Research Labs
A Dynamic Pari-Mutuel
Market for Hedging,
Wagering, and Information
Aggregation
David M. Pennock
Mike Dooley
Previous Version Appears in EC’04, New York
Research Labs
What is a pari-mutuel market?
A B
• E.g. horse racetrack style wagering
• Two outcomes:
A
B
• Wagers:
Research Labs
What is a pari-mutuel market?
A B
• E.g. horse racetrack style wagering
A
• Two outcomes:
B
• Wagers:
Research Labs
What is a pari-mutuel market?
A B
• E.g. horse racetrack style wagering
A
• Two outcomes:
B
• Wagers:
Research Labs
What is a pari-mutuel market?
A B
• E.g. horse racetrack style wagering
A
• Two outcomes:
B
• 2 equivalent
1+ $ on B = 1+ 8 =$3
ways to consider
$ on A
4
payment rule
• refund + share of B
• share of total
total $ = 12 = $3
$ on A
4
Research Labs
What is a pari-mutuel market?
• Before outcome is revealed, “odds” are
reported, or the amount you would win per
dollar if the betting ended now
• Horse A: $1.2 for $1; Horse B: $25 for $1; … etc.
• Strong incentive to wait
•
•
•
•
payoff determined by final odds; every $ is same
Should wait for best info on outcome, odds
 No continuous information aggregation
 No notion of “buy low, sell high” ; no cash-out
Research Labs
Dynamic pari-mutuel market
Basic idea
1
1
Research Labs
Dynamic pari-mutuel market
Basic idea
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Dynamic pari-mutuel market
Setup & Notation
A1 A2 ... Ak
•
•
•
•
•
Outcomes:
Prices (per share):
Payoffs (per share):
Money wagered on i:
# shares purchased of i:
Ai
pi
Pi
Mi
Si
• Total money:
T
= jMj
• Share-weighted total:
• “Other” money:
• “Share-weighted other” $:
W
T-i
W-i
= jSjMj
= T - Mi
= W - S i Mi
Research Labs
What is a share?
• Two alternatives; Share of
• losing money only
Winners get: original money refunded +
equal share of losers’ money
• all money
Winners get equal share of all money
• For standard PM, they’re equivalent
• For DPM, they’re not
Research Labs
How are prices set?
• A price function pi(n) gives the
instantaneous price of an infinitesimal
additional share beyond the nth
• Cost of buying n shares: 0 pi(n) dn
n
• Different reasonable assumptions lead
to different price functions
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“All money” case
• Payoffs: Pi=T/Si
• Trader’s exp payoff/shr for e shares:
Pr(Ai) (E[Pi,tfinal] -pi) + (1-Pr(Ai)) (-pi)
• Assume: E[Pi,tfinal] = Pi
Pr(Ai) (Pi-pi) + (1-Pr(Ai)) (-pi)
!
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Market probability
• Market probability MPr(Ai)
• Probability at which the expected
value of buying a share of Ai is zero
• “Market’s” opinion of the probability
• MPr(Ai) = pi/ Pi
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Price function derivation
• Not unique; assume constraint, e.g.:
pi/pj = Mi/Mj
•  MPr(Ai) = MiSi/W
•  pi = dMi/dSi = MiT/W
• Given current state (IC), number of
shares received for m add. dollars is:
sharesi(m)=
mSi/T - W-i/T-i(m/T+(T+m)/T-i
Log((T+m)Mi/T/(Mi+m)))
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Price function derivation
• Cost of buying n shares is
costi(n) = sharesi-1(n)
• No closed form;
use e.g. Newton’s method
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Buying a set of outcomes
• Let Q be a set of outcomes
• Key simplifications
• dSi = dSj
• dMi/dMj = const
• Buying a set of outcome Q behaves
like buying a single outcome with
• MQ= iQMi
• SQ = iQSi Mi/jQMj
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Combinatorial market
• Outcomes are base states
• Events are sets of outcomes
• 2k possible events arising from k
states
• Use previous derivation to allow
buying/selling arbitrary events
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Selling
• A key advantage of DPM is the ability
to cash out to lock gains / limit losses
• “All money” case
• Traders simply sell back to the market
maker: double sided liquidity
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Selling
•
“Losing money” case: Each share is
different. Composed of:
1. Original price refunded
priI(A)
where I(A) is indicator fn
2. Payoff
PayI(A)
•
Selling do-able, more complicated;
complexity can be hidden from
traders to a degree
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Other price functions
Share type
Constraint/
Assumption
Result
Losing money
p1= P2
p2= P1
Closed form
cost() & shares()
Losing money
pi/pj = Mi/Mj
Closed form
cost() & shares()
All money
pi/pj = Mi/Mj
Closed form
shares() ;
Numeric cost()
All money
pi/pj = Si/Sj
Closed form
cost() & shares()
Research Labs
Initialization
• Price functions are indeterminate when Mi=0
or Si=0
• Need to “seed” the market with money,
shares per outcome; could come from
• Patron
• Ante
• Capital - transaction fees
• Acts like “b” in MSR
Higher seed  more risk, more initial
liquidity
• Unlike MSR, liquidity increases over time as
shares are purchased
Research Labs
Intermediate redistributions
• For tracking repeated statistic
• Interest rate
• Real estate index
• Oil prices
• Redistribute money according to
statistic at repeated intervals
• pi/pj = Mi/Mj
• No loss of money; continuity
• Traditional MM (incl. MSR): requires
additional subsidy
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Market scoring rule
• Hanson 2002, 2003
• Special case of market maker:
Automated, bounded loss
• Market maker always stands willing to
accept an (infinitesimal) trade at current
prices
• Full cost for some quantity is the integral
over instantaneous prices
n
• One example:
prii(n) = e(n -a )/b
cost(n) =
pri (n)dn
j e(n -a )/b
0
i
i
j
j

Research Labs
Market scoring rule
• Market maker’s loss is bounded by b
• Higher b more risk, more “liquidity”
• Level of liquidity (b) never changes as
wagers are made
• Could charge transaction fee, put back into b
(Todd Proebsting)
• Much more to MSR: sequential shared
scoring rule, combinatorial MM “for free”,
... see Hanson 2002, 2003
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Mechanism comparison
CDA
Bounded liquidity dynamic
info
risk
aggreg.
X


payoff
vector
fixed
Increasing
MM
liquidity

N/A
CDAw
MM
X



X
MSR




X
PM


X
X
N/A
DPM



X

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Future work
• DPM call market
• Combinatorial DPM implementation
• Empirical testing
What dist rule & price fn are “best”?
• Real-valued (0-infinity) outcomes