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  1. 1. Markets in Uncertainty: Risk, Gambling, and Information Aggregation a tutorial by David M. Pennock Michael P. Wellman pennockd@yahoo-inc.com [email_address] dpennock.com ai.eecs.umich.edu/people/wellman presented at the 19th National Conference on Artificial Intelligence, July 2004, San Jose, CA, USA MP1-
  2. 2. Outline <ul><li>Overview tour 15 min What is a “market in uncertainty”? </li></ul><ul><li>Background 30 min </li></ul><ul><li>Single agent perspective </li></ul><ul><ul><li>Subjective probability </li></ul></ul><ul><ul><li>Utility, risk, and risk aversion </li></ul></ul><ul><ul><li>Decision making under uncertainty </li></ul></ul><ul><li>Multiagent perspective </li></ul><ul><ul><li>Trading/allocating risk </li></ul></ul><ul><ul><li>Pareto optimality </li></ul></ul><ul><ul><li>Securities: Markets in uncertainty </li></ul></ul>
  3. 3. Outline <ul><li>Mechanisms, examples and empirical studies 45 min </li></ul><ul><ul><li>What & how: Instruments & mechanisms </li></ul></ul><ul><ul><li>Real-money markets: Examples & evaluations </li></ul></ul><ul><ul><ul><li>Iowa Electronic Market </li></ul></ul></ul><ul><ul><ul><li>Options </li></ul></ul></ul><ul><ul><ul><li>TradeSports: Effects of war </li></ul></ul></ul><ul><ul><ul><li>Horse racing, sports betting </li></ul></ul></ul><ul><ul><li>Play-money markets </li></ul></ul>
  4. 4. Outline <ul><li>Lab experiments and theory 20 min </li></ul><ul><ul><li>Laboratory experiments, field tests </li></ul></ul><ul><ul><li>Theoretical underpinnings </li></ul></ul><ul><ul><ul><li>Rational expectations </li></ul></ul></ul><ul><ul><ul><li>Efficient markets hypothesis </li></ul></ul></ul><ul><ul><ul><li>No-Trade Theorems </li></ul></ul></ul><ul><ul><ul><li>Information aggregation </li></ul></ul></ul>
  5. 5. Outline <ul><li>Characterizing information 20 min aggregation </li></ul><ul><ul><li>Market as an opinion pool </li></ul></ul><ul><ul><li>Market as a “composite agent” </li></ul></ul><ul><ul><ul><li>Market belief, utility </li></ul></ul></ul><ul><ul><ul><li>Market Bayesian updates </li></ul></ul></ul><ul><ul><ul><li>Market adaptation, dynamics </li></ul></ul></ul><ul><ul><li>Paradoxes, impossibilities </li></ul></ul><ul><ul><ul><li>Opinion pool impossibilities </li></ul></ul></ul><ul><ul><ul><li>Composite agent non-existence </li></ul></ul></ul>
  6. 6. Outline <ul><li>Computational aspects 60 min </li></ul><ul><ul><li>Combinatorics </li></ul></ul><ul><ul><ul><li>Compact securities markets </li></ul></ul></ul><ul><ul><ul><li>Combinatorial securities markets </li></ul></ul></ul><ul><ul><ul><li>Compound securities markets </li></ul></ul></ul><ul><ul><ul><li>Market scoring rules </li></ul></ul></ul><ul><ul><ul><li>Dynamic pari-mutuel market </li></ul></ul></ul><ul><ul><ul><li>Policy Analysis Market </li></ul></ul></ul><ul><ul><li>Distributed market computation </li></ul></ul><ul><li>Legal issues; miscellaneous 5 min </li></ul><ul><li>Discussion, Q&A 15 min </li></ul>
  7. 7. 1. Overview tour What is a “market in uncertainty” ?
  8. 8. A market in uncertainty <ul><li>Take a random variable, e.g. </li></ul><ul><li>Turn it into a financial instrument payoff = realized value of variable </li></ul>US’04Pres = Bush? 2004 CA Earthquake? = 6 ? = 6 $1 if  6 $0 if I am entitled to:
  9. 9. Aside: Terminology <ul><li>Key aspect: payout is uncertain </li></ul><ul><li>Called variously: asset, security, contingent claim, derivative (future, option), stock, prediction market, information market, gamble, bet, wager, lottery </li></ul><ul><li>Historically mixed reputation </li></ul><ul><ul><li>Esp. gambling aspect </li></ul></ul><ul><ul><li>A time when options were frowned upon </li></ul></ul><ul><li>But when regulated serve important social roles... </li></ul>
  10. 10. Why? Reason 1 <ul><li>Get information </li></ul><ul><li>price  expectation of random variable (in theory, lab experiments, empirical studies, ...more later) </li></ul><ul><li>Do you have a random variable whose expectation you’d like to know? A market in uncertainty can probably help </li></ul>
  11. 11. Why? Reason 1: Information <ul><li>“Information market”: financial mechanism designed to obtain estimates of expectations of random variables </li></ul><ul><li>Easy as 1, 2, 3: </li></ul><ul><ul><li>Take a random variable whose expectation you’d like to know </li></ul></ul><ul><ul><li>Turn it into a financial instrument (payoff= realized value of variable) </li></ul></ul><ul><ul><li>Open a market in the financial instrument </li></ul></ul><ul><ul><li> price(t)  E t [X] (in many cases, ... more later) </li></ul></ul>
  12. 12. Getting information <ul><li>Non-market approach: ask an expert </li></ul><ul><ul><li>How much would you pay for this? </li></ul></ul><ul><li>A: $5/36  $0.1389 </li></ul><ul><ul><li>caveat: expert is knowledgeable </li></ul></ul><ul><ul><li>caveat: expert is truthful </li></ul></ul><ul><ul><li>caveat: expert is risk neutral, or ~ RN for $1 </li></ul></ul><ul><ul><li>caveat: expert has no significant outside stakes </li></ul></ul>= 6 $1 if  6 $0 if I am entitled to:
  13. 13. Getting information <ul><li>Non-market approach: pay an expert </li></ul><ul><ul><li>Ask the expert for his report r of the probability P( ) </li></ul></ul><ul><ul><li>Offer to pay the expert </li></ul></ul><ul><ul><ul><li>$100 + log r if </li></ul></ul></ul><ul><ul><ul><li>$100 + log (1-r) if </li></ul></ul></ul><ul><li>It so happens that the expert maximizes expected profit by reporting r truthfully </li></ul><ul><ul><li>caveat: expert is knowledgeable </li></ul></ul><ul><ul><li>caveat: expert is truthful </li></ul></ul><ul><ul><li>caveat: expert is risk neutral, or ~ RN </li></ul></ul><ul><ul><li>caveat: expert has no significant outside stakes </li></ul></ul>“ logarithmic scoring rule”, a “proper” scoring rule = 6 = 6  6
  14. 14. Getting information <ul><li>Market approach: “ask” the public—experts & non-experts alike—by opening a market: </li></ul><ul><li>Let any person i submit a bid order: an offer to buy q i units at price p i </li></ul><ul><li>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 ) </li></ul><ul><li>Match up agreeable trades (many poss. mechs...) </li></ul>= 6 $1 if  6 $0 if I am entitled to: = 6
  15. 15. Getting information <ul><li>Market approach: “ask” the public—experts & non-experts alike—by opening a market: </li></ul><ul><li>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 } </li></ul><ul><li>In equilibrium (no trades) </li></ul><ul><ul><li>max bid p i < min ask p j = “bid-ask spread” </li></ul></ul><ul><ul><li>bounds aggregate public opinion of expectation </li></ul></ul>= 6 $1 if  6 $0 if I am entitled to:
  16. 16. Aside: Mechanism alternatives <ul><li>This is the continuous double auction (CDA) </li></ul><ul><li>Many other market & auction mechanisms work: </li></ul><ul><ul><li>call market </li></ul></ul><ul><ul><li>pari-mutuel market </li></ul></ul><ul><ul><li>market scoring rules </li></ul></ul><ul><ul><li>CDA w/ market maker </li></ul></ul><ul><ul><li>Vegas bookmaker, others </li></ul></ul><ul><li>Key: Market price = aggregate estimate of expected value </li></ul>[Hanson 2002]
  17. 17. (Real) Great expectations <ul><li>For dice example, no need for market: E[x] is known; no one should disagree </li></ul><ul><li>Real power comes for non-obvious expectations of random variables, e.g. </li></ul>$1 if ; $0 otherwise I am entitled to: $x if interest rate = x on Jan 1, 2004 I am entitled to:
  18. 18. $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:
  19. 19. http://tradesports.com
  20. 20. http://www.biz.uiowa.edu/iem I P O http://www.wsex.com/
  21. 21. Play money; Real expectations http://www.hsx.com/
  22. 22. http://us.newsfutures.com/ Cancer cured by 2010 Machine Go champion by 2020 http://www.ideosphere.com
  23. 23. Does it work? Yes... <ul><li>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.: </li></ul><ul><ul><li>Markets like the Iowa Electronic Market predict election outcomes better than polls [Forsythe 1992, 1999][Oliven 1995][Rietz 1998][Berg 2001][Pennock 2002] </li></ul></ul><ul><ul><li>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] </li></ul></ul><ul><ul><li>Sports betting markets provide accurate forecasts of game outcomes [Gandar 1998][Thaler 1988][Debnath EC’03][Schmidt 2002] </li></ul></ul>
  24. 24. Does it work? Yes... <ul><li>E.g. (cont’d): </li></ul><ul><ul><li>Laboratory experiments confirm information aggregation [Plott 1982;1988;1997][Forsythe 1990][Chen, EC-2001] </li></ul></ul><ul><ul><li>And field tests [Plott 2002] </li></ul></ul><ul><ul><li>Theoretical underpinnings: “rational expectations” [Grossman 1981][Lucas 1972] </li></ul></ul><ul><ul><li>Procedural explanation: agents learn from prices [Hanson 1998][Mckelvey 1986][Mckelvey 1990][Nielsen 1990] </li></ul></ul><ul><ul><li>Proposals to use information markets to help science [Hanson 1995] , policymakers, decision makers [Hanson 1999] , government [Hanson 2002] , military [DARPA FutureMAP, PAM] </li></ul></ul><ul><ul><li>Even market games work! [Servan-Schreiber 2004][Pennock 2001] </li></ul></ul>
  25. 25. Why? Reason 2 <ul><li>Manage risk </li></ul><ul><li>If is horribly terrible for you Buy a bunch of and if happens, you are compensated </li></ul>= 6 = 6 $1 if  6 $0 if I am entitled to: = 6
  26. 26. Why? Reason 2 <ul><li>Manage risk </li></ul><ul><li>If is horribly terrible for you Buy a bunch of and if happens, you are compensated </li></ul>$1 if $0 if I am entitled to:
  27. 27. The flip-side of prediction: Hedging (Reason 2) <ul><li>Allocate risk (“hedge”) </li></ul><ul><ul><li>insured transfers risk to insurer, for $$ </li></ul></ul><ul><ul><li>farmer transfers risk to futures speculators </li></ul></ul><ul><ul><li>put option buyer hedges against stock drop; seller assumes risk </li></ul></ul><ul><li>Aggregate information </li></ul><ul><ul><li>price of insurance  prob of catastrophe </li></ul></ul><ul><ul><li>OJ futures prices yield weather forecasts </li></ul></ul><ul><ul><li>prices of options encode prob dists over stock movements </li></ul></ul><ul><ul><li>market-driven lines are unbiased estimates of outcomes </li></ul></ul><ul><ul><li>IEM political forecasts </li></ul></ul>
  28. 28. Reason 2: Manage risk <ul><li>What is insurance? </li></ul><ul><ul><li>A bet that something bad will happen! </li></ul></ul><ul><ul><li>E.g., I’m betting my insurance co. that my house will burn down; they’re betting it won’t. Note we might agree on P(burn)! </li></ul></ul><ul><ul><li>Why? Because I’ll be compensated if the bad thing does happen </li></ul></ul><ul><li>A risk-averse agent will seek to hedge (insure) against undesirable outcomes </li></ul>
  29. 29. E.g. stocks, options, futures, insurance, ..., sports bets, ... <ul><li>Allocate risk (“hedge”) </li></ul><ul><ul><li>insured transfers risk to insurer, for $$ </li></ul></ul><ul><ul><li>farmer transfers risk to futures speculators </li></ul></ul><ul><ul><li>put option buyer hedges against stock drop; seller assumes risk </li></ul></ul><ul><ul><li>sports bet may hedge against other stakes in outcome </li></ul></ul><ul><li>Aggregate information </li></ul><ul><ul><li>price of insurance  prob of catastrophe </li></ul></ul><ul><ul><li>OJ futures prices yield weather forecasts </li></ul></ul><ul><ul><li>prices of options encode prob dists over stock movements </li></ul></ul><ul><ul><li>market-driven lines are unbiased estimates of outcomes </li></ul></ul><ul><ul><li>IEM political forecasts </li></ul></ul>
  30. 30. Examples <ul><li>I buy MSFT stock at s . I’m 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 </li></ul><ul><li>I’m a farmer. I’m afraid corn prices will go too low. I buy corn futures to lock in a price today. </li></ul>
  31. 31. Examples <ul><li>I own a house in CA. I’m afraid of earthquakes. I pay an insurance premium so that, if an earthquake happens, I am compensated. </li></ul><ul><li>I am an Oscar-nominated actor. I’m afraid I’m 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…  ) </li></ul>
  32. 32. What am I buying? <ul><li>When you hedge/insure, you pay to reduce the unpredictability of future wealth </li></ul><ul><li>Risk-aversion: All else being equal, prefer certainty to uncertainty in future wealth </li></ul><ul><li>Typically, a less risk-averse party (e.g., huge insurance co, futures speculator) assumes the uncertainty (risk) in return for an expected profit </li></ul>
  33. 33. On hedging and speculating <ul><li>Hedging is an act to reduce uncertainty </li></ul><ul><li>Speculating is an act to increase expected future wealth </li></ul><ul><li>A given agent engages in a (largely inseparable) mixture of the two </li></ul><ul><li>Both can be encoded together as a maximization of expected utility , where utility is a function of wealth, ... more later </li></ul>
  34. 34. On trading <ul><li>Why would two parties agree to trade in a “market in uncertainty”? </li></ul><ul><ul><li>They disagree on expected values (prob’s) </li></ul></ul><ul><ul><li>They differ in their risk attitude or exposure – they trade to reallocate risk </li></ul></ul><ul><ul><li>Both (most likely) </li></ul></ul><ul><li>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. </li></ul>
  35. 35. On computational issues <ul><li>Information aggregation is a form of distributed computation </li></ul><ul><li>Agent level </li></ul><ul><ul><li>nontrivial optimization problem, even in 1 market; ultimately a game-theoretic question </li></ul></ul><ul><ul><li>probability representation, updating algorithm (Bayes net) </li></ul></ul><ul><ul><li>decision representation, algorithm (POMDP) </li></ul></ul><ul><ul><li>agent problem’s computational complexity, algorithms, approximations, incentives </li></ul></ul>some 
  36. 36. On computational issues <ul><li>Mechanism level </li></ul><ul><ul><li>Single market </li></ul></ul><ul><ul><ul><li>What can a market compute? </li></ul></ul></ul><ul><ul><ul><li>How fast (time complexity)? </li></ul></ul></ul><ul><ul><ul><li>Do some mechanisms converge faster (e.g., subsidy) </li></ul></ul></ul><ul><ul><li>Multiple markets </li></ul></ul><ul><ul><ul><li>How many securities to compute a given fn? How many secs to support “sufficient” social welfare? (expressivity and representational compactness) </li></ul></ul></ul><ul><ul><ul><li>Nontrivial combinatorics (auctioneer’s computational complexity; algorithms; approximations; incentives) </li></ul></ul></ul>some 
  37. 37. On computational issues <ul><li>Machine learning, data mining </li></ul><ul><ul><li>Beat the market (exploiting combinatorics?) </li></ul></ul><ul><ul><li>Explain the market, information retrieval </li></ul></ul><ul><ul><li>Detect fraud </li></ul></ul>some 
  38. 38. 2. Background <ul><li>Single agent perspective </li></ul><ul><ul><li>Subjective probability </li></ul></ul><ul><ul><li>Utility, risk, and risk aversion </li></ul></ul><ul><ul><li>Decision making under uncertainty </li></ul></ul><ul><li>Multiagent perspective </li></ul><ul><ul><li>Trading/allocating risk </li></ul></ul><ul><ul><li>Pareto optimality </li></ul></ul><ul><ul><li>Securities: markets in uncertainty </li></ul></ul>
  39. 39. Decision making under uncertainty <ul><li>How should agents behave (make decisions, choose actions) when faced with uncertainty? </li></ul><ul><li>Decision theory: Prescribes maximizing expected utility </li></ul>
  40. 40. Why reason about uncertainty? <ul><li>Propositional logic: No uncertainty Could never “explain” seatbelt use </li></ul><ul><li>Decisions: D - drive car S - wear seatbelt </li></ul><ul><ul><li>Events: A - accident occurs </li></ul></ul><ul><ul><ul><li>A   D </li></ul></ul></ul><ul><ul><ul><li> A   S </li></ul></ul></ul><ul><ul><ul><li>Can’t explain DS </li></ul></ul></ul><ul><ul><ul><li>Key: A is uncertain </li></ul></ul></ul>
  41. 41. Why Bayesian uncertainty? <ul><li>E.g. You can buy skis for $b Or you can rent for $b/k, k>1 </li></ul><ul><li>Worst-case analysis: Rent for k days, then buy You’ll spend at most $2b </li></ul><ul><li>But what if you strongly believe you’ll ski more than k times?  Buy earlier </li></ul><ul><li>That k+1st time is your last?  Don’t buy </li></ul><ul><li>Expected (utility) case often more appropriate </li></ul>
  42. 42. Decision making under uncertainty, an example ABC TV’s “Who Wants to be a Millionaire?”
  43. 43. Decision making under uncertainty, an example <ul><li>v 15 = $1,000,000 if correct $32,000 if incorrect $500,000 if walk away </li></ul>
  44. 44. Decision making under uncertainty, an example <ul><li>if you answer: </li></ul><ul><li>E[v 15 ] = $1,000,000 *Pr(correct) + $32,000 *Pr(incorrect) </li></ul><ul><li>if you walk away: </li></ul><ul><li>$500,000 </li></ul>
  45. 45. Decision making under uncertainty, an example <ul><li>if you answer: </li></ul><ul><li>E[v 15 ] = $1,000,000 *0.5 $32,000 *0.5 </li></ul><ul><li>= $516,000 </li></ul><ul><li>if you walk away: </li></ul><ul><li>$500,000 </li></ul><ul><li>you should answer, right? </li></ul>
  46. 46. Decision making under uncertainty, an example <ul><li>Most people won’t answer: risk averse </li></ul><ul><li>U($x) = log($x) </li></ul><ul><li>if you answer: </li></ul><ul><li>E[u 15 ] = log($1,000,000) *0.5 +log($32,000) *0.5 </li></ul><ul><li>= 6/2+4.5/2 = 5.25 </li></ul><ul><li>if you walk away: </li></ul><ul><li>log($500,000) = 5.7 </li></ul>
  47. 47. Decision making under uncertainty, an example <ul><li>Maximizing E[u i ] for i<15 more complicated </li></ul>Q7, L={1,3} walk answer L1  0.4 X 0.6 log($2k) Q7, L={3} Q8, L={1,3} log($1k) walk answer L3  0.8 X 0.2 Q8, L={3} log($1k) log($2k) L3
  48. 48. Decision making under uncertainty, in general  =set of all possible future states of the world
  49. 49. Decision making under uncertainty, in general <ul><li> are disjoint exhaustive states of the world </li></ul><ul><ul><li> i : rain tomorrow & Bush elected & Y! stock up & car not stolen & ... </li></ul></ul><ul><ul><li> j : rain tomorrow & Bush elected & Y! stock up & car stolen & ... </li></ul></ul> 1  2  3    i  |  |
  50. 50. Decision making under uncertainty, in general <ul><li>Equivalent, more natural: </li></ul><ul><ul><li>E i : rain tomorrow E j : Bush elected </li></ul></ul><ul><ul><li>E k : Y! stock up </li></ul></ul><ul><ul><li>E l : car stolen </li></ul></ul><ul><ul><li>|  |=2 n </li></ul></ul> E 1 E 2 E i E j E n
  51. 51. Decision making under uncertainty, in general <ul><li>Preferences, utility </li></ul><ul><ul><li> i >  j  u(  i ) > u(  j ) </li></ul></ul><ul><li>Expected utility </li></ul><ul><ul><li>E[u] =   Pr(  )u(  ) </li></ul></ul><ul><li>Decisions (actions) can affect Pr(  ) </li></ul><ul><li>What you should do: choose actions to maximize expected utility </li></ul><ul><li>Why?: To avoid being a money pump [de Finetti’74] , among other reasons... </li></ul>
  52. 52. Preference under uncertainty <ul><li>Define a prospect ,  = [p,  1 ;  2 ] </li></ul><ul><li>Given the following axioms of  : </li></ul><ul><ul><li>orderability: (  1   2 )  (  1   2 )  (  1 ~  2 ) </li></ul></ul><ul><ul><li>transitivity: (  1   2 )  (  2   3 )  (  1   3 ) </li></ul></ul><ul><ul><li>continuity:  1   2   3   p.  2 ~ [p,  1 ;  3 ] </li></ul></ul><ul><ul><li>substitution:  1 ~  2  [p,  1 ;  3 ] ~ [p,  2 ;  3 ] </li></ul></ul><ul><ul><li>monotonicity:  1   2  p>q  [p,  1 ;  2 ]  [q,   ;  2 ] </li></ul></ul><ul><ul><li>decomposability: </li></ul></ul><ul><ul><ul><li>[p,  1 ; [q,  2 ;  3 ]] ~ [q, [p,  1 ;  2 ]; [p,  1 ;  3 ]] </li></ul></ul></ul><ul><li>Preference can be represented by a real-valued expected utility function such that: </li></ul><ul><ul><ul><li>u([p,  1 ;  2 ]) = p u(  1 ) + (1–p)u(  2 ) </li></ul></ul></ul>
  53. 53. Utility functions <ul><li>(  a probability distribution over  ) </li></ul><ul><li>E[u]:  represents preferences, </li></ul><ul><li>E[u](  )  E[u](  ) iff    </li></ul><ul><li>Let  (  ) = au(  ) + b, a>0. </li></ul><ul><ul><li>Then E[  ](  ) = E[au+b](  ) = a E[u](  ) + b. </li></ul></ul><ul><ul><li>Since they represent the same preferences,  and u are strategically equivalent (  ~ u). </li></ul></ul>
  54. 54. Utility of money <ul><li>Outcomes are dollars </li></ul><ul><li>Risk attitude: </li></ul><ul><ul><li>risk neutral: u ( x ) ~ x </li></ul></ul><ul><ul><li>risk averse (typical): u concave ( u  ( x ) < 0 for all x ) </li></ul></ul><ul><ul><li>risk prone: u convex </li></ul></ul><ul><li>Risk aversion function: </li></ul><ul><li>r ( x ) = – u  ( x ) / u  ( x ) </li></ul>
  55. 55. Risk aversion & hedging <ul><li>E[u]=.01 (4)+.99 (4.3) = 4.2980 </li></ul><ul><li>Action: buy $10,000 of insurance for $125 </li></ul><ul><li>E[u]=4.2983 </li></ul><ul><li>Even better, buy $5974.68 of insurance for $74.68 </li></ul><ul><li>E[u] = 4.2984  Optimal </li></ul> 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)
  56. 56. Securities market s <ul><li>Note that, in previous example, risk-neutral insurance company also profits: E[v] = .01(-5,900) + 0.99(74.68) = $14.93 Both parties gain from bilateral agreement </li></ul><ul><li>Securities market generalizes this to </li></ul><ul><ul><li>arbitrary states </li></ul></ul><ul><ul><li>more than two parties </li></ul></ul><ul><li>Market mechanism to allocate risk among participants </li></ul>
  57. 57. Pareto optimality <ul><li>An allocation is Pareto optimal iff there does not exist another solution that is </li></ul><ul><ul><li>better for one agent and </li></ul></ul><ul><ul><li>no worse for all the rest. </li></ul></ul>… a minimal (and maximal?) condition for social optimality, or efficiency .
  58. 58. What is traded: Securities <ul><li>Specifies state-contingent returns , r = (r 1 ,…,r |  | ) in terms of numeraire (e.g., $) </li></ul><ul><li>Examples: </li></ul><ul><ul><li>(1,…,1) riskless numeraire ($1) </li></ul></ul><ul><ul><li>(0,…,0,1,0,…,0) pays off $1 in designated state (“Arrow security” for that state) </li></ul></ul><ul><ul><li>r i = 1 if  i  E 1 , r i = 0 otherwise </li></ul></ul>$1 if E 1
  59. 59. Terms of trade: Prices <ul><li>Price p <E i > associated with security </li></ul><ul><ul><li>Relative prices dictate terms of exchange </li></ul></ul><ul><li>Facilitate multilateral exchange via bilateral exchange: </li></ul><ul><ul><li>defines a common scale of resource value </li></ul></ul><ul><li>Can significantly simplify a resource allocation mechanism </li></ul><ul><ul><li>compresses all factors contributing to value into a single number </li></ul></ul><ul><li>A “default interface” for multiagent systems </li></ul>$1 if E i
  60. 60. Equilibrium <ul><li>General ( competitive , Walrasian ) equilibrium describes a simultaneous equilibrium of interconnected markets </li></ul><ul><li>Definition: A price vector and allocation such that </li></ul><ul><ul><li>all agents making optimal demand decisions (positive demand = buy; negative demand = sell) </li></ul></ul><ul><ul><li>all markets have zero aggregate demand (buy volume equals sell volume) </li></ul></ul>
  61. 61. Complete securities market <ul><li>A set of securities is complete if rank of returns matrix = |  |  1 </li></ul><ul><li>For example, set of |  |  1 Arrow securities: “Arrow-Debreu securities market” </li></ul><ul><li>Market with complete set of securities guarantees a Pareto optimal allocation of risk, under classical conditions </li></ul>
  62. 62. Incomplete markets <ul><li>Securities do not span states of nature (always the case in practice) </li></ul><ul><li>Equilibria may exist, but may not be Pareto optimal Example: missed insurance opportunity </li></ul><ul><li>More: “Theory of Incomplete Markets”, Magill & Quinzii, MIT Press, 1998 </li></ul>
  63. 63. Why trade securities? <ul><li>Profit from perceived mispricings </li></ul><ul><ul><li>Price p <E 1 > differs significantly enough from trader’s belief Pr(E 1 ) </li></ul></ul><ul><ul><li>speculation </li></ul></ul><ul><li>Insure against risk </li></ul><ul><ul><li>Trader’s marginal value for wealth in E 1 , relative to p <E 1 > , differs from that in other states </li></ul></ul><ul><ul><li>e.g., home fire insurance </li></ul></ul><ul><ul><li>hedging </li></ul></ul>
  64. 64. Societal roles of security markets <ul><li>From speculation: </li></ul><ul><ul><li>Aggregate beliefs </li></ul></ul><ul><ul><li>Disseminate information </li></ul></ul><ul><li>From hedging: </li></ul><ul><ul><li>Allocate risk </li></ul></ul>
  65. 65. Summary: Background <ul><li>General equilibrium framework for market-based exchange </li></ul><ul><li>Incorporate uncertainty through securities </li></ul><ul><li>Agents trade securities in order to optimize expected utility, thereby: </li></ul><ul><ul><li>Allocating risk </li></ul></ul><ul><ul><li>Reaching “consensus” probabilities </li></ul></ul>
  66. 66. 3. Mechanisms, examples & empirical studies <ul><li>What & how: Instruments & mechanisms </li></ul><ul><li>Real-money markets: Examples & evaluations </li></ul><ul><ul><li>Iowa Electronic Market </li></ul></ul><ul><ul><li>Options </li></ul></ul><ul><ul><li>TradeSports: Effects of war </li></ul></ul><ul><ul><li>Horse racing, sports betting </li></ul></ul><ul><li>Play-money markets </li></ul>
  67. 67. Building a market in uncertainty <ul><li>What is being traded? the “good” </li></ul><ul><li>Define: </li></ul><ul><ul><li>Random variable </li></ul></ul><ul><ul><li>Payoff function </li></ul></ul><ul><ul><li>Payoff output </li></ul></ul><ul><li>How is it traded? the “mechanism” </li></ul><ul><ul><li>Call market </li></ul></ul><ul><ul><li>Continuous double auction </li></ul></ul><ul><ul><li>Continuous double auction w/ market maker </li></ul></ul><ul><ul><li>Pari-mutuel </li></ul></ul><ul><ul><li>Bookmaker </li></ul></ul><ul><ul><li>Combinatorial (later) </li></ul></ul>
  68. 68. What is being traded? <ul><li>Underlying statistic / random variable </li></ul><ul><ul><li>Binary: ; Discrete: </li></ul></ul><ul><ul><li>Continuous: “interest rate”, “dividend flow” </li></ul></ul><ul><ul><li>Clarity: e.g., “Saddam out”, “House burns”, “Gore wins”, “Buchanan wins” </li></ul></ul><ul><li>Payoff function </li></ul><ul><ul><li>Arrow: (0,0,0,1,0) ; Portfolio: (2,4,0,1,0) </li></ul></ul><ul><ul><li>Dividends, options: Max[0,s-k], arbitrary (non-linear) fn </li></ul></ul><ul><li>Payoff output </li></ul><ul><ul><li>dollars, fake money, commodities </li></ul></ul>= 6
  69. 69. How is it traded? <ul><li>Call market </li></ul><ul><ul><li>Orders are collected over a period of time; collected orders are matched at end of period </li></ul></ul><ul><ul><li>One-time or repeated </li></ul></ul><ul><ul><li>Pre-defined or randomized stopping time/rule </li></ul></ul><ul><ul><li>Mth price auction </li></ul></ul><ul><ul><li>M+1st price auction </li></ul></ul><ul><ul><li>k-double auction </li></ul></ul><ul><li>lim period  0: Continuous double auction </li></ul>
  70. 70. A note on selling <ul><li>In a securities market, you can sell what you don’t have: you agree to pay according to terms </li></ul><ul><li>Binary case: sell “$1 if A” for $0.3 </li></ul><ul><ul><li>Receive $0.3 (now, or contractually later), pay $1 if A </li></ul></ul><ul><li>Exactly equivalent to buying “$1 if A ” for $0.7 </li></ul><ul><ul><li>sell “$1 if A” @ $0.3 </li></ul></ul><ul><ul><li>buy “$1 if A ” @ $0.7 </li></ul></ul><ul><li>Alternative: Market institution always stands ready to buy/sell exhaustive bundle for $1.00 </li></ul><ul><ul><li>Iowa Electronic Market </li></ul></ul>A occurs A occurs -1+.3 = -.7 0+.3 = .3 0 -.7 = -.7 1 -.7 = .3
  71. 71. Mth price auction <ul><li>N buyers and M sellers </li></ul><ul><li>Mth price auction: </li></ul><ul><ul><li>sort all bids from buyers and sellers </li></ul></ul><ul><ul><li>price = the Mth highest bid </li></ul></ul><ul><ul><li>let n = # of buy offers >= price </li></ul></ul><ul><ul><li>let m = # of sell offers <= price </li></ul></ul><ul><ul><li>let x = min(n,m) </li></ul></ul><ul><ul><li>the x highest buy offers and x lowest sell offers transact </li></ul></ul>
  72. 72. Call market <ul><li>Buy offers (N=4) </li></ul><ul><li>Sell offers (M=5) </li></ul>$0.15 $0.12 $0.09 $0.05 $0.30 $0.17 $0.13 $0.11 $0.08 = 6 $1 if  6 $0 if
  73. 73. Mth price auction <ul><li>Buy offers (N=4) </li></ul><ul><li>Sell offers (M=5) </li></ul>$0.05 $0.08 $0.09 $0.11 $0.12 $0.13 $0.15 $0.17 $0.30 1 2 3 4 5     <ul><li>Matching buyers/sellers </li></ul>price = $0.12 = 6 $1 if  6 $0 if
  74. 74. M+1st price auction <ul><li>Buy offers (N=4) </li></ul><ul><li>Sell offers (M=5) </li></ul>$0.05 $0.08 $0.09 $0.11 $0.12 $0.13 $0.15 $0.17 $0.30 1 2 3 4 5     <ul><li>Matching buyers/sellers </li></ul>price = $0.11 6 = 6 $1 if  6 $0 if
  75. 75. k-double auction <ul><li>Buy offers (N=4) </li></ul><ul><li>Sell offers (M=5) </li></ul>$0.05 $0.08 $0.09 $0.11 $0.12 $0.13 $0.15 $0.17 $0.30 1 2 3 4 5     <ul><li>Matching buyers/sellers </li></ul>price = $0.11 + $0.01*k 6 = 6 $1 if  6 $0 if
  76. 76. Continuous double auction CDA <ul><li>k-double auction repeated continuously </li></ul><ul><li>buyers and sellers continually place offers </li></ul><ul><li>as soon as a buy offer  a sell offer, a transaction occurs </li></ul><ul><li>At any given time, there is no overlap btw highest buy offer & lowest sell offer </li></ul>
  77. 77. http://tradesports.com
  78. 78. http://www.biz.uiowa.edu/iem http://us.newsfutures.com/ I P O
  79. 79. CDA with market maker <ul><li>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 </li></ul><ul><li>Market maker essentially sets prices; others take it or leave it </li></ul><ul><li>While standard auctioneer takes no risk of its own, market maker takes on considerable risk, has potential for considerable reward </li></ul>
  80. 80. CDA with market maker <ul><li>E.g. World Sports Exchange (WSE): </li></ul><ul><ul><li>Maintains $5 differential between bid & ask </li></ul></ul><ul><ul><li>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.” </li></ul></ul><ul><ul><li>“ The depth of markets can vary with the contest.” </li></ul></ul><ul><ul><li>Also, WSE pauses market & adjusts prices (subjectively?) after major events (e.g., goals) </li></ul></ul><ul><ul><li>http://www.wsex.com/about/interactiverules.html </li></ul></ul>
  81. 81. CDA with market maker <ul><li>E.g. Hollywood Stock Exchange (HSX): </li></ul><ul><ul><li>“ Virtual Specialist” automated market maker </li></ul></ul><ul><ul><li>Always willing to buy & sell at a single point price  no bid-ask spread </li></ul></ul><ul><ul><li>Price moves when buys/sells are imbalanced </li></ul></ul><ul><ul><li>Fake money, so it’s OK if Virtual Specialist loses money – in fact it does [Brian Dearth, personal communication] </li></ul></ul><ul><ul><li>http://www.hsx.com/ </li></ul></ul>
  82. 82. http://www.wsex.com/ http://www.hsx.com/
  83. 83. Bookmaker <ul><li>Common in sports betting, e.g. Las Vegas </li></ul><ul><li>Bookmaker is like a market maker in a CDA </li></ul><ul><li>Bookmaker sets “money line”, or the amount you have to risk to win $100 (favorites), or the amount you win by risking $100 (underdogs) </li></ul><ul><li>Bookmaker makes adjustments considering amount bet on each side &/or subjective prob’s </li></ul><ul><li>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 </li></ul>
  84. 84. Pari-mutuel mechanism <ul><li>Common at horse races, jai alai games </li></ul><ul><li>n mutually exclusive outcome (e.g., horses) </li></ul><ul><li>M 1 , M 2 , …, M n dollars bet on each </li></ul><ul><li>If i wins: all bets on 1, 2, …, i -1, i +1, …, n lose </li></ul><ul><li>All lost money is redistributed to those who bet on i in proportion to amount they bet </li></ul><ul><li>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 ) </li></ul>
  85. 85. Pari-mutuel market <ul><li>E.g. horse racetrack style wagering </li></ul><ul><li>Two outcomes: A B </li></ul><ul><li>Wagers: </li></ul>A B
  86. 86. Pari-mutuel market <ul><li>E.g. horse racetrack style wagering </li></ul><ul><li>Two outcomes: A B </li></ul><ul><li>Wagers: </li></ul> A B
  87. 87. Pari-mutuel market <ul><li>E.g. horse racetrack style wagering </li></ul><ul><li>Two outcomes: A B </li></ul><ul><li>Wagers: </li></ul> A B
  88. 88. Pari-mutuel market <ul><li>E.g. horse racetrack style wagering </li></ul><ul><li>Two outcomes: A B </li></ul><ul><li>2 equivalent ways to consider payment rule </li></ul><ul><ul><li>refund + share of B </li></ul></ul><ul><ul><li>share of total </li></ul></ul> A B $ on B 8 $ on A 4 1+ = 1+ =$3 total $ 12 $ on A 4 = = $3
  89. 89. Pari-mutuel market <ul><li>Before race begins, “odds” are reported, or the amount you would win per dollar if betting ended now </li></ul><ul><ul><li>Horse A: $1.2 for $1; Horse B: $25 for $1; … etc. </li></ul></ul><ul><li>Normalized odds = consensus probabilities </li></ul><ul><li>Actual payoffs depend only on final odds, not odds at time of bet: incentive to wait </li></ul><ul><li>In practice “track” takes 17% first, then redistributes what remains </li></ul>
  90. 90. Examples of markets <ul><li>Continuous double auction (CDA) </li></ul><ul><ul><li>Iowa Electronic Market (IEM) </li></ul></ul><ul><ul><li>TradeSports, experimental Soccer market </li></ul></ul><ul><ul><li>Financial markets: stocks, options, derivatives </li></ul></ul><ul><li>CDA with market maker </li></ul><ul><ul><li>World Sports Exchange (WSE) </li></ul></ul><ul><ul><li>Hollywood Stock Exchange (HSX) </li></ul></ul><ul><li>Pari-mutuel: horse racing </li></ul><ul><li>Bookmaker: NBA point spread betting </li></ul>
  91. 91. Example: IEM Iowa Electronic Market http://www.biz.uiowa.edu/iem $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
  92. 92. Example: IEM Iowa Electronic Market http://www.biz.uiowa.edu/iem $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
  93. 93. IEM vote share market as of 4/22/2003 $ 1  vote share o f 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
  94. 94. IEM vote share market as of 7/25/2004 $ 1  vote share o f 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 o f Bush v. Dean $1  vote share of Dean
  95. 95. Example: IEM 1992 [Source: Berg, DARPA Workshop, 2002]
  96. 96. Example: IEM [Source: Berg, DARPA Workshop, 2002]
  97. 97. Example: IEM [Source: Berg, DARPA Workshop, 2002]
  98. 98. Example: IEM [Source: Berg, DARPA Workshop, 2002]
  99. 99. Example: IEM [Source: Berg, DARPA Workshop, 2002]
  100. 100. Speed: TradeSports 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 Alou’s catch. Still 1 out. The Marlins proceed to hit 8 runs in the 8 th inning [Source: Wolfers 2004]
  101. 101. The marginal trader [Forsythe 1992,1999; Oliven 1995; Rietz 1998] <ul><li>Individuals in IEM are biased, make mistakes </li></ul><ul><ul><li>Democrats buy too many Democratic stocks </li></ul></ul><ul><ul><li>Arbitrage is left on the table </li></ul></ul><ul><ul><li>When there are multiple equivalent trades, the cheapest is not always chosen </li></ul></ul><ul><li>Yet market as a whole is accurate, efficient </li></ul><ul><li>Why? Prices are set by “marginal” traders, not average traders </li></ul><ul><ul><li>Marginal traders are: active traders, price setters, unbiased, better performers </li></ul></ul>
  102. 102. Forecast error bounds [Berg 2001] <ul><li>Single market gives E[x] </li></ul><ul><ul><li>IEM winner takes all: P(candidate wins) = P(C) </li></ul></ul><ul><ul><li>IEM vote share: E[candidate vote share] = E[V] </li></ul></ul><ul><li>Can we get error bounds? e.g. Var[x]? </li></ul><ul><li>Yes: combine the two markets </li></ul>P(C)=0.6 <ul><ul><li>WTA gives P(C) = P(V>0.5) </li></ul></ul>E[V]=0.55 <ul><ul><li>Vote share gives mean of dist </li></ul></ul>vote share 0.50 <ul><ul><li>Assume e.g. normal dist of votes </li></ul></ul><ul><ul><li>Report 95% confidence intervals = error bounds </li></ul></ul>
  103. 103. Evaluating accuracy: Recall log scoring rule <ul><li>Logarithmic scoring rule (one of several “proper” scoring rules) </li></ul><ul><li>“Pay an expert approach”: </li></ul><ul><ul><li>Offer to pay the expert </li></ul></ul><ul><ul><ul><li>$100 + log r if </li></ul></ul></ul><ul><ul><ul><li>$100 + log (1-r) if </li></ul></ul></ul><ul><li>Expert should choose r=Pr(A), given caveats </li></ul>= 6  6 X X Note: still works as a “tax”
  104. 104. Evaluating accuracy <ul><li>Log score gives incentives to be truthful </li></ul><ul><li>But log score is also an appropriate measure of expert’s accuracy </li></ul><ul><li>Experts who are better probability assessors will earn a higher avg log score over time </li></ul><ul><li>We advocate: evaluate the “market” just as you would evaluate an individual expert </li></ul><ul><li>For a given market (person), compute average log score over many assessments </li></ul>
  105. 105.  log score =  information <ul><li>Log score dynamics also shows speed of information incorporation </li></ul><ul><li>Expected log score = P(A) log P(A) + P( A ) log P( A ) = - entropy </li></ul><ul><li>Actual log score at time t </li></ul><ul><ul><li>= - amount market is “surprised” by true outcome </li></ul></ul><ul><ul><li>= - # of bits of info provided by revelation of true outcome </li></ul></ul><ul><li>As bits of info flow into market, log score  </li></ul>
  106. 106. Avg log score dynamics FX HSX WSE bball WSE soccer IEM
  107. 107. Avg log score 22 IEM political markets Average log score =  i log (p i )/N p i : i th winner’s normalized price
  108. 108. Example: options <ul><li>Options prices (partially) encode a probability distribution over their underlying stocks </li></ul><ul><ul><li>Arbitrary derivative  P(underlying asset) </li></ul></ul>stock price s 10 20 30 40 50 payoff call 20 = max[0,s-20] call 30 = max[0,s-30] call 40 = max[0,s-40]
  109. 109. Example: options <ul><li>Options prices (partially) encode a probability distribution over their underlying stocks </li></ul><ul><ul><li>Arbitrary derivative  P(underlying asset) </li></ul></ul>stock price s 10 20 30 40 50 payoff - 2*call 30 “ butterfly spread” call 20 + call 40
  110. 110. Example: options <ul><li>Options prices (partially) encode a probability distribution over their underlying stocks </li></ul><ul><ul><li>Arbitrary derivative  P(underlying asset) </li></ul></ul>stock price s 10 20 30 40 50 payoff - 2*call 40 call 30 + call 50
  111. 111. Example: options <ul><ul><li>call 10 - 2 call 20 + call 30 = $2.13 relative </li></ul></ul><ul><ul><li>call 30 - 2 call 30 + call 40 = $5.73 likelihood of falling </li></ul></ul><ul><ul><li>call 30 - 2 call 40 + call 50 = $3.54 near center </li></ul></ul>stock price s 10 20 30 40 50 payoff $2.13 $5.73 $3.54
  112. 112. Example: options <ul><li>More generally, uses prices as constraints E[Max[0,s-10]]=p 10 ; E[Max[0,s-20]]=p 20 ; ... etc. </li></ul><ul><li>Fit to assumed distribution; or maximize {entropy, smothness, etc.} subject to constraints </li></ul>stock price s 10 20 30 40 50 probability [Jackwerth 1996]
  113. 113. Example: TradeSports [Source: Wolfers 2004]
  114. 114. [Source: Wolfers 2004]
  115. 115. [Source: Wolfers 2004]
  116. 116. [Source: Wolfers 2004]
  117. 117. State Price Distribution [Source: Wolfers 2004]
  118. 118. State Price Distribution: War and Peace [Source: Wolfers 2004]
  119. 119. Example: horse racing <ul><li>Pari-mutuel mechanism </li></ul><ul><li>Normalized odds match objective frequencies of winning very closely </li></ul><ul><ul><li>3:1 horses win about twice as much as 6:1 horses, etc. </li></ul></ul><ul><li>Slight favorite-longshot bias (favorites are better bets; extremely rarely E[return] > 0) </li></ul><ul><li>[Ali 77; Rosett 65; Snyder 78; Thaler 88; Weitzman 65] </li></ul>
  120. 120. Example: horse racing <ul><li>Tracks can be biased, e.g., “Winning Colors”, a S Californian horse, 1988 Kentucky Derby: </li></ul><ul><ul><li>$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] </li></ul></ul><ul><li>Some teams apparently make more than a decent living “beating the track” using computer models: e.g., Bill Benter’s team in Hong Kong </li></ul><ul><ul><li>logistic regression standard; now SVMs [Edelman 2003] </li></ul></ul><ul><ul><li>http://www.unr.edu/gaming/confer.asp </li></ul></ul><ul><ul><li>http://www.wired.com/wired/archive/10.03/betting_pr.html </li></ul></ul>
  121. 121. Example: sports betting <ul><li>US NBA Basketball </li></ul><ul><ul><li>Closing lines set by “market” are unbiased estimates of game outcomes  better than opening lines set by experts [Gandar 98] </li></ul></ul><ul><li>Soccer (European football) Experimental market in Euro 2000 Championship [Schmidt 2002] </li></ul><ul><ul><li>Market prediction > betting odds > random </li></ul></ul><ul><ul><li>Market “confidence” statistically meaningful </li></ul></ul>
  122. 122. World Sports Exchange: WSE <ul><li>Online “in-game” sports betting markets </li></ul><ul><li>Trading allowed continuously throughout game: as goals are scored, penalties are called, etc.  i.e. as information is revealed! </li></ul><ul><ul><li>National Basketball Association (NBA) </li></ul></ul><ul><ul><li>Soccer World Cup </li></ul></ul><ul><ul><li>MLB, NHL, golf, others… </li></ul></ul><ul><li>http://wsex.com </li></ul><ul><li>[Debnath, EC-2003] </li></ul>Same concept, better site:
  123. 123. Soccer World Cup 2002 <ul><li>15 Soccer markets (June 7–15, 2002) </li></ul><ul><li>Several 1st round and 2nd round games </li></ul><ul><li>All games ended without penalty shoot-out </li></ul><ul><li>Scores recorded from www.LiveScore.com </li></ul><ul><li>Sampled the stream of price and score information every 10 seconds </li></ul>
  124. 124. Ex: Price reaction to goals <ul><li>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 . </li></ul>
  125. 125. Ex: Price reaction to goals <ul><li>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 </li></ul>
  126. 126. Avg log score & entropy
  127. 127. Delay Calculation Where : Timestamp of scoring : Timestamp of price update : Delay in updating score + network delay : Delay in updating the price + network delay
  128. 128. Reaction time after goals
  129. 129. NBA 2002 <ul><li>18 basketball markets during 2002 Championships (May 6–31, 2002) </li></ul><ul><li>Score recorded from www.SportsLine.com </li></ul><ul><li>Sampled the stream of price and score information every 10 seconds </li></ul>
  130. 130. Correlation between price and score <ul><li>San Antonio vs. LA Lakers (May 07, 2002, Final Score: 88-85, Correlation: 0.93). </li></ul>
  131. 131. Correlation between price and score
  132. 132. Avg log score & entropy
  133. 133. Soccer vs. NBA Soccer World Cup 2002 NBA Championship 2002
  134. 134. Soccer vs. NBA <ul><li>Soccer characteristics </li></ul><ul><ul><li>Price does not change very often </li></ul></ul><ul><ul><li>Price change is abrupt & immediate after goal </li></ul></ul><ul><ul><li>Average entropy decreases gradually toward 0 </li></ul></ul><ul><ul><li>Comebacks less likely  more surprising when they occur </li></ul></ul><ul><li>Basketball characteristics </li></ul><ul><ul><li>Price changes very often by small amounts </li></ul></ul><ul><ul><li>Price is well correlated with scoring </li></ul></ul><ul><ul><li>More uncertainty until late in the games </li></ul></ul><ul><ul><ul><li>entropy > 0.7 for 77% of game; >0.8 for 55.5% of game </li></ul></ul></ul><ul><ul><li>More “exciting” late  outcome is unclear until near end </li></ul></ul>
  135. 135. Basketball as coin flips <ul><li>Model scoring as a series of coin flips </li></ul><ul><ul><li>tails = Boston + 1 </li></ul></ul><ul><ul><li>heads = Detroit + 1 </li></ul></ul><ul><li>Current scores: B t ,D t </li></ul><ul><li>Final scores: B T ,D T </li></ul><ul><li>Compute P(B T -D T > 5.5 | B t ,D t ) </li></ul><ul><li>E[D + B] = 180 </li></ul><ul><li>E[B - D] = 5.5 </li></ul><ul><li>E[B]=92.75;E[D]=87.25 </li></ul><ul><li>p = P(tails) = P(Boston) = 92.75/180 = 0.515 </li></ul>May10 505 DETROIT o/u 180 07:00 506 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
  136. 136. Basketball as coin flips Detroit score Boston score actual price “binomial” price $1 iff B T -D T >5.5
  137. 137. “Explain the market” Parallel IR IEM Giuliani NY Senate 2000 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] “ cancer”, “prostate”, “prostate cancer”, … ny.politics Washington Post “ cancer”, “from prostate”, “is suffering from”, …,“diagnosis”, … “ prostate cancer”, ... “ lazio”, “rick lazio”, ... “rep rick lazio”, … “ lazio”, “rick lazio”, “rick”, …, “ rep rick lazio”, …
  138. 138. “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
  139. 139. Applications & future work <ul><li>Monitoring dynamics </li></ul><ul><ul><li>Automatic explanations </li></ul></ul><ul><ul><li>Low probability event detection </li></ul></ul><ul><ul><li>Sporting events: auto highlights, auto summary, attention scheduling, finding turning points, most exciting games/moments, modeling different sports... </li></ul></ul>
  140. 140. Play-money market games http://www.hsx.com/ http://www.ideosphere.com/ http://www.newsfutures.com/ http://www.ipreo.com/ http://www.100world.com/ http://www.incentivemarkets.com/
  141. 141. Play-money market games <ul><li>Many studies show that prices in real-money markets provide accurate likelihoods </li></ul><ul><li>Researchers credit monetary incentives/risk </li></ul><ul><li>Can play money markets provide accurate forecasts? </li></ul><ul><li>Incentives in market games may derive from entertainment value, educational value, competitive spirit, bragging rights, prizes </li></ul>
  142. 142. Market games analyzed <ul><li>Hollywood Stock Exchange (HSX) </li></ul><ul><ul><li>Play-money market in movies and stars </li></ul></ul><ul><ul><li>Movie stocks; movie options </li></ul></ul><ul><ul><li>Award options (e.g., Oscar options) </li></ul></ul><ul><li>Foresight Exchange (FX) </li></ul><ul><ul><li>Market game to bet on developments in science & technology; e.g., Cancer cured by 2010; Higgs boson verified; Water on moon; Extraterrestrial life verified </li></ul></ul><ul><li>NewsFutures </li></ul><ul><ul><li>Newsworthy events; items of pop interest </li></ul></ul>
  143. 143. Put-call parity <ul><li>stock price s - call price + put price = strike price k </li></ul>stock price s 10 20 30 40 50 payoff put 30 = max[0,20-s] buy stock k=20 call 30 = max[0,s-20] - call 30 = - max[0,s-20]
  144. 144. Internal coherence: HSX <ul><li>Prices of movie stocks and options adhere to put-call parity, as in real markets </li></ul><ul><li>Arbitrage loopholes disappear over time, as in real markets </li></ul>
  145. 145. Internal coherence HSX vs IEM <ul><li>Arbitrage closure for HSX award options </li></ul><ul><li>Arbitrage closure on IEM qualitatively similar to HSX, though quantitatively more efficient </li></ul>
  146. 146. Forecast accuracy: HSX <ul><li>0.94 correlation </li></ul><ul><li>Comparable to expert forecasts at Box Office Mojo </li></ul>
  147. 147. Combining forecasts <ul><li>HSX + Box Office Mojo (expert forecast) </li></ul><ul><li>Correlation of errors: 0.818 </li></ul> corr av err av%err fit HSX 0.940 3.57 31.5 1.16 BOMojo 0.945 3.31 27.5 1.10 avg 0.950 3.16 27.0 1.15 avg-max 0.956 2.90 26.6 1.08
  148. 148. Probabilistic forecasts HSX <ul><li>Bins of similarly-priced options </li></ul><ul><li>Observed frequency  average price </li></ul><ul><li>Analysis similar for horse racing markets </li></ul><ul><li>Error bars: 95% confidence intervals assuming events are indep Bernoulli trials </li></ul>
  149. 149. Avg logarithmic score forecast source avg log score Feb 19 HSX prices -0.854 DPRoberts -0.874 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
  150. 150. Probabilistic forecasts FX <ul><li>Prices 30 days before expiration </li></ul><ul><li>Similar results: </li></ul><ul><ul><li>60 days before </li></ul></ul><ul><ul><li>specific date </li></ul></ul><ul><li>Average logarithmic score </li></ul>FX
  151. 151. Real markets vs. market games HSX IEM average log score arbitrage closure
  152. 152. Real markets vs. market games HSX FX, F1P6 expected value forecasts 489 movies forecast source avg log score F1P6 linear scoring -1.84 F1P6 F1-style scoring -1.82 betting odds -1.86 F1P6 flat scoring -2.03 F1P6 winner scoring -2.32 probabilistic forecasts
  153. 153. Does money matter? Play vs real, head to head <ul><li>Experiment </li></ul><ul><li>2003 NFL Season </li></ul><ul><li>Online football forecasting competition </li></ul><ul><ul><li>Contestants assess probabilities for each game </li></ul></ul><ul><ul><li>Quadratic scoring rule </li></ul></ul><ul><ul><li>~2,000 “experts”, plus: </li></ul></ul><ul><ul><li>NewsFutures (play $) </li></ul></ul><ul><ul><li>Tradesports (real $) </li></ul></ul><ul><ul><ul><li>Used “last trade” prices </li></ul></ul></ul><ul><li>Results: </li></ul><ul><li>Play money and real money performed similarly </li></ul><ul><ul><li>6 th and 8 th respectively </li></ul></ul><ul><li>Markets beat most of the ~2,000 contestants </li></ul><ul><ul><li>Average of experts came 39 th </li></ul></ul>Forthcoming, Electronic Markets , Emile Servan-Schreiber, Justin Wolfers, David Pennock and Brian Galebach
  154. 155. Does money matter? Play vs real, head to head Statistically: TS ~ NF NF >> Avg TS > Avg
  155. 156. Market games summary <ul><li>Online market games can contain a great deal of information reflecting interactions among millions of people </li></ul><ul><ul><li>Naturally attract well-informed and well-motivated players </li></ul></ul><ul><ul><li>Game players tend to be knowledgeable and enthusiastic </li></ul></ul><ul><ul><li>Internet polls - skewed demographic </li></ul></ul><ul><ul><ul><li>Polls typically ask questions of the form “What do you want?” </li></ul></ul></ul><ul><ul><ul><li>Games ask questions of the form “What do you think will happen?” </li></ul></ul></ul>
  156. 157. Market games discussion <ul><li>Are incentives strong enough? </li></ul><ul><ul><li>Yes (to a degree) </li></ul></ul><ul><ul><li>Manifested as price coherence, information incorporation, and forecast accuracy </li></ul></ul><ul><ul><li>Reduced incentive for information discovery possibly balanced by better interpersonal weighting </li></ul></ul><ul><li>Statistical validations show HSX, FX, NF are reliable sources for forecasts </li></ul><ul><ul><li>HSX predictions >= expert predictions </li></ul></ul><ul><ul><li>Combining sources can help </li></ul></ul>
  157. 158. Applications <ul><li>Obtain information from existing games </li></ul><ul><li>Build new games in areas of interest </li></ul><ul><ul><li>Alternative to costly market research </li></ul></ul><ul><ul><li>Easy/inexpensive to setup compared to real markets </li></ul></ul><ul><ul><li>Few regulations compared to real markets </li></ul></ul><ul><ul><li>Worldwide audience </li></ul></ul>
  158. 159. Future work <ul><li>Data mining and fusion algorithms can improve predictions </li></ul><ul><ul><li>Weight users by expertise, reliability, etc. </li></ul></ul><ul><ul><li>Controlling for manipulation </li></ul></ul><ul><ul><li>Merging with other sources </li></ul></ul><ul><ul><ul><li>Box office prediction (market + chat groups, query logs, movie reviews, news, experts) </li></ul></ul></ul><ul><ul><ul><li>Weather forecasting (futures, derivatives + experts, satellite images) </li></ul></ul></ul><ul><li>Privacy issues and incentives </li></ul>
  159. 160. 4. Lab experiments & theory <ul><li>Laboratory experiments, field tests </li></ul><ul><li>Theoretical underpinnings </li></ul><ul><ul><li>Rational expectations </li></ul></ul><ul><ul><li>Efficient markets hypothesis </li></ul></ul><ul><ul><li>No-Trade Theorems </li></ul></ul><ul><ul><li>Information aggregation </li></ul></ul>
  160. 161. Laboratory experiments <ul><li>Experimental economics </li></ul><ul><li>Plott and “decendents”: Ledyard, Hanson, Fine, Coughlan, Chen, ... (and others) </li></ul><ul><li>Controlled tests of information aggregation </li></ul><ul><li>Participants are given information, asked to trade in market for real monetary stakes </li></ul><ul><li>Equilibrium is examined for signs of information incorporation </li></ul>
  161. 162. Plott & Sunder 1982 <ul><li>Three disjoint exhaustive states X,Y,Z </li></ul><ul><li>Three securities </li></ul><ul><li>A few insiders know true state Z </li></ul><ul><li>Market equilibrates according to rational expectations: as if everyone knew Z </li></ul>$1 if X $1 if Y $1 if Z ? Z 1 price of Z time 0
  162. 163. Plott & Sunder 1982 <ul><li>Three disjoint exhaustive states X,Y,Z </li></ul><ul><li>Three securities </li></ul><ul><li>Some see samples of joint; can infer P(Z|samples) </li></ul><ul><li>Results less definitive </li></ul>$1 if X $1 if Y $1 if Z ? P(XYZ) 1 price of Z time 0
  163. 164. Plott & Sunder 1988 <ul><li>Three disjoint exhaustive states X,Y,Z </li></ul><ul><li>Three securities </li></ul><ul><li>A few insiders know true state is not X </li></ul><ul><li>A few insiders know true state is not Y </li></ul><ul><li>Market equilibrates according to rational expectations: Z true </li></ul>$1 if X $1 if Y $1 if Z not X not Y 1 price of Z time 0
  164. 165. Plott & Sunder 1988 <ul><li>Three disjoint exhaustive states X,Y,Z </li></ul><ul><li>One security </li></ul><ul><li>A few insiders know true state is not X </li></ul><ul><li>A few insiders know true state is not Y </li></ul><ul><li>Market does not equilibrate according to rational expectations </li></ul>$1 if Z not X not Y 1 price of Z time 0
  165. 166. Forsythe and Lundholm 90 <ul><li>Three disjoint exhaustive states X,Y,Z </li></ul><ul><li>One security </li></ul><ul><li>Some know not X </li></ul><ul><li>Some know not Y </li></ul><ul><li>As long as traders are sufficiently knowledgeable & experienced, market equilibrates according to rational expectations </li></ul>$1 if Z not X not Y 1 price of Z time 0
  166. 167. Small groups <ul><li>In small, illiquid markets, information aggregation can fail </li></ul><ul><li>Chen, Fine, & Huberman [EC-2001] propose a two stage process </li></ul><ul><ul><li>Trade in a market to assess participants’ risk attitude and predictive ability </li></ul></ul><ul><ul><li>Query participants’ probabilities using the log score; compute a weighted average of probabilities, with weights derived from step 1 </li></ul></ul>
  167. 168. Small groups [Source: Fine DARPA Workshop, 2002]
  168. 169. Field test: Hewlett Packard <ul><li>Plott & Chen [2002] conducted a field test at Hewlett Packard (HP) </li></ul><ul><li>Set up a securities market to predict, e.g. “next months sales (in $) of product X” </li></ul><ul><ul><li>$1 iff $0 < sales < $10K $1 iff $20K < sales < $30K </li></ul></ul><ul><ul><li>$1 iff $10K < sales < $20K $1 iff sales > $30K </li></ul></ul><ul><li>Employees could trade at lunch, weekends, for real $$ </li></ul><ul><li>Market predictions beat official HP forecasts </li></ul>
  169. 170. Why does it work? Rational expectations <ul><li>Theory: Even when agents have asymmetric information, market equilibrates as if all agents had all info [Grossman 1981; Lucas 1972] </li></ul><ul><li>Procedural explanation: agents learn from prices [Hanson 98; Mckelvey 86; Mckelvey 90; Nielsen 90] </li></ul><ul><ul><li>Agents begin with common priors, differing information </li></ul></ul><ul><ul><li>Observe sufficient summary statistic (e.g., price) </li></ul></ul><ul><ul><li>Converge to common posteriors </li></ul></ul><ul><ul><li>In compete market, all (private) info is revealed </li></ul></ul>
  170. 171. Efficient market hypotheses (EMH) <ul><li>Internal coherence prices are self-consistent or arbitrage-free </li></ul><ul><li>Weak form: Internal unpredictability future prices unpredictable from past prices </li></ul><ul><li>Semi-strong form: Unpredictability future prices unpredictable from all public info </li></ul><ul><li>Strong form: Expert-level accuracy unpredictable from all public & private info; experts cannot outperform naïve traders </li></ul><ul><li>More: </li></ul>stronger assump’s http://www.investorhome.com/emh.htm
  171. 172. How efficient are markets? <ul><li>Good question: as many opinions as experts </li></ul><ul><li>Cannot prove efficiency; can only detect inefficiency </li></ul><ul><li>Generally, it is thought that large public markets are very efficient, smaller markets questionable </li></ul><ul><li>Still, strong form is sometimes too strong: </li></ul><ul><ul><li>There is betting on Oscars until winners are announced </li></ul></ul><ul><ul><li>Prices do not converge completely on eventual winners </li></ul></ul><ul><ul><li>Yet aggregating all private knowledge in the world (including Academy members’ votes) would yield the precise winners with certainty </li></ul></ul>
  172. 173. No-trade theorems <ul><li>Why trade? These markets are zero-sum games (negative sum w/ transaction fees) </li></ul><ul><li>For all money earned, there is an equal (greater) amount lost; am I smarter than average? </li></ul><ul><li>Rational risk-neutral traders will never trade [Milgrom & Stokey 1982][Aumann 1976]. Informally: </li></ul><ul><ul><li>Only those smarter than average should trade </li></ul></ul><ul><ul><li>But once below avg traders leave, avg goes up </li></ul></ul><ul><ul><li>Ad infinitum until no one is left </li></ul></ul><ul><ul><li>Or: If a rational trader is willing to trade with me, he or she must know something I don’t know </li></ul></ul>
  173. 174. But... Trade happens <ul><li>Volume in financial markets, gambling is high </li></ul><ul><li>Why do people trade? </li></ul><ul><ul><li>1. Different risk attitudes (insurance, hedging) Can’t explain all volume </li></ul></ul><ul><ul><li>2. Irrational (boundedly rational) behavior </li></ul></ul><ul><ul><ul><li>Rationality arguments require unrealistic computational abilities, including infinite precision Bayesian updating, infinite game-theoretic recursive reasoning </li></ul></ul></ul><ul><ul><ul><li>More than 1/2 of people think they’re smarter than average </li></ul></ul></ul><ul><ul><ul><li>Biased beliefs, differing priors, inexperience, mistakes, etc. </li></ul></ul></ul><ul><li>Note that it’s rational to trade as long as some participants are irrational </li></ul>
  174. 175. A theory of info aggregation Notation <ul><li>Event: A (event negation: A ) </li></ul><ul><li>Security: </li></ul><ul><li>Probability: Pr(A) </li></ul><ul><li>Likelihood: L(A) = Pr(A)/(1-Pr(A)) </li></ul><ul><li>Log-likelihood: LL(A) = ln L(A) </li></ul><ul><li>Price of at time t: p t </li></ul><ul><li>Likelihood price: l t = p t /(1-p t ) </li></ul><ul><li>Log-likelihood price: ll t = ln l t </li></ul>$1 if A $1 if A [Pennock 2002]
  175. 176. Assumptions <ul><li>Efficiency assumption: Let p t be the price of at time t Then </li></ul><ul><li>Pr(A|p t ,p t-1 ,p t-2 ,…,p 0 ) = p t </li></ul><ul><li>(Markov assumpt. + accuracy assumpt.) </li></ul>$1 if A
  176. 177. Consequences <ul><li>E[p t |p t-1 = x] = x expected price at time t is price at t-1 </li></ul><ul><li>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 </li></ul>Pr(ll t =x+  |A,ll t-1 =x) Pr(ll t =x+  | A ,ll t-1 =x) = e 
  177. 178. Consequences <ul><li>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 </li></ul><ul><li>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 </li></ul>E[p t |A,p t-1 =x] = x + Var(p t |p t-1 =x) x y x
  178. 179. 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
  179. 180. Avg log score dynamics FX HSX WSE bball WSE soccer IEM
  180. 181. Applications & future work <ul><li>Better understanding of market dynamics & assumptions required for predictive value </li></ul><ul><li>Closeness of fit to theory is a measure of market forecast accuracy; could serve as an evaluation metric or confidence metric </li></ul><ul><li>Explaining symmetry, power-law dist in IEM </li></ul>
  181. 182. Coin-flip model <ul><li>Previous theory: minimalist assumptions; no explicit notion of evidence </li></ul><ul><li>Coin-flip model of evidence incorporation: </li></ul><ul><ul><li>A  occurrence of n/2 tails out of n flips </li></ul></ul><ul><ul><li>Release of info  revelation of flip outcomes </li></ul></ul><ul><ul><li>At time t: it tails have occurred out of k t flips </li></ul></ul><ul><ul><li>For A to occur, n/2-it more tails are needed </li></ul></ul>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
  182. 183. Avg log score dynamics FX HSX WSE bball WSE soccer IEM coin flip model
  183. 184. 5. Characterizing information aggregation <ul><li>Market as an opinion pool </li></ul><ul><li>Market as a “composite agent” </li></ul><ul><ul><li>Market belief, utility </li></ul></ul><ul><ul><li>Market Bayesian updates </li></ul></ul><ul><ul><li>Market adaptation, dynamics </li></ul></ul><ul><li>Paradoxes, impossibilities </li></ul><ul><ul><li>Opinion pool impossibilities </li></ul></ul><ul><ul><li>Composite agent non-existence </li></ul></ul>
  184. 185. Aggregating beliefs  Y B R B ush wins 2004 Y HOO stock > 30 R ain tomorrow Y B R Y B R
  185. 186. Opinion pools (1959-) <ul><li>Linear (LinOP): weighted arithmetic mean </li></ul><ul><ul><li>Pr 0 (  ) = w 1 Pr 1 (  ) +  + w n Pr n (  ) </li></ul></ul><ul><ul><li>w i are “expert weights” </li></ul></ul><ul><li>Logarithmic (LogOP): wtd geometric mean </li></ul><ul><ul><li>Pr 0 (  )  [Pr 1 (  )] w1  [Pr n (  )] wn </li></ul></ul><ul><li>Supra Bayesian </li></ul><ul><ul><li>Pr 0 (  | Pr 1  Pr n )  Pr sb (Pr 1  Pr n |  )Pr sb (  ) </li></ul></ul>
  186. 187. Subjective probability de Finetti (1937) p <E 1 > + p < Ê 1 > = 1 p <E 1 |E 2 >  p <E 2 > = p <E 1 E 2 > etc... No arbitrage (No Dutch books) (No risk-free profits)  p <E 1 > , p < Ê 1 > , p <E 2 > , p <E 1 |E 2 > , p <E 1 E 2 >
  187. 188. Consensus probability at market equilibrium p <E 1 > + p < Ê 1 > = 1 p <E 1 |E 2 >  p <E 2 > = p <E 1 E 2 > etc... No arbitrage (No Dutch books) (No risk-free profits)  p <E 1 > , p < Ê 1 > , p <E 2 > , p <E 1 |E 2 > , p <E 1 E 2 >
  188. 189. A Market Model $1 if E 1 $1 if E 2 $1 if E S Competitive equilibrium prices p <E 1 > , p <E 2 > ,…  consensus belief Pr 1 , u 1 subjective probability utility for money Pr i , u i Pr n , u n
  189. 190. Advantages <ul><li>Explicit incentives for participation, honesty, and to gather evidence </li></ul><ul><li>No central coordinator </li></ul><ul><li>Well defined protocols </li></ul><ul><li>Library of economic tools to aid in analysis </li></ul><ul><li>Sparse communications </li></ul><ul><li>Allows for limited privacy </li></ul><ul><li>Risk-neutral probabilities agree at equil </li></ul>
  190. 191. Risk-neutral probability <ul><li>Behavior is the product of Pr and u </li></ul><ul><ul><li>max a   Pr(  )  u(a,  ) </li></ul></ul><ul><li>An observer cannot determine Pr or u </li></ul><ul><ul><li>Agent A  with Pr  f(  ) and u/f(  ) is equivalent to agent A with Pr and u </li></ul></ul><ul><li>Pr RN   Pr * u  </li></ul><ul><li>u RN  u/u  </li></ul>
  191. 192. Trading with risk-neutral probability <ul><li>A RN agent would buy if p <E> < Pr(E) </li></ul><ul><li>Any agent would buy if p <E> < Pr RN (E) </li></ul><ul><li>Any agent would sell if p <E> > Pr RN (E) </li></ul><ul><li>If Pr i RN (E)  Pr j RN (E) then i and j would desire to trade </li></ul><ul><li>At equilibrium, all agents’ risk-neutral probabilities agree, & equal prices </li></ul>$1 if E $1 if E $1 if E
  192. 193. <ul><li>Constant absolute risk aversion (CARA): u i (y)=-e -c i y </li></ul><ul><li>Disjoint events </li></ul>Market LogOP If <ul><li>Equilibrium prices compute LogOP </li></ul><ul><li>“ Expert weights” are normalized measure of risk tolerance </li></ul>Then  [Pr i (E j )]  i p <E j >  i=1 N
  193. 194. Market LinOP <ul><li>Generalized log utility for money (GLU): u i (y)=ln(y+b i ) </li></ul><ul><li>Disjoint events </li></ul>If <ul><li>Equilibrium prices compute LinOP </li></ul>Then   i Pr i (E j ) p <E j > = i=1 N
  194. 195. Composite agent <ul><li>CARA or GLU </li></ul><ul><li>Disjoint events </li></ul>If Then Then <ul><li>Total demand for each security equals that of a rational individual </li></ul><ul><li>Beliefs equal the equilibrium prices </li></ul><ul><li>Super-agent is less risk averse than any individual </li></ul>
  195. 196. Market Adaptation <ul><li>Single security </li></ul><ul><li>Multiperiod market </li></ul><ul><li>Agents with GLU </li></ul><ul><li>Fixed beliefs </li></ul>belief time successes:0 trials:0 successes:0 trials:1 successes:1 trials:2 successes:2 trials:3 successes:2 trials:4 successes:3 trials:5 successes:5 trials:10 successes:10 trials:20 successes:17 trials:30 successes:24 trials:40 successes:30 trials:50 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) price frequency wealth Beta(1,1)
  196. 197. Learning from prices Pr 1 (E 1 ), Pr 1 (E 2 ),… Supra Bayesian Pr 2 (E 1 ), Pr 2 (E 2 ),…
  197. 198. Learning from prices Supra Bayesian Pr(E 1 ), Pr(E 2 ),… p <E 1 > , p <E 2 > ,… Pr 1 (E 1 ), Pr 1 (E 2 ),… Pr 2 (E 1 ), Pr 2 (E 2 ),…
  198. 199. Bernoulli trials model s successes in n trials s  successes in n  trials Pr(E|p <E> ) = wPr(E) + (1-w)p <E> where w  “ Market” s  successes in n  trials E Ê Ê E Ê E E n n+n 
  199. 200. Equilibrium with Learning <ul><li>Weighted average update Pr(E|p <E> ) = wPr(E) + (1-w)p <E> and agents with GLU  still LinOP prices confidence-based wts </li></ul><ul><li>Geometric average update Pr(E|p <E> )  Pr(E) w (p <E> ) (1-w) and agents with CARA  still LogOP prices </li></ul>
  200. 201. Market Dynamics <ul><li>Agents with GLU </li></ul><ul><li>Weighted average belief update: </li></ul><ul><li>Pr(E t |p <E t > ) = 0.2 Pr(E t ) + 0.8 p <E t > </li></ul>price discounted frequency belief time 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)
  201. 202. Market Dynamics <ul><li>Agents with CARA </li></ul><ul><li>Mixed populations </li></ul>belief belief wealth GLU wealth CARA wealth Beta(8,4)
  202. 203. Impossibility theorems <ul><li>Combining probabilities: Pr 0 = f(Pr 1 ,Pr 2 ,...,Pr n ) </li></ul><ul><li>Properties / axioms: </li></ul><ul><ul><li>Non-dictatorship (ND) </li></ul></ul><ul><ul><li>Proportional Dependence on States (PDS) Pr 0 (  )  f(Pr 1 (  ), Pr 2 (  ), … , Pr n (  )) </li></ul></ul><ul><ul><li>Independence Preservation Property (IPP) </li></ul></ul>& &  & 
  203. 204. Impossibility theorems <ul><li>Combining probabilities: Pr 0 = f(Pr 1 ,Pr 2 ,...,Pr n ) </li></ul><ul><li>Properties / axioms: </li></ul><ul><ul><li>Non-dictatorship (ND) </li></ul></ul><ul><ul><li>Marginalization property (MP) f(A B) + f(A B ) = f(A) aggregate, marginalize = marginalize, aggregate </li></ul></ul><ul><ul><li>Externally Bayesian (EB) f(A|B) = f(A B) / f(B) condition, aggregate = aggregate, condition </li></ul></ul>
  204. 205. Market impossibilities <ul><li>Market is just another function f </li></ul><ul><ul><li>Sometimes weighted algebraic/geometric avg </li></ul></ul><ul><ul><li>In general, arbitrary non-linear fn </li></ul></ul><ul><li>Still, subject to all the same paradoxes, impossibilities, limitations </li></ul><ul><li>In some cases, a composite agent does not exist [due to Pratt, described in Raiffa 1968] </li></ul><ul><ul><li>market: flipping a coin can help overall utility </li></ul></ul><ul><ul><li>individual: flipping a coin never helps </li></ul></ul>
  205. 206. 6. Computational aspects <ul><li>Combinatorics </li></ul><ul><ul><li>Compact securities markets </li></ul></ul><ul><ul><li>Combinatorial securities markets </li></ul></ul><ul><ul><li>Compound securities markets </li></ul></ul><ul><ul><li>Market scoring rules </li></ul></ul><ul><ul><li>Dynamic pari-mutuel market </li></ul></ul><ul><ul><li>Policy Analysis Market </li></ul></ul><ul><li>Distributed market computation </li></ul>
  206. 207. Complete securities markets <ul><li>A set of securities is complete if rank of returns matrix = |  |  1 </li></ul><ul><li>For example, set of |  |  1 Arrow securities: “Arrow-Debreu securities market” </li></ul><ul><li>Market with complete set of securities guarantees a Pareto optimal allocation of risk, under classical conditions </li></ul><ul><li>For all practical purposes, |  | = 2 n securities is intractable </li></ul>
  207. 208. Complete securities markets <ul><li>Problems </li></ul><ul><ul><li>“ Space complexity”: Can’t even write down all securities, store all prices, quantities, etc. </li></ul></ul><ul><ul><li>Liquidity: Too many securities dividing traders’ attention. Bounded rationality  can’t possible explore, let alone optimize over all securities </li></ul></ul><ul><li>Solution approaches </li></ul><ul><ul><li>Find subset of securities that are (nearly) sufficient for given agents: 1. Compact markets </li></ul></ul><ul><ul><li>Define mechanisms to match expressive bids: 2. Combinatorial mkts 3. Compound markets </li></ul></ul><ul><ul><li>Automated market maker 4. Market scoring rules 5. Dynamic pari-mutuel </li></ul></ul>
  208. 209. Do we really need all these securities? <ul><li>Under what conditions are fewer than |  |-1 securities “sufficient” </li></ul><ul><li>Intuitively, many features of state of nature have nothing to do with each other. </li></ul><ul><li>Idea: maybe we can expoit (conditional) independence among events </li></ul><ul><li>Compact securities markets [Pennock & Wellman 2000] </li></ul>
  209. 210. <ul><li>Joint probability distribution </li></ul><ul><li>Exploiting independence </li></ul><ul><ul><li>Pr(R Y B) = Pr(R)  Pr(Y)  Pr(B) </li></ul></ul><ul><ul><li>8 states  3 prob. values </li></ul></ul>Independence Y R B X X 2 3 = 8 states  7 prob. values R ain tomorrow Y HOO stock > 30 B in Laden captured
  210. 211. Conditional independence <ul><li>Pr(Y I B) = Pr(Y|I)  Pr(I|B)  Pr(B) </li></ul><ul><li>8 states  5 assessments </li></ul>Y HOO stock > 30 B udget surplus > 0 I nterest rate < 1%
  211. 212. Bayesian networks <ul><li>Conditional independence encoded in graph structure. </li></ul><ul><li>Factors joint distribution into product of conditionals. </li></ul><ul><li>Example: 13 rather than 63 prob values. </li></ul>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 )) E 2 E 5 E 3 E 6 E 4 E 1
  212. 213. <ul><li>Conditional security: </li></ul><ul><ul><li>Pays off $1 if E 1 & E 2 occur </li></ul></ul><ul><ul><li>Lose price paid ( p < E 1 |E 2 > ) if Ê 1 & E 2 </li></ul></ul><ul><ul><li>Bet “called off” if Ê 2 </li></ul></ul>Conditional securities $1 if E 1 |E 2
  213. 214. Bayes-net structured markets <ul><li>Securities markets can be structured analogously to a BN </li></ul><ul><li>One (conditional) security for each CPT entry </li></ul><ul><li>Fully connected BN  complete market </li></ul>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 ) $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 E 2 E 5 E 3 E 6 E 4 E 1
  214. 215. Compact markets? <ul><li>Idea: Include securities markets according to conditional probs in Bayesian network </li></ul><ul><li>Problem : Agents may disagree about independence structure </li></ul>E 2 E 5 E 3 E 6 E 4 E 1 $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
  215. 216. Compact markets (II)? <ul><li>Structure market according to unanimously agreed-upon independencies </li></ul>E 2 E 5 E 3 E 6 E 4 E 1 $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
  216. 217. Belief and willingness-to-pay agent E $x if E Pr(E)=0.4 p x =0.4 lim x  0
  217. 218. Prior Stakes $1000 if E E $x if E Pr(E)=0.4 (risk-averse) agent Therefore, trading behavior may not reveal “true” independencies p x =0.3 lim x  0
  218. 219. Risk-neutral probability <ul><li>Behavior is the product of Pr and u </li></ul><ul><ul><li>max a   Pr(  )  u(a,  ) </li></ul></ul><ul><li>An observer cannot determine Pr or u </li></ul><ul><ul><li>Agent A  with Pr  f(  ) and u/f(  ) is equivalent to agent A with Pr and u </li></ul></ul><ul><li>Pr RN   Pr * u  </li></ul><ul><li>u RN  u/u  </li></ul>
  219. 220. Trading with risk-neutral probability <ul><li>A RN agent would buy if p <E> < Pr(E) </li></ul><ul><li>Any agent would buy if p <E> < Pr RN (E) </li></ul><ul><li>Any agent would sell if p <E> > Pr RN (E) </li></ul><ul><li>If Pr i RN (E)  Pr j RN (E) then i and j would desire to trade </li></ul><ul><li>At equilibrium, all agents’ risk-neutral probabilities agree, & equal prices </li></ul>$1 if E $1 if E $1 if E
  220. 221. Compact markets (III)? <ul><li>Structure market according to unanimously agreed-upon risk-neutral independencies </li></ul>$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 E 2 E 5 E 3 E 6 E 4 E 1
  221. 222. Operationally complete securities markets <ul><li>If, in equil, all RN indep agree with market structure  mkt is operationally complete </li></ul><ul><ul><li>Pareto optimal allocation of risk </li></ul></ul><ul><ul><li>supports all desirable trades, but not all conceivable </li></ul></ul><ul><li>RN independencies change out of equilibrium; perhaps more arguable basis for agreement on true independencies </li></ul>
  222. 223. Decomposable networks <ul><li>BN with edge between every pair of nodes with common child. </li></ul>Bayesian network Markov network E 2 E 5 E 3 E 6 E 4 E 1 Bayesian network Fill-in E 2 E 5 E 3 E 6 E 4 E 1 E 2 E 5 E 3 E 6 E 4 E 1 moralization
  223. 224. <ul><li>CARA & Markov indep  risk-neutral indep </li></ul><ul><li>If all agents have CARA, then market structured as TRIANGULATE [  n i=1 MORALIZE (D i )] is op complete </li></ul><ul><li>Can still yield exponential savings </li></ul><ul><li>This example: 19 vs. 63 </li></ul>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 E 2 E 5 E 3 E 6 E 4 E 1
  224. 225. Independence-preserving aggregation <ul><li>Structural unanimity </li></ul><ul><li>Proportional dependence on states </li></ul><ul><li>Pr 0 (  )  f(Pr 1 (  ), Pr 2 (  ), … , Pr n (  )) </li></ul><ul><li>Unanimity </li></ul><ul><li>Nondictatorship </li></ul>& &  & 
  225. 226. Summary <ul><li>Under certain theoretical conditions, structured markets are optimal, with exponentially fewer securities than would otherwise be required </li></ul><ul><li>Could have application in creating new derivatives markets that allow agents to hedge more of their risks, w/o combinatorial explosion of fin instruments </li></ul>
  226. 227. Combinatorial auctions <ul><li>E.g.: spectrum rights, computer system, … </li></ul><ul><li>n goods  bids allowed  2 n combinations Maximizing revenue: NP-hard (set packing) </li></ul><ul><li>Enter computer scientists (hot topic)… </li></ul><ul><li>Survey: [Vries & Vohra 02] </li></ul>
  227. 228. Combinatorial auctions (Some) research issues <ul><li>Preference elicitation [Sandholm 02] </li></ul><ul><li>Bidding languages [Nissan 00] & restrictions [Rothkopf 98] </li></ul><ul><li>Approximation </li></ul><ul><ul><li>relation to incentive compatibility [Lehmann 99] and bounded rationality [Nisan & Ronen 00] </li></ul></ul><ul><li>False-name bidders [Yokoo 01] </li></ul><ul><li>Winner determination </li></ul><ul><ul><li>GVA (VCG) mechs, iterative mechs [Parkes 99, Wurman 00]; “smart markets” [Brewer 99] </li></ul></ul><ul><ul><li>integer programming; specialized heuristics [Sandholm 99] </li></ul></ul><ul><li>FCC spectrum auctions </li></ul><ul><li>Optimal auction design [Ronen 01] </li></ul>More: [Vries & Vohra 02]
  228. 229. Combined value trading <ul><li>Traders are often interested in portfolios (“bundles”) rather than individual assets </li></ul><ul><ul><li>Buy Apple, sell Microsoft </li></ul></ul><ul><ul><li>Sell Dallas $9, Sacramento $33, San Antonio $28, LA $11  Sell Western Division $81  Buy Eastern $19 </li></ul></ul><ul><li>Esp. in thin markets there is “execution risk”: price of one asset may change while others are executed </li></ul><ul><li>CVT: Combinatorial auction mechanism for assets </li></ul><ul><ul><li>Traders can submit conditional orders , that are filled only if other related orders are also filled </li></ul></ul><ul><ul><li>Essentially can request bundles </li></ul></ul>[Bossaerts, Fine, Ledyard 2002]
  229. 230. Combined value trading <ul><li>Form of bids </li></ul><ul><ul><li>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. </li></ul></ul><ul><ul><li>For -$3 (receive $3), sell 4 units of C & buy 3 of D; F=1 All or nothing offer </li></ul></ul><ul><li>Computational problem </li></ul><ul><ul><li>If all F=0, linear programming  polynomial </li></ul></ul><ul><ul><li>If  F>0, mixed linear, integer programming  NP-hard Moreover, prices may not exist  Discriminative pricing </li></ul></ul>[Bossaerts, Fine, Ledyard 2002]
  230. 231. Thick markets [Source: Ledyard, DARPA Workshop, 2002]
  231. 232. Thin markets, no CVT [Source: Ledyard, DARPA Workshop, 2002]
  232. 233. Thin markets, CVT [Source: Ledyard, DARPA Workshop, 2002]
  233. 234. Market combinatorics [Thanks: Wolfers, Fortnow]
  234. 235. <ul><li>What about Pr(CA ^ AZ) ? Pr(CA | AZ) ? Pr(Elec | FL) ? Pr((IL^NJ)  (  IL ^  NJ )) ? </li></ul><ul><li>Not derivable as a linear combinations of base securities </li></ul><ul><li>2 2 50 possible functions </li></ul><ul><li>“ Only” 2 50 securities needed to span space </li></ul>Market combinatorics [Thanks: Wolfers, Fortnow]
  235. 236. Info mkt combinatorics UN action casualties Bin Laden captured Turkey action SARS US leaves Iraq oil prices Afghanistan
  236. 237. Info mkt combinatorics An A2 A1 Ai A6 A4 A3 A5 … … binary variables Note: E[A]=Pr(a)
  237. 238. Market combinatorics <ul><li>In principle, markets in all possible combinations will get you everything you want </li></ul><ul><li>In practice, this is infeasible </li></ul><ul><li>It’s also unnatural </li></ul>$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:
  238. 239. Compound securities [Fortnow EC-2003] <ul><li>A bidding language: write your own security </li></ul><ul><li>For example </li></ul><ul><li>Offer to buy/sell q units of it at price p </li></ul><ul><li>Let everyone else do the same </li></ul><ul><li>Auctioneer must decide who trades with whom at what price… How? (next) </li></ul><ul><li>More concise/expressive; more natural </li></ul>$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: $1 if Boolean_fn | Boolean_fn I am entitled to:
  239. 240. The matching problem <ul><li>There are many possible matching rules for the auctioneer </li></ul><ul><li>A natural one: maximize trade subject to no-risk constraint </li></ul><ul><li>Example: </li></ul><ul><ul><li>buy 1 of for $0.40 </li></ul></ul><ul><ul><li>sell 1 of for $0.10 </li></ul></ul><ul><ul><li>sell 1 of for $0.20 </li></ul></ul><ul><li>No matter what happens, auctioneer cannot lose money </li></ul>$1 if A1 $1 if A1&A2 $1 if A1& A2 trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 0.60 0.60 -0.40 -0.40 -0.90 0.10 0.10 0.10 0.20 -0.80 0.20 0.20 -0.10 -0.10 -0.10 -0.10
  240. 241. The matching problem <ul><li>Another way to look at it: Logical reduction </li></ul><ul><li>| | </li></ul><ul><li>Example: </li></ul><ul><ul><li>buy 1 of for $0.40 </li></ul></ul><ul><ul><li>sell 1 of for $0.10 </li></ul></ul><ul><ul><li>sell 1 of for $0.20 </li></ul></ul><ul><li>|| </li></ul><ul><li>Clear match btw buy and sell | </li></ul>$1 if A1 $1 if A1&A2 $1 if A1& A2 $1 if A1 = sell for $0.3
  241. 242. The matching problem <ul><li>Divisible orders: will accept any q*  q </li></ul><ul><li>Indivisible: will accept all or nothing </li></ul><ul><li>Let  =all possible combinations; |  |=2 n </li></ul><ul><li>Let  i be fraction of order i filled </li></ul><ul><li>Let  i  be payoff for order i in state  </li></ul><ul><li>Div. MP: Does  i  [0,1],  , -  i  i   0 </li></ul><ul><li>Indiv. MP: Does  i  {0,1},  , -  i  i   0 </li></ul><ul><li>Optimizations </li></ul><ul><ul><li>max trade; max percent orders filled </li></ul></ul><ul><ul><li>max auctioneer utility subject to no-risk </li></ul></ul><ul><ul><li>max auctioneer utility -- with risk (“market maker”) </li></ul></ul>(at least 1  i > 0)
  242. 243. Divisible vs. indivisible <ul><li>Sell 1 of A1 at $0.50 </li></ul><ul><li>Buy 1 of (A1&A2) | (A1 || A2) at $0.50 </li></ul><ul><li>Buy 1 of A1| A2 at $0.40 </li></ul>trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 -0.50 -0.50 0.50 0.50 0.50 -0.50 -0.50 0 0 0.60 0 -0.40 0 -0.40 0 0.10
  243. 244. Divisible vs. indivisible <ul><li>Sell 1 of A1 at $0.50 </li></ul><ul><li>Buy 1 of (A1&A2) | (A1 || A2) at $0.50 </li></ul><ul><li>Buy 1 of A1| A2 at $0.40 </li></ul>trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 -0.50 -0.50 0.50 0.50 0.50 -0.50 -0.50 0 0 0.60 0 -0.40 0 -1 0 0.50
  244. 245. Divisible vs. indivisible <ul><li>Sell 1 of A1 at $0.50 </li></ul><ul><li>Buy 1 of (A1&A2) | (A1 || A2) at $0.50 </li></ul><ul><li>Buy 1 of A1| A2 at $0.40 </li></ul>trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 -0.50 -0.50 0.50 0.50 0.50 -0.50 -0.50 0 0 0.60 0 -0.40 -0.50 0.10 0.50 0.10
  245. 246. Divisible vs. indivisible <ul><li>Sell 1 of A1 at $0.50 </li></ul><ul><li>Buy 1 of (A1&A2) | (A1 || A2) at $0.50 </li></ul><ul><li>Buy 1 of A1| A2 at $0.40 </li></ul>trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 -0.50 -0.50 0.50 0.50 0.50 -0.50 -0.50 0 0 0.60 0 -0.40 0.50 0.10 -0.50 -0.40
  246. 247. Divisible vs. indivisible <ul><li>Sell 1 of A1 at $0.50 </li></ul><ul><li>Buy 1 of (A1&A2) | (A1 || A2) at $0.50 </li></ul><ul><li>Buy 1 of A1| A2 at $0.40 </li></ul>trader gets $$ in state: A1A2 A1 A2 A1 A2 A1A2 -0.50 -0.50 0.50 0.50 0.50 -0.50 -0.50 0 0 0.60 0 -0.40 0 0 0 -0.10 3/5 x 3/5 x 1 x divisible match!
  247. 248. Complexity results <ul><li>Divisible orders: will accept any q*  q </li></ul><ul><li>Indivisible: will accept all or nothing </li></ul><ul><li>Natural algorithms </li></ul><ul><ul><li>divisible: linear programming </li></ul></ul><ul><ul><li>indivisible: integer programming; logical reduction? </li></ul></ul>Fortnow; Kilian; Sami # events divisible indivisible O(log n) polynomial NP-complete O(n) co-NP-complete  2 p complete reduction from SAT reduction from X3C reduction from T  BF LP
  248. 249. Open questions <ul><li>Other matching rules </li></ul><ul><ul><li>maximize utility subject to no-risk </li></ul></ul><ul><ul><li>maximize utility (market maker) </li></ul></ul><ul><li>What to do with the surplus </li></ul><ul><ul><li>can be in cash and “leftover” securities </li></ul></ul><ul><ul><li>auctioneer keeps surplus </li></ul></ul><ul><ul><li>surplus is shared back among traders, auctioneer; how? </li></ul></ul><ul><li>Trader optimization problem </li></ul><ul><ul><li>how to choose securities, p’s, q’s, subject to limits/penalties for number, complexity of bids </li></ul></ul><ul><ul><li>ultimately a game-theoretic question </li></ul></ul><ul><li>Approximate algorithms, heuristics </li></ul><ul><li>Incentive properties </li></ul>
  249. 250. The problem of liquidity <ul><li>Too many markets => Too little trade per market (“thin”, “illiquid”, large bid/ask spread) </li></ul><ul><li>Combinatorial markets/CVT: trader attention is limited, each market may get few bids </li></ul><ul><li>Compound markets: may be few matches </li></ul><ul><li>Automated market maker ensures liquidity </li></ul><ul><ul><li>Market scoring rules [Hanson 2002] </li></ul></ul><ul><ul><li>Dynamic pari-mutuel market [Pennock 2004] </li></ul></ul>
  250. 251. Scoring rule <ul><li>Logarithmic scoring rule (there are others) </li></ul><ul><li>Recall “pay an expert approach”: </li></ul><ul><ul><li>Offer to pay the expert </li></ul></ul><ul><ul><ul><li>$100 + log r if </li></ul></ul></ul><ul><ul><ul><li>$100 + log (1-r) if </li></ul></ul></ul><ul><li>Expert should choose r=Pr(A), given caveats </li></ul>= 6  6
  251. 252. Market scoring rule [Hanson 2002] <ul><li>System maintains a complete joint probability distribution over all variables </li></ul><ul><ul><li>Exponential space </li></ul></ul><ul><ul><li>Might use Bayes net or other compact representation, introduces complications </li></ul></ul><ul><li>Anyone at any time who thinks the probabilities are wrong, can change them by accepting a scoring rule payment </li></ul><ul><li>Trader must agree to pay off the previous person who changed the probabilities </li></ul>
  252. 253. Market scoring rule <ul><li>Example </li></ul><ul><li>Requires a “patron”, though only pays final trader, & payment is bounded </li></ul>A1A2 A1 A2 A1 A2 A1A2 0.25 0.25 0.25 0.25 0.20 0.20 0.30 0.30 100+log(.2) 100+log(.2) 100+log(.3) 100+log(.3) 100+log(.25) 100+log(.25) 100+log(.25) 100+log(.25) log(.2/.25) log(.2/.25) log(.3/.25) log(.3/.25) -.10 -.10 +.08 +.08 Trader can change to: Trader gets $$ in state: Trader pays $$ in state: total transaction: current probabilities:
  253. 254. Market scoring rule <ul><li>Note, a trader can change any part of the joint distribution, e.g. P(A 1 |A 3 ); no need to specify all </li></ul><ul><li>Conceptually, to traders it appears as if a market maker always stands willing to accept an (infinitesimal) trade at current prices </li></ul><ul><li>Full cost for some quantity is the integral over instantaneous prices, solvable in closed form for log scoring rule </li></ul>
  254. 255. [Source: Hanson, 2002] Accuracy .001 .01 .1 1 10 100 Estimates per trader Market Scoring Rules Simple Info Markets thin market problem Scoring Rules opinion pool problem
  255. 256. Computational issues <ul><li>Straightforward approach requires exponential space for prices, holdings, portfolios </li></ul><ul><li>Could represent probabilities using a Bayes net or other compact representation; changes must keep distribution in the same representational class </li></ul><ul><li>Could use multiple overlapping patrons, each with bounded loss. Limited arbitrage could be obtained by smart traders exploiting inconsistencies between patrons </li></ul>[Source: Hanson, 2002]        
  256. 257. RIP Policy Analysis Market <ul><li>Real combinatorial markets in Middle East issues </li></ul><ul><li>DARPA, Net Exchange, Caltech, GMU </li></ul><ul><li>Two year field test, starts 2003 </li></ul><ul><li>Open to public, real-money markets </li></ul><ul><li>~20 nations, 8 quarters, ~5 variables each: </li></ul><ul><ul><li>Economic, political, military, US actions </li></ul></ul><ul><li>Want many combos (> 2 500 states) </li></ul><ul><li>Legal: “DARPA & its agents not under CFTC’s regulatory umbrella” (paraphrased) </li></ul><ul><li>http://www.policyanalysismarket.org </li></ul>[Source: Hanson, 2002]
  257. 258. RIP Policy Analysis Market <ul><li>Killed in a single day under congressional/press firestorm </li></ul><ul><li>Misunderstood as betting on terrorism </li></ul><ul><li>After initial outrage, “good side” began to appear in media. Comments & compilations: </li></ul><ul><ul><li>http://hanson.gmu.edu/policyanalysismarket.html </li></ul></ul><ul><ul><li>http://dpennock.com/pam.html </li></ul></ul><ul><li>“ All press is good press”: Has drawn attention to prediction markets, spurned private sector development </li></ul>
  258. 259. Dynamic pari-mutuel market <ul><li>Standard PM: Every $1 bet is the same </li></ul><ul><li>DPM: Value of each $1 bet varies depending on the status of wagering at the time of the bet </li></ul><ul><li>Encode dynamic value with a price </li></ul><ul><ul><li>price is $ to buy 1 share of payoff </li></ul></ul><ul><ul><li>price of A is lower when less is bet on A </li></ul></ul><ul><ul><li>as shares are bought, price rises; price is for an infinitesimal share; cost is integral </li></ul></ul>
  259. 260. Dynamic pari-mutuel market <ul><li>Outcomes: A B </li></ul><ul><li>Current payoff/shr: $5.20 $0.97 </li></ul>$3.27 $3.27 $3.27 A B A B $1.00 $1.25 $1.50 $3.00 sell 100@ sell 100@ sell 35@ buy 4@ buy 52@ $3.25 $3.27 $3.27 $3.27 $0.25 $0.50 $0.75 sell 100@ sell 100@ sell 3@ buy 200@ $0.85 market maker traders
  260. 261. How are prices set? <ul><li>A price function pri(n) gives the instantaneous price of an infinitesimal additional share beyond the nth </li></ul><ul><li>Cost of buying n shares: </li></ul><ul><li>Different assumptions lead to different price functions, each reasonable </li></ul>
  261. 262. Mechanism comparison *Technically has risk, but bounded **One-sided liquidity  N/A N/A N/A damped volatility    payoff vector fixed   **  DPM    * MSR   PM   CDAwMM   CDA dynamic info aggreg. liquidity no risk
  262. 263. An info market model: Computational properties <ul><li>From a computational perspective, we are interested in: </li></ul><ul><ul><li>What can a market compute? </li></ul></ul><ul><ul><li>How fast? (time complexity) i.e., What mechanisms or protocols lead to faster convergence to the rational expectations equilibrium? </li></ul></ul><ul><ul><li>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? </li></ul></ul>
  263. 264. Market computation [Feigenbaum EC-2003] <ul><li>General formulation </li></ul><ul><ul><li>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) </li></ul></ul><ul><ul><li>Represent f( x ) as a circuit </li></ul></ul><ul><ul><li>Questions </li></ul></ul><ul><ul><ul><li>How do we set up a market to compute f? </li></ul></ul></ul><ul><ul><ul><li>How quickly can the market compute f? </li></ul></ul></ul>AND XOR OR x 1 x 2 x 3 x 4 f(x 1 ,x 2 ,x 3 ,x 4 )= (x 1  x 2 )  (x 3  x 4 )
  264. 265. Market model <ul><li>Each participant has some bit of information x i </li></ul><ul><li>There is a security F that pays off $1 if and only if f( x )=1 at some future date, and $0 otherwise. </li></ul><ul><li>Trading occurs in synchronous rounds </li></ul><ul><ul><li>In each round, participants bid their true expectation </li></ul></ul><ul><ul><li>Clearing price is determined using a simplified Shapley-Shubik trading model, yielding mean bid </li></ul></ul><ul><li>Questions we ask/answer: </li></ul><ul><ul><li>Does the clearing price converge to a stable value? </li></ul></ul><ul><ul><li>How fast does it converge (in how many rounds)? </li></ul></ul><ul><ul><li>Does the stable price of F reveal the true value of f? </li></ul></ul>
  265. 266. Theorems <ul><li>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 </li></ul><ul><li>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 ) </li></ul>
  266. 267. <ul><li>In the example, with only a single security on f, the market may not converge </li></ul>Example and interpretation <ul><li>Interpretation of theory: </li></ul><ul><ul><li>1 security supports computation of threshold fn only </li></ul></ul><ul><ul><li>More complex functions must utilize more securities: # of securities required = threshold circuit size of f </li></ul></ul>AND XOR OR x 1 x 2 x 3 x 4 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 <ul><li>With 2 additional securities it will converge in 4 rounds </li></ul>$1 if x 4
  267. 268. Extensions, future work <ul><li>Dynamic information revelation and changes </li></ul><ul><li>Overcoming false information </li></ul><ul><li>Obtaining incentive compatibility </li></ul><ul><li>Modeling agent strategies </li></ul><ul><li>Modeling overlapping information sources </li></ul><ul><li>Characterizing in terms of work/round </li></ul><ul><li>Bayesian network representation of prior </li></ul><ul><li>Dealing with limited-precision prices </li></ul>
  268. 269. Open questions <ul><li>What is the relationship between our model and perceptron (neural network) learning? </li></ul><ul><ul><li>Perceptrons exactly compute threshold functions </li></ul></ul><ul><ul><li>Could envision a system to learn smallest set of threshold functions to approximate desired function f, thereby minimizing the number of securities required </li></ul></ul><ul><li>Can alternate market protocols lead to faster convergence? Can subsidies speed convergence? </li></ul><ul><li>What can other types of securities (e.g., nonbinary securities) compute? </li></ul>
  269. 270. Legal issues <ul><li>Regulatory bodies: Commodity Futures Trading Commission (CFTC), Securities and Exchange Commission (SEC) </li></ul><ul><ul><li>IEM has “no action” letter from CFTC </li></ul></ul><ul><li>Financial institutions regularly create customized derivatives to hedge risks </li></ul><ul><li>Generally setting up markets for hedging is legal; setting up markets strictly for information gathering may be gambling; CFTC regulated  not gambling </li></ul><ul><li>No logical distinction: Trading options  betting on Oscars  Playing Roulette  - sum game </li></ul>
  270. 271. Legal issues <ul><li>Gambling in US </li></ul><ul><ul><li>Legal in some form in 48 states (lotteries, bingo, Indian reservations, riverboat) </li></ul></ul><ul><ul><li>ironically, by far worst E[return] </li></ul></ul><ul><ul><li>Illegal in many forms in all states </li></ul></ul><ul><ul><ul><li>Sports betting legal only in Las Vegas </li></ul></ul></ul><ul><ul><ul><li>Federal Wire Act: “bans the use of telephones to accept wagers on sporting events.” Computers? Non-sports? </li></ul></ul></ul><ul><ul><ul><li>“ The central question—whether Internet gambling is legal, illegal or exists in a legal nether world where no rules apply—is as gray as lawyers can make it.” [MSNBC] </li></ul></ul></ul>
  271. 272. Legal issues <ul><li>Gambling in US (cont’d) </li></ul><ul><ul><li>Several bills are going through Congress, both for outlawing/restricting, legalizing/regulating </li></ul></ul><ul><ul><li>Some states (Nevada, New Jersey) are considering legalizing online gambling </li></ul></ul><ul><ul><li>So-called “skill-based” games are OK! </li></ul></ul><ul><ul><ul><li>WorldWinner.com — includes BlackJack, Trivia </li></ul></ul></ul>
  272. 273. Legal issues <ul><li>Gambling in UK </li></ul><ul><ul><li>online gambling: recently legalized, regulated, tax-free (temporary), growing fast </li></ul></ul><ul><li>Caribbean </li></ul><ul><ul><li>Legal, less well regulated </li></ul></ul><ul><ul><li>WSE co-founder arrested upon return to US; intends to challenge law in court; only arrest for offshore bookmaker accepting bets from US </li></ul></ul><ul><ul><li>No individual US bettor has ever been charged </li></ul></ul><ul><li>Great collection of articles: </li></ul><ul><ul><li>http://www.msnbc.com/NEWS/ARCHROULETTE_FRONT.asp </li></ul></ul>

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