Information and Optimal Trading Strategies with Dark Pools *

Anna Bayona 1 Ariadna Dumitrescu2 Carolina Manzano3

June 2017

Abstract

We study how asymmetric information affects market participants’ choice of trading venue(either an exchange or dark pool), and the optimal submission strategies in a sequential tradinggame. The exchange is organized as a fully transparent limit order book, and the dark pool is anopaque venue where orders are continuously executed at the midpoint of the bid and ask pricesthat prevail in the exchange. We find that when the limit order book conveys no informationrational uninformed traders never trade in the dark pool due to price risk. However, pricerisk may be reduced when the information in the book induces an uninformed buyer (seller) tobelieve that the value of the asset is high (low) since the order was previously submitted byan informed buyer (seller). The optimal strategy of an informed trader takes into account theinformation leakage, and the strategic response of each type of trader in the subsequent stages.Our main finding is that, adding a dark pool alongside an exchange, may divert the informedtrader from the exchange to the dark pool. When the execution risk in the dark pool is low, theinformed trader decides to trade in the dark pool. In this case the adverse selection problem inthe exchange reduces and consequently, the uninformed trader decides to switch from no tradeto placing a limit order in the exchange.

Keywords: trading venues, dark liquidity, limit order book, price risk, adverse selectionJEL codes: G12, G14, G18

*The authors would like to thank seminar participants at Universitat de les Illes Baleares and ESADE. Any errorsare our own responsibility.

1Anna Bayona, ESADE Business School, Av. Pedralbes, 60-62, Barcelona, 08034, Spain. Tel: (34) 932 806 162,e-mail: anna.bayona@esade.edu

2Ariadna Dumitrescu, ESADE Business School, Av. Pedralbes, 60-62, Barcelona, 08034, Spain. Tel: (34) 934 952191, e-mail: ariadna.dumitrescu@esade.edu

3Carolina Manzano: Universitat Rovira i Virgili and CREIP. Av. de la Universitat 1, Reus, 43204, Spain. Tel:(34) 977 758 914, e-mail: carolina.manzano@urv.cat

1

1 Introduction

In today’s financial markets, traders have access to different types of trading venues, which differin their level of transparency. In addition to the traditional exchanges (lit markets), traders alsohave access to a dark pools – trading venues or mechanisms containing anonymous, non-displayedliquidity that is available for execution. Dark pools grew as a result of technological innovationsand Reg NMS regulation in 2007.1 Their current consolidated trading volume in US equity marketsis around 14% while in European equity markets is around 9.1% (Rosenblatt Securities Inc.). Ourpaper studies how the existence of a dark pool affects traders’ optimal submission strategies in asequential trading game with asymmetric information. Informed and uninformed traders face asimultaneous choice of trading venue and order type in the exchange (market order or limit order)or refrain from trading. We model the competition between an exchange (that is organized as alimit order book) and a dark pool in the presence of asymmetric information. Our model allowsus to understand the strategic behavior of traders in the presence of private information but also,unlike other papers in the literature, we study the optimal venue and type of order decision, andhow the leakage of information affects this decision.

Our main finding is that adding a dark pool alongside an exchange may switch the optimalstrategies of each type of traders. Thus, in the first period, when the execution risk in the dark poolis low, the informed trader may divert from the exchange to the dark pool. When doing so, theadverse selection in the exchange decreases and this induces an uninformed trader to switch fromno trade to placing a limit order.

Our model reflects the main characteristics of today’s financial markets. The exchange is orga-nized as a fully transparent limit order book. Despite the fact that there exist many types of darkpools, our modelization captures their main features: (1) No pre-trade transparency. Dark pools arecompletely opaque that do not quote the liquidity that is available and this makes execution uncer-tain; (2) They do not determine prices and derive their price from those prevailing in the exchangeas the midpoint between the best bid and ask prices at this point in time (if the order is executed).This type of pricing is typical of dark pools which are owned by agency brokers or exchanges andrepresent 57% of the consolidated US equity trading volume (Buti et al. 2016). Traders may beof two different types: rational or liquidity traders. Rational traders strategically choose whetheror not to trade, and if they trade they simultaneously choose the venue and the type of order (inthe exchange both market and limit orders are available) given their information at each point intime. Rational traders are informed if they know the liquidation value of the asset, and uninformedtraders if they only know the distribution of the liquidation value of the asset. Since the limitorder book is fully transparent, information in the book is available to all rational traders. Thetiming of the model is as follows. First, the liquidation value of the asset is realized. Second, thereare two periods of trading. In each period, a trader may arrive to the market. Rational traders

1The policy debate on the regulation of dark pools is currently very active (see, for example, U.S. Securities andExchange Commission, 2015).

2

observe the state of the limit order book and strategically choose an order submission strategy thatmaximizes expected profits given the information set at each period. Liquidity traders always tradebased on their exogenous liquidity needs, and only submit market orders to the exchange to ensureexecution. Third, if in the first period a trader had submitted a dark pool order and this order wasnot executed in the dark then the order returns to the exchange as a market order. These resembleImmediate-or-Cancel (IOC) orders that are common in dark pools. Fourth, the liquidation valueof the asset is made public and the trading game is over. Since our model can be represented bya sequential game of incomplete information, the equilibrium concept used is the Perfect BayesianEquilibrium. To the best of our knowledge, we are the first to model the competition between anexchange that is organized as a limit order book and a dark pool in the presence of asymmetricinformation.

To understand the effects of adding a dark pool alongside the exchange, we first discuss theequilibrium in the benchmark model where traders do not have access to the dark pool. We solvethe model backwards. In the last trading period, limit orders are not chosen since they will notbe executed. An informed trader always chooses a market order since it gives positive profits. Anuninformed trader obtains information about the state of the book and updates his beliefs aboutthe value of the asset. An uninformed trader selects a market order if he strongly believes that aninformed trader in the previous period had chosen an order of the same direction, which indicatesthat the value of the asset is favorable. Otherwise, an uninformed trader refrains from trading.In the first trading period, an informed trader chooses between a market order and a limit orderwhen he prefers immediacy to the price improvement provided by a limit order. An uninformedtrader selects a limit order instead of not trading when there is a high probability that his order willbe executed against the order of a liquidity trader instead of an informed trader. Due to adverseselection, uninformed traders do not want to trade with informed traders. If their limit order isexecuted against a market order of the opposite sign submitted by an informed trader then it revealsthat the value of the asset is disadvantageous. Even though the optimal strategies profiles are uniquein the first trading period, multiple equilibria may exist in the case that in the first trading periodinformed traders choose market orders and uninformed traders choose not to trade.

Adding a dark pool not only enlarges traders’ strategies set but also induces a substitutionbetween the strategic selection of market participation, trading venue and order type. Thus, evenif execution risk in the dark pool is high, informed traders tend to replace market orders by darkorders when they can take advantage of the price improvement. The price improvement is largestwhen the discount factor is high and when there is a small probability that the market moves againstthe trader. As the execution risk in the dark reduces, strategy profiles in which informed traderssubmit market orders cannot be an equilibria of the game, and informed traders decide to go to thedark. Despite of the fact that uninformed traders do not select to go to the dark pool due to adverseselection and price risk, adding a dark pool alongside an exchange affects the optimal submissionstrategies of the uninformed trader even in the first period. Uninformed traders may switch from

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no trade to trade in the exchange using limit orders. This is because the mere existence of thedark market offers the possibility to informed traders in the second period to migrate from the litto the dark, and consequently reduce the adverse selection which induces uninformed traders toparticipate in the market submitting limit orders. Despite the fact that optimal strategies profilesare unique in the first trading period, multiple equilibria may exist unless in the first trading periodinformed traders choose dark pool orders and uninformed traders choose limit orders.

Our paper is closely related to the theoretical analysis which analyzes the effects of the addinga dark pool alongside an exchange on market performance. This literature shows that the impactof the dark pool on price discovery is ambiguous. Thus, Zhu (2014), using a Glosten and Milgrom(1985) type model, finds that adding a dark pool alongside an exchange concentrates price-relevantinformation into the exchange and improves price discovery. Improved price discovery coincideswith reduced exchange liquidity. Ye (2011), using a Kyle (1985) type model, finds that a dark poolreduces price discovery and volatility. Note that these two models have a different market structure:Ye (2011) only allows the informed trader to select trading venue, while Zhu (2014) allows bothinformed and liquidity traders to select the venue to trade. Buti et al. (2011) model competitionbetween a fully transparent limit order book and an opaque dark pool. The introduction of thedark pool increases overall trading volume. They find that the market share of the dark pool ishigher when the depth of the limit order book is high, when the spread of the limit order bookis narrow, when the tick size is large, and when traders seek protection from price impact. Butiet al. (2013) show that reserve orders help traders compete for liquidity by reducing the frictionsgenerated by exposure costs. Large traders always benefit from using reserve orders, whereas smalltraders benefit only when the tick size is large. Buti et al. (2016) show that the welfare effects ofadding a dark pool are negative if the initial book is illiquid, while when book liquidity increases,large traders are better off and small traders are worst off. Competition between dealer marketsand other forms of exchange, such as passive crossing networks (similar to dark pool), has alsobeen analyzed by Hendershott and Mendelson (2000), Barclay et al. (2003) and Degryse et al.(2009). Hendershott and Mendelson (2000) find that a crossing network imposes positive liquidityexternalities and negative crowding externalities on each other. Also Barclay et al. (2003) findthat informed traders prefer to trade in electronic communication networks (ECN) than in marketsoperated by market makers. Degryse et al. (2009) show that the order flow dynamics depend onthe degree of transparency.

Our research is also related to two broader strands of literature that the previously mentionedpapers tackle: competition between integrated and segmented markets, and traders’ optimal ordersubmission strategies. On the one hand, early theoretical papers involving multiple trading venues(see, for instance, Pagano, 1989; and Chowdry and Nanda, 1991) show that there is a naturaltendency towards agglomeration since liquidity increases due to scale, and it is beneficial. This ten-dency may be offset by the presence of frictions, trading costs, informational barriers or regulatoryobstacles. Since market fragmentation is associated with the surge of venues with different degrees

4

of transparency, a part of this literature relates to fragmentation and market transparency2. Con-cerning the visibility of market quotes, for instance, Biais (1993) and Frutos and Manzano (2002)compare centralized and fragmented markets and show that the ability to observe price setters’quotes affects spreads and market participants’ welfare. In relation to the disclosure of post-tradeinformation, Madhavan (1995) shows that delaying disclosure benefits large traders who place mul-tiple trades. Frutos and Manzano (2005) in a two-stage trade model show that opaqueness increasescompetition among dealers to attract valuable order flow, leading to better prices for investors inthe first period, while the effect on market participants’ welfare in second period is ambiguous.

On the other hand, there is also a large literature that studies limit order markets and traders’optimal order submission strategies with models of asymmetric information . The early static modelsof limit order markets assume that informed traders only use market orders, while uninformedtraders or liquidity traders only use limit orders. Angel (1994), Easley and O’Hara (1991) andHarris (1998) model an informed investor’s order placement strategy in choosing between marketand limit orders. They argue that informed traders are more likely to use market orders and rarelyuse limit orders. Also, Glosten (1994), Rock (1996), Seppi (1997) and Biais et al. (2000) arguethat informed traders prefer market orders to profit from their private information. In contrast,Chakravarty and Holden (1995) and Kaniel and Liu (2006), who analyze informed traders’ choicebetween limit orders and market orders, find that informed traders may prefer limit orders sincethey may actually convey more information than market orders. Finally, Parlour (1998), Foucault(1999), Foucault, Kadan and Kandel (2005), Goettler et al. (2009), Rosu (2009), Van Achter (2008),Rosu (2012) introduce dynamic models that allow informed and uninformed traders to determinetheir optimal choice of order type in limit order markets. Empirical studies find that both informedand uninformed traders use a mixture of market orders and limit orders (see for example Biais et al.1995; Kavajecz and Odders-White 2004; Anand et al., 2005). Bloomfield, O’Hara and Saar (2005)conduct a laboratory experiment and find that informed traders use both market and limit orders.They use market orders earlier in the trading period to profit from their private information, andthen (as the prices get closer to true value) they use their private information to switch to limitorders to earn the bid-ask spread. Uninformed traders use limit orders early but then switch tomarket orders to meet their liquidity targets (they are liquidity traders). They also find thatinformed traders’ submission patterns are more sensitive to changes in market transparency. Webuild on this literature by developing a model in which rational traders can decide simultaneouslythe venue in which they want to trade and their optimal order submission strategy.

The paper is organized as follows. Section 2 presents the model. Section 3 analyzes the equilib-rium in benchmark model without dark pool. Section 4 presents the equilibrium in the full modelwhere traders have access to the dark pool. Section 5 concludes. Proofs are presented in the variousAppendices.

2See also Gomber et al. (2016) for a review of the consolidation versus fragmentation of markets.

5

2 Model

There is a market where a single risky asset with a liquidation value v is traded. The liquidationvalue of the asset v may take two values: v ∈

vH , vL

with equal probabilities. We denote by µ

the unconditional mean of the liquidation value of the asset. The asset may be traded in two venues:a lit exchange and a dark pool.

The lit exchange where the asset is traded is organized as a limit order book. We assume thatthe initial order book has only two prices on each side of the book A1

1, A21, B1

1 , B21 , and they satisfy

vL < B21 < B1

1 < A11 < A2

1 < vH . For simplicity we assume that the depth of the book at each bidand ask price is equal to 1 (l−11 = l11 = l−21 = l21 = 1). The limit order book follows the price andtime priority rule.3 At any point in time, all the information in the order book is available to allmarket participants (the limit order book is fully transparent).

We assume that the prices are situated on a grid and therefore the following relationships hold:

A11 = µ+ k1τ, A

21 = µ+ k2τ, v

H = µ+ k3τ

B11 = µ− k1τ, B2

1 = µ− k2τ, vL = µ− k3τ

with 1 ≤ k1 < k2 < k3, where k1, k2 and k3 are natural numbers and τ is the tick size, the minimumprice increment that traders are allowed to quote over the existing price.

The dark pool is completely opaque in the sense that an order submitted to the dark pool isnot observable to anyone but the trader who submitted it. We assume that the dark pool has anexecution probability θ ∈ [0, 1] that is exogenous and does not change in time.4 When attended inthe dark pool, at time t, the orders are executed at the price equal to the corresponding midpoint

of the lit exchange at time t,A1t +B1

t

2. If the order is not attended in the dark pool, then it returns

to the lit market at t+ 2.Traders are all risk neutral and want to trade one unit of the risky asset. They are of two

possible types: rational traders and liquidity traders. We assume that a rational trader arrives tothe market with probability λ, a liquidity trader arrives with probability η, and no trader arriveswith probability 1−λ−η (see Figure 1). Rational traders may be either informed (with probabilityπ) – they have perfect information about the liquidation value of the asset, or uninformed (withprobability 1 − π) – they know only the distribution of the liquidation value of the asset. Sincean informed trader is perfectly informed, he buys whenever he observes v = vH (henceforth IH),and he sells whenever he observes v = vL (henceforth IL). The uninformed trader knows only the

distribution of the liquidation value and buys with probability1

2(henceforth UB) and sells with

probability1

2(henceforth US).

3The order with the best price is executed first and among the orders with the same price the orders are executedin the order of their arrival.

4The dark-pool we model here is an agency-broker or exchanged-own type in which the price discovery does nottake place, the price being taken as the midpoint of the National Best Bid and Offer (in this model the lit exchange).

6

v

No Trader

1 − λ− η

LiquidityTrader

SellerΠN

SMO,1SMO

12

BuyerΠN

BMO,1BMO1

2η

RationalTrader

UninformedTrader

Seller

0

NT

SDO

ΠUSDO,1(NA)NA

ΠUSDO,1(A)A

ΠUSLO,1SLO

ΠUSMO,1

SMO12

Buyer

0

NT

BDO

ΠUBDO,1(NA)NA

ΠUBDO,1(A)A

ΠUBLO,1BLO

ΠUBMO,1

BMO

12

1 − π

InformedTrader

0

NT

DO

ΠIDO,1(NA)NA

ΠIDO,1(A)A

ΠILO,1LO

ΠIMO,1

MO

π

λ

Figure 1: Tree of Events

Rational traders always decide whether and how to trade based on their expectations conditionalon the information they posses at each time: information about the liquidation value of the assetand about the state of the book. Note that the information about the state of the book is availableto all traders, while information about the liquidation value is only available to the informed trader.Rational traders select whether to trade or not to trade (NT ). If they choose to trade, traders selectthe trading venue (public exchange or dark pool). If they choose the public exchange (lit market),traders can choose between placing a market order (henceforth MO) or a limit order (henceforthLO). Note that when a trader submits a LO that does not improve the price, the order goes to thequeue, and due to the time priority, it is not executed. Thus a limit order that does not improve theprice is equivalent to a NT order. To be able to differentiate the two cases, therefore, we assumethat the LO always improves the price by (at least) one tick. If the trader selects a dark pool,submits a dark pool order (henceforth DO). We assume for simplicity that, if they trade, the ordersize equals 1 and all rational traders have a discount factor δ ∈ [0, 1], that is common across tradersand periods.

7

t = 0

Nature draws v

v ∈vH , vL

• Traders may arrive

• Rational traderschoose their optimalsubmission strategiesbased on each trader’sinformation set

• Liquidity traderstrade according totheir liquidity needs

t = 1

• Traders may arrive

• Rational traderschoose their optimalsubmission strategiesbased on each trader’sinformation set

• Liquidity traderstrade according totheir liquidity needs

t = 2

If at t = 1 a DOwas submitted andnot attended theorder returns tothe lit marketas a MO

t = 3

Liquidation valueis made publicand the tradinggame is over

t = 4

1

Figure 2: Time line of the trading game when traders have access to the dark pool

Consequently, the possible strategies of a rational trader (both informed and uninformed) are

O = BMO, BLO, BDO, SMO, SLO, SDO, NT , (1)

where B denotes a buy order and S, denotes a sell order. Note however, that the informed traderIH chooses only among the buy orders and NT, while the informed trader IL chooses only amongthe sell orders and NT , so for the informed traders, the direction to trade is endogenous and itdepends on their private information. Liquidity traders trade for liquidity or hedging needs and

they might either buy or sell with probability1

2. They can trade only in the lit market and to

ensure execution they always submits market orders.The sequence of events described also in Figure (2) is as follows:Date t=0: The liquidation value of the asset v is realized.Dates t=1, 2: In each date, a trader may arrive to the market. An informed trader observes

the liquidation value of the asset. The state of the limit order book at the beginning of each dateis public information. A rational trader chooses an order submission strategy in O that maximizeshis expected profits given the information set at each date, It. Liquidity traders always trade basedon their exogenous liquidity needs. All traders may trade one unit of the asset.

Date t=3: If at t = 1 a rational trader submitted a dark pool order and this order was notattended, then the order returns to the lit market as a market order.

Date t=4: The liquidation value of the asset is made public and the trading game is over.

2.1 Strategies’ Payoffs

We denote the profits of a particular order as ΠRO,t, where the superscript R denotes that the order

comes from a rational trader (either informed or uninformed), the subscript O is the order typefrom (1), and the subscript t is the date when the order is submitted. The characteristics of eachtype of order and their corresponding expected profits are explained below. The expected profits ofall traders conditional on their information set in all the possible cases are described in the Internet

8

Appendix I. We assume that there are no transaction costs.

• Market order (MO): Market orders are executed immediately at the given best availableask/bid prices. The expected profit of a buy market order at date t is

E(ΠRBMO,t|It

)= E (v|It)−A1

t ,

and the expected profit of a sell market order is

E(ΠRSMO,t|It

)= B1

t − E (v|It) .

• Limit orders (LO): Improving limit orders may be submitted only by rational traders andmay be executed in the next period. Therefore the expected profit from a limit order isdiscounted by δ. Note that a limit order is executed only when it improves the current marketconditions (i.e., price) and a market order of the opposite sign hits the limit order in thefollowing period. Thus, limit orders provide better prices than market orders, but exhibitexecution risk because of the uncertainty of execution. Note also that when placing the limitorder although the trader could improve the price by more than one tick, he never finds itoptimal to do so, because if he does it, he reduces his profit. The expected profit of a buylimit order at date t is

E(ΠRBLO,t|It

)= pRBLO,t (It) δ(E (v|It)−B1

t − τ),

and of a sell limit order at date t is

E(ΠRSLO,t|It

)= pRSLO,t (It) δ(A

1t − τ − E (v|It)).

• Dark orders (DO): As explained above we have that with probability θ an order submitted tothe dark pool is attended (executed) and with probability (1 − θ) is not attended. Since nonew trader arrives in the market at t = 3, 4 an order that returns to the market from darkpool will be always a MO, as the probability of execution of a LO at t = 3 is equal to 0.Therefore, the expected profit of a buy dark order submitted at time t is:

E(ΠRBDO,t|It

)= θ

(E (v|It)−

A1t +B1

t

2

)+ (1− θ)δ2E

(ΠRBMO,t+2|It

)

and for a sell dark order submitted at time t, the expected profit equals

E(ΠRSDO,t|It

)= θ

(A1t +B1

t

2− E (v|It)

)+ (1− θ)δ2E

(ΠRSMO,t+2|It

).

• No trade (NT ) means that the trader refrains from participating in the market which leads

9

t = 0

Nature draws v

v ∈vH , vL

• Traders may arrive

• Rational traderschoose their optimalsubmission strategiesbased on each trader’sinformation set

• Liquidity traderstrade according totheir liquidity needs

t = 1

• Traders may arrive

• Rational traderschoose their optimalsubmission strategiesbased on each trader’sinformation set

• Liquidity traderstrade according totheir liquidity needs

t = 2

Liquidation valueis made publicand the tradinggame is over

t = 3

1

Figure 3: Time line of the trading game when traders do not have access to the dark pool

to zero profit at time t: E(

ΠRNT,t|It

)= 0.

If profits are strictly positive, then a trader can choose between a MO or LO, and if available,or DO. Otherwise, he chooses not to trade (NT ). In case of equality of profits, we assume that aMO dominates LO and DO; and a LO dominates DO.

Our model can be represented by a sequential game of incomplete information. The equilibriumdefinition used is as follows.

Definition 1 A Perfect Bayesian Equilibrium (henceforth PBE) of the trading game is a strategyprofile for all traders and belief system about other traders types at all information sets such that:

i) Sequential Rationality: Given the belief system, at each information set each trader’s strategyspecifies an optimal order that maximizes traders’ expected profits given his beliefs and the strategiesof other traders.

ii) Consistent beliefs: Given the strategy profile, the beliefs are consistent with Bayes rule (whenappropriate).

We focus on a symmetric Perfect Bayesian Equilibrium in pure strategies.

3 Equilibrium in the benchmark model without dark pool

We first consider the benchmark model without a dark pool where traders can trade only in thelit market and can use only the following orders: OND = BMO, BLO, SMO, SLO, NT. Thesequence of events can be seen in Figure (3). The difference with respect to the timeline in Figure(2) is that the liquidation value in case there is no dark pool is revealed at t = 3.

We solve the game backwards, starting at t = 2, since at t = 3 traders do not take any decision.The expected profits for an informed buyer and seller at t = 2 are summarized in Table 1 and itcan be easily seen that the expected profits of the informed traders from MO are always strictlypositive.

10

IH IL

BMO BLO NT SMO SLO NT

(A11, B

11) (k3 − k1) τ 0 0 (k3 − k1) τ 0 0

(A21, B

11) (k3 − k2) τ 0 0 (k3 − k1) τ 0 0

(A11, B

11 + τ) (k3 − k1) τ 0 0 (k3 − k1 + 1) τ 0 0

(A11, B

21) (k3 − k1) τ 0 0 (k3 − k2) τ 0 0

(A11 − τ,B1

1) (k3 − k1 + 1) τ 0 0 (k3 − k1) τ 0 0

Table 1: Expected profits of an informed buyer (IH) and an informed seller (IL) at t = 2

UB US

BMO BLO NT SMO SLO NT

(A11, B

11) −k1τ 0 0 −k1τ 0 0

(A21, B

11) (Xk3 − k2) τ 0 0 − (k1 +Xk3) τ 0 0

(A11, B

11 + τ) (Y k3 − k1) τ 0 0 − (k1 − 1 + Y k3) τ 0 0

(A11, B

21) − (Xk3 + k1) τ 0 0 (Xk3 − k2) τ 0 0

(A11 − τ,B1

1) − (Y k3 + k1 − 1) τ 0 0 (Y k3 − k1) τ 0 0

Table 2: Expected profits of an uninformed buyer (UB) and an uninformed seller (US) at t = 2

At t = 2 the expected profit of each strategy depends on the state of the book (which on itsturn depends on the chosen strategy at t = 1). Uniformed traders form at t = 2 beliefs about thestrategies and type of player in t = 1. Thus, we define the uninformed traders’ belief at t = 2 aboutthe probability that the MO (observed in the book) was submitted by an informed trader as

X =λπΩ1

η + λπΩ1 + λ (1− π) Σ1, (2)

where Ω1 is the probability that an informed trader at t = 1 submits a MO, Σ1 is the probabilitythat an uninformed buyer at t = 1 submits a MO and η is the probability that the MO comes froma liquidity trader. Similarly, we define the uninformed traders’ belief at t = 2 about the probabilitythat the LO (observed in the book) was submitted by an informed trader as

Y =πΩ2

πΩ2 + (1− π) Σ2, (3)

where Ω2 is the probability that an informed trader at t = 1 submits a LO, and Σ2 is probabilitythat an uninformed buyer at t = 1 submits a LO.

Table 2 presents the expected profits of an uninformed buyer and seller at t = 2 , respectively.Note that the expected profits of the uninformed buyer are strictly negative when the book is(A1

1, B11), (A1

1, B21) or (A1

1 − τ,B11), and the expected profits of the uninformed seller are strictly

negative when (A11, B

11), (A2

1, B11) or (A1

1, B11 + τ).

11

By comparing the expected profits of rational traders in t = 2 we obtain:

Lemma 1 In equilibrium the following results hold:

• at t = 2 an informed trader always chooses a MO.

• at t = 2 an uninformed trader may choose either MO or NT , but never chooses a LO.Moreover, the optimal strategy for an uninformed trader is as presented in Table 3:

State of the book UB US

(A11, B

11) NT NT

(A21, B

11)

MO if X >k2k3

NT if X ≤ k2k3

NT

(A11, B

11 + τ)

MO if Y >k1k3

NT if Y ≤ k1k3

NT

(A11, B

21) NT

MO if X >k2k3

NT if X ≤ k2k3

(A11 − τ,B1

1) NT

MO if Y >k1k3

NT if Y ≤ k1k3

Table 3: Optimal trading strategies of an uniformed buyer (UB) and seller (US) at t = 2

Informed traders at t = 2 always choose aMO since it generates positive profits, while profits ofa LO or NT are always null. In contrast, the choice of uninformed traders at t = 2 depends on thestate of the book since it reveals information to them. Uninformed traders never choose a LO sincethe probability of execution is 0 because no new orders arrive at t = 3, and expected profits are null.Hence, the uninformed trader choose between MO and NT . Without loss of generality, we studynext the case of an uninformed buyer.5 When the state of the book conveys no information to theuninformed buyer, i.e., (A1

1, B11), then the optimal choice is NT since the expected profits of MO

are −k1µ < 0. When the book reveals that at t = 1 there has been a BMO or BLO (i.e. (A21, B

11)

or (A11, B

11 + τ)), the uninformed buyer at t = 2 forms the correct beliefs about the probability

that the buy MO or LO was submitted by an informed trader at t = 1, X and Y , respectively. In5The argument for an uninformed seller follows since there exists the following symmetry. When the state of the

book is (A11, B

11), uninformed sellers and buyers always make the same choice. When the state of the book is (A2

1, B11),

uninformed sellers choose the same as uninformed buyers when the state of the book is (A11, B

21). When the state

of the book is (A11, B

11 + τ), uninformed sellers choose the same as uninformed buyers when the state of the book is

(A11 − τ,B1

1).

12

these two states, a BMO is selected by the uninformed trader at t = 2, when the belief that theorder came from an informed trader at t = 1 is sufficiently strong that it makes the correspondingexpected value of submitting a MO positive. When the state of the book is either (A1

1, B21) or

(A11 − τ,B1

1), which reveals that the trader at t = 1 submitted a SMO or SLO respectively, theuninformed trader at t = 2 refrains from trading since expected profits are negative.

Expected profits of an informed and an uniformed trader at t = 1 are included in Table 4 andTable 5. By comparing the profits for each of the possible strategies, we find the optimal strategieschosen by informed and uninformed traders.

IH IL Expected ProfitBMO SO (k3 − k1) τBLO SLO

δη

2(k3 + k1 − 1) τ

NT NT 0

Table 4: Expected profits of an informed buyer (IH) and seller (IL) at t = 1

UB US Expected ProfitBMO SMO −k1τBLO SLO

δ

2((λπ + η) (k1 − 1)− λπk3) τ

NT NT 0

Table 5: Expected profits of an uninformed buyer (UB) and seller (US) at t = 1

Lemma 2 In equilibrium the following results hold:

• at t = 1 an informed trader never chooses NT .

• at t = 1 an uniformed trader never chooses a MO.

For an informed trader at t = 1, the strategy NT is always dominated and, hence, never chosen.For an uninformed trader at t = 1 never chooses a MO since −k1τ < 0, and it is always dominatedby the NT strategy. Further, the previous lemma implies that the candidate strategy profiles att = 1 that can be sustained as a PBE are:

(BMO, SMO, BLO, SLO), (BMO, SMO, NT, NT ),

(BLO, SLO, BLO, BLO), (BLO, SLO, NT, NT ),

13

where the two first components correspond to strategies of informed traders at t = 1 (IH and IL,respectively) and the two last components correspond to strategies of uninformed traders at t = 1

(UB and US, respectively).We are now in a position to characterize the PBE of this trading game where traders do not

have access to a dark pool.

Proposition 1 If k1 > 1, then a PBE of the game is as follows:

• (BMO,SMO,BLO,SLO) is the optimal strategy profile at t = 1 if

Conditionsk3 − k1 ≥ δ η2 (k3 + k1 − 1) and(λπ + η) (k1 − 1)− λπk3 > 0.

The beliefs of an uninformed trader at t = 2 are: X =λπ

η + λπand Y = 0. The optimal

strategy of an informed trader at t = 2 is to choose MO for all possible states of the book, andthe optimal strategy of an uninformed trader at t = 2 is described in Table A.1 of AppendixA.

• (BMO,SMO,NT,NT ) is the optimal strategy profile at t = 1 if

Conditionsk3 − k1 ≥ δ η2 (k3 + k1 − 1) and0 ≥ (λπ + η) (k1 − 1)− λπk3.

The beliefs of an uninformed trader at t = 2 are: X =λπ

η + λπand Y = p ∈ [0, 1]. The

optimal strategy of an informed trader at t = 2 is to choose MO for all possible states of thebook, and the optimal strategy of an uninformed trader at t = 2 is described in Table A.2 ofAppendix A.

• (BLO,SLO,BLO,BLO) is the optimal strategy profile at t = 1 if

Conditionsδ η2 (k3 + k1 − 1) > k3 − k1 and(λπ + η) (k1 − 1)− λπk3 > 0.

The beliefs of an uninformed trader at t = 2 are: X = 0 and Y = π. The optimal strategyof an informed trader at t = 2 is to choose MO for all possible states of the book, and theoptimal strategy of an uninformed trader at t = 2 is described in Table A.3 of Appendix A.

14

• (BLO,SLO,NT,NT ) is the optimal strategy profile at t = 1 if

Conditionsδ η2 (k3 + k1 − 1) > k3 − k1 and

0 ≥ η(k1 − 1)− λπ (k3 − (k1 − 1)) .

The beliefs of an uninformed trader at t = 2 are: X = 0 and Y = 1. The optimal strategyof an informed trader at t = 2 is to choose MO for all possible states of the book, and theoptimal strategy of an uninformed trader at t = 2 is described in Table A.4 of Appendix A.6

Proposition 1 shows that strategy (BMO,SMO,BLO,SLO) is chosen when the discount factor,δ, is small, meaning that the informed trader finds the LO less attractive since he prefers immediacy.In addition, the uninformed trader prefers a LO instead of NT when there is a higher probabilitythat his order gets executed due to the arrival of a liquidity trader (η) instead of an informed trader(λπ). Uninformed traders do not want to trade with informed traders: if the LO is executed dueto a MO of the opposite sign submitted by informed trader at t = 2 then it reveals that the valueof the asset is low (if a BLO has been submitted) and high (if a SLO has been submitted). Whenthe probability that the LO order is executed due to the arrival of an informed trader is high (λπis high in relation to η) then the uninformed trader prefers not to trade. In addition, if k3 is largein relation to k1 then the equilibrium strategy is (BMO,SMO,NT,NT ). In contrast, when δ ishigh and k3 − k1 is small then the informed trader prefers a LO since it yields higher expectedprofits than a MO (since the price improvement combined with the expected value of the assetoutweighs the execution risk). In addition, if the probability of arrival of a liquidity trader (η) ishigh in relation to the probability of arrival of an informed trader (λπ) then the optimal strategy isthe pooling equilibrium (BLO,SLO,BLO,BLO). If all the other conditions, the optimal strategyis (BLO,SLO,NT,NT ).

Figure 4, Left Panel shows the optimal strategy of a trader that arrives at t = 1 when there isno access to the dark pool as a function of the probability that a liquidity trader arrives, η, and ofthe probability that there is an informed trader, π. Informed traders choose between MO and LO.If the probability that a liquidity trader arrives is not too large then an informed trader at t = 1

chooses a MO, otherwise he chooses a LO. Uninformed traders select between not participatingin the market and participating with limit orders. Due to adverse selection, limit orders are notoptimal when the probability that an informed trader arrives in the next trading period is high inrelation to the probability that trade is against a liquidity trader.

Figure 4, Right Panel shows how the optimal strategy of a trader that arrives at t = 1 changeswith the probability that a liquidity trader arrives, η, and the discount factor, δ. Informed traderschoose LO if the discount factor is high and the probability of execution is high (η is high). Oth-erwise, they submit MO since execution is guaranteed. Uninformed traders submit LO if the

6Proposition 3 in Appendix B characterizes the PBE when k1 = 1.

15

Figure 4: Optimal strategies at t = 1 without dark pool. Parameters values: k1 = 2, k2 = 3, k3 = 4,λ = 0.5. Left Panel δ = 0.9, Right Panel π = 0.6.

probability of execution is high (η is high). Otherwise, they refrain from trading (NT ).

4 Equilibrium in a model with dark pool

We next consider the model with a dark pool where traders have access to the orders in (1). Similarlyto the benchmark model, we solve the model backward, starting at t = 2.

Let use denote by

Ω0 = probability that an informed trader at t = 1 selects NT ,

Ω1 = probability that an informed trader at t = 1 selects a MO,

Ω2 = probability that an informed trader at t = 1 selects a LO,

Ω3 = probability that an informed trader at t = 1 selects a DO,

Σ0 = probability that an uninformed trader at t = 1 selects NT ,

Σ1 = probability that an uninformed trader at t = 1 selects a MO,

Σ2 = probability that an uninformed trader at t = 1 selects a LO

Σ3 = probability that an uninformed trader at t = 1 selects a DO.

16

and note that3∑j=0

Ωj = 1, and3∑j=0

(Σj) = 1.

Like in Section 3, at t = 2, the expected profit of each strategy depends on the state of the book.Additionally, uninformed traders form beliefs about the strategies that have been chosen at t = 1.Let X and Y be defined as in (2) and (3), respectively. Let Z denote the uninformed trader’s beliefat t = 2 about the probability that a DO was submitted by an informed trader as

Z =πΩ3

πΩ3 + (1− π) Σ3, (4)

where Ω3 is the probability that an informed trader at t = 1 selects DO, and Σ3 the probabilitythat an uninformed trader at t = 1 selects DO.

The expected profits for an informed buyer and seller at t = 2 are summarized in Table 6 andTable 7, respectively.

IH BMO BDO BLO NT

(A11, B

11) (k3 − k1) τ θk3τ PIδ (k1 + k3 − 1) τ 0

(A21, B

11) (k3 − k2) τ θ

(k3 − k2−k1

2

)τ 0 0

(A11, B

11 + τ) (k3 − k1) τ θ

(k3 − 1

2

)τ 0 0

(A11, B

21) (k3 − k1) τ θ

(k3 + k2−k1

2

)τ 0 0

(A11 − τ,B1

1) (k3 − k1 + 1) τ θ(k3 + 1

2

)τ 0 0

Table 6: Expected profits of an informed buyer (IH) at t = 2

IL SMO SDO SLO NT

(A11, B

11) (k3 − k1)τ θk3τ PIδ(k1 + k3 − 1)τ 0

(A21, B

11) (k3 − k1)τ θ

(k2−k1

2 + k3

)τ 0 0

(A11, B

11 + τ) (k3 − k1 + 1)τ θ

(k3 + 1

2

)τ 0 0

(A11, B

21) (k3 − k2)τ θ

(k3 + k1−k2

2

)τ 0 0

(A11 − τ,B1

1) (k3 − k1)τ θ(k3 − 1

2

)τ 0 0

Table 7: Expected profits of an informed seller (IL) at t = 2

17

where PI is the probability of execution of a limit order placed by an informed trader at t = 1, andequals

PI = pIHBLO,2 (O1 = ∅) = pILSLO,2 (O1 = ∅) =λ(1− θ)1−π2 Σ3

1− λ− η + λ (πΩ3 + (1− π)(Σ0 + Σ3)).

Similarly, the expected profits of the uninformed buyer and seller at t = 2 are summarized inTable 8 and Table 9, respectively.

UB BMO BDP BLO NT

(A11, B

11) −k1τ 0 PUδ (k1 − k3Z − 1) τ 0

(A21, B

11) (Xk3 − k2) τ θ

(Xk3 − k2−k1

2

)τ 0 0

(A11, B

11 + τ) (Y k3 − k1) τ θ

(Y k3 − 1

2

)τ 0 0

(A11, B

21) − (Xk3 + k1) τ −θ

(Xk3 − k2−k1

2

)τ 0 0

(A11 − τ,B1

1) − (Y k3 + k1 − 1) τ −θ(Y k3 − 1

2

)τ 0 0

Table 8: Expected profits of an uninformed buyer (UB) at t = 2

US SMO SDO SLO NT

(A11, B

11) −k1τ 0 PUδτ(k1 − 1− Zk3) 0

(A21, B

11) − (k1 +Xk3) τ −θ

(Xk3 − k2−k1

2

)τ 0 0

(A11, B

11 + τ) − (k1 − 1 + Y k3) τ −θ

(Y k3 − 1

2

)τ 0 0

(A11, B

21) (Xk3 − k2) τ θ

(Xk3 − k2−k1

2

)τ 0 0

(A11 − τ,B1

1) (Y k3 − k1) τ θ(Y k3 − 1

2

)τ 0 0

Table 9: Expected profits of an uninformed seller (US) at t = 2

where PU is the probability of execution of a limit order placed by an uninformed trader at t = 1,and equals

PU = pUBBLO,2 (O1 = ∅) = pUSSLO,2 (O1 = ∅) =12λ(1− θ)(πΩ3 + (1− π)Σ3)

1− λ− η + λ(πΩ3 + (1− π)(Σ0 + Σ3)).

At t = 1 the expected profits of an informed IH and an uniformed buyer UB are summarizedin Table 10 and Table 11, respectively. Notice that due to the symmetry of the game, the expectedprofits of the informed IL trader and uninformed seller US are similar.

By comparing the profits of each of the possible strategies for each type of trader at t = 2,Lemma 3 states the strategies that are dominated and hence never chosen by a rational player.

Lemma 3 In equilibrium the following results hold:

18

IH Expected profits at t = 1

BMO (k3 − k1) τBLO

ηδ

2(k3 + k1 − 1) τ

BDOθk3τ + (1− θ)δ2τ

(λ (1−π)

2 IU,B,O1=∅BLO,2 +

+(k3 − k1)− (k2 − k1)(λπII,L,O1=∅

SMO,2 + η2

))

NT 0

Table 10: Expected profits of an informed buyer (IH) at t = 1

UB Expected Profits at t = 1

BMO −k1τBLO

δτ

2

(η(k1 − 1)− λπII,H,O1=SLO

BMO,2 (k3 − (k1 − 1))

BDO(1− θ)δ2τ

(λ2 (πII,H,O1=∅

BLO,2 + (1− π)IU,B,O1=∅BLO,2 )−

−(k2 − k1)(η2 + λπ2 I

I,L,O1=∅SMO,2 )− k1

)

NT 0

Table 11: Expected profits of an uninformed buyer (UB) at t = 1

• at t = 2 an informed trader never chooses a LO or NT . The optimal strategy depends on thevalue of the parameters as explained in Table B.1 in the Appendix.

• at t = 2 an uninformed trader at t = 2 never chooses a LO. The optimal strategy depends onthe value of the parameters as explained in Table B.2 in the Appendix.

An informed trader at t = 2 never chooses NT since the strategy is always dominated by aMO. In addition, it is never optimal for an informed trader at t = 2 to choose a LO since it isnever executed: a) if the book has changed then no MO can arrive at t = 3; b) if the book has notchanged then a LO can only be executed if an uninformed seller at t = 1 chooses a DO. However,as it can be seen in Lemma 4 below this is never optimal in equilibrium. Similarly, in equilibriuman uninformed trader at t = 2 never chooses a LO since: a) if the book has changed then no MO

arrives at t = 3 and hence it has zero probability of execution; b) if the book has not changed thenthe LO can only be executed if a trader at t = 1 chooses a DO. By Lemma 4 below, we see that itis never optimal for an uninformed trader at t = 1 to choose a DO. Hence the trader at t = 2 formsthe correct beliefs that if the LO is executed at t = 3 it must have come from an informed traderat t = 1 with probability 1. But this information reveals to an uninformed buyer (seller) that thevalue of the asset must be low (high) and, hence, expected profits of a LO are negative.

By comparing the profits for each type of orders at t = 1, we find the optimal strategies chosenby informed and uninformed traders.

19

Lemma 4 In equilibrium the following results hold:

• at t = 1 an informed trader never chooses NT .

• at t = 1 an uniformed trader never chooses a MO or a DO.

For an informed trader at t = 1, the strategy NT is always dominated by a MO and, hence,never chosen. For an uninformed trader at t = 1 never chooses a MO since the expected profitsare −k1τ < 0. The reason why an uninformed trader at t = 1 never chooses a DO is as follows.Without loss of generality, let us focus on an uninformed buyer (the argument for the seller followsby symmetry). As seen in Appendix A, the profits of an uninformed buyer at t = 1 that choosesa BDO satisfy E

(ΠUBBDO,1

)≤ (1− θ)δ2τ

(λ2 − k1

)≤ 0, and hence the BDO strategy is dominated

by NT . In other words, an uninformed chooses not to go to the dark market since: a) if the ordergets executed at t = 1 then its expected profits are zero; b) if the order does not get executed att = 1 then it returns to the market as a MO at t = 3 and the price risk is so high that its expectedprofits are negative. Hence, a DO leads to negative profits for an uninformed trader at t = 1.

Further, using Lemma 4 the candidate strategy profiles at t = 1 that can be sustained as a PBEare:

(BMO, SMO, BLO, SLO), (BMO, SMO, NT, NT ), (BLO, SLO, BLO, BLO),

(BLO, SLO, NT, NT ), (BDO, SDO, BLO, SLO), (BDO, SDO, NT, NT ),

where the two first components correspond to strategies of informed traders at t = 1 (IH and IL,respectively) and the two last components correspond to strategies of uninformed traders at t = 1

(UB and US, respectively).

The PBE of the trading game where rational traders have access to a dark pool is characterizedas follows.

Proposition 2 If k1 > 1, then a PBE of the game is as follows:

• (BMO,SMO,BLO,SLO) is the optimal strategy profile at t = 1 if

Conditions

θ ≤ k3−k1k3

k3 − k1 ≥ η2δ (k3 + k1 − 1) ,

k3 − k1 ≥ θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(λπ + η

2

)), and

(λπ + η) (k1 − 1)− λπk3 > 0.

The beliefs of an uninformed trader at t = 2 are: X =λπ

η + λπ, Y = 0 and Z = q ∈ [0, 1]. The

optimal strategy of an uninformed and an informed trader at t = 2 are described in Tables B.5and B.8 of Appendix B, respectively.

20

• (BMO,SMO,NT,NT ) is the optimal strategy profile at t = 1 if

Conditions

θ ≤ k3−k1k3

k3 − k1 ≥ η2δ (k3 + k1 − 1) ,

k3 − k1 ≥ θk3 + (1− θ)δ2((k3 − k1)− (k2 − k1)

(λπ + η

2

)), and

0 ≥ (λπ + η) (k1 − 1)− λπk3.

The beliefs of an uninformed trader at t = 2 are: X =λπ

η + λπ, Y = p ∈ [0, 1] and Z = q ∈

[0, 1]. The optimal strategy of an uninformed and an informed trader at t = 2 are describedin Tables B.9 and B.10 of Appendix B, respectively.

• (BLO,SLO,BLO,BLO) is the optimal strategy profile at t = 1 if

Conditions

θ ≤ k3−k1k3

η2δ (k3 + k1 − 1) ≥ θk3 + (1− θ)δ2

((k3 − k1)− (k2 − k1)

(λπ + η

2

)),

η2δ (k3 + k1 − 1) > k3 − k1, and(λπ + η) (k1 − 1)− λπk3 > 0.

k3−k1k3

< θ ≤ k3−k1+1k3+

12

η2δ (k3 + k1 − 1) ≥ θk3 + (1− θ)δ2

((k3 − k1)− (k2 − k1)

(η2

))and

(λπ + η) (k1 − 1)− λπk3 > 0.k3−k1+1k3+

12

< θ η2δ (k3 + k1 − 1) ≥ θk3τ + (1− θ)δ2τ

((k3 − k1)− (k2 − k1)

(η2

)).

The beliefs of an uninformed trader at t = 2 are: X = 0, Y = π and Z = q ∈ [0, 1]. Theoptimal strategy of an uninformed and an informed trader at t = 2 are described in TablesB.11 and B.12 of Appendix B, respectively.

• (BLO,SLO,NT,NT ) is the optimal strategy profile of a trader at t = 1 if

Conditions

θ ≤ k3−k1k3

η2δ (k3 + k1 − 1) ≥ θk3 + (1− θ)δ2

((k3 − k1)− (k2 − k1)

(λπ + η

2

)),

η2δ (k3 + k1 − 1) > (k3 − k1) , and

0 ≥ (λπ + η) (k1 − 1)− λπk3.k3−k1k3

< θ ≤ k3−k1+1k3+

12

η2δ (k3 + k1 − 1) ≥ θk3 + (1− θ)δ2

((k3 − k1)− (k2 − k1)

(η2

))and

0 ≥ (λπ + η) (k1 − 1)− λπk3.

The beliefs of an uninformed trader at t = 2 are: X = 0, Y = 1 and Z = q ∈ [0, 1]. Theoptimal strategy of an uninformed and an informed trader at t = 2 are described in TablesB.13 and B.14 of Appendix B, respectively.

21

• (BDO,SDO,BLO, SLO) is the optimal strategy profile of a trader at t = 1 if

Conditions

θ ≤ k3−k1k3

θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(λπ + η

2

))> k3 − k1,

θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(λπ + η

2

))> η

2δ (k3 + k1 − 1) , and(λπ + η)(k1 − 1)− λπk3 > 0.

k3−k1k3

< θ ≤ k3−k1+1k3+

12

θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(η2

))> η

2δ (k3 + k1 − 1) and((λπ + η)(k1 − 1)− λπk3) > 0.

k3−k1+1k3+

12

< θ θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(η2

))> η

2δ (k3 + k1 − 1) .

The beliefs of an uninformed trader at t = 2 are: X = 0, Y = 0 and Z = 1. The optimalstrategy of an uninformed and an informed trader at t = 2 are described in Tables B.15 andB.16 of Appendix B, respectively.

• (BDO,SDO,NT,NT ) is the optimal strategy profile of a trader at t = 1 if

Conditions

θ ≤ k3−k1k3

θk3 + (1− θ)δ2((k3 − k1)− (k2 − k1)

(λπ + η

2

))> (k3 − k1) ,

θk3 + (1− θ)δ2((k3 − k1)− (k2 − k1)

(λπ + η

2

))> η

2δ (k3 + k1 − 1) ,

0 ≥ (λπ + η) (k1 − 1)− λπk3.k3−k1k3

< θ ≤ k3−k1+1k3+

12

θk3 + (1− θ)δ2((k3 − k1)− (k2 − k1)

(η2

))> η

2δ (k3 + k1 − 1) ,

0 ≥ (λπ + η) (k1 − 1)− λπk3.

The beliefs of an uninformed trader at t = 2 are: X = 0, Y = p ∈ [0, 1] and Z = 1. Theoptimal strategy of an uninformed and an informed trader at t = 2 are described in TablesB.17 and B.18 of Appendix B, respectively.7

Proposition 2 shows that having access to a dark pool changes the conditions that optimalstrategy profiles at t = 1 are required to satisfy. These changes are the following. First, theconditions for a PBE to exist depend on how the execution risk in the dark pool, θ, compares totwo cutoffs, which depend on the prices of the limit order book and the possible realizations of theliquidation value. We notice that the strategy profiles where an informed trader at t = 1 submitsa MO cannot be part of a PBE if k3−k1

k3< θ. In addition, strategy profiles where an uninformed

trader at t = 1 chooses NT in the market cannot be part of an equilibria if k3−k1k3

< k3−k1+1k3+

12

< θ.Second, even though the optimal strategies profiles are unique in the first trading period for givenparameter values, multiple equilibria may exist in all the optimal strategy profiles at t = 1 exceptwhen (BDO,SDO,BLO, SLO) is optimal. This is because when (BDO,SDO,BLO, SLO) isoptimal at t = 1, the beliefs of uninformed trader at t = 2 are uniquely determined. In all the otherequilibria, there is a continuum of uninformed trader’s beliefs at t = 2 that can sustain the PBE

7Proposition 4 in Appendix B characterizes the PBE when k1 = 1.

22

Figure 5: Optimal strategies at t = 1 with dark pool as a function of the probability of a liquiditytrader to arrive, η, and the probability of an informed trader to arrive, π. Parameters values:k1 = 2, k2 = 3, k3 = 4, λ = 0.5, δ = 0.9. Values for the execution probability in the dark, θ, arespecified above each graph.

and hence there are multiple equilibria.Figure 5 shows the optimal strategy of a trader that arrives at t = 1 when a dark pool is available

for different levels of execution risk in the dark pool (lowest in the upper left graph, highest in thelower right graph). The horizontal axis represents the probability that a liquidity trader arrives, η,and the vertical axis the probability that there is an informed trader, π. The top-left panel showsthat when the execution risk is high the optimal strategy of the uninformed trader does not changein relation to when traders do not have access to a dark pool. However, the optimal strategy ofthe informed trader is different if a dark pool is available. Informed traders replace MO in thelit market by DO when large price improvements are expected (i.e. when the probability that aliquidity traders arrive is low since they submit market orders). As the probability of execution inthe dark increases, informed traders gradually replace MO by DO, and as θ is even larger they alsoreplace LO by DO. Adding a dark pool alongside the exchange also changes the optimal strategy of

23

Figure 6: Optimal strategies at t = 1 with dark pool as a function of the discount factor δ.Parameters values: k1 = 2, k2 = 3, k3 = 4, λ = 0.5, π = 0.6. Values for the execution probabilityin the dark, θ, are specified above each graph.

an uninformed trader at t = 1 even if an uninformed trader never goes to the dark due to price riskand adverse selection. When the execution risk in the dark is low (lower right graph), uninformedtraders may switch from NT to trade in the lit market using LO. This is because the mere existenceof the dark market offers the possibility to informed traders in the second period to migrate fromthe lit to the dark, and consequently reduce the adverse selection which induces uninformed tradersto participate in the market submitting limit orders. In Figure 6, the horizontal axis represents theprobability that a liquidity trader arrives, η, and the vertical axis the discount factor, δ. Informedtraders find it optimal to go to the dark even if the execution risk is high (upper left graph) if thediscount factor is high and if the probability that the market moves against the trader is small (i.e.there is a small probability that a liquidity trader). As the probability of execution in the darkincreases (upper right graph and lower left graph), MO and LO are gradually substituted by DO.As stated previously, when the execution risk of the dark pool is very low (lower right graph) thenuninformed traders that were not participating in the market switch to limit orders in the lit market

24

due to the reduction of adverse selection.We analyze which PBE exists in a framework where there is almost no adverse selection since

it is unlikely that an informed traders would trade.

Corollary 1 Suppose that k1 > 1 and that π is low enough. At t = 1, the strategy profiles that are nolonger part of the PBE are: (BMO,SMO,NT,NT ), (BLO,SLO,NT,NT ), and (BDO,SDO,NT,NT ).

The intuition of this corollary is as follows. When the probability that an informed trader arrivesis low enough, uninformed traders expect higher profits by submitting a LO than NT since it ismore likely that they trade with liquidity traders. In contrast, when π is high it is more likely thatan informed trader arrives which makes a LO less attractive. This is because if the LO is executeddue to a MO of the opposite sign submitted by an informed trader then it reveals that the valueof the asset is low (if a BLO was submitted) and high (if a SLO was submitted). Hence, due toadverse selection, LO are less attractive due to the fact that the probability of obtaining negativeprofits is higher.

5 Conclusions

In this paper we study market participants’ strategic choice of trading venue and order type whentraders have access to a dark pool and to a traditional exchange (lit market) that is organized asa limit order book. We model the exchange as a fully transparent limit order book, while the darkpool is an opaque market where orders are executed at the midpoint between the best bid andthe ask prices of the exchange. We build a multi-period model that allows us to understand theinteraction of the limit order book with the dark pool in the presence of asymmetric information.We characterize the Perfect Bayesian Equilibria of the trading game.

We find that adding a dark pool alongside an exchange may shift the optimal strategies of eachtype of rational trader. In the first period, an uninformed trader may switch from no trade tosubmitting a limit order in the exchange, while the informed trader’s strategy diverts from theexchange to the dark pool. This occurs if the execution risk in the dark is low and the best price inthe limit order book is sufficiently attractive. However, uninformed traders may trade in the darkonce they have learnt from observing the state of the limit order book if it conveys the informationthat the value of the asset is favorable. In addition, we find that due to adverse selection, uninformedtraders prefer not to trade with informed traders. The optimal strategy of an informed trader inthe first period reveals information to uninformed traders about the value of the asset. Our findingsshow that, even if execution risk in the dark pool is high, informed traders tend to replace marketorders by dark orders when they can take advantage of the price improvement.

Our paper suggests a few open questions for future research. Future work could explore theeffects of adding a dark pool alongside an exchange on market quality parameters and welfare. Thefindings of our paper call for the development of further empirical and experimental work whichstudy the role of information in the competition between a dark pool and an exchange for liquidity.

25

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28

Appendices

A Model without dark pool

Proof of Proposition 1. The procedure we follow to check if a particular strategy profileconstitutes a PBE is as follows:

1. Specify a strategy profile for rational traders at t = 1.

2. Update the beliefs of the uninformed trader at t = 2 using Bayes’ rule at all information sets,whenever possible.

3. Given their beliefs, find the optimal response for the traders at t = 2.

4. Given the optimal response of traders at t = 2, find the optimal action for the traders at t = 1.

5. Check if the optimal strategy profile for the traders at t = 1 coincide with the profile yousuggested in step 1.

We apply the procedure outlined above to check when each possible strategy profile can be anequilibrium.1. (BMO,SMO,BLO,SLO)

First step. In this case Ω0 = 0, Ω1 = 1, Ω2 = 0, Σ0 = 0, Σ1 = 0, and Σ2 = 1.

Second step. Using Bayes’ rule we obtain that X =λπ

η + λπand Y = 0.

Third step. Applying Lemma 1, we know that at t = 2 the optimal strategy for informed tradersis to choose a MO, while for the uninformed trader is as follows:

State of the book UB US

(A11, B

11) NT NT

(A21, B

11)

MO ifλπ

η + λπ>k2k3

NT ifλπ

η + λπ≤ k2k3

NT

(A11, B

11 + τ) NT NT

(A11, B

21) NT

MO ifλπ

η + λπ>k2k3

NT ifλπ

η + λπ≤ k2k3

(A11 − τ,B1

1) NT NT

Table A.1: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

Fourth step. Given the optimal response of traders at t = 2, we find the optimal action for alltraders at t = 1.

29

Informed traders. If they choose a MO, their profit equals (k3 − k1) τ. If, instead, they deviatetowards a LO, as they anticipate that uninformed traders will choose NT , their expected profitsof a LO will be δ

η

2(k3 + k1 − 1) τ. Hence, informed traders at t = 1 have no incentives to deviate

from the prescribed strategy profile whenever

(k3 − k1) τ ≥ δη

2(k3 + k1 − 1) τ .

Uninformed traders. If they behave as the prescribed profile (LO), then they obtain

δ

2(η(k1 − 1)− λπ (k3 − (k1 − 1))) τ ,

given that they anticipate that uninformed traders at t = 2 will choose NT . If, instead, theydeviate choosing NT , then they obtain zero profits. Therefore, uninformed traders at t = 1 haveno incentives incentive to deviate from the prescribed strategy if and only if

η(k1 − 1)− λπ (k3 − (k1 − 1)) > 0. (5)

Fifth step. From the previous two inequalities, nobody at t = 1 has unilateral incentives to deviatefrom (BMO,SMO,BLO,SLO) whenever

k3 − k1 ≥ δη

2(k3 + k1 − 1) and

(λπ + η) (k1 − 1)− λπk3 > 0.

2. (BMO,SMO,NT,NT )

First step. In this case Ω0 = 0, Ω1 = 1, Ω2 = 0, Σ0 = 1, Σ1 = 0, and Σ2 = 0.

Second step. Using Bayes’s rule we obtain that X =λπ

η + λπand Y is undetermined Y ∈ [0, 1] (as

Bayes’s rule implies Y =0

0).

Third step. Applying Lemma 1, we know that at t = 2 the optimal strategy for informed tradersis to choose a MO, while for the uninformed trader is as follows:

30

State of the book UB US

(A11, B

11) NT NT

(A21, B

11)

MO ifλπ

η + λπ>k2k3

NT ifλπ

η + λπ≤ k2k3

NT

(A11, B

11 + τ)

MO if Y >k1k3

NT if Y ≤ k1k3

NT

(A11, B

21) NT

MO ifλπ

η + λπ>k2k3

NT ifλπ

η + λπ≤ k2k3

(A11 − τ,B1

1) NT

MO if Y >k1k3

NT if Y ≤ k1k3

Table A.2: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,NT,NT ).

Fourth step. Given the optimal response of traders at t = 2, we find the optimal action for thetraders at t = 1.

Informed traders. If they choose aMO, they obtain (k3 − k1) τ. If, instead, they deviate towardsa LO, as they anticipate that uninformed traders will not choose a MO of different sign at t = 2,then their expected profit will be: δ η2 (k3 + k1 − 1) τ . Hence, informed traders at t = 1 have noincentives to deviate from the prescribed strategy profile whenever

k3 − k1 ≥ δη

2(k3 + k1 − 1) .

Uninformed traders. If they behave as the prescribed profile (NT ), then they obtain 0. If,instead, they deviate choosing a LO, as they anticipate that uninformed traders will not choose aMO of different sign at t = 2, and hence, their expected profit will be

δ

2(η(k1 − 1)− λπ (k3 − (k1 − 1))) τ .

Therefore, uninformed traders at t = 1 have no incentives to deviate from the prescribed strategyif and only if

0 ≥ (λπ + η) (k1 − 1)− λπk3. (6)

31

Fifth step. No trader at t = 1 has unilateral incentives to deviate from (BMO,SMO,NT,NT )

if and only if

k3 − k1 ≥ δη

2(k3 + k1 − 1) and

0 ≥ (λπ + η) (k1 − 1)− λπk3.

3. (BLO,SLO,BLO,BLO)

First step. In this case Ω0 = 0, Ω1 = 0, Ω2 = 1, Σ0 = 0, Σ1 = 0, and Σ2 = 1.Second step. Using Bayes’s rule we obtain that X = 0 and Y = π.Third step. Applying Lemma 1, we know that at t = 2 the optimal strategy for informed tradersis to choose a MO, while for the uninformed trader is as follows:

State of the book UB US

(A11, B

11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ)

MO if π >k1k3

NT if π ≤ k1k3

NT

(A11, B

21) NT NT

(A11 − τ,B1

1) NT

MO if π >k1k3

NT if π ≤ k1k3

Table A.3: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,BLO,BLO).

Fourth step. Given the optimal response of traders at t = 2, we find the optimal action for thetraders at t = 1.

Informed traders. If they choose a LO, as they anticipate that uninformed traders will notchoose a MO of different sign at t = 2, then their expected profit will be: δ η2 (k3 + k1 − 1) τ. If,instead, they choose a market order, then they obtain (k3 − k1) τ. Hence, at t = 1 informed tradershave no incentives to deviate from the prescribed strategy profile whenever

δη

2(k3 + k1 − 1) > k3 − k1.

Uninformed traders. If they select a LO, their expected profits are δτ2 (η(k1 − 1)− λπ (k3 − (k1 − 1)))

since they anticipate that traders will not choose a MO of different sign at t = 2. If, instead, theychoose NT , they obtain null profits. Therefore, at t = 1 uninformed traders have no incentives to

32

deviate from the prescribed strategy if and only if

(λπ + η) (k1 − 1)− λπk3 > 0. (7)

Fifth step. No trader at t = 1 has unilateral incentives to deviate from (BLO,SLO,BLO, SLO)

if and only if

δη

2(k3 + k1 − 1) > k3 − k1 and

(λπ + η) (k1 − 1)− λπk3 > 0.

4. (BLO,SLO,NT,NT )

First step. In this case Ω0 = 0, Ω1 = 0, Ω2 = 1, Σ0 = 1, Σ1 = 0, and Σ2 = 0.

Second step. Using Bayes’s rule we obtain that X = 0 and Y = 1.Third step. Applying Lemma 1, we know that at t = 2 the optimal strategy for informed tradersis to choose a MO, while for the uninformed trader is as follows:

State of the book UB US

(A11, B

11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ) MO NT

(A11, B

21) NT NT

(A11 − τ,B1

1) NT MO

Table A.4: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,NT,NT ).

Fourth step. Given the optimal response of traders at t = 2, find the optimal action for thetraders at t = 1.

• Informed traders. as they anticipate that uninformed traders will not choose aMO of differentsign at t = 2, then their expected profit will be: δ η2 (k3 + k1 − 1) τ . If, instead, they deviatetowards a MO, they obtain (k3 − k1) τ. Hence, informed traders have no incentives to deviatefrom the prescribe strategy profile whenever

(ηδ

2

)(k3 + k1 − 1) τ > (k3 − k1) τ .

• Uninformed traders. If they chooses NT , they obtain zero profits. If, instead, an uninformedbuyer (seller) deviate towards a BLO (SLO), then he obtains δτ

2 ((λπ + η) (k1 − 1)− λπk3).Hence, uninformed traders have no incentive to deviate from the prescribe strategy profilewhenever

0 ≥ δ

2(η(k1 − 1)− λπ (k3 − (k1 − 1))) . (8)

33

Fifth step. Nobody at t = 1 has unilateral incentives to deviate from (BLO,SLO,NT,NT )

whenever(δη

2

)(k3 + k1 − 1) > (k3 − k1)

0 ≥ δ

2(η(k1 − 1)− λπ (k3 − (k1 − 1))) .

Lemma 5 When there is no dark pool and k1 = 1 the uninformed trader at t = 1 always choosesNT .

The previous lemma implies that the strategies (BMO,SMO,BLO,SLO) and (BLO,SLO,BLO,BLO)

cannot be equilibria of the game.

Proposition 3 If k1 = 1, then a PBE of the game is as follows:

• (BMO,SMO,NT,NT ) is the optimal strategy profile for traders at t = 1 if

k3 − 1 ≥ δ η2k3.

The beliefs of uninformed traders at t = 2 are: X =λπ

η + λπand Y = p ∈ [0, 1]. The optimal

strategy for informed traders at t = 2 is to choose MO for all possible states of the book andthe optimal strategy of uninformed traders at t = 2 is described in Table A.2 of Appendix A.

• (BLO,SLO,NT,NT ) is the optimal strategy profile for traders at t = 1 if

δ

(η

2+

(1− π)λ

2

)k3 > k3 − 1.

The beliefs of uninformed traders at t = 2 are: X = 0 and Y = 1. The optimal strategy forinformed traders at t = 2 is to choose MO for all possible states of the book and the optimalstrategy of uninformed traders at t = 2 is described in Table A.4 of Appendix A.

Proof of Proposition 3. Note that when we replace k1 = 1 in the Proof of Proposition 1 theconditions (5) and (7) are never satisfied and therefore the strategies (BMO,SMO,BLO,SLO)

and (BLO,SLO,BLO,BLO) cannot be part of an equilibrium of the game.Moreover when k1 = 1, the conditions (6) and (8)are always satisfied.

34

B Model with dark pool

Proof of Lemma 3. By simply inspection of the payoff in Tables (6) and (7) can be seen thatthe informed traders at t = 2 will never choose NT because this order is dominated by placing aMO. However, notice also that according to Lemma 4, the uninformed traders never select a DOand therefore Σ3 = 0 which implies

PI = pIHBLO,2 (O1 = ∅) = pILSLO,2 (O1 = ∅) = 0.

Consequently, the informed traders never choose a LO at t = 2, since this order is also dominatedby a MO.

Let us determine next the optimal strategy for each trader. Depending on the values of theparameters we have 6 possible cases for the informed trader and 16 for the uninformed trader.

First let us state a lemma.Note that since k3 > k2 > k1 ≥ 1, the following inequalities hold

k3 − k2k3 − k2−k1

2

<k3 − k1

k3 + k2−k12

<k3 − k1k3

<k3 − k1k3 − 1

2

<k3 − k1 + 1

k3 + 12

.

We define

BX1 =

BMO if (k3−k1)τ ≥ ΠIH

BLO,2

BLO otherwise,

BX2 =

BDO if θk3τ > ΠIH

BLO,2

BLO otherwise,

SX1 =

SMO if (k3−k1)τ ≥ ΠIL

SLO,2

SLO otherwise,

SX2 =

SMO if (k3−k1)τ ≥ ΠIL

SLO,2

SLO otherwise.

35

Condition Optimal Choice of the Informed Trader at t=2

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL

(A11, B

11) BX1 SX1

(A21, B

11) BMO SMO

(A11, B

11+τ) BMO SMO

(A11, B

21) BMO SMO

(A11−τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL

(A11, B

11) BX1 SX2

(A21, B

11) BDO SMO

(A11, B

11+τ) BMO SMO

(A11, B

21) BMO SDO

(A11−τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤k3−k1k3

IH IL

(A11, B

11) BX1 SX1

(A21, B

11) BDO SDO

(A11, B

11+τ) BMO SMO

(A11, B

21) BDO SDO

(A11−τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤k3−k1k3− 1

2

IH IL

(A11, B

11) BX2 SX1

(A21, B

11) BDO SDO

(A11, B

11+τ) BMO SMO

(A11, B

21) BDO SDO

(A11−τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤k3−k1+1k3+

12

IH IL

(A11, B

11) BX2 SX2

(A21, B

11) BDO SDO

(A11, B

11+τ) BDO SMO

(A11, B

21) BDO SDO

(A11−τ,B1

1) BMO SDO

Case I6k3−k1+1k3+

12

< θ

IH IL

(A11, B

11) BX2 SX2

(A21, B

11) BDO SDO

(A11, B

11+τ) BDO SDO

(A11, B

21) BDO SDO

(A11−τ,B1

1) BDO SDO

Table B.1: Optimal Strategy Informed Traders at t = 2

36

Condition Optimal Choice of the Uninformed Trader at t=2

Case A.1.1Xk3 <

k2−k12 < k2

Y k3 <12(< k1)

UB US(A1

1, B11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ) NT SDO

(A11, B

21) BDO NT

(A11 − τ,B1

1) BDO NT

Case A.1.2Xk3 <

k2−k12 < k2

Y k3 = 12(< k1)

UB US(A1

1, B11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ) NT NT

(A11, B

21) BDO NT

(A11 − τ,B1

1) NT NT

Case A.1.3Xk3 <

k2−k12 < k2

12 < Y k3 ≤ k1

UB US(A1

1, B11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ) BDO NT

(A11, B

21) BDO NT

(A11 − τ,B1

1) NT SDO

Case A.1.4Xk3 <

k2−k12 < k2

12 < k1 < Y k3

UB US(A1

1, B11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ)

BDO θ > Y k3−k1Y k3− 1

2

BMO θ ≤ Y k3−k1Y k3− 1

2

NT

(A11, B

21) BDO NT

(A11 − τ,B1

1) NTSDO θ > Y k3−k1

Y k3− 12

SMO θ ≤ Y k3−k1Y k3− 1

2

Table B.2: Optimal Strategy Uninformed Traders at t = 2

37

Case A.2.1Xk3 = k2−k1

2 < k2Y k3 <

12(< k1)

UB US(A1

1, B11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ) NT SDO

(A11, B

21) NT NT

(A11 − τ,B1

1) BDO NT

Case A.2.2Xk3 = k2−k1

2 < k2Y k3 = 1

2(< k1)

UB US(A1

1, B11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ) NT NT

(A11, B

21) NT NT

(A11 − τ,B1

1) NT NT

Case A.2.3Xk3 = k2−k1

2 < k212 < Y k3 ≤ k1

UB US(A1

1, B11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ) BDO NT

(A11, B

21) NT NT

(A11 − τ,B1

1) NT SDO

Case A.2.4Xk3 = k2−k1

2 < k212 < k1 < Y k3

UB US(A1

1, B11) NT NT

(A21, B

11) NT NT

(A11, B

11 + τ)

BDO θ > Y k3−k1Y k3− 1

2

BMO θ ≤ Y k3−k1Y k3− 1

2

NT

(A11, B

21) NT NT

(A11 − τ,B1

1) NTSDO θ > Y k3−k1

Y k3− 12

SMO θ ≤ Y k3−k1Y k3− 1

2

Optimal Strategy Uninformed Traders at t = 2 (Continuation)

38

Case A.3.1k2−k1

2 < Xk3 ≤ k2Y k3 <

12

UB US(A1

1, B11) NT NT

(A21, B

11) BDO NT

(A11, B

11 + τ) NT SDO

(A11, B

21) NT SDO

(A11 − τ,B1

1) BDO NT

Case A.3.2k2−k1

2 < Xk3 ≤ k2Y k3 = 1

2

UB US(A1

1, B11) NT NT

(A21, B

11) BDO NT

(A11, B

11 + τ) NT NT

(A11, B

21) NT SDO

(A11 − τ,B1

1) NT NT

Case A.3.3k2−k1

2 < Xk3 ≤ k212 < Y k3 ≤ k1

UB US(A1

1, B11) NT NT

(A21, B

11) BDO NT

(A11, B

11 + τ) BDO NT

(A11, B

21) NT SDO

(A11 − τ,B1

1) NT SDO

Case A.3.4k2−k1

2 < Xk3 ≤ k2k1 < Y k3

UB US(A1

1, B11) NT NT

(A21, B

11) BDO NT

(A11, B

11 + τ)

BDO θ > Y k3−k1Y k3− 1

2

BMO θ ≤ Y k3−k1Y k3− 1

2

NT

(A11, B

21) NT SDO

(A11 − τ,B1

1) NTSDO θ > Y k3−k1

Y k3− 12

SMO θ ≤ Y k3−k1Y k3− 1

2

Optimal Strategy Uninformed Traders at t = 2 (Continuation)

39

Case A.4.1k2−k1

2 < k2 < Xk3Y k3 <

12

UB US

(A11, B

11) NT NT

(A21, B

11)

BDO θ > Xk3−k2Xk3− k2−k12

BMO θ ≤ Xk3−k2Xk3− k2−k12

NT

(A11, B

11+τ) NT SDO

(A11, B

21) NT

SDO θ > Xk3−k2Xk3− k2−k12

SMO θ ≤ Xk3−k2Xk3− k2−k12

(A11−τ,B1

1) BDO NT

Case A.4.2k2−k1

2 < k2 < Xk3Y k3 = 1

2

UB US

(A11, B

11) NT NT

(A21, B

11)

BDO θ > Xk3−k2Xk3− k2−k12

BMO θ ≤ Xk3−k2Xk3− k2−k12

NT

(A11, B

11+τ) NT NT

(A11, B

21) NT

SDO θ > Xk3−k2Xk3− k2−k12

SMO θ ≤ Xk3−k2Xk3− k2−k12

(A11−τ,B1

1) NT NT

Case A.4.3k2−k1

2 < k2 < Xk312 < Y k3 ≤ k1

UB US

(A11, B

11) NT NT

(A21, B

11)

BDO θ > Xk3−k2Xk3− k2−k12

BMO θ ≤ Xk3−k2Xk3− k2−k12

NT

(A11, B

11+τ) BDO NT

(A11, B

21) NT

SDO θ > Xk3−k2Xk3− k2−k12

SMO θ ≤ Xk3−k2Xk3− k2−k12

(A11−τ,B1

1) NT SDO

Case A.4.4k2−k1

2 < k2 < Xk3k1 < Y k3

UB US

(A11, B

11) NT NT

(A21, B

11)

BDO θ > Xk3−k2Xk3− k2−k12

BMO θ ≤ Xk3−k2Xk3− k2−k12

NT

(A11, B

11+τ)

BDO θ >Y k3−k1Y k3− 1

2

BMO θ ≤Y k3−k1Y k3− 1

2

NT

(A11, B

21) NT

SDO θ > Xk3−k2Xk3− k2−k12

SMO θ ≤ Xk3−k2Xk3− k2−k12

(A11 − τ,B1

1) NTSDO θ > Y k3−k1

Y k3− 12

SMO θ ≤ Y k3−k1Y k3− 1

2

Optimal Strategy Uninformed Traders at t = 2 (Continuation)

40

Proof of Proposition 2. Because of the symmetry of the model, without any loss of generality,at t = 1 we focus on buyers. We present the proof for one of the possible strategy profile at t = 1

that yields an equilibrium. The proofs of all the other 5 equilibria can be obtained on request fromthe authors.

E1: (BMO,SMO,BLO,SLO)

First step. In this case Ω0 = 0,Ω1 = 1,Ω2 = 0,Ω3 = 0,Σ0 = 0,Σ1 = 0, Σ2 = 1, and Σ3 = 0.Second step. Using Bayes’s rule,

X =λπ

η + λπ, Y = 0 and Z = q ∈ [0, 1] .

Third step. Using steps 1 and 2, the expected profits for uninformed traders at t = 2 are given by

UB BMO BDO BLO NT

(A11, B

11) −k1τ 0 0 0

(A21, B

11)

(λπ

η+λπk3 − k2)τ θ

(λπ

η+λπk3 − k2−k12

)τ 0 0

(A11, B

11 + τ) −k1τ − θ

2τ 0 0

(A11, B

21) −

(λπ

η+λπk3 + k1

)τ −θ

(λπ

η+λπk3 − k2−k12

)τ 0 0

(A11 − τ,B1

1) − (k1 − 1) τ θ2τ 0 0

Table B.3: Expected profits for an uninformed buyer at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

US SMO SDO SLO NT

(A11, B

11) −k1τ 0 0 0

(A21, B

11) −

(k1 + λπ

η+λπk3

)τ −θ

(λπ

η+λπk3 − k2−k12

)τ 0 0

(A11, B

11 + τ) − (k1 − 1) τ θτ

2 0 0

(A11, B

21)

(λπ

η+λπk3 − k2)τ θ

(λπ

η+λπk3 − k2−k12

)τ 0 0

(A11 − τ,B1

1) −k1τ − θτ2 0 0

Table B.4: Expected profits for an uninformed seller at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

Hence, the optimal responses for uninformed traders are:

41

UB US

(A11, B

11) NT NT

(A21, B

11)

NT if k3 ≤ k2−k12

η+λπλπ

BDO

if k2−k12η+λπλπ < k3 ≤ k2 η+ππ or

if k2 η+λπλπ < k3

and θ > k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

BMO if k2 η+λπλπ < k3 and

θ ≤ k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

SDO if k3 < k2−k1

2η+λπλπ

NT if k2−k12η+λπλπ ≤ k3

(A11, B

11 + τ) NT SDO

(A11, B

21)

BDO if k3 < k2−k1

2η+λπλπ

NT if k2−k12η+λπλπ ≤ k3

NT if k3 ≤ k2−k12

η+λπλπ

SDO

if k2−k12η+λπλπ < k3 ≤ k2 η+ππ or

if k2 η+λπλπ < k3

and θ > k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

SMO if k2 η+λπλπ < k3 and

θ ≤ k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

(A11 − τ,B1

1) BDO NT

Table B.5: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

Using step 1, the expected profits for informed traders at t = 2 are given by

42

IH BMO BDO BLO NT

(A11, B

11) (k3 − k1) τ θk3τ 0 0

(A21, B

11) (k3 − k2) τ θ

(k3 − k2−k1

2

)τ 0 0

(A11, B

11 + τ) (k3 − k1) τ θ

(k3 − 1

2

)τ 0 0

(A11, B

21) (k3 − k1) τ θ

(k3 + k2−k1

2

)τ 0 0

(A11 − τ,B1

1) (k3 − k1 + 1) τ θ(k3 + 1

2

)τ 0 0

Table B.6: Expected profits for an informed buyer at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

IL SMO SDO SLO NT

(A11, B

11) (k3 − k1) τ θk3τ 0 0

(A21, B

11) (k3 − k1)τ θ

(k3 + k2−k1

2

)τ 0 0

(A11, B

11 + τ) (k3 − k1 + 1)τ θ

(k3 + 1

2

)τ 0 0

(A11, B

21) (k3 − k2)τ θ

(k3 − k2−k1

2

)τ 0 0

(A11 − τ,B1

1) (k3 − k1)τ θ(k3 − 1

2

)τ 0 0

Table B.7: Expected profits for an informed seller at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

Hence, the optimal responses for informed traders are:

43

Condition Optimal Responses

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.8: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,BLO,SLO).

44

Fourth step. Given the optimal responses of traders at t = 2, find the optimal action for thetraders at t = 1 in each of the 6 cases. However, given the nature of this particular equilibrium, wecan group cases into the following and analyze them:

• Case I1 + I2 + I3

• Case I4 + I5

• Case I6

Case I1 + I2 + I3: θ ≤ k3−k1k3

• Informed traders

Consider an informed buyer at t = 1. If he chooses a BMO, then he will obtain

E(ΠIHBMO,1

)= (k3 − k1) τ .

If instead he deviates towards a BLO, then in the next period the prices will be (A11, B

11 + τ) and,

then, he anticipates the following behavior for potential sellers at t = 2:

1. if there is an uninformed seller, then he will choose SDO, and

2. if there is a liquidity seller, then he will set SMO.

Therefore, the BLO at t = 1 will only be executed if in the next period there is a liquidity seller.Thus, the corresponding expected profits are given by

E(ΠIHBLO,1

)=η

2δ (k3 + k1 − 1) τ .

If instead he deviates towards a BDO, he knows that in the next period the prices in the book willnot change. In this case, he anticipates the following behavior for traders at t = 2:

1. if there is an informed trader, then he will be a buyer and will choose a BMO,

2. if there is an uninformed buyer, then he will choose a NT , and

3. if there is an uninformed seller, then he will choose a NT.

Thus, Table 10 implies that

E(ΠIHBDO,1

)= θk3τ + (1− θ)δ2

(k3 − k1 − (k2 − k1)

(λπ +

η

2

))τ ,

45

since IUS,O1=∅SLO,2 = 0 and IIH,O1=∅

BMO,2 = 1. Hence, informed traders at t = 1 have no incentives todeviate from the prescribed strategy profile whenever

k3 − k1 ≥η

2δ (k3 + k1 − 1) and

k3 − k1 ≥ θk3 + (1− θ)δ2(

(k3 − k1)− (k2 − k1)(λπ +

η

2

)).

• Uninformed traders

Consider an uninformed buyer at t = 1. If instead he deviates towards a BLO, then in the nextperiod the prices will be (A1

1, B11 + τ) and, then, he anticipates the following behavior for potential

sellers at t = 2:

1. if there is an informed seller, then he chooses SMO,

2. if there is an uninformed seller, then he will choose SDO, and

3. if there is a liquidity seller, then he will set SMO.

Therefore, the BLO will be executed only with a liquidity seller or with a informed seller. Hence,the corresponding expected profits will be

E(ΠUBBLO,1

)=δ

2((λπ + η) (k1 − 1)− λπk3) τ ,

since IIL,O1=BLOSMO,2 = 1.

If instead, the uninformed buyer deviates and chooses BDO, then he knows that at t = 2 theprices in the book will not change. In this case, he anticipates the following behavior for traders att = 2:

1. if there is an informed buyer, then he will choose a BMO,

2. if there is an informed seller, then he will choose a SMO,

3. if there is an uninformed buyer, then he will choose NT, and

4. if there is an uninformed seller, then he will choose NT.

Thus, Table 11 implies that

E(ΠUBBDO,1

)= (1− θ)δ2

(λ

2π (k1 − k2) +

η

2(k1 − k2)− k1

)τ < 0,

since IIL,O1=∅SLO,2 = IUS,O1=∅

SLO,2 = 0 and IIH,O1=∅BMO,2 = 1.

46

If instead, the uninformed buyer deviates and chooses NT, then he will obtain

E(ΠUBNT,1

)= 0.

Hence, he will not have incentives to deviate if

(λπ + η) (k1 − 1)− λπk3 > 0.

Fifth step. From step 4, nobody at t = 1 has unilateral incentives to deviate whenever

k3 − k1 ≥η

2δ (k3 + k1 − 1) ,

k3 − k1 ≥ θk3 + (1− θ)δ2(k3 − k1 − (k2 − k1)

(λπ +

η

2

)), and

(λπ + η) (k1 − 1)− λπk3 > 0.

Case I4 +I5: k3−k1k3

< θ ≤ k3−k1+1k3+

12

• Informed traders

Consider an informed buyer at t = 1. If he chooses a BMO, then he will obtain

E(ΠIHBMO,1

)= (k3 − k1) τ.

If instead he deviates towards a BDO, he knows that in the next period the prices in the bookwill not change. In this case, he anticipates the following behavior for traders at t = 2:

1. if there is an informed trader, then he will be a buyer and will choose a BDO,

2. if there is an uninformed buyer, then he will choose a NT , and

3. if there is an uninformed seller, then he will choose a NT.

Thus, Table 10 implies that

E(ΠIHBDO,1

)= θk3τ + (1− θ)δ2τ

(k3 − k1 − (k2 − k1)

η

2

),

since IUS,O1=∅SLO,2 = 0 and IIH,O1=∅

BMO,2 = 0.

However, since k3−k1k3

< θ, then

θk3τ + (1− θ)δ2(k3 − k1 − (k2 − k1)

η

2

)τ > (k3 − k1) τ

is always satisfied and, hence, in this case there is no equilibrium in which (BMO,SMO,BLO,SLO)

is the strategy profile chosen at t = 1.Case I6: k3−k1+1

k3+12

< θ

47

• Informed traders

Consider an informed buyer at t = 1. If he chooses a BMO, then he will obtain

E(ΠIHBMO,1

)= (k3 − k1) τ.

If instead he deviates towards a BDO, he knows that in the next period the prices in the bookwill not change. In this case, he anticipates the following behavior for traders at t = 2:

1. if there is an informed trader, then he will be a buyer and will choose a BDO,

2. if there is an uninformed buyer, then he will choose a NT , and

3. if there is an uninformed seller, then he will choose a NT.

Thus, Table 10 implies that

E(ΠIHBDO,1

)= θk3τ + (1− θ)δ

((k3 − k1)− (k2 − k1)

η

2

)2τ ,

However, since k3−k1k3

< k3−k1+1k3+

12

< θ, then

θk3τ + (1− θ)δ2(k3 − k1 − (k2 − k1)

(η2

))τ > (k3 − k1) τ

is always satisfied and, hence, in this case there is no equilibrium in which (BMO,SMO,BLO,SLO)

is the strategy profile chosen at t = 1.E2: (BMO,SMO,NT,NT )

Following the same procedure we obtain that in this case the optimal responses of uninformedand informed traders are:

48

UB US

(A11, B

11) NT NT

(A21, B

11)

NT if k3 ≤ k2−k12

η+λπλπ

BDO if k2−k12η+λπλπ < k3 ≤ k2 η+λπλπ

BDO if k2 η+λπλπ < k3 and

θ >k3−k2 η+ππ

k3− k2−k12η+λπλπ

BMO if k2 η+λπλπ < k3 and

θ ≤ k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

SDO if k3 < k2−k1

2η+λπλπ

NT if k2−k12η+λπλπ ≤ k3

(A11, B

11 + τ)

NT if p ≤ 12k3

BDO if 12k3

< p < k1k3

BDO if p ≥ k1k3

AND θ > pk3−k1pk3− 1

2

BMO if p ≥ k1k3

AND θ ≤ pk3−k1pk3− 1

2

SDO if p < 1

2k3NT if 1

2k3≤ p

(A11, B

21)

BDO if k3 < k2−k1

2η+λπλπ

NT if k2−k12η+λπλπ ≤ k3

NT if k3 ≤ k2−k12

η+λπλπ

SDO if k2−k12η+λπλπ < k3 ≤ k2 η+λπλπ

SDO if k2 η+λπλπ < k3 and

θ >k3−k2 η+ππ

k3− k2−k12η+λπλπ

SMO if k2 η+λπλπ < k3 and

θ ≤ k3−k2 η+λπλπ

k3− k2−k12η+λπλπ

(A11 − τ,B1

1)

BDO if p < 1

2k3

NT if 12k3≤ p

NT if p ≤ 12k3

SDO if 12k3

< p < k1k3

SDO if p ≥ k1k3

and θ > pk3−k1pk3− 1

2

SMO if p ≥ k1k3

and θ ≤ pk3−k1pk3− 1

2

Table B.9: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,NT,NT ).

49

Condition Optimal Responses

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.10: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BMO,SMO,NT,NT ).

50

UB US

(A11, B

11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ)

NT if k3 ≤ 12π

BDO if 12π < k3 ≤ k1

π

BDO if k3 > k1π and

θ > πk3−k1πk3− 1

2

BMO if k3 > k1π and

θ ≤ πk3−k1πk3− 1

2

SDO if k3 < 1

2πNT if k3 ≥ 1

2π

(A11, B

21) BDO NT

(A11 − τ,B1

1)

BDO if k3 < 1

2πNT if k3 ≥ 1

2π

NT if k3 ≤ 12π

SDO if 12π < k3 ≤ k1

π

SDO if k3 > k1π and

θ > πk3−k1πk3− 1

2

SMO if k3 > k1π and

θ ≤ πk3−k1πk3− 1

2

Table B.11: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,BLO,BLO).

E3: (BLO,SLO,BLO,BLO) Following the same procedure we obtain that in this case theoptimal responses of uninformed and informed traders are:

51

Condition Optimal Response

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.12: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,BLO,BLO).

52

UB US

(A11, B

11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ)

NT if k3 ≤ 12π

BDO if 12π < k3 ≤ k1

π

BDO if k3 > k1π and

θ > πk3−k1πk3− 1

2

BMO if k3 > k1π and

θ ≤ πk3−k1πk3− 1

2

SDO if k3 < 1

2πNT if k3 ≥ 1

2π

(A11, B

21) BDO NT

(A11 − τ,B1

1)

BDO if k3 < 1

2πNT if k3 ≥ 1

2π

NT if k3 ≤ 12π

SDO if 12π < k3 ≤ k1

π

SDO if k3 > k1π and

θ > πk3−k1πk3− 1

2

SMO if k3 > k1π and

θ ≤ πk3−k1πk3− 1

2

Table B.13: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,NT,NT ).

E4: (BLO,SLO,NT,NT )

Following the same procedure we obtain that in this case the optimal responses of uninformedand informed traders are:

53

Condition Optimal Response

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.14: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BLO,SLO,NT,NT ).

54

UB US

(A11, B

11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ) NT SDO

(A11, B

21) BDO NT

(A11 − τ,B1

1) BDO NT

Table B.15: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BDO,SDO,BLO, SLO).

E5: (BDO,SDO,BLO, SLO)

Following the same procedure we obtain that in this case the optimal responses of uninformed anduninformed traders are:

55

Condition Optimal Response

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.16: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BDO,SDO,BLO, SLO).

56

UB US

(A11, B

11) NT NT

(A21, B

11) NT SDO

(A11, B

11 + τ)

NT if Y ≤ 12k3

BDO if 12k3

< Y < k1k3

BDO if Y ≥ k1k3

andθ > Y k3−k1

Y k3− 12

BMO if Y ≥ k1k3

andθ ≤ Y k3−k1

Y k3− 12

NT if Y ≥ 1

2k3SDO if Y < 1

2k3

(A11, B

21) BDO NT

(A11 − τ,B1

1)

NT if Y ≥ 1

2k3BDO if Y < 1

2k3

NT if Y ≤ 12k3

SDO if 12k3

< Y < k1k3

SDO if Y ≥ k1k3

andθ > Y k3−k1

Y k3− 12

SMO if Y ≥ k1k3

andθ ≤ Y k3−k1

Y k3− 12

Table B.17: Optimal responses of uninformed traders at t = 2 when the strategy profile at t = 1 is(BDO,SDO,NT,NT ).

E6: (BDO,SDO,NT,NT )

Following the same procedure we obtain that in this case the optimal responses of uninformed andinformed traders are:

57

Condition Optimal Response

Case I1

θ ≤ k3−k2k3− k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BMO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SMO

(A11 − τ,B1

1) BMO SMO

Case I2k3−k2

k3− k2−k12

< θ

≤ k3−k1k3+

k2−k12

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SMO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BMO SDO

(A11 − τ,B1

1) BMO SMO

Case I3k3−k1

k3+k2−k1

2

< θ

≤ k3−k1k3

IH IL(A1

1, B11) BMO SMO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I4k3−k1k3

< θ

≤ k3−k1k3− 1

2

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BMO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SMO

Case I5k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SMO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BMO SDO

Case I6k3−k1k3− 1

2

< θ

≤ k3−k1+1k3+

12

IH IL(A1

1, B11) BDO SDO

(A21, B

11) BDO SDO

(A11, B

11 + τ) BDO SDO

(A11, B

21) BDO SDO

(A11 − τ,B1

1) BDO SDO

Table B.18: Optimal responses of informed traders at t = 2 when the strategy profile at t = 1 is(BDO,SDO,NT,NT ).

58

Proposition 4 If k1 = 1, then a PBE of the game is as follows.

• (BMO,SMO,NT,NT ) is the optimal strategy profile for traders at t = 1 if

Conditions

θ ≤ k3−k1k3

k3 − 1 ≥ η2δk3,

k3 − 1 ≥ θk3 + (1− θ)δ2(k3 − 1− (k2 − 1)

(λπ + η

2

)).

k3−k1k3

< θ Not an equilibrium.

The beliefs of uninformed traders at t = 2 are: X =λπ

η + λπ, Y = p ∈ [0, 1] and Z = q ∈ [0, 1].

The optimal strategy for uninformed and informed traders at t = 2 are described in Tables B.9and B.10 of Appendix B, respectively.

• (BLO,SLO,NT,NT ) is the optimal strategy profile for traders at t = 1 if

Conditions

θ ≤ k3−k1k3

η2δk3 ≥ θk3 + (1− θ)δ2

((k3 − 1)− (k2 − 1)

(λπ + η

2

)),

η2δk3 > (k3 − 1) .

k3−k1k3

< θ ≤ k3−k1+1k3+

12

η2δk3 ≥ θk3 + (1− θ)δ2

((k3 − 1)− (k2 − 1)

(η2

)).

k3−k1+1k3+

12

< θ δ η2k3 ≥ θk3 + (1− θ)δ2((k3 − 1)− η

2 (k2 − 1)).

The beliefs of uninformed traders at t = 2 are: X = 0, Y = 1 and Z = q ∈ [0, 1]. The optimalstrategy for uninformed and informed traders at t = 2 are described in Tables B.13 and B.14of Appendix B, respectively.

• (BDO,SDO,NT,NT ) is the optimal strategy profile for traders at t = 1 if

Conditions

θ ≤ k3−k1k3

θk3 + (1− θ)δ2((k3 − 1)− (k2 − 1)

(λπ + η

2

))> (k3 − 1) ,

θk3 + (1− θ)δ2((k3 − 1)− (k2 − 1)

(λπ + η

2

))> η

2δk3.k3−k1k3

< θ ≤ k3−k1+1k3+

12

θk3 + (1− θ)δ2((k3 − 1)− (k2 − 1)

(η2

))> η

2δk3.

k3−k1+1k3+

12

< θ θk3 + (1− θ)δ2((k3 − 1)− (k2 − 1)

(η2

))> η

2δk3.

The beliefs of uninformed traders at t = 2 are: X = 0, Y = p ∈ [0, 1] and Z = 1. The optimalstrategy for uninformed and informed traders at t = 2 are described in Tables B.17 and B.18of Appendix B, respectively.

Proof of Corollary 1. The Corollary follows immediately by taking the limit π goes to zero inthe conditions that guarantee the existence of the PBE stated in Proposition 2.

59