Spoofing the Limit Order Book: An Agent-Based Model

Xintong Wangxintongw@umich.edu

Michael P. Wellmanwellman@umich.edu

Computer Science & EngineeringUniversity of Michigan

ABSTRACTWe present an agent-based model of manipulating prices infinancial markets through spoofing : submitting spurious or-ders to mislead traders who observe the order book. Builtaround the standard limit-order mechanism, our model cap-tures a complex market environment with combined privateand common values, the latter represented by noisy observa-tions upon a dynamic fundamental time series. We considerbackground agents following two types of trading strategies:zero intelligence (ZI) that ignores the order book and heuris-tic belief learning (HBL) that exploits the order book topredict price outcomes. By employing an empirical game-theoretic analysis to derive approximate strategic equilibria,we demonstrate the effectiveness of HBL and the usefulnessof order book information in a range of non-spoofing envi-ronments. We further show that a market with HBL tradersis spoofable, in that a spoofer can qualitatively manipulateprices towards its desired direction. After re-equilibratinggames with spoofing, we find spoofing generally hurts mar-ket surplus and decreases the proportion of HBL. However,HBL’s persistence in most environments with spoofing indi-cates a consistently spoofable market. Our model providesa way to quantify the effect of spoofing on trading behaviorand efficiency, and thus measures the profitability and costof an important form of market manipulation.

CCS Concepts•Computing methodologies → Multi-agent systems;

KeywordsSpoofing, agent-based simulation, trading agents

1. INTRODUCTIONElectronic trading platforms have transformed the finan-

cial market landscape, supporting automation of tradingand consequential scaling of volume and speed across ge-ography and asset classes. Automated traders have un-precedented ability to gather and exploit market informa-tion from a broad variety of sources, including transactionsand order book information exposed by many market mecha-nisms. Whereas some of these developments may contribute

Appears in: Proceedings of the 16th International Confer-ence on Autonomous Agents and Multiagent Systems (AA-MAS 2017), S. Das, E. Durfee, K. Larson, M. Winikoff(eds.), May 8–12, 2017, Sao Paulo, Brazil.Copyright c© 2017, International Foundation for Autonomous Agents andMultiagent Systems (www.ifaamas.org). All rights reserved.

to improved price discovery and efficiency, the automationand dissemination of market information may also introducenew possibilities of disruptive and manipulative practices infinancial markets.

Recent years have witnessed several cases of fraud andmanipulation in the financial markets, where traders madetremendous profits by deceiving investors or artificially af-fecting market beliefs. On April 21, 2015, nearly five yearsafter the “Flash Crash”,1 the U.S. Department of Justicecharged Navinder Singh Sarao with 22 criminal counts, in-cluding fraud and market manipulation [3]. Prior to theFlash Crash, Sarao allegedly used an automated programto place orders amounting to about $200 million worth ofbets that the market would fall, and later replaced or mod-ified those orders 19,000 times before cancellation. TheU.S. Commodity Futures Trading Commission (CFTC) con-cluded that Sarao’s manipulative practice was responsiblefor significant order imbalances. Though recent analysis hascast doubt on the causal role of Sarao on the Flash Crash [1],many agree that such manipulation could increase the vul-nerability of markets and exacerbate market fluctuations.

The specific form of manipulation we examine in this pa-per is spoofing. Spoofing refers to the practice of submittinglarge spurious orders to buy or sell some security. The or-ders are spurious in that the spoofer does not intend for themto execute, but rather to mislead other traders by feigningstrong buy or sell interest in the market. Spoof orders maylead other traders to believe that prices may soon rise or fall,thus altering their own behavior in a way that will directlymove the price. To profit on its feint, the spoofer can submita real order on the opposite side of the market and as soonas the real order transacts, cancel all the spoof orders.

In 2010, the Dodd-Frank Wall Street Reform and Con-sumer Protection Act was signed into federal law, outlawingspoofing as a deceptive practice. In its allegations againstSarao, the CFTC notes that “many market participants, re-lying on the information contained in the order book, con-sider the total relative number of bid and ask offers in theorder book when making trading decisions”. In fact spoof-ing can be effective only to the extent that traders actuallyuse order book information to make trading decisions. De-spite detection systems and regulatory enforcement efforts,spoofing is hard to eliminate due to the difficulty in generalof determining the manipulation intent behind placement oforders.

1The Flash Crash was a sudden trillion-dollar dip in U.S.stock markets on May 6, 2010, during which stock indexescollapsed and rebounded rapidly [11].

In this study, we aim to reproduce spoofing in a computa-tional model, as a first step toward developing more robustmeasures to characterize and prevent spoofing. Our modelimplements a continuous double auction (CDA) market witha single security traded. The CDA is a two-sided mecha-nism adopted by most financial and commodity markets [7].Traders can submit orders at any time, and whenever anincoming order matches an existing one they trade at theincumbent order’s limit price. We adopt an agent-basedmodeling approach to simulate the interactions among play-ers with different strategies. The market is populated withmultiple background traders and in selected treatments, onespoofer. Background traders are further divided into agentsusing instances of the zero intelligence (ZI) and heuristicbelief learning (HBL) strategy families. In both strategiestraders employ a noisy observation of the security’s funda-mental value they receive on arrival to trade. The HBLstrategy further considers information about orders recentlysubmitted to the order book. The spoofer in our modelmaintains large buy spoof orders at one tick behind the bestbid, designed to manipulate the market by misleading othersabout the level of demand.

We first address the choice of background traders amongHBL and ZI strategies, through empirical game-theoreticanalysis. In most non-spoofing environments we examine,HBL is adopted in equilibrium and benefits price discoveryand social welfare. By executing a spoofer against the foundequilibria, we show that spoofing can qualitatively manip-ulate price given sufficient HBL traders in the market. Wefinally re-equilibrate games with spoofing and find HBL stillexists in some equilibria but with smaller mixture probabil-ity. Though the welfare benefits of HBL persist, the presenceof spoofing generally decreases market surplus.

The paper is structured as follows. In Section 2, we discussrelated work and our contributions. We describe the mar-ket model, background trading strategies, and the spoofingstrategy in Section 3. In Section 4, we discuss experimentsand present main findings. Section 5 concludes with openquestions.

2. RELATED WORK & CONTRIBUTIONS

2.1 Agent-Based FinanceAgent-based modeling (ABM) takes a simulation approach

to study complex domains with dynamically interacting de-cision makers. ABM has been frequently applied to financialmarkets [12], for example to study the Flash Crash [17] orbubbles and crashes in the abstract [13]. Often the goal ofABM is to reproduce stylized facts of the financial system[18]. Researchers also use ABM to investigate the effectsof particular trading practices, such as market making [22]and high-frequency trading [15, 23]. ABM advocates arguethat simulation is particularly well-suited to study finan-cial markets [2], as analytic models in this domain typicallyrequire extreme stylization for tractability, and pure data-driven approaches cannot answer questions about changingmarket and agent designs.

2.2 Bidding StrategiesThere is a substantial literature on autonomous bidding

strategies in CDA markets [26]. The basic zero intelligence(ZI) strategy [10] submits offers at random offsets from valu-ation. Despite its simplicity, ZI has been shown surprisingly

effective for modeling some cases [6]. In this study, we adoptan extended and parameterized version of ZI to representtrading strategies that ignore order book information.

Researchers have also extended ZI with adaptive featuresthat exploit observations to tune themselves to market con-ditions.2 For example, the zero intelligence plus strategyoutperforms ZI by adjusting an agent-specific profit marginbased on successful and failed trades [4, 5]. Adaptive ag-gressiveness [21] adds another level of strategic adaptation,allowing the agent to control its behavior with respect toshort and long time scales.

Gjerstad proposed a more direct approach to learningfrom market observations, termed GD in its original ver-sion [9] and named heuristic belief learning (HBL) in a sub-sequent generalized form [8]. The HBL model estimates aheuristic belief function based on market observations overa specific memory length. Variants of HBL (or GD) havefeatured prominently in the trading agent literature. GDXcalculates the belief function in a similar manner, but usesdynamic programming to optimize the price and timing ofbids [19]. Modified GD further adapts the original GD tomarkets that support persistent orders [20].

We adopt HBL as our representative class of agent strate-gies that exploit order book information. HBL can be ap-plied with relatively few tunable strategic parameters, com-pared to other adaptive strategies in the literature. Ourstudy extends the HBL approach to a more complex finan-cial market environment than addressed in previous studies.The extended HBL strategy considers the full cycle of anorder, including the times an order is submitted, accepted,canceled, or rejected.

2.3 Spoofing in Financial MarketsThe literature on spoofing and its impact on financial

markets is fairly limited. Some empirical research basedon historical financial market data has been conducted tounderstand spoofing. Lee et al. [14] empirically examinespoofing by analyzing a custom data set, which providesthe complete intraday order and trade data associated withidentified individual accounts in the Korea Exchange. Theyfound investors strategically spoof the stock market by plac-ing orders with little chance to transact to add imbalanceto the order book. They also discovered that spoofing usu-ally targets stocks with high return volatility but low marketcapitalization and managerial transparency. Wang investi-gates spoofing on the index futures market in Taiwan, iden-tifying strategy characteristics, profitability, and real-timeimpact [24]. Martinez-Miranda et al. [16] implement spoof-ing behavior within a reinforcement learning framework tomodel conditions where such behavior is effective.

2.4 ContributionsOur contributions are threefold. First, we adapt HBL

to a complex market environment that supports persistentorders, combined private and fundamental values, noisy ob-servations, stochastic arrivals, and ability to trade multi-ple units with buy or sell flexibility. Second, we demon-strate through empirical game-theoretic analysis the effec-tiveness of our extended HBL and (as corollary) the use-

2Note that to some extent, the adaptive functions of thesestrategies would be implicitly handled by the game-theoreticequilibration process we employ to determine the parametricconfigurations of the (non-adaptive) trading strategies.

fulness of order book information in the absence of spoof-ing, in a range of parametrically different market environ-ments. Third and most importantly, we provide the firstcomputational model of spoofing a dynamic financial mar-ket, and demonstrate the effectiveness of spoofing againstapproximate-equilibrium traders in that model. Our modelprovides a way to quantify the effect of spoofing on tradingbehavior and efficiency, and thus a first step in the design ofmethods to deter or mitigate market manipulation.

3. MARKET MODEL

3.1 Market EnvironmentThe model employs a CDA market mechanism over a sin-

gle security. Prices are fine-grained and take discrete valuesat integer multiples of the tick size. Time is also fine-grainedand discrete, with trading over a finite horizon T . Agents inthe model submit limit orders, which specify the maximum(minimum) price at which they would be willing to buy (sell)together with the number of units to trade.

The fundamental value r of the underlying security changesthroughout the trading period, according to a mean-revertingstochastic process:

rt = max{0, κr + (1− κ)rt−1 + ut}; r0 = r. (1)

Here rt denotes the fundamental value of the security at timet ∈ [0, T ]. The parameter κ ∈ [0, 1] specifies the degree towhich the fundamental reverts back to r. The perturbationin the fundamental at time t is normally distributed: ut ∼N(0, σ2

s).The CDA market maintains a limit order book of out-

standing orders, and provides information about the bookto traders with zero delay. The buy side of the order bookstarts with BIDt, the highest-price buy order at time t, andextends to lower prices. Similarly, the sell side starts withASKt, the lowest-price sell order at time t, and extends tohigher prices. On order cancellation or transaction, the mar-ket removes the corresponding orders and updates the orderbook. Agents may use order book information at their owndiscretion.

The market is populated with multiple background traders,and in selected treatments, a spoofer. Background tradersrepresent investors with preferences for holding long or shortpositions in the underlying security. The spoofer seeks trad-ing profits through its price manipulation actions.

The individual preference of background trader i is definedby its private value Θi, a vector of length 2qmax, where qmax

is the maximum number of units a trader can be long orshort at any time. Private values are subject to diminishingmarginal utility and element θqi in the vector specifies theincremental private benefit foregone by selling one unit ofthe security given a current net position of q.

Θi = (θ−qmax+1i , . . . , θ0i , θ

1i , . . . , θ

qmaxi )

Alternatively, θq+1i can be understood as the marginal pri-

vate gain from buying an additional unit given current netposition q. To reflect diminishing marginal utility, that is

θq′≤ θq for all q′ ≥ q, we generate Θi from a set of 2qmax

values drawn independently from N(0, σ2PV ), sort elements

in descending order, and assign θqi to its respective value inthe sorted list. The agent’s overall valuation for a unit ofthe security is the sum of fundamental and private value.

The entries of a background trader follow a Poisson pro-cess with an arrival rate λa. On each entry, the trader ob-serves an agent-and-time-specific noisy fundamental ot =rt + nt with the observation noise following nt ∼ N(0, σ2

n).Given its incomplete information about the fundamental,the agent can potentially benefit by considering market in-formation, which is influenced by the aggregate observationsof other agents. When it arrives, the trader withdraws itsprevious order (if untransacted) and submits a new single-unit limit order, either to buy or sell as instructed with equalprobability. The trader’s order price is jointly decided by itsvaluation and trading strategy, as discussed in the next sec-tion. The spoofing agent, if present, initially arrives at adesignated intermediate time Tsp ∈ [0, T ] and executes themanipulation strategy described in Section 3.2.4.

3.2 Trading Strategies

3.2.1 Estimation of the Final FundamentalAs holdings of the security are evaluated at the end of a

trading period, traders estimate the final fundamental valuebased on their noisy observations. We assume the marketenvironment parameters (mean reversion, shock variance,etc.) are common knowledge for background agents.

Given a new noisy observation ot, an agent estimates thecurrent fundamental by updating its posterior mean rt andvariance σ2

t in a Bayesian manner. Let t′ denote the agent’spreceding arrival time. We first update the previous poste-riors (rt′ and σ2

t′) by mean reversion for the interval sincepreceding arrival (δ = t− t′):

rt′ ← (1− (1− κ)δ)r + (1− κ)δ rt′ ;

σ2t′ ← (1− κ)2δσ2

t′ +1− (1− κ)2δ

1− (1− κ)2σ2s .

The estimates for t are then given by:

rt =σ2n

σ2n + σ2

t′rt′ +

σ2t′

σ2n + σ2

t′ot ; σ2

t =σ2nσ

2t′

σ2n + σ2

t′.

Based on the posterior estimate of rt, the trader computesrt, its estimate at time t of the terminal fundamental rT , byadjusting for mean reversion:

rt =(1− (1− κ)T−t

)r + (1− κ)T−trt. (2)

3.2.2 ZI as a Background Trading StrategyZI traders compute limit-order prices based on fundamen-

tal observations and private values. The ZI agent shades itsbid from its valuation by a random offset, which is uniformlydrawn from [Rmin, Rmax]. Specifically, a ZI trader i arrivingat time t with position q generates a limit price

pi(t) ∼

{U [rt + θq+1

i −Rmax, rt + θq+1i −Rmin] if buying,

U [rt − θqi +Rmin, rt − θqi +Rmax] if selling.

Our version of ZI also takes into account the current quotedprice, as governed by a strategic threshold parameter η ∈[0, 1]. Before submitting a new limit order, if the agent couldachieve a fraction η of its requested surplus, it would simplytake that quote.

3.2.3 HBL as a Background Trading StrategyHBL agents go beyond their own observations and pri-

vate values by also considering order book information. The

strategy is centered on belief functions that traders formon the basis of observed market data. Agents estimate theprobability that orders at various prices would be acceptedin the market, and choose a limit price maximizing theirown expected surplus at current valuation estimates.

The HBL agent’s probability estimate is based on ob-served frequencies of accepted and rejected buy (bid) andsell (ask) orders during the last L trades, where L, theagent’s memory length, is a strategic parameter. On arrivalat time t, the HBL agent builds a belief function ft(P), de-signed to represent the probability that an order at price Pwill result in a transaction. Specifically, the belief functionis defined for encountered prices P by:

ft(P) =

TBLt(P)+ALt(P)

TBLt(P)+ALt(P)+RBGt(P)if buying,

TAGt(P)+BGt(P)TAGt(P)+BGt(P)+RALt(P)

if selling.

(3)

Here, T and R specify transacted and rejected orders re-spectively; A and B represent asks and bids; L and G de-scribe orders with prices less than or equal to and greaterthan or equal to price P correspondingly. For example,TBLt(P) is the number of transacted bids found in mem-ory with price less than or equal to P up to time t. Agentscompute the statistics upon each arrival and update theirmemory whenever the market receives new order submis-sions, transactions, or cancellations.

Since our market model supports order cancellations andkeeps active orders in the order book, the notion of a rejectedorder is tricky to define. To solve this problem, we introducea grace period τgp and an alive period τal of an order. Wedefine the grace period τgp = 1/λa and the alive period τal ofan order as the time interval from submission to transactionor withdrawal if it is inactive, or to the current time if active.An order is considered as rejected only if its alive period τalis longer than τgp, otherwise it is partially rejected by afraction of τal/τgp.

As the belief function (3) is defined only at encounteredprices, we extend it over the full domain by cubic spline in-terpolation. To speed the computation, we pick knot pointsand interpolate only between those points.

After formulating the belief function, agent i with an ar-rival time t and current holdings q searches for the priceP∗i (t) that maximizes expected surplus:

P∗i (t) =

{arg maxp(rt + θq+1

i − p)ft(p) if buying,

arg maxp(p− θqi − rt)ft(p) if selling.

Under the special cases when there are fewer than L trans-actions at the beginning of a trading period or when one sideof the order book is empty, HBL agents behave the same asZI agents until enough information is gathered to form thebelief function. As those cases are rare, the specific ZI strat-egy that HBL agents adopt does not materially affect overallperformance.

3.2.4 Spoofing StrategyWe design a simple spoofing strategy which maintains

a large volume of buy orders at one tick behind the bestbid. Specifically, upon arrival at Tsp ∈ [0, T ], the spoofingagent submits a buy order at price BIDTsp − 1 with volumeQsp � 1. Whenever there is an update on the best bid, thespoofer cancels its original spoof order and submits a newone at price BIDt−1 with the same volume. As background

traders submit only single-unit orders, they cannot transactwith the spoof order, which is shielded by a higher orderat BIDTsp . If that higher order gets executed, the spooferimmediately cancels and replaces its spoof orders before an-other background trader arrives. We assume in effect thatthe spoofer can react infinitely fast, in which case its spooforders are guaranteed never to transact.

By continuously feigning buy interest in the market, thisspoofing strategy specifically aims to raise market beliefs.Other spoofing strategies such as adding sell pressure or al-ternating between buy and sell pressure can be easily con-structed by extension from the current version.

3.3 SurplusWe calculate market surplus as the sum of agents’ sur-

pluses at the end of the trading period T . An agent’s totalsurplus (i.e., payoff) is its net cash from trading plus thefinal valuation of holdings. The market’s final valuation oftrader i with ending holdings H is rTH +

∑k=Hk=1 θki for long

position H > 0, or alternatively, rTH−∑k=0k=H+1 θ

ki for short

position H < 0.

4. EXPERIMENTS AND RESULTSWe conduct simulation experiments for our market model

over a variety of parametrically defined market environments.Each simulation run evaluates a specified profile of back-ground trading strategies in the designated environment.We sample 20,000 runs for each profile, to account for stochas-tic features such as market fundamental series, agent arrivalpatterns, and private valuations. Given the accumulatedpayoff data from these samples, we estimate an empiricalgame model, and from that derive an approximate Nash equi-librium, where agents have no incentive to deviate to otheravailable strategies, given others’ choices. This allows us toevaluate market performance and the impact of spoofing instrategically stable settings.

We start by specifying our experimental market settingsin Section 4.1. Section 4.2 introduces the empirical game-theoretic analysis (EGTA) [25] methods used to identifyequilibrium solutions. In Section 4.3, we conduct EGTAon agents’ choices among ZI and HBL strategies in gameswithout spoofing. Section 4.4 investigates games with spoof-ing. We first illustrate that the HBL strategy is spoofableand spoofing can cause a rise in market prices and a redis-tribution of surplus between ZI and HBL agents. We thenre-equilibrate the game with spoofing to investigate the im-pact of spoofing on HBL adoption and market surplus.

4.1 Market Environment SettingsOur simulations consider nine environment settings that

differ in the variance defining market shock, σ2s ∈ {105, 5 ×

105, 106}, and observation noise, σ2n ∈ {103, 106, 109}. Shock

variance governs fluctuations in the fundamental time series,and observation variance the quality of information agentsget about the true fundamental. Preliminary explorationsover environment parameters suggested these as the mostsalient for our study. We label the three low, medium, andhigh shock variances as {A,B,C} and noisy observation vari-ances as {1, 2, 3} respectively to describe the nine environ-ments. For instance, A1 represents a market with low shockσ2s = 105, and low observation noise σ2

n = 103.For each environment setting, we consider games with

N ∈ {28, 65} background traders and in selected treatments,

Table 1: Background trading strategies included in empirical game-theoretic analysis.

Strategy ZI1 ZI2 ZI3 ZI4 ZI5 ZI6 ZI7 HBL1 HBL2 HBL3 HBL4

L NA NA NA NA NA NA NA 2 3 5 8Rmin 0 0 0 0 0 250 250 250 250 250 250Rmax 250 500 1000 1000 2000 500 500 500 500 500 500η 1 1 0.8 1 0.8 0.8 1 1 1 1 1

a spoofer. The global fundamental time series is generatedaccording to (1) with fundamental mean r = 105, meanreversion κ = 0.05, and shock variance σ2

s . Each tradingperiod lasts T = 10, 000 time steps. Background traders ar-rive at the market according to a Poisson distribution witha rate λa = 0.005 and upon each arrival, the trader ob-serves a noisy fundamental ot. The maximum number ofunits background traders can hold at any time is qmax = 10.Private values are drawn from a Gaussian distribution withzero mean and a variance of σ2

PV = 5 × 106. The spoofingagent initially arrives at time Tsp = 1000, submits a largebuy order at price BIDTsp − 1 with volume Qsp = 200, andlater maintains spoofing orders at price BIDt − 1.

To provide a benchmark for market surplus, we calculatethe social optimum, which depends solely on the trader pop-ulation size and valuation distribution. From 20,000 samplesof the joint valuations, we estimate mean social optima of18389 and 43526 for markets with 28 and 65 backgroundtraders, respectively. We further calculate the average orderbook depth (on either buy or sell side) in markets withoutspoofing. Throughout the trading horizon, the N = 28 mar-ket has a relatively thin order book with an average depthof 12, whereas the N = 65 market has a thicker one with anaverage depth of 30.

The background trading strategy set (Table 1) includesseven versions of ZI and four versions of HBL.3 Agents areallowed to choose from this restricted set of strategies tomaximize their payoffs.

4.2 EGTA ProcessWe model the market as a symmetric game, which is de-

fined by a market environment and a set of players represent-ing background traders. The spoofer agent, when present,implements a fixed policy so is not considered a strategicplayer. The payoff of a specific strategy in a profile is theaverage over agents playing that strategy in the profile, andthus depends on the number playing each strategy, not onindividual mappings. We are interested in players’ strategicchoices in Nash equilibria of these games.

As game size grows exponentially in the number of play-ers and strategies, it is computationally prohibitive to ana-lyze games with this many traders. We therefore apply ag-gregation to approximate the many-player games as gameswith fewer players. The specific technique we employ, calleddeviation-preserving reduction (DPR) [27], defines reduced-game payoffs in terms of payoffs in the full game as follows.Consider an N -player symmetric game, which we want toreduce to a k-player game. The payoff for playing strategys1 in the reduced game, with other agents playing strate-gies (s2, . . . , sk), is given by the payoff of playing s1 in thefull N -player game when the other N − 1 agents are evenly

3We also considered ZI strategies with larger shading rangesand HBL strategies with longer memory lengths, but theyfail to appear in equilibrium.

divided among strategies s2, . . . , sk. To facilitate DPR, wechoose values for N to ensure that the required aggregationscome out as integers. Specifically, in this study we reducethe market environments with 28 (65) background traders togames with four (five) background traders. With one back-ground player deviating to a new strategy, we can reducethe remaining 27 (64) players to three (four).

To find the Nash equilibrium of a game, we run simula-tions to get payoffs for background strategies and conductEGTA based on those payoffs. Exploration starts with sin-gleton subgames (games over strategy subsets), and spreadsto consider other strategies incrementally. Equilibria foundin a subgame are considered as candidate solutions of the fullgame. We attempt to refute these candidates by evaluatingdeviations outside the subgame strategy set, constructing anew subgame when a beneficial deviation is found. If we ex-amine all deviations without refuting, the candidate is con-firmed. We continue to refine the empirical subgame withadditional strategies and corresponding simulations until atleast one equilibrium is confirmed and all non-confirmed can-didates are refuted.

4.3 Games without SpoofingSince spoofing targets the order book and can be effective

only to the extent traders exploit order book information,we first investigate whether background agents adopt theHBL strategy in games without spoofing. Applying EGTAto the eleven background strategies in Table 1, we found atleast one equilibrium for each market environment.4

As indicated in Figure 4 with blue circles, HBL is adoptedwith positive probability by background traders in most non-spoofing environments. That is, in the absence of spoofing,investors generally have incentives to make bidding decisionsbased on order book information. We find that HBL is ro-bust and widely preferred in markets with more traders,low fundamental shocks, and high observation noise. Intu-itively, a larger population size implies a thick order bookwith more learnable aggregated data; low shocks in funda-mental time series increase the predictability of future priceoutcomes; and high observation noise limits what an agentcan glean about the true fundamental from its own infor-mation. The two exceptions (environments C1 and C2 with28 background traders) where all agents choose ZI can beexplained by the environments’ small population size, largefundamental shocks, and relatively small observation noise.

We further conduct EGTA in games where backgroundtraders are restricted to strategies in the ZI family (ZI1–ZI7 in Table 1). This is tantamount to disallowing learningfrom order book information. To understand the effect oforder book disclosure on market performance, we compare

4Details of the HBL adoption rates and market surpluses ofall found equilibria in games with and without spoofing areavailable in an online appendix at http://hdl.handle.net/2027.42/136123

16500

16700

16900

17100

17300

17500

17700

17900

18100

18300

18500Su

rplu

sHBL and ZI ZI

A1 A2 A3 B1 B2 B3 C1 C2 C3

(a) N = 28

39500

40000

40500

41000

41500

42000

42500

43000

43500

44000

Surp

lus

HBL and ZI ZI

A1 A2 A3 B1 B2 B3 C1 C2 C3

(b) N = 65

120

130

140

150

160

170

180

190

200

RM

SD

HBL and ZI ZI

A1 A2 A3 B1 B2 B3 C1 C2 C3

(c) N = 28

90

100

110

120

130

140

150

160

170

RMSD

HBL and ZI ZI

A1 A2 A3 B1 B2 B3 C1 C2 C3

(d) N = 65

Figure 1: Comparisons of market surplus (Figures 1(a) and 1(b)) and price discovery (Figures 1(c) and 1(d)) for equilibriumin each environment, with and without the HBL strategies available to background traders. Blue circles represent equilibriumoutcomes when agents can choose both HBL and ZI strategies; orange triangles represent equilibrium outcomes when agentsare restricted to ZI strategies. Overlapped markers are outcomes from the same equilibrium mixture, despite the availabilityof HBL. The market generally achieves higher surplus and better price discovery when HBL is available.

equilibrium outcomes for each environment, with and with-out the HBL strategy set available to background traders,on two measures: market surplus and price discovery (Fig-ure 1). Price discovery reflects how well transactions revealthe true value of the security; it is defined as the root-mean-squared deviation (RMSD) of the transaction price from theestimate of the true fundamental (2) over the trading period.Lower RMSD means better price discovery.

Overall in our experiments, background traders achievehigher surplus (Figures 1(a) and 1(b)) and better price dis-covery (Figures 1(c) and 1(d)) when the market provides or-der book information and enables the HBL strategy option.When the equilibrium includes HBL, we find transactions re-veal fundamental estimates well, especially in markets withlower shock and observation variances (i.e., A1, A2, B1, B2).We also notice small exceptions in scenarios with high obser-vation variance and more background traders (environmentA3 and C3 with 65 players) where ZI-only equilibria exhibithigher surplus than equilibria combining HBL and ZI.

4.4 Games with Spoofing

4.4.1 Comparison across Fixed Strategy ProfilesTo examine the effectiveness of the designed spoofing strat-

egy, we play a spoofer against each equilibrium found in Sec-tion 4.3 and perform controlled comparisons with and with-out spoofing. In the paired instances, background agentsplay identical strategies, and are guaranteed to arrive at thesame time, receive identical private values, and observe thesame fundamental values. Therefore, any change in behav-ior is caused by the spoof orders’ effects on HBL traders.For every setting, we simulate 20,000 paired instances, andcompare their transaction price differences (Figure 2), andsurplus differences attained by HBL and ZI traders (Fig-ure 3). Transaction price difference at a specific time isdefined as the most recent transaction price of a game withspoofing minus that of the paired game without spoofing.Similarly, surplus difference of HBL or ZI is the aggregatedsurplus obtained in a game with spoofing minus that of thepaired game without spoofing.

Figure 2 shows5 , across all environments, positive changesin transaction prices subsequent to arrival of a spoofing

5As spoofing has no impact on pure ZI populations, we dis-play transaction price differences only for equilibria includ-ing HBL.

agent at Tsp = 1000. This suggests HBL traders are trickedby the spoof buy orders: they believe the underlying securityshould be worth more and therefore submit or accept limitorders at higher prices. Though ZI agents do not changetheir bidding behavior directly, they may be passively af-fected due to the requested surplus fraction η, and maketransactions at higher prices.

Several other interesting findings are revealed by the trans-action price difference series (Figure 2). First, the averageprice rise caused by spoofing in market with 28 backgroundtraders is higher than that of the 65-background-trader mar-ket. The market with fewer background traders is more sus-ceptible to spoofing, possibly due to the limited pricing infor-mation a thin market could aggregate. Second, for marketspopulated with more HBLs than ZIs, the transaction pricedifferences increase throughout the trading period. This am-plification can be explained by HBLs consistently submittingorders at higher prices and confirming each other’s spoofedbelief. However, for markets with more ZIs, the spoofingeffect diminishes as ZIs who do not change their limit-orderpricing can partly correct HBLs’ illusions. Third, differencesin transaction prices first increase and then tend to stabilizeor decrease as time approaches the end of a trading period.As time approaches T = 10, 000, spoofing wears off in theface of accumulated observations and mean reversion.

Figure 3 demonstrates a redistribution of surplus betweenHBL and ZI agents when we include a spoofer: HBL’s aggre-gated surplus decreases, while ZI’s increases compared to thenon-spoofing baseline. That is, the ZI agents benefit fromthe HBL agents’ spoofed beliefs. Since the decreases in HBLsurplus are consistently larger than the increases for ZI, theoverall market surplus decreases. We also find that marketswith a spoofer against background traders at equilibriumhave statistically significantly higher RMSD, which affirmsthe notion that spoofing, as a deceptive practice, hurts pricediscovery.

4.4.2 Spoofing ProfitabilityTo examine the potential to profit from a successful price

manipulation, we extend the spoofing agent with an ex-ploitation strategy: buying, then (optionally) spoofing toraise the price, then selling. It starts by buying when it findsa limit sell order with price less than the fundamental mean.It then optionally runs the spoofing trick, or alternativelywaits, for 1000 time steps. Finally, the exploiter sells whenit finds a limit buy order with price more than fundamental

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Figure 2: Transaction price differences throughout the trading horizon with and without a spoofer against each HBL-and-ZIequilibrium found in Section 4.3. Multiple lines of the same environment represent different equilibria. A market with HBLagents is spoofable, as the designed spoofing tactic can successfully raise market prices.

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Figure 3: Background trader surplus differences in markets with and without a spoofer against each HBL-and-ZI equilibriumfound in Section 4.3. Repetitions of the same market environment represent outcomes of multiple equilibria. Spoofing generallycauses a decrease in HBL’s aggregated surplus and an increase in ZI’s total surplus.

mean. Note that even without the spoof, this exploitationstrategy is profitable in expectation due to the mean rever-sion, and the reliable arrival of background traders willingto sell at prices better than the fundamental.

In controlled experiments, we find that exploitation profitsare consistently increased when the spoof is also deployed.Specifically, across 28-trader market environments, the ex-ploiter makes an average profit of 206.1 and 201.8 with andwithout spoofing, and the increases in profit range from 1.2to 11.5. For the 65-trader market, the average profits ofthis exploitation strategy with and without spoofing are 50.5and 46.3 respectively, with the increases in profit varyingfrom 1.7 to 9.4 across environments.6

4.4.3 Re-Equilibrating Games with SpoofingTo understand how spoofing affects background-trader in-

teractions, we conduct EGTA again to find Nash equilib-rium in games with spoofing, where background traders canchoose any strategy in Table 1. As indicated in Figure 4 withorange triangles, after re-equilibrating games with spoofing,HBL is generally adopted by a smaller fraction of traders,but may still persist at equilibrium in most market envi-

6Statistical tests show all increases in profit are significantlylarger than zero. Regardless of spoofing, the exploitationstrategy profits more in the thinner market due to thegreater variance in transaction prices.

ronments. HBL’s existence after re-equilibration indicates aconsistently spoofable market: the designed spoofing tacticfails to eliminate HBL agents and in turn, the persistence ofHBL may incentivize a spoofer to continue effectively ma-nipulating the market.

Finally, we investigate the effect of spoofing on marketsurplus. Figure 5 compares the total surplus achieved bybackground traders in equilibrium with and without spoof-ing. Given the presence of HBL traders, spoofing generallydecreases total surplus (as in Figure 5, most filled orange tri-angles are below the filled blue circles). However, spoofinghas ambiguous effect in the thicker market with large obser-vation variance (environment A3 and C3 with 65 backgroundagents). This may be because noise and spoofing simulta-neously hurt the prediction accuracy of the HBL agents andtherefore shift agents to other competitive ZI strategies withhigher payoffs. Finally, we find the welfare effects of HBLstrategies persist regardless of spoofing’s presence: marketspopulated with HBL agents in equilibrium generally achievehigher total surplus than those markets without HBL (as inFigure 5, the hollow markers are below the filled markers).

5. CONCLUSIONWe constructed a computational model of spoofing: the

tactic of manipulating market prices by targeting the order

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Figure 4: HBL adoption rates at equilibria in games with and without spoofing. Each blue (orange) marker specifies the HBLproportion at one equilibrium found in a specific game environment without (with) spoofing.

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Figure 5: Total surplus achieved at equilibria in games with and without spoofing. Each blue (orange) marker specifies thesurplus at one equilibrium found in a specific game environment without (with) spoofing. Surplus achieved at equilibriacombining HBL and ZI and equilibria with pure ZI are indicated by markers with and without fills respectively.

book. To do so, we designed an HBL strategy that uses or-der book information to make pricing decisions. Since HBLtraders use the order book, they are potentially spoofable,which we confirmed in simulation analysis. We demonstratethat in the absence of spoofing, HBL is generally adopted inequilibrium and benefits price discovery and social welfare.Though the presence of spoofing decreases the HBL propor-tion in background traders, HBL’s persistence in equilib-rium indicates a robustly spoofable market. By comparingequilibrium outcomes with and without spoofing, we findspoofing tends to decrease market surplus. Comparisonsacross parametrically different environments reveal factorsthat may influence the adoption of HBL and the impact ofspoofing.

Our agent-based model aims to capture the complex essenceof real-world financial markets and the strategic interactionsamong investors. We acknowledge several factors that limitthe accuracy of our equilibrium analysis in individual gameinstances; these include sampling error, reduced-game ap-proximation, and most importantly, restricted strategy cov-erage. Despite such limitations (inherent in any complexmodeling effort), we believe the model offers a constructivebasis for other researchers, regulators, and policymakers tobetter evaluate spoofing and understand its interplay withother trading strategies.

We close with two open questions for further investigation.First, are there more robust ways for exchanges to dis-

close order book information? Similarly, are there strategiesby which traders can exploit market information but in lessvulnerable ways? Preliminary explorations suggest that amodified version of HBL which constructs the same belieffunction but weights orders based on recency can be lesssusceptible to spoofing. Intuitively, as the spoofing strat-egy relies on frequent order cancellations and modifications,spoof orders canceled recently are weighted more which in-creases the rejected order term in the belief function as spec-ified in (3). As a result, the rejected term in denominatorgenerally decreases the accept probability of orders with acertain price and undermines the spoofed belief. This mod-ified HBL is just one possible approach—more robust learn-ing schemes that ignore skeptical orders but still learn frommarket information can be explored based on our model.

Second, based on a model of this kind, can we design ef-fective spoofing detection methods? An ability to generatedata streams with and without spoofing may provide a use-ful resource for learning telltale signatures of a spoofer.

AcknowledgmentsWe thank Uday Rajan and Erik Brinkman for helpful dis-cussions and the anonymous reviewers for constructive feed-back. This work was supported in part by grant IIS-1421391from the US National Science Foundation.

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