>> Yuval Peres: Hi everyone, so we’re very happy to have Yiling Chen from Harvard to tell us about
Market Making via Convex Optimization.
>> Yiling Chen: Thanks a lot. Thanks for having me here. Today I’m going to talk about prediction
markets. In particular, I want to talk about the new mechanism that we try to design for prediction
market when we care about getting information from a diverse group of individuals when the event of
interest has a really large and underlying outcome space. So we’re going to talk about [indiscernible]
our prediction market today. So this is a joint work with Jake Abernethy who’s currently at Penn and
Jenn Wortman Vaughan who’s actually at MS [indiscernible] LA, so. Yeah, both of them are wonderful.
Jake is on the job market this year.
I’d like to start with something that seems irrelevant, its orange juice. That’s the number one juice in
the United Stated and a lot of them are made from concentration, so it’s concentrated orange juice. So,
concentrated oranges is not only hydro-logically concentrated but it’s also geographically concentrated
too. So most of the oranges that are used to make concentrated orange juice is coming from Central
Florida. So they are near Miami and so that’s ninety-nine percent of those oranges are coming from
that single area. So because in order to make orange juice we need oranges so those manufacturers
who care about oranges they have the concern about the production of the oranges in the future. So
because of that there’s orange future contract being traded in the market. So this type of contract if I’m
purchasing this type of contract it will entitle me the right to purchase some number of orange juice
solid in future date. This particular contract is for April, okay. Of course there’s a price for this contract
depending on peoples projection or prediction about how likely we’re going to have a good orange
production in the future, okay. There’s one property with orange trades they took five to ten years to
grow but they’re very sensitive to temperature. So if the lower temperatures drop below freezing even
for a few hours those trees could die. So as a result you can also think about the prices of this orange
juice futures contract is sensitive to temperatures too. So the production of oranges is related to the
weather in that area.
Researchers actually look at the orange futures contract. What they found out is quite interesting. So
they found that by looking at the market data for this contract they can improve weather forecast,
especially the lowest temperature, okay. So, what I’m trying to tell using this example is that markets do
have some ability to incorporate and aggregate irrelevant information related to the contract being
traded. So in this particular case because weather is an important factor that’s related to the
production of oranges and hence is affecting the price of the futures contract here and the market price
incorporates that information. Another example I like to give if you all remember is the HP researchers
P not equal to NP proof. When the proof come out I’m not a Zeist theorist so I have really no clue about
whether or how likely the proof is correct or incorrect. So if you ask me I can give you a completely
uninformed guess. But when I read Scott Aaronson’s blog this is what he was saying and though he’s
basically saying that if the proof is correct, “then I, Scott Aaronson, will personally supplement his prize
by the amount of $200,000.” I’m sure that Scott knows a lot more about how likely this proof is going to
be correct than me. So after reading this I have a higher confidence that the proof probably is not
correct, okay.
Scott explained it in the following way. So he’s basically saying that, “I have a way of stating my
prediction that no reasonable person could hold against me: I’ve literally bet my house on it.”
>>: People got very mad at him.
>> Yiling Chen: Yes, ha-ha.
[laughter]
>> Yiling Chen: People got very mad with him but that’s a different story. The point I’m trying to make
is that the betting is not necessarily a bad thing. It’s a way for people to give, to state a credible opinion
because I can state incredible opinion when I’m not going to lose anything. So I basically can bluff, I can
say anything because I don’t really have a stake in what I’m going to say whether that’s correct or not.
But if you’re asking me to wager on something I probably will think more carefully and try to give a more
credible opinion. So betting does have a socially beneficial property. That is that it allows people to
back up their opinion with the amount of money that they’re willing to pay, okay. That trumps the idea
we can use the market mechanism for predictions. That’s what we call a prediction market. A
prediction market basically is just a financial market.
So its trace a financial contract but it has a slightly different purpose. So it has an underlying event of
interest and then it tends a designer contract such that the payoff of the contract is related to the
outcome of the event of interest. I’m having a contract that I get from Intrade, Intrade is the online
prediction market platform. So this particular contract the underlying event is whether the movie Argo
will win the Academy Award for best picture or not. So it’s a binary random variable. The way the
contract is designed is that if the movie does win the award every share of the contract pays out ten
dollar to its shareholders. If it doesn’t it pays off zero dollar so it was nothing, okay. The current
prediction is two dollar and fifty-four cents, it’s around two dollar fifty cents. If I think about me as a
trader coming to the market and considering whether I should buy or sell this contract. So I have a
belief about how likely the event that the movie will win the award. Suppose that my belief is
probability P. If I’m a rational trader, meaning that I care about making money, then if the price is lower
than ten dollar times P which is my expected value after security then I should be waiting to buy the
security because in expectation I’m making money. Similarly if the price is higher than ten times P in
expectation I’m making money if I’m selling the security. So I should be waiting to either buy or sell
depending on my belief about how likely the event will happen.
So if everyone is doing this we can think about the market price actually represent the collective
prediction of the population. So there’s some mapping from the price to what the market believe the
probability for the event to happen. So this is the markets prediction that currently is around twentyfive percent, meaning that was probability point two five this movie will win the Academy Award, okay.
This is actually the highest among all the candidate right now. So there’s a lot of evidence showing that
a markets does work. So I’m now going to repeat all this positive evidences. But in general what people
find is that comparing market with alternative prediction methods market generally at least performs
just as well as the alternatives and in most cases better.
>>: Interesting exception was in this election. You could arbitrarily in-trade to get another market and
make money for sure. Further more Intrade they say totally different things on the total election on the
fifty different states.
>> Yiling Chen: Yeah, so the second one actually doesn’t really indicate that Intrade is wrong because
the prediction is probability but the first one it’s [indiscernible] two groups of people that look at the
prices Intrade and the prices back fire which is a UK gambling set. They have exact same security and
they’re consistently priced differently. Actually the two groups they [indiscernible] both groups they did
some experiments trying to take advantage of the arbitrage opportunity and looking to whether and
why the institutions are not interested in doing that. There is quite a few reasons for that. One is that
one market is in U.S. dollars and the other market is in pounds. So there’s some currency risk and also
the amount of money that can be get from doing this exploring the arbitrage opportunity is too small for
institutional traders and the investment is probably to big for individual traders. So it’s kind of
interesting that arbitrage opportunity stays there, yeah.
>>: [indiscernible] markets like Intrade is that they were using source of information [indiscernible] so
then people get motivated to buy a stamp and it’s relatively a low [indiscernible] form of advertising …
>> Yiling Chen: Yeah, exactly. So there are period of evidences, so a period of time both for this election
and for the last two times election. There’s evidence showing that someone trying to manipulate
Intrade price and just try to get a higher price for its favorable candidate. They were successful for a
very short period of time. So you can say that the history is that the price was driven up for a short
period of time but it quickly goes back. So, which means that the manipulation in markets is always
possible, but at the same time if the price is off and think about all the other traders it provides more
incentive for them to crack the price and making money. Also I like to think about prediction market as
one of the possible method rather than they correct a method of making predictions because the
question is not whether that should be the method. The question is, compare it with all other possible
methods. The amount or the method that we have which one gives us better prediction in general.
Okay, so about the mechanism. So we can trade this type of contract using a stock market type of
mechanism which is an outer matching mechanism. So buyers submit buy orders and sellers submit sell
orders. So [indiscernible] buy prices and sell prices and whenever the buy prices is higher than the sell
prices then a transaction happens. The market institution just to matches the buy orders with sell
orders just to do this order matching not barring any risk, okay. So we can do that but there’s one
problem with this type of mechanism, is that if there are not enough traders in the market then even if
someone has information about the underlying event and they want to trade the particular contract
they can not find a counter party to trade with them and that’s problematic. That causes information
not being able to be reviewed. A solution to that is people design automated market maker mechanism.
This is mechanisms that every trader trade directly with the market institution. So they buy from the
market institution and they sell to the market institution. So they don’t really need another trader to
trade with. Of course these market institutions bear some risk and can lose money. The question of
design would be, how can design the mechanism to make sure that the market institution doesn’t lose a
huge amount of money or his possible loss is bounded, okay.
So let me describe the standard market maker mechanism designed by Robin Hanson. So the way that it
works is if we have an event of interest that has mutually exclusive and exhaustive outcome then we will
create one outcome, we will create one contract corresponding to one outcome. So this outcome
mutually exclusive and exhaustive let’s assume that for now. That’s a complete market. So in financial
markets term that’s called a complete market. If I care about tomorrow’s weather in Seattle and
supposing that it either will be sunny or it will be rainy, or it will be snowy and any two of them or three
of them won’t happen on the same day then I can have three contracts looks like this, okay. The
automated market maker which is the market institution per se keeps a cost of potential function. So
what this function is used is the following and so this function trying to capture the total amount of
money that the market maker has already collected from all traders in all past transactions. So this is
the function of this vector Q. So this Q vector represents the total number of shares that have been
purchased all sold to the market maker by all traders in the market. So this vector could be something
like a hundred, negative fifty, or zero. So that could be one potential value of the vector so that
captures the total historical transactions on all to raise securities in the market, okay.
If some trader comes to the market and want to make a purchase we can think about the trader
basically want to change this Q vector. So if the trader comes and saying that I’m interested in buying a
bundle of shares so this bundle could be ten shares for the first contract and zero shares for the second,
and negative five shares for the third one. So can take negative value and active value representing sell,
okay. Then the market maker will use this cost of potential function and calculating how much the
trader needs to pay to the market institution. So the difference in the cost of potential function is the
payment the trader needs to pass by to the market maker to make that transaction. Given that the
partial derivative of the cost of function represents the instantaneous price of the security. That is the
price for purchasing or selling [indiscernible] shares of the security. That is our market prediction. So
that is the prediction in the market.
Okay, now let me give you a very specific example. This is the Logarithmic Market Scoring Rule market
maker proposed by Robin Hanson. The name comes because underlying the mechanism uses allow us
to make proper scoring rule. But let me try to [indiscernible] part completely. So the way it works is
that it uses a cost function like this and a price function like this. But the exact form of the function is
not important for this talk right now. I’d like to show it by picture how the market works. So supposing
I’m in a two outcome market what I plotted here is the price for one outcome as the function of how
many shares has been traded for that particular outcome, okay. Assuming everything else remains to be
the same so traders just to purchase this particular outcome. Then when nothing has been purchased
the price for the security is point five, so and the price for the other security is also point five. So the
market start was uniformed distribution for these two outcomes. If some trader comes to the market
saying that I want to buy a hundred shares and then this shaded red area is the amount of the money
that the trader needs to pay to the market maker to purchase his a hundred shares. If another trader
comes and saying that I want to buy another hundred shares too and because his purchase is after the
first trader so he has to pay the shaded blue area for buying the hundred shares and that is bigger than
the shaded red area. So the price increases, okay.
So let’s take a closer look to the price function here. We just want to get a basic sense of whether a
market maker mechanism like this looks reasonable or not. So does it do something sensible here? So
first if I look the prices for security PI it is increasing function with QI. Meaning that as more and more
people purchasing the security the prices of the security increases so that sounds very reasonable
because things will go weird after I’m buying some security the prices for that security keeps dropping
because I should just keep buying if my belief hasn’t changed at all. That’s going to be really weird. The
second thing is that if I sum the prices; PI a sum across I they equal to one. Why is this important? So
suppose the sum of the prices is less than one is there anything that you can do that appears to be
beneficial to you?
>>: Buy them all.
>> Yiling Chen: Buy them all, right. Because we know that the outcome is mutually exclusive and
exhaustive. Meaning that in the future one and exactly one outcome will happen. One and exactly one
security will pay off one dollar. So if the sum of the prices is less than one I should just buy them all so I
pay less than one dollar and I get a guaranteed one dollar in the future. So there’s arbitrage
opportunities on the table. Similarly if the prices seem to be greater than one I should sell them all and I
still make money. So the prices sum to one is actually really important for the market to not allow
arbitrage opportunities on the table, okay.
Oh, this doesn’t show up correctly. The third thing is that because the market maker now handles every
transaction [indiscernible] risks so he holds shares. He can potentially use money and if we want anyone
to be willing to use the mechanism like this we’d better give the mechanism some theoretical guarantee
in saying that who ever runs the mechanism will not lose everything in their life. So we want the
potential loss of the market institution in the worst case to have a bound. This mechanism has the
worst case loss bound that is B log N where N is the number of outcomes. We want to have three
outcomes and you go to three and B is this particular parameter here that we used here. This basically
is the liquidity parameter that determines how quickly the price changes as people are buying shares.
It’s kind of like the steepness of the price curve. This is the parameter that the market institution can
choose to set ahead of time and any positive parameter will be okay.
Is there any questions so far?
>>: [inaudible] the last bullet is kind of the incentive to be a small as possible. What’s the other?
>> Yiling Chen: Yeah.
>>: There should be other things …
>> Yiling Chen: So …
>>: That you want to be big.
>> Yiling Chen: Yeah, so the bad thing about setting is to be absolutely small is that if I set the B to be
small this curve will become really steep. So what it means is that some trader comes and buying five
shares my prices for the security is already approaching to one. So there would be very little market
activity going on. Also if the price curve is really deep then the [indiscernible] is large. So remember
that we have an instantaneous price so this is the price curve. It appears that buying or selling
[indiscernible] share of the security the prices is the same. But buying one share versus selling one
share there is a difference of the prices there. So there still exist at what’s called a [indiscernible] so
that’s the price of selling versus the price of buying. If the, this curve is really deep that [indiscernible] is
large. We don’t like markets to have large [indiscernible] because that doesn’t incentive wise traders to
trade. If the trader’s belief happen to fall into that spread they won’t trade. So they won’t find it’s
profitable to trade anyway. So that’s kind of the trade off of setting the B.
>>: [inaudible] prohibited and often bad mouthed. How does prohibition affect the market design?
>> Yiling Chen: That’s a great question. So right now this mechanism allows short-sale. So this is, we
just allow short-sale because it has bounded worst case loss and we don’t really care about short-sale.
>>: Okay.
>> Yiling Chen: Short-sale hurts, like restricting or not allowing short-sale hurts information aggregation
and things. We start from the go of thinking that we’d like to get information from participants so we
really like to encourage information aggregation. Okay, so that’s the existing mechanism when we have
mutually exclusive and exhaustive outcomes. So, oh this is the interface how it is actually implemented.
So Inkling Market implement this mechanism. As you can see that the traders can come and saying that
I want to buy fifteen shares and then the market counter price is here. After the transaction the price
will be here and the cost to make this transaction happen is this amount. So this is the interface to the
trader how you can hide all the details of the mechanism from the trader, okay. So this mechanism
works one way, design one contract for each outcome out of the outcome space, what we consider the
outcome space to be mutually exclusive and exhaustive, okay.
What if we have large state spaces? So for example if we consider horse racing if we have un-competing
candidate the outcome space can cancel and affect [indiscernible] possible permutations. If we consider
the presidential election prediction and if we care about the outcome of every state then we have two
to the power of N, where N equal to something roughly equal to fifty, depending on how you count. If
you think about an option which is based on the stock price then the stock price can be any real value in
the future, then we basically have an infinite space. So we have a continuous space that’s infinite. For
this situation we can not just run the market maker we just talk about, the Logarithmic Market Scoring
Rule market maker because calculating the prices of this market becomes intractable. So we just simply
can not calculate the prices. More over when the outcome space is really large we essentially are asking
humans to give probability for some event that has extremely small probabilities, and humans are just
not good at it.
For all these reasons we have to reconsider how can we design markets for large outcome space? So, of
course we can run separate independent markets. So like in the horse racing what they were doing is
that there’s a market for win and a market for place, and a market for show. So win is allowing people
to bet on which horse will run first and place is for the first two positions, the top two positions, and
show the top three positions and the markets are separated. What it means is that if common betting
horse A is going to win I change the odd or the probability for the win market. But nothing has been
changed in the place and show market while logically you’d think that this event is logically dependent,
right. So, if we run independent markets we’re acknowledging the logical dependences for the events.
That’s not something that we like.
So the goal that we try to achieve is that if you give me a small set of securities over a very large or
possibly infinite outcome space, is this possible for us to design a customized market mechanism for this
set of securities that preserves the logical dependences of the events, and at the same time hopefully
can be operated in a computationally efficient way. Question.
>>: I think maybe an odd ball case but if the U.S. was actually a horse race I could bet on Romney to
place or show and not care about the public opinion polls.
>> Yiling Chen: Yeah, ha-ha. You can definitely do that. So that’s why I’m saying that give me a small
set of securities. So probably would not, hoping that everything that can possibly be bet on is
interesting. So if you’re interested in particular things let me start from there, okay. The two that we’re
using is convex optimization and the conjugate duality. That has also been used in early papers too,
okay. Is that clear Ago? Good.
So let me start by explaining what I mean by saying that we’re given a small set of securities. So we
consider the set of securities as a menu table. So basically each row corresponding to a possible state of
the world in the future that can be possibly realized and the number of state is large. Each column
represents a security of interest. We have a relatively small set of securities. So we have K securities.
What this menu table tells us is that I’d like to know for every possible realization in the future. So for
every state what is the payoff of the security? So at the abstract level it means that we want to have an
efficiently computable payoff function such that for every security in every state you can tell me what
the payoff looks like, okay. I’m going to talk about the row vectors of this menu table later on. So the
row vectors that are the important things that I will use later on. But basically this is the set of security
that I’m given to start with.
So one example is subset betting. So for horse racing if I have three competing candidate then these are
all possible states. You can define betting language that defines all allowable securities in the following
way. If you allow people to bet on horse A is going to win at a particular position, for example second
position, like betting on Romney’s going to win at the second position. Similarly, so basically a trader
can pick a candidate and pick a position and bet on that. By buying more than one security it’s possible
for the traders to bet on things like a particular candidate will win at one position in a subset. So that’s
why the subset betting comes into play. Also the inverse a subset of the candidate will win a particular
position, okay. For securities like this, so A finishes at position one. The payoff is one dollar for all those
outcomes that A indeed is at position one and zero dollar otherwise. So that’s the security. This is
proven to be sharp P-hard if we consider running the Logarithmic Market Scoring Rule for this type of
betting languages. So it proves that it’s sharp P-hard.
Another example is pair betting. If they allow people to bet on the relatives trends between two
candidates we can think about that the [indiscernible] natural betting languages that people do have
information about. So similarly it pays off one dollar if this relationship indeed becomes true in the final
outcome, okay. This is again sharp P-hard if a [indiscernible] decide to use the standard Logarithmic
Market Scoring Rule market.
>>: So what’s hard exactly?
>> Yiling Chen: So what’s hard exactly? So basically what we were trying to show there is we want to
compute the price for the securities as if we’re operating a Logarithmic Market Scoring Rule market.
>>: As if you’re operating one on all the possible states?
>> Yiling Chen: Yes. So the naïve way of running the market is running that on all possible state and
then some of those prices that A is ahead of B that will give us the price of this particular security. That’s
the …
>>: [inaudible]
>> Yiling Chen: Naïve way.
>>: [inaudible] security was a bundle of [indiscernible] sets.
>> Yiling Chen: Exactly. So that’s the price that we want to calculate. The question asking here is there
possibly a smart way to do that such that we can circum when actually doing all the computation with all
the [indiscernible]. In this case no we can not do better. There are indeed positive examples that we
can do.
>>: Okay, what you said now is much more stronger than a theorem. You said it’s sharp P-hard and
then now you said you can not do better than the [indiscernible]?
>> Yiling Chen: Oh, so, yeah. So the naïve way of doing that is just running…
>>: Right [inaudible]…
>> Yiling Chen: A market with [indiscernible] contracts.
>>: [inaudible] much more efficient than that which…
>> Yiling Chen: It could be, yeah, yeah. So I shouldn’t say that. So the more precise way is that we
proved that it’s sharp P-hard.
>>: Okay.
>> Yiling Chen: We don’t know that. I’m not making difference between the last computationally
complex classes basically saying that.
>>: [indiscernible] example in one of the share prices [indiscernible] states [indiscernible].
>> Yiling Chen: Tell me the [indiscernible].
>>: [indiscernible] like borrow one of these settings are you assuming that the state [indiscernible] or is
it sufficient to have [indiscernible] function of the state being …
>> Yiling Chen: Yeah, so for this particular case the state space is [indiscernible] because this is for the
permutation [indiscernible].
>>: Right.
>> Yiling Chen: But for the [indiscernible] we have the same result. So if it’s a presidential election
example that I give earlier on I would still have the same sharp P-hard read out. But we do have positive
read out for some structures. Like if we’re betting on the basketball game outcomes and there is a
tournament structure and if it asks if people are interested in betting, if team A meets team B, A is going
to win or team A will win a certain [indiscernible] game. For those types of betting languages we can
actually use Logarithmic Market Scoring Rule efficiently. So there are positive examples. It’s not always
computationally hard. [inaudible]
>>: So when the state, the number of states increases like this, for example [inaudible] blow up. The
number of securities that needs to be traded in the market also grows a lot for the same [inaudible]…
>> Yiling Chen: U-huh.
>>: Right. Are there any foreseeable problems for having a lot of different securities for the same
company?
>> Yiling Chen: U-huh.
>>: For example the number of participants you need in the market to reach the desired [indiscernible].
>> Yiling Chen: That’s an interesting question. So for this particular example actually we don’t have that
large number of securities. So we have [indiscernible] too. The reason that we’re having a market
maker is to make sure that trader comes to the market they can always trade. So the only decision they
need to make is that whether the price is favorable for them to trade or not. So that helps the situation
of saying that oh as the number of securities grow we need to have a large number of traders because
who ever has the information they can still come to the market and review their information. So that’s
one of the advantages of having a market maker because if having, if you’re using the stock market
mechanism for [indiscernible] setting we’re really in trouble because they’re trading different securities
and they’re looking for matching with other traders which are interested in the same security.
Okay, so, what we’re trying to do is we want to design market maker mechanism that’s tailored to the
securities that we’re interested in. At a very high level what we need to do is we need a pricing
algorithm such that given any history of transactions if a trader comes to the market saying that I want
to buy an R bundle of securities the market maker should tell me the cost of doing that. Okay, so that’s
what we need. But the challenge is that how can we ensure the reasonableness of the prices for a
market like this for the pricing algorithm? So if you look at the complete market the price is sum to one
that’s a very natural constraint and when you talk about that that’s important. But if we just look at the
pair betting, what can I do? The pair betting is still something that has a lot of intuition that I can draw
from. So for example if there is the security that’s saying I is going to finish ahead of J and another
security is saying that J is going to finish ahead of I, these two outcomes are mutually exclusive and
exhaustive. So by the same logic I can say that the prices for these two securities should sum to one.
What if I consider three securities? I ahead of J, J ahead of K, and K ahead of I, if I have a cycle I’ve
mastered two outcomes can be true. So the sum of the prices of these three securities should be less
than equal to two. But I really don’t want to continue this process by thinking about whether I can elicit
all possible constraints that I should put on the security and this is just pair betting. If you’re giving me
any menu of securities I generally don’t have any idea about how I should put constraints on the prices.
So that’s the challenge. So we took an axiomatic approach but we’re really back up and saying that let
me not even think about how I want to run the market. Let me start with, what are the desirable
properties I want the market mechanism to have?
So we start with the first one that we call path independence. So intuitively what it means is that if a
trader coming to the market and wants to purchase a bundle R the cost that he needs to pay for this
bundle should be the same compared with another trader trying to split this bundle R equal to different
portion and purchase different portions independently and sum that together. Meaning that if you hold
R bundle at the end no matter how you achieve that R bundles assuming there’s no other traders
trading in between you then you should pay exactly the same amount. So that seems something
reasonable, arguably reasonable at least. So formally what it means is that if given history and a trader
wants to purchase the sum of these two then the cost of the purchase should equal to given the history
the trader purchased the first pair and then purchased the second pair, okay. This one alone will bring
us back to the framework of having a cost potential function.
>>: Shouldn’t there [indiscernible] a [indiscernible] after he buys [indiscernible]?
>> Yiling Chen: So this is the …
>>: Oh, yeah, yeah.
>> Yiling Chen: First pair so the history includes the first pair, exactly, yeah.
>>: [inaudible] very much [indiscernible] fact that the potential energy in a conservative system is
independent path?
>> Yiling Chen: Exactly, so this is really, [indiscernible] property. So we start with not considering having
a cost potential function but by assuming that path independence…
>>: Yes.
>> Yiling Chen: Was just to come back to having a cost potential function which we’re very happy about
because we have a lot of tools to deal with that, okay. So that’s path independence.
>>: So, I understand it’s a very natural condition but is there a particular reason why it’s actually
designed, what if it didn’t have path independence, would it be arbitrage or something?
>>: Yeah…
>> Yiling Chen: [inaudible]…
>>: It would be like two different guys on two different paths and they arbitrage half way between and
both make money.
>> Yiling Chen: Yeah.
>>: I, that doesn’t, I don’t see that.
>> Yiling Chen: So…
>>: So…
>> Yiling Chen: It’s, I wouldn’t say that this is equivalent to arbitrage but this closely related to having
arbitrage. So if the path is not independent in a systematic way then it’s possible for the traders to find
arbitrage opportunities. But you had a great question. Actually this is the condition that I think we
don’t have to have if we think about this is coming from the prediction market mechanism perspective.
So we’re naturally having this property. But if I think about stock market we never have path
independence because we don’t know when other traders coming and what their [indiscernible] is. So I
would say that this only arguably reasonable and this comes from the tradition of the market maker
mechanism for prediction markets.
So other properties would like to have market predictions. We like to have a notion of price. So we
require that there is a notion of instantaneous price. Mathematically this just means that this cost
potential function must be continuous and differentiable. So, and the prices is just the ingredient of this
cost function. Okay, so we like that abstraction. The other desirable property is what we call
information incorporation. Sorry, I guess this decided to not show you at the right places. So what it
says is just the purchase of any bundle R shouldn’t decrease the price for that bundle R. So if we keep
buying the prices can not decrease. What it means mathematically is this cost function is a convex
function.
Okay and the next property is no arbitrage. So I have explained that what actually we’ll relax this
property later on, but what this property means is that if someone comes and buys a bundle R we do
not allow the possibility that this trader will make a positive profit no matter which outcome happens.
So meaning that it’s possible this trader makes a profit in some outcomes but he should potentially lose
money in some other outcomes. So there’s no guaranteed profit. If we combine the first three
properties we can derive a constraint for the prices. So what it means is that the set of allowable prices
actually precisely speaking it should be closure of this. But I dropped the closure because that doesn’t
really give any intuition here. So remember the grading of the cost function is the allowable prices in
the market. The right hand side is a convex hull of the security payoffs. So it’s the manual of the
security that we gave the big table and all those row vectors. This is the convex hull of all those row
vectors, okay. What it says is that if we want these three conditions to all be satisfied then the
allowable prices should be content in the convex hull, okay.
We also have the last property what we call the expressiveness. Because we’re interested in getting
information from trader’s. Expressiveness basically says that if some trader has some belief then they
should always be possible for the trader to review that belief in the market through the trading of the
securities. If we required that we get the other direction it says that the convex hull of the security
payoffs should be content in the visible price regions here, okay. Combining all this together we get a
characterization. So if we want to satisfy all five properties, which we argue that probably natural, then
the market maker must use a convex cost function such that the grading of the cost function, the set of
gradings is exactly equal to the convex hull of the security payoffs. If I think about the complete market
case it actually makes perfect sense because of all complete markets the convex hull is the probability
simplex. What it says is that the allowable prices should be probability distributions in that setting,
okay.
So far so good, we get a characterization. We’ll figure out what should be the constraints that we’d put
on the prices to make sure that there’s no arbitrage opportunities, okay. But this is not constructive at
all. This doesn’t really tell us anything about if I want a market you give me a set of securities how I
should be running the market. So we want to consider how can we design a cost function that satisfies
this characterization. How am I doing time wise?
>>: You’re doing good.
>> Yiling Chen: Okay, good, ha-ha.
>>: Twelve .
>> Yiling Chen: Twelve minutes, okay that’s great, that’s perfect. So we make use of fact from convex
analysis. Because we know that we need a convex cost function. There’s a fact in convex analysis saying
that any convex cost function and also differentiable one can be rated in terms of this convex
optimization using its conjugate function of C. So this R is a conjugate, okay. So far it doesn’t really help
us yet because you’re just giving me a different way of writing my convex function. I still don’t know
how can I have a convex function and make sure that it satisfies that property. But then this also has an
associated property with this fact is that the optimization problem, the optimal solution X, X star equal
to the grading of the cost function. So we need a cost function and we need the market price to be the
grading of the cost function. So combining all this together, what it means is that if we need a convex
cost function to operate a market for the customized securities that we’re considering, we only need to
pick appropriate R function and the domain of the optimization. So basically the domain because the
domain constraints the allowable prices. So those are the only two things that we need to consider and
then we will have a market maker mechanism.
So for complete market our domain is the simplex and the cost, the conjugate function R we can pick for
any strictly convex function. For the Logarithmic Market Scoring Rule market maker that we just saw
this is the negative entropy function. We can check that the Logarithmic Market Scoring Rule the
optimization problem gives the same cost function as function …
>>: [inaudible] log sum X.
>> Yiling Chen: Yeah, so this expression has a lot of applications in other domains too, so.
>>: Yeah it’s real stands in for max pretty easily.
>> Yiling Chen: Yeah…
>>: It’s analytic.
>> Yiling Chen: Yeah.
>>: So, yeah.
>>: [inaudible] constant B come from on this?
>> Yiling Chen: Oh, so this R…
>>: Oh you put that into…
>> Yiling Chen: Function …
>>: Into the R, okay.
>> Yiling Chen: This is the R function.
>>: Alright.
>> Yiling Chen: Yeah, so in, so this also corresponds to the online learning algorithm, the exponential
update, the no regret learning algorithm. This B will be their learning rate. So we actually also show
that for complete market there is one to one correspondence between the online learning algorithm
and the class of market makers that we can design. So there’s actually a one to one mapping between
those two. Then for the complex securities that we consider, so any set of securities that we’re given
we just need to change the domain to be the convex hull of the security payoffs and still use any strictly
convex function, okay. One more advantage of thinking about designing market maker in this frame
work is that they care about the worst case loss of the market maker. The worst case loss is bounded
according to this expression tool. So by choosing an R function we actually also know what is the worst
case loss of the market maker. So that’s a nice property for us to have because we will know that if R is
bounded of course our market maker has a bounded worst case loss, okay.
So let me give an example of how we can use this frame work to design a market maker that we couldn’t
operate before using the Logarithmic Market Scoring Rule, so for subset betting. So the language for
subset betting is, a trader can pick a candidate I and a position J and bet on the candidate I will finish at
position J. The payoff equal to one if that indeed happens and zero if that’s not true. Instead of thinking
about a payoff vector for all the possible securities in the future, we can think about given any particular
outcome which is a permutation of all competing candidates we have a payoff matrix and that matrix
basically has value one when candidate I finishes at position J. So we have rows representing positions
and columns representing candidates. Instead of representing that as a vector we represent that as a
matrix. Now the question is what is the convex hull of all this payoff matrixes? That happens to be all
doubly stochastic matrixes. The advantage of reaching to this pond is we can actually using exponential
number of constraints to represent the convex hull of the security payoffs. So we only need this side of
constraints. So we have a convex problem with some relatively small number of constraints and we can
solve that efficiently. If we just chose this conjugate function it has a formula that looks very similar to
entropy but it’s not because we’re not considering independent or exclusive outcomes here then we
have a market maker. The worst case loss of the market maker is the following depending this
[indiscernible] parameter, okay. We can operate this market efficiently.
What about pair betting? So the other example that Logarithmic Market Scoring Rule can not handle.
Unfortunately we couldn’t handle that too because representing the convex hull is computationally
difficult. But there is still hope here. So originally we required that the allowable prices region has to
equal to the convex hull, so based on the properties that we desire so we required that. But we asked
the question, are both directions necessary? So suppose we only require this direction so that’s bad. So
the market maker will have unbounded loss. But if it only required the other direction so this actually
implies that we relaxed no arbitrage condition. So we allow arbitrage opportunities in the market now
but actually it doesn’t matter. The market makers worst case loss can only improve. So it seems that
I’m lying…
>>: [inaudible] finally makes money. This is great…
>> Yiling Chen: Yeah, so…
>>: [inaudible] arbitrager in the market major make money.
>> Yiling Chen: So it seems that I must be lying or …
>>: [indiscernible]
>> Yiling Chen: I’m saying something not …
>>: [indiscernible]
>> Yiling Chen: Not logical here right.
>>: [indiscernible]
>> Yiling Chen: Because I trust so much that no arbitrage is important but here I’m relaxing that. So let
me try to explain what’s happening here. So what’s happening here is that if the market starts with
some price that’s within the convex hull then the market maker is having a consistent price to start with
and there’s no arbitrage opportunity at that point. But it does allow some trader comes to the market
and changes the prices to be outside of the convex hull and creating the arbitrage opportunity. That
doesn’t matter because the market maker can correct that back and make more money, or the other
traders can correct that back and that doesn’t hurt the market makers loss. That’s why this is possible.
>>: So is the arbitrage being limited to the market makers maximum loss? Because they’re counting
that everybody makes money …
>> Yiling Chen: It’s not …
>>: Arbitrage makes money who loses?
>>: [indiscernible]
>> Yiling Chen: Trader loses.
>>: The trader who created this arbitrage?
>> Yiling Chen: Right…
>>: Yeah, right.
>> Yiling Chen: The trader who created the arbitrage …
>>: [indiscernible]
>>: Okay.
>> Yiling Chen: So this is just saying that I’m a market maker I know that the prices should be allowed
into the convex hull of the security payoffs if there’s no arbitrage. But I’m not preventing you to
changes to the outside and create an arbitrage opportunity. So for the analogy for the complete market
case is that the prices can be sum to be greater than one and the market maker doesn’t have the
constraint. So if some trader really changes the market price to be sum to be greater than one that’s
arbitrage opportunity. The market maker allowed that opportunity to happen during the process of the
market.
>>: But what do you put in as an ultimate procedure the moment it goes out immediately the market
then corrects it back in.
>> Yiling Chen: That could be computationally hard to figure out the arbitrage opportunity. So that’s…
>>: [indiscernible]
>>: So [indiscernible]
>> Yiling Chen: Yeah.
>>: [indiscernible]
>>: So there may be arbitrage that’s computationally impossible to take advantage of and …
>> Yiling Chen: Yeah.
>>: And that’s fine.
>> Yiling Chen: So that’s where this comes into play. So if its computationally easy then the market
maker can consider doing that. But for the pair betting case it’s computationally hard. Given that what I
have been saying is that if we allow the arbitrage opportunity what we need is just, we can allow the
allowable prices region to be bigger than the convex hull and we can use less constraint. We really don’t
need to describe the convex hull. So for the pair betting we can just think about, oh I can come up with
some intuitive constraints and I put there and then I can operate a market. So the prices are not as
accurate as before but I gain computational efficiency. So that’s one way to trade off the accuracy of
the prices and computational efficiency, okay.
So that’s the end of my talk. Just to summarize, so we give a convex optimization frame work for
thinking about how can we design automated market maker mechanism for smaller set of securities
over a large outcome space. I mentioned that the worst case monetary loss can be described in terms
of this conjugate function. Also there’s some other market properties can be described using that
function too that I didn’t tell you. This opens up the possibility of, so if we open up the possibility of
arbitrage we can also improve the computation, so basically we get computational efficiency.
>>: Tractability.
>> Yiling Chen: Tractability by approximating the market price. So we do like a, basically we
approximate the pricing, okay. Thanks a lot.
[applause]
>> Yuval Peres: [inaudible] any questions?
>>: Yes I have one so, the market makers [indiscernible] notes [indiscernible] loss. So there are some
mechanisms to [indiscernible] compensate them like some percent of [indiscernible]. What, which one
do you…
>> Yiling Chen: That’s a…
>>: [inaudible]…
>> Yiling Chen: That’s a great question. So in reality stock market make money by taking transaction
[indiscernible]. You can add transaction [indiscernible] here but having transaction [indiscernible] will
hurt information, its [indiscernible] aggregation. So that’s the trade off. The other part is I’d actually
like to argue if the purpose is to get information it makes sense for the market institution to lose money,
because if I’m the one who’s interested in getting information I’m willing to spend money on getting the
information. So that’s why the market maker accepts it as a market. I also have an economist’s answer
for this type of question. For the stock market it is a zero sum game. So what it means is that there’s no
trade theorem saying that if traders are all rational and the rationality is common knowledge they
shouldn’t trade with each other. Because the simple fact that you’re waiting to trade with me and it’s a
zero sum game means that you believe you can make money from this trade. If I’m a rational trader and
I know you’re smart and you have good information I should just refuse trading with you.
>>: You might have different [indiscernible] information.
>> Yiling Chen: Yes, so why people trading stock market? People saying that there are different
reasons. Some people trade for liquidity they really just need money. They need to sell their shares so,
but that can still not explain the high volume of like traders trading. So of course irrational traders can
trade in the market too. But that also gives a reason, if we believe that rational traders shouldn’t trade
in a zero sum game that gives a reason to consider a market mechanism that actually make the whole
game to be positive some because the market maker subsidize the market.
>>: But, you know, suppose they’re not interested in information. Suppose you’re just interested in
making money but you want to do it by sport betting or political betting…
>> Yiling Chen: Yeah you can [indiscernible] transaction [inaudible]…
>>: [indiscernible]
>> Yiling Chen: And there is also…
>>: But do you have any idea what’s the best way to do it, its percentage on each transaction …
>> Yiling Chen: Yeah.
>>: Or something different?
>> Yiling Chen: That’s a great question. I don’t have a good answer about how to do it. In practice
people so some market charge six percentage, some market charges a monthly account fee. We’re
actually working on something trying to figure out whether we can dynamically change the conjugate
function to have a better property regarding to the potential loss or the potential profit. So that’s, I
really don’t have a perfect answer to that.
>>: Now when traders [indiscernible] they say know the current market price for this security but if they
[indiscernible] say I’m not telling you you’re a bad person you can make [indiscernible] decide to take it
or not. Then you can see what spread you get for offering to buy…
>> Yiling Chen: Yeah.
>>: And charge according to that spread…
>> Yiling Chen: Yeah that is an interesting approach. There are actually people taking that approach in
designing market maker mechanism. So for example I believe Semi-Group they have been starting
market maker mechanism who’s trying to learn from the traders. So basically the trader can submit a
[indiscernible] or decide on whether they accept it or not and they take that as a signal about their
valuation and they’re trying to update the market price according to that. Its, so whenever we get past
independence we won’t have no arbitrage. So it’s almost always like that so that means that in theory
there can possibly be arbitrage opportunities too. But it’s totally open to like what are the set of
properties that at least here are really important in practice. One possible direction that I’m interested
in doing is if you look at market maker in financial markets they’re not doing this. So they behave very
differently. They’re trying to make profit. Here the market maker is trying to minimize the worst case
possible loss and not thinking about making profit at all. How can we possibly think about this two
different paradigms and thinking about connecting them? That’s also a potentially interesting direction
to go.
>>: You know I guess we all know how [indiscernible] mutual systems…
>> Yiling Chen: [indiscernible]
>>: For horse races work or less…
>> Yiling Chen: Yeah.
>>: But it would be interesting to figure out how they price exactism trifectus.
>> Yiling Chen: For trifectors?
>>: Where you actually specify the presentation of the first three horses, mixed in with all these two
dollar tickets and so…
>> Yiling Chen: Yeah.
>>: Being you know win, place, or show.
>> Yiling Chen: Yeah.
>>: So they would rather complicate the pricing problem…
>> Yiling Chen: They…
>>: They’re interested in making money but it would be interesting to look at their algorithm to see
what it is.
>> Yiling Chen: That would be really interesting because fundamentally this is about logically dependent
event …
>>: Yeah.
>> Yiling Chen: And they have exactly the same problem.
>>: They have [indiscernible], well yeah.
>> Yiling Chen: That would be really interesting.
>>: Yeah, [indiscernible] not so large. [indiscernible] is still, I mean twelve horses in a race come on.
>>: [inaudible]
>> Yiling Chen: That’s a lot.
>>: [indiscernible] lot of [indiscernible].
>> Yiling Chen: Yeah.
>> Yuval Peres: Okay let’s thank Yiling again.
[applause]