(preliminary and incomplete)
Too Many Middlemen? Impaired Learning from Trades 1
Albert J. Menkveld and Bart Zhou Yueshen
first version: June 15, 2012
current version: June 15, 2012
1
Albert J. Menkveld, VU University Amsterdam, Tinbergen Institute, and Duisenberg School of Finance; address:
FEWEB, De Boelelaan 1105, 1081 HV, Amsterdam, Netherlands; tel: +31 20 598 6130; fax: +31 20 598 6020;
[email protected]. Bart Zhou Yueshen, VU University Amsterdam, Tinbergen Institute, and Duisenberg
School of Finance; address: FEWEB, De Boelelaan 1105, 1081 HV, Amsterdam, Netherlands; tel: +31 20 598 2895;
fax: +31 20 598 6020; [email protected]. We thank Bengt Holmström, Albert S. Kyle and seminar or conference participants at Erasmus U Rotterdam, Finance Down Under 2012 (U Melbourne) for very helpful comments
and suggestions. Menkveld gratefully acknowledges VU University Amsterdam for a VU talent grant and NWO for a
VIDI grant. Yueshen thanks his MPhil thesis committee members for helpful comments.
(preliminary and incomplete)
Too Many Middlemen? Impaired Learning from Trades
Abstract
This paper studies high-frequency traders as new middlemen in financial markets with heterogeneous reselling opportunities. Inter-HFT trades occur as a means of optimal reallocation within middlemen (for
best reselling opportunity). Price pressure arises to compensate for the expected cost of carry. Without an
inter-HFT flag, an investor cannot tell such a trade apart from an unpressured one: an inference problem. As
the investor mixes probabilities and posts orders following the previous observed trade, the price pressure
is reinforced. We note that the inference problem is a consequence of middlemen multiplicity – it does not
manifest when there is only one (representative) middleman. The inference problem is costly not only to the
investor, but also to the society.
JEL code: G10
keywords: high-frequency trader, electronic market, middleman
1
Introduction
Recent years have seen significant changes in how financial assets are traded. Notably, electronic limit order
trading gradually grow and dominate exchange markets1 . In the meantime, competition among trading
venues increases market fragmentation; for example, SEC (2010) notes that the market share of NYSE
dropped from 79% in 2005 to 25% in 2009. Along with these changes emerge high-frequency traders
(HFTs), who participate more than 50% of U.S. equity trading volume in the past few years (SEC, 2010).
These HFTs frequently rebalance their positions to zero by actively trading between investors, and therefore,
they effectively serve as new middlemen in the market. Contrary to the market structure where a “specialist”
trades between buyers and sellers, new limit order markets are often seen intermediated by multiple HFTs –
a phenomenon we refer to as middlemen multiplicity.
The role of middlemen in a financial market has been well studied; see, for example, Ho and Stoll
(1983), Glosten and Milgrom (1985), Grossman and Miller (1988), Biais (1993), and many others. Yet the
literature mostly focuses on the aggregate or the representative behavior of the middlemen. Little is known
about how individual middleman interacts with each other, and, more importantly, how such interaction
affects other participants as well as the market condition. This paper develops a theoretical model to provide
new insights, from the angle of information-learning, to the consequences of middlemen multiplicity. In
particular, the model predicts “swap trade” within middlemenand explains how such volume could reduce
the well being of both investors and the society.
The key motivation for middlemen to trade with each other is the heterogeneity in reselling opportunities.
The reselling opportunity of a middleman measures how likely it2 can meet investors and rebalance its
inventory position back to zero3 . Intermediaries, especially HFTs, are subject to cost of carry, which is
motivated from their extreme risk aversion and from the expected margin costs as HFTs are usually thincapitaled hedge funds4 . Such carrying costs arise if they cannot resell/rebalance their inventory timely5 . The
1
Advancement in electronic limit order markets in recent years has been documented in, for example, Jain (2005).
We refer to individual middleman as “it”, because in the context of this paper, such middlemen are best thought of as super
computers – high frequency trading algorithms – in modern limit order markets.
3
The simplest reselling opportunity, when a middleman holds positive inventory, is to resell to a potential buyer in the same
market. Alternatively, the middleman can crawl for counterparties in different markets that trade the same asset. Further, the
middleman could try to replicate a portfolio, from markets that trade different assets, to hedge its exposure. For example, an
equity index portfolio (like ETF SPY) could be replicated reasonably well with properly basketed individual component stocks;
and positions in a future contract can be hedged by trading related options. See Menkveld (2011) for an example of how an HFT
crawl for reselling opportunity in two European markets.
4
See, for example, Easley, López de Prado, and O’Hara (2011).
5
Menkveld (2011) shows that, on average, HFTs rebalance their inventory position to zero in minutes. The highest rebalancing
2
1
reselling opportunities can, in the sense discussed above, measure each middleman’s profitability, and they
differ from one middleman to another. Unconditionally, a middleman’s technology – hardware, algorithm,
colocation, and etc. – determines its reselling opportunity, i.e. its profitability. Even if in the long run
competition drives these fundamentals even across all middlemen, their reselling opportunities could still
differ, conditional on the realization of a particular state of the world, where the inventory holdings, quote
positions on the limit order book, and/or others realize to be different.
We show that when the asset is ex post inefficiently allocated within middlemen, trades naturally occur
among them to reallocate the asset from the low reselling opportunity middleman to who has relatively
high reselling opportunity. We refer to such trades as “swap trades”.6 The model shows that within the
intermediary sector, swap trades are Pareto efficient (proposition 1).
The model also predicts price pressure to accompany such swap trades. This is because when swap
trading, the buyer middleman requires a discount on the price to compensate the expected cost of carry in
case that it cannot resell the asset in time. Such price pressure always exists and its size is determined by
the buyer’s reselling opportunity7 . (Only when the buyer middleman has probability one to resell, the price
pressure diminishes to zero.)
On the other hand, when the asset is (ex post) efficiently allocated within middlemen, there will be no
swap trade. Instead, the middleman who has the position may be able to resell the asset to an investor at the
unpressured, fundamental price. From the perspective of other market participants, however, a swap trade
(with pressured price) cannot be distinguished from such a fundamental trade (with unpressured price). An
inference problem hence immediately arises if the fundamental value is not perfectly understood. Note that
the inference problem is the consequence of middlemen multiplicity: When a single middleman intermediates trades, or when the intermediary sector is viewed in aggregation, swap trades do not manifest as
reallocation within the middlemen internalizes, and only the fundamental trade, if any, reflects in the market
history. Although the one-middleman structure has dominated financial markets for centuries, we view such
characterization inappropriate in modern electronic limit order markets, where the role of intermediary is
delegated by multiple HFTs.
frequency reaches seconds.
6
In line with the theory, we conjecture that a significant proportion of HFT trading volume, which accounts for more than 50%
of total U.S. equity trades (SEC, 2010), can be attributed to swap trades.
7
The bargaining power between the two middlemen also matters. In the model, however, we derive a lower bound of the price
pressure, where the seller middleman – the first mover – has the full bargaining power and posts a limit order, i.e. a take-it-or-leave-it
offer, to the buyer middleman.
2
[Figure 1 about here]
Figure 1 illustrates the trade cycle in two scenarios: i) traditional specialist market, and ii) electronic
limit order market. When only one middleman M (e.g. NYSE specialist) exists, after the seller S trades
with it, she knows that any subsequent trade reflects fundamental information as another investor B must
participate. However, when multiple middlemen M1 and M2 (HFTs) exist, the seller has to mix probabilities
for each observed subsequent trade, because it could either be fundamental or be swap: If the reselling
opportunity for M1 (who has just traded with S) is high, the observed trade is between M1 and a fundamental
buyer and reflects fundamental information; but if the reselling opportunity is low, the observed trade is
between the two M and implies price pressure. The seller’s learning, therefore, could be impaired by the
multiplicity of middlemen. Further, the seller’s subsequent trades will be affected by the learning, continuing
the circle. Finally, although the current paper and figure 1 illustrates the learning problem from a seller’s
perspective, the same intuition applies to buyers and even other middlemen.
We formalize the above intuition in a two-stage – a short term and a long term – sequential game. The
asset is misallocated in this economy where a shocked investor holds an undesirably large position and seeks
to trade with the buyers. Standard adverse-selection stands in the way of efficient allocation8 . The seller
tries to learn the fundamental value from the trade activity in the short term and then trades with the large
bulk of buyers in the long term. Trades are executed in a stylized limit order market. Three scenarios are
considered: i) no middleman, ii) one middleman, and iii) multiple (two) middlemen.
We show that the seller learns more precisely about the fundamental value in the two scenarios with
middleman(men) than without. However, the middlemen multiplicity could impair her learning, compared
to the one-middleman scenario (proposition 2). The model notes that as the seller mixes probability between two possible states of the world, she might eventually reinforce the price pressure of the swap trade
(proposition 3). Further, the social welfare – measured in terms of asset allocation efficiency – might also be
reduced by middlemen multiplicity (proposition 4). Finally, we show that a disclosure policy that requires
swap trades be flagged can improve both the seller’s learning and the social welfare (corollary 3)9 .
8
Since Akerlof (1970), adverse selection has come to be seen as a fundamental cause of market failure. Follow-up articles,
Kessler (2001) and Levin (2001), show that realized gains-from-trade (i.e. welfare) are non-monotonic in information asymmetry.
Our setting illustrates this result (see the analysis in section 4).
9
In this spirit, the CFTC recently published weekly trade statistics to “provide the public, academia, and traders with further
insight into market liquidity.” For 35 of its futures contracts it published large traders’ net position changes from January 2009
through May 2011. This enables one to decompose overall volume into into “fundamental” and “non-fundamental” volume. See
“Net Position Changes Data” at http://www.cftc.gov/MarketReports/NetPositionChangesData/index.htm, retrieved April 11, 2012.
3
The finding that while the intermediary sector – populated by sophisticated HFT algorithms – Pareto
improves through swap trades, the investors and the society might suffer recalls the presidential address by
Stein (2009): “. . . a larger number of sophisticated arbitrageurs . . . need not make the world a better place for
those who look to asset prices to provide a reliable reflection of underlying fundamental values”. We think
of our model prediction as an instance of how, in the context of modern electronic limit order markets, the
particular type of arbitrageurs, HFTs, might inflict negative externality on other less sophisticated investors.
Two distinctive features of our model are worth mentioning. First, the information asymmetry between
investors and middlemen is reversed. The new middlemen, high-tech HFTs, are not at an informationdisadvantage relative to investors, especially in the short term, which we interpret as a time interval of only
(milli)seconds. Both Biais, Foucault, and Moinas (2011b) and Jovanovic and Menkveld (2011) take this
same point of view on the information advantage of HFTs. Empirically, Brogaard (2010) and Hendershott
and Riordan (2011) provide evidence that HFTs substantially contribute to price discovery. Second, the
model predicted “swap trades” differ from the risk-sharing “hot potato trades” in the literature (see, among
others, Lyons (1997), Viswanathan and Wang (2004), and Naik, Neuberger, and Viswanathan (1999)) in that
rather than information-driven, swap trades are motivated by the heterogeneous reselling opportunities of
HFTs.
In addition to the aforementioned HFT literature, the paper also fits into a broader category of algorithmic or automated trading. Foucault and Menkveld (2008) study smart routers that investors use to benefit
from liquidity supply in multiple markets. Hendershott, Jones, and Menkveld (2011) show that algorithmic
trading (AT) causally improves liquidity and makes price quotes more informative. Chaboud et al. (2009)
relate AT to volatility and document a rather weak relationship. Hendershott and Riordan (2010) and find
that both AT demanding liquidity and AT supplying liquidity make prices more efficient. Hasbrouck and
Saar (2010) study low-latency trading or “market activity in the millisecond environment” in NASDAQ’s
electronic limit order book in 2007 and 2008 and find that increased low-latency trading is associated with
improved market quality.
The remainder of the paper is structured as follows. Section 2 sets up the model. Equilibrium is analyzed
in section 3. A measure of learning is developed in subsection 3.3.2. Subsection 3.4 discusses swap-trade,
price pressure, and excessive selling. Welfare is analyzed in section 4, with subsection 4.2 focusing on the
impact of a disclosure regulation. Section 5 concludes. The appendix contains a notation summary (section
4
A) and proofs (section B).
2
Primitives
In this paper, we adopt the convention that upper case letters denote random variables, while their realizations as well as other deterministic variables are denoted in lower case. The environment is a static version
of Lagos, Rocheteau, and Weill (2009)10 , henceforth referred to as LRW setting. The motivation of the
primitives is discussed at the end of this section.
Goods and the asset.
There are two goods, a non-tradable fruit and a numéraire good. There is one traded
asset in finite, fixed supply, a0 (> 0). Each unit of the asset pays off 1 unit of the fruit at the end of the time.
Agents.
There are three types of agents: a shocked investor, S (see “incentives to trade” below), infinitely
many unshocked investors, B, and two middlemen11 , M. For exposition clarity, denote the number of unshocked investors by n, which will be taken to infinity at appropriate times, to facilitate the analysis. Each
individual maximizes the expected utility of consumption at the end of the time. The preferences are quasilinear: c + Zui (a), where c and a are, respectively, the numéraire and the fruit consumption (which coincides
with the agent’s asset holdings), Z is the systematic preference shock that the entire economy is subject to,
and i ∈ {S, B, M} is the agent type. For S, uS (a) = u(a), where the function u(·) is twice-differentiable,
strictly increasing, and strictly concave for all a ≥ 0. For B, uB (a) = min{u(a), u(δ)}, where δ := a0 /(n + 1)
is the socially optimal allocation of the asset per investor; that is, each unshocked investor has no incentive
to hold more than δ units of the asset12 . Set ua (δ) = 1. Finally, uM (a) = uB (a) − ka for all a ≥ 0, where
k > 0 is the marginal cost for M to carry a unit of the asset to the time of consumption13 . Z is binomially
distributed, with probability λ0 to be h and with probability (1 − λ0 ) to be l, where 0 < l < h < ∞.
10
Similar settings have been also used in Lagos and Rocheteau (2007), Lagos and Rocheteau (2009), Duffie, Gârleanu, and
Pedersen (2005), Duffie, Gârleanu, and Pedersen (2007), Pagnotta and Philippon (2011) and etc.
11
In LRW (and etc.), the intermediary sector is populated by a continuum of market makers with unit measure. The current
paper features the interaction within the middlemen, and such interaction is most clearly illustrated with two middlemen. Details
of the interaction follows in “trading procedure” below.
12
This restriction emphasizes the homogeneity across all B-type agents by evenly distributing the asset among them. Alternatively, without such a restriction, one can think of B as a representative that maximizes the equally weighted sum of utility of all
such investors. The algebra will be slightly different but the results will not change.
13
To ensure the participation of M, k should be bounded from above by a threshold that will be derived explicitly later in section
3.2 later. (Such cost of carry is motivated in the discussion at the end of this section.)
5
Incentives to trade.
The economy begins with a misallocation of the asset. The extreme case is consid-
ered: The single investor, S, is shocked by an endowment of all a0 units of the asset, and becomes the natural
seller. The buy side constitutes the other investors, each having 0 endowment of the asset.14
We differ from LRW (and others) in that instead of stochastic access to financial markets and bilateral Nash
bargaining, i) information asymmetry is the friction in the way of a direct and efficient asset reallocation,
and ii) there is a stylized limit order market, as in Jovanovic and Menkveld (2011).
Trade environment.
There are three segmented markets. The main market, market 0, begins with S and
the two M. Each M has a private reselling opportunity. We model such private reselling opportunity as
“private markets” 1 and 2 to which only the respective M has access15 . There is one investor, called EB
(early buyer), who arrives early in market i with probability φ · θi , where φ is the probability of his early
P
arrival, and θi is the probability that such uncertain arrival happens in market i ( i θi = 1 and θi ≥ 0 for
i ∈ {0, 1, 2})16 . Formally, define random variable O with P(O = i) = θi as M’s reselling oopportunity, and
random variable E with P(E = 1) = φ and P(E = 0) = 1 − φ for EB’s early arrival. The remaining investors,
called LBs (late buyers), arrive late in all markets with probability one. Prices are discrete; quotes can only
be issued at either l or h.17 All trades are executed in a price-discriminating fashion, consistent with a limit
order market.
Uncertainty. Three random variables determine the state of the world in this economy: the preference
shock Z, the reselling opportunity (i.e. the market EB might arrive in) O, and the actual arrival of EB
E. We fix a probability space (Ω, F , P) such that Ω = {h, l} × {0, 1, 2} × {0, 1}, F = 2Ω , and P measures
P(Z = h) = λ0 , P(Z = l) = 1 − λ0 , P(O = i) = θi for i ∈ {0, 1, 2}, P(E = 1) = φ and, P(E = 0) = 1 − φ.
14
The extreme misallocation does not affect the results of the paper. As will be clear soon, the size of holding endowment does
not qualitatively affect S’s decision. The welfare results are robust to the size of misallocation by the limit argument that the number
of (unshocked) investors is infinite. See the analysis in sections 3.3.1 and 4.
15
An agent without the access to a market cannot retrieve the trading information (e.g.: quotes, trade price and volume) in that
market. (The link between private reselling opportunity and such a “private market” is motivated at the end of this section.)
16
The two events, that the EB arrives and that he might arrive in market i, are independent for i ∈ {0, 1, 2}. Therefore, the
intersection of the two events, that EB arrives in market i, has probability φ · θi . Note also that the EB arrives, if ever, only in one of
the three markets.
17
Discrete price restriction is introduced only for modeling convenience. Numerical procedures can help solve the equilibrium
strategies in a more general framework where prices are continuous and Z is distributed on an arbitrary support.
6
Time line and trading procedures
t=1
t=2
t=3
t=4
S submits supply
[M trade(s) against it]
O and Z realize and are revealed
[M attempt(s) to resell]
[EB buys the asset]
S learns and trades
with LB
t=1
0M
1M
2M
(no middleman)
(one middleman)
(multiple middlemen)
S submits a supply schedule in
S submits a supply schedule in
S submits a supply schedule in
market 0 and then leaves.
market 0 and then leaves.
market 0 and then leaves.
M trades against the supply.
Both M rush to market 0. Nature
tosses a (fair) coin to pick the
winner, called M1.
t=2
Z is revealed to EB and LB.
Z is revealed to M, EB, and LB.
Z is revealed to both M, EB, and
LB.
Nature picks the channel in
Nature picks, and reveals to M,
Nature picks, and reveals to M,
which EB might arrive.
the channel in which EB might
the channel in which EB might
arrive.
arrive.
M submits a supply schedule to
M1 submits a supply schedule to
the main market or its private
either the main market or its
market (but not both).
private market (but not both).
t=3
M2 trades against the supply in
the main market (if there is any).
t=4
Nature decides whether EB
Nature decides whether EB
Nature decides whether EB
arrives.
arrives.
arrives.
Upon arrival, EB trades against
Upon arrival, EB trades against
Upon arrival, EB trades against
the supply (if there is any).
the supply (if there is any).
the supply (if there is any).
M leaves the economy.
Both M leave the economy.
S returns, parses the main
S returns, parses the main market
S returns, parses the main market
market activity, and submits a
activity, and submits a supply.
activity, and submits a supply.
LBs arrive and trade against the
LBs arrive and trade against the
LBs arrive and trade against the
supply (if there is any).
supply (if there is any).
supply (if there is any).
supply.
7
Timing. The difference between the arrivals of EB and of LBs captures the difference between thin and
segmented markets (in the short-term) and densely populated markets (in the long-term). We illustrate the
time line on page 7. In addition, three scenarios are described in the table: no middleman (“0M”), one
middleman (“1M”), and two middlemen (“2M”).18
Motivation for the primitives
Trades are motivated by a misallocation where a seller holds too many of the asset and wants to sell them
to the buyers who have higher marginal utility than S does. The main friction that hampers the efficient
allocation, under which all investors equally share the total supply, is the information asymmetry between
the first mover (S) and the late mover (LB). Assuming infinitely many buyers and restricting their holdings
to δ (= a0 /n), the model focuses on the information friction (see the analysis in section 3). We motivate
such friction by the mere fact that late agents observe more than the early one does, and it is necessary for
the early one, S, to stick her neck out to explicitly show her interest to sell. The adverse-selection cost of
S follows the discriminatory pricing in a limit order market19 . We consider a simplest form of uncertainty
where Z is binomially distributed, because such a distribution can be completely characterized by a single
variable, ensuring the tractability. Empirical evidence suggests that adverse-selection has become a large
component in transaction cost; see, for example, Hendershott, Jones, and Menkveld (2011).
S can try to mitigate the adverse-selection problem by trading, in the short-term, with potentially early
arriving, more informed EB. The reason why informed EB arrive early is that he observes signals about the
fundamentals earlier than other LBs do. For example, it might be that the EB is better equipped than the
majority of buyers and has lower latency to access the market. After the learning, S can then trade, in the
long-term, with LBs with less adverse selection cost. Though the model only allows a one-shot learning
from the short-term trading, in reality, such learning from short-term trades repeats. One can think of the
current model as a simplified version of one with recursive short-term trades from which S can learn about
the fundamental value, before the market clears with the arrival of massive buyers in the long-term. Learning
18
Two variants of “1M” are noted. The first is described and analyzed in this paper, where the M can access its private reselling
opportunity only, not that of the other M. This view is consistent with a traditional specialist market where the middleman can
only make use of its own reselling opportunity. The second is an aggregated/representative M, who can access both M1 and M2’s
reselling opportunities. When considering the intermediary sector as a whole, this second view is appropriate. Because the current
paper aims to explore the interaction within middlemen, the analysis of the second aggregate view is not presented. An extension
with this second view of “1M” can be easily incorporated.
19
Optimal strategies under discriminatory pricing has been studied, for example, by Wang and Zender (2002) and Viswanathan
and Wang (2002), who analyze the adverse-selection cost with continuous prices.
8
from repeated trading has been studied in different contexts by, for example, Kyle (1985) and Glosten and
Milgrom (1985).
Both M observe the realization of Z, mainly because of their superior information processing technology.
For this reason, it is our preferred interpretation to think of M as high-frequency traders, who analyze market
information much more accurately than human investors in very short time horizons (as the short-term in the
model). Biais, Foucault, and Moinas (2011a) model this informational edge by allowing institutions to invest
in algorithm and observe the fundamental value; Jovanovic and Menkveld (2011) allows the middleman to
observe perfectly “hard information” that is “easily processed by machines such as price quotes in the
local index, same industry stocks, foreign exchange, etc”. One can think of the information symmetry
between M and EB after the revelation of Z as an extreme case of Jovanovic and Menkveld (2011) where
the portion of “soft information” is zero20 and in the long-term, the (late) buyers always observe the “hard
information”. We choose to model S as the (only) uninformed agent in the economy, but it is natural to
extend the analysis to allow buyers to be unaware of the “hard information” with some probability (as in
Jovanovic and Menkveld, 2011)21 . Empirically, Brogaard (2010) and Hendershott and Riordan (2010) have
found algorithmic trades, to which HFT is a sub-category, contribute substantially to price discovery.
The two M share the public reselling opportunity in market 0 if EB arrives there. When EB arrives in
either market 1 or 2, such reselling opportunity becomes private to the corresponding M because the other
M cannot access such a private market. One should interpret an M’s private market as its private reselling
opportunity, which is not accessible by the other M because i) ex ante their technologies (e.g. algorithm
and latency) are different, and because ii) ex post their inventories, capitals, and information set might
differ. It is best to liken the connection among the main market and the private markets to those among, to
name some, the underlying market and the derivative markets, different markets that trade the same security,
and the exchange market and dark pools. The M in our context, in a sense, also serves as cross-market
arbitrageurs. Menkveld (2011) identifies a high-frequency trader who arbitraged across Chi-X (an entrant
market) and Euronext (the incumbent) simultaneously. Such high-frequency M usually have thin capital and
20
We argue that this is a reasonable way to model the fact that in extreme short time horizon, for example, in (milli)seconds,
machine-friendly hard information dominates soft information. See Petersen (2004) for more discussion on hard and soft information.
21
We conjecture that with a more general information structure as in Jovanovic and Menkveld (2011), both the buy-side and
the sell-side will have impaired learning in equilibrium. The current model shows that (proposition 3) such impairment leads S to
excessively sell the asset (an outward shift of the supply schedule). Symmetrically, we conjecture that when buyers have impaired
learning, the demand shifts inward.
9
therefore are extremely risk-averse, as their margin accounts are subject to (instantaneous) mark-to-market
requirement. We model such costs by a reduced-form parameter k, proportional to the inventory size of M.
After posting her supply in t = 1, S leaves the market temporarily as her technology, unlike those of M
(HFTs), has latency accessing the market. On the other hand, as super computers, M continuously monitor
the market and therefore stay in the market throughout t = 1, 2, and 3, and are able to trade with each
other, if desired, without being intercepted by EB; that is, the M2 in the “2M” scenario squeezes in between
M1 and EB (if arrives) in the “1M” scenario, and trades with M1’s supply when desirable before EB can.
To focus on how the intermediation of M between S and EB could reveal information to S, we require M
leave the market after time 3. This is admittedly a simplifying assumption, which essentially rules out the
inter-temporal strategic behaviors of M. When M are allowed to intermediate between S and LBs in t = 4,
they profit from adversely selecting S whenever there is still information asymmetry. It is conjectured that
M have incentives to prevent S from perfectly seeing the realization of Z and thus would strategically act so
when trading with potential EB and/or each other in t = 3.22 Switching on the inter-temporal strategies of
M, therefore, will only worsen the inference problem of S in equilibrium.
The interpretation of the fruit is versatile, following footnote 4 (p. 405) of Lagos and Rocheteau (2007);
see also Duffie, Gârleanu, and Pedersen (2005) and Weill (2007). The uncertainty in the model arises from
the systematic preference shock Z, consistent with LRW. An alternative way, which will lead to a different
mathematical derivation but the same results as in the current model, is to consider (aggregate) production
uncertainty, for example, by letting each unit of the asset produce Z units of the fruit.
In the current model, the arrival probabilities of EB is exogenous given for each market. One can generalize this by, for example, considering the arrival probability φi in each market with some joint density
function G(φ0 , φ1 , φ2 ), and the realization of these reselling probabilities is only observable to the middlemen. The optimal decision of M then will be endogenized about which market to choose when attempting
to resell a position. This will give interesting extension of the model about the size of price pressure when
the two M swap-trade with each other (see the analysis in section 3.2 and 3.4).
22
In section 3.3.3 carefully analyzes how the strategies of M in time 3 might generate inference problem to S, preventing S from
observing the true Z.
10
3
Equilibrium
The equilibrium of the economy is characterized by a number of optimal supply and demand schedules that
agents submit to the markets in different time periods. We backwardly solve these strategies for B, M, and
S respectively. A measure of learning is developed in subsection 3.3.2. We conclude this section with a
discussion of the price pressure and how it is amplified.
3.1
Buyers
In t = 4, all investors arrive in the market. They all observe the realization of Z. Each of them initially has
zero position in the asset and is willing to pay at most Z · ua (δ) = Z (recall that ua (δ) has been normalized to
1) for the last marginal unit. We will henceforth refer to Z as the fundamental value of the asset.
In the limit of n → ∞, δ = (a0 /n) → 0, and yet the aggregate demand is (n − 1)δ → a0 .23 Since all LB
in this limit case demand an infinitesimally small amount of the asset at price Z, the aggregate demand in
t = 4 is therefore a flat line segment at price Z of length a0 .
An EB, if arrives in any of the market in t = 3, also observes the fundamental value Z. In the limit, his
maximum holding is δ → 0. Therefore, he is willing to pay at most Z for this marginal unit.
3.2
Intermediaries
We discuss the strategies of M in the two different scenarios respectively.
One middleman(“1M”)
Call the middleman M1. Suppose M1 carries δ units of the asset into t = 3.24
Knowing that the fundamental value is z and that the demand of a potential EB is δ at z, M1 will post an ask
quote at (z, δ) in the market indicated by Nature, in which the EB might arrive with probability φ. However,
if the indicated market is out of reach, then M1 cannot do anything but hold the position, suffering the cost
of carry k · δ before consuming the fruit.
That is, in the 1M scenario, before the information about arrival and that about the fundamental value,
23
Strictly, the number of LBs is either n − 1, i.e. n investors excluding S, or n − 2, i.e. n investors excluding both S and EB. The
difference does not affect the limit aggregate demand when n goes to ∞.
24
The holding of M1 will not exceed δ in equilibrium, because in t = 1 they will bid only δ units of the asset from S in order to
resell to EB later. For amounts more than δ, the asset generates strictly lower marginal utility (by k) to M than to other investors, as
the remaining part cannot be resold.
11
i.e. the realizations of O and Z, is revealed, M1 has (marginal) utility25 equal to
E (θ0 + θ1 )φ · Z + (1 − (θ0 + θ1 )φ) · Z · (1 − k) = z0 · (1 − k · (1 − (θ0 + θ1 )φ))
(1)
for the last unit of the asset, where z0 = λ0 h + (1 − λ0 )l is the unconditional expectation of Z, i.e. the
fundamental value before the true realization is revealed.
In t = 1, therefore, M1 has reservation value for the δ units of the asset as derived in equation (1). To
ensure participation, we make the following assumption:
Assumption 1. (Participation of M.) Assume that
!
1
l
l
1−
.
1− ≤k ≤
h
1 − (θ0 + θ1 )φ
z0
The upper bound says that the cost of carry for M, k, is low enough such that the (marginal) utility for δ
units of the asset (equation 1) is larger than or equal to l, the lower grid of possible prices in the market.
The lower bound ensures the participation of both M when swapping, as we derive below in the multiple
middlemen scenario.26
Multiple middlemen(“2M”) Throughout the paper, we illustrate the “multiple middlemen” scenario with
two middlemen. The general case can be similarly analyzed accordingly. Both rushing to the main market,
the two M are equally likely to win and trade with S in t = 1. Call the winner M1, who has δ units of the
asset27 .
When enters into t = 3, M1 observes the realizations of O and Z, i.e. the fundamental value and the
market where the EB might arrive (with probability φ). The same as in “1M” scenario, if O = 0 or O = 1,
M1 posts an ask quote of (z, δ) in the respective market (because EB has the reservation value z for the asset).
Now, however, M1 can still “swap trade” the asset with M2 (in the main market) in case of O = 2, i.e. when
the EB might arrive in M2’s private market. In a sense, the swap trade mechanism allows M1 to resell to
some EB that it does not have access to directly. Conditional on that EB might arrive in market 2, M2 has
In the limit, the holding δ → 0. Therefore, we directly interpret the δ units of the asset as the last marginal unit of the holding.
The participation of M must be ensured because of the discrete price grid {l, h}. In a more general case where the price is
continuous and the support is wide enough, such an assumption on the size of cost of carry is not needed.
27
See footnote 24 for the size of M1’s holding.
25
26
12
reservation value
((1 − k)z <) φz + (1 − φ) · z · (1 − k) = (1 − (1 − φ)k) z (< z)
for the marginal unit of the asset. The reservation is higher than that for M1 to hold the asset to consume the
fruit, which is (1 − k)z. Note that when Z = l, i.e. the fundamental value is low, the reservation value of M
is below the lower price grid l, and there will be no “swap trade”. As such, a “swap trade” between M1 and
M2 only happens when Z = h, and to achieve so, M1 as the first mover posts (l, δ) in the main market, and
then M2 takes it over. The associated price pressure, measured as a percentage of the fundamental value, is
(z − l)/z = 1 − l/h. It should be understood that the price pressure, although in the current model is presented
in terms of the price grids l and h, does not require such price discreteness in general. In a more generic
setting where Z is continuous, the swap trade will take place at price (1−(1−φ)k)z, implying a price pressure
of size (1 − φ)k, which is intuitively proportional to the cost of carry k.
To sum up, the optimal supply for M1 in t = 3 is i) to post (z, δ) in either the main market or its private
market when the EB might arrive there, ii) to post (l, δ) in the main market to swap with M2 when O = 2
(EB might arrive in M2’s private market) and z = l, and iii) to do nothing if O = 2 but z = h.
Intuitively, when O = 2, a swap trade between the two M helps to reallocate the asset within the intermediary sectors so that the asset is most likely to be resold. The reallocation improves the allocational
efficiency among the middlemen. The following proposition states this result formally.
Proposition 1. (Swap trades improve efficiency among middlemen.) When the asset is misallocated among
middlemen (i.e. when M1 holds the asset but cannot resell to EB), swap trade (if possible) is Pareto improving within the intermediary sector.
Proof. It is easy to show that both M are weakly better off when O = 2 and Z = h. Information is symmetric
between M1 and M2. Therefore, as M1 is the first mover, it can fully exploit the surplus of M2, and M2 is
(weakly) better off28 . M1, however, gains the swap trade revenue is larger than its utility of consuming and
paying the cost of carry: l ≥ h · (1 − k), which we ensure by making assumption 1.
For simplicity, the following analysis assumes that the two M have symmetric private markets in the
following sense:
28
M2 is, in fact, strictly better off in the current model because of the price discreteness, as M1 has to swap at l (lower than M2’s
reservation).
13
Assumption 2. (Symmetric private markets of M.) Assume that the two M have the same private channels
with θ1 = θ2 = (1 − θ0 )/2.
Based on the above strategy, in t = 1 when neither Z or the arriving market of EB has been revealed,
both M have the same reservation utility (recall that, under assumption 2, θ1 = θ2 ):
(θ0 + θ1 )(φEZ + (1 − φ)(1 − k)EZ) + θ2 (λ0 l + (1 − λ0 )(1 − k)l) . . . for M1
=(θ0 + θ2 )(φEZ + (1 − φ)(1 − k)EZ) + θ1 (λ0 l + (1 − λ0 )(1 − k)l) . . . for M2,
which by assumption 1 is higher than l. Therefore, in t = 1, upon seeing S’s supply, both M rush to the
market and bid at most
1 − θ0
1 + θ0
(1 − (1 − φ)k)z0 +
(1 − (1 − λ0 )k)l
2
2
(2)
for δ units of the asset, where z0 = EZ is the unconditional mean of Z. Because they have the same strategy,
Nature picks the winner with equal probability. Note that neither M will bid for more than δ units from S,
because the amount in excess will not be sold (the maximum capacity of EB is δ) and the marginal utility
for these units is lower for M (Z(ua − k)) than for S (Zua (a)).
3.3
Seller
We first derive the optimal supply of S in t = 4, then discuss and develop a measure for her learning, and
finally analyze her strategy in t = 1.
3.3.1 t = 4
Suppose when entering t = 4, S has a > 0 units of the asset and holds the belief that Z = h with probability λ.
The belief λ is, in general, different from the prior λ0 because S can learn and Bayesian-update the posterior
by analyzing the t = 3 main market activities. We defer the discussion on learning to the next subsection.
The strategy of S is a number of sell limit orders, which can each be characterized by a pair (p, q), where
p ∈ {l, h} is the limit price (for execution) and q > 0 is the order size. The total supply s is the aggregation
of all marginal supply q at different prices.
14
Because all LB know the true realization of Z, the supply of S is subject to their adverse-selection. S
chooses her marginal supplies, ql and qh , at prices l and h respectively, to solve:
U(λ, a) := max(1 − λ) lql + lu(a − ql ) + λ lql + hqh + hu(a − ql − qh )
ql ,qh
(3)
subject to
0 ≤ ql , qh ≤ a, and ql + qh ≤ a.
The solution is given by the following lemma. (The proofs of all lemmas are provided in section B in the
appendix.)
Lemma 1. (The optimal supply of S in t = 4.) With a > 0 units of the asset and belief λ ∈ [0, 1], the optimal
(cumulative) supply for S is




−1 l−λh , if 0 ≤ λ ≤ λ̄(a)


a
−
u

a
(1−λ)l

s(l; λ, a) = 
, and s(h; λ, a) = a − δ,





if λ̄(a) < λ ≤ 1
0,
(4)
where the belief threshold is λ̄(a) = l(1 − ua (a))/(h − lua (a)). Fixing a, when λ ≤ λ̄(a), the supply s(l; λ, a)
is strictly decreasing in λ, but the curvature depends on the third order derivatives of u(·).
Figure 2 illustrates the supply function. Given the discrete prices (either l or h), the supply is composed
of only two points, one on the horizontal line price = h and the other on price = l. The graph illustrates two
such supplies given posterior beliefs, λ1 with blue disks and λ2 with red squares, where λ2 > λ1 . Intuitively,
S will supply everything but the last δ units of the asset, after which the marginal utility (weakly) exceeds
the market price, whenever the price turns out to be high, because there is no adverse selection cost for S to
supply at h when the fundamental price is indeed h. The blue disk and the red square overlap at (h, a − δ).
When the price turns out to be low, S in general does not supply her full position, because ex ante she is
subject to the adverse selection cost of size (h − l)s(l; λ), the shaded area in the graph. (We omit the last
argument of a in the supply functions for notation simplicity.) A higher posterior belief implies to S that it is
less likely to be Z = l, and she posts a smaller amount of the asset at l. The increase from λ1 to λ2 , therefore,
reduces the expected adverse selection cost of S; the grayish shaded area shrinks to only the part with dashes.
The lower half of the graph plots the two supply functions s(l; λ) and s(h; λ) against the posterior belief. The
15
supply at h is always the full position (less the δ units), while the supply at l decreases as the belief rises and
flattens when the belief is above the threshold λ̄(a).
Lemma 2 below describes the expected utility, U(λ, a), given the belief λ, as defined in equation 3.
The upper-right quadrant of figure 3 depicts the shape of this conditional expected utility function against
parameter λ.
Lemma 2. (S’s expected utility in t = 4, given belief λ.) The expected utility U(λ, a), under the optimal
supply of lemma 1, is i) strictly increasing in λ, ii) strictly convex for λ ∈ [0, λ̄(a)], and iii) linear for
λ ∈ [λ̄(a), 1]. Fixing λ, U(λ, a) is strictly increasing in a.
An immediate result following the lemma is given in the following corollary:
Corollary 1. (S’s expected utility and her holding.) Ceteris paribus, the more assets S holds in t = 4, the
better off is she. That is, given the same posterior belief in t = 4, S wants to minimize the transfer in t = 1
and to keep the asset to sell later in t = 4 when she has learned more about the fundamental value.
Proof. By lemma 2, U(λ, a) is strictly increasing in a.
As a remark, because in t = 1 the amount that S can sell is at most δ = a0 /n, which in the limit of n → ∞
is virtually zero, corollary 1 above does not impose a restriction on S’s strategy in t = 1. Intuitively, the
corollary states that S will use as few units of the asset as possible to learn the information of the same
quality, and defer selling to t = 4 when the posterior belief about Z is, hopefully, more precise (there are
cases where S learns nothing; see the discussion of “1M” scenario in subsection 3.3.3). This result provides
an alternative explanation on optimal order execution from the informational perspective; see Bertsimas and
Lo (1998), Almgren and Chriss (2000), and Obizhaeva and Wang (2005) for the motivation of minimizing
price impacts of trades.
In equilibrium (see subsection 3.3.3 below), S’s position a in t = 4 is either a0 − δ or a0 , as will appear
in the last argument of U(·) and s(·). In the limit of n → ∞, however, δ → 0, and S’s position in t = 4 is,
effectively, always a0 . For notation simplicity, we drop this last argument of U(·) and s(·) in the following
analysis, simply writing U(λ) and s(z; λ), and it should be understood that this limit argument is implicitly
assumed29 .
When δ → ∞, U(λ, a0 ) = U(λ, a0 − δ) and s(z; λ, a0 ) = s(z; λ, a0 − δ) because both functions are continuous in a, and the limits
exist.
29
16
3.3.2
Learning
We provide a definition of “precision” of S’s learning about Z in t = 3 and a general result of how such
precision affects the t = 1 expected utility of S.
S tries to update her posterior belief of the distribution of Z from the main market activity in t = 3.
The possible activities in market 0 are random and are consequences of her strategy in t = 1 as well as the
market structure – the number of middlemen (see the analysis in 3.3.3 below). The t = 3 activity can be
characterized as a pair of quote and trade. The quote can either be h or l, the trade can be either 1 (for yes)
and 0 (for no), and finally, there can be no activity, which we denote by ∅. Formally, define the t = 3 market
activity by random variable I (because the activities serve as S’s new information about Z). For example,
I can be (l, 1) for a trade at price l, (h, 0) for a quote at h but no hit, or can be simply ∅. Essentially, each
realization of I corresponds to an element of the σ-algebra F . For example, in the “2M” scenario and S has
sold δ units of the asset to M1 in t = 1, I = (h, 1) in t = 3 maps to {(Z = h, O = 0, E = 1)}. We analyze in
more details such mapping from market activity to events about Z, O, and E later.
S’s posterior belief about Z, given a realization of I, is Λ := P(Z = h|I). By construction, all such
random variables Λ have the constant mean λ0 , which is the unconditional probability of Z being high.30
We give two polar examples of such t = 4 beliefs. First, by doing nothing, S learns nothing from t = 3, and
gets a degenerate random variable Λ = λ0 with probability one. Second, without information asymmetry, S
perfectly learns and gets a binomial random variable Λ which realizes to be 1 with probability λ0 (whenever
Z is high) and to be 0 with probability (1 − λ0 ) (whenever Z is low).
Given a particular S’s t = 1 strategy and fixing the number of M, consider the expected variance of Z
conditional on I. With j possible realizations of market activity I, the posterior belief (of this strategy) Λ
has j realizations, λi (∈ [0, 1]) weighted by probability πi > 0, for i = 1, . . . , j. The expected variance of Z
That is, the class of {Λ} is a mean-preserving spread around λ0 . One can show this easily by invoking the law of iterated
expectations.
30
17
(conditional on the possible market event I) is31
E[var(Z|I)] =
X
πi λi (1 − λi )(h − l)2
i=1,..., j


 X

X
= (h − l)2 
πi λi −
πi λ2i 
i=1,..., j
i=1,..., j






2
= (h − l) λ0 · (1 − λ0 ) −varΛ ,
| {z }

(5)
=:σ20
where the last equality follows i) the mean-preserving property of Λ and ii) that EΛ2 = varΛ + (EΛ)2 . For
notation simplicity, denote unconditional variance of Z by (h − l)2 σ20 , i.e. σ20 := λ0 · (1 − λ0 ). We define the
precision of the signals associated with Λ as follows:
Definition 1. (Precision of learning.) The precision of signals following a particular strategy, whose t = 4
posterior beliefs are characterized by Λ, is the (scaled) reciprocal of the expected variance of Z (conditional
on possible market event I) (equation 5). Call the precision ρ(Λ):
ρ(Λ) =
1
(h − l)2
= 2
,
E[var(Z|I)] σ0 − varΛ
where 0 < ρ(Λ) ≤ ∞.
The precision measures, on average, how precise S can learn about Z after learning from the I produced by a
particular strategy of S in t = 1. Slightly abusing the notations, we will often refer it simply as “the precision
of a strategy”, or refer ρ(Λ) as the “precision of Λ”. The above definition is consistent with the intuition
that more precise signals provide smaller posterior variances of Z. The polar examples above illustrates this:
When there is no learning, the precision is ρ = 1/σ20 ; when the learning is perfect, the precision is indeed
infinite using the definition above.
In fact, the precision ρ(Λ) is a sufficient statistic for the S’s expected utility if she chooses the strategy
that generates Λ in t = 1. That is, there is a one-to-one mapping from ρ(Λ) to EU(Λ). The following lemma
summarizes this result.
Lemma 3. (Signal precision and expected utility.) Consider two t = 1 strategies, a and b, which generate
31
A binomial random variable with probability p to be h and probability 1 − p to be l has variance p · (1 − p)(h − l)2 .
18
posterior beliefs Λa and Λb respectively. S prefers strategy a to strategy b if and only if ρ(Λa ) ≥ ρ(Λb ). The
preference is strict when the inequality between the two precisions is strict.
This lemma will be used to help S choose her optimal strategy in t = 1, analyzed in the next subsection.
3.3.3 t = 1
S knows that the demand for the asset in t = 1 is at most δ (see subsections 3.1 and 3.2), which in the limit
of n → ∞ is arbitrarily small (hence cannot be further split). Her strategy is effectively reduced to one of
the following: i) to do nothing, ii) to post a limit sell order (l, δ), or iii) to post a limit sell order (h, δ). (Note
that to post two orders, (l, δ) and (h, δ), is dominated by ii) above, because given the demand of at most δ,
only the lower order will be executed.)
The limit argument of n → ∞ helps to simplify the analysis. Under such a limit, irrespective of successful selling δ (→ 0) units of the asset or not, S still holds a0 units of the asset in t = 4. Therefore, one
can compare the expected utility of U(λ, a0 ) with U(λ, a0 − δ) easily (see also subsection 3.3.1). This limit
argument effectively helps the analysis focus on the information aspect. Henceforth, the second argument
of U(·) will be dropped for notation clarity.
No middleman (“0M”). In this no middleman scenario, posting (l, δ) is equivalent to learning nothing,
because even the EB arrives in the main market and takes the order away, the conditional probability of Z
being high is still λ0 . That is,
Λ0M,l = λ0 , with probability 1.
The posterior precision in this case is just the ex ante precision: ρ(λ0 ) = 1/σ20 , where σ20 = λ0 · (1 − λ0 ).
If S instead posts (h, δ), with probability θ0 φ the EB arrives in the main market (O = 0) and, further,
with probability λ0 he will hit the order (Z = h and E = 1). That is, the main market activity is I = 1
with probability θ0 φλ0 . Therefore, when the order is taken, in t = 4 S updates her posterior to λ = 1, and
if the order is not taken, S updates, by Bayesian rule, her posterior to λ = λ0 (1 − θ0 φ)/(1 − θ0 φλ0 ) (< λ0 ).
Therefore, the strategy of posting (h, δ) gives the binomial posterior, a random variable Λ that takes value
1 with probability θ0 φλ0 and takes value λ0 (1 − θ0 φ)/(1 − θ0 φλ0 ) with the remaining probability. We shall
19
refer to the posterior of this strategy as Λ0M,h :
Λ
0M,h







1,
0M,h
= P(Z = h|I
)=



λ0 (1−θ0 φ)


,
 (1−θ
0 φλ0 )
if I 0M,h = (h, 1)
(6)
otherwise.
Intuitively, the second strategy is preferred to the first one because it offers some learning while the first does
not. We formally show this result in proposition 2 at the end of this subsection.
One middleman (“1M”).
As in the “0M” scenario, S can still post (h, δ) and to acquire the posterior
Λ1M,h = Λ0M,h , as if there were no M. This is because at h, the asset is too costly for M as the price exceeds
M’s reservation utility (see subsection 3.2).
Under assumption 1, if S posts (l, δ), M1 will always take the order. Depending on the realization of
O (as revealed by Nature to M1), M1 then posts (z, δ) in that market, if reachable, to resell the asset (see
the equilibrium strategy for M1 discussed in subsection 3.2). Therefore, with probability θ0 , O = 0 and S
observes the ask quote of M1 in the main market and learns the realization z perfectly (regardless of whether
the EB arrives or not), and with probability (1 − θ0 ), S learns nothing as M1 tries either its private market
or does nothing (if EB is to arrive in the market out of M1’s reach). The posterior of this strategy is Λ1M,l ,
which has realization:
Λ1M,l





1,











1,







= P(Z = h|I 1M,l ) = 
0,











0,









λ0 ,
Multiple (two) middlemen (“2M”).
if I 1M,l = (h, 1);
if I 1M,l = (h, 0);
if I 1M,l = (l, 1);
(7)
if I 1M,l = (l, 0);
if I 1M,l = ∅.
As before, S can still post (h, δ) and to acquire the posterior Λ2M,h =
Λ0M . Neither M will bid this ask quote because the price h is higher than their marginal utility. (See
subsection 3.2).
Under assumptions 1 and 2, if S posts (h, δ), both M will want to buy the δ units of the asset. Call the
winner M1. Depending on which market EB might arrive in (as revealed by Nature to both M), M1 then
20
posts (z, δ) in that market, if reachable (O = 0 or O = 1), to resell the asset. If the EB might arrive in
M2’s private market, the two M swap trade with each other if and only if Z realizes to be high. (See the
equilibrium strategies for both M discussed in subsection 3.2). When returns to the main market, S updates
her posterior according to the market activity I 2M,l . Denote the posterior of this strategy by Λ2M,l , which
takes four possible values:
Λ2M,l





1,











1,







= P(Z = h|I 2M,l ) = 
0,









θ2 λ0



θ2 λ0 +θ0 φ·(1−λ0 ) ,








 λ0 θ1 ,
θ1 +θ2 ·(1−λ0 )
if I 2M,l = (h, 1)
if I 2M,l = (h, 0)
if I 2M,l = (l, 0)
(8)
if I 2M,l = (l, 1)
if I 2M,l = ∅
S learns perfectly that Z is high when there is a high quote in the main market (market 0), irrespective of
whether the quote is hit, because such a quote is from M1 directly targeting the potential EB. Also perfectly
clear is when there is a low quote (l, δ) in market 0, implying a low fundamental value. However, when there
is a trade at the low price, the learning is no longer perfect, because such a trade could correspond to either
a trade between M1 and the EB (implying Z = l, O = 0, and E = 1), or a swap trade between the two M
(implying Z = h and O = 2). Finally, there could be no activity at all in market 0. This could either be that
M1 targets the EB in market 1 (implying Z = h with probability λ0 , i.e. no learning), or that M1 could not
swap with M2 even though EB might arrive in M2’s private market (implying Z = l).
Comparing Λ1M,l and Λ2M,l , it can be seen that S always perfectly learns when the market activity is
I = (h, 0), (h, 1), and (l, 0). However, the learning is no longer perfect when I = (l, 1) is low, because the
swap trade between the two M creates “confusion” to S. On the other hand, when O = 2, in contrast to
no-learning in the “1M” scenario, in the “2M” scenario, S does learn something (though imperfectly) from
the potential swap trade. Depending on the sizes of these two effects, therefore, the preicisions of Λ1M,l may
or may not be higher than that of Λ2M,l . S, therefore, may or may not prefer many middlemen to a single
middleman, in terms of information learning.
We conclude this subsection by the following proposition, which gives the optimal strategies of S in
t = 1 under the three different scenarios.
21
Proposition 2. (S’s decision in t = 1.) In the “0M” scenario, S always posts a limit sell order (h, δ). In
the “1M” scenario, S always posts a limit sell order (l, δ). In the “2M” scenario, S always posts a limit sell
order (l, δ). Further, S prefers “1M” to “0M” and “2M” to “1M”, but the comparison between “1M” and
“2M”, in general, cannot be signed.
Proof. This is a direct result following lemma 3, by first evaluating and then pairwise comparing ρ(Λ0M,l ),
ρ(Λ0M,h ), ρ(Λ1M,l ), ρ(Λ1M,h ), ρ(Λ2M,l ), and ρ(Λ2M,h ).
3.4
Swap-trade, price pressure, and excessive selling: a realization of the “2M” scenario
Consider the following state of the world in the “2M” scenario: M1 has bid δ units of the asset from S (at
price l) in t = 1, Nature reveals that Z = h and that O = 2 in t = 2, and M1 consequently swap trades with
M2 at l in the main market in t = 3, leaving the market activity I 2M,l = (l, 1). See subsection 3.2 above for
the equilibrium strategies of M.
Under such a realization, price pressure manifests after the swap-trade in the sense that the t = 3 trade is
not at the fundamental price h, but at the swapping price l (< h = Z). One can think of such price pressure
as a “mini crash” created by the swap-trade. If such swap-trade repeats (not modeled), the eventual price
pressure could be huge. (See panel (b) of figure 1 for the illustrative loop.)
Further, when S returns in t = 4, she does not know whether the previous trade is pressured or not:
Is it between M1 and EB (fundamental trade), or between the two M (swap trade)? She then updates her
posterior belief to
λ=
θ2 λ0
.
θ2 λ0 + θ0 φ · (1 − λ0 )
(9)
(See equation 8 for the other posterior beliefs of S in the “2M” scenario.) S then posts her supply function,
s(·; λ), according to this belief. By lemma 1, as long as the posterior λ is lower than the threshold λ̄(a0 ),
s(l; λ) > 0. Note that in the current state of the world, the realization of Z is h. Therefore, due to the
“confusion” from the trading activity in t = 3, S willingly trades the asset with LBs and effectively reinforces
the price pressure accompanying the swap trade of the two M, for s(l; λ) units. Such a sale is excessive at
the low price in the sense that S would sell nothing at this price if there is no such a confusion.
Proposition 3. (S’s supply under “confusion”.) The lower is the ex ante probability of no EB showing up
22
in the main market, in the sense of (1 − θ0 φ) being very small, the more S will supply at the pressured price
after seeing a trade at l in the “2M” scenario.
Proof. Define η := θ0 φ. It is easy to verify that ∂λ/∂η < 0 from equation (9). Because the posterior is
0 < λ̄(a0 ) when η is at its maximum η = 1 and is 1 > λ̄(a0 ) when η = 0, there exists some η (large enough)
such that by continuity, λ(η) < λ̄(a0 ). Then by lemma 1, ∂s(l; λ(η))/∂η ≤ 0, and the inequality is strict for η
large enough.
Small values of (1 − θ0 φ) can be interpreted as that events of sudden buy-side liquidity dry-up (in the main
market), like the May 6 Flash Crash (see e.g. CFTC and SEC, 2010), are rare. Under such small ex ante
probability, the above model predicted price pressure might come as a surprise when the seller reinforces the
price pressure by excessively loading the market at the pressured price. In fact, proposition 3 says that the
rarer is such liquidity dry-up, the more S will rationally attempt to sell at the pressured price, even when the
fundamental price is in fact still high. Such excess supply (at the low price) is the result of the ambiguous
interpretation of the t = 3 trade, when swap-trades among middlemen are possible.
4
Welfare
We begin with a general characterization of welfare, defined as the aggregate gains from trade in this economy. All individuals’ utility values are equally weighted. We then proceed with different scenarios, using
the equilibrium strategies derived in the previous section (proposition 2). Subsection 4.1 answers whether
welfare is improved as the number of M increases. Subsection 4.2 shows that a disclosure policy strengthens
the signal precision (hence also S’s expected utility) and boosts welfare (more efficient allocation).
Define the social welfare as the (equally weighted) aggregation of gains from trade in this economy. We
begin with the following lemma that characterizes the welfare function.
Lemma 4. (Welfare as the aggregate gains from trade. Fix a realization of preference z. For a given transfer
of size s of the asset from S to LB, the so-defined welfare is
w(z, s) = z · [s · ua (0) − (u(a0 ) − u(a0 − s))] .
The first term in the square brackets is the aggregate gains from trade for the s/δ LBs, who in the limit each
23
gains the marginal utility of ua (0) for the infinitesimally small amount of the asset. The second term refers
to the loss of S in terms of the fruit consumption: She gains from the trade in terms of the numéraire good
consumption. However, from a social planner’s perspective, the transfers of the numéraire good cancel and
do not affect the aggregate gains from trade. The scalar z is the preference shock that strikes everyone in
the economy. Note that the gains from trade of EB and of M, if any, converge to zero in the limit, because
the size δ is effectively zero. It is easy to verify that w(z, s) is strictly increasing and strictly concave in s,
for 0 ≤ s ≤ a0 . That is, the more are the transfers to LBs, the larger is the social gains from trade, but the
marginal increase is diminishing.
The transfer s to LBs is effectively the t = 4 supply of S, s(z; λ), because in the limit of n → ∞, the
transfer from t = 1 to t = 3 is virtually zero.32 From lemma 1, it can be seen that s(h; λ) = a0 for all λ; that
is, the transfer is always full, conditional on Z = h. Therefore, the equilibrium welfare expression can be
further simplified and linked to each scenario:
wi = λ0 w(h, a0 ) + (1 − λ0 )Ew(l, s(l; Λi )),
(10)
where the superscript i indicates one of the three scenarios studied in this paper: “0M”, “1M”, or “2M”. The
corresponding posterior belief Λi also follows the optimal t = 1 strategy of S as described in proposition 2.
We discuss the welfare expressions under different Λs below.
No middleman (“0M”).
The optimal strategy of S, by proposition 2, gives the posterior Λ0M,h in this
scenario. Evaluating expression (10) with Λ0M,h gives
w0M = λ0 w(h, a0 ) + (1 − λ0 )w(l, s(l; α)),
(11)
where α := λ0 (1 − θ0 φ)/(1 − θ0 φλ0 ) is the posterior belief in case of Z = l in this scenario. (See equation 6
and the discussion there for more details.)
32
The setting of infinitely many buyers (n → ∞) and the restriction on buyer’s holding δ lead to this result that only the transfer
in the long-term matters. This way, the focus of the analysis directs to the learning from the short-term.
24
One middleman (“1M”). S has the posterior Λ1M,l in this scenario. Evaluating expression (10) with Λ1M,l
gives
w1M = λ0 w(h, a0 ) + (1 − λ0 ) [θ0 w(l, a0 ) + (1 − θ0 )w(l, s(l; λ0 ))] .
(12)
In this scenario, the transfer is still full when Z = h as before. When Z = l, with probability θ0 O = 0
and M1 posts quotes, which is fully revealing to S, in market 0, while with probability (1 − θ0 ), there is no
activity (I = ∅) in the main market so that S learns nothing. (See equation 7 and the discussion there for
more details.)
Multiple (two) middlemen (“2M”).
S has the posterior Λ2M,l in this scenario (see equation 8). Evaluating
expression (10) with Λ0M,l gives
w2M = λ0 w(h, a0 ) + (1 − λ0 ) (1 − φ)θ0 w(l, a0 ) + φθ0 w(l, s(l; β)) + (1 − θ0 )w(l, s(l; γ)) ,
(13)
where β := θ2 λ0 /(θ2 λ0 + θ0 φ · (1 − λ0 )) and γ := θ1 λ0 /(θ1 + θ2 · (1 − λ0 )) are S’s updated posterior beliefs
upon, respectively, seeing I = (l, 1) (i.e. trade at l) and seeing I = ∅ (i.e. no activity in the main market).
In this scenario, the transfer is still full when Z = h as before. When Z = l, the transfer depends on the
realization of the main market activity I. (See equation 8 and the discussion there for more details.)
4.1
Comparison across the scenarios
In general, the comparison among the welfare expressions (11), (12), and (13) cannot be signed, even
though the introduction of M does improve the signal precision of S. Put alternatively, helping S to learn
more precisely about the fundamental value does not necessarily prompt S to sell more – resulting in a more
efficient allocation – in expectation.
Consider two posterior beliefs, Λa and Λb (resulting from two strategies), with ρ(Λa ) > ρ(Λb ). By
lemma 3, the order of the precision implies EU(Λa ) > EU(Λb ). This, however, does not necessarily translate
to Es(l; Λa ) > Es(l; Λb ), and consequently, not necessarily to Ev(s(l; Λa )) > Ev(s(l; Λb )), either. Figure 3
illustrates how a more precise posterior Λ1 improves S’s expected utility (from no-learning, λ0 ), increases
the expected supply s(l; Λ), and yet lowers the welfare. The shape of the U(·) function follows lemma 2.
25
The supply s(l; λ) is decreasing in belief λ, but the curvature depends on the third-order derivative of the
u(·) function; see lemma 1 for more details on the shape of s(l; λ). The shape of w is determined by that of
v(·). The no-learning signal λ0 serves as the benchmark, while Λ1 (as a result of some particular strategy)
provides S with the posterior with realization either Λ1 = 0 or Λ1 = λ̄(a0 ) (> λ0 ).33 As can be seen from
the graph, though the expected utility increases from U 0 to U 1 , the expected supply Es(l; Λ) might increase
or decreases, depending on the curvature of s(l; λ) (the graph illustrates an increase from q0 to q1 ). Further,
welfare might lower from ŵ0 to w1 , or increase from w0 to w1 , depending the degree of concavity of w(·).
Intuitively, this is because when choosing her supply, S only cares about her well-being by maximizing
the expected utility, which is composed of both the gain in the numéraire good and the reduction in the
utility of fruit consumption. She is unwilling to transfer the asset to other investors (LBs) if the gain in the
numéraire good is not sufficient. From a social planner’s perspective, however, only the efficient allocation
of the asset matters, while the transfers of the numéraire good always offset in aggregation. In a sense, S
overlooks the positive externality of transferring the asset at unfavorable prices when making her decisions.
Proposition 4. (Learning and welfare.) A more precise signal for S (see definition 1) does not necessarily
lead to a higher welfare.
An immediate consequence of the above proposition is that even though M provide(s) more precise signals
to S, the welfare does not necessarily improve. The following corollary summarizes this result:
Corollary 2. (Intermediaries and welfare.) The introduction of the middlemen provides more precise signals
(about Z) to S: ρ(Λ1M,l ) > ρ(Λ0M,h ) and ρ(Λ2M,l ) > ρ(Λ0M,h ); equivalently, S is better off in either the “1M”
or the “2M” scenario than in the “0M” scenario. (See proposition 2 and lemma 3.) However, the welfare
might be reduced by the introduction of M. Further, it is also possible that the introduction of a second M
lowers both the expected utility of S and the welfare.
Proof. We prove the proposition and its corollary by a numerical example. See the discussion below about
figure 4.
The upper panel of figure 4 plots the expected utility of S in the three scenarios against φ, the arrival
probability of EB (irrespective of the destination market). In the lower panel, the social welfare is plotted
33
For the purpose of this illustration (that increasing S’s expected utility does not necessarily leads to an increase in social
welfare), the choice of the two can be arbitrary. We chose the no-learning Λnl and Λ1 simply for graphical clarity.
26
against the same range of φ. Four series are plotted in each panel: “0M”, “1M”, “2Mnd ”, and “2Mdi ”;
the scripts “nd” and “di” refer to the regulation of “no-disclosure” or “with disclosure”, the implication of
which is discussed in the next subsection. The optimal t = 1 strategy (either posting l or h) is omitted in
the superscript. (So far we have only discussed the “no-disclosure” regime, and let us focus only on the first
three series now.) The shaded area corresponds to the region where the addition of a second M to the market
not only reduces S’s expected utility but also the welfare. There is a salient tick of w2M
nd when φ exceeds
about 0.72. Note from equation (13) that β as a function of φ decreases (see also section 3.4 and proposition
3) – if the arrival probability of EB is higher, upon seeing a trade at low price in the main market, S weighs
more on the event that the trade was fundamental, and the updated belief (β) lowers accordingly. As soon
as β, the posterior belief, falls below the threshold λ̄(a0 ) (see lemma 1), the supply at low (s(l; β)) begins to
increase, and that is when the welfare, as a function of how many S supplies, ticks up. (When β ≥ λ̄(a0 ), the
supply at low is 0; see lemma 1.) The parametrization used in this numerical procedure is: u(a) = ln(a + 1),
a0 = 10, h = 10, l = 5, λ0 = 0.4, and θ0 = θ1 = θ2 = 1/3. The cost of carry k is chosen to satisfy assumption
1.
4.2
Disclosure
Let us revisit the “2M” scenario. Consider a regulation that adds to the market history an “M-M” flag; that
is, in addition to the pair of price and quantity, investors also observes the type of each trade: either it is a
trade involving (at least one) investor (unflagged), or it is a trade between two middlemen (flagged).
With such a flag, it is not hard to see that the signal precision of S improves. Essentially, a third
dimension, flag, is added to the possible market activity I. For example, I can now be (l, 1, 1), where the last
argument indicates the trade at price l is flagged, i.e. inter-middlemen.
Denote in this case the posterior (of posting (h, δ) in t = 1) of S by Λ2M
di , and to contrast, call the
posterior under the no-disclosure regime Λ2M
nd , where the subscripts “di” and “nd” refer to disclosure and
27
no-disclosure respectively. It is easy to derive that
Λ2M,l
di






1,










1,












0,
2M,l
= P(Z = h|Idi ) = 





0,











1,








λ0


,
 θ1 +θθ21·(1−λ
0)
2M,l
if Idi
= (h, 0);
2M,l
if Idi
= (h, 1, 0);
2M,l
if Idi
= (l, 0);
if
2M,l
Idi
(14)
= (l, 1, 0);
2M,l
if Idi
= (l, 1, 1);
if I = ∅.
That is, S can now perfectly learn Z from the main market activity, as long as there is any, thanks to the
“M-M” flag, which tells S whether the trade at l is between M and EB (implying Z = l) or between the two
2M,l
M (“swap trade”, implying Z = h). Essentially, the market activity (without disclosure) Ind
= (l, 1) has
2M,l
2M,l
been further split into Idi
= (l, 1, 0) and Idi
= (l, 1, 1). (See equation (8) for comparison.) When there is
2M,l
no activity at all in the main market, the learning remains the same. Therefore, overall, ρ(Λ2M,l
di ) > ρ(Λnd ),
as can easily be proved by evaluating the definition of precision by formula (5). Further, the 2M welfare
expression with disclosure becomes
w2M
di = λ0 w(h, a0 ) + (1 − λ0 ) θ0 w(l, a0 ) + (1 − θ0 )w(l, s(l; γ)) ,
(15)
which, compared with w2M
nd (equation (13)), is (weakly) higher. (The parameter γ is also defined after
equation 13.) Intuitively, this is because the disclosure solves the “confusion” of S seeing a trade at l in the
main market – instead of mixing between two possibilities, S supplies all a0 units of the asset as soon as
she sees no “M-M” flag on the trade, which tells her that the trade is between M1 and the EB and that the
fundamental value is indeed Z = l. Thus, the first part of the following corollary has been proved:
Corollary 3. (Flagging trades between middlemen.) In the “2M” scenario, both S and the society benefits
from a disclosure regulation, which tags an “M-M” flag onto each trade between middlemen. The flag also
makes the “2M” scenario (weakly) better than the “1M” scenario for both S and the society.
Proof. The first half of the corollary has been proved in the analysis proceeding the corollary. The second
1M,h ) which can be directly shown by lemma 3, and ii) that w2M,l ≥ w1M,l
half states i) that ρ(Λ2M,l
di ) ≥ ρ(Λ
di
28
which is true because the difference between equation (14) and equation (7) is positive: Recall that w(·) is
increasing in the supply, while s(l; λ) is decreasing in the posterior, and note that θ1 λ0/(θ1 + θ2 · (1 − λ0 )) <
λ0 .
The intuition of the second half of the corollary is as follows. Whenever there is some trading activity in the
main market in t = 3, S perfectly infers about Z in both “1M” and “2M-disclosure” scenarios. The difference
is that when there is no trading activity in the main market, S learns nothing in the “1M” scenario, while she
learns in the “2M-disclosure” scenario that such no-activity could partly be due to that M1 could not swap
trade with M2, implying a low realization of Z. Such extra learning makes the signal in the “2M-disclosure”
more precise and improves S’s expected utility. By the same reasoning, S supplies more when she sees no
activity in t = 3 under the “2M-disclosure” scenario than under the “1M” scenario, because with a lower
posterior (θ1 λ0/(θ1 + θ2 · (1 − λ0 )) < λ0 ) S supplies more at l, increasing the asset transfer to LB.34 We
illustrate both the welfare and S’s expected utility in the “2M-disclosure” regime in figure 4.
5
Conclusion
High frequency traders (HFT), as the middlemen in modern exchanges, not only intermediate between
fundamental investors (as in Grossman and Miller, 1988) but also trade with each other, “crawling for
counterparties”, to resell the asset. When the asset owner HFT finds that its reselling opportunities are small,
i.e. when there is an ex post misallocation of the asset within middlemen, swap trades occur to reallocate the
asset to another HFT who have better access to investors. Unlike “hot potato trades” previously documented
in the literature (see, for example, Lyons (1997)), such swap trades are motivated simply from HFTs’ (ex
post) heterogeneous reselling opportunities.
Price pressure accompany swap trades because the HFT buyer needs compensation for the expected cost
of carry. An inference problem for investors might arise because when the fundamental value of the asset
is not perfectly understood, such price-pressured trades are not distinguishable from (unpressured) trades
between a middleman and an informed investor. A seller with such an inference problem sells more at the
34
The result stated in the second half of corollary 3 is not unique due to discrete prices, which prevents M1 from swap trading
with M2 when Z = l. In a more general setting, where Z has a continuous support and agents can trade at any prices in R+ , M1
will be able to swap trade with M2 whenever it finds EB might arrive in M2’s private channel. The no-activity (in t = 3) will
only correspond to the case where M1 goes to its private market, from which S learns nothing. Therefore, under this more general
setting, S is indifferent between “1M” and “2M-disclosure” in terms of information learning. Similarly, the welfare will be the same
between the two scenarios, too.
29
pressured price than she would if she had known the trade is pressured. This way, the seller reinforces and
accentuates the price pressure accompanying the swap trade of middlemen.
A theoretical model characterizes the above features. A seller holding a large position of the asset
seeks to trade with buyers, but a standard information asymmetry problem is standing in the way of socially
efficient allocation. Scenarios of one middleman (“1M”) and multiple middlemen (“2M”) are compared
with the benchmark scenario where there is no middleman (“0M”). We interpret the “1M” scenario as the
traditional specialist market structure, like the NYSE specialist. In view of the modern electronic limit order
markets populated by HFTs, a multiple-middleman characterization – the “2M” scenario in the model –
captures the reality better.
The model predicts that the introduction of a middleman does help the seller to better learn about the
fundamental value. However, adding a second middleman does not necessarily improve the learning. Further, we identify with a numerical procedure that a second middleman might also be socially costly. It is
possible to resolve the problem by flagging the trades between middlemen. Both the seller and the society
benefit from such transparency.
30
Appendix
A
Notation summary
It is a convention in this paper that letters in upper case denote random variables, while those in lower case
represent their realizations or refer to other deterministic variables.
• a, the amount of the asset. It is the same as the
consumption of the fruit because each unit of
the asset produces one unit of the fruit.
• a0 , the aggregate endowment of the asset in the
economy. It is also the endowment to S.
• c, the amount of the numéraire good.
• δ, the efficient allocation amount of the asset
per investor. That is, δ = a0 /n.
• E, the arrival of an early buyer (see below). It
takes Boolean values 1 for arrival and 0 for no
arrival. P(E = 1) = φ and P(E = 0) = 1 − φ.
• EB, the early buyer, who arrives in early in
t = 3.
• h, the high realization of Z (0 < l < h < ∞).
• k, the carrying cost per unit of the asset for M.
• l, the low realization of Z) (0 < l < h < ∞).
• λ and Λ, S’s belief of Z = h, usually refers
to the posterior belief of S in t = 4. Such beliefs depend on the market activity in t = 3,
and therefore are random ex ante (in t = 1),
written in the upper case as Λ.
• LB, the later buyer(s), who arrive late in t = 4.
There are infinitely many of them.
• M, the middleman. Depending on the scenario, there is either one or two such M.
• n, the number of investors in this economy.
31
• O, the reselling opportunity of a middleman.
It indicates the market (0, 1, or 2) in which EB
might arrive in. P(O = j) = θ j , for j ∈ 0, 1, 2.
O and E are independent.
• φ, the probability of the event of the EB arrives. The event is independent of which market, the main market (market 0) or the two M’s
private markets, EB arrives in.
• ρ, the precision of a posterior belief λ, defined
as the (scaled) reciprocal of the expected variance of Z, given the belief λ.
• S, the shocked seller.
• σ20 , the scaled unconditional variance of Z, i.e.
σ20 = varZ/(h − l)2 = λ0 · (1 − λ0 ).
• θi , the probability of the event that the EB
might arrive in market i, for i ∈ 0, 1, 2. The
event is independent of whether EB indeed arP
rives. i θi = 1.
• w, the welfare, defined as the (equally
weighted) aggregate gains from trade of all
agents in the economy.
• Z, the systematic preference shock that scales
the utility function u(·). It is a binomially distributed random variable, with probability λ0
to be h and with probability 1 − λ0 to be l.
• z0 , the unconditional expectation of Z, i.e. z =
λ0 h + (1 − λ0 )l.
B
Proofs
B.1
Proof of lemma 1 and 2
Proof. The first-order conditions with respect to ql and qh of the maximization problem (3) are:



(1 − λ)l · (1 − ua (a − ql )) + λh · (1 − ua (a − ql − qh )) = 0
.


λh · (1 − ua (a − ql − qh )) = 0
Off the corner, the above two equations solve for:

−1 l−λh
∗


ql = a − ua (1−λ)l


l−λh
q∗ = u−1
− δ.
a
(A1)
(1−λ)l
h
Note that the inverse function of ua (·) exists because uaa (·) < 0 by assumption. However, ua · > 0 for all
a ≥ 0, and the above solved q∗l and q∗h must be non-negative. These conditions together require:
a − u−1
a
!
l − λh
≥ 0,
(1 − λ)l
which, after some simplification gives the threshold:
0 ≤ λ ≤ λ̂(a) :=
l · (1 − ua (a))
.
h − lua (a)
Therefore, to sum up, for 0 ≤ λ ≤ λ̄(a), the optimal (cumulative) supply is simply s(l; λ) = q∗l and s(h; λ) =
q∗l + q∗h , where q∗l and q∗h are solved above in equation (A1). For λ̄(a) ≤ λ ≤ 1, the solution is cornered by
q∗l = 0 and q∗h = a − δ, that is, s(l; λ) = 0 and s(h; λ) = a − δ. This proves lemma 1.
Lemma 2 is easily proved by envelope theorem:
∂U
= hq∗h + hu(δ) − lu(q∗h + δ)
∂λ
= hq∗h + hu(δ) − l · [u(δ) + ua (δ)q∗h − C]
> 0,
where the last equality follows the fundamental theorem of calculus and the constant C is such that u(δ) +
ua (δ)qh − C = u(qh + δ). The second order partial derivative is then:


∂q∗h
∂q∗h 
if λ ≥ λ̄a
∂2 U
0,
∗
∗
=
h
−
lu
(q
+
δ)
=
.

a
∂q
h

h−l h
∂λ
∂λ  1−λ
∂λ2
,
if
λ
≤
λ̄a
∂λ
∂q∗
It is easy to verify that ∂λh > 0 by using the construction that ua (·) > 0 and uaa (·) < 0. The shape of U with
respect to λ can thus be easily illustrated. Finally, again by envelope theorem,
∂U
= (1 − λ)lua (a − q∗l ) + λh > 0.
∂a
The proof is complete.
32
B.2
Proof of lemma 3
Proof. Given two posterior beliefs, Λ1 and Λ2 with the same mean λ0 , we want to show that EU(Λ1 ) ≥
EU(Λ2 ) if and only if ρ(Λ1 ) ≥ ρ(Λ2 ). Note from lemma 2 that U(·) is (piece-wise) convex on λ ∈ [0, 1]. It
is well-known that for a convex function f (·), E f (X) ≥ E f (Y) if and only if the two random variables X and
Y satisfy EX = EY and varX ≥ varY, that is, if and only if X is a mean-preserving spread of Y 35 . We use
this result to facilitate the proof.
From the definition of ρ(Λ) (see the equation in definition 1), it is clear that ∂ρ(Λ)/∂varΛ > 0. Hence,
varΛ is a equivalent statistic of the precision ρ(Λ). The above property of convex functions then immediately
completes the proof.
B.3
Proof of lemma 4
In autarky, the aggregate utility of the n investors and the two middlemen, conditional on Z = z, is
zu(a0 ) + (n − 1 + 2)zu(0) = z[u(a0 ) + (n + 1)u(0)].
(The initial endowment in the numéraire good is, without loss of generality, normalized to zero.) Consider
the following transfers from S (as a result of certain equilibrium trades): i) δ̂ ∈ {0, δ} units to EB, ii) δ̃ ∈ {0, δ}
units to M (if there is any, depending on the scenario), and iii) s < a0 (very large, relative to δ) units to s/δ
LBs. Note that in equilibrium, the asset holdings of EB and M are, summed together, at most δ, depending
on the realization and the scenario. Then the aggregate utility, conditional on a realization z of Z, is
M
S
z }| { s
z }| {
s
zu(a0 − s) + zu(δ̂) + z(u(δ̃) − kδ̃) + u(δ) + n − 1 − zu(0) .
|{z}
δ
|
{z δ
}
EB
LB
Note that the transfers of the numéraire good offset each other in the aggregation because it is equally
appreciated by any pair of agents that constitute a trade. Take the difference between the post trade aggregate
utility and the autarky aggregate utility to get the gains from trade:
"
#
s
w(z, s; n) = z · (u(δ̂(n)) − u(0)) + (u(δ̃(n)) − u(0)) +
(u(δ(n)) − u(0)) − (u(a0 ) − u(a0 − s)) ,
δ(n)
where δ is written as a function of n to remind that δ = a0 /n which is the social optimal amount of allocation
per investor. Take limit of w(n) and apply l’Hôpital’s rule to derive the limit welfare expression (the argument
of n → ∞ is omitted):
w(z, s) = lim w(z, s; n) = z · [s · ua (0) − (u(a0 ) − u(a0 − s))] .
n→∞
This completes the proof.
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35
Figure 1: Trade cycle – seller (S), middleman (M), and buyer (B)
The upper graph illustrates the trade cycle when there is only one middleman: The seller transfers the asset to the
buyer via the middleman (the solid arrows), and then the seller learns from the trade (the dashed arrow). The lower
graph illustrates the trade cycle when there are multiple (two) middlemen: After the seller has sold the asset to one
of the middleman, the middleman might either resell to the buyer, or the swap trade with another middleman (solid
arrows). In the multiple-middlemen scenario, upon seeing a trade, the seller tries to learn from the market activity
(dashed arrow) but cannot tell whether it is a swap trade, and an inference problem about the fundamental value arises
if the information is asymmetric.
(a) Single middleman
S
B
(fundamental price)
M
(b) Multiple (two) middlemen
S
B
M2
(fundamental price)
(pressured price)
M
M1
36
Figure 2: S’s supply function in t = 4 and her posterior belief λ
The upper part of the graph plots S’s optimal supply function in t = 4 for two posterior beliefs, λ1 with red squares and
λ2 with blue disks, where λ2 > λ1 . Due to the discrete prices, the supply function consists only two points, one on the
horizontal line price = h and the other on price = l. The lower part of the graph plots the two supply functions, s(l; λ)
and s(h; λ), against the posterior belief λ. S always supplies her full position at h, while her supply at l reduces as the
posterior belief increases. The shaded area ((h − l)s(l; λ)) in the upper part represents the adverse selection cost to S
when Z realizes to be h. An increase in the belief, reducing S’s supply at l, lowers her (expected) adverse selection.
See more discussion in section 3.3.1 and lemma 1.
price
h
l
0
s(l; λ2 )
s(l; λ1 )
a−δ
λ1
λ2
λ̄(a)
s(l; λ)
s(h; λ)
supply at l
1
posterior belief, λ
37
supply at h
quantity
Figure 3: S’s expected utility, the supply when Z = l, and the welfare
This graph illustrates S’s expected utility, her supply when Z = l, and the social welfare. The upper-right quadrant
depicts how the belief affects the expected utility. The shape of U(·) is described in lemma 2. The lower-right quadrant
plots the optimal supply function at price l (lemma 1) against the (posterior) belief λ. The supply function is decreasing
in λ, but the curvature is not necessarily as illustrated. Finally, the lower-left quadrant plots the welfare against the
supply at Z = l (equation 10). Consider two signals (see section 3.3.2 for terminologies): Λ0 = λ0 , i.e. no learning,
and Λ1 which updates the belief either to λ̄ (> λ0 ) or to 0. By construction, all signals are mean-preserving, i.e.
EΛ0 = EΛ1 = λ0 . As can be seen from the graph, the new signal Λ1 improves S’s expected utility from U 0 to U 1 ,
but the effect on the supply depends on the curvature of the supply function. Further, depending on the degree of the
concavity, the social welfare might be improved (from w0 to w1 ) or worsened (from ŵ0 to w1 .)
expected utility, U(λ)
U1
U0
welfare, w
ŵ 0
w1 w0
λ̄(a0 )
0
q0
λ0
q1
a0
supply at l , s(l; λ)
38
1
belief (of Z = h), λ
Figure 4: Welfare and S’s expected utility in three scenarios
The upper panel plots S’s expected utility, which is an equivalent statistic for signal precision (see definition 1 and
lemma 3) in the four scenarios: “0M” – no middleman, “1M” – one middleman, “2Mnd ” and “2Mdi ” – many (two)
middlemen with and without disclosure respectively (see the time line on page 7 and also section 4.2). The lower
panel plots the welfare (defined in section 4) in these scenarios. The series are all plotted against φ ∈ (0, 1), the arrival
probability of EB. It can be seen from the plots that the comparison of S’s expected utility and the welfare among in
the first three scenarios cannot be signed in general. The shaded area shows that under certain parameters, adding a
second M into the economy not only reduces S’s expected utility, but also reduces welfare. The disclosure – flagging
trades between middlemen – does improve both the expected utility of S and the social welfare. The parametrization
used in this numerical procedure is: u(a) = ln(a + 1), a0 = 10, h = 10, l = 5, λ0 = 0.4, and θ0 = θ1 = θ2 = 1/3. The
cost of carry k is chosen to satisfy assumption 1.
(a) S’s expected utility against φ
64
Expected utility of S
62
60
58
56
EU0M
54
EU1M
EU2M
nd
52
EU2M
di
50
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Early buyer (EB) arrival probability, φ
0.8
0.9
1
(b) Welfare against φ
54
53
Welfare
52
51
50
w0M
49
w1M
w2M
nd
48
w2M
di
47
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Early buyer (EB) arrival probability, φ
39
0.8
0.9
1