Download Automated Market Makers - A Survey

Automated Market Makers - A Survey
Greg Pastorek
(Dated: May 4, 2016)
I.
INTRODUCTION
Prediction markets allow participating agents, called
traders, to be on the outcome of events, which pay off
if correct. For example, in sports betting markets, the
traders can be for or against a variety of outcomes, such
as a particular team winning.
Prediction markets are useful in information aggregation, since when each trader quotes a price or makes
trades, it is revealing its (sometimes private) information about its belief on the outcome. These are superior
to other methods, such as polls, since the participants
are incentived financially to reveal their own information. One example of a successfully information aggregation prediction market is the Iowa Electronic Market,
a University run market that lets traders bet on a variety of outcomes ranging from presidential elections to
weather events. The information gathered from the market is available for students to use in research. Another
example is the New York Stock Exchange, which helps
aggregate company valuations through their stock prices.
Formally, prediction markets are continuous double
auction markets, with both buyers and sellers, where orders are matched if the buyer’s price is higher than the
sellers, matched at the price of the person who made the
offer first. Traders buy and sell these bets continuously,
called securities.
The aforementioned market introduces some problems.
One, is that a trader will often have difficulty finding a
counter party for there offer, depending on the level of
activity. Since the market is zero-sum, if a trader wants
to make a trader it belives is profitable, it must assume
another trader has the opposite opinion. Thus finding a
counter party is difficult and the trader must wait. Even
worse, traders will not want to wait, and thus it will not
even want to enter the market. This creates a vicious
cycle of low activity and low liquidity. Another problem
is that while a trader quotes a price and waits, it has
revealed its private information by providing a quote.
This may tip off other traders and prevent them from
trading with the current quote. [7]
A solution to these problems is to introduce a market
making agent. Automated market makers are algorithmic agents that provide liquidity in prediction markets,
by offering quoted prices for buying and selling at all
times. In our sports betting example, the bookie acts
as a market maker since they always offer both sides of
every bet. These agents will inherently run at a loss,
although we will later discuss ways to collect profit.
II.
MATHEMATICAL FORMULATION
We are in a market betting on the outcome of an event
{ω}n , with possible outcomes ω1 , ..., ωn . Assume we have
one market maker in the market. Its current position
can be describe by the payout vector x ∈ Rn , where each
element of the vector x describes the payout of a potential
event. For example, if ωi occurs, then the market maker
will receive xi . When a trade is made with the market
maker, the payout vector is transformed x → x0 .
For example, suppose we are in a market for the outcome of a coin flip. ω1 is heads, ω2 is tails. Traders
can buy and sell two securities, s1 and s2 . s1 pays $1
if ω1 (heads) occurs, and s2 pays $2 if ω2 (tails) occurs.
Suppose the market maker holds nothing, and then buys
one tails contract s2 from a trader. It’s payout vector
changes from {0, 0} → {0, 1}
1.
Cost Functions
A market maker’s behavior is defined by how it sets
its prices based on its past actions and beliefs about
the future. Securities are priced according to the market
maker’s cost function (also referred to as a risk measure).
A cost function is a function C : Rn → R, that takes the
market makers payout vector as an input. We use the difference in cost functions between two payout vectors to
price the trade. For example, suppose the market maker
has payout vector q and cost function C, and a trade
comes in an wants to buy r and thus change the market
maker’s payout vector to q − r. The transaction will cost
the trader C(q) − C(q − r).
Since the change in the cost function determines the
price the market charges, the instantanious price for any
security is the partial derivative of the cost function with
respect to the quantity of the security. That is the price
for security i is:
pi (x) = −∂C(x)/∂xi
2.
Desired Properties of Cost Functions
Othman et al [7] outlines 5 desired properties of cost
functions as the basis for the types we will build. They
are outlined below:
• Convexity:
2
That is C(ax + (1 − a)y) ≤ aC(x) + (1 − a)C(y) for
all a ∈ [0, 1].
This desired property encourages diversification.
That is, the cost of two portfolios held by the same
entity should be less than or equal to the sum of
the cost of the two portfolios held separately. A
risk measure that is not convex can produce bizarre
degeneracies. For instance, Artzner et al. (1999)
point out that a non-convex risk measure can produce cases of intra-firm arbitrage. If a firm uses
a non-convex risk measure, it can be advantageous
for two traders to report their specific portfolios x
and y separately, rather than to report their desks
common exposure as x+y. Since the firm itself exists as (and is exposed to) the sum total of its risks,
this division should be discouraged.
• Monotonicity:
For all x and y such that xi ≤ yi ∀i, C(x) ≤ C(y).
This property prevents arbitrage. Since y will always produce a greater payout than x, C(y) must
be greater than C(x) so that buying y costs more.
Suppose it didn’t, then the trade x → y would have
a negative cost to the trader (they would be selling to the market maker), yet the trader would be
gauranteed a better payout than previously. Monotonicity can prevent this type of arbitrage.
• Bounded loss
Bounded loss means supx [maxi (xi ) − C(x)] < ∞.
We desire bounded loss to prevent bankruptcy, and
unbounded worst-case loss is clearly undesirable.
• Translation invariance
For all x and scalar α, C(x + α1) = C(x) + α. That
is, adding α to every payout in the world will cost
α.
• Positive homogeneity
For all x and scalar γ > 0, C(γx) = γC(x). Positive homogeneity ensures a scale-independent price
response. One way to intuitively think of this is
that the cost function is currency independent.
Show by Arrow (Arrow 1963), it is impossible to have
all 5 of these properties besides one cost function. That
function is C(x) = maxi (xi ). However, max is a poor
cost function for many reasons. Consider the following
example, a trader wishes to make a bet and move market
maker’s payout vector from {5, 3} to {5, 3} costs nothing!
We will relax the 5 desired properties. We prioritize
the bounded loss requirement, thus we compromise to
relax positive homogeneity or translation invariance.
A.
Convex Risk Measures
Convex risk measures relax the positive homogeneity
requirement. Thus, they are translation invariance, have
bounded loss, are monotonic, and convex. This section
will cover two instances of convex cost functions, the
Logarthmic Market Scoring Rule (LMSR) and Constantutility cost-functions.
1.
LMSR
The Logarithmic Market Scoring Rule (LMSR) [4] was
originally developed by Hanson et al. It is popular because of its simplicity - its cost and first derivatives have
a simple closed-form.
C(x) = b log(
P
i
exp( xbi ))
And the prices (∂C(q)/∂q) are:
pi (x) =
Pexp(xi /b)
j exp(xj /b)
The LMSR has worst-case bounded loss of b log(n). It
has one parameter, b, referred to as the liquidity parameter. This parameter adjusts the sensitivity of the price
changes with respect to trade size. The higher b, the less
sensitivty the market.
For example:
With b = 10, C({6, 3}) − C({5, 3}) ≈ 0.56
With b = 1, C({6, 3}) − C({5, 3}) ≈ 0.92
2.
Constant-Utility Cost Functions
Constant-utility cost functions are a more general representation of convex cost functions developed by Chen
and Pennock [3]. They work by keeping the market
makers expected utility constant for all payout vectors.
To define these, we first define a utility function, some
strictly increasing concave function u : U → R+ , where
U is an open inerval on the positive real line.
We define the cost cost utility function as follows, let u
be some utility function and let x0 ∈ domu. Let πi be the
market makers belief on the probability of ωi occuring.
The cost function C is thus the implicit solution to:
P
i
πi u(C(x) − xi ) = u(x0 )
Computation is done by running a binary-search over
the possible values of C(x). Chen and Pennock [3]
note that this is a general representation of convex
cost functions. For instance. the popular LMSR is
the constant-utility cost function with πi = 1/n and
u(x) = −exp(−x/b)
3
3.
Extensions
The market makers discussed so far run at a loss, and
they are also liquidity insensitive. Liquidity insensitivity means that the pricing influence of trades does not
depend on the level of activity in the market. This is
not desirable, consider the following: should buying one
security when no other trades have occurred move the
market just as much when 10,000 trades have occurred?
Most will disagree. Othman and Sandholm [6] propose
an extension to constant-utility cost functions that make
them profit-taking and liquidity sensitive.
Let f and g be non-decreasing functions R+ → R, with
f (0) = g(0) = 0. Call f the liquiduty function and g the
profit function.
Also, let d be some distance function Rn Rn → R
Finally, maintain some state s, where s = 0 initially.
If a trader wants to move the market maker’s payout
vector from x to y, it will be charged M (x, y). This is
defined below:
thus increases (becomes more liquidity) with trading activity. No expressivenenss is lost, since we can just double the set of securities to include one security for each
sale. Under this scheme, we redefine how pricing works,
rather than paying C(q)−C(q 0 ) to move the market from
q 0 → q, the trader now pays C(q) − mean(q) − C(q 0 ).
Thus, the market maker is always buying securities. Note
that this constraint also adds profit taking, consider a
round trip with two securities, where one is the shortversion of the other. C({1, 1}) − 1 − C({0, 0}) will be
positive, even though {1, 1} and {0, 0} are equivalent.
This not only adds liquidity sensitivity, but also removes the non-intuitive liquidity parameter. While there
is a still a parameter α, in [5] Othman et al show that α
relates to the profit taking by the market maker.
The α parameter affects the profit taking, and it can be
v
related to the maximum commission taken. If α = n logn
,
the commission will not exceed v.
The closed form solution for the price function is:
pi (x)
P
Solve for C(y) (we know C(x) from the last iteration)
0
i πi u(C(y) − yi + f (s + d(x, y))) = u(x + f (s +
d(x, y)))
j
=
αlog(
P
j
exp(xj /b(x)))
+
xj exp(xi /b(x))−sumj xj exp(xj /b(x))
P
j xj exp(xj /b(x))
P
C.
Kelly-Criterion Market Maker
The total cost is then:
M (x, y) = C(y) − C(x) + f (s + d(x, y)) − f (s) + g(s +
d(x, y)) − g(s)
Also, update the state s ← s + d(x, y)
There are a few restrictions on the choice of f , g, and
d outlined by Othman and Sandholm [4]. For example, it is required that lims→∞ f 0 (s) = lims→∞ f 00 (s) =
lims→∞ g 00 (s) = 0
B.
Homogeneous Cost Functions
Homogeneous Cost Functions are another family,
where we relax the translation invariance requirement
and add positive homogeneity. Recall, that is C(γx) =
γC(x) for scalar γ > 0. One type of homogeneous market maker with some desirable properties is the OPRS
Market Maker developed by Othman et al. [5].
The OPRS Market Maker is a homogeneous analgoue
to the LMSR function. It is liquidity sensitive and profit
taking.
The cost function is defined below:
P
C(x) = b(x) log( i exp(xi /b(x))), where b(x) =
P
α i xi
We constrain that all elements of x are positive, thus
b(x) can capture the level of activity in the market and
Othman describes in his thesis a way to design a superior market maker when we have a good prior on the
future. This is realistic for many applications, such as
prop trading desks, where extensive research is put in
to predictive models. This market maker is base on the
Kelly-criterion, which is a way to make bets that maximizes expected log-utility.
In this model, traders are anonymous, and at each time
step one is selected to trade with at arandom. The state
of the market is (t, w), where t is the current traders
index and w is the market maker’s payoff vector. Let π
be the market makers pricing function, p be the market
makers probability distribution on the outcomes, and let
V be the expected utility.
At termination, based on the Kelly-Criterion, the expected utility is:
V (tf , w) =
P
i
pi log(wi )
The optimal pricing function π ∗ must be solved for as
follows:
Let δ(t)betheprobabilitythatthesessionterminatesattimet(whiletr
π) be the probability that the trader takes bet x at price
π(x) (note that traders are anonymous so is does not
depend on t).
P
V (t, w) = (1 − δ(t)) x P (x, π)V (t + 1, w − x + π(x)) +
δ(t)V (tf , w)
π ∗ (t, w) = argmaxπ V (t, w)
4
from the market is show in figure 1.
FIG. 1. Distribution of probabilities time series
III.
This experiment showed some problems with the automated market maker. In particular, the prices has a
”spikiness” element to them. That is, prices for very
similar securities were wildly inconsistent. This is nonsense, because there is no reason that a certain day should
be significantly less likely than the days immediately before and after it (weekends and holidays aside). Figure
2 shows the ”spikiness” of the prices. It turns out that
the LMSR actually induces spikey prices. It is because
Oi pi (x) > Oj pj (x) for some x whenevr pi (x) > pj (x),
thus the prices are pushed up higher as trades are made
for higher prices faster than lower prices, so the spikiness
grows.
APPLICATIONS
The experiment analyzed successful trader strategies.
Most involved spike-dampening. With the realization
that these spikes were nonsense, traders could enter a
A.
Gates-Hillman Prediction Market
The Gates-Hillman Prediction Market was an experimental prediction market run at Carnegie Mellon University [1]. The prediction market was run to predict
the opening date of a new building. The participants
used raffle tickets instead of real money, and they could
be redeemed for prizes. The market used an automated
market maker (LMSR with b = 32).
The event space was the set of days in a year (n = 365).
Each day has one security that pays out if the building
opens on that day. Trades would trade using an interface
that let them bet for or against spans of days, thus they
would buy securities in bundles.
FIG. 2. Prices across securities showing spikiness
statistical arbitrage by betting against spikes and betting for undervalued spans, and waiting for the spikes to
even out, or otherwise getting positive expected value on
the trades made against the spikes.
The implicit probability distribution of opening days
[1] Othman, Sandholm. The Gates Hillman Prediction
Market
https://www.cs.cmu.edu/~sandholm/Gates-Hillman\
%20prediction\%20market.RevEconDesign13.pdf
[2] Das, Brahma, Chakraborty, Lavoie, Magdon-Ismail. A
Bayesian Market Maker.
http://www.cse.wustl.edu/~allenlavoie/papers/bmm.
pdf
[3] Chen, Pennock. A Utility Framework for Bounded-Loss
Market Makers.
https://arxiv.org/ftp/arxiv/papers/1206/1206.
5252.pdf
[4] Hanson. Logarithmic Market Scoring Rules for Modular
Combinatorial Information Aggregation
http://mason.gmu.edu/~rhanson/mktscore.pdf
[5] Othman, Sandholm, Reeves. A Practical LiquiditySensitive Automated Market Maker
http://www.eecs.harvard.edu/cs286r/courses/
fall12/papers/OPRS10.pdf
[6] Othman, Sandholm. Automated Market Makers That
Enable New Settings: Extending Constant-Utility Cost
Functions
http://repository.cmu.edu/cgi/viewcontent.cgi?
article=3781&context=compsci
[7] Othman. Automated Market Making: Theory and Practice