Discussion Paper No. 1050

STRATEGIC SPEED CHOICE
BY HIGH-FREQUENCY TRADERSUNDER SPEED BUMPS

Jun Aoyagi

February 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Strategic Speed Choice by High-Frequency Traders
under Speed Bumps⇤
Jun Aoyagi†
February 17, 2019

Abstract
We study how high-frequency traders (HFTs) strategically decide their speed levelin a market with a random speed bump. If HFTs recognize the market impact of their
speed decision, they perceive a wider bid-ask spread as an endogenous upward-sloping
cost of being faster. We find that the speed elasticity of the bid-ask spread (slope of theendogenous cost function) negatively depends on the expected length of a speed bumpsince a longer delay makes market makers insensitive to HFTs’ speed increment. Hence,
speed bumps promote the investment of HFTs in high-speed technology by reducing
the marginal cost of getting faster, undermining their intended purpose of protecting
market makers. Depending on the expected length of a bump, an arms race among HFTs
exhibits both complementarity and substitution. These findings explain the ambiguous
empirical results regarding speed bumps and adverse selection for market makers.
Keywords: High-frequency trading, market structure, speed bumps, adverse selection,
strategic speed decision.
JEL Classification: D40, D47, G10, G18, G20⇤ First draft: June 2018.

† Department of Economics, University of California at Berkeley. E-mail jun.aoyagi@berkeley.edu.

I appreciate the constructive comments from Michael Brolley, Nicolae Garleanu, Terry Hendershott, Ryo Horii,
Ulrike Malmendier, Hayden Melton, Emi Nakamura, Christine Parlour, Yoko Shibuya, David Sraer, Yingge
Yan, Haoxiang Zhu, and seminar participants at Osaka University (ISER), UC Berkeley and University ofTokyo. I am grateful to ClaireValgardson for copy-editing. This paper was awarded the 2018 Moriguchi Prizeby the Institute of Social and Economic Research, Osaka University.

11

Introduction“Never before in human history have people gone to so much trouble andspent so much money to gain so little speed.”—Flash Boys: A Wall Street
Revolt by Lewis (2014).

The ever-increasing speed of electronic financial markets pushes traders to be lightning
fast. They are obsessed with being the first to acquire information for trading purposes,
spending significant amounts of money on high-speed technologies. These include custombuilt fiber-optic cables, microwave/millimeter-wave transmissions, and even lasers thatcan send signals from one place to another. With these sophisticated tools, high-frequency
traders (HFTs) can extract information from massive layers of signals at the speed of light.
Regulators are concerned about how quickly HFTs can access and act on information.
It is argued that the informational advantage that HFTs obtain through increasing speed
exposes market makers to the cost of adverse selection in the sense of Glosten and Milgrom(1985). That is, HFTs engage with market makers only if they receive news that is not yet
publicly available and find market makers’ orders outdated and mispriced.1 By exploiting
their speed advantage, HFTs “snipe” stale quotes provided by market makers (Budish et al.,2015).
The speed race by HFTs has prompted some exchange platforms to slow down HFTinvolved transactions by introducing speed bumps. A speed bump imposes a delay on thearrival or execution of orders at a market, aiming to protect traders from exposure to theabove-mentioned risks. For example, the Investors Exchange (IEX) adopts a 350-microsecond
speed bump on incoming orders and outgoing information from the exchange. The Aequitas NEO Exchange and TMX Group, both Canadian exchanges, also apply a few milliseconds of random delay to non-cancellation orders.2 Specifically, the speed bump in thelatter markets aims to slow down only HFT-involved orders by classifying traders into highfrequency (latency-sensitive) and non-high-frequency categories.3
1 One of the most frequently cited market benefits is liquidity provision by high-frequency market makers.

However, extremum events, such as the May 2010 “flash crash,” make regulators increasingly concerned thatthe liquidity provided by HFTs is likely to evaporate when it is most needed. See, Conrad et al. (2015) for theempirical study of this liquidity evaporation.
2 See Appendix F of Baldauf and Mollner (2017) which provides a comprehensive summary of institutional
details.
3 Depending on the institutional details, the types of traders (or orders) to be protected may change. For

2

This paper analyzes the effect of speed bumps on speed decision of HFTs and on theadverse selection cost for market makers. We focus on a non-cancellation delay aimed athampering sniping behavior of HFTs. As mentioned in Subsection 1.1, the empirical results
are ambiguous: in some studies, less (or more) frequent executions by platforms do notaffect a spread, while others find that they mitigate the problem. In this study, these results
can be reconciled by considering a strategic speed decision by HFTs with an endogenously
derived cost.
In our model, a speed bump can increase the speed of HFTs. Specifically, once we allow
HFTs to strategically choose their speed level, a speed bump increases the marginal benefit
of being faster. The strategic choice of speed means that each HFT decides her speed level,
anticipating the reaction of price setters (i.e., market makers). That is, just as the monopolistic trader of Kyle (1985) knows the price impact of her trading behavior, each HFT in ourmodel knows the price impact of her speed decision.4
These conditions arise because major high-speed financial institutions have significant
shares in the trading volume in markets.5 For example, Table I shows the top five highfrequency financial institutions and their shares in the BrokerTec platform, through whichmore than half of the U.S. Treasury is traded. Typically, HFTs benefit from a huge number ofsmall, short-lived transactions, and each trading decision does not impact the equilibrium
price. However, when an institution decides a speed technology, she becomes aware of
the market impact of her choice because a sizable number of transactions involve the same
speed technology and affect the equilibrium price.
Once HFTs understand how their speed influences the price, they perceive the reactionof the bid-ask spread set by market makers as an endogenous cost of being faster. As the literature points out, the faster the HFTs, the more severe adverse selection the market makers
instance, the speed bump in the IEX is more likely to protect pegged orders from non-HFTs, but not market
makers on a lit LOB, from being sniped by HFTs. The non-cancellation delay and the HFT-specific delay
adopted by the two Canadian exchanges are more likely to save market makers from adverse selection cost.
4 The term “speed” includes the choice of the geographical location of the firm’s information server. For
example, a spot in the mid-Atlantic ocean is the optimal point to exploit the price difference between the
NYSExchange and the London Stock Exchange.
5 There is anecdotal evidence for HFTs being aware of the market impact of their speed choice. For example,
clients who purchase a speed device from a trade technology company often try to hide it by asking to peel
corporate logos from shipments due to confidentiality clauses. See, for example, https://www.wsj.com/ andLewis (2014).

3

Table I: Top 5 Firms by Volume on BrokerTec
Firm
Jump Trading
Citadel LLC
Teza Technologies
KCG
JP Morgan

Volume($ millions)
2,291,000
1,004,000
905,000
798,000
649,000

Market Share
28%
12%
11%
10%
8%

Note: It tabulates shares in May-June, 2015. Data regarding top-10
HFTs is also available and indicates a similar result. Source: Risk.com,
October 2015, Issue 10.

face. Thus, a higher speed puts positive pressure on the bid-ask spread and, in turn, reduces
the sniping profit of HFTs. Therefore, the spread works as an endogenous upward-sloping
cost of being faster. To our knowledge, the existing models of the speed decision of HFTs,such as those by Foucault et al. (2003, 2016), and Delaney (2018), deal only with the exogenous sunk cost of speed. By contrast, we propose endogenous cost as a means throughwhich a speed bump affects the speed decision of HFTs. This is because market makers’
adverse selection risk, as well as the equilibrium spread, is affected by a speed bump, while
the traditional exogenous sunk cost is independent of it. Specifically, the sensitivity of the
spread to HFTs’ speed, which is an endogenous marginal cost, determines the optimal speed
level, and it varies with the length of a speed bump.
To separate this intuition, we consider a simple benchmark structure: homogeneous
markets with a single HFT, having a random speed bump of a d-period with l = E[d]. Ifl increases, market makers know that they are less likely to be picked off by the HFT. As
a result, they do not care much about a marginal increase in the HFT’s speed, and theirpricing behavior, i.e., the bid-ask spread, becomes less responsive. This induces a lower
endogenous marginal cost of speed investment for the HFT, providing her with a stronger
incentive to be faster.
As an extension, we consider a speed competition among multiple HFTs—an “armsrace”—and allow them to serve not only as snipers but also as high-frequency market makers. In the literature, such as Foucault et al. (2003), traders’ speed levels interact with eachother because one trader’s speed affects other traders’ probability of being picked off. Importantly, unlike the traditional exogenous (sunk) cost, our endogenous cost of speed provides a new channel for the interaction because a bid-ask spread is an equilibrium variable.
4

We find that a speed bump has the same positive impact on the equilibrium speed as in thebenchmark model, and, depending on the relative significance of the exogenous and endogenous costs, the arms race can exhibit both strategic complementarity and substitution.
If the exogenous sunk cost is relatively insignificant compared to the endogenous cost,an arms race creates strategic complementarity. In this case, a speed-up by a competitor(trader B) makes an HFT (trader A) respond tit for tat to keep herself faster. This is because
the speed-up by the opponent as a market maker reduces the sensitivity of a spread to
trader A’s speed-up. Also, a faster opponent decreases the sniping probability of traderA, who cares less about an adverse price movement caused by her speed increase. As aresult, a faster opponent reduces trader A’s marginal cost of being faster and enhances her
investment in speed.
When an arms race exhibits strategic complementarity, the introduction of a speed bumpor a longer delay can backfire. By the same arguments as in the benchmark, a higher lmakes each HFT willing to be faster, triggering a fiercer speed competition and positive
externality due to complementarity. Although a speed bump protects market makers andmitigates adverse selection via its direct effect, the equilibrium speed increases substantiallyand dominates the direct protection, worsening adverse selection risk.
On the other hand, if the exogenous cost is more significant, playing tit for tat is costlier,leading to a negative reaction of trader A to a speed-up by her opponent. Intuitively, trader
A needs to pay a sunk cost anyway, while it rewards her only if she can snipe trader B’s limit
order. If trader B becomes faster, a decline in the sniping probability of trader A makes the
sunk cost more salient, leaving her reluctant to increase her speed. This reasoning follows
the existing literature, in which an arms race involves strategic substitution. In this case,a speed bump or a longer delay can achieve its primary purpose: HFTs slow down andadverse selection is mitigated.
We can compare our strategic model with the endogenous cost of speed to a traditionalmodel with an exogenous sunk cost. These models can be seen as two extreme cases. On
the one hand, traditional models show that an arms race involves strategic substitution andthat a speed bump reduces the equilibrium speed, mitigating adverse selection both by theprotection of market makers (the direct effect) and by slowing HFTs down. These effects are
summarized by the solid arrows in Figure I. On the other hand, a speed bump in our strategic model triggers an intensive arms race, dramatically increases the equilibrium speed, and5

Figure I: Effects of a speed bump

Note:The right box summarizes the effect of speed bumps on market makers (MMs)’ behavior, and the left
box represents those on the HFTs’ speed choice.

imposes more severe adverse selection (the red dashed arrow in Figure I). By adjusting thecost parameter for an exogenous sunk cost and by changing the expected length of delay,
our model navigates between these extremes. Thus, our results propose a way to reconcile the existing empirical findings that report positive, negative, and insignificant effects ofspeed bumps.

1.1

Literature Review

This paper contributes to the literature on high-frequency trading and market structure (seeJones, 2013; O’Hara, 2015; Menkveld, 2016 for reviews). Biais et al. (2015) analyze the effectof an arms race and show that a higher speed triggers more severe adverse selection forslow traders. Delaney (2018) describes the speed decision of HFTs as a model of irreversible
investment with an optimal stopping time, while Bongaerts and Van Achter (2016) viewit from a perspective of high-frequency market making.6 However, the speed decision inthese models is discrete (i.e., being fast or not), and they abstract away from addressing theimplications of the equilibrium level of speed. Based on Foucault et al. (2003), Liu (2009)
6 Aı̈t-Sahalia and Saglam (2013), Hoffmann (2014), Foucault et al. (2016) construct models with HFTs toaddress the effect of high-frequency market making. See Conrad et al. (2015) for the empirical study of highfrequency quoting.

6and Foucault et al. (2016) investigate a continuous choice of speed based on the monitoring
intensity of traders.7 However, traders decide on the speed level simultaneously with othertypes of players (e.g., market makers), which requires them to focus on the exogenous costof speed investment. Our model differs from theirs since the speed decision is continuousand bears an endogenous cost due to the strategic motive of HFTs. Our results are uniquesince these two modifications empower us to analyze how speed choice is affected by speed
bumps.
As traders get faster, questions arise regarding the speed and frequency of executions
by a trading platform. By altering the trading frequency of the Kyle-type model, Du andZhu (2017) show that a low-frequency platform works better to reallocate assets, though
it limits the ability to react to new information promptly. Pagnotta and Philippon (2018)
also consider platforms’ decisions regarding execution frequency and fees to attract speedsensitive traders. Menkveld and Zoican (2017) also explore the effect of latency on HFTs’
strategy and spread, citing risk aversion as a key to generating the result.8 In their analyses,
which pays little attention to the speed choice of HFTs, the frequency of transactions isdetermined at a market level and applies to all investors.
Our model shares the same interests as the studies on the impact of slow market structures, such as frequent batch auctions (Budish et al., 2015; Haas and Zoican, 2016) and speed
bumps (Baldauf and Mollner, 2017; Brolley and Cimon, 2017; Aldrich and Friedman, 2018),
on HFTs’ behavior and adverse selection for market makers. However, they do not consider a continuous optimal speed decision by HFTs with a delay-sensitive endogenous cost.
Thus, they conclude that these mechanisms mitigate adverse selection for market makers,
an assertion that will be overturned in our model.9
The scope of the literature extends to other empirical findings regarding the HFT and the
7 Foucault et al. (2013) consider the optimal choice of the monitoring intensity by high-frequency snipers

and market makers. It involves the exogenous cost but is not strategic. Both snipers and market makers
obtain a positive profit from trading due to heterogeneous private values of an asset, generating strategic
complementarity in an arms race.
8 Menkveld and Zoican (2017) obtain a hump-shaped equilibrium spread against a delay. This stems fromthe switch from the pure-strategy to mixed-strategy equilibrium, and it depends on the risk aversion parameter.
9 Moreover, these models do not study the coexistence of slow and fast markets, which is analyzed inAppendix A.2. In independent work, Brolley and Cimon (2017) explore this coexistence and find a result
consistent with ours, although it stems from a completely different mechanism.

7

effect of bumps.10 Hu (2018) analyzes the SEC approval of the IEX as a national securities
exchange, making traders route their orders under the “Order Protection Rule,” and finds
a net improvement in market quality measured by the spreads. Shkilko and Sokolov (2016)
exploit interruptions of messaging via microwave communication caused by precipitation(i.e., rain or snow) to find a reduction in quoted spreads.11 Chen et al. (2017) investigate
the effect of a bump in the TMX Alpha, reporting an increase in quoted spreads. In ourmodel, we can reconcile these results because, depending on the relative significance of theexogenous cost and the level of expected delay, speed bumps will affect a spread negatively,
positively, or not at all.

2

The Benchmark Model

This section proposes a simple benchmark model to separate the main mechanism. Consider a one-shot exchange of an asset, in which a short-lived HFT tries to snipe stale limit
orders. This single asset has a stochastic liquidation value v = ±s with equal probability. v

is publicly announced at a stochastic time T, which occurs as a Poisson arrival with intensity
g. With the public announcement, the asset is liquidated. It is traded during t < T due toliquidity needs or the arrival of private information, as in Glosten and Milgrom (1985) andBudish et al. (2015). Following the convention of market microstructure, we assume thateach trader can hold only a unit position.
10 Hendershott and Moulton (2011) analyze the impact of the hybrid market at the NYSE and show thatthe faster market structure increases quoted and effective spreads and adverse selection cost. Riordan andStorkenmaier (2012) focus on the system upgrade in the Deutsche Boerse, Frino et al. (2014), Boehmer et al.
(2015), and Brogaard et al. (2015) study the colocation as an example of latency reduction, and Hasbrouck
and Saar (2013) construct a measure of low-latency in the NASDAQ to find subsequent shrinkages in spreads.
On the other hand, Ye et al. (2013) analyze the importance of the tick-size constraint and report that latency
declines at the NASDAQ did not significantly alter spreads (except for the smallest stocks).
11 Although the interruption by precipitation may have a similar effect to a speed bump, they mentionedthat this phenomenon is not paid much attention by financial institutions, while traders anticipate a speed
bump and take it into their decision making.

8

2.1

Traders

There is a continuum of competitive slow, uninformed market makers with a unit mass.
At the beginning of the trading game (t = 0), all market makers submit a single-unit limit
order with a half spread s to commit to trade at this price. The order will disappear from thelimit order book if there is a taker or if the market maker cancels it based on public news.
To focus on the short-horizon behavior, we assume that market makers do not return to themarket once they exit. Cancellation is immediate and incurs no cost.
There is one (N = 1) risk-neutral high-frequency trader (HFT). Before t = 0, the HFT
invests in a technology that provides the speed f.12 Equipped with the speed device withf, she can observe private news regarding v and react to it with a Poisson probability withintensity f. We denote TH as the arrival time of this Poisson news. Upon the arrival of thenews, the HFT immediately submits market orders to “snipe” stale limit orders provided
by the continuum of market makers.13,14
In addition, there is a continuum of liquidity traders who are exposed to a liquidity
shock. The shock exogenously makes them submit buy or sell market orders with equal
probability. We can think of them as noise traders, and trading against them conveys no
information to market makers. Let TL be the timing of the Poisson shock which arrives withintensity b(g ).

Finally, as in Haas and Zoican (2016) and Brolley and Cimon (2017), assume that trading information, including traders’ identity, becomes public immediately after an order is
executed, i.e., the market is perfectly transparent.
12 In the benchmark model with N = 1, imposing a sunk cost on the speed investment does not change ourresult. In the extension with N

2, we need a positive and convex cost to hamper the strategic complemen-tarity and to derive an equilibrium.
13 If we give an index i 2 [0, 1] to each market maker, the HFT submits marketable limit orders to obtainsniping profit s

si from each i. Since all the market makers quote a competitive homogeneous spread, theR1
HFT’s aggregate gain is 0 (s s)di = s s.
14 We can show that the HFT does not intentionally delay the timing of the order submission: if she gets
information at t, she immediately sends the order at t. Putting a time lag between obtaining the informationand submitting the order can reduce a spread and increase sniping profit. However, without a commitment
device, this cannot be an equilibrium since it is always optimal for the HFT at the information arrival time TH
to snipe immediately given the lag she announces at t = 0, i.e., there is a time inconsistency.

9

2.2

Market Structure

A continuous market imposes a random speed bump on incoming orders from the HFT.
Specifically, an order submitted to the market at date t arrives at t + d, where d is a random delay. Orders from liquidity traders and cancellation requests from market makers are
executed promptly.15 Thus, during t 2 (t, t + d), outstanding limit orders can be illusory
for the HFT if liquidity traders trade against them or if market makers cancel due to publicnews.
For notational simplicity, we assume that d follows an exponential distribution with aparameter b, and the expected length of a delay is denoted by l ⌘ E[d] = b 1 .16
Alternative Market Structures
As an extension, we analyze a situation with multiple HFTs, N

2, in Section 3, in which

each HFT serves not only as a sniper but also as a high-frequency market maker. Thissetting sheds light on a strategic property of an “arms race.” Appendix A.1 considers a casewhere market makers can continuously update (cancel and resubmit) their limit orders. The
coexistence of slow and fast markets is analyzed in Appendix A.2. This shows that a bumptriggers a shift of adverse selection from slow markets with a bump to fast markets with nobumps, consistent with the empirical result by Chen et al. (2017).

2.3

Equilibrium

We conceptualize our model as a sequential game with two stages, as depicted in Figure II.
In the first stage, the HFT decides the level of f.17 Given this, each market maker submits
a competitive limit order, anticipating a confrontation with the informed HFT and liquidity
traders. In the trading stage, the HFT looks for an opportunity to snipe.
15 For simplicity, we assume that there are no other sources for a latency, while the primitive parameters,such as b, can be seen as the potential latency that characterizes the speed of each type of trader.
16 The randomness of d does not significantly affect our result, while it makes the solution simpler. The casewith a deterministic d is available on request.
17 Of course, the setting of the speed decision used in our model does not comprehensively reflect real-world
conditions. As Dugast et al. (2014) suggest, some components of speed choice may occur simultaneously withmarket makers’ behavior. However, we believe that the ex-ante strategic speed decision is still significant
because HFTs would not invest in speed ex-ante if they did not exploit it in an ex-post trading game.

10

Figure II: Timeline

The equilibrium concept is a subgame perfect equilibrium, and the HFT chooses the
optimal level of speed f in light of the optimal reaction of market makers. That is, the HFT
knows the price impact of her speed choice, as the monopolist in Kyle (1985) knows the priceimpact of her trading behavior. In contrast to Kyle (1985), however, the trading stage int 2 (0, T ) is competitive, and the HFT behaves as if her trading strategy does not have a

price impact. This is because she splits her orders and sends them to an infinitely large
number of market makers given the outstanding limit orders. This follows the literatureand captures the real-world behavior of HTFs, who send and cancel a massive number ofsmall orders within a very short time frame.
First, we analyze the behavior of market makers given f. Then, we formulate the expected profit function of the HFT in the trading stage given market makers’ behavior andthe speed. Finally, we determine the optimal speed for the HFT.
2.3.1

Optimal Behavior of Market Makers

In a perfect competition, a limit order sent by a market maker yields zero expected profit,
as in Glosten and Milgrom (1985). Without a loss of generality, let us consider how an ask
price s is determined when v = s.18
Given that a market order arrives at date t, it is possible that the taker is information
or liquidity driven. As a result, the spread is set so that s = E[v|buy order at t], where theexpectation is over d and the timing of the trade. The key effect of a bump is to reduce theprobability of being picked off by the HFT or, put differently, to increase the probability that
18 Results for the opposite case can be given by a symmetric argument.

11market makers observe public news to cancel their limit orders.
Suppose that a trade takes place at date t. If t < d, there is no fear of facing an informationdriven HFT because of the speed bump. Put differently, the fastest possible arrival of the
HFT occurs at d. During this “safe interval,” liquidity traders arrive before the public news
with a density be ( b+g)t . Otherwise, market makers can cancel their orders with densityge ( b+g)t at period t.
If a trade occurs at t

d, on the other hand, it bears an adverse selection cost: the HFT

buys an asset only if the limit order is mispriced given the true information. The HFT getsto trade if she becomes informed at t
events during (td, and there are no liquidity shocks or public newsd, t). In this case, a market maker obtains ss. Market makers can also

trade with liquidity traders if there is a liquidity shock at t, and the HFT becomes informed
after td. In this case, the trading profit is s

E[v] = s. Since d is stochastic, the expected

return for a market maker is
V = Ed

Z d
0

sbe

( b+g)t

dt +

Z •
d

( bs + f(ss))e y(t d) e ( b+g)d dt ,(1)

where the expectation relates to d, and y ⌘ f + b + g. The first integral in (1) shows the
trading profit in t < d, while the second describes the case with td.

This formulation is the result of the following probabilities: given d,Pr(HFT arrives at t) = fe f(t d) e ( b+g)t ,Pr(Liq. traders arrive at t) = be f(t d) e ( b+g)t ,Pr(cancellation at t) = ge f(t d) e ( b+g)t ,which lead to the second term in (1). It is then possible to get the equilibrium spread fromthe break-even condition:
Proposition 1. The equilibrium (half) spread is given bys=

fEd [e ( b+g)d ](f + b) Ed [e ( b+g)d ] + bby+ g (1

Ed [e

f
1+ly
= f
s.
( b+g)d ])
+
b
1+ly

(2)

A few remarks on s are in order. First, a direct effect of the speed bump appears in theform of the discount on the arrival rate of the HFT, which is given by (1 + ly) 1 . This term
12

mitigates adverse selection risk by generating a safe interval.
If f is fixed, a lower l induces a higher spread since the expected delay becomes shorter.
Also, an infinitely small expected delay (l ! 0) makes s converge to the traditional equilibrium spread of Glosten and Milgrom (1985). Therefore, as Budish et al. (2015) and Baldauf

and Mollner (2017) point out, the direct effect of a bump mitigates the adverse selection costfor market makers.
This argument is built on the assumption that f is fixed, i.e., the HFT’s speed decision
is not influenced by the bump. When the speed choice by the HFT is considered, the speed
bump affects s via the fluctuation of the optimal speed as well. Namely, the optimal f
depends on the expected length of the delay, l. The existing models argue that the incentive
to be faster diminishes as the bump gets longer, i.e., a higher l reduces s not only by the
direct effect but also by making f lower. Our model proposes the opposite effect of l on s
through f.
The following properties of s are useful to understand the mechanism. First, note thatthe price impact of the speed is positive:
∂s
> 0.
∂f
We call this the “sensitivity” of a spread (price) to a speed-up by the HFT. It turns out that
this represents the slope of the endogenous cost of being faster. At the same time, we havethe following:
Lemma 1. The sensitivity of the price to the speed is decreasing in l, i.e.,
∂
∂l

✓

∂s
∂f

◆

< 0.

Therefore, the longer the expected delay, the less sensitive the spread becomes. A market
with a higher l is protected by a longer (expected) safe interval, and market makers behave
as if the share (arrival rate) of the HFT is small. Hence, market makers care less about the
speed investment by the HFT, making their pricing behavior less sensitive to f.

13

2.3.2

Profit of the HFT

When the HFT becomes informed and submits market orders at t, they will be executed at
t + d if (i) there is no liquidity shock during (t, t + d) and (ii) no public news arrives in thesame interval. This happens withpt (f, d) ⌘ Pr( TH = t, min{ T, TL } > t + d) = fe yt e ( b+g)d .
Thus, if the random delay is d, the profit from sniping at t is pt (f, d)(s

(3)

s ).

The first coefficient in (3) represents the probability that the HFT obtains the information
at t. The sniping probability involves an additional exponential coefficient, e ( b+g)d , whichshows the disadvantage of the HFT that stems from a speed bump, i.e., front-running byliquidity traders or cancellation by market makers due to the d-delay. Therefore, a longer
delay directly reduces the expected profit of the HFT in the second stage.
The objective function of the HFT in the first stage takes a simple form:
W (f) ⌘ EdZ •0s)dt =

pt (f, d)(sf1(sy 1 + ( b + g)l

s ).

(4)

Note that the HFT always submits unit orders since she exits the market once her orders are
executed, i.e., she is a short-term investor.19
2.3.3

Optimal Speed

We move on to the speed choice by the HFT in the first stage. To obtain an interior solution,
we make the expected length of the delay relatively short:l< p1b( b + g).

(5)

The intuition behind this condition will be provided after offering our main propositions.
19 See Appendix A.1 for a more general setting with continuous updating by market makers and time-dependent st .

14

Under (5), the optimization problem of the HFT is
max W (f) ⌘ EdZ •pt (f, d)(s0f1+lys.t. s = fs.
1+ly + bfs)dt ,(6)

This indicates that the HFT decides f knowing the price impact of her speed decision, i.e.,
she is strategic. In this case, being faster pushes up the price charged by market makers
and saps her sniping profit. For this reason, we can think of the equilibrium spread as an
endogenous cost of speed. Importantly, Lemma 1 suggests that the slope of this endogenous
cost is affected by l, and, in turn, affects the marginal cost of being faster for the HFT.
To analyze how l alters the optimal decision, consider a marginal gain of being faster:
dW
= Ed
df

= (s

Z • ⇢

dpt (f, d)d+ pt (f, d) (s
df
df
0
Z •
dpt (f, d)s(f)) Ed
dt (1 #(f)),
df
0

(s

where
#⌘

s(f))

s(f)) dt .

(7)

d log(s s(f))⇥R •⇤ > 0.
d log Ed 0 pt (f, d)dt

# is the sensitivity of the sniping profit (s s) to a change in the expected sniping probability⇥R •⇤of the HFT (Ed 0 pt (f, d)dt ). We call this the elasticity, and Appendix B.1 provides an

explicit formula. Note that obtaining a higher f affects W through two competing channels
and exhibits a price-liquidity tradeoff: it increases the sniping probability (the first term in[7]), while reducing the sniping profit via the adverse price movement (the second term in[7]).
When the equilibrium spread s is more sensitive to f than the sniping probability p (f),being faster harms the profit of the HFT, and incentive to increase f dwindles. On the other
hand, if the HFT knows that the price impact of her speed choice is small, it is more likelythat an improvement in sniping probability ( dp
df ) dominates a decline in profit due to thewider spread (d(s s)df ), luring her to be faster.

In other words, for the strategic HFT, themarginal cost captured by the sensitivity of the spread matters considerably.
The following results guarantees the concavity of the problem (see Appendix B.1 forproofs):15

Lemma 2. The elasticity is increasing in the speed: d#(f)/df > 0.
When the HFT is fast, market makers estimate that the economy is inhabited by a relatively large measure of the HFT in terms of the arrival rate. Therefore, a marginal increasein f reduces market makers’ expected profit, and they charge a wide spread to compensate for the expected loss. That is, as the HFT becomes faster, market makers grow moreconcerned about facing the HFT, and their pricing behavior is more sensitive to changes in
speed. Thus, as f increases, it is more likely that the steeper endogenous marginal cost of
being faster will outweigh the higher marginal benefit from a higher p, making the objectivefunction (6) concave.20
As a result, the optimal speed f⇤ is derived by solving for the FOC.
Proposition 2. (i) The optimal speed is given by⇤f =(ii) f⇤ is increasing in l.

pb + g(1 + l( b + g))p.
1 l b( b + g)

Proof. See Appendix B.1.

(8)⇤

In contrast to the traditional models, Proposition 2 demonstrates that, if the speed choice
is strategic, a speed bump increases the equilibrium speed of the HFT. This modification of
the speed decision is natural given that several high-frequency financial institutions control
significant shares, as discussed in Section 1 and shown in Table I. When the HFT knows how
her speed investment affects the pricing behavior of market makers, an equilibrium spread
generates an endogenous cost of being faster. This not only guarantees a bounded solution
even without an exogenous cost of speed, but also overturns the traditional result.
The key mechanism is the negative impact of a speed bump on the sensitivity of the pricein Lemma 1. Since a bump intends to slow down the HFT and protect market makers, thespread becomes insensitive to a change in the speed. That is, an intentional delay endogenously reduces the marginal cost of being faster. Hence, a speed bump does not prevent a
speed race but promotes it, as Proposition 2 attests.
20 The condition in (5) is required to make Lemma 2 hold. If l is sufficiently large, market makers becomed#too insensitive to make df> 0. Thus, the HFT can be infinitely fast, and we need an exogenous cost to makethe problem well defined.

16

This surprising finding highlights the main difference of our results from the literature,such as Budish et al. (2015), Haas and Zoican (2016), and Baldauf and Mollner (2017). As in
their models, if we assume that the HFT does not care about the effect of her speed choice on
the spread, the second effect in (7) disappears. In this case, some exogenous costs of speed
are required to make f⇤ bounded, and the effect of l on f⇤ becomes reversed. We compare
our model to the traditional ones in more detail in Subsection 3.4.

2.4

Effect on Adverse Selection

We are interested in how a speed bump affects adverse selection cost for market makers. It
is straightforward that there are two competing effects. First, as the literature suggests, aspeed bump reduces adverse selection cost because it dampens the probability for market
makers of confronting the HFT. However, our strategic model adds an opposing channel:
a speed bump promotes speed investment by the HFT since it endogenously reduces themarginal cost of being faster. In the following, we take the equilibrium half spread s as ameasure of adverse selection and investigate its equilibrium behavior.
Proposition 3. The equilibrium spread is independent of the expected delay, i.e., ds/dl = 0.
Proof. See Appendix B.2.

⇤

This result shows that a speed bump cannot mitigate (or worsen) adverse selection formarket makers. Together with (ii) in Proposition 2, this highlights how our model differsfrom the existing ones.
With f fixed, a speed bump reduces the profit of the HFT since the first-d periods become safe intervals for market makers and the HFT cannot snipe. To compensate for thisdisadvantage, the strategic HFT gets faster. Anticipating the price impact of her speed investment, she chooses the level of f that eliminates the cost from the speed bump, thereby
muting its effect. As a result, the two competing consequences of a speed bump cancel eachother out.
Put differently, the speed-up due to a bump or a longer delay is an indirect effect of adschange in the price sensitivity, df. Since the reduction of a spread by l is a direct effect,

the speed-up cannot predominate. In the following sections, however, we show that thisfinding regarding adverse selection significantly changes if we consider different market
structures.
17

3Multiple HFTs and High-Frequency Market Making

In the real world, HFTs serve not only as takers (snipers) but also as liquidity providers.
We modify the benchmark model to capture this fact. Assume that there are two HFTs(i = 1, 2), both of whom provide limit orders at t = 0 at a competitive price. At a random
date, Ti ⇠ exp(fi ), HFT i obtains private news about v. When the news arrives, it is optimalfor HFT i to immediately send market orders to snipe the stale limit orders of her opponent(HFT j) and to simultaneously cancel her limit orders.
The behavior of liquidity traders is the same as in the benchmark, but we ignore public
news at T ⇠ exp(g), since it only adds complexity. The other structures of the game staythe same as in the benchmark. Note that the results with N = 2 can be easily extended to
3 with an additional parameter N that measures the (inverse of) market power. For

N

technical reasons, assume that b

3.1

1 and focus on the symmetric equilibrium.

Optimal Behavior of HFTs

Consider HFT j as a high-frequency market maker (HFMM). Her behavior is the same asthat of ordinary market makers in the benchmark model except that she can cancel her limit
orders at Tj ⇠ exp(fj ). Thus, the break-even condition provides the following equilibrium
spread:

sj =

fi
s.
fi + b(1 + l(fi + fj + b))

This spread has the same structure as s in (2): it reflects an expected value of v conditionalon the trade. Note that the symmetric equilibrium makes the spreads set by both HFTs thesame; s = s1 = s2 .
We turn to the optimal speed decision of HFT i. Since the competition drives her total
profit from market making to zero, her gains come only from the sniping profit. Thus, theoptimization problem is analogous to (4):1fi
( s s j ),1 + l( b + fj ) fi + fj + bfis.t., s j =.
fi + b(1 + l(fi + fj + b))

max Wi (f) =fi

Since this is exactly the same as the benchmark case if we substitute fj for g, the best re-18

sponse function of HFT i is a modified (8):BRi (fj ) =as long as 1 > lqp

b + fj [1 + l( b + fj )]q,
1 l b( b + fj )(9)b( b + fj ). Otherwise, fi = • is the best response. In this section, we

focus on bounded responses, while Subsection 3.2 analyzes all possible symmetric equilibria.21 The following property of the best response function helps explain the mechanism:
Proposition 4. The best response function exhibits strategic complementarity, i.e.,dBRi (fj )
> 0.
dfj

The intuition behind this proposition should be clear if we analyze the marginal gain for
HFT i:
wi ⌘

∂Wi
= (s
∂fi

sj )

∂(s s j )
∂pi
+ pi
,
∂fi
∂fi

(10)

where
pi ⌘

fi
[1 + l( b + fj )]yrepresents her sniping probability. The first term is the marginal improvement in the sniping
probability, and the second stands for a decline in the profit. These terms can be seen as themarginal benefit and cost of being faster. The structure of the marginal gain of increasing fi
is the same as in the benchmark case, while it depends on the speed of the competitor.
Furthermore, we need a cross derivative to obtain the reaction of BRi .
"

∂wi
= (s
∂fj

}|
# z"
#{
2 (s
∂
(
s
s
)
∂
(
s
s
)
∂
s
)
∂p
∂pi
j
j
j
i
sj )
+ i
+
+ pi
.
∂fj ∂fi
∂fi
∂fj
∂fj
∂fi
∂fj ∂fi∂2 p(11)

When the opponent (HFT j) increases her speed, it affects both the marginal benefit andcost of being faster for HFT i. The first component of (11) is a change in the marginal benefit
that stems from a marginal improvement in pi . A faster opponent (i) increases or decreases
the marginal improvement in the sniping probability and (ii) raises the sniping profit, making it more worthwhile to have a higher pi . These are the first and second terms in the firstbrackets in (11). At the same time, a faster opponent reduces the (endogenous) marginal
cost of being faster for HFT i. Intuitively, (iii) since a faster opponent makes HFT i less
21 Technically, we can avoid the unbounded equilibrium if we introduce a positive exogenous sunk cost forthe speed.

19likely to snipe, she does not need to care much about the adverse price movement. Moreover, (iv) a faster opponent becomes more insensitive to HFT i’s speed-up due to the same
logic that states that a higher l makes s less sensitive to f in the benchmark case. As a result,the second term is positive, as is the total effect of fj on wi , i.e., a faster opponent renders
speeding up more profitable for HFT i.
Moreover, “tit for tat” due to complementarity can be strong enough when the opponent
is sufficiently fast.
Lemma 3. There is a unique fj = f0 such thatd2 BRi (fj )df2j> 0 , fj > f0 .

Proof. See Appendix B.3.

(12)⇤

If fj is sufficiently high, the negative effect of fi on the expected profit becomes minimal.
This is because a very fast opponent makes it extremely difficult for HFT i to snipe. Thus,
she barely cares about the negative impact of fi on the price. In addition, a fast opponent
as a market maker tends to be highly insensitive to a change in fi because she estimatesthat being sniped by HFT i is not likely to happen. Both of these effects strongly prompt an
incentive of HFT i to be faster, making the best response function convex.

3.2

Equilibrium Speed

To see if (9) has a symmetric solution, we first observe that BRi (0) > 0, i.e., when facing azero-speed opponent, HFT i still maintains a positive speed. This is because fi > 0 yields a
positive profit, while fi = 0 keeps it at zero. Together with (12), this implies that multiple
equilibria can arise. We focus on the symmetric equilibria.
Proposition 5. (i) There is a unique l = l0 . If l > l0 , no bounded solution exists. If l  l0 ,there are two bounded solutions to BR(f) = f. The low-f solution is stable and the high-f solution
is unstable.
(ii) In the stable equilibrium, f⇤ ⌘ fi = fj is increasing in l.
Proof. See Appendix B.3.

⇤

20

Figure III: Best Response Functions

Note that a higher l has the same implication as a higher fj for the sensitivity of s tofi and for the improvement in the sniping probability pi . Thus, due to the same logic asin Lemma 3, a sufficiently high l makes the complementarity strong enough to eliminate abounded solution, i.e., f = • is always optimal. On the other hand, when l is small, we
obtain bounded symmetric equilibria.
Following the convention (Hendershott and Mendelson, 2000; Zhu, 2014), we use stability as an equilibrium selection criterion. The unstable equilibrium is not robust to a small
perturbation in a parameter value, whereas the stable one does not diverge even if a parameter changes slightly. Figure III provides the best response functions for different values ofl. In multiple equilibria, a small-f solution is stable, while a higher f makes “tit for tat”
strong and the solution can explode.
As Figure III indicates, a longer delay has the same effect on the optimal speed as in thebenchmark, i.e., it increases the marginal benefit of being faster. Thus, the best response
function shifts upward, leading to a higher speed in the stable equilibrium.

3.3

Effect of Speed Bumps on the Spread

The effect of a bump on the spread and adverse selection can be derived analytically. Dueto “fast market making,” adverse selection risk is mitigated by fast market makers, and it
helps l protect them. However, in the symmetric equilibrium, the increase in market makers’ speed occurs identically for snipers. Then, the above-mentioned effect of fast market
making is offset by the increase in the snipers’ speed.
21Since the strategic complementarity is sufficiently strong, this arms race outweighs the
direct protection of the speed bump, expanding the spread.
ds
Proposition 6. A longer speed bump widens the spread; dl> 0.

Proof. See Appendix B.4.

⇤

The introduction of a speed bump or a longer expected delay can backfire not only inds
terms of speed but also of adverse selection. In the benchmark model, we have dl= 0because the speed-up by the single HFT is an indirect consequence of the speed bump andcannot offset the direct protection of market makers.
By contrast, multiple HFTs generate a positive externality through strategic complementarity (Proposition 4). In this situation, an increase in l indirectly affects the best response
functions of both HFTs, shifting them upward, as shown in Figure III. This triggers an arms
race with positive externality, amplifying the first indirect effect. As a result, the speed-up
in the symmetric equilibrium dominates the direct protection of market makers, leading tomore severe adverse selection.

3.4

Comparison with Traditional Models with an Exogenous Cost

The results in the previous subsection run counter to traditional models with an exogenous
cost of speeding up. To illustrate this, consider a model with non-strategic HFTs; Insteadof an endogenous cost, we introduce an exogenous sunk cost of being faster denoted by
C (fi ) = 2c fi2 , as in Foucault et al. (2016). To make the comparison clearer, we call our modelin Subsection 3.1 the strategic model.
If the strategic motive is absent, the FOC in (10) and cross derivative in (11) are modified
as follows:
wi ⌘
∂wi
=(s
∂fj

∂Wi
= (s
∂fi

∂pi
cfi ,
∂fi
∂2 p i∂p ∂(s s j )
sj )
+ i
.
∂fj ∂fi
∂fi
∂fj
sj )

(13)

The second term of (10) that represents an endogenous marginal cost is replaced by theexogenous marginal cost, cfi , and the effect of the opponent’s speed via the strategic motive,
denoted by the second set of brackets in (11), disappears from (13). We focus on a symmetricequilibrium and obtain the following results.
22

Proposition 7. (i) Around the symmetric equilibrium, the best response function exhibits strategic
substitution;

dBRi (fj )
< 0.
dfj

(ii) The equilibrium speed and spread are decreasing in l.
Proof. See Appendix B.5.

⇤

The third column (panels A3, B3, and C3) of Figure IV shows these results. As we haveestablished, if HFTs are strategic, HFT j’s speed-up improves wi through the second brackets
in (11), while this effect is absent in the traditional models.
To understand this intuition, note that the profit function of an HF sniper is roughly
given by
Vi = max pi (fi , fj , l)(sfis)

C (fi ),

where pi is the sniping probability of HFT i. In this formulation, the interaction between
HFT i and j occurs only through pi .
A marginally faster opponent in a traditional model affects HFT-i’s decision by making
sniping more difficult, i.e., fj changes Vi only through pi . Since the exogenous cost is sunk,
HFT i must pay it anyway. By contrast, her speed investment pays out only if her sniping
attempt is fulfilled. Therefore, if HFT i thinks she is less likely to snipe due to a faster opponent (a lower pi ), the exogenous cost becomes more salient (C/pi increases), hampering her
speed investment. This logic applies to a speed bump as well: a bump makes sniping lesslikely, leaving HFTs reluctant to pay the sunk cost. This is why traditional models conclude
that a speed bump is effective to slow HFTs down and mitigate adverse selection, which is
replicated by Proposition 7.
Once HFTs become aware of the price impact of their speed choice, they perceive theprice as s = s(fi , fj , l), generating a new channel through which fj affects Vi . In the previous subsection, we showed that this modification overturns the result of the benchmark.
This is because the pricing behavior of market makers becomes insensitive to an increase infi (the endogenous marginal cost declines) when the opponent becomes faster or a bump is
expected to be longer. In our strategic model, this endogenous cost channel works against
the traditional exogenous cost.

23

3.5

Strategic Complementarity or Substitution

As Proposition 7 and the following discussion suggest, an exogenous sunk cost tends tomake an arms race exhibit strategic substitution, while an endogenous cost in our model
promotes complementarity (Proposition 4). These results imply that introducing an exogenous sunk cost in our strategic model allows us to explain both complementarity and substitution in an arms race, as well as the positive and negative reaction of the spread to a
bump, by changing the parameter values.
Consider a model that is the same as in Subsection 3.1, except that HFT i solves
1fi(s s j ) C (fi ),1 + l( b + fj ) yfis.t., s j =,fi + b(1 + l(fi + fj + b))

max Wi (f) =fiwhere C (f) = 2c f2 is the exogenous sunk cost of speed.
The best response functions, equilibrium speed, and spread are provided in Figure IV.
The strategic model in Subsection 3.1 corresponds to c = 0. As expected, a small exogenous
cost (panels A1 and A2 with c = 0.002 and 0.01) offers strategic complementarity and anincreasing f⇤ against l (panel B1). This leads to the increasing s against l (panel C1).
As long as c > 0, f⇤ becomes flatter as l rises and can be hump shaped (panels B1 and
B2). This is because a higher l makes C more salient. That is, a longer delay makes sniping
more difficult, and HFTs become more reluctant to pay the sunk cost. In this situation, anarms race exhibits both complementarity and substitution (Panel A2 with c = 0.04), i.e.,the BR curves become hump shaped. When l is small enough, the model is close to ourstrategic model with complementarity, while a large l makes it similar to the traditionalmodel with substitution. As a result, the direct protection of market makers by the bumpdominates the speed increase in the high-l region, and s slopes downward, as shown inPanels C1 with c = 0.002 and C2.
Finally, if the exogenous cost is sufficiently large, the economy reverts to the traditional world with no strategic speed choice (panels A3, B3, and C3). The competition exhibits global substitution, and the equilibrium speed and the spread are downward sloping
against l, i.e., a speed bump mitigates adverse selection.
Overall, we can think of the strategic model with complementarity and the traditional24

Figure IV: Effect of Exogenous Cost

Note: The first row plots the best response functions with different (cl), the second and third rows plot the
optimal speed and the equilibrium spread, respectively, against the expected length of a speed bump withdifferent values of c. In each row, c = 0 corresponds to the strategic model with no exogenous costs. The
figures in the third column replicate the traditional results.

models with substitution as two extremes. By changing the significance of the exogenous
cost, c, we can explain intermediate cases. Moreover, we have established that a speed
bump, l, also works to adjust the relative significance of the exogenous cost because alonger delay makes it more salient. This premise is summarized in Figure V and has some
empirical implications.

25

Figure V: Summary

4

Empirical Implications and Policy Discussion

Our model provides some testable implications for the strategic nature of an arms race
among HFTs and for the effect of speed bumps on a spread and adverse selection for market
makers.
As Subsection 3.5 demonstrates, an arms race exhibits strategic complementarity or substitution depending on the exogenous cost of speed investment and the expected lengthof a speed bump. As summarized in Figure V, a long (resp. short) expected speed bumpor a significant (resp. slight) exogenous cost of speed creates strategic substitution (resp.
complementarity), and the introduction of a bump and a marginally longer delay would
effectively reduce (resp. widen) a spread.
In the real world, the speed cost for HFTs involves two factors. The first is a large sunkcost to develop a high-speed communication technology or investigate the optimal locationof an information servers to exploit an arbitrage between segmented markets. For example, it is well known that Spread Networks LLC invested about $300 million to reconstruct
a fiber-optic network between the Chicago and New York exchanges. The second cost canbe a relatively small subscription fee to access these technologies developed by communication service companies or to colocate an information server to an exchange platform.22
Depending on the condition of the financial markets, our model suggests different implications. If HFTs take the first investment approach, the traditional model fits better: an arms
race involves strategic substitution and a bump is effective. In contrast, if the exogenous
cost is relatively small compared to the total profit, our strategic model is more appropriate, suggesting the complementarity and detrimental effect of a bump. This comparison
22 For example, monthly payments to plug into NSYSE and NASDAQ amount to $10,000⇠$22,000.

26

can also be applied to the market power of a high-frequency financial institution because
the endogenous cost stems from the strategic motives of HFTs. In other words, when N islarge, the adverse price impact of increasing fi diminishes, leaving the model close to the
traditional one.
Also, we can vary the length of a speed bump or the distribution of a random delay
to test the implications of l on a spread. If a bump is expected to be long, a marginallylonger delay can be effective, slowing HFTs down due to the substitution in the traditionalmodel. On the other hand, if a platform tries to avoid a delay cost by keeping the lengthof a bump minimal, the introduction of a bump can aggravate adverse selection since HFTs
become faster. Although we can compare the length of bumps in each platforms (i.e., bumps
in the ANEO and TMX are longer than those in the IEX or Chicago Stock Exchange), ourmodel shows that the level of l also matters. Deriving the critical (c, l) involves numericalcalculation in our model, and estimating it requires further data on the cost of the speed
technologies, speed levels, profit, and the market power of high-frequency financial institutions.

4.1

Policy Implications

Recently, exchange platforms have experienced declines in their revenue from transaction
fees. Instead, they charge an increasingly high price to the fast access to their data, suchas colocation of data servers. In other words, one of the suppliers of the speed technology,
which we did not specify in the model, can be the exchange platforms. That is, on the one
hand, exchange platforms introduce innovation, such as a speed bump or frequent batch
auction (FBA) to slow traders down, while, on the other hand, they offer a speed technology
at a positive fee.
Importantly, our model suggests that the introduction of a delay does not necessarily
conflict with a provision of the expensive speed technologies by an exchange platform.
That is, a speed bump can increase HFTs’ demand for fast information access, allowing
an exchange platform to charge a higher price.
In this situation, we can analyze whether “the market will fix the market,” as Budish etal. (2018) put forth. They argue that a platform does not have an incentive to introduce FBA,
as long as competing platforms can copy the innovation with a low cost. In our model, abump does not always mitigate adverse selection, while an exchange platform may have27

an incentive to introduce a bump to obtain the higher demand for the speed technology.
More detailed analyses can be our future research topics, but aforementioned mechanism
can provide another explanation for why “the market cannot fix the market.”5Conclusion

A speed bump, which seeks to mitigate adverse selection for market makers, can backfire.
When HFTs strategically choose their speed level by considering the impact of their speed
decision on market behavior, a bid-ask spread charged by market makers works not only
as a trading cost but also as an endogenous cost of speeding up, since it widens as HFTs
get faster. This endogenous cost tends to be insensitive to an increase in speed by HFTs
when a speed bump is expected to be longer. This is because market makers behave as if
the share of HFTs is small under a longer expected speed bump and do not care much about
a speed increase. Then, if a bump is introduced or the length of a delay becomes longer, themarginal cost of speed diminishes, leading to a higher equilibrium speed of HFTs.
We also consider an arms race among multiple HFTs who serve both as snipers and as
market makers. When the significance of the exogenous speed cost is not high comparedto the endogenous cost, or when the expected length of a delay is relatively short, a speed
competition can show strategic complementarity. In this situation, a longer speed bumptriggers a positive externality among HFTs, leading to a very fast equilibrium speed. As aresult, the increase in HFTs’ trading speed dominates a direct reduction of an arrival rateof HFTs by a speed bump. That is to say, a longer bump exacerbates adverse selection andwidens the equilibrium spread. The opposite holds in the case of substitution.
Thus, our model, which incorporates strategic speed choice and the endogenous cost(i.e., the equilibrium spread), generates results that are opposite to the traditional models, inwhich an arms race exhibits strategic substitution, the speed and adverse selection decrease
with the length of the delay, and a bump is effective. By adding a traditional exogenous costto our strategic model, we can derive a rich description of the characteristics of an arms race
among HFTs and can explain the somewhat ambiguous effects of speed bumps on spreads
and adverse selection.

28

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Studies, 2017, 30 (4), 1188–1228.
O’Hara, Maureen, “High frequency market microstructure,” Journal of Financial Economics, 2015, 116(2), 257–270.
Pagnotta, Emiliano S and Thomas Philippon, “Competing on speed,” Econometrica, 2018, 86 (3),1067–1115.
Riordan, Ryan and Andreas Storkenmaier, “Latency, liquidity and price discovery,” Journal of Financial Markets, 2012, 15 (4), 416–437.
Shkilko, Andriy and Konstantin Sokolov, “Every cloud has a silver lining: Fast trading, microwave
connectivity and trading costs,” SSRN Electronic Journal, 2016.
Ye, Mao, Chen Yao, and Jiading Gai, “The externalities of high frequency trading,” SSRN Electronic
Journal, 2013.
Zhu, Haoxiang, “Do dark pools harm price discovery?,” The Review of Financial Studies, 2014, 27 (3),
747–789.

30

A

Different Market Structures

A.1

Continuous Updating by Market Makers

We modify the benchmark case by allowing each market maker to update (cancel and resubmit)
limit orders continuously before HFTs move. Other structures of the model is the same as the benchmark.23
Consider a market maker who updates her limit order by resubmitting competitive st . The competition drives st = E[v|trade at t]. Since d is stochastic,
st =

=

Z •
0

Z •
t

be bd E[v|trade at t,d]dd
bebd

E[v|trade at t,d]dd +

= 0⇥

Z •

= (1

e bt )t

be bd dd +
f
s.
f+b

Z t
0

be bd

Z t
0

be bd E[v|trade at t,d]dd

f
sdd
f+b

(14)
(15)

In (14), the first term represents the case that t is in the “safe interval,” i.e., 0 < t < d. Conditional on
trade occurs, the market maker expects that E[v] = 0 because the trade must be against a liquidity
trader, and it does not convey any information. This is why the first term in (15) bears 0. The second
is the case that t is outside of the “safe interval,” leading the conditional expected return to be theprobability of the HFT arrival (times s) in (15).
A speed bump has the same effect on the endogenous marginal cost as in the benchmark (theproof is omitted as it is straightforward):
⇣ ⌘
∂st
∂
t
Proposition 8. (i) ∂s
>
0,
and
(ii)
∂b
∂b ∂f > 0.
Since b = l 1 represents the inverse of the expected length, a longer delay (i) directly mitigates
adverse selection for market makers, but (ii) it makes the marginal cost (spread) less sensitive tospeed-up by the HFT.
We impose an exogenous sunk cost to make the model well-defined. The optimization problemof the HFT regarding the speed is given byZ •c 2
max W (f) = Edfe yt e hd (s st+d )dt
f ,
f
2
0
f
s.t., st = (1 e bt )s.
f+b

The sniping profit from sending market orders at t is given by s st+d since they possibly arriveand executed at t + d. Note that, given that the trade occur, there is no price uncertainty since st isdeterministic.
The behavior of the optimal speed and spread is hard to show analytically. However, the numerical solutions in Figure VI can be discussed by using the ingredients we have already analyzed. In
the single HFT economy, a speed bump positively affects the optimal speed f⇤ through (i) decline
in the marginal cost and (ii) increase in the sniping profit, while (iii) it reduces f⇤ by magnifying the23 We can eliminate the possibility that an informed HFT splits her orders across the time because executions

in a part of the markets let other market makers know the arrival of the HFT and true information. This event
triggers the cancellation of outstanding limit orders. We also abstract away from the possibility of a mixed
strategy since each order from the HFT does not have price impact. As mentioned in Footnote 14, we canshow that the mixed strategy is not an equilibrium because waiting is not credible.

31

Figure VI: Effect of l on f⇤ and st

Note: The panel A plots the optimal speed against l with different values of c. The panels B and C plot the
dynamics of the spread, st with different values of l. The panel B is the case with the increasing optimal speed,df⇤
df⇤dl > 0, and the panel C is the decreasing optimal speed, dl < 0.

exogenous sunk cost. When l is small, the execution risk is relatively small, leaving effect (iii) lesssignificant compared to (i) and (ii), while a longer delay in the high-l region makes (iii) more salient.
As a result, f⇤ takes the hump-shaped curve as depicted by Panel A in Figure VI. Given l, a larger
cost slows HFT down as we can see from a parallel shift in the curves.
As (14) suggests, the spread is time dependent and increasing in t. This is because the market
makers expect that they are less likely to be in the safe interval as t increases. The effect of l = b 1 is
dst
=
dl

=

1 ∂st
df ∂st
+
2
l ∂b
dl ∂f
1
f
df
te bt
+
(1
l2
b + f dle bt )b.
( f + b )2

(16)

First, a longer delay (higher l) reduces the spread since market makers are directly protected by thelonger safe interval. This is represented by the first term in (16). However, as a higher l may push f
df
up or down, it increases or decreases the spread, as the second term in (16) suggests. When dl > 0,
the second effect competes with the first effect: the safe interval gets longer, while the HFT becomes
faster. The result is provided by Panel B in Figure VI.
Whether the first effect dominates the second effect depends on t. From (16), the first negative
effect is increasing in t  b 1 and then starts decreasing. On the other hand, the second one istmonotonically increasing in t and concave. There is a unique t = t, such that ds
dl > 0 if and only ift > t, i.e., a longer speed bump increases the spread and worsens the adverse selection problem inthe long-run.
The intuition is straightforward. When the current period t is t > t, the probability that t is in32the safe interval is relatively small. Then, market makers think that an increase in l has only a small
effect to mitigate the adverse selection, while it increases the speed of the HFT.
df
When dl < 0, the result is given by Panel C in Figure VI. Since a bump reduces the speed inthis case, the second effect helps the first effect reduce the spread and adverse selection cost. Onceagain, whether or not a speed bump increases the equilibrium speed depends on the exogenous
cost, c, competing with the endogenous cost effect. Hence, the effect on the spread is governed by
the market structure c and the time frame t.

A.2

Coexistence of Fast and Slow Markets

In the real economy, the major market structure is still the continuous limit market with no speed
bumps. Thus, the introduction of a speed bump inevitably makes these market structures coexist.
However, analyses provided by the literature deal only with homogeneous markets. We extend ourmodel to relax this limitation.

A.2.1

Environment

Consider the benchmark economy in Section 2. q 2 (0, 1) fraction of market makers are in the market
with a delay d > 0, which is stochastic, and the rest of them are in the market with no delay, d = 0.
We call the first market the slow market and the latter one the traditional fast market. Each market iscompetitive. Market makers in the slow market submit limit orders with the (half) spread sl , while
those who in the traditional fast market provide s0 . As in the benchmark, l is the expected lengthof the delay. In contrast to the literature (Biais et al., 2015), we impose no restrictions on the venue
choice by the HFT. Transactions information in each market becomes public right after an order
execution, i.e., the markets are perfectly transparent.

A.2.2

Strategies of HFTConsider a strategy of the HFT who becomes informed of v = s at date t. There are two possible (pure) trading strategies for the HFT. First, if she submits orders into the fast market, they are
immediately executed, fulfilling 1 q of her total buying attempts. This market activity is publicly observable, allowing all market makers to realize that the transactions are information driven.24
Based on this premise, market makers in the slow market can cancel their limit orders in the intervalt 2 (t, t + d) which is protected by the speed bump. We call this the “strategy one” and denote it by
A = 1.
Second, the HFT who becomes informed at t can immediately send market orders for q shares
into the slow market, anticipating the execution with the d-delay. She refrains from sending orders
to the fast market at t and waits until the orders sent to the slow market are executed. By observing
the execution in the slow market, she sends orders to the traditional fast market at t + d, which incur
no delay by the construction. In this case, all of her orders arrive at the markets at the same time,(t + d), and she can conceal her identity. Hence, none of the market makers can cancel their quotes.
This “wait-and-grasp-all” strategy is denoted as A = 2.25 Overall, taking A = 2 bears the execution
risk, though the return from it is larger than A = 1 if accomplished.
The mixed strategy is the probability distribution over the set of actions A 2 A = {1, 2}, andlet qt 2 [0, 1] be the probability that the HFT takes the action A = 2. For A 2 A, let w A (t) be theexpected profit from taking A 2 A when the information arrives at date t. Figures VII and VIII
illustrate the timing of the executions when the HFT becomes informed at t.
24 This is because orders from liquidity traders will be fulfilled at the slow and the fast markets simultane-

ously.
25 Note that making other lengths of strategic time lag is not optimal for the HFT, as any other intentional
delay than d tells that the orders are not from liquidity traders but from the HFT.

33

Figure VII: Strategy 1

Figure VIII: Strategy 2First, A = 1 does not bear the execution risk because the HFT can immediately snipe limit orders
in the fast market. However, this behavior becomes public immediately, allowing market makers inthe slow market to cancel their orders. Thus, q fraction of quotes disappear, and the expected profit
is given by
w1 (t) = (1 q)(s s0 ).
On the other hand, A = 2 can snipe all outstanding liquidity at the same time, while it bears theexecution risk that stems from the d-delay. If there is a liquidity shock or the public news during(t, t + d), the HFT cannot exploit her information and speed. Given that the HFT gets informed atdate t, she obtains the profit with probability Pr( TL > d, T > d) = e ( b+g)d . Moreover, since the HFT
is a price taker regarding her trading behavior, her expected return is
w2 ( t ) =

=

Z •
0

q(s

be (b+ b+g)d [q(ssl ) + (1 q)(s1 + l( b + g)s l ) + (1
s0 )

q)(s

s0 )] dd,.

Note that both of {w j (t)} j2A are time independent due to the memoryless property of the exponential distribution. This implies that the optimal decision of A 2 A is also time independent. No matter
when the HFT becomes informed, a timing of private news does not matter—only the delay can be
her concern.

A.2.3

Behavior of Market Makers

We take q an exogenous parameter in our model and assume that each market maker is randomly
assigned the structure of the market. Given an assigned market, each market maker earn zero profit
and does not have an incentive to move to another market with a different structure.

34

In the Fast Market
The market makers in the fast market suffer from adverse selection cost no matter what strategy the
HFT takes. In other words, They are not given any chances to cancel orders by observing marketbased information before the HFT snipes them. The expected return is ✓Z d◆Z •( b+g)ty(t d) ( b+g)dV0 = Ed q
s0 be
dt +
e
( bs0 + f(s0 s))dt
(17)
0

+(1

q)

Z •
0

d

e yt ( bs0 + f(s0s))dt .

The first line is the case that the HFT takes A = 2. In this case, the fast market is under the protection of the speed bump even though the speed bump is not applied to the fast market.26 This isbecause the HFT takes “wait-and-grasp-all” strategy, and she does not snipe the fast market until she
accomplishes her trading attempt in the slow market. The expected profit, in this case, is identical tothose in the benchmark with homogeneous markets. In the second line, the possibility of A = 1 ischaracterized. In this case, the speed bump does not matter for the fast market, and the conditional
expected return is the same as a model with l = 0.

In the Slow Market
In contrast to the fast market, the strategy of the HFT determines whether or not market makers inthe slow market are protected. If q = 0, the slow market is perfectly protected by the speed bump:they can cancel their limit orders to avoid the HFT for sure. On the other hand, if q , 0, it is possiblethat the HFT arrives at the slow market to trade.
The expected return is ✓Z d◆Z •( b+g)ty(t d) ( b+g)d
Vl = E q
s0 be
dt +
e
( bs0 + f(s0 s))dt
(18)
0

+(1

q)

Z •
0

d

sbe yt dt .

The intuitions behind the first line are the same as those in (17). As mentioned above, when the HFT
takes A = 1, there is no chance for the HFT to snipe in the slow market. On the other hand, liquidity
traders arrive at the slow market at t if TL = t, TH > t, and T > t, which gives the integrand in thesecond line.

A.2.4

Equilibrium in the Trading StageLet Q ⌘ (1q)/q. We first solve for the equilibrium spread given q:

Proposition 9. The equilibrium spread in fast and slow markets are given by8f>
for j = 0,
< f+ b+ byql
1+l( b+g)(1 q )
sj =
fq
>
: b+fq +lb( b+g)⇣1+ fq ⌘ for j = l.

(19)

b+g

Proof. Solving Vj = 0 yields the result.
26 Liquidity⇤

traders arrive at t if (i) the HFT becomes informed after t or (ii) the HFT becomes informed before t and takes A = 2.
Given that the quote remains alive at t ,Pr( TH < t, A TH = 2|quote is alive at t ) = Pr( TH < t). Since, otherwise, the HFT arrives immediately at Th
Rt
and snipes the stale quotes. Therefore, Pr(Liq. trade at t) = be yt + be ( b+g)t 0 fe ft dt = be ( b+g)t .

35

These formulae show the following:
Corollary 1. q affects s0 and sl in an opposite way, that is,∂s0∂s< 0, l > 0.
∂q∂qWith f fixed, s0 is decreasing in q, while sl is increasing in q. For market makers in the fast
market, a higher q (i.e., the probability of A = 2) implies that the fast market is more likely to beprotected by the speed bump in the slow market due to the HFT’s “wait-and-grasp-all” strategy.
This mitigates the adverse selection risk for the market makers in the fast market, making s0 lower.
On the other hand, a higher q (or fq) has a negative impact on market makers in the slow market.
This is because a higher probability of A = 2 reduces the chance for market makers to observesniping activity in the fast market to cancel the quote. Thus, a higher q exposes the slow market tomore severe adverse selection and pushes the spread sl up.
Now, the (mixed) strategy is characterized by the following, in which q must satisfy the indifference condition, w1 = w2 .
Proposition 10. The optimal trading strategy for the HFT is
8
f+ b>
if lQ( b + g) > b ,><0
q = q ⇤ 2 [0, 1] if lQ( b + g) 2 [1, f+b b ],>>:1
if lQ( b + g) < 1,
with

q⇤ =

(f + b) b( b + g) Ql
1 + lh
.
1 + lQ( b + g) bl(1 lhQ) f

Proof. See Appendix B.6.

(20)

(21)
⇤

With f fixed, the strategy q of the HFT in the second stage game crucially depends on (i) the
expected length of delay l and (ii) the share of the slow market q. As proposed by (20), a higher land smaller q make the HFT reluctant to take A = 2 because both negatively impact the expected
profit of A = 2 by imposing a higher execution risk and lower profit in the slow market, respectively.
Thus, as l or Q increases, q ⇤ declines and converges to 0. On the other hand, the HFT sticks to thestrategy A = 2 (i.e., q ! 1) when l or Q is sufficiently small.
A higher speed f has two effects on the behavior of q ⇤ . First, it is straightforward that a higher fwidens the region for q = q ⇤ 2 (0, 1). Also, under the mixed strategy, we have the following:⇤

Corollary 2. Ceteris paribus, ∂q∂f > 0.

When the HFT becomes faster, the spreads in the fast and slow markets are differently affected.
Since the slow market is more likely to be protected by the speed bump, a higher f has a stronger
effect on s0 than sl . Moreover, as mentioned earlier, the slow market will face the HFT only if she
takes A = 2 with probability q. Hence, as we can see from (19), the effect of f on sl is discounted
by q (i.e., f affects sl via fq). Thus, when f is high, the profit from the fast market shrinks morecompared to the profit from the slow market. This induces the stronger incentive for the HFT toshift her priority towards the gain from the slow markets. Therefore, she tends to refrain from taking
A = 1, and q increases.

A.2.5

Strategic Speed Choice

Given the equilibrium in the trading stage, the HFT decides her speed level. Her objective function
is denoted byZ
W (f) =•

0fe yt w A (f)dt

36

subject tow A (f) =(w1 (t) = (1 h q)(sw2 (t) = Ed es0 (f, q ))i
( b+g)d ( ssl (f, q ))if q 2 [0, 1)
if q = 1,(22)

the spreads in (19) as functions of (f, q ), and the equilibrium strategy q given by (20). Note that themixed strategy q ⇤ 2 (0, 1) makes it indifferent for the HFT to take A = 1 and A = 2, leading to thefirst line in (22). Furthermore, when q = 1, the economy converges to the benchmark case since theeffect of the speed bump in the slow market encompasses the fast market too. Thus, s0 = sl , and we
obtain w2 in (22).

A.2.6

Short expected delay

As we have established in (20), a sufficiently short expected delay, such that l < ( b + g) 1 Q 1 ,does not hamper the incentive of the HFT to take A = 2. She always takes “wait-and-grasp-all”
strategy, resulting in q = 1. This makes the economy, as well as the equilibrium results, same as thebenchmark case in Section 2. Therefore, a longer expected speed bump increases the speed level f⇤ ,which completely offsets the reduction in the adverse selection cost due to the longer safe interval(Proposition 3). This region is depicted by the left region of the shaded area in Figure X.

A.2.7

Long expected delay

When a delay is sufficiently long, the HFT becomes reluctant to take A = 2 because of the higher
execution risk. She starts adopting the mixed strategy (q = q ⇤ ) or immediately snipes in the fast
market (q = 0). The switch between these two cases occurs atf̂ ⌘ b [l( b + g) Q

1] .

(23)

When f < f̂ (resp. f > f̂), the strategy of the HFT is q = 0 (resp. q = q ⇤ ). As we have discussed, thisthreshold is increasing in l and decreasing in q since both of them reduce the expected profit fromsniping in the slow market.
pLemma 4. When q = 0, the optimal speed level is given by f0⇤ =b( b + g). The speed and the spread are
independent of the expected length of the speed bump l.
Proof. Plugging q = 0 into (22) and taking derivative immediately derive the result.

⇤

Since the objective function W switches at f̂, we have a couple of candidates for f⇤ dependingon f̂ ? f0⇤ , and this is crucially affected by the values of l and Q.
Figure IX plots the objective function W against f with various parameter values for l, in which
the effect of f on q is taken into account. This function is not smooth due to the switch at f = f̂.
When l is relatively small, we have q = q ⇤ 2 (0, 1), and the optimal f is higher than f̂. Then the
speed is positively affected by l, i.e., the longer the expected delay, the faster the HFT. As shown by
Corollary 2, this pushes q ⇤ up. However, because the longer expected delay escalates the execution
risk and the expected return starts waining, W (q ⇤ ) dips below W (q = 0) at some l. Thus, there is al that makes q = q ⇤ and q = 0 indifferent.
As a result, the optimal speed plummets as l increases when q switches from q = q ⇤ to q = 0.
Intuitively, the optimal speed must incorporate the execution risk by the speed bump only if the
HFT snipes in the slow market with a strictly positive probability. Otherwise (if q = 0), the speed
bump has nothing to do with the HFT’s expected profit. The speed decision cares only about theendogenous cost at the fast market, which is more sensitive to the change in f compared to theendogenous cost that stems from the slow market. As a result, the optimal speed with q = q ⇤ is too
fast if q = 0 is the optimal strategy, leading to a dive of f⇤ at the switch. See Figure X for the visual
illustration of the effect of l.

37

Figure IX: W (f) with different l

Figure X: Effect of l

A.2.8

Adverse Selection Cost

Figure X shows the level of the optimal speed f⇤ , the mixed strategy q ⇤ , and the spreads in bothmarkets as functions of l. First, if the delay is sufficiently small, so that lQ( b + g) < 1, the execution
risk for the HFT is sufficiently low, and she takes A = 2 for sure (q ⇤ = 1). The result is the same as
Section 2, and the optimal speed is increasing in l, while the adverse selection cost, measured by thehalf spread, is constant. Note that there is no difference between the slow and the fast markets.
Second, if lQ( b + g) > 1 but l is intermediate, the HFT finds it not attractive to take A = 2 with
100% probability because of the relatively high execution risk. Thus, she starts to mix A = 2 with
A = 1, so that she stochastically snipes at the timing of information revelation. This is represented by
the shaded area in Figure X. In this case, if she keeps f constant, the welfare declines as l increases.
However, a longer expected delay makes the endogenous cost (i.e., the spread) insensitive to anincrease in the speed because market makers set s as if the HFT arrives with a lower probability.
This promotes the investment in the speed. In contrast to the speed, the probability of taking A = 2
declines, although a higher f⇤ has a positive impact on q ⇤ . This is due to the dominating negative
effect of l on q ⇤ : a longer expected delay makes A = 2 a less attractive choice for the HFT.
Finally, if l becomes sufficiently large, as in the right region of the shaded area, the HFT no longer
figures that taking A = 2 pays out because the execution risk becomes sufficiently high. Thus, sheswitches to taking A = 1 with 100% probability, making the optimal level of the speed insensitive
to l. The optimal level f⇤ jumps down from the middle region of l as mentioned in the previous
subsection.
Regarding the adverse selection cost in both markets, the first region with a small l provides the38constant and same level of s. This is natural because both markets face the same risk of the HFT
arrival. The intermediate region of l makes them behave differently. A higher l directly mitigates
adverse selection for market makers, while it increases the optimal speed and the probability of
the HFT-confrontation. The fast market bears the risk of the HFT no matter what strategy the HFT
takes, though A = 2 is discounted by the delay l. On the other hand, the slow markets are exposedto the HFT only if she takes A = 2, and this is protected by l. As shown by Corollary 1, thisasymmetry makes sl decreasing and s0 increasing—a longer speed bump protects the slow markets
at the expense of the traditional fast markets.
Once the delay becomes sufficiently long (right side of the shaded area), the risk of HFT completely diminishes in the slow market because the HFT takes A = 1 for sure. Hence sl = 0. In thefast market, the spread drops as well because the speed of the HFT is humbled. The fast market stillbears the risk of the HFT and keeps the spread strictly positive.

B

Appendix: Proofs

B.1

Proof of Lemma 2 and Proposition 2

Let h ⌘ b + g. The explicit formula for W is given by
W (f) =

1 f b (1 + ly),
1 + lh y f + b (1 + ly)

wherep ⌘ Ed
s

Z •
0

s=

pt (f, d)dt =1 f,
1 + lh yb (1 + ly).
f + b (1 + ly)

Therefore,
W 0 (f) = p0 (ss) + p(s= p0 (s
with
#⌘

p (s
p0 s

s)(1

s)0

#)

s)0
(1 + lh )yf=.
sh (1 + ly) f + b (1 + ly)

It is obvious that d#/df > 0. This implies that the optimization problem satisfies the SOC.
The solution is derived by solving the FOC, which is reduced to
1 = # ( f ).
Note that #(0) = 0, #0 (f) > 0, andlim #(f) =f!•

1 + lh.
hl(1 + b)

Thus, as long as limf!• #(f) > 1, there is a unique solution. We can easily check that this conditionis expressed as (5). If this is not satisfied, we have f⇤ = •.
When (5) holds, the f⇤ > 0 solves 1 = #(f), and some tedious calculations show that the solution
is given by (8). The second statement is obvious from (8).

39

B.2

Proof of Proposition 3

By taking a derivative, we haveds
df
⇠ fy +(1 + lh ).
dldlMoreover, by the implicit function theorem,

(24)

df
hg2 f2 g= 2dl
hg + by2 (1 + hl)2(25)

where g ⌘ f + b(1 + ly). By substituting (25) for the one in (24),ds⇠ yb(1 + lh )(1 + ly)dl= hb(1 + ly)2 f2 .
p

Therefore, at the optimal speed f⇤ =B.3

h (1+lh )

1 l

p

bh

fg

, we have ds/dl = 0.

Proof of Lemma 3 and Proposition 5

Let r ⌘

pb + fj . The second order derivative of BRi is
d2 BRi (fj )
df2j

with
R⌘
0

dr R
=
dfj 2r2✓

1 + lr2p+(1 l br )2 1
l (1 + lpb )r 2

1 ,

2lr
p .
l br

We can check that r RR > 1 is identical to Z (r ) < 0 withZ (r ) ⌘ 2bl3 r3

◆

R0
r
R

p

b(3 + 2l)r + 1.
p1Note that we are focusing on the bounded solution, that is 1 > l br. Since Z ( p) < 0 andllb

Z (0) > 0, there is a unique r ⇤ such that r > r ⇤ , Z (r ) < 0. Then, we can define f0 be the solution ofr = r ⇤ and obtain the result.
The symmetricequilibrium is given by solving f = BR(f), which is rewritten as X (r, l) = 0pwith r ⌘b + f andX (r, l) = l(1 +pb )r 3r 2 + (1This function has the following properties:lbpb)r + b.

∂X (r, l)> 0, 8r > 0,∂lX (r, 0) = r2 + r + b, lim X (r, l) = •.
l!•Therefore, as l increases, X shifts up from X (r, 0) and eventually explodes for all r. At l = 0, X = 0
has a unique solution in the positive r region. By the continuity of X regarding l, if l & 0, then1
X = 0 attains three solutions, two in the positive region (a larger one can be greater than p).
l

Let r + and r

be these two solutions.

∂X (r + ,l)
Since
∂r

40

>

∂X (r ,l)
0 and
∂r

b

< 0, the implicit function

theorem implies drdl < 0 and drdl > 0, which means that the stable solution is increasing in l. By themonotonicity of X regarding l, there is a unique l = l0 such that r (l0 ) = r + (l0 ), and X (r, l) > 0
for all r if l > l0 , i.e., there are no solutions.
+

B.4

Proof of Proposition 6

First, by letting y ⌘ Âi fi + b and h ⌘ fj + b, the FOC for HFT i can be expressed as
1=

fi
Y (y)
,
fi + b(1 + ly) Y (h )withY(x) =x.
1 + lxUnder the symmetric equilibrium, it reduces to
1 = s(f, l)

Y (y),Y (h )

with y ⌘ 2f + b, h ⌘ f + b, and f is the equilibrium speed. We can check thatds
⇠ fy[y(1 + ly)dl

2h (1 + lh )]lhfy(1 + lb).

(26)

Since the symmetric equilibrium solves
f2 = bh (1 + ly)2 ,we know that f =26 is positive.

B.5pbh (1 + ly)b

1. By using these conditions, we can check that the RHS of

Proof of Proposition 7

First of all, the traditional model satisfies the SOC: By letting G ⌘ fi + b(1 + ly),
dwi
= (s
dfi

s)

∂(s s) ∂pi∂2 p i+2∂fi ∂fi∂fi⇠ Gb(1 + ly)< 0.
Then, the FOC to solve is
cfi =

y(G

fi (1 + bl))b Y (h )⌘ K (fi , fj , l),yG Y (y)(27)

with y ⌘ Âi fi + b and h ⌘ fj + b. We can easily check that the RHS of (27) is decreasing in fi .
i
Since K is decreasing in fj and l around the symmetric equilibrium, we can prove that dBR
dfj < 0 anddfdl< 0. Since the form of the equilibrium spread is identical to the strategic model, the oppositedfds
effect of dl in Proposition 6 shows that dl
< 0.

B.6

Proof of Proposition 10

The comparison is
w1 ? w2 , lhQ(s41s0 ) ? ssl .

By plugging the formulae for the equilibrium spreads into the inequality above,withL(q ) ⌘ lhQ(ss0 ) = lhQR(q ) ⌘ s

J (q ),
f + bJ (q )sl =K (q ),
f + bK (q )✓ylh
K (q ) = 1 + q, J (q ) = 1 + lh 1
h 1 + (1 q )lh

y
q+q
h

These functions have the following properties:

◆

.

dL
lhQ
lhQ(1 + ly)> 0, L(0) =, L (1) =,dq
f+b
f + b(1 + ly)dR
1 + ly< 0, R(0) = b 1 , R(1) =.
dq
f + b(1 + ly)
Thus, if lhQ < 1, we have L(1) < R(1), indicating that R > L for all q 2 [0, 1]. Therefore, q ⇤ = 1
is the optimal. When lhQ 1, the result depends on L(0) ? R(0). If lhQ < ( b + f)/b, then R(0) >L(0), which implies that there is a unique interior solution q ⇤ that solves the indifference condition.
The solution solves L(q ) = R(q ), and tedious calculation gives (21). Finally, if lhQ > ( b + f)/b, wehave R < L for all q 2 [0, 1]. Thus, q = 1 is the optimal strategy.

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