A Unified Framework for Dynamic
Pari-mutuel Information Market
Yinyu Ye
Stanford University
Joint work with Agrawal (CS), Delage (EE),
Peters (MS&E), and Wang (MS&E)
Outline
•
•
•
•
•
•
Information Market
Pari-mutuel Information Market
LP Pari-mutuel Mechanism
Dynamic Pari-mutuel Mechanism
Sequential Convex Pari-mutuel Mechanism
Desired Properties of SCPM and New
Mechanism Design
• Extensions to General Trading Market
What is Information Market
• A place where information is aggregated via
market for the primary purpose of forecasting
events.
• Why:
– Wisdom of the Crowds: Under the right conditions
groups can be remarkably intelligent and possibly
smarter than the smartest person.
James Surowiecki
– Efficient Market Hypothesis: financial markets are
“informationally efficient”, prices reflect all
known information
Sport Betting Market
• Market for Betting the World Cup Winner
– Assume 5 teams have a chance to win the World Cup:
Argentina, Brazil, Italy, Germany and France
Options for the Market
• Double Auction: Let participants trade directly with one another
– Requires participants to find someone to take the other side of their
order (i.e.: the complement of the set of teams which they have selected)
– Appropriate method for markets with small number of states and large
number of participants
• Centralized Market Maker
– Introduce a market maker who will accept or reject orders that he
receives from market participants
– Market organizer may be exposed to some risk
– This approach works better in thinly traded markets
• Greater liquidity can be induced by allowing multi-lateral order
matching
• Lower transaction costs (no search costs for the participants)
• Problem: How should the market organizer fill orders in
such a manner that he is not exposed to any financial risk?
Central Organization of the Market
• Belief-based
• Central organizer will determine prices for each
state based on his beliefs of their likelihood
• This is similar to the manner in which fixed
odds bookmakers operate in the betting world
• Generally not self-funding
• Pari-mutuel
• A self-funding technique popular in horseracing
betting.
Pari-mutuel Market Model I
• Definition
–Etymology: French pari-mutuel, literally, mutual stake
A system of betting on races whereby the winners
divide the total amount bet in proportion to the sums
they have wagered individually (after deducting
management expenses).
• Example: Parimutuel Horseracing Betting
Horse 1
Horse 2
Horse 3
Bets
Total Amount Bet = $6
Outcome: Horse 2 wins
Winners earn $2 per bet plus stake back: Winners
have stake returned then divide the winnings among
themselves
World Cup Betting Market
• Market for World Cup Winner
– We’d like to have a standard payout of $1 per share if
a participant has a winning order.
• Combinatorial Orders
Order
Price
Limit 
Quantity
Limit q
Argentina
Brazil
Italy
1
0.75
10
1
1
1
2
0.35
5
3
0.40
10
1
4
0.95
10
1
5
0.75
5
Germany
France
1
1
1
1
1
1
1
1
Pari-mutuel Market Model II
• Combinatorial Betting Language in the Market
– N possible states of the world (one will be realized)
– n participants who, trader k, submit orders to a
market maker containing the following information:
• ai,k - state bid (either 1 or 0)
• πk – bid price per share
• lk – limit on share quantity
• Market maker will determine the following:
• xk – order fill/# of awarded shares
• pi – state price/beliefs/probabilities
• Call or dynamic auction mechanism is used.
• If an order is accepted and correct state is
realized, market maker will pay the winning
order a fixed amount $1 per share.
Research Evolution
Call Auction Mechanisms
2002 – Bossaerts, Fine, and Ledyard
Issues with double auctions that
can lead to thinly traded markets
Call auction mechanism can solve
this problem
2003 – Fortnow, Killian, Pennock and
Wellman
Solution technique for the call
auction mechanism
2005 – Lange and Economides
Non-convex call auction
formulation with unique state
prices
2005 – Peters, So and Y
Convex programming of call
auction with unique state prices
Dynamic Market Makers
2003 – Hanson
Combinatorial information
market design
2004, 2006 – Pennock, Chen, and
Dooley
Dynamic Pari-mutuel market
2007 – Chen and Pennock
Cost function based market
2007 – Peters, So and Y
Dynamic market-maker
implementation of call auction
mechanism
LP Pari-mutuel Market Mechanism
Boosaerts et al. [2001], Lange and Economides [2001],
Fortnow et al. [2003], Yang and Ng [2003], Peters et al. [2005], etc
max

k
xk  z ( max i { aik xk })
k
s.t.
a
k
x z
ik k
iS
k
xk  l k
kN
xk  0
kN
An LP pricing mechanism for the call auction market
World Cup Betting Results
Orders Filled
Order
Price
Limit
Quantity
Limit
Filled
Argentina
Brazil
Italy
1
0.75
10
5
1
1
1
2
0.35
5
5
3
0.40
10
5
1
4
0.95
10
0
1
5
0.75
5
5
Germany
France
1
1
1
1
1
1
1
1
State Prices
Price
Argentina
Brazil
Italy
Germany
France
0.20
0.35
0.20
0.25
0.00
Other Issues
• How to make state prices unique
• How to create initial funding to the market
• How to incorporate the market maker’s own
belief into the market
Non-convex formulation with unique state
prices/beliefs by Lange and Economides [2005]
Belief-Based and Risk Neutral
Expected extra
profit
Worst-case
Profit
max

k

k
xk  z    i ( z  aik xk )

k
k
xk   i ( aik xk )
k
s.t.
a
ik
k
xk  z
iS
k
xk  l k
kN
xk  0
kN
This mechanism has a fixed price i for all i
Monetary
profit
retained by
market
maker on
state i
Market
maker’s
belief on
state i
Belief-Based and Risk Averse
Expected extra
profit
Worst-case
Profit
max

k
xk  z  b  i ( z  aik xk )
k
s.t.
a
k
x z
ik k
iS
k
xk  l k
kN
xk  0
kN
b is a combination weight factor in [0 1]
Convex Pari-mutuel Market Mechanism
Peters et al. [2005], etc
Expected total
objective
max

k
xk  z  b  i ln( z  aik xk )
k
s.t.
a
i
x z
ik k
k
iS
k
xk  l k
 jN
xk  0
 jN
Monetary
profit
retained
by market
maker
Market
maker’s
belief on
state i
Theorem (Peters et al. 2005) Convex
programming of call auction has unique state
prices p(b) that are identical to those of the
non-convex formulation of Lange and
Economides (2005).
Utility-Maximization Interpretation
Let the concave and increasing utility function be
ln(.), and i be the market maker’s probability belief
on state i. Then, the objective of the market maker is
the worst case profit combined with an expected
utility value of the contingent state realization. Here,
b is a positive combination weight factor:

k
k
xk  z  bi ln( z  aik xk )
i
k
Dynamic Pari-mutuel Market Model
• Traders come one by one with order (a, , l)
• Market maker has to make an order-fill
decision as soon as an order arrives
– may need to accept bets that do not have a
matching bet yet.
• Market maker still hopes
– to pay the winners almost completely from the
stakes of losers
– to update state prices reflect the traders'
aggregated belief on outcome states
Desirable Properties of Mechanisms
• Efficient computation for price update
• Truthfulness (in myopic sense)
– Bidding true value of a bet should be dominant strategy for each
trader (if he or she is a one-time trader)
• Properness
– A dominant strategy for traders is to place bets on outcome states so
that resulting price reflects his or her true belief
– stronger condition than truthfulness
• Bounded worst-case loss
– Net amount the market maker may have to pay the winners at the
end
• Risk attitude of the market-maker
– Market organizer takes certain risk when accepting bets that are not
matched by the current bets in the system
– The risk attitude of market maker determines the dynamics of
market
– extreme risk averseness implies that no bet will ever get accepted.
Background: Existing Mechanisms I
• Market Scoring Rule
– Traders report their beliefs/prices, p, on outcome states
directly
– Payment is determined by a scoring rule, si( p ), on
reported price vector p in the probability simplex
S={ p  0: ∑ pi=1 }
For some positive constant b:
Logarithmic Market Scoring Rule (LMSR) Hanson [2003]
si (p)  bln (pi )  1 i
Quadratic Market Scoring Rule (QMSR)
si (p)  b(2 pi - || p ||2 ) i
Market Scoring Rule
Suppose constant b=0.1 and you bet the distribution
p=(0.2, 0.3, 0.2, 0.25, 0.05)
on the five teams. Then, if Brazil wins, your reward for
each share under (LMSR) is
0.1ln(.3) + 1 = .87
Background: Existing Mechanisms II
• Cost-Function Based Scoring Rule (Chen and
Pennock 2007)
– Trader submits an order quantity characterized by the
vector v, where vi represents the number of shares that
the trader desires over state i
– The total fee charge to the trader
C(q  v)  C(q)
where C( q ) is a cost function of the current outstanding
share quantity vector q.
– Instantaneous price vector would be ∇C( q ) reflecting
aggregated beliefs/probabilities.
Background: Existing Mechanisms III
Theorem (Chen and Pennock 2007) Every scoring rule
admits a cost-function representation, C(q), where
si (p)  qi - C (q), i
p  C (q) ,  pi  1
i
• LMSR:
C (q)  b ln ( e
qi / b
)
i
• QMSR:
eT q 1 T
1 T
C(q) 

q (1  ee )q
N
4b
N
Note that the quadratic cost scoring rule cannot guarantee the
price/probability vector nonnegative
Background: Existing Mechanisms IV
• Sequential Convex Pari-mutuel Mechanism (Peters et
al. 2007) for an arrival order (a,, l )
max { x,s,z} x  (  i b ln( si ))  z
i
s.t.
ax  s  ze  q,
0  x l
where q is the current outstanding share quantity vector,
e is the vector of all ones, and x it the order fill variable.
• Prices are the optimal Lagrange multipliers of the
convex optimization problem
Background: Existing Mechanisms V
It turns out that one can use the KKT optimality
conditions to create a quick update scheme to solve the
SCPM model for an arrival order, instead of needing to
solve the full convex program each time.
Theorem (Peters et al. 2007) The SCPM problem
can be solved in double-logarithmic time, that is,
log log(1/ε) arithmetic operations.
The computational complexity of the three
described mechanisms are essentially identical.
Questions
• What are the common features and
differences among these mechanisms?
• Why some properties are satisfied or
unsatisfied by a mechanism?
– What type of cost-functions imply a valid
scoring rule?
• How to compare and rationalize different
mechanisms
• Is there new and better mechanism yet to be
discovered?
In this Work
A unified framework is developed that
• subsumes existing mechanisms
• establishes necessary and sufficient conditions
for satisfying certain desirable properties
• provides a tool for designing new mechanisms
with all desirable properties
Unified Pari-mutuel Market Mechanism
• Generalized Sequential Convex Pari-mutuel
Mechanism for an arrival order (a,, l )
max { x,s,z} x  u (s)  z
s.t.
ax  s  ze  q,
0  x l
where q is the current outstanding share quantity vector,
e is the vector of all ones, x it the order fill variable, and
u(s) is any (expected) concave and increasing value
function of slack shares retained by the market maker.
Prices in SCPM
• Market maker maximization principle: the unified
framework is to balance market maker’s revenue
from the arrival trader and (future) value
• Prices/beliefs are the optimal Lagrange multipliers of
the convex optimization problem with maximizers
(x*,s*,z*), and they are
p*  u (s*)  0
with
p
*
i
i
1
The Main Results
• Every scoring rule or cost function mechanism is the
SCPM corresponding to a specific concave and
increasing value function.
• Conversely, every concave and increasing value function
in SCPM induces a scoring rule or cost function
mechanism and can be truthfully implemented.
• The properties of the value function and its derivatives,
such as boundedness, smoothness, span, etc, determine
other desired or undesired properties of the mechanism,
such as the worst-case loss, properness, risk-attitude, etc.
Value Functions of Existing Mechanisms
• LMSR:
u(s)  b ln ( e  si / b )
i
• QMSR*:
• Log-SCPM:
eT s 1 T
1 T
u(s) 
 s (1  ee )s
N 4b
N
u (s)  b ln( si )
i
Other Utilities
• Linear-SCPM:
• Min-SCPM:
• Exp-SCPM:
u (s)  c s
T
u (s)  min( s)
u (s)  b e
i
 si / b
Truthfulness
• Our unified framework is an affine maximizer of the
form
max {x, z}  max{ x, l}  u( ze  q  ax)  z
so that the general VCG scheme can be applied: let
(x*,z*) be the maximizer of above, charge the trader by
(max {z} u ( ze  q)  z )
 (u ( z * e  q  ax*)  z*)
• Corollary (Agrawal et al. 2009) For fixed a and l, the
one time trader will truthfully bet , his or her
valuation of one share of a, in general SCPM.
Efficient Implementation
• The VCG scheme involves solving a convex
optimization problem
max {x, z}  max{ x, l}  u( ze  q  ax)  z
as mentioned earlier, it can be solved efficiently
in double-logarithmic time.
Properness I
• Definition The scoring rule s(.) is proper if the
optimal strategy for a selfish trader is to report his or
her private belief r, that is,
r  arg max pS  ri si (p)
i
In the cost-function market model, C(.) is proper if
q*  arg max q  0  ri (qi  C (q)) and r  C (q*)
i
The scoring rule is strictly proper if r is the only
maximizer.
Properness II
• Proper Market Scoring Rule → SCPM
Theorem (Agrawal et al. 2009) Any proper market
scoring rule with cost function C( q ) can be
formulated as SCPM with u( s ) = - C( - s )
• SCPM → Proper Market Scoring Rule
Theorem (Agrawal et al. 2009) The SCPM gives a
proper market scoring rule if ∇u(.) spans the simplex S;
and a strictly proper rule if u(.) is smooth. The SCPM
also gives an implicit cost function:
c(q)  min { z} z  u ( ze  q)
Properness III
• LMSR: Strictly proper
• QMSR: Strictly proper
• Log-SCPM: Strictly proper
• Linear-SCPM: Not proper
• Min-SCPM: Proper but not strictly
• Exp-SCPM: Strictly proper
Bounds on Worst-Case Loss I
• Definition Worst-case net amount that market maker
may have to pay the winners.
For outstanding share quantities q, the traders have paid
C (q)  C (0)
Thus, the worst case loss of the market maker is given by
a convex optimization problem
max i {max q
0
qi  C (q)  C (0)}
 C (0)  max i {max s u (s)  si }
Bounds on Worst-Case Loss II
• LMSR:
WCL  b ln( N )
• QMSR:
N 1
WCL  b
N
•
•
•
•
Log-SCPM:
unbounded
Linear-SCPM: unbounded
Min-SCPM: 0 (extreme risk averse)
Exp-SCPM:
WCL  b ln( N )
Controllable Risk Measure I
• The return for market maker is random depending on the
actual realization of states in question. Let c be the
money collected so far and qi be the number of shares
already sold on state i . Then, on accepting new order
(a,, l ) with x shares, the return in state i is
z (i)  c  qi  x  ai x
Theorem (Agrawal et al. 2009) The SCPM with concave
and increasing value function is equivalent to choosing x
in order to minimize a convex risk measure on random
return z(i). Moreover, any convex risk measure can be
used to construct an SCPM model with a corresponding
concave value function.
Controllable Risk Measure II
• The risk measure used is of form
min pS Ep [ z(i)]  L(p)
which considers the worst distribution p in terms of
tradeoff between expected return and a penalty function
L(p).
• For many popular mechanisms, penalty function L(p)
represents divergence from a prior distribution, which
presents an learning interpretation of various value
functions used in SCPM.
Controllable Risk Measure III
• LMSR:
L(p)  bLKL (p | U )
• QMSR: has no controllable risk measure! This is due to
the fact that the function is not monotone and it leads to
negative prices
• Log-SCPM:
L(p)  bL (p | U )
ll
• Linear-SCPM: unbounded
• Min-SCPM: 0
• Exp-ECPM:
L(p)  bLKL (p | U )
Design of New Mechanism I
Neither existing mechanism is “perfect”: LMSR
has a unbounded worst-case loss if the number of
states is large, and Log-SCPM is even worse.
QMSR has no controllable risk measure and it even
leads to negative “probabilities”, while Min-SCPM
is overly conservative.
To illustrate this point, we consider an information
market where the number of states is exponentially
large.
Permutation Betting Market
(Agrawal et al. 2008)
Ranks
Horses
Horses
Realized
Permutation Matrix
Horses
Bid Matrix
0
0

0

0
0
1
0
0
0
1
1
0
0
0
0
0
0
0
1
1
Ranks
0
0
0

0
1
0
0

1

0

0
1
0
0
0
0
1
0
0
0
0
0
0
0
1
0
0
0

0

1
0

Ranks
Proportional Betting Market
Reward = $3
Design of New Mechanism II
Is there a “perfect” market mechanism?
The answer is “yes” and the design tool is the
unified SCPM.
Quad-SCPM:
u(s)  max v  s
1 T
1
2
e v  || v ||
N
4b
Desirable Properties of Quad-SCPM
(Agrawal et al. 2009)
•
•
•
•
Efficient computation for price update: yes
Truthfulness (Myopic): yes
Properness: yes and strictly
Bounded worst-case loss: identical to QMSR
N 1
WCL  b
N
• Controllable risk measure of market-maker: yes
L(p)  b || p  U ||
2
General Trading Market: LP Mechanism
max

k
xk
k
s.t.
a
x  bi
ik k
iS
k
xk  l k
kN
xk  0
kN
bi: initial supply quantity of resource i;
aik: demand rate of trader k on resource i;
k: revenue per share from trade k;
xk: decision variable of order fill for trader k.
Sequential or Dynamic Trading Market
• Traders come one by one; buy or sell, or
combination, with combinatorial bid(s)
(A,, l )
• Market maker has to make an order-fill
decision as soon as an order arrives
• Market maker still hopes
– to maximize revenue or minimize regret
– to enforce truthful bid prices 
– to control “risk”
Sequential Trading Market Mechanism
• Arrival multiple bids (A,, l ) from a trader
max {x,s} πx  u (s)
s.t.
Ax  s  b  q,
0  x l
where q is the resource quantity committed/sold to
earlier traders, x it the order fill variable vector for the
new orders, and u(s) is any concave and increasing value
function of slack resource quantity, s, retained by the
market maker.
Prices in Trading Market Mechanism
• Market maker maximization principle: to balance the
immediate earning, x, and future revenue, u(s), with
reserve prices p=∇u ( b – q ).
• New (reserve) prices are the optimal Lagrange
multipliers of the convex optimization problem with
maximizers (x*,s*), and they are
p*  u (s*)
Property Design
Choose u(s) such that
• to approximate future revenue and set reserve
prices
• to bound the worst case regret/revenue loss
• to learn resource prices with risk measures
• to establish a proper scoring rule
Charge the trader such that
• traders would bid  truthfully
(Work in Process, Agrawal et al. 2009)
Contribution Summaries
• An SCPM model with necessary and sufficient
conditions for desired properties, such as (myopic)
truthfulness, properness, worst case loss bounds, risk
measure, etc.
• Unify and subsume existing mechanisms: LMSR,
cost-function based market makers including “utility
based market makers” of Chen et al. [2007]
• Belong to affine maximization framework where the
general VCG scheme is applicable
• Provide an efficient and systematic tool for designing
new mechanisms: Quad-SCPM
• Applications to general trading markets and resource
allocation
Open Questions
• Process of multiple combinatorial bids
simultaneously while maintaining truthfulness
and solution efficiency
• Bounds on market maker’s revenue loss and/or
regret for general trading markets
• Mechanism to markets with large number of
states/goods
• Mechanism for general dynamic programming
such as revenue or inventory management.