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Markets in Uncertainty: Risk, Gambling, and Information Aggregation a tutorial by David M. Pennock Michael P. Wellman 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

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AAAI04 July 2004MP1-2Pennock/Wellman Outline 1. Overview tour15 min What is a market in uncertainty? 2. Background30 min A. Single agent perspective 1. Subjective probability 2. Utility, risk, and risk aversion 3. Decision making under uncertainty B. Multiagent perspective 1. Trading/allocating risk 2. Pareto optimality 3. Securities: Markets in uncertainty

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AAAI04 July 2004MP1-3Pennock/Wellman Outline 3. Mechanisms, examples and empirical studies 45 min A. What & how: Instruments & mechanisms B. Real-money markets: Examples & evaluations 1. Iowa Electronic Market 2. Options 3. TradeSports: Effects of war 4. Horse racing, sports betting C. Play-money markets

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AAAI04 July 2004MP1-4Pennock/Wellman Outline 4. Lab experiments and theory20 min A.Laboratory experiments, field tests B.Theoretical underpinnings 1. Rational expectations 2. Efficient markets hypothesis 3. No-Trade Theorems 4. Information aggregation

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AAAI04 July 2004MP1-5Pennock/Wellman Outline 5. Characterizing information20 min aggregation A. Market as an opinion pool B. Market as a composite agent 1. Market belief, utility 2. Market Bayesian updates 3. Market adaptation, dynamics C. Paradoxes, impossibilities 1. Opinion pool impossibilities 2. Composite agent non-existence

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AAAI04 July 2004MP1-6Pennock/Wellman Outline 6. Computational aspects60 min A. Combinatorics 1. Compact securities markets 2. Combinatorial securities markets 3. Compound securities markets 4. Market scoring rules 5. Dynamic pari-mutuel market 6. Policy Analysis Market B. Distributed market computation 7. Legal issues; miscellaneous5 min 8. Discussion, Q&A15 min

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1. Overview tour What is a market in uncertainty ?

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AAAI04 July 2004MP1-8Pennock/Wellman A market in uncertainty Take a random variable, e.g. Turn it into a financial instrument payoff = realized value of variable = 6 ? = 6 $1 if 6 $0 if I am entitled to: US04Pres = Bush? 2004 CA Earthquake?

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AAAI04 July 2004MP1-9Pennock/Wellman 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...

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AAAI04 July 2004MP1-10Pennock/Wellman 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

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AAAI04 July 2004MP1-11Pennock/Wellman Why? Reason 1: Information Information market: financial mechanism designed to obtain estimates of expectations of random variables Easy as 1, 2, 3: 1.Take a random variable whose expectation youd like to know 2.Turn it into a financial instrument (payoff= realized value of variable) 3.Open a market in the financial instrument price(t) E t [X] (in many cases,... more later)

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AAAI04 July 2004MP1-12Pennock/Wellman Getting information Non-market approach: ask an expert –How much would you pay for this? A: $5/36 $ –caveat: expert is knowledgeable –caveat: expert is truthful –caveat: expert is risk neutral, or ~ RN for $1 –caveat: expert has no significant outside stakes = 6 $1 if 6 $0 if I am entitled to:

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AAAI04 July 2004MP1-13Pennock/Wellman 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 = 6 6 logarithmic scoring rule, a proper scoring rule

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AAAI04 July 2004MP1-14Pennock/Wellman 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 q i units at price p i Let any person j submit an ask order: an offer to sell q j units at price p j (if you sell 1 unit, you agree to pay $1 if ) Match up agreeable trades (many poss. mechs...) = 6 $1 if 6 $0 if I am entitled to: = 6

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AAAI04 July 2004MP1-15Pennock/Wellman 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, p i > p j, then i&j trade min(q i,q j ) units at price {p j,p i } In equilibrium (no trades) –max bid p i < min ask p j = bid-ask spread – bounds aggregate public opinion of expectation = 6 $1 if 6 $0 if I am entitled to:

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AAAI04 July 2004MP1-16Pennock/Wellman 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]

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AAAI04 July 2004MP1-17Pennock/Wellman $1 if ;$0 otherwise I am entitled to: (Real) Great expectations For dice example, no need for market: E[x] is known; no one should disagree Real power comes for non-obvious expectations of random variables, e.g. $x if interest rate = x on Jan 1, 2004 I am entitled to:

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AAAI04 July 2004MP1-18Pennock/Wellman $1 if ;$0 otherwise I am entitled to: Bin Laden captured $max(0,x-k) if MSFT = x on Jan 1, 2004 I am entitled to: call option $f(future weather) I am entitled to: weather derivative $1 if Kansas beats Marq. by > 4.5 points; $0 otherw. I am entitled to:

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AAAI04 July 2004MP1-20Pennock/Wellman IPOIPOIPOIPO

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AAAI04 July 2004MP1-21Pennock/Wellman Play money; Real expectations

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Cancer cured by 2010 Machine Go champion by 2020

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AAAI04 July 2004MP1-23Pennock/Wellman 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 polls [Forsythe 1992, 1999][Oliven 1995][Rietz 1998][Berg 2001][Pennock 2002] –Futures and options markets rapidly incorporate information, providing accurate forecasts of their underlying commodities/securities [Sherrick 1996][Jackwerth 1996][Figlewski 1979][Roll 1984][Hayek 1945] –Sports betting markets provide accurate forecasts of game outcomes [Gandar 1998][Thaler 1988][Debnath EC03][Schmidt 2002]

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AAAI04 July 2004MP1-24Pennock/Wellman Does it work? Yes... E.g. (contd): –Laboratory experiments confirm information aggregation [Plott 1982;1988;1997][Forsythe 1990][Chen, EC-2001] –And field tests [Plott 2002] –Theoretical underpinnings: rational expectations [Grossman 1981][Lucas 1972] –Procedural explanation: agents learn from prices [Hanson 1998][Mckelvey 1986][Mckelvey 1990][Nielsen 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 2004][Pennock 2001]

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AAAI04 July 2004MP1-25Pennock/Wellman Why? Reason 2 Manage risk If is horribly terrible for you Buy a bunch of and if happens, you are compensated = 6 $1 if 6 $0 if I am entitled to: = 6

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AAAI04 July 2004MP1-26Pennock/Wellman Why? Reason 2 Manage risk If is horribly terrible for you Buy a bunch of and if happens, you are compensated $1 if$0 if I am entitled to:

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AAAI04 July 2004MP1-27Pennock/Wellman 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

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AAAI04 July 2004MP1-28Pennock/Wellman 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

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AAAI04 July 2004MP1-29Pennock/Wellman 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

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AAAI04 July 2004MP1-30Pennock/Wellman Examples I buy MSFT stock at s. Im afraid it will go down. I buy a put option that pays Max[0,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.

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AAAI04 July 2004MP1-31Pennock/Wellman 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… )

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AAAI04 July 2004MP1-32Pennock/Wellman 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

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AAAI04 July 2004MP1-33Pennock/Wellman 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

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AAAI04 July 2004MP1-34Pennock/Wellman On trading Why would two parties agree to trade in a market in uncertainty? 1.They disagree on expected values (probs) 2.They differ in their risk attitude or exposure – they trade to reallocate risk 3.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.

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AAAI04 July 2004MP1-35Pennock/Wellman On computational issues Information aggregation is a form of distributed computation Agent level –nontrivial optimization problem, even in 1 market; ultimately a game-theoretic question –probability representation, updating algorithm (Bayes net) –decision representation, algorithm (POMDP) –agent problems computational complexity, algorithms, approximations, incentives some

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AAAI04 July 2004MP1-36Pennock/Wellman On computational issues 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) some

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AAAI04 July 2004MP1-37Pennock/Wellman On computational issues Machine learning, data mining –Beat the market (exploiting combinatorics?) –Explain the market, information retrieval –Detect fraud some

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2. Background A. Single agent perspective 1. Subjective probability 2. Utility, risk, and risk aversion 3. Decision making under uncertainty B. Multiagent perspective 1. Trading/allocating risk 2. Pareto optimality 3. Securities: markets in uncertainty

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AAAI04 July 2004MP1-39Pennock/Wellman Decision making under uncertainty How should agents behave (make decisions, choose actions) when faced with uncertainty? Decision theory: Prescribes maximizing expected utility

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AAAI04 July 2004MP1-40Pennock/Wellman Why reason about uncertainty? Propositional logic: No uncertainty Could 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

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AAAI04 July 2004MP1-41Pennock/Wellman Why Bayesian uncertainty? E.g. You can buy skis for $b Or you can rent for $b/k, k>1 Worst-case analysis: Rent for k days, then buy Youll spend at most $2b But what if you strongly believe youll ski more than k times? Buy earlier That k+1st time is your last? Dont buy Expected (utility) case often more appropriate

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AAAI04 July 2004MP1-42Pennock/Wellman Decision making under uncertainty, an example ABC TVs Who Wants to be a Millionaire?

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AAAI04 July 2004MP1-43Pennock/Wellman Decision making under uncertainty, an example v 15 = $1,000,000if correct $32,000 if incorrect $500,000 if walk away

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AAAI04 July 2004MP1-44Pennock/Wellman Decision making under uncertainty, an example if you answer: E[v 15 ] = $1,000,000 *Pr(correct) + $32,000 *Pr(incorrect) if you walk away: $500,000

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AAAI04 July 2004MP1-45Pennock/Wellman Decision making under uncertainty, an example if you answer: E[v 15 ] = $1,000,000 *0.5 $32,000 *0.5 = $516,000 if you walk away: $500,000 you should answer, right?

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AAAI04 July 2004MP1-46Pennock/Wellman Decision making under uncertainty, an example Most people wont answer: risk averse U($x) = log($x) if you answer: E[u 15 ] = log($1,000,000) *0.5 +log($32,000) *0.5 = 6/2+4.5/2 = 5.25 if you walk away: log($500,000) = 5.7

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AAAI04 July 2004MP1-47Pennock/Wellman Decision making under uncertainty, an example Maximizing E[u i ] for i<15 more complicated Q7, L={1,3} walkanswerL1 0.4 X 0.6 log($2k) Q7, L={3} Q8, L={1,3} log($1k) walkanswerL3 0.8 X 0.2 Q8, L={3} log($1k) log($2k) L3

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AAAI04 July 2004MP1-48Pennock/Wellman Decision making under uncertainty, in general =set of all possible future states of the world

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AAAI04 July 2004MP1-49Pennock/Wellman Decision making under uncertainty, in general are disjoint exhaustive states of the world i : rain tomorrow & Bush elected & Y! stock up & car not stolen &... j : rain tomorrow & Bush elected & Y! stock up & car stolen & i |

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AAAI04 July 2004MP1-50Pennock/Wellman Decision making under uncertainty, in general Equivalent, more natural: E i : rain tomorrow E j : Bush elected E k : Y! stock up E l : car stolen | |=2 n E1E1 E2E2 EiEi EjEj EnEn

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AAAI04 July 2004MP1-51Pennock/Wellman Decision making under uncertainty, in general Preferences, utility i > j u( i ) > u( j ) Expected utility E[u] = Pr( )u( ) Decisions (actions) can affect Pr( ) What you should do: choose actions to maximize expected utility Why?: To avoid being a money pump [de Finetti74], among other reasons...

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AAAI04 July 2004MP1-52Pennock/Wellman 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: p. 2 ~ [p, 1 ; 3 ] substitution: 1 ~ 2 [p, 1 ; 3 ] ~ [p, 2 ; 3 ] monotonicity: 1 2 p>q [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 ) + (1–p)u( 2 )

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AAAI04 July 2004MP1-53Pennock/Wellman Utility functions ( a probability distribution over ) E[u]: represents preferences, E[u]( ) E[u]( ) iff Let ( ) = au( ) + b, a>0. –Then E[ ]( ) = E[au+b]( ) = a E[u]( ) + b. –Since they represent the same preferences, and u are strategically equivalent ( ~ u).

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AAAI04 July 2004MP1-54Pennock/Wellman Utility of money Outcomes are dollars Risk attitude: –risk neutral: u(x) ~ x –risk averse (typical): u concave (u (x) < 0 for all x) –risk prone: u convex Risk aversion function: r(x) = – u (x) / u (x)

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AAAI04 July 2004MP1-55Pennock/Wellman Risk aversion & hedging E[u]=.01 (4)+.99 (4.3) = Action: buy $10,000 of insurance for $125 E[u]= Even better, buy $ of insurance for $74.68 E[u] = Optimal 1 : car stolen u( 1 ) = log(10,000) 2 : car not stolen u( 2 ) = log(20,000) u( 1 ) = log(19,875)u( 2 ) = log(19,875) u( 1 ) = log(15,900)u( 2 ) = log(19,925)

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AAAI04 July 2004MP1-56Pennock/Wellman Securities markets Note that, in previous example, risk-neutral insurance company also profits: E[v] =.01(-5,900) (74.68) = $14.93 Both parties gain from bilateral agreement Securities market generalizes this to –arbitrary states –more than two parties Market mechanism to allocate risk among participants

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AAAI04 July 2004MP1-57Pennock/Wellman 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.

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AAAI04 July 2004MP1-58Pennock/Wellman What is traded: Securities Specifies state-contingent returns, r = (r 1,…,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) –r i = 1 if i E 1, r i = 0 otherwise $1 if E 1

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AAAI04 July 2004MP1-59Pennock/Wellman 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 E i

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AAAI04 July 2004MP1-60Pennock/Wellman 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)

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AAAI04 July 2004MP1-61Pennock/Wellman 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

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AAAI04 July 2004MP1-62Pennock/Wellman Incomplete markets Securities do not span states of nature (always the case in practice) Equilibria may exist, but may not be Pareto optimal Example: missed insurance opportunity More: Theory of Incomplete Markets, Magill & Quinzii, MIT Press, 1998

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AAAI04 July 2004MP1-63Pennock/Wellman Why trade securities? Profit from perceived mispricings –Price p differs significantly enough from traders belief Pr(E 1 ) –speculation Insure against risk –Traders marginal value for wealth in E 1, relative to p, differs from that in other states –e.g., home fire insurance –hedging

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AAAI04 July 2004MP1-64Pennock/Wellman Societal roles of security markets From speculation: –Aggregate beliefs –Disseminate information From hedging: –Allocate risk

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AAAI04 July 2004MP1-65Pennock/Wellman 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

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3. Mechanisms, examples & empirical studies A. What & how: Instruments & mechanisms B. Real-money markets: Examples & evaluations 1.Iowa Electronic Market 2.Options 3.TradeSports: Effects of war 4.Horse racing, sports betting C. Play-money markets

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AAAI04 July 2004MP1-67Pennock/Wellman 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)

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AAAI04 July 2004MP1-68Pennock/Wellman 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: Max[0,s-k], arbitrary (non-linear) fn Payoff output –dollars, fake money, commodities = 6

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AAAI04 July 2004MP1-69Pennock/Wellman 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 –M+1st price auction –k-double auction lim period 0: Continuous double auction

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AAAI04 July 2004MP1-70Pennock/Wellman 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 $0.3 –buy $1 if $0.7 Alternative: Market institution always stands ready to buy/sell exhaustive bundle for $1.00 –Iowa Electronic Market A occursA occurs = = = =.3

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AAAI04 July 2004MP1-71Pennock/Wellman 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 <= price –let x = min(n,m) –the x highest buy offers and x lowest sell offers transact

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AAAI04 July 2004MP1-72Pennock/Wellman Call market $0.15 $0.12 $0.09 $0.05 $0.30 $0.17 $0.13 $0.11 Buy offers (N=4) Sell offers (M=5) $0.08 = 6 $1 if 6 $0 if

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AAAI04 July 2004MP1-73Pennock/Wellman $0.05 $0.08 $0.09 $0.11 $0.12 $0.13 Mth price auction $0.15 $0.17 $0.30 Buy offers (N=4) Sell offers (M=5) Matching buyers/sellers price = $0.12 = 6 $1 if 6 $0 if

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AAAI04 July 2004MP1-74Pennock/Wellman $0.05 $0.08 $0.09 $0.11 $0.12 $0.13 M+1st price auction $0.15 $0.17 $0.30 Buy offers (N=4) Sell offers (M=5) Matching buyers/sellers = 6 $1 if 6 $0 if price = $0.11 6

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AAAI04 July 2004MP1-75Pennock/Wellman $0.05 $0.08 $0.09 $0.11 $0.12 $0.13 k-double auction $0.15 $0.17 $0.30 Buy offers (N=4) Sell offers (M=5) Matching buyers/sellers = 6 $1 if 6 $0 if price = $ $0.01*k 6

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AAAI04 July 2004MP1-76Pennock/Wellman Continuous double auction CDA 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

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IPOIPOIPOIPO

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AAAI04 July 2004MP1-79Pennock/Wellman 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

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AAAI04 July 2004MP1-80Pennock/Wellman 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

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AAAI04 July 2004MP1-81Pennock/Wellman 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/

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AAAI04 July 2004MP1-83Pennock/Wellman 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

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AAAI04 July 2004MP1-84Pennock/Wellman Pari-mutuel mechanism Common at horse races, jai alai games n mutually exclusive outcome (e.g., horses) M 1, M 2, …, M n dollars bet on each If i wins: all bets on 1, 2, …, i-1,i+1, …, 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 gets: $1 + $1/M i * (M 1, M 2, …,M i-1, M i+1, …, M n ) = $1/M i * (M 1, M 2, …, M n )

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AAAI04 July 2004MP1-85Pennock/Wellman Pari-mutuel market E.g. horse racetrack style wagering Two outcomes: A B Wagers: AB

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AAAI04 July 2004MP1-86Pennock/Wellman AB Pari-mutuel market E.g. horse racetrack style wagering Two outcomes: A B Wagers:

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AAAI04 July 2004MP1-87Pennock/Wellman AB Pari-mutuel market E.g. horse racetrack style wagering Two outcomes: A B Wagers:

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AAAI04 July 2004MP1-88Pennock/Wellman AB Pari-mutuel market E.g. horse racetrack style wagering Two outcomes: A B 2 equivalent ways to consider payment rule –refund + share of B –share of total $ on B 8 $ on A 4 1+ = 1+ =$3 total $ 12 $ on A 4 = = $3

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AAAI04 July 2004MP1-89Pennock/Wellman 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

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AAAI04 July 2004MP1-90Pennock/Wellman 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

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AAAI04 July 2004MP1-91Pennock/Wellman Example: IEM Iowa Electronic Market $1 if Gephardt wins $1 if H. Clinton wins $1 if Kerry wins $1 if Lieberman wins $1 if other wins price=E[C]=Pr(C)=0.056 as of 4/22/2003 US Democratic Pres. nominee 2004

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AAAI04 July 2004MP1-92Pennock/Wellman Example: IEM Iowa Electronic Market $1 if Democrat votes > Repub $1 if Republican votes > Dem price=E[R]=Pr(R)=0.494 as of 7/25/2004 US Presidential election 2004

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AAAI04 July 2004MP1-93Pennock/Wellman IEM vote share market as of 4/22/2003 $1 vote share of Kerry $1 2-party vote share of Bush v. other $1 2-party vote share of other Dem US Pres. election vote share 2004 price=E[VS for K]=0.148 $1 vote share of Bush v. Kerry

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AAAI04 July 2004MP1-94Pennock/Wellman IEM vote share market as of 7/25/2004 $1 vote share of Bush v. Kerry US Pres. election vote share 2004 price=E[VS for B v. K]=0.508 $1 2-party vote share of Kerry $1 vote share of Bush v. Dean $1 vote share of Dean

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AAAI04 July 2004MP1-95Pennock/Wellman Example: IEM 1992 [Source: Berg, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-96Pennock/Wellman Example: IEM [Source: Berg, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-97Pennock/Wellman Example: IEM [Source: Berg, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-98Pennock/Wellman Example: IEM [Source: Berg, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-99Pennock/Wellman Example: IEM [Source: Berg, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-100Pennock/Wellman Contract: Pays $100 if Cubs win game 6 (NLCS) Price of contract (=Probability that Cubs win) Cubs are winning 3-0 top of the 8 th 1 out. Time (in Ireland) Fan reaches over and spoils Alous catch. Still 1 out. The Marlins proceed to hit 8 runs in the 8 th inning [Source: Wolfers 2004] Speed: TradeSports

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AAAI04 July 2004MP1-101Pennock/Wellman 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

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AAAI04 July 2004MP1-102Pennock/Wellman P(C)=0.6 –WTA gives P(C) = P(V>0.5) Forecast error bounds [Berg 2001] Single market gives E[x] –IEM winner takes all: P(candidate wins) = P(C) –IEM vote share: E[candidate vote share] = E[V] Can we get error bounds? e.g. Var[x]? Yes: combine the two markets E[V]=0.55 –Vote share gives mean of dist vote share 0.50 –Assume e.g. normal dist of votes –Report 95% confidence intervals = error bounds

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AAAI04 July 2004MP1-103Pennock/Wellman Evaluating accuracy: Recall 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 r=Pr(A), given caveats = 6 6 X X Note: still works as a tax

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AAAI04 July 2004MP1-104Pennock/Wellman 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

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AAAI04 July 2004MP1-105Pennock/Wellman 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

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AAAI04 July 2004MP1-106Pennock/Wellman Avg log score dynamics FX HSX WSE bball WSE soccer IEM

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AAAI04 July 2004MP1-107Pennock/Wellman Avg log score 22 IEM political markets Average log score = i log (p i )/N p i : i th winners normalized price

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AAAI04 July 2004MP1-108Pennock/Wellman Example: options Options prices (partially) encode a probability distribution over their underlying stocks –Arbitrary derivative P(underlying asset) stock price s call 20 = max[0,s-20] payoff call 30 = max[0,s-30] call 40 = max[0,s-40]

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AAAI04 July 2004MP1-109Pennock/Wellman Example: options Options prices (partially) encode a probability distribution over their underlying stocks –Arbitrary derivative P(underlying asset) stock price s call 20 payoff - 2*call 30 + call 40 butterfly spread

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AAAI04 July 2004MP1-110Pennock/Wellman Example: options Options prices (partially) encode a probability distribution over their underlying stocks –Arbitrary derivative P(underlying asset) stock price s call 30 payoff - 2*call 40 + call 50

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AAAI04 July 2004MP1-111Pennock/Wellman Example: options call call 20 + call 30 = $2.13 relative call call 30 + call 40 = $5.73 likelihood of falling call call 40 + call 50 = $3.54 near center stock price s payoff $2.13$5.73$3.54

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AAAI04 July 2004MP1-112Pennock/Wellman Example: options More generally, uses prices as constraints E[Max[0,s-10]]=p 10 ; E[Max[0,s-20]]=p 20 ;... etc. Fit to assumed distribution; or maximize {entropy, smothness, etc.} subject to constraints stock price s probability [Jackwerth 1996]

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AAAI04 July 2004MP1-113Pennock/Wellman Example: TradeSports [Source: Wolfers 2004]

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AAAI04 July 2004MP1-114Pennock/Wellman [Source: Wolfers 2004]

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AAAI04 July 2004MP1-115Pennock/Wellman [Source: Wolfers 2004]

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AAAI04 July 2004MP1-116Pennock/Wellman [Source: Wolfers 2004]

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AAAI04 July 2004MP1-117Pennock/Wellman State Price Distribution [Source: Wolfers 2004]

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AAAI04 July 2004MP1-118Pennock/Wellman State Price Distribution: War and Peace [Source: Wolfers 2004]

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AAAI04 July 2004MP1-119Pennock/Wellman Example: horse racing Pari-mutuel mechanism Normalized odds match objective frequencies of winning very closely –3:1 horses win about twice as much as 6:1 horses, etc. Slight favorite-longshot bias (favorites are better bets; extremely rarely E[return] > 0) [Ali 77; Rosett 65; Snyder 78; Thaler 88; Weitzman 65]

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AAAI04 July 2004MP1-120Pennock/Wellman 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_pr.html

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AAAI04 July 2004MP1-121Pennock/Wellman 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

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AAAI04 July 2004MP1-122Pennock/Wellman 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… [Debnath, EC-2003] Same concept, better site:

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AAAI04 July 2004MP1-123Pennock/Wellman Soccer World Cup Soccer markets (June 7–15, 2002) Several 1st round and 2nd round games All games ended without penalty shoot-out Scores recorded from Sampled the stream of price and score information every 10 seconds

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AAAI04 July 2004MP1-124Pennock/Wellman Ex: Price reaction to goals Sweden vs. Nigeria (Final score 2-1, goals scored at 31 st (0-1), 39 th (1-1) and 83 rd (2-1) minutes. Yellow bars indicate goals.

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AAAI04 July 2004MP1-125Pennock/Wellman Ex: Price reaction to goals Denmark vs. France (Final Score: 2-0, goals scored at the 22 nd (1-0) and 85 th (2-0) minute of the game) Yellow bars indicate goals

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AAAI04 July 2004MP1-126Pennock/Wellman Avg log score & entropy

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AAAI04 July 2004MP1-127Pennock/Wellman Delay Calculation Where : Timestamp of scoring : Timestamp of price update : Delay in updating score + network delay : Delay in updating the price + network delay

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AAAI04 July 2004MP1-128Pennock/Wellman Reaction time after goals

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AAAI04 July 2004MP1-129Pennock/Wellman NBA basketball markets during 2002 Championships (May 6–31, 2002) Score recorded from Sampled the stream of price and score information every 10 seconds

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AAAI04 July 2004MP1-130Pennock/Wellman Correlation between price and score San Antonio vs. LA Lakers (May 07, 2002, Final Score: 88-85, Correlation: 0.93).

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AAAI04 July 2004MP1-131Pennock/Wellman Correlation between price and score

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AAAI04 July 2004MP1-132Pennock/Wellman Avg log score & entropy

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AAAI04 July 2004MP1-133Pennock/Wellman Soccer vs. NBA Soccer World Cup 2002NBA Championship 2002

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AAAI04 July 2004MP1-134Pennock/Wellman 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

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AAAI04 July 2004MP1-135Pennock/Wellman Basketball as coin flips Model scoring as a series of coin flips –tails = Boston + 1 –heads = Detroit + 1 Current scores: B t,D t Final scores: B T,D T Compute P(B T -D T > 5.5 | B t,D t ) E[D + B] = 180 E[B - D] = 5.5 E[B]=92.75;E[D]=87.25 p = P(tails) = P(Boston) = 92.75/180 = May DETROIT o/u : BOSTON -5.5 = ( ) p j (1-p) (180-B t -D t -j) j=93-B t 180-B t -D t 180-B t -D t j

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AAAI04 July 2004MP1-136Pennock/Wellman Basketball as coin flips Detroit score Boston score actual price binomial price $1 iff B T -D T >5.5

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AAAI04 July 2004MP1-137Pennock/Wellman Explain the market Parallel IR IEM Giuliani NY Senate 2000 cancer, prostate, prostate cancer, … ny.politicsWashington Post cancer, from prostate, is suffering from, …,diagnosis, … prostate cancer,... lazio, rick lazio,... rep rick lazio, … lazio, rick lazio, rick, …, rep rick lazio, … Use expected entropy loss to determine the key words and phrases that best differentiate between text streams before and after the date of interest [Pennock 2002]

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AAAI04 July 2004MP1-138Pennock/Wellman Explain the market Parallel IR IEM Gore US Pres 2000 florida, ballots, recount, palm beach, ballot, beach county, palm beach county… us.politics FX Extraterrestrial Life meteorite, life, evidence, martian meteorite, primitive, gibson, organic, of possible, martian, life on mars,... sci.space.news

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AAAI04 July 2004MP1-139Pennock/Wellman 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...

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AAAI04 July 2004MP1-140Pennock/Wellman Play-money market games

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AAAI04 July 2004MP1-141Pennock/Wellman 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

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AAAI04 July 2004MP1-142Pennock/Wellman 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

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AAAI04 July 2004MP1-143Pennock/Wellman Put-call parity stock price s - call price + put price = strike price k stock price s put 30 = max[0,20-s] payoff buy stock k=20 call 30 = max[0,s-20] - call 30 = - max[0,s-20]

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AAAI04 July 2004MP1-144Pennock/Wellman 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

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AAAI04 July 2004MP1-145Pennock/Wellman Internal coherence HSX vs IEM Arbitrage closure for HSX award options Arbitrage closure on IEM qualitatively similar to HSX, though quantitatively more efficient

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AAAI04 July 2004MP1-146Pennock/Wellman Forecast accuracy: HSX 0.94 correlation Comparable to expert forecasts at Box Office Mojo

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AAAI04 July 2004MP1-147Pennock/Wellman Combining forecasts HSX + Box Office Mojo (expert forecast) Correlation of errors: corrav errav%errfit HSX BOMojo avg avg-max

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AAAI04 July 2004MP1-148Pennock/Wellman Probabilistic forecasts HSX 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

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AAAI04 July 2004MP1-149Pennock/Wellman Avg logarithmic score forecast sourceavg log score Feb 19 HSX prices DPRoberts Fielding-1.04 expert consensus-1.05 Feb 18 HSX prices-1.08 Tom-1.08 John-1.22 Jen-1.25 HSX Oscar options 2000

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AAAI04 July 2004MP1-150Pennock/Wellman Probabilistic forecasts FX Prices 30 days before expiration Similar results: –60 days before –specific date Average logarithmic score FX

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AAAI04 July 2004MP1-151Pennock/Wellman Real markets vs. market games HSX IEM average log score arbitrage closure

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AAAI04 July 2004MP1-152Pennock/Wellman Real markets vs. market games HSXFX, F1P6 probabilistic forecasts expected value forecasts 489 movies forecast sourceavg 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

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AAAI04 July 2004MP1-153Pennock/Wellman 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 –6 th and 8 th respectively Markets beat most of the ~2,000 contestants –Average of experts came 39 th Forthcoming, Electronic Markets, Emile Servan-Schreiber, Justin Wolfers, David Pennock and Brian Galebach

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AAAI04 July 2004MP1-154Pennock/Wellman

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AAAI04 July 2004MP1-155Pennock/Wellman Does money matter? Play vs real, head to head Statistically: TS ~ NF NF >> Avg TS > Avg

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AAAI04 July 2004MP1-156Pennock/Wellman 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?

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AAAI04 July 2004MP1-157Pennock/Wellman 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

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AAAI04 July 2004MP1-158Pennock/Wellman 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

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AAAI04 July 2004MP1-159Pennock/Wellman 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

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4. Lab experiments & theory A. Laboratory experiments, field tests B. Theoretical underpinnings 1.Rational expectations 2.Efficient markets hypothesis 3.No-Trade Theorems 4.Information aggregation

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AAAI04 July 2004MP1-161Pennock/Wellman 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

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AAAI04 July 2004MP1-162Pennock/Wellman 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 $1 if X$1 if Y$1 if Z ?Z 1 price of Z time 0

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AAAI04 July 2004MP1-163Pennock/Wellman Plott & Sunder 1982 Three disjoint exhaustive states X,Y,Z Three securities Some see samples of joint; can infer P(Z|samples) Results less definitive $1 if X$1 if Y$1 if Z ?P(XYZ) 1 price of Z time 0

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AAAI04 July 2004MP1-164Pennock/Wellman 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 $1 if X$1 if Y$1 if Z not Xnot Y 1 price of Z time 0

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AAAI04 July 2004MP1-165Pennock/Wellman 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 $1 if Z not Xnot Y 1 price of Z time 0

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AAAI04 July 2004MP1-166Pennock/Wellman 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 Xnot Y 1 price of Z time 0

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AAAI04 July 2004MP1-167Pennock/Wellman Small groups In small, illiquid markets, information aggregation can fail Chen, Fine, & Huberman [EC-2001] propose a two stage process 1. Trade in a market to assess participants risk attitude and predictive ability 2. Query participants probabilities using the log score; compute a weighted average of probabilities, with weights derived from step 1

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AAAI04 July 2004MP1-168Pennock/Wellman Small groups [Source: Fine DARPA Workshop, 2002]

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AAAI04 July 2004MP1-169Pennock/Wellman 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 < sales < $10K $1 iff $20K < sales < $30K $1 iff $10K $30K Employees could trade at lunch, weekends, for real $$ Market predictions beat official HP forecasts

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AAAI04 July 2004MP1-170Pennock/Wellman 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

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AAAI04 July 2004MP1-171Pennock/Wellman Efficient market hypotheses (EMH) Internal coherence prices are self-consistent or arbitrage-free Weak form: Internal unpredictability future prices unpredictable from past prices Semi-strong form: Unpredictability future prices unpredictable from all public info Strong form: Expert-level accuracy unpredictable from all public & private info; experts cannot outperform naïve traders More: stronger assumps

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AAAI04 July 2004MP1-172Pennock/Wellman 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

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AAAI04 July 2004MP1-173Pennock/Wellman 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 1982][Aumann 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

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AAAI04 July 2004MP1-174Pennock/Wellman 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

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AAAI04 July 2004MP1-175Pennock/Wellman A theory of info aggregation Notation 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: p t Likelihood price: l t = p t /(1-p t ) Log-likelihood price: ll t = ln l t $1 if A [Pennock 2002]

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AAAI04 July 2004MP1-176Pennock/Wellman Assumptions Efficiency assumption: Let p t be the price of at time t Then Pr(A|p t,p t-1,p t-2,…,p 0 ) = p t (Markov assumpt. + accuracy assumpt.) $1 if A

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AAAI04 July 2004MP1-177Pennock/Wellman Consequences E[p t |p t-1 = x] = x E[p t |p t-1 = x] = x expected 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 Pr(ll t =x+ |A,ll t-1 =x) Pr(ll t =x+ |A,ll t-1 =x) = e = e

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AAAI04 July 2004MP1-178Pennock/Wellman Consequences Pr(p t =y|A,p t-1 =x) = Pr(p t =y|p t-1 =x) Pr(p t =y|A,p t-1 =x) = Pr(p t =y|p t-1 =x) 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 E[p t |A,p t-1 =x] = x + Var(p t |p t-1 =x) x yxyxyxyx

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AAAI04 July 2004MP1-179Pennock/Wellman Empirical verification Distribution of changes in log-likelihood price over 22 IEM markets, consistent with theory Distribution of changes in log-likelihood price of winning candidates divided by losing candidates. Line is e, as predicted by theory

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AAAI04 July 2004MP1-180Pennock/Wellman Avg log score dynamics FX HSX WSE bball WSE soccer IEM

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AAAI04 July 2004MP1-181Pennock/Wellman 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

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AAAI04 July 2004MP1-182Pennock/Wellman 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 k t flips –For A to occur, n/2-it more tails are needed p t =Pr(A|i t,k t ) = (1/2) n-k t n-k t j j=n/2-i t n-k t

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AAAI04 July 2004MP1-183Pennock/Wellman Avg log score dynamics FX HSX WSE bball WSE soccer IEM coin flip model

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5. Characterizing information aggregation A. Market as an opinion pool B. Market as a composite agent 1. Market belief, utility 2. Market Bayesian updates 3. Market adaptation, dynamics C. Paradoxes, impossibilities 1. Opinion pool impossibilities 2. Composite agent non-existence

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AAAI04 July 2004MP1-185Pennock/Wellman Y B R Aggregating beliefs Bush wins 2004 YHOO stock > 30 Rain tomorrow Y B R Y B R

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AAAI04 July 2004MP1-186Pennock/Wellman Opinion pools (1959-) Linear (LinOP): weighted arithmetic mean –Pr 0 ( ) = w 1 Pr 1 ( ) + + w n Pr n ( ) –w i are expert weights Logarithmic (LogOP): wtd geometric mean –Pr 0 ( ) [Pr 1 ( )] w1 [Pr n ( )] wn Supra Bayesian –Pr 0 ( | Pr 1 Pr n ) Pr sb (Pr 1 Pr n | )Pr sb ( )

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AAAI04 July 2004MP1-187Pennock/Wellman p + p = 1 p p = p etc... No arbitrage (No Dutch books) (No risk-free profits) p, p, p, p, p Subjective probability de Finetti (1937)

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AAAI04 July 2004MP1-188Pennock/Wellman p + p = 1 p p = p etc... No arbitrage (No Dutch books) (No risk-free profits) p, p, p, p, p Consensus probability at market equilibrium

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AAAI04 July 2004MP1-189Pennock/Wellman $1 if E 1 $1 if E 2 $1 if E S A Market Model Pr 1, u 1 subjective probability utility for money Pr i, u i Pr n, u n Competitive equilibrium prices p, p,… consensus belief

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AAAI04 July 2004MP1-190Pennock/Wellman Advantages Explicit incentives for participation, honesty, and to gather evidence No central coordinator Well defined protocols Library of economic tools to aid in analysis Sparse communications Allows for limited privacy Risk-neutral probabilities agree at equil

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AAAI04 July 2004MP1-191Pennock/Wellman Risk-neutral probability Behavior is the product of Pr and u –max a Pr( ) u(a, ) An observer cannot determine Pr or u –Agent A with Pr f( ) and u/f( ) is equivalent to agent A with Pr and u Pr RN Pr * u u RN u/u

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AAAI04 July 2004MP1-192Pennock/Wellman Trading with risk-neutral probability A RN agent would buy if p < Pr(E) Any agent would buy if p < Pr RN (E) Any agent would sell if p > Pr RN (E) If Pr i RN (E) Pr j RN (E) then i and j would desire to trade At equilibrium, all agents risk-neutral probabilities agree, & equal prices $1 if E

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AAAI04 July 2004MP1-193Pennock/Wellman Constant absolute risk aversion (CARA): u i (y)=-e -c i y Disjoint events If Equilibrium prices compute LogOP Expert weights are normalized measure of risk tolerance Then Market LogOP [Pr i (E j )] i p i=1 N

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AAAI04 July 2004MP1-194Pennock/Wellman If Market LinOP Generalized log utility for money (GLU): u i (y)=ln(y+b i ) Disjoint events Equilibrium prices compute LinOP Then i Pr i (E j ) p = i=1 N

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AAAI04 July 2004MP1-195Pennock/Wellman Composite agent If CARA or GLU Disjoint events Then Total demand for each security equals that of a rational individual Beliefs equal the equilibrium prices Super-agent is less risk averse than any individual Then

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AAAI04 July 2004MP1-196Pennock/Wellman Market Adaptation Single security Multiperiod market Agents with GLU Fixed beliefs belief time successes:0 trials:0successes:0 trials:1successes:1 trials:2successes:2 trials:3successes:2 trials:4successes:3 trials:5successes:5 trials:10successes:10 trials:20successes:17 trials:30successes:24 trials:40successes:30 trials:50 price frequency wealth Beta(1,1) wealth Beta(1,2) wealth Beta(2,2) wealth Beta(3,2) wealth Beta(3,3) wealth Beta(4,3) wealth Beta(6,6) wealth Beta(11,11) wealth Beta(18,14) wealth Beta(25,17) wealth Beta(31,21)

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AAAI04 July 2004MP1-197Pennock/Wellman Learning from prices Pr 1 (E 1 ), Pr 1 (E 2 ),… Supra Bayesian Pr 2 (E 1 ), Pr 2 (E 2 ),…

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AAAI04 July 2004MP1-198Pennock/Wellman Learning from prices Supra Bayesian Pr(E 1 ), Pr(E 2 ),… p, p,… Pr 1 (E 1 ), Pr 1 (E 2 ),… Pr 2 (E 1 ), Pr 2 (E 2 ),…

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AAAI04 July 2004MP1-199Pennock/Wellman s successes in n trials Bernoulli trials model s successes in n trials E Ê Ê E Ê E E Pr(E|p ) = wPr(E) + (1-w)p where w Market s successes in n trials n n+n

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AAAI04 July 2004MP1-200Pennock/Wellman Equilibrium with Learning Weighted average update Pr(E|p ) = wPr(E) + (1-w)p and agents with GLU still LinOP prices confidence-based wts Geometric average update Pr(E|p ) Pr(E) w (p ) (1-w) and agents with CARA still LogOP prices

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AAAI04 July 2004MP1-201Pennock/Wellman Market Dynamics Agents with GLU Weighted average belief update: Pr(E t |p ) = 0.2 Pr(E t ) p price frequency n t=1 n-t (1 E t ) n t=1 n-t (1 E t )+ n t=1 n-t (1 Ê t ) wealth Beta(52,62) price discounted frequency belief time

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AAAI04 July 2004MP1-202Pennock/Wellman Market Dynamics Agents with CARA Mixed populations belief wealth GLU wealth CARA wealth Beta(8,4)

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AAAI04 July 2004MP1-203Pennock/Wellman Impossibility theorems Combining probabilities: Pr 0 = f(Pr 1,Pr 2,...,Pr n ) Properties / axioms: –Non-dictatorship (ND) –Proportional Dependence on States (PDS) Pr 0 ( ) f(Pr 1 ( ), Pr 2 ( ), …, Pr n ( )) –Independence Preservation Property (IPP) & &

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AAAI04 July 2004MP1-204Pennock/Wellman Impossibility theorems Combining probabilities: Pr 0 = f(Pr 1,Pr 2,...,Pr n ) Properties / axioms: –Non-dictatorship (ND) –Marginalization property (MP) f(A B) + f(A B) = f(A) aggregate, marginalize = marginalize, aggregate –Externally Bayesian (EB) f(A|B) = f(A B) / f(B) condition, aggregate = aggregate, condition

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AAAI04 July 2004MP1-205Pennock/Wellman Market impossibilities Market is just another function f –Sometimes weighted algebraic/geometric avg –In general, arbitrary non-linear fn Still, subject to all the same paradoxes, impossibilities, limitations In some cases, a composite agent does not exist [due to Pratt, described in Raiffa 1968] –market: flipping a coin can help overall utility –individual: flipping a coin never helps

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6. Computational aspects A. Combinatorics 1. Compact securities markets 2. Combinatorial securities markets 3. Compound securities markets 4. Market scoring rules 5. Dynamic pari-mutuel market 6. Policy Analysis Market B. Distributed market computation

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AAAI04 July 2004MP1-207Pennock/Wellman Complete securities markets 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 For all practical purposes, | | = 2 n securities is intractable

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AAAI04 July 2004MP1-208Pennock/Wellman Complete securities markets Problems –Space complexity: Cant even write down all securities, store all prices, quantities, etc. –Liquidity: Too many securities dividing traders attention. Bounded rationality cant possible explore, let alone optimize over all securities Solution approaches –Find subset of securities that are (nearly) sufficient for given agents: 1. Compact markets –Define mechanisms to match expressive bids: 2. Combinatorial mkts 3. Compound markets –Automated market maker 4. Market scoring rules 5. Dynamic pari-mutuel

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AAAI04 July 2004MP1-209Pennock/Wellman Do we really need all these securities? Under what conditions are fewer than | |-1 securities sufficient Intuitively, many features of state of nature have nothing to do with each other. Idea: maybe we can expoit (conditional) independence among events Compact securities markets [Pennock & Wellman 2000]

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AAAI04 July 2004MP1-210Pennock/Wellman Joint probability distribution Exploiting independence –Pr(R Y B) = Pr(R) Pr(Y) Pr(B) –8 states 3 prob. values Independence Y R B XX 2 3 = 8 states 7 prob. values Rain tomorrow YHOO stock > 30 Bin Laden captured

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AAAI04 July 2004MP1-211Pennock/Wellman Conditional independence YHOO stock > 30 Budget surplus > 0 Interest rate < 1% Pr(Y I B) = Pr(Y|I) Pr(I|B) Pr(B) 8 states 5 assessments

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AAAI04 July 2004MP1-212Pennock/Wellman Pr(E 6 |E 3 E 5 ) Pr(E 6 |E 3 Ê 5 ) Pr(E 6 |Ê 3 E 5 ) Pr(E 6 |Ê 3 Ê 5 ) Pr(E 6 |pa(E 6 )) Bayesian networks E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 Conditional independence encoded in graph structure. Factors joint distribution into product of conditionals. Example: 13 rather than 63 prob values.

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AAAI04 July 2004MP1-213Pennock/Wellman Conditional security: –Pays off $1 if E 1 & E 2 occur –Lose price paid (p ) if Ê 1 & E 2 –Bet called off if Ê 2 $1 if E 1 |E 2 Conditional securities

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AAAI04 July 2004MP1-214Pennock/Wellman Bayes-net structured markets Pr(E 6 |E 3 E 5 ) Pr(E 6 |E 3 Ê 5 ) Pr(E 6 |Ê 3 E 5 ) Pr(E 6 |Ê 3 Ê 5 ) E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 $1 if E 6 |E 3 E 5 $1 if E 6 |Ê 3 Ê 5 $1 if E 6 |E 3 Ê 5 $1 if E 6 |Ê 3 E 5 Securities markets can be structured analogously to a BN One (conditional) security for each CPT entry Fully connected BN complete market

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AAAI04 July 2004MP1-215Pennock/Wellman Compact markets? Idea: Include securities markets according to conditional probs in Bayesian network Problem: Agents may disagree about independence structure E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 $1 if E 6 |E 3 E 5 $1 if E 6 |Ê 3 Ê 5 $1 if E 6 |E 3 Ê 5 $1 if E 6 |Ê 3 E 5

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AAAI04 July 2004MP1-216Pennock/Wellman Compact markets (II)? Structure market according to unanimously agreed-upon independencies E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 $1 if E 6 |E 3 E 5 $1 if E 6 |Ê 3 Ê 5 $1 if E 6 |E 3 Ê 5 $1 if E 6 |Ê 3 E 5

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AAAI04 July 2004MP1-217Pennock/Wellman agent Belief and willingness-to-pay E $x if E Pr(E)=0.4 pxpx =0.4 lim x 0

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AAAI04 July 2004MP1-218Pennock/Wellman Prior Stakes $1000 if E E $x if E Pr(E)=0.4 pxpx =0.3 lim x 0 (risk-averse) agent Therefore, trading behavior may not reveal true independencies

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AAAI04 July 2004MP1-219Pennock/Wellman Risk-neutral probability Behavior is the product of Pr and u –max a Pr( ) u(a, ) An observer cannot determine Pr or u –Agent A with Pr f( ) and u/f( ) is equivalent to agent A with Pr and u Pr RN Pr * u u RN u/u

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AAAI04 July 2004MP1-220Pennock/Wellman Trading with risk-neutral probability A RN agent would buy if p < Pr(E) Any agent would buy if p < Pr RN (E) Any agent would sell if p > Pr RN (E) If Pr i RN (E) Pr j RN (E) then i and j would desire to trade At equilibrium, all agents risk-neutral probabilities agree, & equal prices $1 if E

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AAAI04 July 2004MP1-221Pennock/Wellman Compact markets (III)? Structure market according to unanimously agreed-upon risk-neutral independencies $1 if E 6 |E 3 E 5 $1 if E 6 |E 3 Ê 5 $1 if E 6 |Ê 3 Ê 5 $1 if E 6 |Ê 3 E 5 E2E2 E5E5 E3E3 E6E6 E4E4 E1E1

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AAAI04 July 2004MP1-222Pennock/Wellman Operationally complete securities markets If, in equil, all RN indep agree with market structure mkt is operationally complete –Pareto optimal allocation of risk –supports all desirable trades, but not all conceivable RN independencies change out of equilibrium; perhaps more arguable basis for agreement on true independencies

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AAAI04 July 2004MP1-223Pennock/Wellman Decomposable networks E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 Bayesian network Fill-in BN with edge between every pair of nodes with common child. E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 Bayesian network E2E2 E5E5 E3E3 E6E6 E4E4 E1E1 Markov network moralization

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AAAI04 July 2004MP1-224Pennock/Wellman CARA & Markov indep risk-neutral indep If all agents have CARA, then market structured as TRIANGULATE [ n i=1 MORALIZE (D i )] is op complete Can still yield exponential savings This example: 19 vs. 63 Compact markets (IV) Independency markets $1 if E 6 |E 3 E 5 $1 if E 6 |E 3 Ê 5 $1 if E 6 |Ê 3 Ê 5 $1 if E 6 |Ê 3 E 5 E2E2 E5E5 E3E3 E6E6 E4E4 E1E1

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AAAI04 July 2004MP1-225Pennock/Wellman Independence-preserving aggregation & & Structural unanimity Proportional dependence on states Pr 0 ( ) f(Pr 1 ( ), Pr 2 ( ), …, Pr n ( )) Unanimity Nondictatorship

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AAAI04 July 2004MP1-226Pennock/Wellman Summary Under certain theoretical conditions, structured markets are optimal, with exponentially fewer securities than would otherwise be required Could have application in creating new derivatives markets that allow agents to hedge more of their risks, w/o combinatorial explosion of fin instruments

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AAAI04 July 2004MP1-227Pennock/Wellman Combinatorial auctions E.g.: spectrum rights, computer system, … n goods bids allowed 2 n combinations Maximizing revenue: NP-hard (set packing) Enter computer scientists (hot topic)… Survey: [Vries & Vohra 02]

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AAAI04 July 2004MP1-228Pennock/Wellman Combinatorial auctions (Some) research issues Preference elicitation [Sandholm 02] Bidding languages [Nissan 00] & restrictions [Rothkopf 98] Approximation –relation to incentive compatibility [Lehmann 99] and bounded rationality [Nisan & Ronen 00] False-name bidders [Yokoo 01] Winner determination –GVA (VCG) mechs, iterative mechs [Parkes 99, Wurman 00]; smart markets [Brewer 99] –integer programming; specialized heuristics [Sandholm 99] FCC spectrum auctions Optimal auction design [Ronen 01] More: [Vries & Vohra 02]

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AAAI04 July 2004MP1-229Pennock/Wellman Combined value trading Traders are often interested in portfolios (bundles) rather than individual assets –Buy Apple, sell Microsoft –Sell Dallas $9, Sacramento $33, San Antonio $28, LA $11 Sell Western Division $81 Buy Eastern $19 Esp. in thin markets there is execution risk: price of one asset may change while others are executed CVT: Combinatorial auction mechanism for assets –Traders can submit conditional orders, that are filled only if other related orders are also filled –Essentially can request bundles [Bossaerts, Fine, Ledyard 2002]

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AAAI04 July 2004MP1-230Pennock/Wellman Combined value trading Form of bids –For $33, buy 6 units A & 2 units B; fill fraction F=1/3 Means will accept any fraction of the portfolio 1/3 I.e., if F=1/3, will accept $11 for 2 of A & 2/3 of B or $22 for 4 of A & 4/3 of B, etc. –For -$3 (receive $3), sell 4 units of C & buy 3 of D; F=1 All or nothing offer Computational problem –If all F=0, linear programming polynomial –If F>0, mixed linear, integer programming NP-hard Moreover, prices may not exist Discriminative pricing [Bossaerts, Fine, Ledyard 2002]

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AAAI04 July 2004MP1-231Pennock/Wellman Thick markets [Source: Ledyard, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-232Pennock/Wellman Thin markets, no CVT [Source: Ledyard, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-233Pennock/Wellman Thin markets, CVT [Source: Ledyard, DARPA Workshop, 2002]

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AAAI04 July 2004MP1-234Pennock/Wellman Market combinatorics [Thanks: Wolfers, Fortnow]

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AAAI04 July 2004MP1-235Pennock/Wellman [Thanks: Wolfers, Fortnow] What about Pr(CA ^ AZ) ? Pr(CA | AZ) ? Pr(Elec | FL) ? Pr((IL^NJ) ( IL^ NJ)) ? Not derivable as a linear combinations of base securities possible functions Only 2 50 securities needed to span space Market combinatorics

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AAAI04 July 2004MP1-236Pennock/Wellman Info mkt combinatorics UN action casualties Bin Laden captured Turkey action SARS US leaves Iraq oil prices Afghanistan

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AAAI04 July 2004MP1-237Pennock/Wellman Info mkt combinatorics An A2 A1 Ai A6 A4 A3 A5 … … binary variables Note: E[A]=Pr(a)

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AAAI04 July 2004MP1-238Pennock/Wellman Market combinatorics In principle, markets in all possible combinations will get you everything you want In practice, this is infeasible Its also unnatural $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to: $1 if A1&A2&…&An I am entitled to:

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AAAI04 July 2004MP1-239Pennock/Wellman Compound securities [Fortnow EC-2003] A bidding language: write your own security For example Offer to buy/sell q units of it at price p Let everyone else do the same Auctioneer must decide who trades with whom at what price… How? (next) More concise/expressive; more natural $1 if Boolean_fn | Boolean_fn I am entitled to: $1 if A1 | A2 I am entitled to: $1 if (A1&A7)||A13 | (A2||A5)&A9 I am entitled to: $1 if A1&A7 I am entitled to:

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AAAI04 July 2004MP1-240Pennock/Wellman The matching problem There are many possible matching rules for the auctioneer A natural one: maximize trade subject to no-risk constraint Example: –buy 1 of for $0.40 –sell 1 of for $0.10 –sell 1 of for $0.20 No matter what happens, auctioneer cannot lose money $1 if A1 $1 if A1&A2 trader gets $$ in state: A1A2 A1A2 A1A2 A1A

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AAAI04 July 2004MP1-241Pennock/Wellman The matching problem Another way to look at it: Logical reduction| Example: –buy 1 of for $0.40 –sell 1 of for $0.10 –sell 1 of for $0.20 || Clear match btw buy and sell| $1 if A1 $1 if A1&A2 $1 if A1 = sell for $0.3

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AAAI04 July 2004MP1-242Pennock/Wellman The matching problem Divisible orders: will accept any q* q Indivisible: will accept all or nothing Let =all possible combinations; | |=2 n Let i be fraction of order i filled Let i be payoff for order i in state Div. MP: Does i [0,1],, - i i 0 Indiv. MP: Does i {0,1},, - i i 0 Optimizations –max trade; max percent orders filled –max auctioneer utility subject to no-risk –max auctioneer utility -- with risk (market maker) (at least 1 i > 0)

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AAAI04 July 2004MP1-243Pennock/Wellman Divisible vs. indivisible Sell 1 of A1 at $0.50 Buy 1 of (A1&A2) | (A1 || A2) at $0.50 Buy 1 of A1|A2 at $0.40 trader gets $$ in state: A1A2 A1A2 A1A2 A1A

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AAAI04 July 2004MP1-244Pennock/Wellman Divisible vs. indivisible Sell 1 of A1 at $0.50 Buy 1 of (A1&A2) | (A1 || A2) at $0.50 Buy 1 of A1|A2 at $0.40 trader gets $$ in state: A1A2 A1A2 A1A2 A1A

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AAAI04 July 2004MP1-245Pennock/Wellman Divisible vs. indivisible Sell 1 of A1 at $0.50 Buy 1 of (A1&A2) | (A1 || A2) at $0.50 Buy 1 of A1|A2 at $0.40 trader gets $$ in state: A1A2 A1A2 A1A2 A1A

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AAAI04 July 2004MP1-246Pennock/Wellman Divisible vs. indivisible Sell 1 of A1 at $0.50 Buy 1 of (A1&A2) | (A1 || A2) at $0.50 Buy 1 of A1|A2 at $0.40 trader gets $$ in state: A1A2 A1A2 A1A2 A1A

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AAAI04 July 2004MP1-247Pennock/Wellman Divisible vs. indivisible Sell 1 of A1 at $0.50 Buy 1 of (A1&A2) | (A1 || A2) at $0.50 Buy 1 of A1|A2 at $0.40 trader gets $$ in state: A1A2 A1A2 A1A2 A1A /5 x 1 x divisible match!

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AAAI04 July 2004MP1-248Pennock/Wellman Complexity results Divisible orders: will accept any q* q Indivisible: will accept all or nothing Natural algorithms –divisible: linear programming –indivisible: integer programming; logical reduction? # eventsdivisibleindivisible O(log n)polynomialNP-complete O(n) co-NP-complete 2 p complete reduction from SAT reduction from X3C reduction from T BF Fortnow; Kilian; Sami LP

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AAAI04 July 2004MP1-249Pennock/Wellman Open questions Other matching rules –maximize utility subject to no-risk –maximize utility (market maker) What to do with the surplus –can be in cash and leftover securities –auctioneer keeps surplus –surplus is shared back among traders, auctioneer; how? Trader optimization problem –how to choose securities, ps, qs, subject to limits/penalties for number, complexity of bids –ultimately a game-theoretic question Approximate algorithms, heuristics Incentive properties

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AAAI04 July 2004MP1-250Pennock/Wellman The problem of liquidity Too many markets => Too little trade per market (thin, illiquid, large bid/ask spread) Combinatorial markets/CVT: trader attention is limited, each market may get few bids Compound markets: may be few matches Automated market maker ensures liquidity –Market scoring rules [Hanson 2002] –Dynamic pari-mutuel market [Pennock 2004]

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AAAI04 July 2004MP1-251Pennock/Wellman Scoring rule Logarithmic scoring rule (there are others) Recall pay an expert approach: –Offer to pay the expert $100 + log r if $100 + log (1-r) if Expert should choose r=Pr(A), given caveats = 6 6

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AAAI04 July 2004MP1-252Pennock/Wellman Market scoring rule [Hanson 2002] System maintains a complete joint probability distribution over all variables –Exponential space –Might use Bayes net or other compact representation, introduces complications Anyone at any time who thinks the probabilities are wrong, can change them by accepting a scoring rule payment Trader must agree to pay off the previous person who changed the probabilities

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AAAI04 July 2004MP1-253Pennock/Wellman Market scoring rule A1A2 A1A log(.2) 100+log(.2) 100+log(.3) 100+log(.3) 100+log(.25) 100+log(.25) log(.2/.25) log(.2/.25) log(.3/.25) log(.3/.25) Trader can change to: Trader gets $$ in state: Trader pays $$ in state: total transaction: current probabilities: Example Requires a patron, though only pays final trader, & payment is bounded

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AAAI04 July 2004MP1-254Pennock/Wellman Market scoring rule Note, a trader can change any part of the joint distribution, e.g. P(A 1 |A 3 ); no need to specify all Conceptually, to traders it appears as if a market maker always stands willing to accept an (infinitesimal) trade at current prices Full cost for some quantity is the integral over instantaneous prices, solvable in closed form for log scoring rule

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AAAI04 July 2004MP1-255Pennock/Wellman Accuracy Estimates per trader Market Scoring Rules Simple Info Markets thin market problem Scoring Rules opinion pool problem [Source: Hanson, 2002]

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AAAI04 July 2004MP1-256Pennock/Wellman Computational issues Straightforward approach requires exponential space for prices, holdings, portfolios Could represent probabilities using a Bayes net or other compact representation; changes must keep distribution in the same representational class Could use multiple overlapping patrons, each with bounded loss. Limited arbitrage could be obtained by smart traders exploiting inconsistencies between patrons A B C F E D H G [Source: Hanson, 2002]

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AAAI04 July 2004MP1-257Pennock/Wellman RIP Policy Analysis Market Real combinatorial markets in Middle East issues DARPA, Net Exchange, Caltech, GMU Two year field test, starts 2003 Open to public, real-money markets ~20 nations, 8 quarters, ~5 variables each: –Economic, political, military, US actions Want many combos (> states) Legal: DARPA & its agents not under CFTCs regulatory umbrella (paraphrased) [Source: Hanson, 2002]

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AAAI04 July 2004MP1-258Pennock/Wellman RIP Policy Analysis Market Killed in a single day under congressional/press firestorm Misunderstood as betting on terrorism After initial outrage, good side began to appear in media. Comments & compilations: –http://hanson.gmu.edu/policyanalysismarket.html –http://dpennock.com/pam.html All press is good press: Has drawn attention to prediction markets, spurned private sector development

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AAAI04 July 2004MP1-259Pennock/Wellman Dynamic pari-mutuel market Standard PM: Every $1 bet is the same DPM: Value of each $1 bet varies depending on the status of wagering at the time of the bet Encode dynamic value with a price –price is $ to buy 1 share of payoff –price of A is lower when less is bet on A –as shares are bought, price rises; price is for an infinitesimal share; cost is integral

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AAAI04 July 2004MP1-260Pennock/Wellman $3.27 Dynamic pari-mutuel market Outcomes: A B Current payoff/shr:$5.20$0.97 ABAB $1.00 $1.25 $1.50 $3.00 sell sell buy buy $3.25 $3.27 $0.25 $0.50 $0.75 sell sell buy $0.85 market maker traders

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AAAI04 July 2004MP1-261Pennock/Wellman How are prices set? A price function pri(n) gives the instantaneous price of an infinitesimal additional share beyond the nth Cost of buying n shares: Different assumptions lead to different price functions, each reasonable

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AAAI04 July 2004MP1-262Pennock/Wellman Mechanism comparison no riskliquiditydynamic info aggreg. payoff vector fixed damped volatility CDA N/A CDAwMM N/A PM N/A DPM ** MSR * *Technically has risk, but bounded **One-sided liquidity

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AAAI04 July 2004MP1-263Pennock/Wellman An info market model: Computational properties From a computational perspective, we are interested in: –What can a market compute? –How fast? (time complexity) i.e., What mechanisms or protocols lead to faster convergence to the rational expectations equilibrium? –Using how many securities? (expressivity and representational compactness) i.e., What market structures require a minimum of securities yet still aggregate information quickly and accurately?

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AAAI04 July 2004MP1-264Pennock/Wellman Market computation [Feigenbaum EC-2003] General formulation –Set up the market to compute some function f(x 1,x 2,…,x n ) of the information x i available to each market participant (e.g., we want the market to compute future interest rates given other economic variables) –Represent f(x) as a circuit –Questions How do we set up a market to compute f? How quickly can the market compute f? AND XOR OR x1x1 x2x2 x3x3 x4x4 f(x 1,x 2,x 3,x 4 )= (x 1 x 2 ) (x 3 x 4 )

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AAAI04 July 2004MP1-265Pennock/Wellman Market model Each participant has some bit of information x i There is a security F that pays off $1 if and only if f(x)=1 at some future date, and $0 otherwise. Trading occurs in synchronous rounds –In each round, participants bid their true expectation –Clearing price is determined using a simplified Shapley-Shubik trading model, yielding mean bid Questions we ask/answer: –Does the clearing price converge to a stable value? –How fast does it converge (in how many rounds)? –Does the stable price of F reveal the true value of f?

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AAAI04 July 2004MP1-266Pennock/Wellman Theorems For any prior distribution on x, if f(x) takes the form of a weighted threshold function (i.e., f(x) = 1 iff i w i x i > 1 for some weights w i ), then the market price will ultimately converge to the true value of f(x) in at most n rounds If f(x) cannot be expressed as a weighted threshold function (i.e., f(x) is not linearly separable), then there is some prior on x for which the price of F is stuck at $0.5 indefinitely, and does not reveal the true value of f(x)

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AAAI04 July 2004MP1-267Pennock/Wellman Interpretation of theory: –1 security supports computation of threshold fn only –More complex functions must utilize more securities: # of securities required = threshold circuit size of f In the example, with only a single security on f, the market may not converge Example and interpretation AND XOR OR x1x1 x2x2 x3x3 x4x4 f(x 1,x 2,x 3,x 4 ) $1 if (x 1 x 2 ) (x 3 x 4 ) $1 if x 3 x 4 With 2 additional securities it will converge in 4 rounds $1 if x 4

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AAAI04 July 2004MP1-268Pennock/Wellman Extensions, future work Dynamic information revelation and changes Overcoming false information Obtaining incentive compatibility Modeling agent strategies Modeling overlapping information sources Characterizing in terms of work/round Bayesian network representation of prior Dealing with limited-precision prices

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AAAI04 July 2004MP1-269Pennock/Wellman Open questions What is the relationship between our model and perceptron (neural network) learning? –Perceptrons exactly compute threshold functions –Could envision a system to learn smallest set of threshold functions to approximate desired function f, thereby minimizing the number of securities required Can alternate market protocols lead to faster convergence? Can subsidies speed convergence? What can other types of securities (e.g., nonbinary securities) compute?

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AAAI04 July 2004MP1-270Pennock/Wellman Legal issues Regulatory bodies: Commodity Futures Trading Commission (CFTC), Securities and Exchange Commission (SEC) –IEM has no action letter from CFTC Financial institutions regularly create customized derivatives to hedge risks Generally setting up markets for hedging is legal; setting up markets strictly for information gathering may be gambling; CFTC regulated not gambling No logical distinction: Trading options betting on Oscars Playing Roulette - sum game

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AAAI04 July 2004MP1-271Pennock/Wellman Legal issues Gambling in US –Legal in some form in 48 states (lotteries, bingo, Indian reservations, riverboat) ironically, by far worst E[return] –Illegal in many forms in all states Sports betting legal only in Las Vegas Federal Wire Act: bans the use of telephones to accept wagers on sporting events. Computers? Non-sports? The central questionwhether Internet gambling is legal, illegal or exists in a legal nether world where no rules applyis as gray as lawyers can make it. [MSNBC]

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AAAI04 July 2004MP1-272Pennock/Wellman Legal issues Gambling in US (contd) –Several bills are going through Congress, both for outlawing/restricting, legalizing/regulating –Some states (Nevada, New Jersey) are considering legalizing online gambling –So-called skill-based games are OK! WorldWinner.com includes BlackJack, Trivia

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AAAI04 July 2004MP1-273Pennock/Wellman Legal issues Gambling in UK –online gambling: recently legalized, regulated, tax-free (temporary), growing fast Caribbean –Legal, less well regulated –WSE co-founder arrested upon return to US; intends to challenge law in court; only arrest for offshore bookmaker accepting bets from US –No individual US bettor has ever been charged Great collection of articles:

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