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 stakeA 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

Winners earn $2 per bet plus stake back: Winners have stake returned then divide the winnings among themselves

Bets

Total Amount Bet = $6

Outcome: Horse 2 wins

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 Germany France

1 0.75 10 1 1 1

2 0.35 5 1

3 0.40 10 1 1 1

4 0.95 10 1 1 1 1

5 0.75 5 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 EvolutionCall Auction Mechanisms Dynamic Market Makers

2002 – Bossaerts, Fine, and LedyardIssues with double auctions that can lead to thinly traded marketsCall auction mechanism can solve this problem

2003 – Fortnow, Killian, Pennock and WellmanSolution technique for the call auction mechanism

2005 – Lange and EconomidesNon-convex call auction formulation with unique state prices

2005 – Peters, So and YConvex programming of call auction with unique state prices

2003 – HansonCombinatorial information market design

2004, 2006 – Pennock, Chen, and DooleyDynamic Pari-mutuel market

2007 – Chen and PennockCost function based market

2007 – Peters, So and YDynamic market-maker implementation of call auction mechanism

LP Pari-mutuel Market MechanismBoosaerts et al. [2001], Lange and Economides [2001],

Fortnow et al. [2003], Yang and Ng [2003], Peters et al. [2005], etc

Nkx

Nklx

Sizxa

xazx

k

kk

kkik

kkiki

kkk

0

s.t.

max max }){(

An LP pricing mechanism for the call auction market

World Cup Betting Results

Orders Filled

Order Price Limit

Quantity Limit

Filled Argentina Brazil Italy Germany France

1 0.75 10 5 1 1 1

2 0.35 5 5 1

3 0.40 10 5 1 1 1

4 0.95 10 0 1 1 1 1

5 0.75 5 5 1 1

Argentina Brazil Italy Germany France

Price 0.20 0.35 0.20 0.25 0.00

State State PricesPrices

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

Nkx

Nklx

Sizxa

xax

xazzx

k

kk

kkik

kkiki

kkk

kkiki

kkk

0

s.t.

(

( max

)

)

This mechanism has a fixed price i for all i

Monetary profit retained by market maker on state i

Worst-case Profit

Expected extra profit

Market maker’sbelief on state i

Belief-Based and Risk Averse

Nkx

Nklx

Sizxa

xazbzx

k

kk

kkik

kkiki

kkk

0

s.t.

( max )

b is a combination weight factor in [0 1]

Worst-case Profit

Expected extra profit

Convex Pari-mutuel Market MechanismPeters et al. [2005], etc

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).

Expected total objective

Njx

Njlx

Sizxa

xazbzx

k

kk

kkik

kkik

ii

kkk

0

s.t.

)ln( max Monetary profit retained by market maker

Market maker’sbelief on state i

Utility-Maximization Interpretation

k

kiki

ik

kk xazbzx )ln(

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:

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]

Quadratic Market Scoring Rule (QMSR)

i)(pbs ii 1ln )(p

ipbs ii )||p||-2( )( 2p

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

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.

) C()C( qvq

Background: Existing Mechanisms IIITheorem (Chen and Pennock 2007) Every scoring rule

admits a cost-function representation, C(q), where

• LMSR:

• QMSR:

Note that the quadratic cost scoring rule cannot guarantee the price/probability vector nonnegative

qeeqqe

q )( 11

4

1 TTT

NbN) C(

1,)(

),()(

i

i

ii

p C

i- Cqs

qp

qp

)( /ln)( i

bqiebC q

Background: Existing Mechanisms IV• Sequential Convex Pari-mutuel Mechanism (Peters et

al. 2007) for an arrival order (a,, 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

lx

zx

zsbxi

ii,zx,

0

, s.t.

)ln( max )(}{

qesa

s

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 )

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.

lx

zx

zux,zx,

0

, s.t.

)( max }{

qesa

ss

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

iip

u

1

with

0*)(*

*

sp

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:

• QMSR*:

• Log-SCPM:

)( /ln)( i

bsiebu s

seesse

s )( 11

4

1 TTT

NbN) u(

i

isbu )ln()(s

Other Utilities

• Linear-SCPM:

• Min-SCPM:

• Exp-SCPM:

scs Tu )(

)min()( ss u

i

bisebu

/)(s

Truthfulness• Our unified framework is an affine maximizer of the

form

so that the general VCG scheme can be applied: let (x*,z*) be the maximizer of above, charge the trader by

• 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.

zxzulxzx )(},max{max },{ aqe

)(

)(

**)*(

)(max }{

zxzu

zzuz

aqe

qe

Efficient Implementation• The VCG scheme involves solving a convex

optimization problem

as mentioned earlier, it can be solved efficiently in double-logarithmic time.

zxzulxzx )(},max{max },{ aqe

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,

In the cost-function market model, C(.) is proper if

The scoring rule is strictly proper if r is the only maximizer.

*)( and ))((max arg * qrqq 0q CCqri

ii

)(maxarg i

iisrS pr p

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:

)(min }{ qeq zuz)c( z

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

Thus, the worst case loss of the market maker is given by a convex optimization problem

)()( 0q CC

})({maxmax)(

)}()({maxmax 0

i

ii

suC

CCq

i

s0

0q

s

q

Bounds on Worst-Case Loss II

• LMSR:

• QMSR:

• Log-SCPM: unbounded• Linear-SCPM: unbounded• Min-SCPM: 0 (extreme risk averse)• Exp-SCPM:

)ln(WCL Nb

N

Nb

1WCL

)ln(WCL Nb

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

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.

xaxqciz ii )(

Controllable Risk Measure II

• The risk measure used is of form

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.

)()]([min ppp LizES

Controllable Risk Measure III

• LMSR:

• 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:

• Linear-SCPM: unbounded• Min-SCPM: 0 • Exp-ECPM:

)|( UbL)L( KL pp

)|( UbL)L( ll pp

)|( UbL)L( KL pp

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

Horses Ranks

11100

00000

00000

01100

00010

Bid Matrix

Ho

rse

s

Ranks

00100

10000

00001

01000

00010

Ranks

Ho

rse

s

Realized Permutation Matrix

Proportional Betting Market

Reward = $3

(Agrawal et al. 2008)

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:

2||||4

11max vves sv bN

) u( T

Desirable Properties of Quad-SCPM

• Efficient computation for price update: yes• Truthfulness (Myopic): yes• Properness: yes and strictly • Bounded worst-case loss: identical to QMSR

• Controllable risk measure of market-maker: yes

N

Nb

1WCL

2|||| Ub)L( pp

(Agrawal et al. 2009)

General Trading Market: LP Mechanism

Nkx

Nklx

Sibxa

x

k

kk

kikik

kkk

0

s.t.

max

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

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.

lx0

qbsAx

sπxsx

, s.t.

)( max }{ u,

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

*)(* sp u

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.