Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Computational Aspects of Prediction Markets
David M. Pennock and Rahul Sami
December 5, 2012
Presented by: Rami Eitan
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Table of contents
1
Prediction Markets
2
Partition model of knowledge
3
Distributed information markets
4
Convergence time bounds
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Prediction markets
Many traders, each holds some partial information.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Prediction markets
Many traders, each holds some partial information.
Goal: to aggregate all the traders’ information and learn
about the true state of the world.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Prediction markets
Many traders, each holds some partial information.
Goal: to aggregate all the traders’ information and learn
about the true state of the world.
Ideally the price of a security should converge to a value that
reflects all the information about it known to the traders.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Prediction markets
Many traders, each holds some partial information.
Goal: to aggregate all the traders’ information and learn
about the true state of the world.
Ideally the price of a security should converge to a value that
reflects all the information about it known to the traders.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Computational aspects
Total number of possible securities.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Computational aspects
Total number of possible securities.
Complexity of updating the price after each round.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Computational aspects
Total number of possible securities.
Complexity of updating the price after each round.
How many iterations before the price converge, if at all?
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Computational aspects
Total number of possible securities.
Complexity of updating the price after each round.
How many iterations before the price converge, if at all?
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Partition model of knowledge
Let Ω be a set of possible states of the world. At any point of
time, the world is in exactly one state ω ∈ Ω.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Partition model of knowledge
Let Ω be a set of possible states of the world. At any point of
time, the world is in exactly one state ω ∈ Ω.
Each agent i has partial information about the state of the
world represented by a collection of subsets of Ω.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Partition model of knowledge
Let Ω be a set of possible states of the world. At any point of
time, the world is in exactly one state ω ∈ Ω.
Each agent i has partial information about the state of the
world represented by a collection of subsets of Ω.
Agents can distinguish between different subsets of Ω but not
between states within the same subset.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Partition model of knowledge
Let Ω be a set of possible states of the world. At any point of
time, the world is in exactly one state ω ∈ Ω.
Each agent i has partial information about the state of the
world represented by a collection of subsets of Ω.
Agents can distinguish between different subsets of Ω but not
between states within the same subset.
Given n agents, their combined information π̂ is the coarsest
common refinement of the partitions π1 , π2 ..., πn .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Partition model of knowledge
Let Ω be a set of possible states of the world. At any point of
time, the world is in exactly one state ω ∈ Ω.
Each agent i has partial information about the state of the
world represented by a collection of subsets of Ω.
Agents can distinguish between different subsets of Ω but not
between states within the same subset.
Given n agents, their combined information π̂ is the coarsest
common refinement of the partitions π1 , π2 ..., πn .
We assume a common prior distribution P ∈ ∆(Ω) shared by
all agents before receiving any information.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
let x = {0, 1}n ∈ Ω be the state of the world.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
let x = {0, 1}n ∈ Ω be the state of the world.
Common prior distribution P : {0, 1}n → [0, 1] over the values
of x.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
let x = {0, 1}n ∈ Ω be the state of the world.
Common prior distribution P : {0, 1}n → [0, 1] over the values
of x.
n agents, each has a single bit of information xi .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
let x = {0, 1}n ∈ Ω be the state of the world.
Common prior distribution P : {0, 1}n → [0, 1] over the values
of x.
n agents, each has a single bit of information xi .
Agents are rational, truthful, risk neutral and myopic.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
We would like to know the value of f (x) : {0, 1} → {0, 1}.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
We would like to know the value of f (x) : {0, 1} → {0, 1}.
Agents trade in a Boolean security F which pays 1$ if
f (x) = 1 and 0$ otherwise.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
We would like to know the value of f (x) : {0, 1} → {0, 1}.
Agents trade in a Boolean security F which pays 1$ if
f (x) = 1 and 0$ otherwise.
Trading is done synchoronously with agents placing bids each
round r .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
We would like to know the value of f (x) : {0, 1} → {0, 1}.
Agents trade in a Boolean security F which pays 1$ if
f (x) = 1 and 0$ otherwise.
Trading is done synchoronously with agents placing bids each
round r .
Agents value F according to their expectation of payoff, and
bid truthfully: bir = Ei [f (x)]. For simplicity we assume all
traders place a bid for exactly one security each round.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market
We would like to know the value of f (x) : {0, 1} → {0, 1}.
Agents trade in a Boolean security F which pays 1$ if
f (x) = 1 and 0$ otherwise.
Trading is done synchoronously with agents placing bids each
round r .
Agents value F according to their expectation of payoff, and
bid truthfully: bir = Ei [f (x)]. For simplicity we assume all
traders place a bid for exactly one security each round.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market cont.
At thePend of each round the market clears a new price
p r = i bi /n.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market cont.
At thePend of each round the market clears a new price
p r = i bi /n.
S r denotes the set of commonly known possible states after
round r .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market cont.
At thePend of each round the market clears a new price
p r = i bi /n.
S r denotes the set of commonly known possible states after
round r .
Sir denotes the set of states agent i consider possible after
round r .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Model of an information market cont.
At thePend of each round the market clears a new price
p r = i bi /n.
S r denotes the set of commonly known possible states after
round r .
Sir denotes the set of states agent i consider possible after
round r .
After each round, each agent i can update their Sir by
eliminating all x’s that would result a different price than
what was anounced.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Note that:
S 0 = Ω.
Si0 = {y ∈ Ω|yi = xi }
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Note that:
S 0 = Ω.
Si0 = {y ∈ Ω|yi = xi }
S 1 = {y ∈ S 0 |price1 (y ) = p 1 }, where price1 (x) is the function
that relates each state to a clearing price. Even though x is
unknown, any observer can still calculate the clearing price for
hypothetical x and eliminate those which are different than
what was declared.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Note that:
S 0 = Ω.
Si0 = {y ∈ Ω|yi = xi }
S 1 = {y ∈ S 0 |price1 (y ) = p 1 }, where price1 (x) is the function
that relates each state to a clearing price. Even though x is
unknown, any observer can still calculate the clearing price for
hypothetical x and eliminate those which are different than
what was declared.
Sir = {y ∈ S r |yi = xi }
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Note that:
S 0 = Ω.
Si0 = {y ∈ Ω|yi = xi }
S 1 = {y ∈ S 0 |price1 (y ) = p 1 }, where price1 (x) is the function
that relates each state to a clearing price. Even though x is
unknown, any observer can still calculate the clearing price for
hypothetical x and eliminate those which are different than
what was declared.
Sir = {y ∈ S r |yi = xi }
Sir ⊆ Sir −1 and S r ⊆ S r −1
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Note that:
S 0 = Ω.
Si0 = {y ∈ Ω|yi = xi }
S 1 = {y ∈ S 0 |price1 (y ) = p 1 }, where price1 (x) is the function
that relates each state to a clearing price. Even though x is
unknown, any observer can still calculate the clearing price for
hypothetical x and eliminate those which are different than
what was declared.
Sir = {y ∈ S r |yi = xi }
Sir ⊆ Sir −1 and S r ⊆ S r −1
A convergence is reached after a finite number of steps. We
denote: S ∞ , Si∞ and p ∞ .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Example: OR function
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Example: Carry bit function
the function Cn takes 2n bits as input - (x, y ) and returns the carry
bit for x + y . Let us examine the function C2 with the following
distribution:
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Example: Carry bit function
the function Cn takes 2n bits as input - (x, y ) and returns the carry
bit for x + y . Let us examine the function C2 with the following
distribution:
The pair (x1 , y1 ) takes on the values (0, 0), (0, 1), (1, 0), (1, 1)
uniformly.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Example: Carry bit function
the function Cn takes 2n bits as input - (x, y ) and returns the carry
bit for x + y . Let us examine the function C2 with the following
distribution:
The pair (x1 , y1 ) takes on the values (0, 0), (0, 1), (1, 0), (1, 1)
uniformly.
(x2 , y2 ) has a distribution conditional on (x1 , y1 ). If
(x1 , y1 ) = (1, 1), then (x2 , y2 ) takes the values
(0, 0), (0, 1), (1, 0), (1, 1) with probabilities 12 , 16 , 16 , 16
respectively. Otherwise with probabilities 16 , 16 , 16 , 12
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Rational expectations
Rational expectations equilibrium
A rational expectations equilibrium is a mapping P ∗ : Ω → ℜ s.t in
every state ω, if every trader conditions her demand (or supply) on
her private information πi and the price P ∗ (ω), the market will
clear at the price of exactly P ∗ (ω).
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Stochastically monotone function
A function g : ℜn → ℜ is calledPStochastically monotone if it can
be written in the form g (x) = i gi (xi ) where each gi : ℜ → ℜ is
strictly increasing.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Equilibrium price characterization
Stochastically monotone function
A function g : ℜn → ℜ is calledPStochastically monotone if it can
be written in the form g (x) = i gi (xi ) where each gi : ℜ → ℜ is
strictly increasing.
Stochastically regular function
A function g : ℜn → ℜ is called Stochastically regular if it can be
written in the form g = h ◦ g ′ where g ′ is stochastically monotone
and h is invertible on the range of g ′ .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Nielsen et al. 1990
Shared knowledge at equilibrium
Suppose that, at equilibrium, the n agents have a common prior,
but a possibly different information about the value of F . For all i ,
let pi∞ = E (F |x ∈ Si∞ ). if g is stochastically monotone and
g (p1∞ , p2∞ ...pn∞ ) is common knowledge then:
p1∞ = p2∞ = ... = pn∞ = E (F |x ∈ Si∞ ) = p ∞
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Nielsen et al. 1990
Shared knowledge at equilibrium
Suppose that, at equilibrium, the n agents have a common prior,
but a possibly different information about the value of F . For all i ,
let pi∞ = E (F |x ∈ Si∞ ). if g is stochastically monotone and
g (p1∞ , p2∞ ...pn∞ ) is common knowledge then:
p1∞ = p2∞ = ... = pn∞ = E (F |x ∈ Si∞ ) = p ∞
This means that at equilibrium all agents must have exactly the
same expectation of the value of the security and that this mus
agree with the expectation of an uninformed observer. Is
equilibrium enough for the purpose of information aggregation?
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Nielsen et al. 1990
Shared knowledge at equilibrium
Suppose that, at equilibrium, the n agents have a common prior,
but a possibly different information about the value of F . For all i ,
let pi∞ = E (F |x ∈ Si∞ ). if g is stochastically monotone and
g (p1∞ , p2∞ ...pn∞ ) is common knowledge then:
p1∞ = p2∞ = ... = pn∞ = E (F |x ∈ Si∞ ) = p ∞
This means that at equilibrium all agents must have exactly the
same expectation of the value of the security and that this mus
agree with the expectation of an uninformed observer. Is
equilibrium enough for the purpose of information aggregation?
No. We also want that p ∞ = f (x)
Example: XOR with a uniform distribution.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Characterizing computable aggregates
weighted threshold function
A function f : {0, 1}n → {0, 1} is a weighted threshold function iff
there are w1 , w2 ...wn ∈ ℜ s.t
f (x) = 1 iff
n
X
wi xi ≥ 1
i =1
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Characterizing computable aggregates
weighted threshold function
A function f : {0, 1}n → {0, 1} is a weighted threshold function iff
there are w1 , w2 ...wn ∈ ℜ s.t
f (x) = 1 iff
n
X
wi xi ≥ 1
i =1
Equilibrium of weighted threshold functions
If f is a weighted threshold function, then for any prior probability
distribution, the equilibrium price of F is equal to f (x).
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof:
p ∞ = E (F |x ∈ Si∞ ) thus, if p ∞ = 0 or 1 then f (x) = p ∞
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof:
p ∞ = E (F |x ∈ Si∞ ) thus, if p ∞ = 0 or 1 then f (x) = p ∞
Assume towards contradiction that 0 < p ∞ < 1. from Nielsen et
al. we get:
P(f (y ) = 1|y ∈ S ∞ ) = p ∞
∀i P(f (y ) = 1|y ∈
David M. Pennock and Rahul Sami
Si∞ )
=p
∞
Computational Aspects of Prediction Markets
(1)
(2)
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof:
p ∞ = E (F |x ∈ Si∞ ) thus, if p ∞ = 0 or 1 then f (x) = p ∞
Assume towards contradiction that 0 < p ∞ < 1. from Nielsen et
al. we get:
P(f (y ) = 1|y ∈ S ∞ ) = p ∞
∀i P(f (y ) = 1|y ∈
Si∞ )
=p
∞
(1)
(2)
since Si∞ = {y ∈ S ∞ |yi = xi } Equation (2) can be written as:
∀i P(f (y ) = 1|y ∈ S ∞ , yi = xi ) = p ∞
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
(3)
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Define:
Ji+ =P(yi = 1|y ∈ S ∞ , f (y ) = 1)
Ji− =P(yi = 1|y ∈ S ∞ , f (y ) = 0)
n
X
wi Ji+
J+ =
J− =
i =1
n
X
wi Ji−
i =1
Since p ∞ 6= 0, 1 both Ji+ , JI− are well defined for all i .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Define:
Ji+ =P(yi = 1|y ∈ S ∞ , f (y ) = 1)
Ji− =P(yi = 1|y ∈ S ∞ , f (y ) = 0)
n
X
wi Ji+
J+ =
J− =
i =1
n
X
wi Ji−
i =1
Since p ∞ 6= 0, 1 both Ji+ , JI− are well defined for all i .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Claim: Equations (1) and (3) imply that Ji+ = Ji− for all i
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Claim: Equations (1) and (3) imply that Ji+ = Ji− for all i
Proof: Assume xi = 1, we then have:
P(f (y ) = 1|y ∈ S ∞ ) · Ji+
P(f (y ) = 1|y ∈ S ∞ ) · Ji+ + P(f (y ) = 0|y ∈ S ∞ ) · Ji−
= P(f (y ) = 1|yi = 1, y ∈ S ∞ ) (Bayes’ law)
p ∞ · Ji+
= p ∞ by Eqs. (1) and (3)
p ∞ · Ji+ + (1 − p ∞ )Ji−
⇒ Ji+ = p ∞ Ji+ + (1 − p ∞ )Ji−
⇒ Ji+ = Ji−
The proof is symmetrical for xi = 0
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Using linearity of expectation we can write:
#
!
" n
X
wi yi y ∈ S ∞ , f (y ) = 1
Ji+ = E
i =1
#
!
" n
X
∞
−
wi yi y ∈ S , f (y ) = 0
Ji = E
i =1
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof cont.
Using linearity of expectation we can write:
#
!
" n
X
wi yi y ∈ S ∞ , f (y ) = 1
Ji+ = E
i =1
#
!
" n
X
∞
−
wi yi y ∈ S , f (y ) = 0
Ji = E
i =1
Since f is a weighted threshold function, f (y ) = 1) iff
and thus Ji+ ≥ 1, similarly Ji− < 1 which implies
And so we have a contradiction. David M. Pennock and Rahul Sami
n
P
i =1
Ji+ 6= Ji−
wi yi ≥ 1
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Upper bound on the number of iterations
In the worst case, at most n rounds of trading are required to
reach an equilibrium.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Upper bound on the number of iterations
In the worst case, at most n rounds of trading are required to
reach an equilibrium.
Each set S 0 , S 1 . . . S n has a strictly lower dimension than the
previous on until the market reaches an equilibrium.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Upper bound on the number of iterations
In the worst case, at most n rounds of trading are required to
reach an equilibrium.
Each set S 0 , S 1 . . . S n has a strictly lower dimension than the
previous on until the market reaches an equilibrium.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Upper bound on the number of iterations
In the worst case, at most n rounds of trading are required to
reach an equilibrium.
Each set S 0 , S 1 . . . S n has a strictly lower dimension than the
previous on until the market reaches an equilibrium.
Dimension
The dimension of a set S ⊆ {0, 1}n is the dimension of the smallest
affine subset of ℜn that contains all points in S Denoted dim(S).
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Upper bound on the number of iterations
In the worst case, at most n rounds of trading are required to
reach an equilibrium.
Each set S 0 , S 1 . . . S n has a strictly lower dimension than the
previous on until the market reaches an equilibrium.
Dimension
The dimension of a set S ⊆ {0, 1}n is the dimension of the smallest
affine subset of ℜn that contains all points in S Denoted dim(S).
Lemma
If S r 6= S r −1 , then dim(S r ) < dim(S r −1 ).
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof
Let k = dim(S r −1 ). Consider the bids in round r :
bir = E (f (y ) = 1|y ∈ S r −1 , yi = xi )
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof
Let k = dim(S r −1 ). Consider the bids in round r :
bir = E (f (y ) = 1|y ∈ S r −1 , yi = xi )
(0)
(1)
Depending on the value of xi , bir is either hi or hi . These
depend only on S r −1 which is common knowledge.
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
Proof
Let k = dim(S r −1 ). Consider the bids in round r :
bir = E (f (y ) = 1|y ∈ S r −1 , yi = xi )
(0)
(1)
Depending on the value of xi , bir is either hi or hi . These
depend only on S r −1 which is common knowledge.
(1)
(0)
Denote di = hi − hi and we get:
n
1 X (0)
(hi + di xi )
p =
n
r
i =1
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
(4)
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
proof cont.
Equation (4) defines a hyperplane the intersection of which with
S r −1 is the resulting common knowledge S r .
(0)
This is because all agents already know all hi and di and so they
use the linear equation (4) to rule out any possibility that would
not have resulted the price p r .
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
proof cont.
Equation (4) defines a hyperplane the intersection of which with
S r −1 is the resulting common knowledge S r .
(0)
This is because all agents already know all hi and di and so they
use the linear equation (4) to rule out any possibility that would
not have resulted the price p r .
If S r 6= S r −1 , this intersection defines a linear subspace of
dimension (k-1) that contains S r and so S r has a dimension of at
most (k − 1).
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
Prediction Markets
Partition model of knowledge
Distributed information markets
Convergence time bounds
proof cont.
Upper bound on the number of iterations
Let f be a weighted threshold function, and P an arbitrary
distribution. Then after at most n rounds of trading, the price
reaches its equilibrium value p ∞ = f (x)
This follows directly from the Lemma as dim(S 0 ) = n and once we
have a round r s.t S r = S r −1 then no trader has gained any new
knowledge and thus p ∞ = f (x)
David M. Pennock and Rahul Sami
Computational Aspects of Prediction Markets
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