A Utility Framework for
Bounded-Loss Market Makers
Yiling Chen (Yahoo! Research),
David M. Pennock (Yahoo! Research)
2007
Presentation by: Yonatan Herzig
21/11/12
What will we see?
Outline
Introduction
 Background
 Utility-Based Market Makers
 Relationship Between MSR and UtilityBased Market Makers
 Cost-Function Formulation of Market
Makers
 Liquidity and Market Maker Loss

Introduction - 1 / 2
In prediction markets price discovery is the
market designer’s end goal and trading is a
means to that end.
 Liquidity is needed for price discovery.
 Prediction market are expected to be thinly
traded, since meeting trader demand is not the
market’s primary goal.

Introduction - 2 / 2
An automated market maker can improve liquidity.
 The market maker continually announces prices offering
both to buy and to sell some quantity of the security,
adjusting his prices in programmatic response to trader
demand.
 a prediction market designer may well subsidize a
market maker that expects to lose some money.
 In return for improving trader incentives, liquidity, and
price discovery.
 This paper presents a framework for Bounded-Loss
Market Makers

Background

1.
2.
3.
Scoring rules:
let v represent a discrete random variable with N mutually
exclusive and exhaustive outcomes.
let r  ( r1 , r2 ,..., rN ) be a probability estimate for variable v.
A scoring rule is a list of scoring functions, S  s1 (r ), s2 (r ),..., sN (r )
such that a score si (r ) is assigned to r if outcome i of the
random variable v is realized.

MSR (Hanson [2003, 2007] ):
1.
A patron subsidizes an automated market maker in order to
improve liquidity. The patron loses no more than a fixed constant.
The market maker begins by setting an initial probability estimate,
r 0 . Every trader can change this probability to a new one if he is
willing to pay the appropriate scoring rule.
The scoring rule: si (r )  ai  b log(ri )
(b  0)
2.
3.
Utility-Based Market Makers
What is the utility of money?


Risk averse
The utility function, u(m), of money is a measure for the
satisfaction associated with a particular level of wealth.
The marginal utility, u’(m), depicts how much more satisfied an
agent will be with gaining more wealth. It indicates the agent’s
desire to take risks for gaining more – his risk aversion.
Risk indifferent
Risk loving
Utility-Based Market Makers
Risk neutral probability






u(m) – utility function of money.
 - Subjective probability estimate for the forecast variable.
m - Wealth vector across outcome results.
The agent’s risk neutral probabilities:
Risk-neutral probabilities are the price levels that the agent is
indifferent between buying or selling an infinitely small number of
shares.
Equal to subjective probability when agent is indifferent to risk.
Utility-based market makers
The main idea: The market maker equates the
instantaneous security prices to his current risk-neutral
probabilities , and is always willing to accept infinitely
small buy or sell orders of Arrow-Debreu contracts at
these prices.
 A trader comes and buy an infinitely small number of
security i,  i . Then, the market maker’s wealth in state j  i
increases by pi i , and for state i, decreases by (1  pi ) i
 Let
be the vector of the total number
of outstanding shares.
 Then:

Utility-based market makers

Thus, this equation system defines a utility
based market maker:
Loss of utility based market makers

At any time of the market, a utility-based market maker satisfies:
where k is constant.

Proof: Lets derivate by qi:
 (   j u( m j ))
j
qi
   j u '( m j ) pi   i u '(mi )( pi  1) 
j i
 pi   j u '( m j )   i u '( mi ) 
j
 i u '(mi )  j u '(m j )

j
j
u '( m j )
  i u '(mi ) 
j
0

Thus, the expected utility is constant during all process of trading. A
market maker can choose initial wealth vector,
, to start the
market, and k equals
Loss of utility based market makers

A market maker has bounded loss if and only if there exists a real
constant l such that for any m that meets
, l ≤ mj is
satisfied for all j.

For a real-valued, continuous and strictly increasing utility function u(m),
any fixed subjective probability estimate π whose elements are nonzeros,
and a utility-based market maker who sets prices according to its rules:

the necessary and sufficient condition for the market maker to
have bounded loss for any feasible expected utility level k is that at
least one of the following conditions is satisfied:
The domain of u(m) is bounded below
The range of u(m) is bounded above but not bounded below
1.
2.
Loss of utility based market makers
Semi intuitive proof:
 The domain of u(m) is bounded below
If the market maker plays by the defined rules, u(m) is always
defined, so m is always bounded below.
 The range of u(m) is bounded above but not bounded
below
lets assume m is not bounded below, so at a certain point for
some i: mi   , then:
  u ( m )  k     k    u ( m )   u ( m )  
j
j
j
j i
j
j
j i
j
But u(m) is bounded above: a contradiction.
j
Hyperbolic Absolute Risk Aversion
(HARA)








The hyperbolic absolute risk aversion (HARA) class of utility
functions contains most popular parametric families of utilities.
Generic form:
Where M is a real number, γ is an extended real number, and α > 0.
The utility function is defined on the domain
:
: family of constant relative risk aversion (CRRA) utility
functions
:
:
It can be verified that a utility-based market maker using a nonlinear HARA utility function is guaranteed to have bounded loss
Weighted Pseudospherical
Scoring Rules

The scoring rule function of this class is (Jose et al. [2006] ):

Where b>0 and the reported probability estimate r is weighted by
a base-line probability estimate  .
Weighted pseudospherical scoring rules are strictly proper for any
real β.
β→1 :
:
β = 2 and π is uniform : the spherical scoring rule
The quadratic scoring rule (2), however, does not belong to this
class.
A MSR market maker’s loss is bounded when using any one of
these rules (the payment when probability 1 is reported for the
true outcome is finite).





Expected-Utility-Maximization
Problem






show that the weighted pseudospherical scoring rules can arise from
the solution of an expected-utility-maximization problem of a
forecaster against a betting opponent (Jose et al. [2006] ).
A market with a market maker can be viewed as a sequence of twoperson betting, each happening between a trader (A) and the market
maker (B).
This kind of bet can be seen as selecting a vector m , s.t. the wealth
vectors for A and B are m andm , and they both accept according to
their subjective probability estimates on v.
In a utility based market, a (risk neutral) agent will change
into f
for the market maker such that the trader’s expected wealth is
maximized:
The market maker is willing to accept only if the expected utility stays
constant.
We get the next optimization problem:
Relationship Between MSR and
Utility-Based Market Makers
The Market Maker Equivalence Theorem:
 A utility-based market maker who has a subjective
probability estimate  and a HARA utility function with
γ = 0 is equivalent to a market scoring rule market
maker who utilizes a π-weighted pseudospherical
scoring rule with β = 1− 1/γ .
 More specifically, for example:
 A negative exponential utility market maker (γ = ±∞) is
equivalent to an MSR market maker with a weighted
logarithmic scoring rule.
 If starting the market with uniform prior probabilities, a
negative exponential utility market maker is equivalent
to a logarithmic MSR market maker.
Cost-Function Formulation




The market contains a total of N securities, each paying $1 per
share if its corresponding outcome happens.
q is the vector of all quantities of shares held by traders.
Market utilizes a cost function,
, that records the total amount
of money traders have spent as a function of the total number of
shares held of each security.
Agent willing to change
to
, pays the market maker

the going price of security i is: pi (q)  C / qi
the sum of prices always equals 1 to ensure no arbitrage.

Thus, these properties hold:

Cost Functions and UtilityBased Market Makers


utility-based market maker has a cost function that
is defined by:
, where k is constant.
since C ( q ) is the total money collected by the
market maker and qj is the total money that the
market maker needs to pay if outcome j happens.
According to the implicit function theorem in
mathematical analysis, we can see that if the utility
function u(m) is continuous, differentiable, and
strictly increasing, there exists a unique function
C (q)
Cost Functions and UtilityBased Market Makers

Actually what we are trying to solve is (assuming uniform  ):
pi CF  piUB
C ( q )
u '(mi )
u '(C ( q )  qi )


qi
 u '(m j )  u '(C (q )  q j )

j
j
For example, for the logarithmic utility function with b>0:
,u(m)  log(b  m), the cost function for two outcome event is:
Verify it for homework ;)
 And for the negative exponential utility function:


The price functions can be calculated by differentiation.
Cost Functions and Market
Scoring Rules





For scoring rules, such as quadratic coring rule, that do not belong
to the weighted pseudospherical scoring rule class, we describe
how to make the translation directly
In MSR, when a trader wants to change p to p ' gets profit
Under the cost function formulation, A trader with a p ' estimation,
will buy or sell shares, until the market becomes p '.
He changes the quantity to q ', and when outcome i happens, his
profit is
For MSR market maker, and cost function formulation to be
equivalent, the trader should profit the same amount for all i:
Cost Functions and Market
Scoring Rules

Solving the last equation system for LMSR leads to:

For the quadratic scoring rule, we get:

Price functions can be obtained by differentiation.
What did we see?
we have established equivalence translations for restricted classes.
 The cost function formulation seems most natural for implementation purposes.
 Market scoring rules make many analysis such as loss of market makers
straightforward, however directly converting them to cost-function formulations is
not always easy.
 Utility-based market makers, whose cost functions are easy to find, connect the
MSR with cost-formulation for a large class of scoring rules.

Liquidity and Market Maker
Loss

To address liquidity in prediction markets (no bid-ask spread or
market depth), we examine The instantaneous liquidity for security
i at q :

The slope of the price function, ∂pi(q)/∂qi, approximates the bid-ask
spread for the differentiable price function pi(q).
It approximates the price difference between buying one share of
security i from the market maker and selling one share to the
market maker.
Moreover, ∂pi(q)/∂qi inversely relates to the instantaneous depth of
the market for security i. A smaller ∂pi(q)/∂qi means that it requires
more shares to drive price up or down by one unit, hence implying
a deeper market.


Loss and Instantaneous
Liquidity


We consider a class of market makers whose cost functions are
symmetric and second-order differentiable (so price function is
differentiable ).
Symmetric cost functions: if q ' is a permutation of q , then
C ( q )  C ( q ')




Same goes for price functions: They have the same form for every i.
This means that if all qi’s are equal, all pi’s are equal to 1/N.
Lets assume
and
.
a market maker’s worst-case loss is bounded by:
Loss and Instantaneous
Liquidity



Theorem: For all market makers who utilize a second-order
differentiable and symmetric cost function over N outcomes and
maintain an instantaneous liquidity level no lower than ρ for all q ,
the minimum worst-case loss is
Proof: Minimum loss is when liquidity is the lowest. So this happens
i
when: pi ( q )  1 / N  qi / 
Lets calculate the market loss as we’ve learned:
N 1

N
N 1
Lmax   (1  pi ( q ))dq  
(
  qi )dq i 
0
0
N
 ( N  1)2  ( N  1) 2  ( N  1) 2



2
2
N
2N
2N 2

i
i
Questions?
Thank you!