Comparing Market Structures: Allocative and Informational
Efficiencies of Continuous Trading, Periodic Auctions, and
Dark Pools
Romans Pancs⇤
November 19, 2012
Abstract
This paper compares three market structures in a dynamic dealer-mediated market with
asymmetric information about an asset’s value. The compared market structures are (i) the
continuous protocol of Glosten and Milgrom (1985), where a just-arrived trader trades immediately at a bid or an ask price publicly posted by the dealer, (ii) the periodic auction, wherein
trades are cleared in batches, publicly, and at regular intervals, and (iii) the dark pool, wherein
a just-arrived trader trades immediately, and the dealer obfuscates past transaction prices. The
comparison focuses on allocative efficiency (i.e., the total surplus from trade), informational efficiency (i.e., the payoff of a market observer who learns from prices), and unravelling (i.e., an
entrant dealer’s ability to profit by offering a contract that does not conform with the prevailing
market structure). The dark pool is found to have first-best allocative efficiency (under conditions), but it is informationally inefficient compared to the periodic auction. The continuous
protocol is more allocatively efficient than the periodic auction when asymmetric information
is small and traders are impatient. Among the three market structures, the dark pool is the
least susceptible to unravelling (under conditions). The continuous protocol is most susceptible to unravelling; it unravels as long as there is any asymmetric information about the asset’s
value. When the asymmetric information is absent or sufficiently small, the periodic auction
unravels.
⇤ rpancs@gmail.com, Department of Economics, University of Rochester. For comments, I thank Andrew Baynes,
Boyan Jovanovic, Matthew Knowles, Arina Nikandrova, Colin Rowat, David Slichter, Andrzej Skrzypacz, and Gabor
Virag.
1
1
Introduction
In centralized exchanges, assets are traded in (a) periodic auctions, (b) continuously, or (c) in dark
pools. Periodic auctions and continuous protocols immediately publicize past transactions. Dark
pools do not. The comparative analysis of these three market structures is this paper’s focus.
The analysis addresses a positive and a normative question. The normative question is: Which
features of the economic environment make either market structure most efficient (allocatively
or informationally, as will be defined)? The positive question is: When is each of these market
structures stable? A market structure is stable if it can be sustained at equilibrium with free entry
of dealers who can offer contracts that do not conform with the prevailing market structure. These
questions are important for a regulator deciding which market structure to encourage.
All three market structures are ubiquitous in practice. Continuous protocols are used to trade
equity. Periodic auctions are used to trade metals. For instance, at the London Metal Exchange,
each traded metal is allocated two five-minute trading slots per day.1 Broadly defined as “dark
markets” (Duffie, 2011), dark pools include over-the-counter markets for government and corporate bonds, and equity. For the week of August 6, 2012, more than half of FTSE All-Share index
trading volume was accounted for by dark pools.2
Merely by observing that, in practice, a particular asset is traded in a particular market structure, one cannot conclude this market structure’s optimality from a regulator’s point of view.
Economic theory gives little reason to believe that competition shall favor the most efficient market structure. This is because, in practice, exchanges are characterized by fixed operating costs
and network externalities, both of which advantage the incumbent exchange. Hence, room for
regulatory policy remains.
Existing literature on comparative normative analysis of market structures is scarce. Madhavan (1990) compares continuous and periodic trading mostly from the price-efficiency perspective. Price efficiency indicates whether, from past prices alone, an observer can learn each trader’s
private information, or only their shared information, or neither. The present paper introduces
a related measure. This measure, informational efficiency, corresponds to the payoff of a ficti1 See
http://www.lme.com/who_how_ringtimes.asp for details.
The Future of Computer Trading in Financial Markets (2012, Footnote 26): “Regarding the FTSE All-Share
index for the week beginning 6th August, 2012: 47.9% of trading volume occurred on lit venues [...]. Even the lit class
of venues such as the LSE allows so-called iceberg orders that are only partially revealed to the public. [...]”
2 Foresight:
2
tious market observer whose unmodelled investment strategy is informed by the trades that he
observes. Informational efficiency is higher when observed trades reveal traders’ private information earlier or with greater accuracy.
This paper compares market structures also (and mainly) according to allocative efficiency,
defined as the expected total surplus generated by trading. Brusco and Jackson (1999) also pursue a normative, mechanism-design approach to market structure using allocative efficiency as
the objective. Both in their paper and the present one, by lingering in a market, a trader may
generate a positive externality, and well-designed market structures exploit this externality. The
two papers differ, however, in the source of the externality and in the aspects of market structure
that the models rationalize. In the model of Brusco and Jackson (1999), the externality arises from
a trader’s intertemporal intermediation, and rationalizes the emergence of designated intermediaries. In the present model, the externality is informational, and sometimes calls for delayed
execution of trades, thereby rationalizing the periodic auction.
The approach in the present paper is short of full-fledged mechanism design. Instead of designing an optimal market structure subject to minimal feasibility constraints, the paper is concerned with the comparison of particular market structures. In this respect, the approach resembles that of Fuchs and Skrzypacz (2012), who, in a dynamic lemons problem, consider the choice
between continuous and periodic trade. What distinguishes the present model is its focus on a
canonical financial environment.
The paper’s central questions are explored in a dynamic trading environment that is essentially
a special case of the environment studied by Glosten and Milgrom (1985). Time is continuous.
Market participants exchange cash for an asset that pays a stochastic amount v at a stochastic
stopping time. A competitive dealer values the asset at v, but does not know v. Each informed
trader knows v and values the asset at v. Each uninformed trader does not know v and values the
asset at v + u, where u can be positive or negative and is independent across traders and of v.3
Traders arrive stochastically and independently of each other and of v. Each trader supplies 1, 0,
or
1 units of the asset—which is a standard, but restrictive, assumption.
In order to accommodate delayed execution of trades (in the periodic auction), the environ-
3 A minor difference from the model of Glosten and Milgrom (1985) is that here the dependence of each uninformed
trader’s valuation on v and u is additive, not multiplicative.
3
ment of Glosten and Milgrom (1985) is extended by assuming that, even though a trader must
submit his trading order immediately upon arrival, he remains in the market until his order is
filled or until, for exogenous reasons, he vanishes, cancelling his order. The Poisson intensity with
which a trader vanishes is called demand for immediacy. It is a form of impatience.4
Addressing the normative question of efficiency, the paper finds that the periodic auction (PA)
is more allocatively efficient than the continuous protocol (CP) when asymmetric information
about v is high or when the demand for immediacy is low. (This efficiency ranking is consistent
with the common practice of using an auction in order to open trading in an otherwise continuous
market, at the beginning of the day, presumably when uncertainty about the asset’s value is the
greatest.) In order to see the intuition for the determinants of the ranking of PA and CP, observe
that, even though PA is not a solution to a mechanism-design problem, it is constructed to possess
a compelling feature; it eliminates the bid-ask spread. The spread is eliminated because the dealer
conditions a trader’s price on others’ submitted orders. Informed and uninformed traders have
different beliefs about the probability distribution of others’ orders, and hence expect different
prices. These expected prices are constructed so that an uninformed trader expects no bid-ask
spread, whereas an informed trader expects the spread that extracts his entire surplus from trade.5
PA’s inevitable disadvantage is the delay in trade; a just-arrived trader must wait for another
trader to arrive (with a sufficiently high probability).
The occasionally superior allocative efficiency of PA relative to CP agrees with experts’ cautious calls for a switch to periodic auctions and away from continuous trading, in order to curb
high-frequency trading (Foresight: The Future of Computer Trading in Financial Markets, 2012, Section
6.11). There are also calls to merely slow down trades while retaining continuous trading (Biais et
al., 2011). This alternative policy cannot be justified by the present model. While the switch from
CP to PA entails less frequent trades, it also requires a change in the pricing rule. In practice, there
is no reason to believe that mandating less frequent trades alone would automatically lead to the
switch to the appropriate pricing rule. If this switch fails to occur, the imposition of higher latency
4 The importance of the demand for immediacy in financial markets has been noted by Demsetz (1968). Demand for
immediacy has been confirmed empirically by Tkatch and Kandel (2006), and has been a central element in the models
of Foucault et al. (2005) and Rosu (2009).
5 The full-extraction of the informed traders’ surplus does not rely on the so-called “shoot-the-agent” technique
developed by Crémer and McLean (1988). In particular, an informed trader is not punished (i.e., is not “shot”) if his
order “contradicts” the order submitted by another informed trader. A variant of the shoot-the-agent technique would
have to be used, however, if the dealer had to provide a trader with strict incentives to submit the intended order.
4
may even lead to inferior allocative-efficiency outcomes, especially when high-frequency traders
play the role of dealers in (dealerless) order-book driven markets (as documented by Jovanovic
and Menkveld (2010) and Menkveld (2011)).
The model’s normative comparison further implies that the dark pool (DP), whenever feasible,
is more allocatively efficient than CP and PA. (DP is feasible when the obfuscation of past trades
is assumed to be feasible and when informed traders are assumed to arrive at a sufficiently high
rate.) Indeed, DP achieves the first-best level of allocative efficiency. This is so because orders
submitted in DP are executed immediately (as in CP), and hence no surplus is lost due to unfulfilled demand for immediacy. Moreover, the prices that uninformed traders face in DP have
no bid-ask spread for uninformed traders (as in PA) because the dealer exploits the information
that he has privately learned from past trades in order to induce different expected prices for informed and uninformed traders. These disparate expectations can be induced because DP does
not immediately publicize past trades.
A policy implication is that the imposition of post-trade transparency by a regulator would
reduce allocative efficiency by rendering DP infeasible. By contrast, post-trade transparency leads
to an increase in informational efficiency in cases when the infeasible DP is replaced by PA, which
can be shown to be always more informationally efficient than DP. Whether informational efficiency increases further if PA is replaced by CP depends on parameter values.
Addressing the positive question of unravelling, the paper finds that, among the three market
structures, CP is most susceptible to unravelling. CP unravels as long as there is any asymmetric
information about v, which occurs when the dealer does not know v, and informed traders arrive
at a positive rate. Unravelling occurs as an entrant dealer offers a better price to a trader willing
to delay his transaction. The price improvement and the delay are set so that only uninformed
traders choose to delay, so the entrant dealer does not suffer from adverse selection.6 This unravelling result is proved by exploiting uninformed traders’ inability to actively manage their orders
by choosing the timing of their trades. A plausible interpretation of CP’s unravelling is not that
any continuous trading is liable to be supplanted by, say, a periodic auction, but that intermediaries and off-the-shelf algorithms shall emerge in order to actively manage traders’ orders on their
6 Unravelling towards later contracting is a feature of the standard screening models of education (see, e.g., MasColell et al. (1995, Chapter 13)) if interpreted so that education consumes time and therefore signals greater patience,
which is valued by the employer.
5
behalf.
The conditions for PA’s unravelling are roughly converse to those for CP’s. PA unravels when
the asymmetric information about v is sufficiently small. Then, an entrant dealer offers to trade
immediately. All informed traders accept; only some uninformed ones do. Nevertheless, adverse
selection is small (because the asymmetric information about v is small, by hypothesis), and so the
dealer profits.7
Among the three market structures, DP is least susceptible to unravelling. Whenever DP is
feasible, it does not unravel. Because all trades in DP are already immediate, no entrant dealer
can profit by offering to trade immediately. Also, because each informed trader’s profit in DP is
zero, no entrant dealer can avoid attracting informed traders if he offers to improve the price in
exchange for a delay.
DP’s superlative allocative efficiency and immunity to unravelling are consistent with dark
pools’ prevalence in practice. In practice, dark pools vary in their implementation. When looking
at dark pools in the data, one should seek DP’s salient features, such as the obfuscation of past
trades and—relative to traditional market structures—a narrower bid-ask spread (or smaller price
impact) and a smaller expected payoff for informed traders. Among the three market structures,
DP gives the lowest (i.e., zero) expected payoff to informed traders, consistent with the received
view that uninformed traders prefer dark pools, whereas informed traders prefer “lit” markets.
This received view has been articulated by Zhu (2012), who also proposes an explanation, which
is related to, but is distinct from, the present paper’s explanation.
In the model of Zhu (2012), the dark pool does not guarantee order execution. Because their
information is correlated, informed traders tend to cluster on the same, “heavy,” side of the market
and are rationed. Uninformed traders are equally likely to be on either side of the market, expect
to be rationed less, and hence gain from trading in the dark pool more than informed traders do.
By contrast, in this paper, DP guarantees the execution of each order. Instead of exploiting the
correlation of informed traders’ orders through rationing, DP exploits it through prices.
The paper’s normative implications must be qualified. The factors that determine the tradeoff between continuous trading and the agglomeration of trades in the model need not be the
dominant ones in practice. In the model, agglomeration mitigates adverse selection. In practice,
7 In
labor markets, unravelling towards earlier contracting has been studied by Roth and Xing (1994).
6
agglomeration may also help traders satisfy demand for multiway trades, curtail traders’ market
power, or coordinate traders on a trading time. In the model, continuous trading satisfies demand
for immediacy. In practice, continuous trading may also help alleviate congestion externalities associated with agglomeration and accommodate re-trades if multiway trades fail to be satisfactorily
executed in an agglomerated market. Nevertheless, the paper’s normative analysis is not moot.
It explores the normative implications of some salient features of economic environment. These
implications can either corroborate the implications of alternative models or, if controverted, can
alert the regulator to the countervailing determinants of an optimal market structure.8
The paper proceeds thus: Section 2 introduces the trading environment, allocative and informational efficiency, and the three market structures, whose relative efficiencies are compared in
Section 3 and whose propensity to unravel is analyzed in Section 4. Section 5 discusses the model’s
assumptions. Omitted proofs are in Appendix A. Appendix B contains extensions that motivate
some of the paper’s assumptions.
2
Model
2.1
Environment
Time is continuous and is indexed by t 2 [0, T ]. Players exchange money for an asset that pays
amount v at some time T, where v is a random variable in {0, 1}, with a commonly known probability Pr {v = 1} = g0 2 (0, 1). Time T is the arrival time of an event of a Poisson intensity
r > 0.
Players comprise a dealer and traders. The dealer values the asset at v, which he does not
observe. He faces no inventory cost, and can supply or demand an arbitrary amount of the asset.
Each trader can be informed or uninformed about v. An informed trader arrives with a positive Poisson intensity l I and values the asset at v, which he privately observes. An uninformed
trader arrives with a positive Poisson intensity lU and values the asset at v + u, where u is pri8 The
normative economics of market structure resembles geriatrics. Consider a hundred-year-old patient who is
dying from a dozen of deceases. If the doctor treats them all, the patient will die within a week, from exhaustion. If
the doctor treat some of them, the patient may live for five more years. If left untreated, the patient shall die within
twenty-four hours. For the doctor, the first step is to understand how to treat each decease optimally. This is what this
paper accomplishes, for a particular “decease.” The paper is silent about the second step, which consists in optimally
prioritizing treatments.
7
vately observed and is distributed independently across traders, with a strictly increasing c.d.f. G
with the mean zero and a positive and bounded p.d.f. g whose derivative g0 is bounded. Assume
that G is symmetric in the sense that u and
which implies G (0) =
1
2
u are equal in distribution—i.e., G ( u) = 1
G ( u ),
and g (u) = g ( u). Let ( ū, ū) denote the support of u, where ū is
positive and possibly infinite. Each trader can supply 1, 0, or
1 units of the asset. Denote the
total arrival intensity by l ⌘ l I + lU and the probability that an arrival is informed by a ⌘ l I /l.
Immediately upon arrival, each trader must either submit a trading order or forfeit his opportunity to do so. With a Poisson intensity r > 0, each trader who has submitted an order loses his
opportunity to trade and vanishes, with his order cancelled. (A higher r is interpreted as a higher
demand for immediacy.) Also cancelled are any orders revealed to be scheduled for or after T.
The dealer and all traders are expected-utility maximizers, with their utilities quasilinear in
money, additive over time, and lacking discounting. Each trader has risk tolerance L > 0, meaning that he can commit only to those trades in which the support of transaction prices is within an
interval of length L.9
All random variables and stochastic processes defined so far are mutually independent.
Some results shall rely on the distributional assumption:
Condition 1. Uninformed traders’ private-valuation component u is distributed uniformly on
⇥ 1 1⇤
2, 2 .
Discussion: Interpreting the Payoff Structure
In order to motivate the payoff structure, assume that the asset is a bond issued by an unmodelled borrower (e.g., a bank or a corporation). At time T, the borrower either repays each bond
at its face value v = 1 or defaults, repaying v = 0. Until then, each bond continuously pays an
(unmodelled) floating-rate dividend, whose expected discounted value is normalized to zero. The
dividend process is independent of v.
Neither the dealer nor uninformed traders have any inside knowledge about the prospect
of the borrower’s default. An informed trader is an insider, who knows whether the default
9 The risk stemming from the uncertainty about transaction prices is related to the liquidity risk, which Garbade
and Silber (1979) define as “the variance of the difference between the equilibrium value of an asset at the time a
market participant decides to trade and the transaction price ultimately realized.” Garbade and Silber (1979) find (in a
non-strategic model without asymmetric information)—as this paper does—that traders’ aversion to the liquidity risk
affects the optimal frequency of transactions. Restrictions on L shall be imposed.
8
shall occur. The dealer and informed traders value the bond at the amount that it pays, v. An
uninformed trader is willing to pay for the bond more than v if the bond’s dividend process is
negatively correlated with his (unmodelled) endowment process, in this case, u > 0. If the bond’s
dividend process is positively correlated with his endowment process, then u < 0 . Thus, a higher
|u| indicates a better quality of the hedge that the bond provides.
2.2
Allocative Efficiency and Informational Efficiency
CP, PA, and DP shall be compared according to their allocative and informational efficiencies.
Allocative efficiency is the expected total surplus—i.e., the expected sum of the dealer’s and
traders’ payoffs from trades realized until time T. Trading between the dealer and an informed
trader does not contribute to efficiency, as both value the asset the same, at v. Trading between the
dealer and an uninformed trader contributes to efficiency amount |u|, the private component of
the uninformed trader’s valuation. The first-best allocative efficiency, denoted by V ⇤ , is attained
when each arriving uninformed trader immediately trades with the dealer, buying when u > 0
and selling if u  0:
V⇤ ⌘
lU E [|u|]
.
r
(1)
The independence of V ⇤ from g0 captures the fact that efficient allocation of the asset is independent of v, and hence of the precision of information about v.
Often, allocative efficiency shall be referred to as simply “efficiency.”
Informational efficiency is the expected payoff of an unmodelled outside observer who bases
his investment decisions on the prices and quantities revealed during trade.10,11 In order to operationalize the concept of informational efficiency, assume that an unmodelled investor observes
announced trades and invests continuously. At each time t, the investor (along with each uninformed trader) assigns probability gt to the event v = 1. The investor’s instantaneous loss due to
uncertainty about v is denoted by l (gt ), where l (0) = l (1) = 0 and l (gt ) > 0 for gt 2 (0, 1). For
instance, l (gt ) ⌘ gt (1
gt ) identifies the loss with the variance of v conditional on the informa-
10 It is assumed that the outsiders’ investments do not affect the asset’s fundamental value, v—by contrast to the
recent literature that focuses on the feedback between market prices, affecting investment decisions, affecting market
prices (see, e. g., Bond et al. (2010) and Dow et al. (2011).
11 Chen et al. (2007) provide evidence that managers make their investment decisions based on the information (e.g.,
about product demand or about competition) revealed by stock prices. Information relevant for investment is contained
also in bonds and credit-default swaps.
9
tion revealed by past trades. The investor’s expected total loss is defined by:
E
"ˆ
0
T
#
l (gt ) dt | g0 .
(2)
The loss stops at T, when the value of v is realized, and no uncertainty remains. Informational
efficiency is defined as the negative of the loss in (2).
Allocative efficiencies for CP, PA, and DP shall be denoted, respectively by V c , V p , and V d . The
corresponding informational efficiencies shall be denoted by I c , I p , and I d .
Discussion: Interpreting Allocative and Informational Efficiencies
In order to validate allocative efficiency, one must interpret the diversity in u as arising from
the heterogeneity of uninformed traders’ hedging needs, not from the heterogeneity of their beliefs about v that admit no common prior. Otherwise, the identified efficiency gains from trade
would be spurious. In the absence of common priors, welfare analysis is problematic, as has been
explained by Gilboa and Schmeidler (2012), and by Kreps (2012, Section 8.6). In particular, it is unclear why traders’ diverse beliefs should be recognized as a legitimate source of gains from trade;
after all, both traders cannot be right. Even if this recognition is granted, the utilitarian criterion
adopted by allocative efficiency has poorer justification in the heterogeneous prior setting than in
the common-prior setting.12
The definition of allocative efficiency requires that r be an exogenous rate at which a trade
opportunity vanishes, not impatience that, in a richer model, might emerge from traders’ desire
to pre-empt each other.13 If r used in efficiency calculations is inflated by the pre-emptive motive,
the implied losses due to delayed trade are spurious. Nor can r be interpreted as a discount
rate, common to all traders and the dealer. If r were the common discount rate, then allocative
efficiency would increase every time the dealer transferred a dollar to a just-arrived trader, since
the just-arrived trader enjoys the dollar at its face value, whereas the dealer, who has been there
since time t = 0, discounts.
12 In the common-prior setting, Harsanyi (1955) justifies the utilitarian welfare function by assuming that not only
individual preferences but also the social preference must satisfy the (objective) expected-utility axioms. By contrast, in
the subjective expected-utility setting, Hylland and Zeckhauser (1979) show that no compelling social welfare function
exists, let alone a utilitarian one.
13 The analogy between the behavioral implications of impatience and strategic pre-emption has been first identified
by Wilson (1986).
10
In contrast to allocative efficiency, informational efficiency remains legitimate under alternative interpretations of the model’s parameters, as long as players’ behavior is unaffected. So, if one
believed that most trade were driven by heterogeneous priors and that demand for immediacy
stemmed from strategic pre-emption, then one might prefer to rank market structures according
to the informational efficiency and neglect the spurious allocative efficiency.
2.3
Three Market Structures
The focus shall be on three market structures: a continuous protocol (CP), a periodic auction (PA),
and a dark pool (DP). Each possesses some desirable features, even though none is constructed
to solve a mechanism-design problem. (Nevertheless, DP will turn out to maximize allocative
efficiency, under conditions.) In each of these market structures, a trader is restricted to submitting
a single market order to buy or sell.14 Unmodelled dealer competition motivates setting prices so
that the dealer’s expected payoff is zero.
Upon arrival, a trader observes the dealer’s pricing rule, which reveals all publicly available
information about v. Occasionally, the analysis shall rely on the assumption that each newly arrived trader is oblivious to context, meaning that he disregards context, defined as the timing
of past trades, the timing of the dealer’s past announcements, and the time when trading began.
Context contains no information about v that cannot be inferred from the dealer’s pricing rule.
Condition 2. Each trader is oblivious to context.
Condition 2 can be interpreted as a restriction on a trader’s attentiveness; the trader does not
follow the market before he arrives and decides to trade. Condition 2 is invoked only in the
analyses of DP and unravelling.
The mechanisms’ pricing rules are described below.
The continuous protocol (CP) is a special case of the protocol of Glosten and Milgrom (1985).15
Let gt denote the probability that the dealer and uninformed traders assign to the event v = 1 at
time t. The expected value of v is also gt . The dealer posts the smallest ask At and the largest bid
14 Market
orders are used in the seminal models of Glosten and Milgrom (1985) and Kyle (1985), and are ubiquitous
in practice.
15 Zachariadis (2011) and the papers referenced therein discuss dynamic analyses of Glosten and Milgrom (1985).
These papers focus on the limit behavior of prices, not on efficiency, and do not compare alternative market structures.
11
Bt for one unit of the asset, at which he breaks even on each transaction:
(1
a ) (1
(1
G (A
a) G ( B
g)) ( A
g) (g
g) + ag min {0, A
B ) + a (1
g) min {0,
1} = 0
(3)
B} = 0,
(4)
where time indices have been omitted—as they often shall be—for parsimony. Conditions (3) implies that A is the dealer’s revised probability that v = 1 conditional on a trader buying. Similarly,
(4) implies that B is the revised probability that v = 1 conditional on a trader selling. By continuity, it can be verified that one can always find an A and a B in [0, 1] that solve (3) and (4); that is,
the market never breaks down, even though sometimes all uninformed traders can refrain from
buying, selling, or both.
All trades are executed immediately and are publicly observed.
It is assumed (as has been customary in the literature following Glosten and Milgrom (1985))
that after trade has been consummated, the dealer cannot make the trader reveal his type—even
though such a revelation is a matter of indifference to the trader. This assumption follows as a
conclusion if one postulates that a trader will not report his type unless he strictly gains from
doing so. Such strict incentives cannot be provided in CP.16
In the periodic auction (PA), upon arrival, each trader privately submits a buy or a sell market order to the dealer. In addition, each trader privately reports his type—i.e., whether he is
an uninformed trader or an informed trader privy to a certain v. This report is a shorthand for
the assumption that each trader can choose from different types of sell and buy orders, thereby
revealing his type to the dealer. Such orders can be constructed (by letting traders bet on the order flow), and each trader type’s strict preference for the designated order can be assured.17 The
dealer executes the submitted orders periodically, with period t. All trades are publicly observed.
Transaction prices are personalized. (Posted prices are not; the dealer cannot observe traders’
0
types.) Pick any trader i. Let pg denote the transaction price of trader i given the dealer’s belief g0
16 The
need for strict incentives can be motivated by a trader’s minimal concern for privacy.
17 Indeed, assume that each trader can choose either order T , which is a simple (buy or sell) market order, or order T ,
U
I
which is TU plus a discount or a surcharge, depending on whether another trader trades on the same or on the opposite
side of the market, respectively. These discounts and surcharges can be constructed so that (i) each uninformed trader
uniquely prefers TU , (ii) each informed trader uniquely prefers TI , and (iii) the dealer breaks even on TU and loses an
arbitrarily small amount on TI . The details are in Section B.1 of the Supplementary Appendix.
12
about v, where belief g0 is conditional on the publicly observed history of past trades and on the
privately inferred types of trader i’s opposing traders, defined as the traders who have arrived
since the last trade, except trader i. If one of the opposing traders is informed, then g0 = v; if not,
then g0 = g (g is the public belief). Thus, trader i’s transaction price is either p0 , pg , or p1 , and is
independent of his order and report.
Let # ⌘ e
lI t
denote the probability that trader i assigns to the event that no opposing trader
is informed. By setting
p0 =
g#
1
#
,
pg = g,
and
p1 =
1 g#
,
1 #
(5)
the dealer breaks even in expectation and eliminates the bid-ask spread faced by uninformed
traders, thereby maximizing the expected gains from trade with each transacting trader.18 In particular, an uninformed trader’s expected transaction price is:
#pg + (1
⇣
#) gp1 + (1
⌘
g) p0 = g,
regardless of whether he buys or sells. This expected price g can be verified to maximize the
instantaneous surplus from trading with an uninformed trader:
g 2 arg max E [[u + g > p] u + [u + g < p] ( u)] .
p
Furthermore, because each trader transacts at the price that equals the asset’s expected value to the
dealer (given the opposing traders’ reports), the dealer’s expected payoff (conditional on publicly
observed history) is zero.
Each uninformed trader faces the execution risk:19
p1
p0 =
18 Pagano and Röell (1992) corroborate:
1
1
#
.
(6)
“the auction market is on average cheaper for the uninformed trader than the
[continuous] dealer market.”
19 When g 2 (0, 1), p0 < 0 and p1 > 1. Thus, a trader may end up paying for the asset more than the asset can be
possibly worth. One can show that p0 and p1 can be confined to [0, 1] at the cost of a positive bid-ask spread, which
nonetheless is smaller than the spread in CP.
13
(Because p1
pg
p0 , each informed trader’s execution risk is smaller, either p1
g or g
p0 .)
p1
p0 )
Any period t honoring traders’ tolerance L for the execution risk (i.e., honoring L
satisfies t
t ⇤ , where:
t⇤ ⌘
1
L
ln
.
lI
L 1
(7)
In a dark pool (DP), the dealer does not publicize trades immediately, but announces them
with some Poisson intensity f. Each trader’s transaction price depends on past order submissions,
privately observed by the dealer.20 Each trader’s prices are determined as in PA except that PA’s
traders who arrive during the deterministic period t are replaced by DP’s traders who arrive
within last t units of time, where t is the minimum of (i) the time since trading has begun and
(ii) the time since the dealer’s last announcement. Condition 2 implies that an arriving trader
believes that t is distributed exponentially with the parameter r + f. Hence, the probability that
no informed trader has arrived since the last announcement or since trading has begun is:
#⌘
ˆ
0
•
e
lI t
(r + f ) e
(r + f ) t
dt =
r+f
.
r + f + lI
(8)
The transaction prices are thus given by (5) with # derived in (8) and with g corresponding to the
prevailing public belief about v.
The probability # in (8) leads to the execution risk:
p1
p0 = 1 +
r+f
,
lI
which does not exceed traders’ risk-tolerance if and only if f  f⇤ , where f⇤ solves
L = 1+
r + f⇤
lI
(9)
and is nonnegative, which occurs if and only if the following condition holds.
20 Among
practitioners, two classes of market structures are referred to as dark pools: market structures that take
transaction prices exogenously from another market (Duffie (2011, Section 1.2) adopts this definition), and market
structures that determine transaction prices internally, but do not immediately publicize them. The dark pool of this
paper belongs to the latter class, which Duffie (2011) calls “dark markets.” For example, in practice, post-trade transparency properties of DP are shared by over-the-counter (OTC) markets for corporate bonds, in which past trades are
reported within minutes through the Trade Reporting and Compliance Engine (Maxwell and Bessembinder, 2008).
14
Condition 3.
L
1+
r
.
lI
Henceforth, assume that Condition 3 is satisfied.
3
3.1
Efficiency Comparison of Market Structures
Continuous Protocol (CP)
Allocative Efficiency: Allocative efficiency shall be computed by first computing the instantaneous gain from trading with an uninformed trader, and then adding up those gains. Define the
markups a (g) ⌘ A (g)
g and b (g) ⌘ g
B (g). The instantaneous gain from trade realized
upon encountering an uninformed trader are
S (g) ⌘
=
ˆ
•
a(g)
ˆ •
a(g)
ug (u) du +
ˆ
ug (u) du +
ˆ
b(g)
•
•
a (1 g )
( u) g (u) du
ug (u) du,
(10)
where the equality is by g (u) = g ( u) and by (3) and (4), which imply b (g) = a (1
inspection of (10), S is symmetric in the sense that S (g) = S (1
g). By
g), and is maximized at g 2
{0, 1}, when both markups are zero.
Aggregating the instantaneous gains from trade in (10) yields allocative efficiency:
V c (g) ⌘
ˆ
0
•
0
B
B
B
( l +r ) t B
le
B (1
B
@
(1
1
a) (S (g) + G ( b) V c ( B))
C
C
C
C
a) (( G ( a) G ( b)) V c (g) + (1 G ( a)) V c ( A)) C dt,
C
A
c
c
+a (gV ( A) + (1 g) V ( B))
where the dependence of A, B, a, and b on g has been suppressed. Solving the above displayed
equation for V c (g) gives:
V c (g) =
(1
a) S (g) + V c ( A) (ag + (1 a) (1 G ( a))) + V c ( B) (a (1
1 + lr (1 a) ( G ( a) G ( b))
g ) + (1
a) G ( b))
.
(11)
15
S
0.25
Vc
0.165
0.24
0.160
0.23
0.155
0.22
0.150
0.21
0.145
0.20
0.140
0.19
0.135
g
0.2
0.4
0.6
0.8
1.0
(a) S, instantaneous gains from trade with an uninformed trader.
0.2
0.4
0.6
0.8
1.0
g
(b) V c , allocative efficiency; r = l = 1
Figure 1: Allocative efficiency under Condition 1 with a = 0.333.
Let T (V c ) (g) denote the mapping defined by the right-hand side of (11). This mapping is a
contraction and inherits the symmetry of S:
Lemma 1. T is a contraction mapping. Hence, V c defined in (11) is the unique fixed point of T and
satisfies V c (g) = V c (1
g) for all g 2 [0, 1] .
Proof. See Appendix A.
Just like S, V c is maximized when g 2 {0, 1}, in which case both markups are zero (i.e., a =
b = 0), so that no uninformed trader is excluded due to a bid-ask spread. As g approaches 12 ,
the asymmetric information between informed traders and the dealer increases in the sense that
the dealer’s uncertainty about v increases. Therefore, one could have conjectured that allocative
efficiency is minimized at g =
1
2 —i.e.,
that V c is quasi-convex. This need not be so, as will be
shown.
Establishing the shape of V c is important because this shape determines the qualitative features of the comparison of CP with PA. The shape of V c is also of independent interest because it
determines the social value of public information, defined as the expected change in allocative
efficiency associated with the policy of publicly releasing a given signal about v.
In order to understand the shape of V c , it is instructive to study the properties of S, some
of which are inherited by V c . These properties are summarized in Lemma 2 and illustrated in
Figure 1a.
16
Lemma 2. S is symmetric (i.e., S (g) = S (1
g)), is maximized at g 2 {0, 1}, is concave at g 2 {0, 1},
and, for some parameter values, is concave also at g = 12 .
Proof. See Appendix A.
The local concavity of S means that, for some common belief g of the dealer and uninformed
traders, one can find a signal about v such that the policy of making this signal public decreases
the expected instantaneous gains from trade. The following corollary (illustrated in Figure 1b)
establishes that, for a sufficiently high r, V c inherits the concavity properties of S. Intuitively,
when r is high, efficiency is dominated by the gains from trade with the first uninformed trader.
Corollary 1. There exists r̄ such that for any r > r̄ there exists a prior belief g and a signal about v such
that the social value of public information is negative.
Proof. See Appendix A.
In order to see why public information can be of little social value when all traders are well
informed about v, consider the following informal argument justifying the local concavity of S
near g = 0.21 Assuming that the markups can be approximated using Taylor expansion near
g = 0, each markup is of order g. The probability measure of excluded uninformed traders is of
the order of the markups, and hence also of order g. The lost surplus is of the order of the product
of the measure of excluded traders times the markups (which are of the order of excluded traders’
private valuations). Hence, the lost surplus is of order g2 .
The possibility of concavity of S near g =
1
2
is subtler, and is also economically more interest-
ing, as it implies that S is not minimized at g = 12 , when asymmetric information is the greatest.
Function S is concave near g =
1
2
when the bid-ask spread (A (g)
B (g) = a (g) + b(g)) changes
little in g near 12 , and when the surplus lost due to markups is convex in each markup. In this
case, a slight deviation of g from
1
2
increases one markup approximately as much as it decreases
the other one, thereby increasing the surplus lost due to the markups. The intuition for why the
surplus lost due to each markup is convex in that markup is illustrated in Figure 2.
21 Levin
(2001) analyzes a lemons market in which the realized gains from trade are non-monotone in the amount of
asymmetric information. The observation that public information can have negative value at a competitive equilibrium
of an exchange economy is due to Hirshleifer (1971).
17
|u|g(u)
R1
b
R3 R4
R2
a
0
½
"
𝛾
Figure 2: Assume Condition 1 with a = 13 . The fourth (Southeastern) quadrant plots markup
a as a function of belief g. The first (Northeastern) quadrant translates the markup (which is
also the private valuation of the marginal excluded uninformed trader) into the surplus lost from
excluding buyers with that marginal valuation. The area R3 + R4 is the surplus lost from excluding
the buyers with the smaller than marginal valuation corresponding to the markup at belief g = 12 .
Analogously, the third (Southwestern) quadrant depicts markup b as a function of belief g, and
the second (Northwestern) quadrant translates this markup into the surplus lost from excluding
sellers with the marginal valuation, equal to the markup. The area R2 is the surplus lost from
excluding the sellers with the smaller than the marginal valuation corresponding to the markup
at belief g = 12 . At g = 12 , a 12 = b 12 . As g rises from 12 to 23 , a falls, b rises, and the bid-ask
spread a + b remains unchanged. As a result, the surplus lost due to a falls by R4 , and the surplus
lost due to b rises by R1 . Because |u| g (u) is increasing, the surplus reduction R1 R4 is positive.
Graphically, R1 R4 is twice the area of the shaded triangle in the first quadrant.
18
Further intuition for the concavity of S near g =
1
2
uses the Taylor expansion of S near g =
(assuming that the relevant derivatives exist). Differentiating (10) once gives S0
1
2
1
2
= 0, and
differentiating (10) once more gives:
✓ ◆
1
S
=
2
00
2 a0
2
∂
( ag ( a))
∂a
2a00 ag ( a) ,
where a, a0 , and a00 are all evaluated at g = 12 . Then, assuming that second-order approximation
of S using Taylor expansion applies at g = 12 :22
✓ ◆
1
S (g) = S
2
S is concave near g =
•
∂
∂a
1
2
if S00
a
1
2
0 2
∂
a00 ag ( a)
( ag ( a)) +
∂a
( a 0 )2
!✓
g
◆2
1
2
+ O g3 ;
(12)
< 0, which is more likely to hold when:23
( ag ( a)) is positive, meaning that that the surplus loss ag ( a) D due to excluded traders
with valuations between a and a + D (who are of probability measure g ( a) D) increases as
the markup a falls by D > 0 and markup b rises by D;
•
a00 ag( a)
,
( a 0 )2
if negative, is not much less than zero 0, meaning that when a falls by a D, b rises by
not much less than D—that is, the bid-ask spread a + b does not decrease by much as the
dealer’s uncertainty about v decreases, captured by g’s small deviation from 12 .
The dependence of the sign of S00
1
2
on a can be illustrated by letting a ! 1 and assuming that
equilibrium markups exist. Then, for (3) to hold, it must be that a !
1
2
. Furthermore (assuming
that the relevant derivatives exist),
lim
a !1
so that, by (12), S00
1
2
a00 ag ( a)
( a 0 )2
=0
< 0 if and only if
∂
∂a
and
lim a0
a !1
2
= 1,
( ag ( a)) > 0. Intuitively, when a ⇡ 1, the dealer’s
posterior belief that v = 1 conditionally on the buyer’s purchase is 1 for any prior, and hence
a (g) ⇡ 1
g, which is linear in g.
22 Here,
O is the “big-O” notation; O g3 signifies terms of order g3 and higher.
⇣
⌘2
1
23 The term ( a0 )2 ∂ ( ag ( a )) g
in (12) corresponds to twice the area of the shaded triangle in the first quadrant
2
∂a
⇣
⌘2
in Figure 2. Term a00 ag ( a) g 12 in (12) cannot be illustrated in that figure because the figure is based on an example
in which a00 = 0.
19
The discussion has focused on the local concavity of S and V c near g 2 {0, 1} (because it is
easy to establish analytically), and near g =
1
2
(because it is economically interesting, implying
that efficiency need not be minimized when asymmetric information is maximized). One can
construct examples, however, in which V c is not only locally concave at some g, but also has more
than two local maxima in g. Hence, little can be said formally about V c in general, except that V c
is symmetric and is maximized at g 2 {0, 1}.
Informational Efficiency: The informational efficiency at belief g satisfies the following functional equation:
I c (g) =
+
ˆ
•
0
le
ˆ
•
0
( l +r ) t
le
0
rt
B (1
B
@
ˆ
0
t
e
rs
l (g) dsdt
a) ( G ( b) I c ( B) + ( G ( a)
G ( b)) I c (g) + (1
+a (gI c ( A) + (1
g) I c ( B))
1
G ( a)) I c ( A)) C
C dt,
A
where the first line captures the observer’s expected loss until the first trader arrives, and the
second line incorporates the change in the public belief that follows trade, if trade occurs.
3.2
Periodic Auction (PA)
Allocative Efficiency: Normalize the time of the most recent auction to 0. Let t 0 denote the arrival
(stopping) time of a trader about whom it is only known that he has arrived within the auction’s
period t. Then, t 0 is distributed uniformly on [0, t ], implying that the expected discount factor of
´t
such a trader is 0 e rt dt/t = (1 e rt ) /rt.24 In expectation, lU t uninformed traders arrive,
each of whom contributes amount E [|u|] to allocative efficiency. Hence, the allocative efficiency,
denoted by V p , is independent of g and solves:
Vp = e
rt
✓
lU
1
e
r
rt
◆
E [|u|] + V p ,
implying:
Vp =
24 This
r (1 e rt ) ⇤
V ,
r (ert 1)
argument does not invoke Condition 2.
20
(13)
where V ⇤ is the “first-best” efficiency, defined in (1).
Informational Efficiency: The expression for informational efficiency captures the fact that the
observer’s loss is positive only until the first informed trader arrives, after which the loss is zero:
p
I (g) =
ˆ
t
0
e
rt
l (g) dt + e
(r + l I ) t p
I (g) .
The above displayed equation can be solved for I p (g):
I p (g) =
l (g) 1 e rt
.
r 1 e (r + l I ) t
(14)
The following lemma shows that V p is decreasing in t (capturing the intuition that a delay in
trade is costly to traders, who demand immediacy) and that I p is decreasing in t (capturing the
intuition that the observer prefers earlier revelation of information).
Lemma 3. Allocative efficiency and, for g 2 (0, 1), informational efficiency are strictly decreasing in the
inter-trade period t.
Proof. See Appendix A.
On efficiency grounds, Lemma 3 justifies the focus on PA with the smallest period consistent
with traders’ risk tolerance. Therefore, henceforth it shall be assumed that t = t ⇤ , where t ⇤ is
the smallest period consistent with traders’ risk tolerance, defined in (7) and reproduced here for
convenience:
t⇤ ⌘
1
L
ln
.
lI
L 1
Because t ⇤ is decreasing in a, l, l I , and L, V p is increasing in a, l, l I , and L.
3.3
Dark Pool (DP)
Allocative Efficiency: In DP, all trades are immediate and uninformed traders face no bid-ask
spread, buying when u > 0 and selling when u < 0. Hence, allocative efficiency is at the first-best
level:
Vd = V⇤ ⌘
lU E [|u|]
,
r
21
where V ⇤ is defined in (1) and is reproduced above for convenience.
Informational Efficiency: For a given announcement intensity f, informational efficiency I d
satisfies:
d
I (g) =
ˆ
0
•
fe
ft
✓
ˆ
t
0
e
rs
l (g) ds + e
( l I +r ) t d
◆
I (g) dt,
where the inner integral is the observer’s loss until the first announcement, after which the loss terminates if and only if the announcement reveals that an informed trader has arrived. Integrating
and expressing I d (g) from the above equation gives:
I d (g) =
l ( g ) (r + l I + f )
.
(r + l I ) (r + f )
Because V d is independent of f and I d is increasing in f (by inspection), on efficiency grounds,
the focus shall be on DP with the largest f consistent with traders’ risk tolerance—i.e., f = f⇤ ,
where f⇤ is defined in (9). In this case,
I d (g) =
3.4
l (g) L
.
r + lI L 1
(15)
Efficiency Comparisons of CP, PA, and DP
The only unambiguous efficiency ranking concerns DP. DP is more allocatively efficient than CP
and PA because DP achieves the first-best efficient outcome. DP will be shown to be less informationally efficient than PA. In expectation, information is released less frequently in DP than in PA,
which favors informational efficiency in PA.25 The countervailing effect is that DP releases information stochastically. Informational efficiency increases in this stochasticity because gains due to
an early release of information are more salient than losses due to a late release of information; the
latter are more likely to be nullified by an exogenous revelation of v at time T. The stochasticity
effect happens to be dominated by the mean effect, however, and hence DP is less informationally
efficient than PA.
The comparison between CP and PA is of interest when DP is infeasible (e.g., because the reg25 Formally,
t⇤ =
1
L
1
ln
<
lI
L 1
l I (L
1)
where the first equality is by (7), and the last equality is by (9).
22
<
1
l I (L
1)
r
=
1
,
f⇤
ulator mandates pre-trade transparency). The relative allocative efficiency of CP and PA depends
on parameters in a complex way. The complexity stems from the highly irregular shape of V c in
(1b). For the special case when Condition 1 holds, Figure 3 compares allocative efficiency in CP
and PA as one varies the share of informed traders a, the uninformed traders’ posterior belief g,
and traders’ demand for immediacy r. In the figure, the efficiency ranking of CP and PA is non-
Figure 3: Condition 1 holds. For the values of parameters a, g, and r inside the shaded solid, PA
is allocatively more efficient than CP. The efficiency ranking of PA and CP is non-monotone in g
and in a.
monotone in a and in g. The nonmonotonicity in g is due to the instances of non-quasiconcavity of
V c that have been discussed in Section 3.1, and the nonmonotonicity in a arises for similar reasons.
The robust feature of this special case in Figure 3 is that CP dominates PA when a is small, or
when g is close to 0 or 1, or when r is high. When a is small, or when g is close to 0 or 1, the bid-ask
23
spread is small, and hence CP achieves efficiency that is close to first-best and thus dominates PA.
As r increases, the necessary delay in PA becomes so costly that CP dominates PA.
Whether PA or CP is more informationally efficient depends on parameters.
The following theorem summarizes the comparison of market structures:
Theorem 1. Suppose that g 2 (0, 1) and that Conditions 2 and 3 hold. Then,
(i) DP is more allocatively efficient than PA and CP. Formally, for all g 2 (0, 1), V d > max {V p , V c (g)}.
(ii) DP is less informationally efficient than PA. Formally, I d < I p .
(iii) CP is more allocatively efficient than PA when r is large, or when a is small, or when g is close
to 0 or 1. Formally, for all (r, l I , lU , L, G, g) such that V c (g) > 0, there exists a finite r̄ such that
r > r̄ implies V c (g) > V p ; for all (r, l, L, G, g, r), there exists a positive ā such that a < ā implies
1
2
V c > V p ; for all (r, l I , lU , L, r) and G with G (z)
or g > 1
ḡ implies V c (g) > V p .

|z| 26
, there exists a positive
2
ḡ such that g < ḡ
Proof. See Appendix A.
It has been a maintained assumption that traders’ arrival rates (l I and lU ) are independent of
the prevailing market structure. One can speculate, however, that in practice, traders’ arrival rates
may be increasing in their expected payoffs, which differ across market structures. Suppose that
this is indeed so. Then, it will be argued that the statement in part (i) of Theorem 1 still remains
true; DP is the most allocatively efficient market structure.
For each market structure, Table 1 reports the traders’ expected payoffs.
PA
DP
CP
informed trader
0
0
uninformed trader
FE u [|u|]
E u [|u|]
E v,g [max {0, v
E u,g [max {0, u
A (g) , B (g)
a (g) , b (g)
v}]
u}]
Table 1: Traders’ expected payoffs in each of the three market structures. The discount factor
F 2 (0, 1) depends on r, r, and t ⇤ , and is computed in (16).
For each market-structure index i 2 {c, d, p} (designating CP, DP, and PA, respectively), let liI
i denote informed and uninformed traders’ arrival intensities, and let V i l , l
and lU
( I U ) denote the
26 The
inequality is satisfied if, for instance, u is distributed uniformly on ( ū, ū) for ū
the requisite restriction on G can be weakened to G (z)
1
2
24
 |z|
1+ a
4a .
1. Moreover, if a is known,
allocative efficiency evaluated at some arrival intensities l I and lU . Because DP achieves first-best
c
c . Because the first-best efficiency is increasing
efficiency, it follows that V d (lcI , lU
) > V c (lcI , lU
)
in lU (as one can ascertain by inspecting (1)), and because the ranking of the traders’ expected
d
payoffs in DP and CP implies lU
c , it follows that V d lc , ld
lU
I
U
c . Finally, because
V d (lcI , lU
)
d = V d lc , ld . Hence,
the first-best efficiency is independent of l I , V d ldI , lU
I
U
⇣
⌘
d
c
V d ldI , lU
> V c (lcI , lU
).
The argument for PA is analogous and establishes:
⇣
⌘
p
p
d
V d ldI , lU
> V p l I , lU .
Hence:
Corollary 2. Suppose that informed and uninformed traders’ arrival intensities l I and lU are weakly
increasing in these traders’ expected payoffs, which differ across market structures. Then, DP is still more
allocatively efficient than CP and PA.
Even though it does not affect the allocative superiority of DP, acknowledging the arrival intensities’ dependence on traders’ expected payoffs could, in principle, affect the relative allocative
efficiency of CP and PA. The reason is that, depending on parameter values, V c can be increasing
p
or decreasing in l I . Therefore, depending on parameter values, lcI > l I can alter the ranking of
V c and V p relative to the case in which l I is the same in both market structures. (The inequality
p
lcI > l I holds because informed traders’ expected payoff is higher in CP than in PA.) In order
to see the intuition for the non-monotonicity of V c in l I , note that when l I approaches 0, V c approaches the first-best efficiency because the bid-ask spread approaches zero. For intermediate
values of l I , V c is depressed due to adverse selection. When l I approaches infinity, the bid-ask
spread is prohibitive for uninformed traders at first, but soon the informed traders’ information
is reflected in prices, and the bid-ask spread vanishes.27 Fuller exploration of market structures’
27 The
dynamic effect of the increase in l I on the evolution of the bid-ask spread has been identified by Glosten
and Milgrom (1985, p. 91): “For example, an increase in the frequency of insider [informed-trader] arrivals has the
immediate effect of increasing the spread. However, as long as trade continues, the increase in insider activity means
more information will be conveyed by transaction prices. This in turn may mean that spreads in the future will be
smaller because the informational differences between insiders and outsiders [the dealer] will be decreased.”
25
relative merits when traders’ intensity depends on their expected payoffs is left for future research.
4
Unravelling of Market Structures
This section describes when CP, PA, or DP each can prevail at equilibrium with free entry of dealers. Let D denote an incumbent dealer, and let E denote an entrant dealer. Free entry is modelled
by assuming that E can offer a contract not conforming to the prevailing market structure. An
admissible contract is a promise to either sell or buy a unit of the asset at some price, with some
(possibly zero) delay. The promise is void if and only if v is revealed (with Poisson intensity r)
before the promised trade is due. E observes all past public trades.
A market structure is said to unravel if a contract that is profitable for E exists. Susceptibility
to unravelling may alert a regulator to infeasible market structures. If a market structure does not
unravel, it is said to be stable.
Theorems 2, 3, and 4 of this section are concerned with conditions for unravelling, illustrated
in Figure 4 for the case when Condition 1 holds. The shaded areas are the parameter constellations
for which some market structure is stable. For some parameter constellations, all market structures
unravel. A stable market structure need not be more efficient than one that unravels.
4.1
Continuous Protocol
A bid-ask pair ( B, A) is prohibitive at a belief g if no uninformed trader is willing to sell at B nor
buy at A—i.e., if G ( B
g) = 0 and G ( A
g) = 1. The following lemma characterizes the unique
bid-ask pair that is prohibitive at equilibrium.
Lemma 4. An equilibrium bid-ask pair ( B, A) is prohibitive if and only if ( B, A) = (0, 1).
Proof. See Appendix.
A necessary and sufficient condition for CP’s unravelling can now be stated:
Theorem 2. The continuous protocol unravels if and only if the bid-ask spread is positive, but not prohibitive—
i.e., if and only if B < A and ( B, A) 6= (0, 1). When unravelling occurs, an entrant dealer attracts uninformed traders by offering to buy or sell with a delay and at prices that improve upon the incumbent dealer’s
prices.
26
1.0
0.8
a
0.6
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
g
Figure 4: CP is stable if g 2 {0, 1} or a 2 {0, 1} (the blue square frame); PA is stable for g in an
interval that is increasing in a (the pink hull-shaped area); DP is stable when a is sufficiently high
(the shaded green rectangle). Condition 1 holds, and r = r = l = 1 and L = 4.
Proof. See Appendix.
Theorem 2 can be restated in an equivalent way: CP unravels if and only if the bid-ask spread
excludes some, but not all, uninformed traders. When some uninformed traders are excluded,
an entrant dealer can propose a trade at a slightly lower markup attracting some of the excluded
traders and none of the informed ones. Informed traders can be excluded if they earn positive
profits with D, and if E sets the delay to be so large that each informed trader prefers to trade
immediately. An informed trader’s payoff with D is positive (for some v) if the bid-ask pair is not
prohibitive.
The condition on the bid-ask in Theorem 2 can be interpreted to mean that, as long as informa-
27
tion about v is asymmetric—but is not too asymmetric—unravelling occurs. One can show that
when uninformed traders’ private valuation components can be sufficiently dispersed, asymmetric information is never so large as to preclude unravelling. In particular, because ( B, A) = (0, 1) is
the only candidate for a prohibitive bid-ask pair (by Lemma 4), an equilibrium with a prohibitive
bid-ask pair is impossible if, for any g 2 (0, 1), either G ( g) > 0 or G (1
g) < 1, or both. This
condition is equivalent to the requirement that the (symmetric, by assumption) support of u be at
⇥ 1 1⇤
least as large as
2 , 2 , which is so for the distribution described in Condition 1. Thus, with sufficiently dispersed private valuation components, any amount of asymmetric information about v
leads to unravelling.
In contrast to the unravelling literature on matching in labor markets (e.g., the market for law
clarks), in which unravelling often leads to deviations to premature transactions, here, unravelling
leads to deviations to delayed transactions. Examples of non-financial markets in which unravelling towards late trades may occur are the dating market, in which courtship signals patience, and
the labor market, in which education signals patience.28 In the present model, a trader’s readiness to wait also signals patience, and thus identifies an uninformed trader. Some uninformed
traders are more patient than informed ones not because the uninformed discount future payoffs
less (indeed, r is the same for all traders), but because the uninformed traders’ opportunity cost
of waiting (which is their payoff from trading with D) can be zero or close to zero.
Even though fragile in the model, continuous trading may be more robust in practice than the
model predicts for at least two reasons that are omitted from the model: (i) E may face entry costs,
and (ii) traders may be able to delay the submission of orders, thereby leaving E’s deviant contracts
unclaimed. Conversely, unravelling would be only reinforced if the model had an additional
reason for informed traders’s impatience: the fear of losing informational advantage should his
private information become public.
4.2
Periodic Auction
In PA, each trader faces a delay, as trade occurs at t ⇤ -long intervals. Because arrivals follow a
Poisson process, a trader who has just arrived believes that his delay is distributed uniformly on
28 An
employer may care about patience owing to its correlation with some productive characteristic. For instance,
Dohmen et al. (2010) report that patience is positively correlated with intelligence.
28
[0, t ⇤ ]. As he waits, he can lose interest in trading (at rate r) or the value of v can be realized (at
rate r). Hence, the trader discounts his instantaneous expected payoff from trade by the discount
factor:
F=
ˆ
0
t⇤
e
(r +r)t dt
t⇤
=
1
⇤
e (r + r ) t
,
(r + r ) t ⇤
(16)
where F 2 (0, 1). Holding fixed the contract offered by E, an uninformed trader is more eager to
desert PA for E when F is lower, so that the trader’s expected payoff from PA is lower.
In order to determine whether PA unravels, it is convenient to consider two cases, either of
which prevails, depending on parameter values. In one case, E can exclude all informed traders
and can profit by trading with only uninformed traders. In another case, this selective exclusion
is impossible, but E can still be able to profit from a contract in which the profit from trading
with uninformed traders exceeds the loss from trading with informed traders. In either case, E
cannot benefit from introducing a delay because an informed trader’s acceptance of E’s contract is
independent of the delay (he earns zero from D’s contract), whereas an uninformed trader is less
likely to accept E’s contract if the delay is positive.
For a sufficient condition for unravelling, consider the case in which E can profitably exclude
all informed traders. If some uninformed traders are willing to buy immediately at an ask A0 > 1,
then E profits because all informed traders are excluded. Such willing uninformed can be found
⇣
⌘
g
if and only if G 11 F
< 1, or equivalently, 1 g < ū (1 F). Similarly, if some uninformed
traders are willing to sell immediately at a bid B0 < 0, then E profits because all informed traders
are excluded. This occurs if and only if G
g
1 F
> 0, or equivalently, g < ū (1
F). Combining
the above two cases gives a sufficient condition for unravelling; PA unravels if E can exclude either
all informed buyers or all informed sellers and still trade:
min {g, 1
g} < ū (1
F) .
(17)
Inequality (17) implies that PA unravels for any g if the support of u is sufficiently large (e.g.,
unbounded). The uninformed with more extreme valuations have more to lose from waiting, and
hence are more willing to accept E’s contract, which promises immediate trade. PA is more likely
to unravel also if the discount factor F is low, which makes waiting in PA relatively unattractive to
29
an uninformed trader. Finally, a g that is sufficiently close to either 0 or 1 implies that E’s markups
are low, and so the uninformed are eager to accept E’s contract. The described qualitative sufficient
conditions for unravelling will be shown to be also necessary.
When (17) is violated,
g 2 [ū (1
F) , 1
ū (1
F)] ,
(18)
informed traders cannot be excluded, and E’s problem resembles the dealer’s problem in CP, with
the difference that each uninformed trader’s outside option, instead of being zero, is his payoff
with D. When E asks amount A0 , an uninformed trader of type u accepts if
A0 > Fu.
g+u
Therefore, the maximal expected payoff of E who intends to sell is
j A (g) ⌘ max
A0 2[g,1]
⇢
ag A
0
✓
1 + (1
a) 1
G
✓
A0 g
1 F
◆◆
A0
.
g
(19)
The maximal expected payoff of E who intends to buy is computed analogously:
j B (g) ⌘ max
B0 2[0,g]
⇢
a (1
0
g ) B + (1
a) G
✓
g
1
B0
F
◆
g
B0
.
(20)
PA is stable at g if and only if (18) holds and E can find neither a profitable ask nor a profitable
bid: j A (g)  0 and j B (g)  0.
The following theorem shows that PA is stable if and only if g is inside a certain interval, which
is increasing in F in the inclusion order.
Theorem 3. PA is stable if and only if g 2 [g⇤ , 1
[g⇤ , 1
g⇤ ], for some g⇤ , which can be such that the interval
g⇤ ] is empty. Moreover, the interval [g⇤ , 1
(and hence in l, L, and
g⇤ ] is weakly (strictly if nonempty) increasing in F
r) in the inclusion order.
Proof. See Appendix A.
In contrast to CP, which unravels towards later trades, PA unravels towards early trades.29
29 Unravelling towards early trades has been documented in labor markets; see, e.g., Roth and Xing (1994). Li and
Suen (2000) explain unravelling by risk-aversion. Du and Livne (2011) give a combinatorial argument that does not
30
PA’s unravelling towards early trades suggests a reason for the prevalence of high-frequency trading in practice. Exchanges sponsoring high-frequency trading may be eager to attract uninformed
traders whose extreme valuations make it costly for them to wait. Note that the fact that PA unravels does not imply that PA is less efficient than, say, CP.
4.3
Dark Pool
As long as DP is feasible (i.e., Condition 3 holds), DP does not unravel. In order to see this, let g
be uninformed traders’ and E’s belief about v. Suppose that E proposes to sell at an ask A with
no delay. (A delay would be suboptimal in DP for the same reason it is suboptimal in PA.) An
uninformed trader accepts E’s contract if and only if A < g, in which case E loses money. An
informed trader accepts if and only if A < v, in which case E, again, loses money. Hence, E who
intends to sell has no profitable entry opportunity. The argument for E who intends to buy is
analogous. The following theorem summarizes the discussion.
Theorem 4. If DP is feasible (i.e., Condition 3 holds), DP does not unravel.
PA unravels as E tempts uninformed traders with immediate trades. This strategy is impotent
in DP, where trades are already immediate. CP unravels as E tempts uninformed traders with
lower markups. This strategy is impotent in DP, where markups are absent.
5
Conclusions
This paper concludes with the discussion of assumptions motivating the choice of the analyzed
market structures.
Synchronous Exchange: Each trader exchanges the asset and money synchronously. A trader
cannot, for instance, obtain the asset immediately and pay later, after v has been observed or after
more orders have been submitted. Such contingent mechanisms are not observed in practice—
perhaps, because they can lead to incentives to manipulate v or future orders, or because the
realization of v cannot be verified and (and hence is uncontractible), or because of traders’ and
rely on risk-aversion.
31
the dealer’s commitment problems.30 If trades could be contingent on v, the first-best allocatively
efficient and the first-best informationally efficient outcomes would be easy to implement.
Privacy Preservation: CP assumes that the dealer cannot learn a trader’s type by simply asking
the trader to report it after his trade has been consummated.
31
In order to motivate this assump-
tion, note that, after his trade has been consummated, a trader is indifferent between reporting
truthfully and lying. So assume that, if asked, each informed trader reports v = 0 regardless of the
true value of v. The dealer learns nothing. This uninformative outcome can be motivated by each
trader’s unmodelled desire to conceal his type—e.g., because he may believe that, with a small
probability, he may benefit from his private information in future, in an unmodelled way.
In PA and DP, the situation is different. Because the dealer can condition a trader’s payment
on the information gleaned from others’ orders and unknown to the trader in question, the dealer
can provide strict incentives to the trader to reveal his type. This observation (which can be stated
rigorously) motivates the assumption that, in PA and DP, each informed trader fully reveals his
type; the incentives to reveal can be made strict by perturbing slightly the pricing rules in PA and
DP.
Feasibility of Personalized Prices: In auctions commonly used in practice, traders transact at
a single price that equates supply and demand. By contrast, in PA, the dealer need not equate
supply and demand (instead, he carries inventories), and different traders, even on the same side
of the market, may transact at different prices. PA nevertheless captures the essential property
of periodic auctions used in practice, which is this: By aggregating orders from multiple traders,
the dealer can better estimate the asset’s value, thereby narrowing down the bid-ask spread. PA
has been constructed so as to facilitate the comparison with CP, which, too, is incompatible with
equating supply and demand.
The point that agglomeration of traders can reduce informational frictions and narrow down
the bid-ask spread is quite general. It can be made also in the standard double-auction environment, even when traders’ valuations are purely private and independent. In this environment, a
simple auction rule proposed by McAfee (1992) equates the supply and demand, and generates
an ask (the same for all buyers) and a weakly lower bid (the same for all sellers) so that it is a
30 Converse
31 The
contracts are observed; a futures contract guarantees a future delivery at a price specified today.
same implicit assumption is made by Glosten and Milgrom (1985) and their followers.
32
dominant strategy for each trader to ask or bid his valuation. As the number of traders grows, the
bid-ask spread converges to zero.
Restriction on the Timing of the Placement of Orders: In the model, a trader must place
his market order immediately upon arrival or else forego the opportunity to trade. A trader’s
limited attention or cost of dynamically managing his order motivate this assumption. So does
tractability. This assumption is restrictive in CP, but not in PA and DP. For instance, in CP, an
uninformed trader who is excluded by the bid-ask spread prevailing at the time of his arrival may
prefer to submit an order later, if prices move in his favor. By contrast, in PA and DP, at the time of
his arrival, each trader expects the same payoff conditionally on trade, independently of the time
when the order is submitted. Because of demand for immediacy, he prefers (weakly, in the case of
an informed trader) to submit his order immediately.
It is a priori unclear whether the allocative efficiency of CP would increase or decrease if traders
could choose when to submit their orders. The direct, non-equilibrium effect of an excluded uninformed trader’s decision to wait is the increase in allocative efficiency due to enabling an otherwise
excluded trader to trade with positive probability in future. The equilibrium effect is ambiguous,
though. Waiting will be not only excluded uninformed traders, but also those for whom the option value from waiting for a better price exceeds the positive gain from trading immediately. This
option-value induced waiting reduces allocative efficiency by jeopardizing efficient trades, due to
demand for efficiency. In addition, early on, the option-value induced waiting exacerbates adverse
selection and widens the bid-ask spread, further reducing allocative efficiency. Equilibrium analysis must also take into account strategic order-submission also by informed traders. Equilibrium
analysis is left for future research.32
32 Back and Baruch (2004) have identified an equilibrium in a model that modifies the model of Glosten and Milgrom
(1985) by assuming that there is a single informed trader who (as in the model of Kyle (1985)) can choose the timing of
his trade. In that model, uninformed traders are not strategic (and thus, among all other things, they do not choose the
timing of trade). Smith (1997) studies the choice of a timing of trade by informed traders in a variant of the model of
Glosten and Milgrom (1985) in which uninformed traders are not strategic.
33
A
Appendix: Omitted Proofs
A.1
Efficiency Comparison of Market Structures
Proof of Lemma 1
Let B ([0, 1]) denote the space of all bounded functions from [0, 1] to R.
Because ag + (1
a ) (1
0 and a (1
G ( a))
well’s monotonicity condition— i.e., W (g)
g ) + (1
a) G ( b)
0, T satisfies Black-
V (g) for all g implies T (W ) (g)
all g, where W, V 2 B ([0, 1]).
For Blackwell’s discounting condition, note that, for all V 2 B ([0, 1]), all a
[0, 1]:
1
1+
 V (g) + ab,
T (V + a ) ( g ) ⌘ V ( g ) + a
r
l
where
b⌘
(1
T (V ) (g) for
0, and all g 2
a) ( G ( A (g) g) G ( B (g) g))
(1 a) ( G ( A (g) g) G ( B (g) g))
l
2 (0, 1) .
r+l
Thus, T satisfies the discounting condition.
Thus, Blackwell’s sufficient conditions are satisfied, and T is a contraction.
Then, the Contraction Mapping Theorem implies that one can compute V c by starting from an
arbitrary function V 2 B ([0, 1]) and iterating T .
In order to show V c (g) = V c (1
metric function. That is, V (1
g), observe that T maps a symmetric function into a sym-
g ) = V ( g ), 1
G (u) = G ( u), and A (g) + B (1
g) = 1
imply T (V ) (1 g) = T (V ) (g). Because the set of symmetric functions is closed, the corollary to the Contraction Mapping Theorem implies that the fixed point of T is symmetric—i.e.,
V c ( g ) = V c (1
g ).
Proof of Lemma 2
In order to prove Lemma 2, it remains to show the concavity of S at the indicated points.
In order to establish concavity at g 2 {0, 1}, differentiate (10) twice:
S00 (g) =
a00 (g) a (g) g ( a (g))
a00 (1
g ) a (1
g ) g ( a (1
a0 (g)
g))
2
g ( a (g)) + a (g) g0 ( a (g))
a 0 (1
g)
2
g ( a (1
g)) + a (1
g ) g 0 ( a (1
g)) .
Using a (0) = a (1) = 0 and the maintained assumptions about g (i.e., that g, g0 < •):
S00 (0) = S00 (1) =
⇣
⌘
g (0) a 0 (0)2 + a 0 (1)2 ,
(A.1)
provided | a00 (0)| < • and | a00 (1)| < •. In order to establish the requisite conditions on a00 ,
34
implicitly differentiate (3) with respect to g, twice, to obtain:
a0 =
a00 =
ag + (1
a (1 2g a)
a) (1 G ( a) aG 0 ( a))
2
2a (1 + a0 ) (1 a) ( a0 ) (2G 0 ( a) + aG 00 ( a))
.
ag + (1 a) (1 G ( a) aG 0 ( a))
Substituting a0 into a00 , and evaluating both at g 2 {0, 1} gives:
a 0 (0) =
2a
1
a
and
2a
1+a
a 0 (1) =
and
4a (1 + a (1
a00 (0) =
4g (0)))
2
4 (1
a00 (1) =
(1 a )
a ) a (1 + a (1
(1 + a )3
4g (0)))
.
whence | a00 (0)| , | a00 (1)| < •, as required. In addition, | a0 (0)| , | a0 (1)| > 0 and g (0) > 0 imply
S00 (0) = S00 (1) < 0 by (A.1).
In order to illustrate the possibility of S00
1
2
< 0, assume Condition 1 with a = 13 . From (3)
and (4):


1
a (g) = g <
g+ g
3


2 g
b (g) = g <
+ g
3 2
1 1 g
3
2
2
(1 g ) .
3
Substituting the above markups into (10) gives:


1 2 5g2
1
2 1 + 2g
S (g) = g <
+
g
3
8
3
3
8
which has S0
1
2
2g2

+ g>
2 2
3
5 (1
8
g )2
,
< 0, as desired.
Proof of Corollary 1
V c (g) is bounded above by V̄ c (g) and below by V c (g), where
V̄ c (g) =
lU
l lU E [|u|]
S (g) +
l+r
l+r
r
and
V c (g) =
lU
S (g) .
l+r
The upper bound V̄ c (g) assumes that after the dealer’s first encounter with a trader, the bid-ask
spread disappears. The lower bound V c (g) assumes that the dealer’s first encounter with a trader
is the only one.
35
Pick a prior belief g 2 (0, 1) and a signal that equiprobably leads to either posterior g + e or
posterior g e, both in (0, 1), for some e > 0, and pick them so that 12 S (g + e) + 12 S (g e) <
S (g). Such a g and a posterior exist by the local concavity of S established in Lemma 2. Because
the inequality is strict, one can find r̄ such that, for all r > r̄:
1
1
S (g + e) + S (g
2
2
e) +
lU E [|u|]
< S (g) .
r
Combining the above inequality with the bounds on V c gives:
1 c
1
V (g + e) + V c (g
2
2
e) < V c (g) ,
implying that the public signal’s value is negative also with multiple arrivals if r is sufficiently
high.
Proof of Lemma 3 For an auxiliary result, define
w ( x, y) ⌘ ( x + y) e x
Note that
∂w ( x, y)
= ( y + (1 + x ) (1
∂x
y
xe x+y .
ey )) e x <
xye x < 0,
implying, for all x > 0 and all y, w ( x, y) < w (0, y) = 0.
By differentiation:
dV p
dt
=
dI p (g)
dt
=
lU E [|u|] e
(rt, rt )
<0
1)2
l (g) el I t w (rt, l I t )
 0,
2
rt e(r+l I )t 1
rt w
rt (ert
with the second inequality being strict for g 2 (0, 1).
Proof of Theorem 1
For part (i), note that DP is first-best efficient, PA is never first-best efficient, and CP is first-best
efficient only for g 2 {0, 1}. In order to see that PA is never first-best efficient, recall that trades
in PA involve a delay, which is costly to traders, who are impatient. In order to see that CP is
first-best efficient at g 2 {0, 1}, recall that then the bid-ask spread is zero, and hence all efficient
trades are consummated, immediately. Therefore, CP dominates PA at g 2 {0, 1} , whereas PA
may dominate CP for other values of g, as can be shown in examples.
36
For part (ii), subtract (15) from (14) to obtain:
p
I (g)
d
I (g) =
=
l (g) L
r ( L 1)
l (g) L
r ( L 1)
1
0
B
B
B
@
lI
r
1+
1
1+
lI
r
!
(L 1) (1 e rt )
L 1 e (r + l I ) t
✓
◆ lr
I
L
1
L 1
✓
◆
L
L
L 1
L 1
r
lI
1
C
C
C > 0.
A
where the second equality is from (7), and the inequality follows from
1
1
1+
b
b
1
a
a
a
b
1
=
1+
0
b
for any a > 1 and b > 0,
which itself follows from
1
lim
a !1 a
a
a
b
∂
∂a
and
1
b
✓
1
a
a
a
b
b
◆
< 0.
The proof of part (iii) shall use the following lemma.
1
2
Lemma 5. Suppose G (z)
+a
 |z| 14a
. Define
Ā (g) =
1+a
g
1 a
and
Then, for all g 2 [0, 1], A(g)  Ā (g) and B (g)
B̄ (g) =
1 a
g.
1+a
B̄ (g).
Proof. If the dealer posts ask Ā(g), his payoff is
(1
a ) (1
G ( Ā (g)
g)) ( Ā (g)
g)
ag (1
✓
g (1 + a )
Ā (g)) = ag 1 +
1 a
2G
✓
2ag
1 a
◆◆
.
z (1+ a )
This payoff is nonnegative for all g 2 [0, 1] if, for all z 0, G (z) 12  4a . By continuity of the
dealer’s expected payoff in the ask, the ask at which the dealer breaks even is weakly lower than
Ā (g).
Similarly, if the dealer posts bid B̄ (g), his payoff is
(1
a) G ( B̄ (g)
g) (g
B̄ (g))
a (1
1 a
g) B̄ (g) = ag
1+a
This payoff is nonnegative for all g 2 [0, 1] if, for all z  0, G (z)
which the dealer breaks even is weakly higher than B̄ (g).
Now turn to the proof of part (iii) of Theorem 1.
37
✓
2G
1
2
✓
2ag
1+a
z (1+ a )
4a .
◆
◆
1+g .
Hence, the bid at
In order to show that V c > V p for a sufficiently large r, note that because V p is continuous in
r, because limr!• V p = 0, and because V c (g) is independent of r, for all (r, l I , lU , L, G, g), one
can find a r̄ such that V c (g) > 0 implies V c (g) > V p for all r > r̄.
In order to show that V c > V p for a sufficiently small a, note that V p is increasing in a and
lima!0 V p = 0, and that V c (g)|a=0 = V ⇤ for all g. It remains to show that V c is continuous in a at
a = 0. There exists a positive a0 such that, for all a < a0 , the conditions of Lemma 5 are satisfied,
and hence
ˆ
S (g) =
•
a(g)
ˆ
ˆ
=
•
2ag
1 a
•
2ag
1 a
ug (u) du +
ˆ
ug (u) du +
ˆ
ug (u) du +
ˆ
b(g)
( u) g (u) du
•
2ag
1+ a
•
•
2ag
1+ a
( u) g (u) du
ug (u) du ⌘ S̄ (g) .
By Lemma 5 again, at the belief g0 that succeeds belief g, the markups a (g0 ) and b (g0 ) are bounded
by
2ag0
1 a
and
2ag0
1+ a ,
respectively. Each of the bounds is increasing in g0 . Belief g0 is maximal when a
trader buys, in which case g0 = A (g)  Ā (g). Thus, S (g0 ) is bounded below by S̄ ( Ā (g)). By
analogy, S after the arrival of the kth trader is bounded below by
S̄
g
✓
1+a
1 a
◆k
1
!
.
Let tk denote the stochastic arrival time of kth trader. It has Gamma distribution:
f (t) = le
lt
(lt)k 1
,
( k 1) !
0.
t
The lower bound on the expected contribution of kth trader is
ˆ
0
•
e
rt
le
lt
(lt)k 1
(1
( k 1) !
a) S̄ (gk
1 ) dt
= (1
l
r+l
◆k
a)
✓
l
r+l
◆k
S̄
✓
1+a
1 a
◆k
1
!
g .
Thus, the lower bound on V c is:
V̄ ⌘
•
 (1
k =1
a)
✓
S̄
✓
1+a
1 a
◆k
1
!
g .
(A.2)
k
The kth element in the series is bounded by Mk ⌘ (1 a) r+ll E [|u|]. The geometric series
•
k =1 Mk converges. Hence, the series in V̄ satisfies the⇣Weierstrass⌘ M-test and converges uniformly. Moreover, because each element (1
a)
l k
r +l
S̄
1+ a k 1
1 a
g of the sum V̄ is continuous
in a, the sum V̄ is also continuous in a by the uniform limit theorem. Because V̄ |a=0 =
38
lU E [|u|]
r
=
V ⇤ and because V c
V̄, lima!0 V p = 0 implies that, for any (r, l, g, L, G, r), there exists an
ā 2 (0, a0 ) such that, for all a < ā, V̄ c > V p .
The above proof for a small a can be adapted to conclude that V c > V p for g near 0 or 1 when
1
2
 12 |z|.33 In particular the lower bound V̄ on V c , defined in (A.2), is continuous in
l E [|u
g and satisfies V̄ |g=0 = U r |] = V ⇤ . Because V p < V ⇤ and V p is independent of g, for all
(r, r, l I , lU ) and G such that G (z) 12  12 |z|, there exists a positive ḡ such that, for all g < ḡ ,
V c (g) > V p . Moreover, because V c is symmetric, for all g > 1 ḡ, V c (g) > V p .
G (z)
A.2
Unravelling of Market Structures
Proof of Lemma 4
Proof. Part 1 of the proof shows that ( B, A) is prohibitive only if g 2 (0, 1). Part 2 establishes the
theorem’s conclusion.
Part 1
An immediate implication of the definition of a prohibitive ( B, A) is B < A.
Furthermore, B < A () g 2 (0, 1). First, it will be shown that g 2 {0, 1} =) B = A.
Indeed, g = 1, by (3), implies A = 1. Similarly, g = 1, by (4), implies B = 1. Thus, g = 1 implies
A = B = 1. Analogously one can show that g = 0 implies A = B = 0. In order to see the
converse—that g 2 (0, 1) =) B < A—note that (3) and g > 0 imply A > g, and (4) and g < 1
imply B < g. Hence, B < g < A.
Part 2
Suppose that ( B, A) is prohibitive. By Part 1, g 2 (0, 1). Then, (3) and g > 0 imply A = 1, and
(4) and g < 1 imply B = 0, as desired.
Suppose that ( B, A) = (0, 1). By Part 1, B < A implies g 2 (0, 1). Then, (3) and g < 1 imply
G (1 g) = 1, and (4) and g > 0 imply G ( g) = 0—i.e., ( B, A) is prohibitive, as desired.
Proof of Theorem 2
Part 1 of the proof considers the case g 2 {0, 1}. Part 2 considers the case g 2 (0, 1).
Part 1
Suppose that g 2 {0, 1}. If g = 1, then A = B = 1. If g = 0, then A = B = 0. In either case, E
can attract traders only by pricing at a loss. Hence, CP is stable.
Part 2
Henceforth, consider the case in which g 2 (0, 1). Multiple cases for the values of A and B will
be considered.
Suppose that A < 1. Then, by g > 0, a > 0. (As before, a ⌘ A
g and b ⌘ g
B are
the markups.) Let E offer to sell at price g + D after delay T, where D is any mark-up satisfying
33 The
inequality is satisfied if u is distributed uniformly on [ 1, 1]. Moreover, if a is known, the requisite restriction
on G can be weakened to G (z)
1
2
 |z|
1+ a
4a .
39
D 2 (0, a). Given D, pick T that is sufficiently large for every informed trader to reject the offer:
1
A>e
( r +r ) T
(1
D) ,
g
1
1 g D
ln
. Such a T exists by A < 1.
r+r
1 A
An uninformed trader of type u accepts the contract ( T, D) if e
or T >
( r +r ) T
(u D) > max {0, u a}.
Thus, the contract attracts some uninformed traders who would not trade otherwise (those with
u 2 (D, a)), as well as some uninformed traders who would. Because D > 0, E profits from his
offer.34
Suppose that 0 < B. (The argument parallels the argument for A < 1.) Then, by g < 1, b > 0.
Let E offer to buy at price g D after delay T, where D is a mark-up satisfying D 2 (0, b). Given
D, set T to be sufficiently large for each informed trader to reject the offer:
B>e
( r +r ) T
(g
D) ,
1
g D
ln
. Such a T exists by B > 0.
r+r
B
An uninformed trader accepts the contract ( T, D) if his u is such that e rT (g D u) >
max {0, B u}. Thus, the offer attracts some traders who would not trade otherwise (those with
or T >
u 2 ( B, g D)), as well as some traders who would (those with u 2
Because D > 0, E profits from his offer.
B
e
rT
(g
D) / 1
e
rT
Suppose that A = 1 and B = 0. In this case, g 2 (0, 1) implies that no uninformed trader
trades, or else D would profit from uninformed traders without losing money from informed
traders, thus contradicting the zero-profit condition. Each informed trader is indifferent between
trading and not, and would wait if E offered any price improvement on D’s prices. No uninformed
trader finds D’s prices acceptable, and hence if an uninformed trader is willing to wait to trade at
E’s price, he is willing to wait regardless of the delay. Hence, if E can profit from some contract,
he can profit from a contract that has no delay. The existence of such a contract contradicts D’s
inability to break even at an interior ask or bid.
Therefore—when A = 1, B = 0, and g 2 (0, 1)—E cannot offer a profitable deviant contract.
Once an informed trader arrives, the value of v is revealed. Thereafter, the posterior public belief
is in {0, 1}. In that case, E cannot offer a profitable deviant contract, as has been shown in Part 1.
Proof of Theorem 3
PA unravels if condition (17) holds:
min {g, 1
g} < ū (1
34 Instead
F) .
of trading with a delay, E can commit to trading immediately, but with probability e (r+r)T . This interpretation is not pursued because in practice it may be easier to commit to a delay than to a randomization. In addition,
in a richer model, an uninformed trader could manipulate the contract by requesting to trade with a randomizing E
repeatedly until randomization favors trade.
40
⇤
, B ).
Henceforth, assume that (17) fails:
min {g, 1
F) .
ū (1
g}
(A.3)
The goal is to investigate whether PA still unravels, and under what conditions.
⇥
⇤
Suppose that g 2 12 , 1 and that E offers an ask A0 . In order to attract at least some informed
traders (which is a necessary condition for E to profit), A0 must satisfy â ⌘
A0 g
1 F
denotes a normalized markup.
2 (0, ū), where â
One can rewrite (19):
j A (g) ⌘ max {ag â (1
â2[0,ū]
F)
ag (1
g ) + (1
a ) (1
G ( â)) â (1
F)} .
(A.4)
The remainder of the proof proceeds in steps.
Step 1 By inspection of (A.4), whenever g < 12 , j A (g) > 0 implies j A (1 g) > 0.
Step 2 By the symmetry of G (i.e., G (u) = 1 G ( u) for all u) and by inspection of (19) and
(20), j B (g) = j A (1
g ).
Step 3 By Steps 1 and 2, when g <
1
2,
j A (g) > 0 implies j B (g) > 0, and when g
1
2,
j B (g) > 0 implies j A (g) > 0. Thus, when checking for unravelling, it suffices to verify whether
E has a profitable bid at g <
1
2
or a profitable ask at g
1
2.
Step 4 By inspection of (A.4), j A is strictly increasing in g for g
1
2
Step 5 By Steps 3 and 4, PA unravels at g
if g > g⇤ for some
1
2.
⇤
g .
Step 6 By symmetry (captured in Steps 2 and 3), PA unravels at g <
1
2
if g < g⇤ , where g⇤ is
the same one as in Step 5.
Step 7 Steps 5 and 6 imply that PA is stable if and only if g 2 [g⇤ , 1 g⇤ ], for some g⇤ . In
order to ascertain that the setp[g⇤ , 1 g⇤ ] can be empty, it suffices to consider the case in which
Condition 1 holds and a <
1
2
F
.
Step 8 By inspection of (A.4), if j A > 0, then â 2 (0, ū), implying that j A is also strictly
decreasing in F.
Step 9 Steps 4 and 8 imply that the interval defined in Step 7 is increasing in F in the inclusion
order.
35
References
Back, Kerry and Shmuel Baruch, “Information in Securities Markets: Kyle Meets Glosten and
Milgrom,” Econometrica, 2004, 72 (2), 433–465.
Biais, Bruno, Thierry Foucault, and Sophie Moinas, “Equilibrium High Frequency Trading,”
2011.
35 In
numerical examples, one can ascertain that g⇤ need not be monotone in a.
41
Bond, Philip, Itay Goldstein, and Edward Simpson Prescott, “Market-Based Corrective Actions,” The Review of Financial Studies, 2010, 23 (2), 781–820.
Brusco, Sandro and Matthew O. Jackson, “The Optimal Design of a Market,” The Journal of Economics Theory, 1999, 88 (1-39).
Chen, Qi, Itay Goldstein, and Wei Jiang, “Price Informativeness and Investment Sensitivity to
Stock Price,” The Review of Financial Studies, 2007, 20 (3), 619–650.
Crémer, Jacques and Richard P. McLean, “Full Extraction of the Surplus in Bayesian and Dominant Strategy Auctions,” Econometrica, 1988, 56 (6), 1247–1257.
Demsetz, Harold, “The Costs of Transacting,” Quarterly Journal of Economics, 1968, 82 (1), 33–53.
Dohmen, Thomas, Armin Falk, David Huffman, and Uwe Sunde, “Are Risk Aversion and Impatience Related to Cognitive Ability?,” American Economic Review, 2010, 100 (3), 1238ñ–1260.
Dow, James, Itay Goldstein, and Alexander Guembel, “Incentives for Information Production in
Markets where Prices Affect Real Investment,” 2011.
Du, Songzi and Yair Livne, “Unraveling and Chaos in Matching Markets,” 2011.
Duffie, Darrell, Dark Markets: Asset Pricing and Information Transmission in Over-the-Counter Markets (Princeton Lectures in Finance), Princeton University Press, 2011.
Foresight: The Future of Computer Trading in Financial Markets
Foresight: The Future of Computer Trading in Financial Markets, Final Project Report, The Government Office for Science, London 2012.
Foucault, Thierry, Ohad Kadan, and Eugene Kandel, “The Limit Order Book as a Market for Liquidity,” Review of Finacial Studies, 2005, 18, 1171ñ–1217.
Fuchs, William and Andrzej Skrzypacz, “Costs and Benefits of Dynamic Trading in a Lemons Market,”
2012.
Garbade, Kenneth D. and William L. Silber, “Structural Organization of Secondary Markets: Clearing Frequency, Dealer Activity andLiquidity Risk,” The Journal of Finance, 1979, 34 (3), 577–593.
Gilboa, Itzhak and David Schmeidler, “A Difficulty with Pareto Dominance,” 2012.
Glosten, Lawrence R. and Paul R. Milgrom, “Bid, ask and transaction prices in a specialist market
with heterogeneously informed traders,” Journal of Financial Economics, 1985, 14, 71–100.
Harsanyi, John C., “Cardinal Welfare, Individualistic Ethics, and Interpersonal Comparisons of Utility,”
Journal of Political Economy, 1955, 63 (4), 309–321.
42
Hirshleifer, Jack, “The Private and Social Value of Information and the Reward to Inventive Activity,”
The American Economic Review, 1971, 61 (4), 561–574.
Hylland, Aanund and Richard Zeckhauser, “The Impossibility of Bayesian Group Decision Making
with Separate Aggregation of Beliefs and Values,” Econometrica, 1979, 47 (6), 1321–1336.
Jovanovic, Boyan and Albert J. Menkveld, “Middlemen in Limit-Order Markets,” 2010.
Kreps, David M., Microeconomic Foundations I: Choice and Competitive Markets, Princeton University
Press, 2012.
Kyle, Albert S, “Continuous Auctions and Insider Trading,” Econometrica, 1985, 53 (6), 1315–1335.
Levin, Jonathan, “Information and the Market for Lemons,” The RAND Journal of Economics, 2001, 32
(4), 657–666.
Li, Hao and Wing Suen, “Risk Sharing, Sorting, and Early Contracting,” Journal of Political Economy,
2000, 108 (5), 1058–1091.
Madhavan, Ananth N., “Trading Mechanisms in Securities Markets,” 1990.
Mas-Colell, Andreu, Michael D. Whinston, and Jerry R. Green, Microeconomic Theory, Oxford
University Press, 1995.
Maxwell, William and Hendrik Bessembinder, “Transparency and the Corporate Bond Market,”
Journal of Economic Perspectives, 2008.
McAfee, Preston, “A Dominant Strategy Double Auction,” Journal of Economic Theory, 1992, 56 (2),
434–450.
Menkveld, Albert J., “High Frequency Trading and the New-Market Makers,” Tinbergen Institute Discussion Paper, 2011.
Pagano, Marco and Ailsa Röell, “Auction and dealership markets: What is the difference?,” European
Economic Review, 1992, 36, 613–623.
Rosu, Ioanid, “A Dynamic Model of the Limit Order Book,” Review of Financial Studies, 2009, 22 (11),
4601–4641.
Roth, Alvin E. and Xiaolin Xing, “Jumping the Gun: Imperfections and Institutions Related to the
Timing of Market Transactions,” American Economic Review, 1994, 84 (4), 992–1044.
Smith, Lones, “Do Rational Traders Frenzy?,” Mimeo, 1997.
Tkatch, Isabel and Eugene Kandel, “Demand for the Immediacy of Execution: Time is Money,” 2006.
Wilson, Robert, Arrow and the ascent of economic theory: Essays in honor of Kenneth J. Arrow, The
Macmillan Press Ltd., 1986.
43
Zachariadis, Konstantinos E., “A Baseline Model of Price Formation in a Sequential Market,” 2011.
Zhu, Haoxiang, “Do Dark Pools Harm Price Discovery?,” 2012.
44
B
Supplementary Appendix
B.1
Indirect Mechanism
It will be shown that, in PA, each trader can be given a strict incentive to identify himself as either
an informed trader or an uninformed trader, at an arbitrarily small cost to the dealer. Market
orders of two types shall be defined: TU and TI .
When a trader submits a TU order, his transaction price (regardless of whether he buys or sells)
is
p1
if any other trader submits a buy TI order, is p0 if any other trader submits a sell TI order, and
is pg if all other traders (if any) submit TU orders. The prices p0 , pg , and p1 are as specified in (5)
in Section 2.3, and are reproduced here for convenience:
g#
p0 =
1
#
pg = g,
,
and
p1 =
1 g#
.
1 #
When a trader submits a TI order, his transaction price is as with TU ; in addition, the trader
#) or pays a surcharge D0 / (1
receives a discount D/ (1
#) if any other trader submits a TI
order, respectively, on the same or on the opposite side of the market.
Lemma 6. Pick any D > 0 and any D0 satisfying
D0 > D max
⇢
1
g
g
g
,
1
g
,
g 2 (0, 1) .
Then, there exists an equilibrium in which each informed trader strictly prefers submitting a TI order, and
each uninformed trader strictly prefers submitting a TU order.
Proof. The result is established by comparing traders’ payoffs from submitting different orders,
under the lemma’s assumptions about D and D0 . These payoffs are presented in the following two
tables.
Order
v
Informed Trader’s Payoff
TI buy
1
TU buy
1
TI buy
0
TU buy
0
TI sell
1
TU sell
1
#g + (1
# ) p1
TI sell
0
#g + (1
# ) p0 + D = D
TU sell
0
1
1
0
#g
#g
0
(1
# ) p1 = 0
# ) p0
(1
#g
#g + (1
# ) p1 + D = D
(1
#g
(1
# ) p1
#g + (1
D0 =
D0
# ) p0 = 0
1
D0 =
D0
1=0
# ) p0 = 0
1
Order
TI buy
Uninformed Trader’s Payoff
u + gD
TU buy
TI sell
(1
g) D0
u
u + (1
TU sell
g) D
gD0
u
In Lemma 6, for any g 2 (0, 1) and any # 2 (0, 1), the dealer can set D/ (1 #) and D0 / (1 #)
arbitrarily close to 0. Then, dealer (in expectation) breaks even on each TU order and loses an
arbitrarily small amount D on each TI order.
When g 2 {0, 1}, the dealer can remove TI orders, retaining only TU orders.
The expected price of a trader submitting a TU order is independent of the number of traders
submitting TU orders, is weakly increasing in the number of traders submitting TI buy orders, and
is weakly decreasing in the number of traders submitting TI sell orders. In this sense, the pricing
mechanism is “conventional”; the trader’s price deteriorates when he faces more competition on
the same side of the market. This property is inherited by the expected price (which incorporates
discounts and surcharges) applied to a trader submitting a TI order, provided D and D0 are set to
be sufficiently small.
2