Chapter 22
Market Microstructure
1. Limit Order Markets
2. Oligopoly
3. The Direction of Trade
1. Limit Order Markets
We now introduce limit order markets as a real
world institution describing the financial sector,
and as a paradigm for trading mechanisms. We
define the general solution to a trading game
and, then demonstrate that many auctions,
retail sectors and stock markets fit into a limit
order framework.
Financial Markets
In the second half of this lecture we examine
electronic limit order markets, which are amongst
the fastest growing markets within the financial
sector.
Instead of dealers mediating between buyers and
sellers anyone in good standing can submit and
sell limit and market orders.
What can we say about the portfolio
management when financial assets are traded on
a limit order market?
General description of a trading game
Consider the problem a trader faces in a limit order
market. First we describe the trading mechanism in a
limit order market.
Then we define his preferences, then the constraints
on his trading problem. These constraints determine
his choice set.
The description is completed by explaining the
trader’s optimization problem, and showing how the
solution to the problem is part of a Nash equilibrium.
Trading in limit order markets
Anyone seeking to trade in the market must submit a
market order or a limit order.
Each order is for a given quantity, negative quantities
standing for units for sale, positive for units
demanded.
Limit orders also specify a transaction price.
All market transactions match a market order with
one or more limit orders, and take place at the limit
order price(s).
Precedence
Market orders to buy are matched against the
lowest price limit order(s) to sell.
If two limit orders to buy are submitted at the
same price, the order submitted first is matched
against a market sell order before the more
recently submitted buy order.
Similarly lower priced limit orders to sell have a
higher priority than higher priced limit sell orders,
and if two bidders seeking to sell a unit at the
same price the person who bid first will be
matched before his rival seller.
Trading window
The trader in this market has just placed a sell
order for 9 units at price 5,800, with an expiry
time of 60,000 seconds (that is 16 hours 40
minutes). There are 6 limit orders to buy already in
the books (2 at 3800 and 4 at 200), and 4 other
limit orders to sell at 6,000.
The Spread
The spread is defined as the difference between
the highest priced limit buy order ask price,
called the bid price, and the lowest priced limit
sell order called the ask.
In the example above, the ask price is 5,800 and
the bid price is 3,800, so the spread is 2,000.
Observe that the trader whose display screen is
illustrated reduced the spread from 2,200 by
placing an order inside the previous bid ask
quotes.
Market orders are executed immediately
A market order to buy (sell) one unit is defined by a
price which is greater (less) than or equal to the
lowest (highest) outstanding limit order to sell (buy).
Therefore market orders transact instantaneously,
market buy (sell) orders reducing the number of
outstanding limit orders to sell (buy).
Market orders to sell (buy) are matched with the
highest (lowest) priced limit order to buy (sell) and
executed at the price of the matching limit buy (sell)
order.
Limit orders are not always executed
All other orders (sell orders priced higher than the
best bid price, buy orders priced lower than the best
ask price) are entered in the book as limit orders.
A limit order can be withdrawn at any time before it
is matched with an incoming market order as part of
a transaction.
The outcomes of the random variables that
determine asset valuation, the strategies of all the
players, and the past history of the game, determine
the execution probability of a limit order.
Examples of limit order markets
In a first price auction, bidders submit a limit order to
buy without seeing the trading window and the
auctioneer submits a market order once all the limit
orders have been placed
In a Dutch auction the auctioneer submits a sequence
of increasingly attractive limit orders to sell until a buyer
submits a market order.
In an English auction bidders submit compete with each
other by submitting limit orders to buy until the bidding
stops and the auctioneer submits a market order.
In retail markets stores submit limit order to sell and
buyers submit market orders to buy.
A trader’s optimization problem
Consider some markets for financial assets on a
stock exchange where traders make bids and
offers at prices they choose throughout the
duration of the game to maximize their expected
subjective value of asset holdings at the end of
the game.
It might be convenient to imagine that the length
of the game is the difference between the
opening and closing time on a typical trading day.
Initial endowment
At the beginning of a game, players receive an initial
endowment of :
- each stock
- money
The endowment might be:
- fixed by the moderator
- the realization of a random variable drawn from a
probability distribution that the moderator decides.
All players belonging to the same player type receive the
same fixed amount in the first case, and get independent
draws from the same distribution in the second case.
Market access
In the market module there is only one medium of
exchange called money: all trades involve a transaction
between money and a stock. (For example trading in
more than one currency are excluded.)
A player might be allowed to submit limit and/or
market orders to buy and/or sell in an asset market.
Depending on the game, money might be regarded
purely as a medium of exchange, or stand for
generalized purchasing power as well.
Preferences
Let u(c) denote a concave increasing utility function.
If bj and cjt enter multiplicatively in the trader’s preferences
for the stock, bjcjT denotes the subjective value of the trader for
asset j at the end of the game. Then the overall objective that
the trader uses to assess her trading and portfolio decisions is:

u mT   j 1 b j c jT x jT
J

Alternatively if (bj + cjT) denotes the subjective value of the
trader for asset j at the end of the game the objective is:

u mT   j 1 b j  c jT x jT
T

Information
Information about the factors, the stocks
and the state of each market is determined
at each player type:
- information at the start of the game
- how the information is updated.
Subjects
might
have
information about:
-valuations,
- assets,
-order book,
- history of transactions.
different
Uncertainty
When the jth asset is a stock, cjT is a random
variable that represents the ex-post return on the
common component at the end of the game,
It is therefore impossible to base trading decisions
at time t < T on cjT because it is unknown at time t.
Accordingly denote by lt the information available to
the trader at time t.
Assume that at each instant t the trader maximizes
the expected value of the utility at T from her
portfolio.
Expected value
Then in the multiplicative case her expected utility
may be expressed as:

 
E u mT   j 1 b j c jT x jT | lt
J
where:
b j c jT x jT
is the liquidation value of her stock holdings
in the jth stock at time T.
The additive case is similar.
Solvency
To survive, a financial institution must enforce
traders to honor contracts between themselves.
While the fiduciary rules vary across institutions, the
market module captures the essence of many, if not
most.
In the market module the trader is constrained at
each point in time by how much she can offer for sale
within each market and how much she can buy in total.
This implies there are J constraints for placing sell
orders but only 1 constraint for placing buy orders.
Constraints on sell orders
We denote the set of buy and sell prices by
 

pk k 1
 p1, p2 , p3 , 
For convenience we assume there are no short sales.
Therefore the total amount of each asset up for sale
cannot exceed her holdings.
Let sjkt denote the quantity of the jth asset for sale at
price k at time t. We require:

k 1 pk s jkt

 x jt
Constraint on buy orders
An overall budget constraint on buy orders prevents
the trader from placing orders that exceed her money
holdings.
It effectively constrains the seller from exchanging
(selling) more money for assets than she holds.
Let djkt denote the quantity of the jth asset
demanded at price k at time t. We require:

k 1 pk d jkt
 
J
j 1
 mt
Choices at time t
Players exploit their trading opportunities defined in the
choice set to make decisions throughout the game.
At each successive instant t  [1, T] the trader may do
nothing, or take an action in one market j  {1, . . . , J}
subject to the constraints defined in the three previous
er
slides:
1.
Delete an existing limit order to buy or sell a
quantity q in market j.
2.
Submit a market buy or a market sell
order for quantity q.
3.
Submit a limit buy or a limit sell order for
quantity q in market j at price pk.
The trader’s optimization problem
In the multiplicative case the trader
sequentially makes the choices that at each
successive instant t  [1, T] maximize:

 
E u mT   j 1 b j c jT x jT | lt
J
subject to the rules prescribing the orders he is
permitted to place, the J budget constraints
preventing short sales, and 1 overall budget
constraint preventing borrowing.
Interdependence between players
The actions of the other players affect the trading
opportunities of the nth player.
Consequently the probability distributions the nth
player uses to take expectations over future events are
partly determined by the trading strategies of the other
players.
Therefore the previous two slides provide an
incomplete description of the trader’s optimization
problem, because they does not fully describe how to
take the expectations over future trading opportunities.
The strategy space
Let dj(lnt) be an indicator function showing which market
the nth trader will choose to act in at time t when his
information is lnt, where djl) =1 if he picks market j and
dj(lnt) = 0 otherwise.
Let q(lnt) denote the quantity he picks (where negative
quantities indicate sell orders), and p(lnt) the price (which
is constrained to be the market price if the trader is
permitted to make only market orders are in this market).
Then a strategy for the nth trader is the vector function
s(lnt) = (dj(lnt), q(lnt), p(lnt)) for each t  [1, T].
The valuation function of trader n
Let sn0(lnt)) denote the optimal order strategy, and
define the value of correctly solving at time t by:



W lnt   E u mnT   j 1 bnj cnjT xnjT | lnt , s 
J
o
Then for all t < r < T, the value function W(lnt)
solves the recursion:
W lnt   EW lnr  lnt 
This equation says that optimized value should
behave like a random walk throughout the game.
The strategic solution
to limit order market game
Suppose each trader n = 1, . . . . ,N picks a
strategy to solve their own optimization
problem, and calculates the expectation knowing
the strategy the other traders picked.
The resulting strategy choice se(lnt) for each
trader n = 1, . . . . ,N is a symmetric Nash
equilibrium for the limit order market game. It
corresponds to the solution of the strategic form
for this game.
Summary
In the second half of this lecture we defined a
limit order market, and described the
maximization problem of a day trader, nesting
it within the solution to the game.
Then we showed that in simple trading
settings, limit order markets have a rich
enough strategy space for players to
communicate their valuations to each other
and exhaust almost all the gains from trade.
2. Oligopoly
How is the solution to a trading game
affected by relaxing the assumption that
there is only a single supplier in each market?
We explore the extent to which monopoly
power is diluted by rival producers.
Resale as a potential source of
competition with the monopolist
Some economists believe that the possibility of
resale by consumers limits the monopolist’s ability
to engage in price discrimination.
They say a low valuation consumer can buy the
good at a low price and then compete with the
monopolist to supply a high valuation demander.
Price competition between the monopolist and low
valuation consumers supposedly drives down the
price that high valuation consumers pay.
Primary versus secondary suppliers
Similarly lawyers for Alcoa once argued that although
the company had a monopoly on the supply of primary
aluminum ingots, it did not have a monopoly on the
market, because there were secondary supplies from
scrap.
Judge Learned Hand rejected Alcoa’s arguments in
1945, finding them guilty of violating the Sherman Act
which prohibits collusion. He noted that resale by
secondary suppliers does not prevent Alcoa from
restricting the total supply of aluminum
Do secondary suppliers, such as consumers of primary
ingot reduce Alcoa’s profits from selling secondary or
aluminum ingot as scrap?
Monopoly with re-trading
We consider the three cases, where trading occurs:
1. on a half open time interval of [0,1) with
consumption at t =1
2. on a closed time interval of [0,1] with
consumption at t =1.
3. on a closed interval of [0,1] with consumption
from the services of a durable good over the
time phase [0,1].
Now assume consumers may themselves sell the
product they purchase from the monopolist.
Competition
from repairs and maintenance
The monopolist faces “extra supply” from consumers
lengthening product life to postpone the cost of
replacement.
One way of mitigating this competition is to void all
product warranties if an unauthorized mechanic
maintains it.
Companies such as Xerox have circumvented this
competition in the past by leasing its goods rather than
selling them outright.
Collusion as a contract between rivals
Collusion is outlawed by the Sherman Act.
It restricts opportunities to establish a long term
relationship of mutually sustaining collusive prices
with the threat of a price war.
Furthermore the larger the number of rivals the more
difficult it is to reach and enforce tacit agreements on
price.
Thus increasing the number of suppliers tends to
foster competition between them.
Price competition from rival suppliers
If demand for a product is a linear function of price
then:
q =  - p
Suppose rivals compete on price. As a function of
(p1,p2), the net profit to the first firm is:


p 1 
p 1 cif p 1 p 2
p1 
1
2


p 1 
p 1 cif p 1 p 2
0 if p 1 p 2
The profit function for the second firm is derived in
a similar fashion.
Quantity competition by rival suppliers
Rather than compete on price, both firms could
choose their own production and let the market price
adjust to clear the market.
For example suppose that in a two stage game, firms
construct capacity for production in the first stage,
and market their produce in a second stage.
Accordingly denote by q1 and q2 the quantities chosen
by the firms.
The industry price is derived from the demand curve
as:
p = ( - q1 – q2)/
Net profit to the ith firm
Net profit to the ith firm:
  q1  q2
li (q1 , q2 )  



 c  qi

The marginal profit with respect to qi:
li ( q1 , q2 )   2q1  q2

c
q1

The competitive limit
If firms compete on price then (in a one period
model where there are no opportunities for
colluding), then in the absence of monopoly, price
equals marginal cost.
If there are N firms which compete on quantity,
then in a symmetric equilibrium each firm produces:
q 
N 11 
c
As N diverges, the industry quantity converges to
the point where price equals marginal cost.
The strategy space
If firms compete on price, then two firms suffice to
reduce price to marginal cost.
If firms compete on capacity quantity, then as the
number of firms increase, price converges to
marginal cost.
But what happens if each firm can credibly commit
to match the lowest competing offer?
These scenarios illustrate the importance of the
strategy space in determining the final allocations
and the realized gains from trade.
Monopolistic Competition
Monopolistic competition describes a product
differentiated market in which each supplier
sells a good with no perfect substitutes, but
several imperfect substitutes.
To what degree do features of monopoly
appear in differential product markets?
Brand loyal customers can be charged
higher prices that deter the switchers. Each
firm delineates its customer base partly
through its price strategy.
Free entry
When marginal costs are constant or increasing,
entry occurs sequentially until it is unprofitable to do
so. This corresponds to a competitive market where
rival suppliers compete for demanders.
When marginal are declining, or when there are
fixed costs and constant marginal costs, the result is
harder to predict.
One possibility is a trigger strategy equilibrium in
which a monopolist or cartel maximizes producer
surplus by threatening to compete by dropping the
price to zero if another firm enters: this may be
illegal under the Robinson-Patman Act.
3. The Direction of Trade
In the experiments above we have taken as given
which traders are buyers and which ones are
sellers. This is not a useful assumption to make in
all markets, particularly markets for financial
securities. What happens when the direction of
trade for each player is determined by the terms
of trade? Can we devise trading games that have
desirable properties in which the identities of
buyers and sellers depend on the configuration of
valuations?
A trading game
Consider the following two stage game of price
determination for three (or any odd number of
players), each of whom can buy or sell at most one
unit of a good.
Every player knows their own valuation for trading a
unit, but not necessarily the valuations of anyone else.
Then players simultaneously or sequentially indicate a
reservation price at which they are indifferent between
buying or selling a unit.
In the second stage of the game every player places
limit or market orders but only at the median
reservation price announced in the first stage. They
cannot place sell (buy) orders if their announcement is
higher (lower) than the median price.
Incentive compatibility
We now prove that the weakly dominant strategy is for
each player to truthfully announce his valuation.
Note first that only lies which affect whether the player
becomes the median valuation are payoff relevant.
Suppose he would have been the median price if he
had been truthful (and not traded) but instead he
announces a higher valuation. Consequently a lower
valuation becomes the trading price. The lying median
valuation player now must buy at a higher price than
his true valuation, and thus incur a loss.
If a person becomes the median valuation by lying
about his type then he foregoes gains from trade.
Summarizing the properties
of the trading mechanism
The trading mechanism is:
1. self financing because it requires no outside funding
to implement.
2. satisfies incentive compatibility because each player
has a weakly dominant strategy to truthfully
announce his valuation.
3. Satisfies the participation constraint because each
player prefers to participate in the trading game
rather than consume his own endowment
4. Efficient, as all the gains from trade are exploited.
A generalization
When there are an even number of players the trading
mechanism is modified slightly.
The first stage proceeds exactly as before. In the
second stage the players announcing the two median
valuations are excluded from participating in the
second round of trading, and must consume their
respective endowments. The other players are
permitted to trade any single price selected between
the two valuations
This mechanism is self financing, satisfies the
participation and incentive compatibility constraints,
but is not fully efficient because the two median
players are prevented from trading with each other
and must consume their endowments instead.
Limit orders as language
We revisit the simple trading game where each trader
is endowed with one unit, only wants one extra unit at
most, valuing each of the first two units the same.
All players with valuations above the median submit a
limit buy order at their valuation less the median, and
all players with valuations below the median submit a
sell order at their valuation plus the median.
Then all players withdraw their valuations and submit
bids only at the revealed median valuation.
Can this collective behavior be rationalized?
Summary
Resale does not limit monopoly power unless the
secondary producers can affect the characteristics of the
product.
A competitive fringe typically reduces but does not
necessarily eliminate monopoly rent.
When there are several comparably sized suppliers, the
strategies space in which rivals compete can greatly affect
the rent they collectively make.
When the direction of trade is endogenous, and players
have private valuations, there are trading games whose
solution outcomes converge to the efficient allocation as
the number of traders increases.