Prediction Market Alternatives for Complex Environments€¦ · Decentralization Conference Tulane...
Transcript of Prediction Market Alternatives for Complex Environments€¦ · Decentralization Conference Tulane...
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Prediction Market Alternatives for ComplexEnvironments
Paul J. Healy (OSU) John Ledyard (Caltech)Sera Linardi (Caltech) Richard Lowery (CMU)
Decentralization ConferenceTulane University
Apr. 5, 2008
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The Success of Prediction Markets
Wall St. market: 1848–1940 (Rhode & Strumpf 2004)
11/15 correct in mid-October, only 1 very wrong (Wilson 1916)
Iowa Electronic Markets (Berg et al. 2003)
See figure...
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The Success of Prediction Markets
Avg. Error: 1.5% vs. 2.1%. Source: Berg,Forsythe, Nelson & Rietz (2003)
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The Success of Prediction Markets
Wall St. market: 1848–1940 (Rhode & Strumpf 2004)
11/15 correct in mid-October, only 1 wrong (W. Wilson)
Iowa Electronic Markets (Berg et al. 2003)
See figure...But... Erikson & Wlezien use trends in polls
TradeSports (Tetlock, Wolfers, Zitzewitz, others...)
Trade volume during Davidson vs. Kansas ≈ 7,700 $10 tickets
NewsFutures, Hollywood Stock Exchange (Pennock et al. 2001)
See figure...
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The Success of Prediction Markets
Source: Wolfers & Zitzewitz (2004)
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Corporate Applications
Predicting printer sales at Hewlett-Packard (K-Y Chen & Plott 2002)
Companies claiming to use prediction markets:
Abbot Labs Arcelor Mittal Best Buy ChryslerCorning Electronic Arts Eli Lilly Frito Lay
General Electric Google Hewlett-Packard IntelInterContinental Hotels Masterfoods Microsoft Motorola
Nokia Pfizer Qualcomm SiemensTNT
Are they doing it ‘right’? Volume? Complexity??
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The Policy Analysis Market (PAM)
2001–2003 DARPA (DoD) => NetExchange (Ledyard, Polk, Hanson)
Goal: Predict events the DoD might care about
NetExchange focus: political instability in Middle East
A subset of the issues:
Correlation blows up the state spaceManipulation? (Camerer 98, Strumpf & Rhode 07)Moral Hazard? (Hanson et. al 07)Moral repugnance & P.R. (Roth 07, Hanson 07)
Aug 03: Shut Down, DARPA audited, Poindexter ‘retired’
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This Paper
Questions:
1 Can markets actually work when the environment gets ‘complicated’?
2 Would other mechanisms do better?
Answers:
Test markets vs. 3 other mechs in complex lab environments
1 Market performs poorly; incentived, iterated polls perform better
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Behavioral Mechanism Design
Methodology of comibining experiments & theory to design bettermechanisms for real-world use
Short run goal: find a better mechanism
1 Propose alternative mechanisms
Existing theory & behavioral data as guides
2 Testbed proposed mechanisms
The control of the laboratory
3 Tweak if necessary
Long run goal: improve the design process
1 Identify general principles while testbedding2 Add new constraints, etc., to the design problem
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This Paper
Questions:
1 Can markets actually work when the environment gets complicated?
2 Would other mechanisms do better?
Answers:
Test markets vs. 3 other mechs in complex lab environments
1 Market falls apart, simple iterated polls perform better
2 Why the poll seems to do better in this environment
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Easy vs. Hard Environments
Example similar to our experiment:
1 Simple: Will UNC beat Kansas tonight?
Two states: UNC,Kansas, one security
2 Hard: Who will win each of the last 3 games (2 semi’s and final)?
Three events, not independentEight states: UNC,Kansas×Memphis,UCLA×East,West“Eastern finalist wins” is correlated with other 2 eventsIncomplete set of securities is typically used
TradeSports offers 6 securities (1+1+4)
We will use a complete set of 8
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2-State Environment
Two coins: θ ∈ Θ = X ,Y
Two flip outcomes: ω ∈ Ω = H,T
State of the world = (θ, ω)
θ and ω are correlated, but we care only about ω
p (H |X ) = 0.2p (H |Y ) = 0.4
One coin θ is randomly drawn (50/50)
Each subject sees flips of chosen coin
si = (H , T , H , H)#si ∈ 2, 3, 4
Observe sample ω’s, infer about true θ, predict true ω
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Distributions: 2 States
p0
0
1
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Distributions: 2 States
p2
p1
p0
0
1
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Distributions: 2 States
p2
p1
p0p2
p1
p3
0
1
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Distributions: 2 States
p2
p1
p0p2
p1
p3
pFI
0
1
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Distributions: 2 States
p2
p1
p0p2
p1
p3
pFIh
0
1
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Distributions: 2 States
p2
p1
p0p2
p1
p3
pFIh
0
1
E
R
R
O
R
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8-State Environment
Three coins, ordered: θ ∈ Θ = XYZ ,XZY ,YXZ ,YZX ,ZXY ,ZYX
Eight flip outcomes:ω ∈ Ω = TTT ,TTH,THT ,THH,HTT ,HTH,HHT , HHH
Pr [X = H ] = 0.2Pr [Y = H ] = 0.4Pr [Z = H ] = 0.4Pr [Y = X ] = 2/3 (correlation distinguishes Y , Z )
One ordering θ is randomly drawn (uniformly)
Each subject sees flips from chosen coin ordering
si = (THT , THT , HHH , TTT , HTH)#si ∈ 3, 5, 7
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
p0
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
p0
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
p3
p1
p2
p0
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
pFI p3
p1
p2
p0
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
pFI p3
p1
p2
h
p0
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Distributions: 8 States
p6
p1
p2
p3
p4
p5
pFI
h
p0
ERROR
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The Mechanisms
1 Double Auction (prediction market)
2 Pari-mutuel (horse track)
3 Iterated Poll (‘Delphi method’: RAND/USAF)
4 Market Scoring Rule (Hanson 2003)
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Alternative Mechanisms: Pari-Mutuel
Bettors buy tickets on each event
nj = # of tickets purchased on event j
Payoff odds of event-j tickets = (nj/ ∑k nk)−1
Still need 2k securities
Still have a no-trade theorem
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Alternative Mechanisms: Poll
Players announce a belief distribution P i over the 8 events
P = (1/n) ∑i Pi is shown
Repeat 5 times
Everone paid based on final average distribution P
Incentive compatible scoring rule:
Everyone receives(
ln[
Pj
]
− ln [1/8])
event-j securitiesIf event k is true, event-k security pays $1.
There exist many seq. equil. with full info aggregation
There exist babbling seq. equil. with “almost” no aggregation
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Alternative Mechanisms: Market Scoring Rule (Hanson)
A public distribution is shown: (1/8, . . . , 1/8)
Individuals may ‘move’ the distribution to(
P i1, . . . ,P i
8
)
Move from (Q1, . . . ,Q8) to(
P i1, . . . ,P i
8
)
=⇒
Receive(
ln[
P ij
]
− ln[
Q ij
])
event-j securities for each j
Moving Pj up means buying, down means sellingIf event k is true, event-k security pays $1Incentive compatible: you should move to your best guess
Subsidized =⇒ avoids no-trade theorem
Incentive compatible =⇒ myopic players reveal truthfully
Incentive to misrepresent? Depends on move timing...
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Methodology
Run experiments using Caltech undergrads paid ≈ $35
No experience
Crossover design: DA-Poll, Poll-DA, MSR-Pari, Pari-MSR
3 subjects per group
8 periods with each mechanism
No rematching
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Period & Order Effects
1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Period
2 States
1 2 3 4 5 6 7 80
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Period
8 States
1 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Order
2 States
1 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Order
8 States
No significant period or order effects (good!)
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2 States: Error
Comparison of l2 distances with 2 states:
Avg Wilcoxon p-valuesDist. DblAuctn MSR Parimutuel Poll
Avg Dist. − 0.262 0.419 0.295 0.266
DblAuctn 0.262 − 0.092 0.646 0.663MSR 0.419 − − 0.225 0.098
Parimutuel 0.295 − − − 0.519Poll 0.266 − − − −
MSR ≥ Parimutuel ≥ Poll ≥ DblAuctnMSR > Poll ≥ DblAuctn
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2 States: Confusion & Mirages
p2
p1
p0
pFI
h
0
1
C
O
N
F
U
S
E
D
C
O
N
F
U
S
E
D
p2
p1
p0
pFI
h
0
1
M
I
R
A
G
E
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2 States: Catastrophes
Periods with catastrophes:
(32 pers. total) DblAuc MSR Pari Poll
No Trade 0 1 4 0Confusion 5 7 6 11
Mirage 13 14 10 12Confused Mirage 0 1 1 3
None 14 12 13 12
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2 States: Summary of Results
2 States 8 StatesMech Err NoTrd Mirg Conf Err NoTrd Mirg Conf
DblAuc X X X X
MSR × X × X
Pari X × X X
Poll X X X ×
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8 States: Error
Comparison of l2 distances with 8 states:
Avg Wilcoxon p-valuesl2 Dist. DblAuc MSR Parimutuel Poll
Avg l2 Dist. − 0.696 0.527 0.605 0.418
DblAuc 0.696 − 0.002 0.093 < 0.001
MSR 0.527 − − 0.083 0.324Parimutuel 0.605 − − − 0.001
Poll 0.418 − − − −
DblAuc > Parimutuel > MSR ≥ Poll
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8 States: Catastrophes: No Trade
DblAuc MSR Parimutuel Poll
Periods w/ No Trade 0 0 9/32 0
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8 States: Confusion
p6
p1
p2
p3
p4
p5
pFI
h
p0
CONFUSED
Pr (TTT |θ) = 24/75 & Pr (HHH|θ) = 4/75 ∀θ
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8 States: Confusion
p6
p1
p2
p3
p4
p5
pFI
p0
hCONFUSED
Confusion occurs in every period of every mechanism
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8 States: Catastrophes: Confusion
l2 distance to convex hull, conditional on trade occurring:
Avg Dist. DblAuc MSR Pari. Poll
Avg. Dist. 0.447 0.362 0.398 0.312# Trade Pers. 32 32 23 32
DblAuc 0.447 − 0.001 0.107 < 0.001
MSR 0.362 − 0.180 0.257Pari 0.398 − 0.008
Poll 0.312 −
DblAuc ≥ Pari ≥ MSR ≥ PollDblAuc > MSR ≥ PollDlbAuc ≥ Pari > Poll
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8 States: Catastrophes: Mirages
Frequency of Mirages:
Pers. w/ No. ofTrade Mirages Frequency
DblAuc 32 13 0.406MSR 32 7 0.219Pari. 23 7 0.304Poll 32 3 0.094
DblAuc > MSR > PollPari > Poll
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8 States: Summary
2 States 8 StatesMech Err NoTrd Mirg Conf Err NoTrd Mirg Conf
DblAuc X X X X × X × ×MSR × X × X X X X X
Pari X × X X × × X ×Poll X X X × X X X X
Increased complexity: Double auction fails, MSR & Poll work
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Declaring a Winner?
Poll’s only failing: confusion in 2-states. How bad is it?
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
Pr(H)
Fre
quen
cyPoll Output: 2 States
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Beating the Prior
Percentage of periods where mechanism outperformed the “informed”prior:
2 States 8 States
DblAuc 0.375 0.000MSR 0.355 0.250Pari 0.393 0.044Poll 0.406 0.313
Poll looks good (relatively)...
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Observations
Why does the poll out-perform the market?
Observation 1: Preferences are aligned in the poll, so traders haveno incentive to misrepresent
‘Misrepresentor’: Move away from full info, then move toward
Number of misrepresentors per mechanism:
DblAuc MSR Pari Poll
14 5 12 3
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Observations
Observation 2: Traders have an incentive to participate in the poll
No-trade theorem in DblAuc and Parimutuel
MSR and poll are subsidized
25.9 cents/trader/period in 2-state35.0 cents/trader/period in 8-state
Pari-mutuel no trade: 4/32 and 9/32 pers.
DblAuc: 1 inactive trader in 4/64 periods
MSR: 1 period of no trade (1st period confusion?)
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Observations
Observation 3: Attention is ‘spread thin’ in the DblAuc
States Txns/Min. Vol./Min.
2 5.00 6.488 2.60 14.47
% of txns on 2 most active securities: 46%
% of txns on 2 least active securities: 8%
Low-hanging fruit is missed:
p (TTT ) = 24/75 and p (HHH) = 4/75 regardless of pvt infoAvg |p (TTT )− 24/75| and Avg |p (HHH)− 4/75| are greater thanany other mechanismSignificantly greater than MSR and Poll
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Observations
Observation 4: Poll averages traders’ announcements, mitigatingeffects of a single aberrant trader
Frequency of worse-than-average final reports & predictions
2 States 8 StatesMech Last Report Prediction Last Report Prediction
DblAuc 11 11 24 24MSR 18 18 9 9
Pari-mutuel 11 11 9 9Poll 28 8 21 8
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Summary
Double auction works fine with 2 states, not 8
Observation: think markets problem (focus on 2 securities)Note: not market power problem
Pari-mutuel hurt by delay and no trade
MSR helps ‘unfocus’ attention, but prone to bad outcomes
Single ‘bad’ player can damage performance
Poll performs best
Aligned incentives, participation incentives, averaging smoothsbehavior, completely ‘unfocused’
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The End
The End
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p(TTT) and p(HHH)
Recall that p (TTT ) = 24/75 and p (HHH) = 4/75 regardless of signals
DblAuc MSR Pari Poll
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4D
ista
nce
Box Plot of | p(TTT) − 24/75 |
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p(TTT) and p(HHH)
Recall that p (TTT ) = 24/75 and p (HHH) = 4/75 regardless of signals
DblAuc MSR Pari Poll
0
0.05
0.1
0.15
0.2D
ista
nce
Box Plot of | p(HHH) − 4/75 |
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Double Auction: Market Thinness
8 states vs. 2 states:
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Double Auction: Market Thinness
8 states vs. 2 states:
Fewer transactions per minute (2.60 vs. 5.00), despite more markets!
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Double Auction: Market Thinness
8 states vs. 2 states:
Fewer transactions per minute (2.60 vs. 5.00), despite more markets!More total volume per minute (14.47 units vs. 6.48 units)
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Double Auction: Market Thinness
8 states vs. 2 states:
Fewer transactions per minute (2.60 vs. 5.00), despite more markets!More total volume per minute (14.47 units vs. 6.48 units)
Fraction of trades in each market:
Session TTT TTH THT THH HTT HTH HHT HHH
1 0.18 0.13 0.15 0.04 0.01 0.15 0.12 0.232 0.25 0.17 0.07 0.03 0.15 0.05 0.14 0.123 0.34 0.04 0.26 0.07 0.14 0.02 0.02 0.114 0.16 0.07 0.23 0.12 0.17 0.09 0.07 0.09
All 0.27 0.11 0.16 0.05 0.14 0.05 0.09 0.12
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Double Auction: Market Thinness
8 states vs. 2 states:
Fewer transactions per minute (2.60 vs. 5.00), despite more markets!More total volume per minute (14.47 units vs. 6.48 units)
Fraction of trades in each market:
Session TTT TTH THT THH HTT HTH HHT HHH
1 0.18 0.13 0.15 0.04 0.01 0.15 0.12 0.232 0.25 0.17 0.07 0.03 0.15 0.05 0.14 0.123 0.34 0.04 0.26 0.07 0.14 0.02 0.02 0.114 0.16 0.07 0.23 0.12 0.17 0.09 0.07 0.09
All 0.27 0.11 0.16 0.05 0.14 0.05 0.09 0.12
‘FOCUS’ = standard deviation of these distributions
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Double Auction: Market Thinness
8 states vs. 2 states:
Fewer transactions per minute (2.60 vs. 5.00), despite more markets!More total volume per minute (14.47 units vs. 6.48 units)
Fraction of trades in each market:
Session TTT TTH THT THH HTT HTH HHT HHH
1 0.18 0.13 0.15 0.04 0.01 0.15 0.12 0.232 0.25 0.17 0.07 0.03 0.15 0.05 0.14 0.123 0.34 0.04 0.26 0.07 0.14 0.02 0.02 0.114 0.16 0.07 0.23 0.12 0.17 0.09 0.07 0.09
All 0.27 0.11 0.16 0.05 0.14 0.05 0.09 0.12
‘FOCUS’ = standard deviation of these distributions
Focus, # transactions, and trading volume don’t predict accuracywell.
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MSR: Market Thinness
Is MSR ‘trading’ focused on a small number of securities?
Session TTT TTH THT THH HTT HTH HHT HHH
1 0.23 0.12 0.11 0.09 0.17 0.12 0.09 0.072 0.36 0.14 0.17 0.02 0.14 0.06 0.02 0.083 0.22 0.16 0.13 0.08 0.10 0.03 0.07 0.214 0.20 0.13 0.06 0.10 0.16 0.10 0.10 0.16
All 0.24 0.13 0.11 0.08 0.14 0.08 0.08 0.14
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MSR: Market Thinness
Is focus predictive of market accuracy?
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
Is activity in a security predictive of its price accuracy?
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
Is activity in a security predictive of its price accuracy?
ERRORt,ω = 0.659− 0.053× NUMMOVESt,ω
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
Is activity in a security predictive of its price accuracy?
ERRORt,ω = 0.659− 0.053× NUMMOVESt,ω
p-value = 0.002, R2 = 0.03
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
Is activity in a security predictive of its price accuracy?
ERRORt,ω = 0.659− 0.053× NUMMOVESt,ω
p-value = 0.002, R2 = 0.03Avg. # moves/period in MSR = 2.03
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MSR: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.007 + 0.202× FOCUSt
p-value < 0.0001, R2 = 0.52Avg. focus in MSR = 0.290Avg. focus in Dbl Auction = 0.475
Is activity in a security predictive of its price accuracy?
ERRORt,ω = 0.659− 0.053× NUMMOVESt,ω
p-value = 0.002, R2 = 0.03Avg. # moves/period in MSR = 2.03Avg. # txns/period in DA = 1.63
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problem
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securities
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securitiesCompetitive =⇒ may hide info
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securitiesCompetitive =⇒ may hide info
Parimutuel is okay when people trade
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securitiesCompetitive =⇒ may hide info
Parimutuel is okay when people trade
No-trade theorem is most obvious here?
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securitiesCompetitive =⇒ may hide info
Parimutuel is okay when people trade
No-trade theorem is most obvious here?
Poll works best with 8 states, good with 2 states
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Mechanism Comparison: Summary
DblAuc’n works with 2 states, fails with 8
Thin markets problemNOT a market power problem
MSR performs poorly for 2 states, OK with 8
Less focusing on securitiesCompetitive =⇒ may hide info
Parimutuel is okay when people trade
No-trade theorem is most obvious here?
Poll works best with 8 states, good with 2 states
Incentives are aligned
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Conclusions
Dbl Auction is not best for all environments
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
Want to ‘move’ many prices at once
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
Want to ‘move’ many prices at once
No-trade theorem and subsidies
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
Want to ‘move’ many prices at once
No-trade theorem and subsidies
Parimutuel vs. Dbl Auction??
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
Want to ‘move’ many prices at once
No-trade theorem and subsidies
Parimutuel vs. Dbl Auction??MSR and Poll are subsidized
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Conclusions
Dbl Auction is not best for all environments
Thin market problem
Want to ‘move’ many prices at once
No-trade theorem and subsidies
Parimutuel vs. Dbl Auction??MSR and Poll are subsidizedBetter performance worth the cost?
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameter
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distribution
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
p0 (ω): Prior distribution on Ω
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
p0 (ω): Prior distribution on Ω
p (ω|θ) = pθ (ω): Conditional distribution
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
p0 (ω): Prior distribution on Ω
p (ω|θ) = pθ (ω): Conditional distribution
Signal: s = (s1, . . . , sn)
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
p0 (ω): Prior distribution on Ω
p (ω|θ) = pθ (ω): Conditional distribution
Signal: s = (s1, . . . , sn)
si =(
si1, . . . , siKi
)
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General Environment
States of the world: (θ, ω) ∈ Θ × Ω
θ is an unknown parameterω is randomly drawn, given θ
Prior beliefs (given)
f (θ, ω): Joint distributionq0 (θ): Prior distribution on Θ
p0 (ω): Prior distribution on Ω
p (ω|θ) = pθ (ω): Conditional distribution
Signal: s = (s1, . . . , sn)
si =(
si1, . . . , siKi
)
sik ∼ p (·|θ) (sik ∈ Ω)
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
sik ∼ p (ω|θ∗) (sik ∈ Ω)
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
sik ∼ p (ω|θ∗) (sik ∈ Ω)
Step 3: Use Bayes’s Law to form various beliefs about θ given signals
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
sik ∼ p (ω|θ∗) (sik ∈ Ω)
Step 3: Use Bayes’s Law to form various beliefs about θ given signals
Individual: q (θ|si ) ‘Full-info’: q (θ|s1, . . . , sn)
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
sik ∼ p (ω|θ∗) (sik ∈ Ω)
Step 3: Use Bayes’s Law to form various beliefs about θ given signals
Individual: q (θ|si ) ‘Full-info’: q (θ|s1, . . . , sn)
Step 4: Form posteriors on Θ given signals:
pi (ω) = ∑θ∈Θ
p (ω|θ) q (θ|si )
pFI (ω) = ∑θ∈Θ
p (ω|θ) q (θ|s1, . . . , sn) .
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General Environment
States: (ω, θ) ∈ Θ × Ω with pdf f (ω, θ) (correlated)
Care about ω, but only have information on θ
Step 1: Draw ‘true’ state (ω∗, θ
∗)
Step 2: Draw signals si = (si1, . . . , siKi) of ω
∗ given θ∗
sik ∼ p (ω|θ∗) (sik ∈ Ω)
Step 3: Use Bayes’s Law to form various beliefs about θ given signals
Individual: q (θ|si ) ‘Full-info’: q (θ|s1, . . . , sn)
Step 4: Form posteriors on Θ given signals:
pi (ω) = ∑θ∈Θ
p (ω|θ) q (θ|si )
pFI (ω) = ∑θ∈Θ
p (ω|θ) q (θ|s1, . . . , sn) .
Step 5: Run the mechanism, which generates h (ω) over Ω.
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Order Effects
1 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Order
2 States
1 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
l 2 Dis
tanc
e
Order
8 States
Pairwise Wilcoxon tests: p = 0.82 or 0.93
By mechanism: p ≥ 0.39 or ≥ 0.07 (Pari. worse)
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2 States: Catastrophes: Confusion
Average l2 distance to convex hull, conditional on confusion:
DblAuc 0.0011MSR 0.0340Pari 0.0185Poll 0.0050
Double Auction & Poll don’t get as ‘badly’ confused
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2 States: Catastrophes: Confusion
Average l2 distance to convex hull, conditional on confusion:
DblAuc 0.0011MSR 0.0340Pari 0.0185Poll 0.0050
Double Auction & Poll don’t get as ‘badly’ confused
Is slight confusion any better than bad confusion??
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2 States: Mechanism Error Without Confusion
Average mechanism error, conditional on NO confusion:
No Confusion All Periods
DblAuc 0.128 0.131MSR 0.136 0.210Pari 0.110 0.148Poll 0.093 0.133
No significant differences in pairwise tests
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8 States: Catastrophes: Confusion
p6
p1
p2
p3
p4
p5
pFI
h
p0
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8 States: Catastrophes: Confusion
p6
p1
p2
p3
p4
p5
pFI
h
p0
CONFUSED
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8 States: Catastrophes: Confusion
p6
p1
p2
p3
p4
p5
pFI
p0
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8 States: Catastrophes: Confusion
p6
p1
p2
p3
p4
p5
pFI
p0
h
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8 States: Catastrophes: Confusion
p6
p1
p2
p3
p4
p5
pFI
p0
hCONFUSED
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8 States: Catastrophes: Mirages
p6
p1
p2
p3
p4
p5
pFI
p0
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8 States: Catastrophes: Mirages
p6
p1
p2
p3
p4
p5
pFI
hp0
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8 States: Catastrophes: Mirages
p6
p1
p2
p3
p4
p5
pFI
hp0
MIRAGE
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8 States: Catastrophes: Mirages
An alternative definition:
Of the 8 probabilities, 6 should move from p0
How many (out of 6) move the right way?
Mean MSR Pari Poll
DblAuc 3.03 0.049 0.046 0.034
MSR 3.69 0.798 0.239Pari 3.70 0.467Poll 3.97
Wilcoxon p-values
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8 States: Mirages
Wilcoxon test on ‘angles’ between(
h − p0)
and(
pFI − p0)
:
MSR Pari Poll
DblAuc 0.025 0.490 0.290MSR 0.180 0.773Pari 0.286Poll
Wilcoxon test p-values
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.218− 0.070× FOCUSt , p-value = 0.222
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.218− 0.070× FOCUSt , p-value = 0.222
Is trading volume in a security predictive of its price accuracy?
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.218− 0.070× FOCUSt , p-value = 0.222
Is trading volume in a security predictive of its price accuracy?
ERRORt,ω = 0.954− 0.003× VOLUMEt,ω, p-value = 0.200
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.218− 0.070× FOCUSt , p-value = 0.222
Is trading volume in a security predictive of its price accuracy?
ERRORt,ω = 0.954− 0.003× VOLUMEt,ω, p-value = 0.200
Is # of txns predictive of price accuracy?
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Double Auction: Market Thinness
Is focus predictive of market accuracy?
ERRORt = 0.218− 0.070× FOCUSt , p-value = 0.222
Is trading volume in a security predictive of its price accuracy?
ERRORt,ω = 0.954− 0.003× VOLUMEt,ω, p-value = 0.200
Is # of txns predictive of price accuracy?
ERRORt,ω = 0.941− 0.011× NUMTXNSt,ω, p-value = 0.367
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