Payment for Order Flow
1
Christine A. Parlour
GSIA, Carnegie Mellon University,
And Securities and Exchange Commission
Mail Stop 1105
450 5th Street
Washington, DC 20549
E-mail: [email protected]
Uday Rajan
GSIA, Carnegie Mellon University
Pittsburgh, PA 15213.
E-mail: [email protected]
1 We
are grateful to Leslie Marx, W. Atkinson, D. Gray, B. Hollield, L. Lightfoot, M. Ready,
and seminar participants at the NBER Microstructure meetings for helpful discussions.
2 The US Securities and Exchange Commission, as a matter of policy, disclaims any responsibility
for any private publication or statements by any of its employees. The views expressed herein are
those of the authors and do not necessarily reect the views of the Commission, or of the author's
colleagues on the sta of the Commission.
1
Payment for Order Flow
Abstract
We develop a dynamic model of price competition in broker and dealer
markets. Competing market makers quote bid-ask spreads, and competing
brokers choose a commission to be paid by an investor. Investors, who submit
either market or limit orders, choose a broker to minimize total transaction
costs. In this environment, payment for order ow unambiguously increases
spreads and increases the ratio of limit to market orders. It can lead to lower
brokerage commissions, but higher market-maker spreads, thereby increasing
the total transactions cost for market makers. We provide testable predictions
on the mix of limit to market orders.
1 Introduction and Institutional Arrangements
Retail order ow is a valuable commodity. It is purchased by market makers either
for direct cash payments, by payment in services (for example, Merrill has a clearing
arrangement with Knight Securities) or through aÆliation arrangements (e.g., Schwab
and Mayer{Schweizer).1
The issue of payment for order ow is inextricably bound with that of \Best
Execution." Briey, the latter arises out of the common law duciary responsiblity
of an agent to obtain the best terms for a customer. In securities markets, this
is interpreted as obtaining the best possible price for the customer. Typically, in
payment for order ow contracts, irrespective of the method of payment, the market
maker guarantees a price to the customer no worse than the best price currently
available.2 Such an agreement is equivalent to a price matching agreement.
It is an open policy question whether payment for order ow impedes competition
in a dealer market. One intuition is that consumers only care about the total price
they pay for executing a trade: that is, the spread they are exposed to and the
intermediation cost they have to pay to brokers. If payment for order ow is simply
a transfer from market makers to brokers, then, given erce competition in the retail
broker market, such transfers should ow back to the consumer either through lower
commission costs or through trading enhancements.3 Payment for order ow, thus,
may be an innocuous side payment that exercises no deleterious eects on consumers.
A competing intuition (dating back to Salop, ????) is that price-matching agreements obviate Bertrand competition. That is, in any one{shot price game, if there is
a price{matching agreement, there is no incentive to cut prices, and hence we should
not expect to observe price equal to marginal cost. Therefore, given that payment
for order ow agreements include explicit price{matching commitments, we should
be wary of their eects on competition.
The purpose of this paper is to provide a theoretical framework to examine the
eect of payment for order ow on the net costs paid by retail investors to execute
1A
survey of such arrangements is presented in a 1991 report to the NASD Board of Governors
\Inducements for Order Flow," and in the SEC document \Report on the Practice of Preferencing."
2 Formally, the market maker guarantees weak price improvement over the best bid and oer
currently posted (NBBO). The inclusion of such provisions allows brokers to full their best execution
obligations.
3 For example, easy to navigate web sites with links to market relevant information can make
trading easier.
1
trades. Thus, we seek to understand how such contracts aect competition in the secondary market and competition in the brokerage market. To examine these questions,
we provide a dynamic model of price competition in the broker and dealer markets.
We nd that payment for order ow can be a way in which brokers capture surplus
from the market makers, without passing it on to consumers. Further, we show that
while intermediation costs may be lower under payment for order ow, the total cost
paid by retail investors (including the spread) is higher.
The central intuition of our paper is that brokers and market makers value limit
orders dierently. Specically, for market makers a limit order represents a competiting liquidity supplier. Thus, a limit order on the books is a potential foregone trade.
The value to him of an incoming market order depends on the state of his limitorder book. Such states are unobservable, and thus non{contractible. Brokers, by
contrast, receive commissions on both limit and market orders (albeit with diering
probabilities), and thus view both as a source of revenue.
Theoretical and empirical work to date has focused on the eects of payment for
order ow on quoted spreads at the NYSE: for example, Battalio, Greene and Jennings
(1997, 1998), Battalio (1997), Easley, Keifer and O'Hara (1996), and Chordia and
Subrahmanyam (1995). Competition between the NYSE and regionals is one between
a large dominant specialist and oor brokers and smaller specialists. The outcomes
that one might expect in this market are dierent from those on Nasdaq, which is
characterized by competing market makers who set quotes.
Several studies have specically looked at the eect of payment for order ow in
dealer markets such as Nasdaq. Bloomeld and O'Hara (1998) examine the question
in an experimental context, and nd that, the more order ow that is preferenced,
the wider the spreads. Our work is closely related to Dutta and Madhavan (1997),
who also consider an innitely repeated game. One point of departure from their
work is that we explicitly model the order routing strategies of brokers and consider
the eect of payment for order ow on the total price paid by consumers. Kandel
and Marx (1999) consider long run equilibrium in a dealer market in which there is
payment for order ow. They consider the eects of such payment and the existence
of dierent order execution systems on the number of market makers in a particular
stock. We are interested in the interaction between the broker market and the market
maker in the presence of payment for order ow, and thus we model competition for
clients between brokers. We also include competition between limit order suppliers
2
and market makers. Our work then, is a complement in that it does not consider
entry and exit of maker makers, but rather takes the market participants as given
and then determines the outcome of competition.
Battalio and Holden (2001) consider a variant of Glosten and Milgrom (1985) with
payment for order ow. They consider the total cost paid by consumers, that is, broker
commission fees plus a liquidity premium. They impose a zero prot (or competitive)
condition on quoted spreads, and thus do not examine how such contracts aect
competition in either market.
We describe the model in Section 2. Section 3 develops properties of the value
functions for both brokers and market makers. In Section 4 we consider the outcomes
of the game with no payment for order ow. We contrast this with the outcomes
that obtain in the presence of payment for order ow in Section 5, and determine the
equilibrium level of this payment in Section 6. Section 7 concludes. All proofs appear
in the appendix, Section 8.
2 The Model
We consider an innite horizon model with three types of players: investors, brokers
and market makers. Market makers supply liquidity by quoting spreads. Brokers are
intermediaries between investors and market makers, and charge a fee for access to
the market. Investors may either demand or supply liquidity; that is, they submit
either a market order or a limit order.
There is one asset being traded. At each of times t = 0; 1; : : :, the following stage
game is played. First, each market maker, m = 1; : : : ; M , quotes a spread, sm . Since
we focus on liquidity provision, we assume that the underlying asset value is known
to all agents. Therefore, rather than model bid and ask prices directly, we examine
the spread. In the secondary market, a market buy order pays the underlying value
plus s2m , and a market sell order receives this value minus s2m . Thus, s2m is the cost of
demanding liquidity in the secondary market.
Next, each broker, b = 1; : : : B , quotes a commission cb , and an order routing
strategy. The commission is paid by an investor whenever a trade is executed. An
P
order routing strategy is denoted by rb : M ! [0; 1], where M
m=1 rb (m) = 1. For each
broker b, rb (m) denotes the probability that the broker will send an order to market
maker m.
3
Collectively, the order routing strategies imply a probability that any particular
retail order will be routed to a specic market maker. Let pm denote this probability.
P
Then, pm = Bb=1 rb (m).
Finally, an investor arrives, and submits either a market order, a limit order or
no order. In the former two cases, she also chooses which broker to submit an order
to. Let qb denote the probability that she chooses broker b. In any given period,
let pm denote the total probability that market maker m receives an order. Then,
P
pm = B
b=1 qb rb (m).
A time line of events in the stage game is presented in gure 1.
Market-makers quote
spreads, sm
Brokers choose commission, cb ,
order-routing strategy, rb
Investor arrives:
chooses order, broker
Figure 1: Events at each stage
In Section 6 we add a contracting stage to the model, before the stage game at
time 0 is played. A contract is an agreement by a broker to route a proportion of his
order ow to a specic market maker, and species a fee, , for each market order
that the broker sends to the market maker.
As we are focusing only on liquidity provision, we assume that buy orders and sell
orders are equiprobable. In each period t, a single investor arrives. The investor has
a personal valuation for liquidity. Suppose he sends his order to broker b, who routes
it to market maker m. If the order is a market order, it is executed immediately, and
the total cost of execution to the investor is (cb + s2m ). If he submits a limit order,
then, if it is executed, he pays cb to the broker, as before. However, the investor
also chooses a price which may be dierent from the true underlying value. Suppose
that the limit order submitter buys at the Bid and sells at the Ask, then this price
improvement decreases his overall cost of trading. Indeeed, if a limit order to buy
(sell) trades at the corresponding ask (bid), the investor obtains an additional benet
of s2m . Therefore, his total execution cost is (cb s2m ).
To focus on the costs to each of the traders, we dene the trading cost of liquidity
demanders, T d (s; c) = c + s2m , the trading cost of liquidity suppliers, T s (s; c) = c s2m
and the expected trading cost, T(s; c) = T s(s; c)+ T d(s; c), of a retail investor. (All
4
these are dened with reference to a stationary equilibrium which we dene below.)
Let denote the probability that the investor submits a market order, and the
probability that she submits a limit order. These probabilities are modeled exogenously as functions of s and c, the spread and commission faced by the investor. We
assume that + 1, with decreasing in (c + 2s ), and decreasing in (c 2s ).
That is, is decreasing in both c and 2s , whereas decreases in c and increases in 2s .
Hence, s ; c < 0, s > 0 and c < 0.
This reects the fact that limit order submitters are supplying liquidity, and therefore benet from a large spread, whereas market order traders demand liquidity.
Hence, in comparing two equilibria, the one with the weakly higher spread will have
a higher ratio of limit to market orders. We assume that these probabilities are
functions only of the current set of intermediation fees and spreads.4
Limit orders are not executed immediately: they stay on the books for one period.
In the next period, if a matching market order comes in, the limit order on the books
is executed. If not, it simply drops out. Therefore, there are two possible `limit book'
states which a market maker can face: either his book has a limit order on one side,
or not.
All brokers and market makers have a common discount factor, Æ . We focus
on pure strategy, stationary equilibria of this innitely-repeated game. In such an
equilibrium, each market maker quotes the same spread, s, at each t, and each broker
b chooses the same c and r at each t. We term such equilibria stationary equilibria.
In the presence of competition, both market makers and brokers wish to charge lower
prices than their competitors, in an attempt to capture a higher share of the market.
This naturally leads to consideration of equilibria in which they choose equal prices.
Equilibria with positive prots for brokers and market makers are sustained in our
model by the threat of deviating to zero-prot strategies for ever. That is, following
any deviation by a market maker, all market makers earn zero prots for the remainder
of the game.
This punishment strategy has an impact on limit order submitters. Though the
choice between market and limit orders is exogenous in our model, our limit order
traders are strategic: they choose a spread at which to submit their competing liquidity supply. Suppose a limit order submitter undercuts the prevailing spread in
4 This
can be justied, for example, with a countable number of investors, out of which one is
randomly drawn each period.
5
the market. Then, she will represent the lowest posted quote (on that side of the
market), and will thus receive any incoming market order, increasing her execution
probability.
However, given that she has undercut the spread, the market maker who received
her order is quoting a lower spread on that side of the market, and thus, next period,
will have to report a `lower quote'. This will entail punishment from other market
makers in subsequent periods, leading to zero prot for the rest of the game. Hence,
this market maker will maximize his one-shot prot by undercutting the existing limit
order, so that the limit order submitter is faced with a zero execution probability.
Suppose, instead that the limit order submitter quotes at the stationary spread.
In this case, the market maker will not automatically face a punishment phase the
next period and thus will not have an incentive to undercut the posted limit order.
Rather, he will incorporate it into his quote, with the limit order having priority in
execution.5 Thus, limit order submitters optimally quote at the stationary spread.
3 Payos in the Innite Game
It is well known that, in repeated games, prices above competitive levels can be
sustained in equilibria with punishment phases. Indeed, the folk theorem asserts that
any payos above minmax levels can be sustained with the appropriate discount rate.
While this is a state-dependent game (as there are dierent possible states of the
limit order book), it suggests that there are potentially a wide range of outcomes.
To simplify the model, we restrict attention to strategies with innite punishment
phases. That is, deviations are punished by reversion to the competitive outcome in
perpetuity.6
First, we characterize payos to a market maker, when all brokers charge a constant fee of c and route to him so that the probability of an order arriving is p. Let
J (0; s; p; c) denote the expected present value of the payos to this market maker if
he has an empty book and let J (1; s; p; c) denote the payos to him when he has a
5 The legislation covering limit
orders in the OTC market are governed by the 1996 Order Handling
rules and the Manning Rules. Although over-the-counter market makers are not required to accept
limit orders, they also are not permitted to refuse to accept certain limit orders in a manner that
unfairly discriminates among customers. Securities Exchange Act Release No. 35751 (May 22,
1995), 60 FR 27997 ("Manning II"), at n.19.
6 An innite punishment phase will ensure that the highest possible prices can be maintained.
6
limit order on his book. In both cases, he, and all other market makers are charging
a spread of s. Further, we write and for the probability of a market and limit
order respectively, suppressing the dependence on (s; c).
If either no order or a market order arrives, he continues to have an empty book
in the next period. In the latter case, he earns 2s with probability p (i.e., if the market
order is routed to him). If a limit order arrives, then, with probability p, his state
next period is 1, and with probability (1 p), it is 0. Hence,
s
J (0; s; p; c) = p( ) + ÆJ (0; s; p; c) + Æ [pJ (1; s; p; c) + (1
+(1
2
)ÆJ (0; s; p; c)
s
p)J (0; s; p; c) + Æ pJ (1; s; p; c) + p( ):
J (0; s; p; c) = Æ (1
p)J (0; s; p; c)]
2
(1)
Next, consider J (1; s; p; c), the expected present value of payos when the market
maker has a previous limit order on his books. If either no order or a market order
arrives, his state next period is 0: the existing limit order is either lled, or expires
unlled. In addition, with probability p2 , there is a market order on the same side
of the market as the limit order. Such an order is lled by the market maker, who
earns 2s on it. Finally, if there is a new limit order, the old one expires unlled, but
the state remains J (1; s; p; c). Hence,
J (1; s; p; c) = p s
( ) + ÆJ (0; s; p; c) + Æ [pJ (1; s; p; c) + (1
2 2
+(1
)ÆJ (0; s; p; c)
J (1; s; p; c) = Æ (1
p)J (0; s; p; c) + Æ pJ (1; s; p; c) +
ps
4
:
p)J (0; s; p; c)]
(2)
From these expressions for J (0; s; p; c) and J (1; s; p; c), we can solve for the respective value functions in closed form.
Lemma 1 Suppose all brokers charge c, all marker makers charge s, and each broker
routes to a particular market maker with probability p. Then, for that market maker,
Æ p)
;
4(1 Æ )
p s(1 + Æ (1
p))
J (1; s; p; c) =
:
4(1 Æ )
J (0; s; p; c) =
p s(2
Consider, analogously, the prots to a broker who charges c to clients, when all
other brokers charge the same amount. There are two possible situations that a
7
broker can face. She either does, or does not, have a limit order awaiting execution
from the previous period. Let 0 denote the state that the broker does not have an
outstanding limit order awaiting execution and let 1 denote the state where she does.
As with the market makers, these are idiosyncratic states that reect the past history
of order ow.
Suppose the state is 0, that is, there is no limit order on the books. Then, with
probability (s; c), a market order comes in, and broker b earns qc. Further, with
probability q (s; c), next period's state is 1, and with probability (1 q (s; c)), it is
0. Hence,
H (0; s; c; p; q ) = cq + Æ
fq
H (1; s; c; p; q ) + (1
q )H (0; s; c; p; q ) g (3)
Next, consider the 1 state. Broker b has an outstanding limit order with market
maker m. With probability q , a market order comes in to broker b. She earns c
on this order. With conditional probability 2p , this order trades against her previous
limit order (with probability 12 , it is on the right side of the market, and with overall
probability p, it goes to the same market maker), which also earns c. Finally, the
state next period is 1 with probability q and 0 with probability 1 q . Hence,
H (1; s; c; p; q ) = cq +
cp
2
+Æ
fq
H (1) + (1
q )H (0)g
(4)
Lemma 2 Suppose all brokers charge c, all marker makers charge s, and each broker
routes to a particular market maker with probability p. Then, for that broker
q (s; c)c[1 + pÆ 2(s;c) ]
H (0; c; s; q; p) =
1h Æ
(s; c)c q (1 + pÆ 2(s;c) ) + p(1
H (1; c; s; q; p) =
1 Æ
Æ)
i
;
It is immediate that H (0; c; s; q; p) H (1; c; s; q; p) (for c 0). A broker who has
an outstanding limit order can earn a commision fee on that limit order with positive
probability this period.
In our formulation of the problem, the probability that a particular market maker
gets a market order is independent of the state of the book. Intutively, this amounts
to the assumption that a broker (such as E*Trade), in the absence of payment for
8
order ow, does not keep track of unlled limit orders that have been posted at market
makers. One can interpret this as an assumption that monitoring the market makers
book to determine if the unlled limit orders belong to one of their clients is overly
burdensome.
Finally, observe that we constructed the value functions under the implicit assumption that limit order submitters do not undercut the spread. Could a limit
order submitter undercut the posted spread and take the whole market? Suppose
that he does so, then, in stationary equilibrium as there is a punishment phase whenever the quote spread deviates from the equilibrium one, the limit order trader will
be undercut before the next order arrives and thus his order would never execute.
So, for a limit order trader the choice is to set a spread of zero or obtain a strictly
positive payo by pricing at the current spread. The latter is optimal.
4 Oligopolistic Behavior with no Payment for Order Flow
We now turn to the case of M 2 and B 2, so that there is competition between
each type of agent. In this section, we assume that contracts for order ow cannot
be signed. In the next section, we turn to payment for order ow.
First, we show that it is a subgame-perfect equilibrium for each market maker to
set s = 0, and for each broker to set c = 0. This competitive outcome, clearly, denes
the best outcome for the consumer: the expected trading cost is zero for market
order submitters. Therefore, this is a natural benchmark against which to compare
the eects of payment for order ow.
Proposition 1 In the absence of payment for order ow, there is a subgame-perfect
equilibrium with the following features:
(i) each market maker, m = 1; : : : ; M , charges sm = 0,
(ii) each broker, b = 1; : : : ; B , charges cb = 0,
(iii) each broker routes orders to market maker m with probability
(iv) all investors who arrive choose broker b with probability B1 .
M,
1
In this rst best outcome, liquidity is provided free of charge. Hence, there is no
role for competing liquidity provision from investors and so all investors are indierent
9
between submitting market orders and limit orders. Thus, trading costs for all parties
s;c)
are zero, or T d (s; c) = T s (s; c) = 0. The ratio of limit to market orders, ((s;c
)
unambiguously increases in the spread. Thus, the ratio in this competitive case
provides a benchmark for other cases. Hence, the ratio of limit to market orders is a
proxy for the degree of competition in the marketplace. 7
Each market maker and broker earns zero prot in this subgame-perfect equilibrium. This is, therefore, the worst subgame-perfect equilibrium for them. Following
the intuition of Abreu (1988), we show that other subgame-perfect equilibria are sustained by the threat of punishing deviators by playing the zero-prot equilibrium for
ever, following any deviation. In characterizing these other subgame-perfect equilibria, we consider only innite punishment phases, following a deviation by any player.
That is, a deviation results in the play of the zero-prot equilibrium for ever. Such
punishment phases can be interpreted as periods of `erce competition.'
Given the punishment strategies, deviation by any market maker or broker leads
to positive prot in the deviating period, followed by zero prot for the rest of the
game. Any spread s and intermediation fee c that satisfy the no{deviation conditions
can be supported in subgame perfect equilibrium.
The maximium spread that can be supported, for a given intermediation fee c,
is the largest one so that market orders are still submitted. If no market orders
are submitted, the market makers cannot make any prots. For a given c, dene
s(c) = minfs j (s; c) = 0g. Since () is continuous, (s; c) > 0 for all s < s(c). We
know that in any equilibrium, s s(c). If s > s(c) no market orders are submitted
and a market maker can unilaterally deviate by decreasing his spread to increase
his prot. Analagously, for a given s, dene c(s) = minfc j (s; c) = 0g. Then,
(s; c) > 0 for all c < c(s). Thus, if brokers charge c they eectively drive all trade
out of the market. Hence, any fee that they charge must be less than c(s).
In equilibrium, if the discount factor is high enough (so that the no deviation
conditions are satised), then any spread between 0 and s(c), and any commission
between 0 and c(s), is sustainable.
7 Other,
more complicated, zero prot equilibria can be sustained. In particular, the brokers could
subsidize market orders, cm < 0 and make money from limit orders, cl > 0. In any such dierential
fee equilibria, the fee for market orders must be less than the fee for limit orders. This is because
brokers only receive a payment when market orders arrive, either directly through the fee or if the
market order executes against an outstanding limit order.
10
Proposition 2 For every s; c such that s 2 [0; s(c)) and c 2 [0; c(s)), there exists a
Æ suÆciently close to 1 such that there is a subgame-perfect equilibrium in which:
(i) each market maker charges s in each period
(ii) each broker charges c in each period
(iii) each broker routes to each market maker with probability
(iv) each investor chooses each broker with probability 1b .
m,
1
and
Hence, in equilibrium, both competition and collusion are sustainable. In particular, the spread charged by the market makers can even exceed the single-period
monopoly spread, and the spread chosen by the brokers can exceed the single{period
monopoly spread.
Proposition 3 For any economy in which s; c are sustainable as an equilibrium, then
there is an M ? and a B ? so that for
(i) M > M ? it is more diÆcult to sustain s as an equilibrium spread.
(ii) For M > M ? it is more diÆcult to sustain c as the equilibrium commission and
for > B ? , it is more diÆcult to sustain c as the equilibrium commission.
Thus, increasing the number of market makers has two eects. It makes collusive
outcomes in the market making market and the brokerage market more diÆcult to
sustain. Essentially, increasing the number of market makers reduces the payos to
brokers as it reduces the probability of execution of limit orders. Hence, prots from
limit order trading are reduced. Of course, in this stylized model, we do not consider
added frictions such as switching costs (among brokers). Such frictions would give
more local monopoly power to both brokers and MMs and would make collusion easier
to sustain.
Corollary 3.1 In the absence of payment for order ow, increases in the number of
Market Makers lead to weakly lower spreads and intermediation fees, hence weakly
lower transaction costs for consumers.
What eects do the collusive outcomes have on order submission? Recall, that the
payo to limit orders is increasing in the spread. That is, higher spreads mean that
limit order trading is more protable and thus any situation that is characterised
11
by high spreads will have ceteris paribus more limit orders. Second, all trade is
deterred when intermediation costs are high. Thus, equilibria with wider spreads are
characterised by more limit orders and fewer market orders. While, equilibria with
higher transaction costs are characterised by a lower volume of trade.
Corollary 3.2 In the absence of payment for order ow,
1. Increases in the number of market makers, decrease the number of limit orders
and increase the number of market orders. Thus, the ratio of limit orders to
market orders decreases.
2. Increases in the number of brokers, increase the volume of trade.
5 Payment for Order Flow
We now consider how payment for order ow contracts change the set of equilibria in
the market place. First, we consider an environment in which all brokers and all market makers have signed contracts. All contracts specify a payment, , that a market
maker pays to a broker on reciept of a market order. The brokers, who have signed
contracts with all market makers, route to each market maker with probability M1 .
Later, in section 6 we consider how such an outcome could arise and the equilibrium
level of order ow payments.
After the introduction of the 1996 Order Handling Rules, market makers are
required to post limit orders in their quotes and to give precedence to limit orders.8
Thus, if the market maker has a limit order on his book and if that limit order goes
o against a purchased market order, the market maker has incurred the cost of the
market order, but has not reaped any of the benet. Further, undercutting the limit
order, if it triggers a price war, can be very costly. To monitor the limit order book
of any particular market maker would be time consuming and thus the order ow
payment is not conditional on the idiosyncratic state of the book. Thus, a market
maker must recoup enough money from each market order execution to reduce the
cost incurred when he executes a limit order against a purchased market order. So,
8 Under
the 1996 Order Handling rules, Securities Exchange Act Release No. 37619A (Sept. 6,
1996), 61 FR 48290 ("Order Handling Rules Release"), market maker that holds a customer limit
order on one side of the market, priced better than the market maker's own quote, and a customer
market order on the other side of the market cannot execute both orders as principal, rather than
crossing the two orders, and thereby deprive the market order customer of the better price)
12
while it is immediate that if payment for order ow is positive, ( > 0), then spreads
must be non{zero, a stronger result obtains. the maintained spreads in any stationary
equilibrium must be larger than the order ow payment, that is 2s > .
Lemma 3 In the presence of payment for order ow, in any stationary equilibrium,
s
2
> .
Such large spreads do not imply that consumers are worse o under payment
for order ow. One argument that has been promulated is that, even with non{zero
spreads it is possible that brokers commissions decrease suÆciently so that the savings
are passed on to consumers. To examine this possibility we consider the lowest prot
strategies for brokers. Recall, that limit order submitters benet from large spreads
as this reduces their transactions costs, whereas market order submitters are hurt by
increases in the spread as this increases their transaction costs. To determine if all the
payment for order ow could be redistributed to consumers, we consider the lowest
prot best responses of brokers that gives the lowest cost to market orders and the
highest cost to limit orders. That is, we consider strategies that reduce transaction
costs to market orders and increase those to limit orders to see if the net eect of
payment for order ow could be transaction cost neutral.
Proposition 4 Suppose that there is payment for orderow, so that
> 0 then
brokers are playing a best response if
(i) Each broker receives > 0 for each market order.
(ii) Brokers charge cl = 0 for limit orders and cm = for each market order.
(iii) Brokers make zero prots.
The dierence in fees for the two types of orders reects the dierent valuation
for the two types of orders. When there is payment for order ow the total payment
received by brokers for a market order is cm + : setting this to zero implies a subsidy
for market orders.
An immediate implication of Proposition (4) and lemma (3) is that, in constrast to
an economy in which there is payment for orderow, there cannot be an equilibrium
in which there are zero transaction costs for market order submitters.
13
Proposition 5 In the presence of payment for order ow, there does not exist a zero
transaction cost equilibrium for market order submitters.
Further, the fact that wide spreads are automatically generated by payment for
order ow contracts and that the most that brokers will subsidize market order submitters is , payment for order ow represents a transfer from the retail investors
who demand liquidity to the retail investors who supply liquidity. That is, market
order submitters suer and limit order submitters gain. Thus, if both brokers and
market makers are in erce competition and earning their minimum payos, limit
order submitters are better o under payment for order ow, as their transaction
costs are simply T S = 0 2s < 0, where s > 0, while market order submitters are
worse o, as T D = + 2s > 0 and 2s > .
It is an open policy question wether such a redistribution is benecial or desirable.
We cannot address such questions in the context of the model. However, we do stress
that low brokerage fees or even subsidies do not imply that consumer costs are lower
than they would be in the absence of payment for order ow.
Corollary 5.1 In any market in which there is positive payment for order ow,
spreads are non{zero and much have a higher proportion of limit orders to market
orders than non{payment for order ow markets.
Payment for order ow contracts also have a strategic eect in the market place in
that, for the same set of exogenous parameters, such contracts aect the equilibrium
set of intermediation fees and spreads that can obtain. The possibility that such
contracts aect the set of equilibrium spreads is a notion missing from the popular
debate on payment for order ow.9 While cross{sectional eects can be tested in
data, equilibrium eects cannot be.
As in the previous section, we consider stationary equilibria that are sustained by
innite punishment strategies. For brokers, we use proposition (4) that guarantees
zero continuation prots and for market makers, they punish with the lowest spread
that is consitent with a subgame perfect punishment strategy. To formalize this
notion, let
s? (c) = minf (s; c) s>2
9 See
2M (s
Æs
2 )
g
for example, A Penny for your Trade? Barron's 01/01/2001 page 43.
14
(5)
Lemma 4 For all market makers in the market, with contracted payment for order
ow , with an NBBO requirement, playing s? (c) is the minimum sustainable spread
in the worst subgame perfect equilibrium.
In characterising the equilibria that obtain, we distinguish between contracts that
have a price matching or NBBO requirement and contracts that do not contain such
a requirement. In single period games, it is well known that such price matching
agreements are collusive. That is, in a one{shot Bertrand game such an agreement
allows equilibria other than the zero prot one to be sustained. In repeated games,
however as players have a wide range of punishment strategies it is unclear to what
extent this allows collusive outcomes to be sustained. With an NBBO requirement,
the incentive of market makers to reduce spreads to increase the business of the
brokers with whom they are linked is removed. Decreasing spread cannot increase
the volume of trade, it will only decrease their payos conditional on their order
ow. So, the more order ow in the market has been assigned to particular market
makers, the less incentive each market maker has to undercut price. In the extreme,
in which all order ow has been dedicated, there is no incentive for a market market
to reduce price as he will not attract any order ow as a result and will only reduce
his prots on his next trade. Thus, if all order ow has been contracted, the no{
deviation constraint of the market makers will not bind and they will always be able
to sustain the monopoly spread. Observe that for brokers, the NBBO requirement
does not have an eect on their ability to sustain collusive outcomes.
For any pair (s; c), we can determine what eect payment for order ow has on
the ability of market participants to enforce collusive equilibria. The range of spreads
that are sustainable for brokers are cm 2 [ ; c] and cl 2 [0; c], where c is dened
such that (s; c) = 0.
For market makers, the range of permissible spreads comes from the condition
that J (0) 0. This can also be expressed as the condition that
m
2
Æ
4 s
We dene
s(c) =
s(c) =
fmin s j satises equation (6)g
fmax s j satises equation (6)g
15
(6)
Thus, s(c) is the mimimun spread that gives market makers zero prots and s(c)
is the maximum individually rational spread that they would charge.
Observe, that the minimum spread sustainable is strictly positive and greater than
in the no payment for order ow case and the largest sustainable spread is weakly
smaller than the largest sustainable in the no payment for order ow case. For brokers
as there is payment for order ow, the lower bound on intermediation fees is strictly
lower if there is payment for order ow (i.e., the lowest intermediation fee can be
negative), while the upper bound is the same. Finally, notice that in the proposition,
we consider single intermediation fees for both market and limit orders.
Proposition 6 For every s(c) 2 [s(c); s(c)] and c so that cm 2 [
; c] and cl
2 [0; c],
there is a Æ suÆciently close to 1 such that there is a subgame perfect equilibrium in
which:
(i) Each market maker charges s in each period.
(ii) Each broker charges c in each period.
(iii) Each broker routes to each market maker with probability
(iv) Each investor chooses each broker with probability
1
M
and
B.
1
For a particular pair, (c; s) that are sustainable under both regimes, does payment
for order ow make it more diÆcult or easier to sustain collusive outcomes?
Consider intermediation fees and spreads that are attainable under both regimes.
That is, consider pairs (c; s) that are individually rational under both payment for
order ow and no payment for orderow. Then,
Proposition 7 Take any spread and intermediation fee (s; c) that are sustainable in
an economy with no payment for orderow then there is a B ? and a M ? , so that
(i)For M > M ? it is easier to sustain s under payment for order ow.
(ii) There is a B ? so that for B > B ? it is more diÆcult to sustain c.
If there is an NBBO or price matching agreement then the set of parameter val16
ues that support high market maker spreads is higher, while there is more likely to
be erce competition in the broker market. Payment for order ow, thus tends to
reduce intermediation fees paid by consumers but increases spreads by eliminating
any incentive to deviate from a collusive equilibrium.
6 Equilibrium Payment for Order Flow Contracts
Is the consumer worse o under payment for order ow? The answer to this question depends on the total transaction costs incurred, which in turn depends on the
equilibrium contracted payment for order ow, . Recall, that in a payment for order
ow contract, Mb market makers sign with broker b and recieve M1b of the order ow.
The contracting state is at time 0, before the game begins. Each broker is permitted to oer a contract, which may be accepted by one and only one market maker.
These contracts, therefore, are exclusive arrangements between a broker and market
maker. Under the terms of the contract, the broker agrees to devote all her orders to
only the market maker who has accepted her contract. In return, the market maker
agrees to pay the broker 0 for each market order he receives. The contracts,
therefore, are parameterized by . The contracts are signed before an investor arrives
at time 0, and are non-negotiable for the remainder of the game.10
First, we show that, in any situation in which the spread and intermediation fee
is positive, signing a payment for order ow contract is a dominant strategy for a
market maker and broker.
Lemma 5 Suppose all market makers choose spread
s > 0 in each period, and all
brokers choose an intermediation fee c > 0 in each period. Suppose market maker m
~
and broker ~b do not have a contract. Then, regardless of any other contracts signed
by other market makers or brokers, there is a ^ such that market maker m
~ and broker
~b both earn higher prots by signing a payment for order ow contract at ^.
Once introduced, such contracts must become pervasive. Consider a broker who
does not receive payment for order ow. If there is competition for intermediation
services, he is unable to compete against those who receive money from market makers. He would, therefore, be driven out of the market. A market maker who does
10 In
reality, such contracts are of course renegotiable. However, note that as each period in our
model corresponds to transaction time it is reasonable to view such agreements as long term.
17
not have such contracts would receive no order ow and therefore has an incentive to
enter into one.
Brokers oer contracts to market makers that market makers will accept and that
maximize prots. This of course, depends on what intermediation fees they believe
they will be able to sustain in equilibrium. Thus, for a specic discount factor, Æ ,
suppose that brokers believe that a pair of spreads and intermediation fees, s(Æ ) and
c(Æ ) are sustainable. Brokers will choose the payment for order ow that maximizes
their prot subject to a participation constraint of the market makers.
Proposition 8 In an equilibrium with endogenous payment for order ow with stationary punishment strategies then payment for order ow is given by
(c; s) =
s
2
"
#
1
(s; c)
Æ
>0
2M
Thus, we can provide comparative static properties on the level of payment for
order ow.
Corollary 8.1 In equilibrium, payment for order ow
(i) Will be larger if the number of Market Makers is larger, or is increasing in M .
(ii) The larger the probability of a limit order arriving,the smaller the payment for
order ow.
(iii) The higher the level of intermediation fees, the larger the payment for order ow.
Thus, payment for order ow contracts have two distinct eects. First, by guaranteeing order ow to market makers, they eliminate incentives for market makers to
compete on spread. Second, they redistribute rents from brokers to market makers,
depending on their respective bargaining power in the intermediation market.
7 Conclusion
We have developed a simple model to examine the eects of payment for order ow on
competition in the retail broker market and the market making market. Payment for
order ow changes the submitted order mix, and can hurt consumers. The equilibrium
18
is characterized by lower intermediation fees, but higher market making spreads.
Lower commission fees should therefore not be taken as evidence that payment for
order ow is innocuous. Further, by explicitly incorporating limit and market orders,
the model provides testable implications on the order ow mix.
In the non-payment for order ow equilibria, while wide or monopoly spreads can
be supported in equilibrium, they might not be. Payment for order ow contracts
essentially commit the MM's to the widest possible spreads. Consumer surplus, of
course depends on both the intermediation fee charged by brokers and the spread.
However, if the price formation mechanism provides a social externality | if it is used
to price other contracts or for allocation of resources, then the fact that liquidity
spreads are wide mean that it is less precise or `informative.' This, therefore is a
potential cause for concern.
Since spreads are wider when payment for order ow is positive, such contracts
increase the proportion of limit to market orders. This observation ows directly from
the decision process of agents. Payos to limit orders are increasing in the spread,
therefore any equilibrium with a higher spread will have a higher percentage of limit
orders in the order ow. Therefore, payment for order ow agreements should see an
increase in liquidity provision by retail investors.
Empirically, the model suggests that lower intermediation fees do not imply that
the consumer is better o. Neither, however, do wider spreads imply that consumers
are worse o. Rather, an appropriate test is the ratio of limit to market orders as
this is a proxy for the degree of competition in the marketplace for liquidity.
19
8 Appendix
Proof of Lemma 1
Subtract equation (2) from equation (1). This yields J (0; s; p; c) J (1; s; p; c) =
, or J (0; s; p; c) = J (1; s; p; c) + 4p s . Substituting for J (0; s; p; c) in the RHS of
4
equation (1), we have
ps
J (1; s; p; c) = Æ (1
p)J (1; s; p; c) + Æ (1
p)
p s(1 + Æ (1
= ÆJ (1; s; p; c) +
p s(1 + Æ (1
J (1; s; p; c) =
4(1 Æ )
p))
ps
p))
4
4
+ Æ pJ (1; s; p; c) +
ps
4
:
Now, J (0; s; p; c) = J (1; s; p; c) + 4p s , which yields
J (0; s; p; c) =
p s(2
4(1
Æ p)
:
Æ)
Proof of Lemma 2
It is immediate that
H (0; c; s; q; p) = H (1; c; s; q; p)
cp
2
(7)
;
so that H (0; c; s; q; p) < H (1; c; s; q; p). Further, from equation (4), we have
Æ (1
q )H (0; c; s; q; p) = (1
Æq )H (1; c; s; q; p)
p
c(q + )
2
Putting these last two equations together, we have
Æ (1
q ) H (1; c; s; q; p)
cp
H (1; c; s; q; p) f1
2
= (1
c
Æq )H (1; c; s; q; p)
p
Æ g = c q + (1
2
Æ + Æq ))
Further, from (7), we have
H (0; c; s; q; p)f1
cp
f1
Æ g = H (1)f1 Æ g
(
)2
Æ pq
= c q +
2
20
p
(q + )
2
2
Æg
Therefore,
H (0; c; s; q; p) =
H (1; c; s; q; p) =
1
1
(
c
c
Æ
q+
Æ
q+
Æ
p
2
pq
)
2
(1
Æ (1
q ))
Proof of Proposition 1
(i) Consider time t of the game. Suppose all market makers except market maker m
choose s = 0 at each time > t, after ever possible history. Then, if market maker
chooses any sm > 0, he does not receive any orders from brokers, so his prot is zero.
Hence, choosing sm = 0, which also yields zero prots, is a best response for market
maker m.
(ii) Similarly, if all brokers except b choose c = 0 after every history, then broker
b receives zero orders, and hence zero payo, by choosing cb > 0. Hence, choosing
cb = 0 at time > t, regardless of history, is a best response.
(iii) Since all market makers quote the same spread, any routing of orders is a best
response by broker b. In particular, sending orders to each market maker with probability M1 is a best response.
(iv) Since s = 0, and c = 0, = 1. All investors submit a market order. Since T d = 0
regardless of which broker is chosen, it is a best response to choose each broker with
probability B1 .
Proof of Proposition 2
The proof proceeds by a series of claims. First, we characterise the optimal deviation of a broker and a market maker if all other participants are playing the stationary
strategy (s; c). Then, we establish the condition under which there is no gain from
one shot deviation.
For a given value of c, and a stationary spread of s, if market makers
deviate they will do so in their 1 state to s~(c) = minfs; s^(c)g, where s^ is the
spread which maximizes (s;c2 ) s .
)s
. If a marketGiven a value of c, dene s^(c) as the spread which maximizes (s;c
2
maker were a monopolist, this spread maximizes its single-period prot. In equilibrium, if the spread which the market makers are sustaining, s > s^, then the optimal
21
deviation by market maker i is to si = s^. However, if s < s^, the market maker
cannot charge more than s, else it receives no orders. Instead, it would like to charge
si = s , for some close to zero. This ensures that it captures all orders. Taking
the limit as ! 0, its maximal prot is at si = s. Dene s~(c) = minfs; s^(c)g. This
then, is the spread to which a market maker would deviate.
When he has an empty book, he will not deviate from the equilibrium spread s
if J (0; s; m1 ; c) (~s; c)~s. Similarly, when he has an unlled limit order in his book,
he will not deviate if J (1; s; m1 ; c) (~s; c)~s. Since J (1; s; p; c) < J (0; s; p; c) for all
p 2 (0; 1], the latter is a tighter condition.
For a given value of s, and a stationary spread of c if a broker deviates he
will do so in the 0 state to c~ = maxfc; c^g, where c^ maximizes the monopolist
one shot prot.
Next, consider the behavior of the brokers. In equilibrium, any deviation by broker
j is punished by brokers B nfj g playing c = 0 forever. Hence, following a deviation
at time t, broker j earns zero prot from any orders submitted in periods (t + 1)
onwards.
Suppose broker j deviates and plays c0 < c at time t.11 Then, at time t, she
earns c0 with probability (s; c0 ). Further, she obtains a limit order with probability
(s; c0 ). At time t + 1, she is punished and all other brokers play c = 0. Market
makers benet from any decreases in the intermediation costs and thus do not seek
to sustain high intermediation costs. We assume that all market makers continue to
play s. For the broker who undercuts the existing prices, the probability of a market
order at time t + 1 that can trade against
her unlled limit order
is (s;2 0) . Hence, the
broker's prot from the deviation is c0 (s; c0 ) + Æ (s;c2m) (s;0) .
Given a spread
s, dene c^(s) as the commission that maximizes her prots from
) (s;0)
deviating or, c (s; c) + Æ (s;c2m
. For broker j , conditional on deviating, the
maximal prot attainable is at c~ = maxfc; c^g.12
The payo to cooperating by a broker in a stationary equilibrium is just:
0
q (s; c)
1 Æ
q (s; c)
H (1; s; c; p; q; ) = H (1; s; c; p; q ) +
1 Æ
H (0; s; c; p; q; ) = H (0; s; c; p; q ) +
11 We
can ignore deviations to c > c, because the broker will not get any orders at this c .
the optimal deviation to c < c is undened, so we take the limit of c as ! 0.
12 Again,
0
0
0
0
22
The dierence in equilibrium payos between the two states is simply
If a broker deviates in the zero state, his prot will be
(c + ) (s; c) +
(s;c)pc
2
.
cÆp (s; 0) (s; c)
2
If a broker deviates in the one state, then his prot will be
c
(c + ) (s; c) + (s; c)p +
2
cÆp (s; 0) (s; c)
2
Thus, conditional on deviating the continuation payos are the same and the dierence in prots if he deviates in the two states is simply, 2c (s; c)p. Thus, while his
prots from deviating are higher in the 1 state, so are his continuation payos. Hence,
brokers may deviate in either state. We assume that they will deviate in the 0 state.
If Æ is suÆciently high then there is no gain from one shot deviation for
Market Makers and Brokers
Suppose there is a symmetric stationary equilibrium, in which all market makers
charge s, all brokers charge c, p = m1 for all m, and q = 1b for all b.
Suppose market maker i deviates. As argued, he will choose si = s~ = maxfs; s^g,
for a prot in the deviating period (~s2;c)~s .
In this equilibrium,
(s; c) s 1 + Æ (1
J (1; s; p; c) =
4(1
(s;c)
Æ) m
m
)
Hence, the no-deviation condition for market maker i is
(s; c) s 1 + Æ (1
4(1
(s;c)
m
Æ) m
)
(~
s; c)~
s
2
(8)
If s 2 (0; s(c), the LHS is positive. Since the RHS converges to 0 as Æ ! 1, it
is clear that, for Æ suÆciently close to 1, the LHS is greater than the RHS, and the
no-deviation condition is satised. If s = 0, then, by Proposition 1, there exists a
subgame-perfect equilibrium for all Æ .
Next, consider broker j . She has no incentive to deviate if
H (0; c; s; q; p)
c~( (s; c~) + Æ (s; c~)
23
(s; 0)
2
)
In the stationary equilibrium prescribed, q = 1b , and p =
M.
1
Hence,
c (s; c) (1 + Æ 2(Ms;c) )
H (0; c; s; q; p) =
(1 Æ ) B
Therefore, the no-deviation condition for broker j is
c (s; c) (1 + Æ 2(ms;c) )
(1 Æ ) b
Æ (s; c)
c (s; c) (1 +
)
2m
c~ ( (s; c~) + Æ (s; c~)
(1
(s; 0)
2
)
Æ ) b c~ ( (s; c~) + Æ (s; c~)
(s; 0)
)
(9)
2
For any c 2 (0; c(s)), the LHS is strictly positive. As Æ ! 1, the RHS converges to
zero. Hence, for any c in this region, there exists a Æ suÆciently close to 1 such that
the no-deviation condition is satised. Again, Proposition 1 covers the c = 0 case.
Combining the no-deviation conditions for the market makers and brokers, there
exists a Æ suÆciently close to 1 such that parts (i) and (ii) of the Proposition hold.
Parts (iii) and (iv) are immediate, given that all market makers charge the same s
and all brokers the same c.
Proof of Proposition 3 The Æ above which there is no deviation is implictly dened
by equation (16). Dene
(s; s~; c) =
(s; c)s
:
(~
s; c)~
s
Since s~(c) = minfs; s^(c)g and s^ is the monopoly outcome, it is clear that (s; s~; c) 1,
with strict equality if s < s^(c). Thus, equation (16) can be reexpressed as:
Æ (c) 2M (s; s~; c)
2M + (s; s~; c) + (s;s~;cM)
(s;c)
Thus, this is increasing in M .
The no{deviation condition for brokers is given by equation (??). Dene
Æ (s; c)
f (Æ; s; c) = c (s; c) 1 +
2M
g (Æ; s; c~) = (1
!
Æ )B ce (s; ce) + Æ (s; ce)
Thus, the condition can be expressed as
f (Æ; s; c) g (Æ; s; c~)
24
(s; 0)
2
!
Clearly, f (Æ; s; c) is linear and increasing in Æ , while g (Æ; s; c~) is concave in Æ .
Further, observe that by denition, c (s; c) bce( (s; ce)).
Further, f (Æ; s; c) is decreasing in m and g (Æ; s; c~) is increasing in b, thus the
intersection is increasing in M and increasing in B .
Proof of Lemma 3
In stationary equilibrium,
J (0) =
M
s
2
+Æ
(
)
M
J (1) + (1
M
) J (0) :
(10)
Further, with stationary spreads, J (0) J (1) = 4s
M 0. From equation (10), if
s = 2 so that the per period payo to market orders is zero, then from equation (10)
J (0) is a linear combination of J (0) and something strictly less than J (0). Hence,
J (0) < J (0), a contradiction, thus 2s > , as 2s < implies that prots to market
makers are strictly negative.
Proof of Proposition 4
If brokers charge cl = 0 for limit orders and cm = for market orders then they
make zero prot on each transaction.
If a broker deviates by increasing his fees, he will not get any order ow. If he
deviates by decreasing his fees, then he will make negative prots.
All brokers and all market makers are charging the same fees and thus investors
face the same prices irrespective of where they send their orders and so randomizing
is a best response.
Proof of Proposition 5
We rst establish that there is no subgame perfect equilibrium in which brokers
make zero prots and subsidize market orders by 2s . Suppose that there exists such
an equilibrium, then market order submitters pay zero transaction costs and limit
order submitters must be paying cl > 0, else brokers make negative prots. Consider
a broker who deviates from this equilibrium in state 0 by charging cem = 0 and
cel = cl > 0.
25
The payos to such a deviation are:
f(0) = 0 + Æ
H
h
i
H (0) + pcel + Æ [1
H (0)] > 0
As we are positive a zero payo equilibrium, H (0) = 0. As ( 2s ; 2s ) > 0, this
deviation will always yield a strictly positive payo. Thus, there cannot exist such a
subgame perfect equilibrium.
Thus, in our candidate zero prot equilibrium, liquidity demanders pay T D (s; c) =
s
+ cm . Since, 2s > , T D > 0.
2
Proof of Corollary (5.1) Immediate
Proof of lemma 4
Suppose all market makers charge s (c; ) in the minimum payo stationary equilibrium. The strongest stationary punishment that can be imposed on a market maker
who deviates is that other market makers play s (c; ) for ever. That is, deviations
are followed by reversion to s (c; ) forever. Deviation can be protable as a lower
spread implies an increased order ow | however as this also increases the arrival
rate of limit orders. For notational compactness we denote the expected discounted
payos when the book is empty by J (0) and the corresponding value function when
the book is full by J (1).
(i) First, we show that no market maker has an incentive to deviate. Along the
equilibrium path, under our symmetry assumptions market maker m obtains a payo
of
J (0) =
M
s
4
(
+Æ
)
M
J (1) + (1
M
) J (0)
(11)
As in the no payment for order ow state, if a market maker deviates, he will
deviate in state 1, the low payo state. Suppose market maker m deviates in state
1, to s~. If he deviates, and posts a lower spread, then the broker with whom he has
signed a contract,( as there is no NBBO requirement), is posting a lower spread and
will get all the order ow if he routes to the deviating broker. Given a deviation by
market maker m, it is rational for the broker to route all incoming orders to him. As
26
s (c; ) is less than or equal to the single period monopoly payo, else it would not
be the minimum sustainable spread, the optimal deviation is to a spread less than
s (c; ). The maximal payo obtained by this deviation occurs when s~ = s (c; ),
and market maker m obtains all orders in that period.
Therefore, the payo after deviation is
s
~
J (1) = 2
+ Æ f J (1) + (1
The no-deviation condition, therefore, is J~(1)
(1
1 s
)
2M 2
(1
1
M
1
) + Æ f (1
M
) J (0)g
J (1) 0, or
) J (1)
Æ (1
1
M
) J (0)g
0
As argued earlier, with stationary spreads, J (0) J (1) = 4s
M . Substituting this
into the above expression, the no-deviation condition becomes
(s; c)
(2M 1)2M 4M
Æ (2(M 1)
Æs
"
#
2M 2M 1
2
Æ 2(M 1)
s
s (c; ) is the minimum spread that satises this inequality. As the probability
of observing a limit order is bounded below by a strictly positive number, such a
spread exists for s 2 . Further, since payos are increasing in s for s less than the
monopoly level, s (c; ) denes the minimum payo stationary equilibrium. That is,
in the class of equilibria in which deviations are followed by reversion to the same
strategy, this oers lowest payo to market makers.
Proof of Proposition 6
The proposition proceeds with a series of claims. First, we establish the range
of permissible spreads and intermediation fees. Second, we determine the optimal
deviations for both market makers and brokers. Finally we present conditions to rule
out such deviations.
The range of Permissible spreads and intermediation fees
27
The lowest sustainable spread is the one that is ex ante rational, that is that has
J (0) 0, as the market makers start out with empty books. If J (0) = 0, then
J (0)
=
0
=
s
(
M 2
s
(
M 2
s
=) (
2
) + Æ
)
)
4M
sÆ
Æ
M
Æ
M
J (1) + 1
s (c)
M 4M
!
J (0) = 0
=
For brokers, the lower bound guarantees them zero prots and the upper bound
is the maximum so that there is no trade.
Given a spread s, a payment for order ow, , and a sustained broker
fee, c, a broker will optimally deviate to c~( ) = min[^c( ); c], where c^( ) =
(cm ; cl ) that maximize monopoly payos.
Given the payment for order ow agreements, , the payo to cooperating by a
broker in a stationary equilibrium is just:
q (s; c)
1 Æ
q (s; c)
H (1; s; c; p; q; ) = H (1; s; c; p; q ) +
1 Æ
H (0; s; c; p; q; ) = H (0; s; c; p; q ) +
The dierence in equilibrium payos between the two states is simply
If a broker deviates in the zero state, his prot will be
(c + ) (s; c) +
(s;c)pc
2
.
cÆp (s; ) (s; c)
2
If a broker deviates in the one state, then his prot will be
c
(c + ) (s; c) + (s; c)p +
2
cÆp (s; ) (s; c)
2
Thus, conditional on deviating the continuation payos are the same and the difference in prots if he deviates in the two states is simply, 2c (s; c)p. Thus, while
his prots from deviating are higher in the 1 state, so are his continuation payos.
Hence, brokers may deviate in either state.13 We assume that they will deviate in
the 0 state. Let c^( ) = (cm ; cl ) that maximise deviation payos. Hence, a deviating
13 Put
something in the text to say that the optimal deviation is to a single
28
c
rather than a
(cm ; cl ).
broker will deviate to c~( ) = min[^c( ); c], where c~( ) is the pointwise minimum of the
two intermediation fees. Then, a broker who deviates will do so to c~.
For a xed c and maintained spread s, a market maker will deviate to
s~( ) = min[^
s( ); s] if there is no NBBO requirement. If there is an NBBO
requirement, if he deviates, he will deviate sm , the spread a monopolist
market maker would charge.
The payo to cooperating by a market maker in a stationary equilibrium is just:
J (0; s; p; c; ) = J (0; s; p; c)
J (1; s; p; c; ) = J (1; s; p; c)
(s; c)p
1 Æ
(s; c)p
1 Æ
If a market maker deviates, he will do so in the state in which there is an order
on his limit order book (state 1). If he deviates under a no NBBO regime his payo
will be:
(s; c)(
s
2
) + Æ (s; c)J (1; s?(c); p; c; ) + Æ (1
p (s; c))J (0; s?(c); c; )
Let s^( ) be the spread that maximizes this. Then, he will deviate to s~( ) =
min[^s( ); s].
If a market maker deviates in an NBBO regime, his payo is strictly negative
unless s > sm .
Conditions under which market makers and brokers will not deviate
1. A broker will not deviate if the payo to cooperating is greater than the payo
to optimal one shot deviation or if,
H (0; s; c; p; q ) +
(~c + ) (s; c~) +
q (s; cm )
1 Æ
cÆp (s; ) (s; c~)
2
Recall,
q (s; c)c[1 + pÆ
H (0; c; s; q; p) =
1 Æ
29
(s;c)
2
]
(12)
(13)
Thus, the condition can be written as:
q (s; c)c[1 +
pÆ (s; c)
2
] + q (s; cm )
(14)
cÆp (s; ) (s; c~)
(15)
2
Clearly, the RHS ! 0 as Æ ! 1, while if c is in the feasible set, the LHS is
positive, or if it is zero, then there is a subgame perfect equilibrium for all Æ .
(1
Æ )(~
c + ) (s; c~) +
2. Under a no NBBO requirement, or an NBBO requirement when the maintained
spread is greater than the spread a one{shot monopolist would charge, a Market
maker will not deviate if
(s; c)p
J (1; s; p; c)
1 Æ
(~s; c)(~s ) + Æ (~s; c)J (1; s?(c); p; c; ) + Æ(1 (s; c))J (0; s?(c); p; c; )
Recall,
J (1; s; p; c) =
p s(1 + Æ (1
4(1 Æ )
p))
Thus, the condition can be rewritten as:
p s(1 + Æ (1
(1
4
Æ ) ( (~
s; c)(~
s
Clearly, the RHS
strictly positive.
p))
(s; c)p
) + Æ (~
s; c)J (1; s? (c); p; c; ) + Æ (1
(s; c))J (0; s? (c); p; c; ))
! 0 as Æ ! 1, while if s is in the feasible set, the LHS is
3. Under an NBBO requirement a market maker will not deviate if the maintained
spread is less than the monopoly spread. In this case, the payo to deviating is
zero,as he does not increase his order ow.
For limit and market order submitters, as all brokers and market makers are
charging the same fees, randomizing across brokers with probability B1 is a best
response.
Proof of Proposition 7
30
If there is no payment for order ow, then the condition for no deviation for a
market maker is
(s; c)
(s; c) s 1 + Æ (1
m
!
)
2 (1
Æ ) (~
s; c) s~ M;
(16)
where s~ is the optimal deviation. If there is payment for order ow and no NBBO
requirement or with an NBBO requirement if the spread is larger than the monopoly
spread, then denoting the optimal deviation by s~( ), the condition is simply
p s(1 + Æ (1
(1
p))
4
Æ ) ( (~
s( ); c)(~
s
(s; c)p
))
+ Æ (~s( ); c)J (1; s? (c); p; c; ) + (1
Æ ) (Æ (1
(s; c))J (0; s?(c); p; c; ))
Observe that s^ > s^( ), if is concave in s. If s s^( ), then under both regimes,
they deviate to the same spread. If s > s^( ), then under payment for order ow,
the optimal deviation is to s^( ), whereas if there is no payment for order ow, the
optimal deviation is to s. If s > s^, the optimal deviation under no payment for order
ow is to s^ while the optimal deviation under payment for order ow is to s^( ).
Let
F (Æ ) = (1
s~
Æ ) (~
s; c) M
G(Æ ) = (1
Æ )[ (~
s( ); c)(~
s
(17)
2
) + Æ (~
s( ); c)J (1; s?(c); p; c; )
(s; c))J (0; s?(c); p; c; )] + (s; c)p
+ Æ (1
(18)
(19)
Clearly, F (Æ ) is decreasing in Æ , given that the continuation payos under the punishment strategies are negative, G(Æ ) is also decreasing in Æ . Thus, we need to show
that F (Æ ) G(Æ ) 0.
F (Æ )
G(Æ )
=
0
(1
(20)
Æ )M
[ (~s; c)~s (~s( ); c)~s( )] + [(1 Æ ) (~s( ); c)M (s;(21)
c)]
2
MÆ ( (~
s( ); c)J (1; s?(c); p; c; ) + (1
(s; c))J (0; s?(c); p; c; )) (22)
Now, as s is concave in s, then [ (~s; c)~s
J (0) 0 =)
(~
s( ); c)~
s( )] 0. Further, given
M
4 2
Æ
s
31
Recall, that s? (c) is dened so that
s? (c) = minf (s; c) 2M (s
Æs
s>2
2 )
g
(23)
Given that J (0) > J (1), the continuation payos are negative. Further, (~s( )) (s; c). Hence, if M is suÆciently large, the expression is positive. This establishes
part (i).
If there is no payment for order ow, the condition for no deviation for broker is
c (s; c) (1 +
Æ (s; c)
)
2m
(1
Æ ) B c~ ( (s; c~) + Æ (s; c~)
(s; 0)
2
)
while if there is payment for order ow, the condition is simply
(s; c)c[1 +
(1
Æ (s; c)
]+
2M
Æ )B (~
c( ) + ) (s; c~( )) +
cÆ (s; ) (s; c~( ))
2M
(s; c)
We restrict attention to deviations to a single spread, c. Observe that as is
concave and not too convex in c, c^( ) > c^ . If the maintained spread c < c^, then
under payment for order ow and no payment for order ow, the optimal deviation
is to the same intermediation fee. If c^( ) > c > c^, then under no payment for order
ow, the optimal deviation is to c^, while under payment for order ow, the optimal
deviation is to c. If c^( ) > c, the optimal deviation under no payment for order ow
is to c^, whereas under payment for order ow the optimal deviation is to c^( ).
Dene
(s; 0)
H (Æ ) = (1 Æ )B c~ ( (s; c~) + Æ (s; c~)
)
2
cÆ (s; ) (s; c~( ))
I (Æ ) = (1 Æ )(~
c( ) + ) (s; c~( )) + (1 Æ )B
(s; cm )
2M
We need to show that for B suÆciently large,
H (Æ )
I (Æ )
0
The condition is equivalent to:
"
(1
[(1
Æ )B c~ ( (s; c~) + c~Æ (s; c~)
Æ )B (~
c( ); s)
(c; s)]
(s; 0)
2
)
32
c~( ) (s; c~( ))
c~( )Æ (s; ) (s; c~( ))
2M
#
As c~ c, it is immediate that for suÆciently large B , the RHS is positive. For
c~ = c~( ), as c < 0, and < 0, we know that the LHS is negative. For c~ < c~( ) the
LHS is also negative as c~ maximizes the concave function, thus for higher c it must
be less. The result follows.
Proof of Lemma 5
Suppose ^b brokers and m
^ market makers have not
signed contracts, where ^b 1
P^b 1 1 and m
^ 1. For unsigned market makers, p = j =1 b m .
If market maker m
~ signs a contract with broker ~b at a price
of , then the probP^b 1 1 1 1
ability that it gets an order improves to p~ = b + j =1 b
m . Each period, the
expected payment made by market maker m
~ is b , so that the present value of contractual payments is b(1Æ) .
;s;p;c)
Now, for s; c > 0, J (0; s; p; c) is strictly increasing in p, since @J (0@p
= 4(1 sÆ) 2(1
Æ p) > 0. Hence, there exists a ^ > 0 such that J (0; s; p~; c) b(1^Æ) > J (0; s; p; c).
Next, consider broker ~b. If broker b signs a contract, its payo increases by b(1^Æ) >
0.
Proof of Proposition 8 Contracts are oered by brokers and the market makers
earn reservation prots. Recall,
J (0) =
!
=
s
(
M" 2
s
2
1
)
Æ
Proof of Corollary 8.1 Immediate
33
2M
s
Æ
=0
# M 4M
References
[1] Reprt to the NASD Board Of Governors (1991), \Inducements for Order Flow."
[2] Abreu, D. (1988), \Towards a theory of discounted Repeated Games," Econometrica 56: 383-396.
[3] Battalio, Robert and Craig W. Holden (1997), \Why Doesn't Decimal Trading Eliminate Payment for Order Flow and Internalization?" Working Paper,
Indiana University.
[4] Battalio, Robert and R. Jennings (1999), \Dealer Revenue, Payment for Order
ow and Trading Costs for Market Orders at Knight Securities," Working Paper,
Georgia State University.
[5] Battalio, Robert, J. Greene and R. Jennings (1997), \Do Competing Specialist
and Preferencing Dealers aect Market Quality?" Review of Financial Studies 10(4): 969{993.
[6] Battalio, Robert (1997), \Third Market Broker Dealers: Cost Competitors or
Cream Skimmers?" Journal of Finance 52(1): 341{352.
[7] Battalio, Robert, J. Greene, and R. Jennings (1998), \Order Flow Distribution, Bid-Ask Spreads and Liquidity Costs: Merrill Lynch's Decision to Cease
Routinely Routing Orders to Regional Stock Exchanges," Journal of Financial
Intermediation 7: 338{358.
[8] Bruno Biais, P. Hillion, C. Spatt An Empirical Analysis of the Limit Order Book
and the Order Flow in the Paris Bourse," Journal of Finance 50: 1655-1689
[9] Chordia, Tarun and Avanidhar Subramanyam (1995), \Market Making, the Tick
Size, and Payment-for-Order Flow: Theory and Evidence," Journal of Business ,
68(4): 544{575.
[10] Dennert, Jurgen (1993), \Price Competition between Market Makers," The Review of Economic Studies, 60(3) 735-751.
[11] Dutta, Prajit, and Ananth Madhavan (1997), \Competition and Collusion in
Dealer Markets," Journal of Finance 52(1): 245{276.
34
[12] Easley, David, Nicholas Keifer and Maureen O'Hara (1996), \Cream Skimming
or Prot Sharing? The Curious Role of Purchased Order Flow," Journal of
Finance 51(3): 811{833.
[13] Hollield, Burton, Miller A, Sand
as, P and J. Slive (2000), Liquidity Supply and
Demand: Empirical Evidence from the Vancouver Stock Exchange, Working
Paper.
[14] Kandel, Eugene, and Leslie Marx (1999), \Paments for Order Flow on Nasdaq,"
Journal of Finance 49(1): 35{66.
[15] Securities and Exchange Commission, \Report on the Practice of Preferencing."
35