Computational Issues in Information Markets

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Computational Issues in Information Markets

• Lance Fortnow
• University of Chicago

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ACM Electronic Commerce 2003

• Computation in a Distributed Information Market
• Joan Feigenbaum, Lance Fortnow, David
  PennockRahul Sami
• Betting Boolean Style A Framework for Trading
  Securities Based on Logical Formula
• Lance Fortnow, Joe Kilian, David Pennock,Michael
  Wellman

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Terrorism Markets?

• Pentagon Prepares A Futures Market On Terror
  Attacks - New York Times 7/29/03
• Swiftly, Plan for Terrorism Futures Market Slips
  Into Dustbin of Idea Without a Future NYT
  7/30/03
• Poindexter Will Be Quitting Over Terrorism
  Betting Plan NYT 8/1/03

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After some thought

• A market in terrorism indicators was a good idea
  it just got bad publicity.
• Economic Scene, New York Times 7/31/03
• Betting on Terror What Markets Can Reveal
• Ideas and Trends, New York Times, 8/3/03
• Economics Can't Solve Everything, Can It?
• Economic View, New York Times, 8/3/03

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Information Markets

• Take some future potential event The Cubs will
  win the pennant.
• Create security NL.Pennant.CHC
• Pays 100 if Cubs win.
• Pays 0 otherwise
• Set up a market for trading NL.Pennant.CHC.
• Bids and asks. Long and short selling.

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Market Pricing

• If you believe Prob(Cubs will win)p and are risk
  neutral then you would be willing to buy or sell
  NL.Pennant.CHC for 100p.
• Price (tradesports.com, 4/3/05) 13.0
• Probability 0.130
• Empirical studies have shown better predictive
  value than experts or polls.

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Questions

• Why do these markets aggregator information so
  well and so efficiently?
• How do you handle many different future
  non-independent events?

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Why do markets perform well?

• Information markets efficiently aggregate
  individual information.
• Rational Expectations Equilibrium Nash
  Equilibrium with players having different partial
  information will remain an equilibrium if all
  players have union of all information.
• When can rational equilibrium be achieved?

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Simple Model

• Let f be a Boolean function on N-variables.
• f0,1N?0,1
• We have one security that pays off 1 if f(x) 1
  and 0 if f(x)0.
• We have n players.
• Distribution x drawn from is common knowledge.
• Player i is given xi.

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Betting Rules

• Modified Shapley-Shubik Scheme
• Players make a bid bi of their expected value
  given known information.
• Trading price p set at average of bids bi.
• Player i buys (bi/p)-1 shares at price p.
• Price p is public but bis not revealed.
• New bids are made given updated information.

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Example OR

• Let f(x,y) x OR y.
• Initial distribution uniform. Input x1 and y0.
• Player 1 Knows f(x,y)1 so bids 1.
• Player 2 Prob(f(x,y)1)1/2 so bids 50.
• Price set at 75.
• After looking at bid player 2 realizes player 1
  must have had 1 and will bid 1 next round.

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Example Parity

• f(x,y) x XOR y.
• Initial distribution uniform. Input x1 and y0.
• Player 1 Prob(f(x,y)1)1/2 so bids 50.
• Player 2 Prob(f(x,y)1)1/2 so bids 50.
• Price set at 50.
• No trading occurs and no new information gained.

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When do prices converge to f?

• For what functions do the prices converge to the
  correct value of f(x)?
• Theorem The following are equivalent
• For any initial distribution, the prices for f(x)
  eventually converge to f(x).
• The function f is a weighted threshold function,
  i.e., f(x) 1 if ? wixi gt v for fixed real wi
  and v.

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Inputs as Hypercube Vertices
000
001
101
100
011
010
111
110
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Threshold Function
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110
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Separating Hyperplane
001
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011
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Non-Threshold Function
000
001
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011
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110
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Threshold Function
000
001
101
100
011
010
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110
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Threshold Function
000
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Threshold Function
000
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011
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110
Under uniform distribution player with 1st bit
bids 0.25, 2nd bit bids 0.75, 3rd bit bids 0.5
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Threshold Function
000
001
101
100
011
010
111
110
Because different solutions lead to different
bids, trading price can give information.
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If Trading Price Does Not Converge
001
000
101
100
011
010
111
110
Expected value of threshold function for zero
inputswill be same for one inputs.
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If Trading Price Does Not Converge
001
000
101
100
011
010
111
110
Expected value of threshold function for zero
inputswill be same for one inputs. CONTRADICTION
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Non-Threshold Function
000
001
101
100
011
010
111
110
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Non-Threshold Function
000
001
101
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011
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111
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Pick point in intersection of convex hull of 0
inputs andconvex hull of 1 inputs.
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Non-Threshold Function
000
001
101
100
011
010
111
110
Use that point to create a distribution where
players bidsdo not distinguish 0 inputs from 1
inputs.
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Efficiency Concerns

• In most natural cases, these markets converge to
  correct answer very quickly.
• We show that if market converges, it converges in
  at most n rounds.
• We give an example where market requires n/2
  rounds to converge.

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Other Directions

• Specific Distributions
• Dani Modk function for kgt2 converges over the
  uniform distribution.
• Future Research Questions
• Show quick updates to small change of
  information.
• Can one use a circuit of threshold functions
  (neural net) to make markets more efficient?

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Experimental Results

• Experiments performed at Penn State by Yiling
  Chen and Tony Kwasnica
• Using five agents (inputs) each gets A/B signal
• Uniform Distribution
• Majority (at least three A)
• Parity (odd)

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Parity Treatment
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Majority Treatment
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How do we handle many securities?

• What if we have lots of securities that reflect
  different states of the future, yet have some
  dependence among them?

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Tradesports.com 8/31/04
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Compound Securities

• (IL and NJ) or (not IL and not NJ)
• This security pays off 100 if Bush wins or loses
  both Illinois and New Jersey.
• Not derivable as a linear combination of base
  securities.

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Example from Tradesports

• Ohio Bid 63.4 Ask 66.9
• Florida Bid 57.5 Ask 59.5
• OH AND FL Bid 55.0 Ask 56.8
• FL AND (NOT OH) FL (FL AND OH)
• Buy FL at 59.5
• Sell OH AND FL for 55.0
• Cost of FL AND (NOT OH) is 4.5

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Large Number of Securities

• 2250 possible functions over the fifty base
  securities corresponding to the states.
• Only 250 securities needed to span the space of
  all possible functions.
• Wont be enough liquidity for nearly all possible
  securities.

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Trading different securities

• Bid of 40 for (ME and not IN)
• Ask of 30 for (not IN)
• Market maker can sell (ME and not IN)and buy
  (not IN), pocketing 10
• If Bush wins Maine and loses Indiana
• Both securities payoff
• Maker nets 10

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Trading different securities

• Bid of 40 for (ME and not IN)
• Ask of 30 for (not IN)
• Market maker can sell (ME and not IN)and buy
  (not IN), pocketing 10
• If Bush wins Indiana
• Both Securities do not payoff
• Maker nets 10

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Trading different securities

• Bid of 40 for (ME and not IN)
• Ask of 30 for (not IN)
• Market maker can sell (ME and not IN)and buy
  (not IN), pocketing 10
• If Bush loses both Indiana and Maine
• (ME and not IN) does not require payment
• Maker receives 100 for (not IN)
• Maker makes 110

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Slightly More Complicated Example

• Ask of 40 for TX
• Bid of 10 for (TX and not FL)
• Bid of 20 for (TX and FL)
• Maker can sell TX and buy (TX and not FL) and (TX
  and FL).
• Maker pockets 10.
• In every case payoffs will cancel out.

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Complexity of Matching

• What is the computational complexity of finding a
  matching in a set of buy and ask orders?
• The answer depends on two factors
• The number of base securities.
• Whether we allow orders to be partially filled.

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Indivisible Orders

• Ask of HI at 30
• Ask of CT at 30
• Ask of (HI xor CT) at 30
• Bid of (HI or CT) at 50
• A match is impossible if can only buy or sell
  zero or one unit of each security.

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Divisible Orders

• Ask of HI at 30
• Ask of CT at 30
• Ask of (HI xor CT) at 30
• Bid of (HI or CT) at 50

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Divisible Orders

• Ask of HI at 30 Maker buys ½
• Ask of CT at 30 Maker buys ½
• Ask of (HI xor CT) at 30 Maker buys ½
• Bid of (HI or CT) at 50 Maker sells 1

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Divisible Orders

• Ask of HI at 30 Maker buys ½
• Ask of CT at 30 Maker buys ½
• Ask of (HI xor CT) at 30 Maker buys ½
• Bid of (HI or CT) at 50 Maker sells 1
• Maker pays 3x(30/2)45 and receives 50.
• Maker clears 5.
• Securities payoffs cancel.

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Divisible Orders

• Ask of HI at 30 Maker buys ½
• Ask of CT at 30 Maker buys ½
• Ask of (HI xor CT) at 30 Maker buys ½
• Bid of (HI or CT) at 50 Maker sells 1
• Suppose Bush wins Hawaii and Connecticut
• Owes 100 for full share of (HI or CT)
• Receives 50 for each ½ share of HI and CT

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Complexity of Finding a Match

• Suppose we have n securities.
• Large number of base securities (about n)
• Small number of base securities (about log n)
• Are securities allowed to be divisible?

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Complexity of Matching Problem
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Complexity of Matching Problem
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Complexity of Matching Problem
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Complexity of Matching Problem
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Complexity of Matching Problem
Reduction from True ??-Formulas
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Complexity of Matching Problem
Remains complete even if securities are of form
(WI and not TN)
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Extensions

• Conditional Securities
• Bush.2004DEM.2004.VP.Edwards
• Far more complicated payoffs and securities
  described by circuits instead of formula.
• Same complexity bounds hold for these extensions.

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Future Directions

• Other matching rules
• Maximize utility subject to no risk
• Maximize expected utility perhaps with risk
• How to distribute surplus made by maker?
• Trader Optimization Issues
• How to choose securities, prices
• Complexity of bids
• Approximation and Heuristic Algorithms

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A Physics Approach to Research

• Our models overly simplify the real world.
• Any model that takes into account all of the
  factors of a financial market will be too hard to
  properly analyze.
• Approach
• Make unrealistic assumptions of the world.
• Test results in experimental economics labs.
• Repeat.