This document summarizes an approach to analyzing prediction markets using convex optimization. It discusses both offline and online formulations of the problem. The offline formulation involves accepting bids to maximize profit, which can be modeled as a linear program. Uniqueness of the solution is analyzed. The online formulation updates the model sequentially as bids arrive in real time. Various choices for the objective function are discussed, balancing risk aversion with expected gain. Truthfulness of bids is also addressed.

1. Real Time Information Reconstruction:
The Prediction Market.
MS&E 211
Omede Firouz 05547809
Xuechen Jiang 05836587
Yixin Kou 05796061
Raghav Ramesh 05835723
December 8, 2012
Abstract
A survey of the prediction market is given and analyzed in the framework of convex
optimization. Online and offline approaches are compared. Conditions for unique-
ness of solution are given. Computational results are shown for both simulated data
and student bets on football games. Comparisons are made to traditional approaches
such as regression. The methods analyzed are in general theoretically sound with few
weaknesses.
1

2. 1 OFFLINE PROBLEM
Introduction
Prediction is a well studied field with many statistical and machine learning techniques
available. Recently, the crowd-sourcing of prediction markets has become of interest. In
many cases, auctions are the source of such distributed information. Methods to analyze the
large and diverse data that arises have been developed in the context of convex optimization.
1 Offline Problem
Consider the problem of bid acceptance for a centralized market maker. Given a set of bids
on various states, the market maker would like to accept bids in order to maximize his/her
own profit. We will consider the state price security. Let π be the bid prices for each bid, x
be the number of shares sold for each bid, and y be the worst case loss. Then the following
linear program will maximize the worst case profit.
Primal: max π T x − y Dual: min qT s
AT −e x 0 A I p ≥ π
≤
I 0 y q −eT 0T s = −1
x ≥ 0, y free p, s ≥ 0
We can also analyze the dual to see that s are the shadow prices for our share limit con-
straints, the amount our objective would increase if we could sell more shares (ceteris
paribus). p are the state prices, and Ai,∗ p are the internal cost of state i occuring per
share sold (unit cost).
We can further analyze the system through complementarity conditions. It is clear that:
1. If we do not sell any shares, the internal cost is strictly higher than the bid price.
2. If we do sell shares, the bid price is at least as high as the internal cost.
3. If we do not max out the shares we sell, the bid price and internal price are equal.
4. If we max out the shares we sell, the bid price is strictly greater than the internal price.
The dual can be stated as the problem of minimizing the money won qT s subject to, for
all possible winning states i, the beliefs/bets made by the winners are fully utilized. (The
winners’ bets imply their beliefs) [2].
Solving the first two games gives the following State Prices:
State: 1 2 3 4 5 6 7 8
Notre Dame 0 0 .27 .43 .3 0 0 0
California .001 .01 0 .119 .68 .12 .06 .01
In these cases, Stanford beat Notre Dame 20-13 (state 5), and California 21-3 (state 7).
These state price distributions are quite bad since state prices of zero imply any bet will be
accepted. Clearly however, there is some nonzero probability on these outcomes. We explain
such a discrepancy by the small number of students (on the order of 100), undervaluing the
extreme outcomes of the game, and a lack of understanding of the market design. Indeed,
2

3. 3 UNIFIED FRAMEWORK
in later games, the state prices more accurately resembled what an expected probability
distribution would be.
2 Uniqueness
However, the question arises as to whether the state prices are necessarily unique. In fact,
using only the current formulation, the state prices are not guaranteed to be unique. Consider
the case of 2 teams and 1 bid: (1,1) with any price π1 .
max π1 x1 − y min q1 s1
     
1 −1 0 p1
s.t.  1 −1 
x1
≤  0  s.t.
1 1 1  p2  ≥ π1
y −1 −1 0 = −1
1 0 q1 s1
x1 ≥ 0, y f ree p1 , p2 , s1 ≥ 0
Note the first two columns of the dual are identical. This is an ’entanglement’ of the
states 1,2 and the dual variables p1 , p2 . With a linear objective, every point on p1 + p2 = 1,
p1 , p2 ≥ 0 is optimal. To see why, for simplicity, let π1 < 1. Then the primal and dual
objectives are 0 by strong duality and s1 = 0. We have p1 + p2 ≥ π1 , p1 + p2 = 1. The first
constraint is redundant since pi1 < 1, and clearly all p1 + p2 = 1 are optimal.
Theorem. If there is a set of states S such that every bet that includes one element in
S includes all of S, the dual has infinite solutions.
Proof. Every row corresponding to S will have identical rows in the primal. Therefore,
the dual will have identical columns corresponding S. Since the objective is linear, every
combination of pi∈S such that i∈S pi = 1 − j∈S , p ≥ 0 are optimal solutions to the dual.
It is natural to break the above symmetry by choosing the midpoint of all possible
solutions since a corner point’s sensitivity will be higher than the midpoint with slight
changes in the parameters. The midpoint is in a sense the most ’robust’ state price.
We can choose the midpoint similar to choosing the analytic center in an interior point
method. Since we are maximizing, we will use a nondecreasing and strictly concave function
of the slack variables.
3 Unified Framework
Here we formulate bid acceptance with the added function u(s) which is nondecreasing and
strictly concave. Since the feasible region remains unchanged and convex, and the objective
is concave maximization, we still have a linearly constrained convex optimization problem
which can be solved efficiently.
3

4. 3 UNIFIED FRAMEWORK
n
max πj xj − y + u(s)
j=1
n
s.t. aij xj + si = y, ∀i = 1, 2, ..., m Dual Var: p
j=1
0 ≤ xj , ∀j = 1, ..., n Dual Var: µ
xj ≤ qj , ∀j = 1, ..., n Dual Var: ν
si ≥ 0, ∀i = 1, ..., m Dual Var: η
Letting u(s) = i u(si ) we write down the KKT conditions.
m
δf
KKT Conditions Stationarity: : πj x j = aij pi − µj + νj ∀j = 1, ..., n
δxj i=1
m
δf
: −1= −pi
δy i=1
δf δu(s)
: = pi − ηi ∀i = 1, ..., m
δsi δsi
Primal Feasibility: 0 ≤ xj ≤ qj , ∀j = 1, ..., n
0 ≤ si , ∀i = 1, ..., m
Dual Feasibility: pi f ree ∀i = 1, ..., m
µj , νj ≥ 0 ∀j = 1, ..., n
ηi ≥ 0 ∀i = 1, ..., m
Complementary Slackness: µj xj = 0 ∀j = 1, ..., n
νj (qj − xj ) = 0 ∀j = 1, ..., n
ηi si = 0 ∀i = 1, ..., m
From [3] we see the interpretation of u is a cost to ’slack’. That is, u is the value lost
by not matching the worst case with each case. It is worth noting that a strictly concave
increasing function is stable in the sense that the maximal point is towards the center of the
optimal set. That is, it has diminishing returns or risk aversion ’built in’.
From [3] we see that the choice of u corresponds to different levels of risk aversion. The
most risk averse option is to let u = min(s), which recovers our original formulation (since
min(s) = 0 of the original formulation, and s ≥ 0) and cannot lose money in the worst case.
By letting u = θ(s) where θ is a probability distribution of our beliefs, our problem becomes
that of maximizing the expected value.
Furthermore, a strictly concave u generates a unique solution.
Theorem: If u(s) is strictly concave, the state prices are unique.
Proof. Suppose there were 2 different state prices corresponding to 2 different optimal so-
lutions to the above optimization problem, z 1 , z 2 . Since the constraints are linear equalities
4

5. 3.1 Parallels to Underconstrained Equations 4 ONLINE PROBLEM
and inequalities, the feasible region is convex and z ∗ = αz 1 + (1 − α)z 2 for 0 ≤ α ≤ 1 is
feasible. Since the objective function is the sum of convex terms and a strictly convex term,
it is strictly convex and f (z ∗ ) > αf (z 1 ) + (1 − α)z 2 , violating our optimality assumption.
Therefore, the state prices must be unique.
3.1 Parallels to Underconstrained Equations
We can draw a parallel here to the problem of underconstrained systems of equations. If
such a solution is consistent, there are infinite solutions. Then to find a unique solution, the
minimum norm solution is selected. This is also an example of convex optimization since
the objective and feasible region are convex.
Theorem. The norm is a convex function.
Proof. By triangle inequality we have α|x| + (1 − α)|y| = |αx| + |(1 − α)y| ≤ |αx + (1 − α)y|.
Furthermore, triangle inequality is tight if and only if x = cy for some nonnegative c.
Theorem. The set of solutions to a linear system Ax = b is an affine set.
Proof. The solution set is a vector space, and all vector spaces are affine.
Although the norm is not strictly convex, we can still establish uniqueness.
Theorem. Minimizing the norm gives a unique solution to a system of equations.
Proof. The solution set is N (A)+ x where Aˆ = b and N (A) is the null space. If b = 0 then x
ˆ x ˆ
is not in the null space. Then, without loss of generality we can choose x to be perpendicular
ˆ
to the null space. In this case, triangle inequality is strict and minimized when we choose 0
as the null space vector. Since this is a unique choice, the minimum norm solution is unique.
In the case that b = 0, x = 0 is the unique minimum norm solution.
4 Online Problem
Up to this point we have assumed all the bids have arrived before our decision has to be
made. However, in practice, bettors would like to know whether their bids are accepted or
rejected in real time. Then, we can modify our formulation to form an online program.
max πk xk − y + u(si )
i
k
s.t. aij xj + si = y, ∀i = 1, 2, ..., m Dual Var: p
j=1
0 ≤ xk Dual Var: µ
xk ≤ q k Dual Var: ν
si ≥ 0, ∀i = 1, ..., m Dual Var: η
5

6. 4 ONLINE PROBLEM
Following [2], for simplicity let bk−1 = k−1 ai xi . That is, b is the outstanding shares in
i=1
each state up to the current time. Then our formulation simplifies to:
k−1
max πk xk − y + u(y − aik xk − bi )
i
s.t. 0 ≤ xk Dual Var: µ
xk ≤ q k Dual Var: ν
There are only two variables, xk and y. We can write down the KKT conditions:
δf
KKT Stationarity: : π k xk − aik u (y − aik xk − bk−1 ) − νk + µk = 0
i
δxk i
δf
: −1+ u (y − aik xk − bt−1 ) = 0
i (1∗ )
δy i
Primal Feasibility: 0 ≤ xk ≤ q k
Dual Feasibility: µk , νk ≥ 0
Complementary Slackness: xk µ k = 0
→ xk (πk xk − aik u (y − aik xk − bk−1 ) − νk ) = 0
i (2∗ )
i
(qk − xk )νk = 0
The problem can now be solved as a series of individual optimizations of two variables,
by updating b repeatedly. From [2] we see pk = u (y t − ait xi − bt−1 represents the state prices
i i
after each iteration. From [2], we can solve this as follows. 1. If the internal price is strictly
higher than the bid price, we can immediately reject a bid. If not, go to the next step.
2.Update the state prices. If the bid price is higher than the new internal price accept it up
to qk (from complementarity) and update b, p. Otherwise, 3. find the quantity needed so
the new internal price exactly matches the bid price. This can be solved as a system of two
equations (1∗ ) and (2∗ ) in the KKT conditions. This is explained thoroughly in [2] lecture
notes 15.
Moreover, let O(f (m, n)) be the complexity of solving the offline formulation for m states
and n bets. Then the online formulation takes O(nf (m, 2) + nm) time, where nm is the
cost of updating the vector b, n times. Such a speed is comparable to the offline problem.
Moreover, the memory required is O(m) and quite small if a huge number of bids are made.
In fact, it was already mentioned in [3] that online linear programs can be used as a way to
solve extremely large linear programs. The online formulation can therefore be used to solve
problems where the number of columns is far larger than the memory available.
It is interesting that depending on the choice of u, the worst case is bounded differently.
In the most risk averse case, u = min(s) and the worst case is 0, but the online formulation
will never accept any realistic bids. By varying the choice of u a tradeoff can be made
between expected gain and risk aversity.
6

7. 5 TRUTHFULNESS
5 Truthfulness
It is a desirable property of our system to have truthful bets. Let a player make a bet and
(x∗ ) be the number of shares sold. We will charge: χ(0) − χ(x∗ ). where
k k
χ(x) = max −y + u(s)
y,s
s.t. si = y − qi − aik x, ∀i = 1, 2, ..., m
We will maximize the profit for an arbitrary bidder over πk and show that at optimality it
is equal to his/her real valuation πk .
ˆ
πk x∗ − ck = πk x∗ − (χ(0) − χ(x∗ ))
ˆ k ˆ k k
= πk x∗ + χ(x∗ ) − χ(0)
ˆ k k
≡ πk xk + χ(x∗ ) since χ(0) is constant with respect to πk
ˆ ∗
k
But note χ has the same constraints as the online problem:
k
aij xj + si ≡ q + aik xk + si = y ∀i = 1, 2, ..., m
j=1
At optimaltiy, χ(x∗ ) has the same x as the optimum of the online problem z ∗ . The only
k
difference is in a single term of the objective function. It follows that χ(x∗ ) = zk − πk xk .
k
∗
Then we have:
πk x∗ − ck = πk x∗ + χ(x∗ )
ˆ k ˆ k k
= π k x ∗ + z ∗ − πk x k
ˆ k
= π k x∗ +
ˆ k πt xt − y + u(s)
t=k
So, if the player bids truthfully and πk = πk , we can replace the above to get:
ˆ
πk x ∗ − ck = π k x ∗ +
ˆ k ˆ k πt xt − y + u(s)
t=k
= πt xt − y + u(s)
In other words, if the player bids truthfully, then his own profit maximization problem
becomes exactly aligned with the pricing and online optimization problems. Therefore, to
bet differently would be suboptimal.
This is a special case of the VCG mechanism, where the price assigned is equal to the
lost value to the rest of the players.[5]
7

8. 6 RECOVERY OF THE GRAND TRUTH
6 Recovery of the Grand Truth
As we have seen, the formulations as given return state prices. These state prices can be seen
as the bettor’s collective belief distribution. If there is a hidden probability distribution the
betters are aware of, a grand truth, we hope to converge to this grand truth after enough bets.
Theorem. Convergence to the grand truth is a necessary condition to bound loss.
Proof. Assume a method does not converge to the grand truth, then there will always be a
bet according to the grand truth that is accepted. In other words, the bid price is less than
the internal price π(x) < pT x. Consider the series of loss due to bets of this type. Such a
series diverges if the bid price does not converge to the grand truth.
From [1] we know the choice of u(s) puts different bounds on the worst case losses. This
worst case loss can be seen as the integral of the difference between the state prices and the
grand truth. The choice of u(s) can be seen to influence the rate of the convergence. Here
we will investigate different u. For this section we will use the data from Stanford students
betting on outcomes again. We use parameters w = 1, w = 10 where b = w/m, and m is the
total
For logarithmic scoring, the worst case loss is unbounded [1]. For exponential scoring,
the worst case loss is b log N [1]. Using w = 1, w = 10 for both logarithmic and exponential
scoring functions we get the following results for the online simulation of the first two games:
Log, w = 1 Exp, w = 1 Log, w = 10 Exp, w = 10
Notre Dame, Total Shares 135 48 2012 221
Notre Dame, Total Bids 58 23 59 59
Berkeley, Total Shares 7070 6488 7606 6695
Berkeley, Total Bids 123 127 122 127
We considered a bid ’accepted’ if we granted more than 0.001 shares. Although the
total number of bids accepted is quite similar (at least for Berkeley), the total shares sold
follows a clear trend. Clearly, b = 10 is less risk averse and accepts more bids. Similarly,
logarithmic scoring seems to be less risk averse than exponential. This makes sense in light
of the fact that exponential can bound the worst case. There is a clear tradeoff then between
risk aversity and total shares accepted.
On the state price convergence, we see the following trend for state 1 in Stanford vs.
Notre Dame (the offline problem has solution p1 = 0).
8

9. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
All the functions converge to the original offline solution of p1 = 0. However, the trend
as to which function works best is not clear in this case. Logarithm initially converges faster,
but takes longer after bid 40. We also note that smaller b appears to converge faster for this
case, although it is not clear what will happen in general.
6.1 Regression
Using a model of grand truth with gaussian peturbations (noise), we note that least squares
regression can be shown to return the grand truth [4].
Conclusion
The prediction market is an interesting emerging field of research. Although many traditional
approaches to prediction have been used such as regression, new approaches can be nearly
self-funded as well as a source of information. More theoretical results are needed, for
example to prove recovery of a grand truth. In comparison, least squares regression can
provably recover the grand truth.
9

10. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
References
[1] S. Agrawal, E. Delage, M. Peters, Z. Wang and Y. Ye. A Unified Framework for
Dynamic Pari-mutuel Information Market Design. The 10th ACM Conference on Electronic
Commerce, 2009.
[2] Y. Ye. MS&E 211 Lecture Notes, 2012.
[3] S. Agrawal, Z. Wang, Y. Ye. A Dynamic Near-Optimal Algorithm for Online Linear
Programming, 2009.
[4] A. Ng. CS 229 Lecture Notes, 2012.
[5]en.wikipedia.org/wiki/Vickrey-Clarke-Groves auction
10

11. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
Appendix A: Code for Problem 1
% Read the excel data into matrices
num1 = xlsread(’NotreDame.xlsx’);
num2 = xlsread(’California.xlsx’);
A1 = num1(:,3:10);
A1 = A1’;
[m1 n1] = size(A1);
A1 = [A1 -1*ones(m1,1)];
f1 = num1(:,2);
f1 = [f1; -1];
f1 = -f1;
b1 = zeros(m1,1);
lb1 = zeros(n1,1);
ub1 = num1(:,1);
[x1, fval1, exitflag1, output1, lambda1] = linprog(f1,A1,b1,[],[],lb1,ub1);
%Print the state prices
lambda1.ineqlin
A2 = num2(:,3:10);
A2 = A2’;
[m2 n2] = size(A2);
A2 = [A2 -1*ones(m2,1)];
f2 = num2(:,2);
f2 = [f2; -1];
f2 = -f2;
b2 = zeros(m2,1);
lb2 = zeros(n2,1);
ub2 = num2(:,1);
[x2, fval2, exitflag2, output2, lambda2] = linprog(f2,A2,b2,[],[],lb2,ub2);
% Print the state prices
lambda2.ineqlin
11

12. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
Appendix B: Code for Problem 6, Online Version
clc
clear all
% Either of Type 1 or Type 2 Data Generation Modes to be chosen
% and other commented
% Type 1 - Data from Auction Games
% num = xlsread(’NotreDame.xlsx’);
% num = xlsread(’California.xlsx’);
% numStates = 8;
% b = 1;
%Type 2 - Random Data Generation
numBids = 100;
numStates = 8;
b = 1;
%num is the overall matrix
num = zeros(numBids,numStates+2);
for i = 1 :numBids
num(i,1) = randi([2,10],1);
num(i,2) = rand(1);
for j = 1 : numStates
num(i,j+2) = randi([0 1],1);
end
end
A = num(:,(3:numStates+2));
A = A’;
[m n] = size(A);
f = num(:,2);
q = num(:,1);
cvx_begin
variable xt
variable y
variable s(m)
dual variable P
maximize( f(1,1)*xt - y + (b/m)*sum(1-exp(-s)));
subject to
P: A(:,1)*xt + s - y*ones(m,1) == 0;
xt >= 0;
xt <= q(1,1);
s >= 0;cvx_end
if (xt < 10^-5)
xt = 0;
end
x_old = xt;
12

13. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
A_old = A(:,1);
Price = P;
for i = 2 : n
cvx_begin
variable xt
variable y
variable s(m)
dual variable P
maximize( f(i,1)*xt - y + (b/m)*sum(1-exp(-s)));
subject to
P: A(:,1)*xt + s - y*ones(m,1) == -A_old * x_old;
xt >= 0;
xt <= q(i,1);
s >= 0;
cvx_end
if (xt < 10^-5)
xt = 0;
end
x_old =[x_old; xt];
A_old =[A_old, A(:,i)];
Price =[Price P];
end
Price = - Price’;
Appendix C: Code for Problem 6, Offline Version
clc
clear all
num = xlsread(’NotreDame.xlsx’);
% num = xlsread(’California.xlsx’);
A = num(:,3:10);
A = A’;
[m n] = size(A);
f = num(:,2);
q = num(:,1);
cvx_begin
variable x(n)
variable y
variable s(m)
dual variable P
b = 0.1;
maximize (f’*x - y + (b/m)*sum(log(s)))
subject to
P: A*x + s -y*ones(m,1) == 0
13

14. 6.1 Regression 6 RECOVERY OF THE GRAND TRUTH
x >= 0
x <= q
s >= 0
cvx_end
14