

u(?2) log(19,925)

Securities market s

- Note that, in previous example, risk-neutral

insurance company also profitsEv

.01(-5,900) 0.99(74.68) 14.93Both parties

gain from bilateral agreement - Securities market generalizes this to
- arbitrary states
- more than two parties
- Market mechanism to allocate risk among

participants

Pareto optimality

- An allocation is Pareto optimal iff there does

not exist another solution that is - better for one agent and
- no worse for all the rest.

a minimal (and maximal?) condition for social

optimality, or efficiency.

What is traded Securities

- Specifies state-contingent returns, r

(r1,,r?) in terms of numeraire (e.g., ) - Examples
- (1,,1) riskless numeraire (1)
- (0,,0,1,0,,0) pays off 1 in designated state

(Arrow security for

that state) - ri 1 if ?i?E1, ri 0 otherwise

1 if E1

Terms of trade Prices

- Price p associated with security
- Relative prices dictate terms of exchange
- Facilitate multilateral exchange via bilateral

exchange - defines a common scale of resource value
- Can significantly simplify a resource allocation

mechanism - compresses all factors contributing to value into

a single number - A default interface for multiagent systems

1 if Ei

Equilibrium

- General (competitive, Walrasian) equilibrium

describes a simultaneous equilibrium of

interconnected markets - Definition A price vector and allocation such

that - all agents making optimal demand decisions

(positive demand buy negative demand sell) - all markets have zero aggregate demand(buy

volume equals sell volume)

Complete securities market

- A set of securities is complete if rank of

returns matrix ? ?1 - For example, set of ? ?1 Arrow securities

Arrow-Debreu securities market - Market with complete set of securities guarantees

a Pareto optimal allocation of risk, under

classical conditions

Incomplete markets

- Securities do not span states of nature (always

the case in practice) - Equilibria may exist, but may not be Pareto

optimalExample missed insurance opportunity - More Theory of Incomplete Markets, Magill

Quinzii, MIT Press, 1998

Why trade securities?

- Profit from perceived mispricings
- Price p differs significantly enough from

traders belief Pr(E1) - speculation
- Insure against risk
- Traders marginal value for wealth in E1,

relative to p, differs from that in other

states - e.g., home fire insurance
- hedging

Societal roles of security markets

- From speculation
- Aggregate beliefs
- Disseminate information
- From hedging
- Allocate risk

Summary Background

- General equilibrium framework for market-based

exchange - Incorporate uncertainty through securities
- Agents trade securities in order to optimize

expected utility, thereby - Allocating risk
- Reaching consensus probabilities

3. Mechanisms, examples empirical studies

- What howInstruments mechanisms
- Real-money marketsExamples evaluations
- Iowa Electronic Market
- Options
- TradeSports Effects of war
- Horse racing, sports betting
- Play-money markets

Building a market in uncertainty

- What is being traded?the good
- Define
- Random variable
- Payoff function
- Payoff output

- How is it traded?the mechanism
- Call market
- Continuous double auction
- Continuous double auction w/ market maker
- Pari-mutuel
- Bookmaker
- Combinatorial (later)

What is being traded?

- Underlying statistic / random variable
- Binary Discrete
- Continuous interest rate, dividend flow
- Clarity e.g., Saddam out, House burns,Gore

wins, Buchanan wins - Payoff function
- Arrow (0,0,0,1,0) Portfolio (2,4,0,1,0)
- Dividends, options Max0,s-k, arbitrary

(non-linear) fn - Payoff output
- dollars, fake money, commodities

6

How is it traded?

- Call market
- Orders are collected over a period of time

collected orders are matched at end of period - One-time or repeated
- Pre-defined or randomized stopping time/rule
- Mth price auction
- M1st price auction
- k-double auction
- lim period?0 Continuous double auction

A note on selling

- In a securities market, you can sell what you

dont have you agree to pay according to terms - Binary case sell 1 if A for 0.3
- Receive 0.3 (now, or contractually later), pay

1 if A - Exactly equivalent to buying 1 if A for 0.7
- sell 1 if A _at_ 0.3
- buy 1 if A _at_ 0.7
- Alternative Market institution always stands

ready to buy/sell exhaustive bundle for 1.00 - Iowa Electronic Market

A occurs A occurs-1.3 -.7 0.3 .3 0 -.7

-.7 1 -.7 .3

Mth price auction

- N buyers and M sellers
- Mth price auction
- sort all bids from buyers and sellers
- price the Mth highest bid
- let n of buy offers price
- let m of sell offers
- let x min(n,m)
- the x highest buy offers and x lowest sell offers

transact

Call market

- Buy offers (N4)

- Sell offers (M5)

0.30

0.15

0.17

0.12

0.13

0.09

0.11

0.05

0.08

Mth price auction

- Buy offers (N4)

- Sell offers (M5)

1

0.30

0.17

2

0.15

3

?

0.13

4

price 0.12

0.12

5

?

0.11

?

0.09

0.08

?

0.05

- Matching buyers/sellers

M1st price auction

- Buy offers (N4)

- Sell offers (M5)

1

0.30

0.17

2

0.15

3

?

0.13

4

0.12

5

?

price 0.11

0.11

?

6

0.09

0.08

?

0.05

- Matching buyers/sellers

k-double auction

- Buy offers (N4)

- Sell offers (M5)

1

0.30

0.17

2

0.15

3

?

0.13

4

price 0.11 0.01k

0.12

5

?

0.11

?

6

0.09

0.08

?

0.05

- Matching buyers/sellers

Continuous double auctionCDA

- k-double auction repeated continuously
- buyers and sellers continually place offers
- as soon as a buy offer ? a sell offer, a

transaction occurs - At any given time, there is no overlap btw

highest buy offer lowest sell offer

http//tradesports.com

http//www.biz.uiowa.edu/iem

http//us.newsfutures.com/

CDA with market maker

- Same as CDA, but with an extremely active, high

volume trader (often institutionally affiliated)

who is nearly always willing to buy at some price

p and sell at some price q p - Market maker essentially sets prices others take

it or leave it - While standard auctioneer takes no risk of its

own, market maker takes on considerable risk, has

potential for considerable reward

CDA with market maker

- E.g. World Sports Exchange (WSE)
- Maintains 5 differential between bid ask
- Rules Markets are set to have 50 contracts on

the bid and 50 on the offer. This volume is

available first-come, first-served until it is

gone. After that, the markets automatically move

two dollars away from the price that was just

traded. - The depth of markets can vary with the contest.
- Also, WSE pauses market adjusts prices

(subjectively?) after major events (e.g., goals) - http//www.wsex.com/about/interactiverules.html

CDA with market maker

- E.g. Hollywood Stock Exchange (HSX)
- Virtual Specialist automated market maker
- Always willing to buy sell at a single point

price ? no bid-ask spread - Price moves when buys/sells are imbalanced
- Fake money, so its OK if Virtual Specialist

loses money in fact it does Brian Dearth,

personal communication - http//www.hsx.com/

http//www.wsex.com/

http//www.hsx.com/

Bookmaker

- Common in sports betting, e.g. Las Vegas
- Bookmaker is like a market maker in a CDA
- Bookmaker sets money line, or the amount you

have to risk to win 100 (favorites), or the

amount you win by risking 100 (underdogs) - Bookmaker makes adjustments considering amount

bet on each side /or subjective probs - Alternative bookmaker sets game line, or

number of points the favored team has to win the

game by in order for a bet on the favorite to

win line is set such that the bet is roughly a

50/50 proposition

Pari-mutuel mechanism

- Common at horse races, jai alai games
- n mutually exclusive outcome (e.g., horses)
- M1, M2, , Mn dollars bet on each
- If i wins all bets on 1, 2, , i-1,i1, , n

lose - All lost money is redistributed to those who bet

on i in proportion to amount they bet - That is, every 1 bet on i gets1 1/Mi

(M1, M2, ,Mi-1, Mi1, , Mn) 1/Mi (M1,

M2, , Mn)

Pari-mutuel market

- E.g. horse racetrack style wagering
- Two outcomes A B
- Wagers

Pari-mutuel market

- E.g. horse racetrack style wagering
- Two outcomes A B
- Wagers

?

Pari-mutuel market

- E.g. horse racetrack style wagering
- Two outcomes A B
- Wagers

?

Pari-mutuel market

- E.g. horse racetrack style wagering
- Two outcomes A B
- 2 equivalentways to considerpayment rule
- refund share of B
- share of total

?

Pari-mutuel market

- Before race begins, odds are reported, or the

amount you would win per dollar if betting ended

now - Horse A 1.2 for 1 Horse B 25 for 1 etc.
- Normalized odds consensus probabilities
- Actual payoffs depend only on final odds, not

odds at time of bet incentive to wait - In practice track takes 17 first, then

redistributes what remains

Examples of markets

- Continuous double auction (CDA)
- Iowa Electronic Market (IEM)
- TradeSports, experimental Soccer market
- Financial markets stocks, options, derivatives
- CDA with market maker
- World Sports Exchange (WSE)
- Hollywood Stock Exchange (HSX)
- Pari-mutuel horse racing
- Bookmaker NBA point spread betting

Example IEMIowa Electronic Market

http//www.biz.uiowa.edu/iem

US Democratic Pres. nominee 2004

1 if other wins

1 if Kerry wins

1 if Lieberman wins

1 if Gephardt wins

1 if H. Clinton wins

priceECPr(C)0.056

as of 4/22/2003

Example IEMIowa Electronic Market

http//www.biz.uiowa.edu/iem

US Presidential election 2004

1 if Democrat votes Repub

1 if Republican votes Dem

priceERPr(R)0.494

as of 7/25/2004

IEM vote share market

US Pres. election vote share 2004

1 ? 2-party vote share of Bush v. other

1 ? 2-party vote share of other Dem

1 ? vote share of Bush v. Kerry

1 ? vote share of Kerry

priceEVS for K0.148

as of 4/22/2003

IEM vote share market

US Pres. election vote share 2004

1 ? 2-party vote share of Kerry

1 ? vote share of Bush v. Kerry

priceEVS for B v. K0.508

1 ? vote share of Dean

1 ? vote share of Bush v. Dean

as of 7/25/2004

Example IEM 1992

Source Berg, DARPA Workshop, 2002

Example IEM

Source Berg, DARPA Workshop, 2002

Example IEM

Source Berg, DARPA Workshop, 2002

Example IEM

Source Berg, DARPA Workshop, 2002

Example IEM

Source Berg, DARPA Workshop, 2002

Speed TradeSports

Source Wolfers 2004

Contract Pays 100 if Cubs win game 6 (NLCS)

Price of contract (Probability that Cubs win)

Fan reaches over and spoils Alous catch. Still

1 out.

Cubs are winning 3-0 top of the 8th1 out.

The Marlins proceed to hit 8 runs in the 8th

inning

Time (in Ireland)

The marginal trader Forsythe 1992,1999 Oliven

1995 Rietz 1998

- Individuals in IEM are biased, make mistakes
- Democrats buy too many Democratic stocks
- Arbitrage is left on the table
- When there are multiple equivalent trades, the

cheapest is not always chosen - Yet market as a whole is accurate, efficient
- Why? Prices are set by marginal traders, not

average traders - Marginal traders are active traders, price

setters, unbiased, better performers

Forecast error bounds Berg 2001

- Single market gives Ex
- IEM winner takes all P(candidate wins) P(C)
- IEM vote share Ecandidate vote share EV
- Can we get error bounds? e.g. Varx?
- Yes combine the two markets

Evaluating accuracyRecall log scoring rule

- Logarithmic scoring rule(one of several proper

scoring rules) - Pay an expert approach
- Offer to pay the expert
- 100 log r if
- 100 log (1-r) if
- Expert should choose rPr(A), given caveats

Evaluating accuracy

- Log score gives incentives to be truthful
- But log score is also an appropriate measure of

experts accuracy - Experts who are better probability assessors will

earn a higher avg log score over time - We advocate evaluate the market just as you

would evaluate an individual expert - For a given market (person), compute average log

score over many assessments

? log score ? information

- Log score dynamics also shows speed of

information incorporation - Expected log score P(A) log P(A) P(A) log

P(A) - entropy - Actual log score at time t
- - amount market is surprised by true outcome
- - of bits of info provided by revelation of

true outcome - As bits of info flow into market, log score ?

Avg log score dynamics

IEM

FX

WSE bball

HSX

WSE soccer

Avg log score22 IEM political markets

Average log score ?i log (pi)/N pi ith

winners normalized price

Example options

- Options prices (partially) encode a probability

distribution over their underlying stocks - Arbitrary derivative ? P(underlying asset)

payoff

10

20

30

40

50

stock price s

Example options

- Options prices (partially) encode a probability

distribution over their underlying stocks - Arbitrary derivative ? P(underlying asset)

butterfly spread

payoff

10

20

30

40

50

stock price s

- 2call30

Example options

- Options prices (partially) encode a probability

distribution over their underlying stocks - Arbitrary derivative ? P(underlying asset)

payoff

10

20

30

40

50

stock price s

- 2call40

Example options

- call10 - 2 call20 call30 2.13

relative - call30 - 2 call30 call40 5.73 likelihood

of falling - call30 - 2 call40 call50 3.54 near

center

payoff

2.13

5.73

3.54

10

20

30

40

50

stock price s

Example options

- More generally, uses prices as constraintsEMax0

,s-10p10 EMax0,s-20p20 ... etc. - Fit to assumed distribution or maximize

entropy, smothness, etc. subject to constraints

Jackwerth 1996

probability

10

20

30

40

50

stock price s

Example TradeSports

Source Wolfers 2004

Source Wolfers 2004

Source Wolfers 2004

Source Wolfers 2004

State Price Distribution

Source Wolfers 2004

State Price Distribution War and Peace

Source Wolfers 2004

Example horse racing

- Pari-mutuel mechanism
- Normalized odds match objective frequencies of

winning very closely - 31 horses win about twice as much as 61 horses,

etc. - Slight favorite-longshot bias (favorites are

better bets extremely rarely Ereturn 0) - Ali 77 Rosett 65 Snyder 78 Thaler 88

Weitzman 65

Example horse racing

- Tracks can be biased, e.g., Winning Colors,a S

Californian horse, 1988 Kentucky Derby - 1 paid in MA 10.60, ..., in FL 10.40,

...,KY 8.80,..., MI 7.40, ..., N.CA 5.20,

..., S.CA 4.40 Wong 2001 - Some teams apparently make more than a decent

living beating the track using computer models

e.g., Bill Benters team in Hong Kong - logistic regression standard now SVMs Edelman

2003 - http//www.unr.edu/gaming/confer.asp
- http//www.wired.com/wired/archive/10.03/betting_p

r.html

Example sports betting

- US NBA Basketball
- Closing lines set by market are unbiased

estimates of game outcomes?better than opening

lines set by experts Gandar 98 - Soccer (European football)Experimental market in

Euro 2000 Championship Schmidt 2002 - Market prediction betting odds random
- Market confidence statistically meaningful

World Sports Exchange WSE

- Online in-game sports betting markets
- Trading allowed continuously throughout game as

goals are scored, penalties are called, etc. ?

i.e. as information is revealed! - National Basketball Association (NBA)
- Soccer World Cup
- MLB, NHL, golf, others
- http//wsex.com
- Debnath, EC-2003

Same concept,better site

Soccer World Cup 2002

- 15 Soccer markets (June 715, 2002)
- Several 1st round and 2nd round games
- All games ended without penalty shoot-out
- Scores recorded from www.LiveScore.com
- Sampled the stream of price and score

information every 10 seconds

Ex Price reaction to goals

- Sweden vs. Nigeria (Final score 2-1, goals scored

at 31st (0-1), 39th (1-1) and 83rd (2-1)

minutes. Yellow bars indicate goals.

Ex Price reaction to goals

- Denmark vs. France (Final Score 2-0, goals

scored at the 22nd (1-0) and 85th (2-0) minute of

the game) Yellow bars indicate goals

Avg log score entropy

Delay Calculation

Where Timestamp of scoring Timestamp of

price update Delay in updating score

network delay Delay in updating the price

network delay

Reaction time after goals

NBA 2002

- 18 basketball markets during 2002 Championships

(May 631, 2002) - Score recorded from www.SportsLine.com
- Sampled the stream of price and score

information every 10 seconds

Correlation between price and score

- San Antonio vs. LA Lakers (May 07, 2002, Final

Score 88-85, Correlation 0.93).

Correlation between price and score

Avg log score entropy

Soccer vs. NBA

Soccer World Cup 2002

NBA Championship 2002

Soccer vs. NBA

- Soccer characteristics
- Price does not change very often
- Price change is abrupt immediate after goal
- Average entropy decreases gradually toward 0
- Comebacks less likely?more surprising when they

occur - Basketball characteristics
- Price changes very often by small amounts
- Price is well correlated with scoring
- More uncertainty until late in the games
- entropy 0.7 for 77 of game 0.8 for 55.5 of

game - More exciting late?outcome is unclear until

near end

Basketball as coin flips

- Model scoring as a series of coin flips
- tails Boston 1
- heads Detroit 1
- Current scores Bt,Dt
- Final scores BT,DT
- ComputeP(BT-DT 5.5 Bt,Dt)

- ED B 180
- EB - D 5.5
- EB92.75ED87.25
- p P(tails) P(Boston) 92.75/180

0.515

May10 505 DETROIT o/u 180 0700 506 BOSTON -5.5

180-Bt-Dt

180-Bt-Dtj

? ( )pj (1-p)(180-Bt-Dt-j)

j93-Bt

Basketball as coin flips

Explain the marketParallel IR

Pennock 2002

IEM Giuliani NY Senate 2000

Use expected entropy lossto determine the key

wordsand phrases that bestdifferentiate between

textstreams before and after thedate of interest

Explain the marketParallel IR

us.politics

IEM Gore US Pres 2000

florida, ballots, recount, palm

beach, ballot, beach county, palm beach

county

FX Extraterrestrial Life

sci.space.news

meteorite, life, evidence, martian

meteorite,primitive, gibson, organic, of

possible,martian, life on mars, ...

Applications future work

- Monitoring dynamics
- Automatic explanations
- Low probability event detection
- Sporting events auto highlights, auto summary,

attention scheduling, finding turning points,

most exciting games/moments, modeling different

sports...

Play-money market games

http//www.hsx.com/

http//www.newsfutures.com/

http//www.100world.com/

http//www.ideosphere.com/

http//www.ipreo.com/

http//www.incentivemarkets.com/

Play-money market games

- Many studies show that prices in real-money

markets provide accurate likelihoods - Researchers credit monetary incentives/risk
- Can play money markets provide accurate

forecasts? - Incentives in market games may derive from

entertainment value, educational value,

competitive spirit, bragging rights, prizes

Market games analyzed

- Hollywood Stock Exchange (HSX)
- Play-money market in movies and stars
- Movie stocks movie options
- Award options (e.g., Oscar options)
- Foresight Exchange (FX)
- Market game to bet on developments in science

technology e.g., Cancer cured by 2010 Higgs

boson verified Water on moon Extraterrestrial

life verified - NewsFutures
- Newsworthy events items of pop interest

Put-call parity

- stock price s - call price put price strike

price k

payoff

10

20

30

40

50

stock price s

Internal coherence HSX

- Prices of movie stocks and options adhere to

put-call parity, as in real markets - Arbitrage loopholes disappear over time, as in

real markets

Internal coherenceHSX vs IEM

- Arbitrage closure for HSX award options
- Arbitrage closure on IEM qualitatively similar to

HSX, though quantitatively more efficient

Forecast accuracy HSX

- 0.94 correlation
- Comparable to expert forecasts at Box Office Mojo

Combining forecasts

- HSX Box Office Mojo (expert forecast)
- Correlation of errors 0.818

Probabilistic forecastsHSX

- Bins of similarly-priced options
- Observed frequency? average price
- Analysis similar for horse racing markets
- Error bars 95 confidence intervals assuming

events are indep Bernoulli trials

Avg logarithmic score

HSX Oscar options 2000

forecast source avg log score Feb 19 HSX

prices -0.854 DPRoberts -0.874 Fielding -1.04 e

xpert consensus -1.05 Feb 18 HSX

prices -1.08 Tom -1.08 John -1.22 Jen -1.25

Probabilistic forecastsFX

- Prices 30 days before expiration
- Similar results
- 60 days before
- specific date
- Average logarithmic score

FX

Real marketsvs. market games

IEM

HSX

averagelog score

arbitrageclosure

Real marketsvs. market games

HSX

FX, F1P6

forecast source avg log score F1P6 linear

scoring -1.84 F1P6 F1-style scoring -1.82 betting

odds -1.86 F1P6 flat scoring -2.03 F1P6 winner

scoring -2.32

expectedvalueforecasts489 movies

Does money matter? Play vs real, head to head

- Experiment
- 2003 NFL Season
- Online football forecasting competition
- Contestants assess probabilities for each game
- Quadratic scoring rule
- 2,000 experts, plus
- NewsFutures (play )
- Tradesports (real )
- Used last trade prices

- Results
- Play money and real money performed similarly
- 6th and 8th respectively
- Markets beat most of the 2,000 contestants
- Average of experts came 39th

Forthcoming, Electronic Markets, Emile

Servan-Schreiber, Justin Wolfers, David Pennock

and Brian Galebach

(No Transcript)

Does money matter? Play vs real, head to head

StatisticallyTS NFNF Avg TS Avg

Market games summary

- Online market games can contain a great deal of

information reflecting interactions among

millions of people - Naturally attract well-informed and

well-motivated players - Game players tend to be knowledgeable and

enthusiastic - Internet polls - skewed demographic
- Polls typically ask questions of the form What

do you want? - Games ask questions of the form What do you

think will happen?

Market games discussion

- Are incentives strong enough?
- Yes (to a degree)
- Manifested as price coherence, information

incorporation, and forecast accuracy - Reduced incentive for information discovery

possibly balanced by better interpersonal

weighting - Statistical validations show HSX, FX, NF are

reliable sources for forecasts - HSX predictions expert predictions
- Combining sources can help

Applications

- Obtain information from existing games
- Build new games in areas of interest
- Alternative to costly market research
- Easy/inexpensive to setup compared to real

markets - Few regulations compared to real markets
- Worldwide audience

Future work

- Data mining and fusion algorithms can improve

predictions - Weight users by expertise, reliability, etc.
- Controlling for manipulation
- Merging with other sources
- Box office prediction (market chat groups,

query logs, movie reviews, news, experts) - Weather forecasting (futures, derivatives

experts, satellite images) - Privacy issues and incentives

4. Lab experiments theory

- Laboratory experiments, field tests
- Theoretical underpinnings
- Rational expectations
- Efficient markets hypothesis
- No-Trade Theorems
- Information aggregation

Laboratory experiments

- Experimental economics
- Plott and decendents Ledyard, Hanson, Fine,

Coughlan, Chen, ... (and others) - Controlled tests of information aggregation
- Participants are given information, asked to

trade in market for real monetary stakes - Equilibrium is examined for signs of information

incorporation

Plott Sunder 1982

- Three disjoint exhaustive states X,Y,Z
- Three securities
- A few insiders know true state Z
- Market equilibrates according to rational

expectations as if everyone knew Z

Plott Sunder 1982

- Three disjoint exhaustive states X,Y,Z
- Three securities
- Some see samples of joint can infer P(Zsamples)
- Results less definitive

Plott Sunder 1988

- Three disjoint exhaustive states X,Y,Z
- Three securities
- A few insiders know true state is not X
- A few insiders know true state is not Y
- Market equilibrates according to rational

expectations Z true

Plott Sunder 1988

- Three disjoint exhaustive states X,Y,Z
- One security
- A few insiders know true state is not X
- A few insiders know true state is not Y
- Market does not equilibrate according to rational

expectations

Forsythe and Lundholm 90

- Three disjoint exhaustive states X,Y,Z
- One security
- Some know not X
- Some know not Y
- As long as traders are sufficiently knowledgeable

experienced, market equilibrates according to

rational expectations

1 if Z

not X

not Y

1

price of Z

0

time

Small groups

- In small, illiquid markets, information

aggregation can fail - Chen, Fine, Huberman EC-2001 propose a two

stage process - Trade in a market to assess participants risk

attitude and predictive ability - Query participants probabilities using the log

score compute a weighted average of

probabilities, with weights derived from step 1

Small groups

Source Fine DARPA Workshop, 2002

Field test Hewlett Packard

- Plott Chen 2002 conducted a field test at

Hewlett Packard (HP) - Set up a securities market to predict, e.g. next

months sales (in ) of product X - 1 iff 0 20K
- 1 iff 10K 30K
- Employees could trade at lunch, weekends, for

real - Market predictions beat official HP forecasts

Why does it work?Rational expectations

- Theory Even when agents have asymmetric

information, market equilibrates as if all agents

had all info Grossman 1981 Lucas 1972 - Procedural explanation agents learn from prices

Hanson 98 Mckelvey 86 Mckelvey 90 Nielsen 90 - Agents begin with common priors, differing

information - Observe sufficient summary statistic (e.g.,

price) - Converge to common posteriors
- In compete market, all (private) info is revealed

Efficient market hypotheses (EMH)

- Internal coherenceprices are self-consistent or

arbitrage-free - Weak form Internal unpredictabilityfuture

prices unpredictable from past prices - Semi-strong form Unpredictabilityfuture prices

unpredictable from all public info - Strong form Expert-level accuracyunpredictable

from all public private infoexperts cannot

outperform naïve traders - More

stronger assumps

http//www.investorhome.com/emh.htm

How efficient are markets?

- Good question as many opinions as experts
- Cannot prove efficiency can only detect

inefficiency - Generally, it is thought that large public

markets are very efficient, smaller markets

questionable - Still, strong form is sometimes too strong
- There is betting on Oscars until winners are

announced - Prices do not converge completely on eventual

winners - Yet aggregating all private knowledge in the

world (including Academy members votes) would

yield the precise winners with certainty

No-trade theorems

- Why trade? These markets are zero-sum games

(negative sum w/ transaction fees) - For all money earned, there is an equal (greater)

amount lost am I smarter than average? - Rational risk-neutral traders will never trade

Milgrom Stokey 1982Aumann 1976. Informally - Only those smarter than average should trade
- But once below avg traders leave, avg goes up
- Ad infinitum until no one is left
- Or If a rational trader is willing to trade with

me, he or she must know something I dont know

But... Trade happens

- Volume in financial markets, gambling is high
- Why do people trade?
- 1. Different risk attitudes (insurance, hedging)

Cant explain all volume - 2. Irrational (boundedly rational) behavior
- Rationality arguments require unrealistic

computational abilities, including infinite

precision Bayesian updating, infinite

game-theoretic recursive reasoning - More than 1/2 of people think theyre smarter

than average - Biased beliefs, differing priors, inexperience,

mistakes, etc. - Note that its rational to trade as long as some

participants are irrational

A theory of info aggregationNotation

Pennock 2002

- Event A (event negation A)
- Security
- Probability Pr(A)
- Likelihood L(A) Pr(A)/(1-Pr(A))
- Log-likelihood LL(A) ln L(A)
- Price of at time t pt
- Likelihood price lt pt/(1-pt)
- Log-likelihood price llt ln lt

1 if A

1 if A

Assumptions

- Efficiency assumptionLet pt be the price of

at time tThen - Pr(Apt,pt-1,pt-2,,p0) pt
- (Markov assumpt. accuracy assumpt.)

1 if A

Consequences

- Eptpt-1 x xexpected price at time t

is price at t-1 - log-likelihood price is e? as likely to go up

by ? in worlds where A is true, as it is to go up

? in worlds where A is false

Consequences

- Pr(ptyA,pt-1x) Pr(ptypt-1x)price

is y/x times as likely to go from x to y in

worlds where A is true - given A is true, expected price at time t is

greater than price at t-1 by an amount prop. to

the variance of price

EptA,pt-1x x

Empirical verification

Distribution of changes e in log-likelihood price

over 22 IEM markets, consistent with theory

Distribution of changes e in log-likelihood price

of winning candidates divided by losing

candidates. Line is ee, as predicted by theory

Avg log score dynamics

IEM

FX

WSE bball

HSX

WSE soccer

Applications future work

- Better understanding of market dynamics

assumptions required for predictive value - Closeness of fit to theory is a measure of market

forecast accuracy could serve as an evaluation

metric or confidence metric - Explaining symmetry, power-law dist in IEM

Coin-flip model

- Previous theory minimalist assumptions no

explicit notion of evidence - Coin-flip model of evidence incorporation
- A ? occurrence of n/2 tails out of n flips
- Release of info ? revelation of flip outcomes
- At time t it tails have occurred out of kt flips
- For A to occur, n/2-it more tails are needed

Avg log score dynamics

IEM

FX

WSE bball

coin flip model

HSX

WSE soccer

5. Characterizin