Investment Behavior under
Uncertainty in Number of Competitors
Taejin Kim and Vishal Mangla∗
August 12, 2014
Abstract
One explanation for why investors crowd into a given strategy, as in the Quant Crisis of 2007,
even when they understand its negative implication is that they are often simply not aware of
the extent of crowding. In this paper, we build a simple model that formalizes this intuition.
To derive excessive crowding, our model relies on two ingredients: (i) an investor’s investment
decision imposes a negative externality on other investors, and (ii) investors are uncertain about
the amount of competing capital at play. The model then allows us to analyze regulations
regarding disclosure of capital committed to a strategy. Interestingly, we find that it is suboptimal
to disclose the amount of capital perfectly to the investors to mitigate the crowding, and that
there is a case for strategic blocking of the information. We show that the implementation of the
optimal disclosure policy requires a commitment device without which there occurs a policy trap.
Keywords: Strategic Substitutability, Uncertainty in Crowding, Institutional Herding, Optimal
Information Revelation.
∗
Taejin Kim is at The Chinese University of Hong Kong (phone: +852-3943-1776, email: [email protected]).
Vishal Mangla is at Moody’s Analytics (e-mail: [email protected]). We are deeply indebted to Arvind
Krishnamurthy. We are grateful for helpful comments from Snehal Banerjee, Eddie Dekel, Michael Fishman, Craig
Furfine, Kathleen Hagerty, Guido Lorenzoni, and Robert McDonald. We also thank participants at Northwestern
University. All errors are our own.
Investment Behavior under
Uncertainty in Number of Competitors
Abstract
One explanation for why investors crowd into a given strategy, as in the Quant Crisis of 2007,
even when they understand its negative implication is that they are often simply not aware of
the extent of crowding. In this paper, we build a simple model that formalizes this intuition.
To derive excessive crowding, our model relies on two ingredients: (i) an investor’s investment
decision imposes a negative externality on other investors, and (ii) investors are uncertain about
the amount of competing capital at play. The model then allows us to analyze regulations
regarding disclosure of capital committed to a strategy. Interestingly, we find that it is suboptimal
to disclose the amount of capital perfectly to the investors to mitigate the crowding, and that
there is a case for strategic blocking of the information. We show that the implementation of the
optimal disclosure policy requires a commitment device without which there occurs a policy trap.
2
The issue of excessive crowding of financial institutions over investment strategies is a central
concern for regulators as it has direct implications for systemic risk. Individually rational investors
take correlated positions that may be socially suboptimal. Stein (2012) provides a formal model
showing that crowding can be welfare reducing in a setting that involves fire-sales. In this paper,
we study a particular mechanism that leads to investment crowding and propose a policy to
mitigate this crowding when it is not feasible to directly regulate the investments made by the
financial institutions.
One potential explanation for why investors crowd into a given strategy even when they understand its negative implications is that they are often simply not aware of the extent of crowding. In
his explanation of the stock market crash of 1987, Grossman (1998) emphasizes that there was no
price based mechanism to mediate the demand for synthetic put options by the investors seeking
portfolio insurance. We build a simple model that formalizes this intuition that uncertainty about
the amount of capital invested in a given strategy can lead to excessive crowding. The model
then allows us to analyze regulations regarding disclosure of capital committed to a strategy.
To derive excessive crowding, our model relies on two ingredients: (i) an investor’s investment
decision imposes a negative externality on other investors, and (ii) investors are uncertain about
the amount of competing capital at play.
These two conditions are prevalent in financial markets. For instance, if more investors hold
the same asset or investment strategy and want to unwind their positions in an illiquid market
over a short time period, the losses will be positive related to the number of investors in that
strategy. Moreover, the market participants’ investment decisions is complicated by the fact that
they cannot observe the exact size of competition in a given strategy due to the opaqueness
and high volatility of financial markets. Stein (2009) argues that these two phenomena are very
relevant for arbitrageurs – “...he will be forced to liquidate some of the commonly held stocks to
meet margin calls, potentially creating a fire-sale effect in prices and inflicting losses on the other
trader” and “...an important consideration for each individual arbitrageur is that he cannot know
in real time exactly how many others are using the same model and taking the same position as
him”.
As a first step, we build a simple model to derive the profit of an arbitrage strategy with
3
a chance of suffering fire sales. Since the probability of a fire sale is positively related to the
size of capital invested in the strategy, the profit is decreasing in the capital amount and exhibits
negative externality.1 Next, we determine the equilibrium portfolio choice of arbitrageurs between
the strategy above and another strategy with no possibility of fire sales. The return of the second
strategy is independent of the capital invested in it. We solve this portfolio choice problem using
an investment game among arbitrageurs for analytical convenience and model the uncertainty
about the amount of capital available for investment by making the number of funds that enter
the game stochastic.2
First, we show that the concentration in the strategy with externality is higher when the
capital size is uncertain than when it is certain. In other words, a player invests more in the
negative-externality strategy when she knows that there are n rivals on average than when she
knows there are exactly n rivals. The key observation for this result is that the return of the
strategy with negative externality takes the shape of convex function exactly when the strategy
faces a high chance of fire sales.3 Intuitively, if an investment suffers from this type of negative
externality, the graph of its expected return as a function of the amount of capital should have an
inverted S-shape. For a small amount of capital, fire sales are unlikely and the return is relatively
flat. As more capital is invested in the strategy, the chance of fire sales become non-trivial and
the expected profit starts to drop. However, due to limited liability, the profit is bounded below
and becomes flat again. In this paper, we focus on the right section of the graph in which fire
sale is a real possibility.4 With convex payoff, the profit from the possibility of few competitors
in the strategy is more in magnitude than the loss from the possibility of many competitors in
the strategy.
1
We assume that the initial price of the arbitrage investment does not cancel the dependence completely. This is
a reasonable assumption in a market with mispricing. As Grossman (1988) points out, there is sometimes no pricebased mechanism. Also, since arbitrageurs learn a specific arbitrage strategy over a period of time, the impact of their
investments on the price does not reflect the amount of capital invested in the strategy, given the presence of liquidity
traders. The price impact of arbitrageurs investments is visible only when they trade in the same direction within a
short period of time such as fire sales. It is very unlikely that arbitrageurs enter into a mispriced strategy at the same
time.
2
In the same spirit of footnote 1, the arbitrageurs do not learn the amount of capital from prices when it is uncertain.
Stein (2009) illustrates that arbitrage capacity cannot be learned when fundamentals are not observed.
3
Mathematically, the result is an application of Jensens inequality, but the convexity of the profit function has
economic meaning in relation to the probability of fire sales.
4
In our model, the profit function is convex for the whole range of the amount of capital.
4
Empirically, we have witnessed several stark episodes consistent with our finding. For example,
in the stock market crash of 1987, a large number of investors engaging in portfolio insurance
liquidated their positions simultaneously bringing down the prices sharply (Grossman (1988)).
In the ‘Quant Meltdown’ of August 2007, the quantitative hedge funds employing the long/short
equity market-neutral strategies sold at the same time getting caught in a fire-sale in an otherwise
calm market (Khandani and Lo (2007)).5 In both of these episodes, large sophisticated investors
presumably understood the adverse effect of the crowding and faced uncertainty in the size of
the crowding. In this paper, using simple intuition, we argue that these conditions induced the
investors to tilt their portfolio more towards the strategy with negative externality. Next, we
introduce a regulator who perfectly observes the amount of capital available for investment. A
real world example that comes close to such a regulator is Securities Exchange Commission. In
this case, even though each individual hedge fund itself may not observe its competitors’ capital
base, SEC with which each hedge fund is required to register, observes this information albeit
with noise. Given that a direct regulation controlling the investment portfolio of the investors is
not feasible, we ask if there exists a disclosure policy that can improve the welfare. Interestingly,
we find that it may not be optimal for the regulator to perfectly disclose her information to all
the market participants and that there is a case for strategic blocking of the information. There
are two key features of our model leading to a non-trivial disclosure policy: (i) the negative
externality of investments makes the decentralized equilibrium under full information inefficient,
leaving a room for Pareto improvement, and (ii) by changing the extent of information disclosure,
the regulator can influence the level of investment in the negative-externality strategy.
In the absence of the knowledge of the exact size of competition, the investors invest in the
negative-externality strategy based on their prior on the average size of competition. If the
realized scale of capital turns out to be too large, there is excessive crowding in the negativeexternality strategy. Similarly, if the realized scale of capital turns out to be too small, there
are too few investors investing in the negative-externality strategy.6 The informed regulator
mitigates this inefficiency by granting each investor a chance to get informed of the realized scale
5
The welfare implications of the episodes are not clear. It should be understood that we cite the episodes because
the environments were very close to our setting and the evolution of events were dramatic.
6
When the level of investment in the negative-externality strategy is sufficiently small, it is more profitable than the
unity return strategy.
5
of capital with a probability chosen by the regulator.7 The investors who actually come to know
the size of the competition then either back out (if the realization is large) or invest more (if the
realization is small) in the negative-externality strategy. For the intermediate scale of capital,
the regulator simply withholds her information. Therefore, under the optimal policy not all the
investors become informed, and exactly how many become informed depends on the realized scale
of capital.
Under the partial disclosure policy described above, ex-post, some investors are perfectly
informed of the aggregate capital size while the other investors are completely uninformed. A
natural alternative to this policy is that the regulator generates a noisy public signal about the
aggregate capital size. Under this policy, all the investors would have the same information expost. However, such a policy is detrimental to the welfare. This is because the investors invest
more aggressively in the negative-externality strategy when they face uncertainty about the total
capital size as described above. And therefore, the welfare when they receive a noisy public signal
is lower than the welfare in the full information economy. In our setting, the welfare reduces as
we move from the (optimal) partial disclosure policy to the perfect disclosure policy to a noisy
public signal policy.
The partial information disclosure policy as the way we describe above suffers a ‘policy trap’
(borrowed from Angeletos, Hellwig, and Pavan (2006)). When the realized capital size of the
investors is large, the regulator decongests the negative-externality strategy by revealing this
large size to some investors which then back out of the strategy. The uninformed investors
internalize this implicit insurance provided by the regulator against the large capital size, and
invest aggressively in the negative-externality strategy. This limits the welfare enhancements
the regulator can achieve. One plausible way for the regulator to get around this trap is to
not intervene (and thus disclose her information) sometimes. When the uninformed investors
know that they are not perfectly insured against the possibility of large crowding in the negativeexternality strategy, they invest less aggressively in the strategy. However, when the realized
capital size is either too small or too large, it is suboptimal for the regulator to not intervene expost. Therefore, to implement the partial intervention policy the regulator needs a commitment
7
This probabilistic approach implies that the regulator treats all investors equally, even though ex-post asymmetry
can emerge. In Section IV, we suggest a feasible implementation of this policy.
6
device. In our setting, the welfare increases as we move from the case of no regulator to the case
of a regulator who fails to commit to intervene partially, to the case of a regulator who intervenes
with the optimal frequency.
In an extension of our model, we analyze the investment behavior of the investors when there
are two rounds of investment. In this setting, the investors who invest in the negative-externality
strategy in the first round perfectly infer the total size of the competition from their realized
profit. These investors then use this information to make their investment in the second round.
To the extent that the knowledge of the total capital size improves the investors’ profit, we would
expect that this ‘learning by doing’ strengthens the investors’ incentives to invest in the negativeexternality strategy in the first round. We show that when the investors are uncertain about
the size of the competing capital, they invest even more aggressively in the negative-externality
strategy when there are multiple rounds of investment.
There is a vast literature on fire-sales that create inefficiency or negative externality on other
investors.8 Morris and Shin (2004) explain a liquidity spiral using loss limits and a downward
sloping demand curve. In their model, a trader internalizes the fact that her payoff negatively
depends on the measure of selling competitors. Wagner (2011) presents a model in which investors minimize the risk of joint liquidation by holding diverse portfolios. Our basic setup of the
investment game is similar to Dixit and Shapiro (1986) who address the entry game in Cournot
oligopoly (although they do not consider the uncertainty in the number of potential competitors).
Uncertainty in the number of funds (“population uncertainty”) has been discussed in the
study of auctions (McAfee and McMillan (1987), Matthews (1987)) and voting games (Myerson
(1998, 1998b, 2000)). In particular, Myerson introduces Poisson games to explain a low turnout in
elections. Uncertainty in the number of market participants has also been discussed in the contexts
of the 1987 market crash and the 1998 hedge fund crisis. Stein (2009) uses population uncertainty
to explain the market inefficiency that arbitrageurs are unable to eliminate a mispricing even with
a large amount of capital available to them. This paper studies the effect of population uncertainty
on the crowding behavior of investors and its implications for policy.
8
A few notable examples of fire-sales are Shleifer and Vishny (1992, 1997), Gromb and Vayanos (2002), and Brunnermeier and Pedersen (2008). The standard mechanism that generates fire-sales is a financial/margin constraint on
intermediaries/arbitrageurs. Also, the primary interest of this literature has been the efficiency of prices, while this
paper focuses on the crowding behavior towards a strategy with negative externality.
7
I
Model
A
Environment
There are three dates, t = 0, 1, 2, and the risk-free discount rate is zero. There are two types of riskneutral investors in the economy: n + 1 funds that are expert at investing in a given specialized
strategy, labeled strategy A, and a generalist who has no particular expertise in investing in
strategy A. The investment strategy A can be imagined as a dynamic trading strategy involving
adjusting positions in multiple securities simultaneously as in, for example, a long/short equity
market-neutral strategy or a synthetic put option replication for portfolio insurance, etc. The
strategy A pays off at t = 2 and the cashflow depends upon the investor in the strategy. The
cashflow is Ṽ for a fund and Ṽ (1−δ) for the generalist, where δ (> 0) captures the loss in efficiency
in switching the investor in strategy A from a fund to the generalist. Other than strategy A there
exists another investment strategy, labeled strategy F , that generates unity cashflow at t = 2
irrespective of the investor in the strategy. The funds choose their strategy at t = 0 (as described
later).
Each fund is subject to a stop-loss covenant under which the fund has to liquidate its position
if the observed market price of its position at t = 1 is below a prespecified limit denoted by
L.9 Moreover, at t = 1, each fund is exposed to an exogenous liquidity shock which hits each
fund independently with probability 1 − ξ and which requires the shocked funds to liquidate their
positions immediately. The generalist assumes the role of a market maker and is willing to buy
the liquidating funds’ positions at t = 1 at a price equal to her expected cashflow: one for strategy
F and v(1 − δ) for strategy A, where v ≡ E[Ṽ ]. We assume the high search cost prohibits the
funds to trade among themselves. We impose the restriction v(1 − δ) < L to make the analysis
interesting. This restriction on the model parameters implies that even if one fund invested in
strategy A liquidates its position, in which case the observed market price of the position is
v(1 − δ), the stop-loss covenant for all funds invested in strategy A gets invoked forcing all these
funds to liquidate at v(1 − δ).
Suppose m + 1 funds are invested in strategy A before the liquidity shock hits at t = 1. Then
9
See Morris and Shin (2004) for motivation behind this assumption.
8
with probability ξ m+1 none of these funds faces the liquidity shock and their expected payoff is
v. However, with probability 1 − ξ m+1 the stop-loss covenant is invoked forcing these funds to
liquidate and receive a payoff of v(1 − δ). Therefore, the expected payoff at the beginning of t = 1
for a fund invested in strategy A when there are m other funds adopting strategy A is given by
Π(m) = vξ m+1 + v(1 − δ)(1 − ξ m+1 )
= v(1 − δ) + vδξ m+1 .
As the number of funds investing in strategy A increases the funds’ expected payoff decreases –
negative externality. Also note that Π(m) is convex in m.
At t = 0, the n + 1 funds randomize between the strategies A and F with a probability of
their choice. Two remarks are in order. First, instead of randomization one can very well imagine
a setting in which each fund splits its investment between two strategies optimally. We do not
follow this approach here because the additional complexity introduced by portfolio optimization
makes the analysis unwieldy. Instead we view the probability assigned by a fund to a strategy
as the weight of that strategy in the fund’s portfolio. Using a simpler form of the profit function
Π we show in Appendix that our main results continue to hold even when we use the portfolio
approach. Second, we do not consider the case in which a fund can exert market power (all the
funds have the same scale of investment in our setting). Although, such an analysis is interesting
in its own right the setting in this paper is rich enough to develop insights about the effect of
uncertainty in the amount of competing capital at play and the corresponding disclosure policy.
In our setting, n + 1 captures the total amount of capital potentially available for investment
in a given specialized strategy at a point in time. We model the uncertainty in the amount
of available capital by making the total number of ‘rivals’, n, unobservable to the participating
funds. The participants only know that n follows the Poisson distribution with mean λ. We use
the Poisson distribution because it has several nice properties that facilitate our analyses and
discussions of the effects of population uncertainty.10
10
See Myerson (1998) for a discussion on the relevant properties of the Poisson distribution.
9
B
Expected Payoff
In the following we express the expected payoff for a fund from investing in strategy A when
each of the other funds chooses strategy A with probability p (and therefore strategy F with
probability 1 − p). We consider two cases: (i) each fund knows the number of other funds to be
exactly n (no population uncertainty); denote the expected profit in this case by E d [Π(m); n, p],11
and (ii) each fund knows the number of other funds follows the Poisson distribution with parameter λ (population uncertainty); denote the expected profit in this case by E s [Π(m); λ, p]. We
use superscript d to refer to the deterministic population size and superscript s to refer to the
stochastic population size. The two expected payoffs are expressed as
n X
n m
E [Π(m); n, p] ≡
p (1 − p)n−m Π(m)
m
m=0
∞ −λ n X
e λ
s
E [Π(m); λ, p] ≡
E d [Π(m); n, p]
n!
d
(1)
(2)
n=0
Poisson distribution has the following useful property that allows a simplification of the expected
payoff in the case of population uncertainty:
Lemma 1. (Poisson Decomposition Property) If the number of rivals follows Pois(λ) distribution
and each rival chooses strategy A with probability p, then the total number of rivals choosing A
follows Pois(λp) distribution.
This lemma immediately leads to the simplification:
s
E [Π(m); λ, p] ≡
∞ −λp
X
e (λp)m
m=0
m!
Π(m)
The following excess payoff functions will be instrumental in characterizing the equilibrium:
Gdn (p) ≡ E d [Π(m); n, p] − 1
Gsλ (p) ≡ E s [Π(m); λ, p] − 1
11
n is the total number of other funds that have access to strategy A and could have potentially invested in A.
However, m is the number of rivals out of the n rivals that actually invested in A in a given draw of the investment
game.
10
In words, given the symmetric strategy of the rivals: invest in A with probability p and in F
with probability 1 − p, the functions Gdn (p) and Gsλ (p) express the expected excess payoff from
choosing strategy A over strategy F , with and without population uncertainty, respectively.
C
Equilibrium
Equilibrium is defined as a Bayesian Nash equilibrium in the standard sense. As stressed by Myerson (1998), in any game with population uncertainty there is essentially no loss of generality in
assuming that only symmetric equilibria exist. Intuitively, in a game with population uncertainty,
all identical funds must have identical predicted behavior in equilibrium.
In stark contrast, if the number of funds is known to everyone, we can in general construct an
asymmetric equilibrium in which funds with identical payoff-relevant characteristics invest differently. The asymmetric equilibria may generate unrealistic results, as in the examples of voting
games. In the following, in order to analyze the effect of population uncertainty on equilibrium
behavior, we focus only on the symmetric equilibria even in the case of no population uncertainty.
II
Symmetric Equilibrium
Let us define a scalar µ ≡ Π(0) = v(1 − δ + δξ) and a function π(m) ≡ Π(m)/µ. Therefore, we
have the normalization π(0) = 1. Since strategy F always yields a payoff of one, we assume µ > 1
so that strategy F does not dominate strategy A. We begin our discussion with the benchmark
case where the number of funds is common knowledge.
A
No Population Uncertainty
Suppose the number of rivals is n, known to all the n + 1 funds. As discussed above, we focus
on the symmetric equilibrium in which each player randomizes between the two strategies with
the same probability. If each of the n rivals chooses strategy A with probability p, the expected
11
excess payoff of the (n + 1)th player from choosing strategy A over strategy F is,
Gdn (p)
d
= E [µπ(m); n, p] − 1 = µ
n X
n
m=0
m
pm (1 − p)n−m π(m) − 1
The symmetric equilibrium is the probability p that sets this excess payoff equal to zero (if possible). If Gdn (1) > 0, it means that even if all the n + 1 funds choose A with probability one,
they earn more than unity. In this case, strategy A dominates strategy F and the equilibrium is
pdn = 1. The following proposition characterizes the equilibrium.
Proposition 1.
1. The equilibrium exists and is unique, denoted by pdn .
2. The concentration in the more profitable strategy is weakly decreasing in the population size:
pdn+1 ≤ pdn (the inequality is strict when Gdn (1) < 0).
As the probability of choosing strategy A increases, the number of funds in A increases in
expectation, making A less profitable. If the probability of choosing A is too small, strategy
A is more profitable than strategy F and vice versa. At the (unique) equilibrium value of this
probability, pdn , A has the same expected payoff as that of F . Similarly, for a given probability
of choosing strategy A, A becomes less profitable as the total number of funds increases (since
there are more funds in A). This means that for a larger population base, a smaller probability
of choosing strategy A is enough to make the expected payoff of A equal to one, the payoff from
choosing strategy F . That is, the value of equilibrium probability falls with the total number of
funds.
B
Social Welfare
The society in our economy consists of the funds and the generalist. Since the generalist is not
able to obtain the full payoff from strategy A, a transaction between a fund invested in strategy A
and the generalist creates inefficiency. A similar inefficiency in a setting with fire-sales is discussed
in Shleifer and Vishny (1992). When many investors holding the same portfolio try to liquidate
their positions at the same time, short term illiquidity may arise due to slow moving capital.
12
In this situation, the assets sold in fire sales may end up with the ‘outsiders’ who are not the
first-best user of the assets. This means that the fire sale transactions result in inefficiency. In
this sense the crowding of investors into a strategy creates a dead-weight loss for the society.
Since the generalist always break even, the (utilitarian) social welfare is simply the funds’
expected aggregate payoff which, when the funds choose the strategy A with probability p, is
given by the expression
Sn (p) =
n+1
X
m=0
n+1 m
p (1 − p)n+1−m [mµπ(m − 1) + n + 1 − m]
m
= (n + 1)(1 + pGdn (p)),
This is because, if there are m funds investing in A, each fund in A earns the payoff µπ(m − 1)
(since there are m − 1 rivals) and the rest earn unity (the equality follows after some algebraic
manipulation).
The first-best or the socially optimal probability, denoted by p∗n , is the probability of choosing
strategy A that maximizes the utilitarian social welfare: p∗n ∈ arg maxp Sn (p). The following
result tells us that the competitive investors end up choosing strategy A more frequently than
efficient.
Proposition 2. The concentration in the more profitable strategy is higher in the competitive
equilibrium than under the social optimum: p∗n < pdn .
When a player chooses strategy A over strategy F , she imposes a negative externality on the
other funds choosing A by making strategy A more crowded. However, each player’s individually
rational investment decision does not consider this social cost. This leads to a “tragedy of the
commons” in which each player overinvests in strategy A.
This fire-sale type externality has been discussed extensively in the literature. An example
is Krishnamurthy (2010) in which the social cost of debt is higher than the private cost of debt
because any single hedge fund does not take into account the fact that increasing its debt implies
that the fund has to liquidate more of its asset when the price of the asset falls, pushing down
13
the price further, which in turn results in greater liquidations by other hedge funds.
C
Population Uncertainty
The total number of rivals follows Pois(λ). If each of the rivals chooses strategy A with probability
p, the expected excess payoff of choosing strategy A over strategy F is
Gsλ (p)
s
= E [µπ(m); λ, p] − 1 = µ
∞ −λp
X
e (λp)m
m=0
m!
π(m) − 1
The symmetric equilibrium is the probability p that sets this excess payoff equal to zero (if possible). As in the case of no population uncertainty, the probability delivering zero excess payoff
does not exist when strategy A dominates strategy F , leading to the corner solution, p = 1, for
the equilibrium probability. The following proposition characterizes the equilibrium, denoted by
psλ .
Proposition 3. The equilibrium psλ is of the form
psλ
γ(µ)
= min 1,
λ
for some function γ(·) satisfying γ(·) > 0, γ 0 (·) > 0.
Intuitively speaking, the higher is the probability with which the funds choose strategy A,
the higher is the expected number of funds in A and lower is the expected payoff of A. In
equilibrium, the funds choose strategy A often enough that the expected payoff of A equals one.
Therefore, if strategy A is made more profitable (i.e., if µ increases), the funds need to choose A
more frequently to make its expected payoff equal to one (i.e., psλ increases). Analogously, as the
average total number of funds increases, the probability of choosing A must fall to ensure that
the expected number of funds in A stays constant (at the level such that the expected payoff of
A equals one).
Next, we compare this equilibrium with the equilibrium of the previous subsection, the case
of no population uncertainty. It turns out that the comparison relies on the shape of the payoff
14
function π(·), in conjunction with the following useful lemma.
Lemma 2. P ois(λ) is a mean preserving spread of B(n, λ/n).
The next proposition, which is one of our main results, states that the funds invest more
aggressively when they do not know the exact population size.
Proposition 4. The concentration in the more profitable strategy is higher when the population
size is stochastic than when it is deterministic, i.e. pdn < psn ∀n.
Proof. The previous lemma and the convexity of π(·) imply Gdn (p) < Gsn (p) ∀n, p. Since Gdn (p) is
decreasing in p and Gdn (psn ) < Gsn (psn ) = 0, we have pdn < psn .
For a given probability of choosing strategy A, the number of funds in A is more likely to
take extreme values when the population size is stochastic than when it is deterministic. This
has two opposite but unequal effects on the expected payoff of A in the stochastic case relative
to the deterministic case – the rise in payoff due to the smaller number of funds in A exceeds
the fall in payoff due to the larger number of funds in A. This is because the strategy A’s payoff
function saturates as the number of funds in A increases. The upshot is that strategy A becomes
more profitable with the variance of the population size. This generates the incentive for any
given fund to bet on the likelihood of the number of rivals in strategy A being small when the
population size is stochastic, making the fund aggressive in its choice of A.
This proposition is based on a simple argument, but affords a counterintuitive insight into
investment crowding. As explained earlier, it is very likely that π(·) is convex in the number of
funds when the expected number of funds is large. Therefore, when people think that a strategy
with negative externality is already crowded, population uncertainty pushes the investors into
crowding even more in the strategy. Needless to say, the absolute level of crowding goes down as
λ increases. The point is that population uncertainty only aggravates the situation and summons
policy intervention we will discuss in Section IV.
15
We have the following two important corollaries of the previous proposition,
Corollary 1. Sn (psn ) < Sn (pdn ) < Sn (p∗n ) ∀n
We have already discussed the second inequality of this relation. Since the funds become more
aggressive in their choice of A when the population size is stochastic, the negative externality they
impose on each other by choosing A worsens in the stochastic case, yielding the first inequality.
This corollary implies that reducing investors’ uncertainty about n is conducive to social welfare
under symmetric information. We discuss the asymmetric case in the next section, a stepping
stone to a policy proposal.
Corollary 2. p∗n , pdn , psn → 0 as n → ∞
As the total population size approaches infinity, for any positive probability of choosing A,
there are infinite number of funds in A driving the expected payoff of A to zero. Therefore, the
equilibrium probability of choosing A approaches zero in the limit.
III
Asymmetrically Informed Players
So far, we have assumed all the funds have symmetric information – either all funds know the total
number of funds or no fund knows the total number of funds. But in reality, it is likely that some
funds know more about their competition than the others. In this section, we characterize the set
of equilibria that emerge when some funds are completely informed and the others are completely
uninformed. This setting lets us analyze the welfare aspect of information. Interestingly, we find
that more information does not always to lead to higher social welfare.
The number of rivals follows Pois(λ) and n is a realization. χ is the probability that a fund
is informed (independent across funds). The average rival population size λ and the probability
χ are common knowledge and an uninformed fund’s strategy is a function only of these two –
16
probability pU (λ, χ) of choosing strategy A. On the other hand, an informed fund’s strategy
depends also on the realization n – probability pI (λ, χ, n) of choosing A.12
A
Equilibrium under Information Asymmetry
In order to characterize the equilibria in this setting, we will first find the best response of the
informed funds to any arbitrary strategy of the uninformed funds. Then, given this best response
correspondence, we will find the optimal strategy of the uninformed funds. Accordingly, let pU
and pI be arbitrary strategies of the uninformed and the informed funds respectively. Then, the
(unconditional) probability q that a given fund chooses strategy A is
q = P {fund is uninformed}P {fund chooses A| fund is uninformed}
+ P {fund is informed}P {fund chooses A| fund is informed}
= (1 − χ)pU + χpI
The excess payoff ΠI of an informed fund from choosing strategy A over strategy F is13
n X
n m
Π (pI , pU ) = µ
q (1 − q)n−m π(m) − 1 = Gdn (q)
m
I
m=0
For pI to be the best response to the uninformed funds’ strategy pU , we must have14
• If ΠI (1, pU ) > 0, then pI = 1.
• If pI < 1, then ΠI (pI , pU ) ≤ 0. If pI > 0, then ΠI (pI , pU ) ≥ 0.
12
Again, we focus only on equilibria in which all funds of the same type invest identically.
It is important to note the meaning we attach to χ – it is the ex-ante probability of being informed, not the actual
fraction of informed funds. As it turns out, this interpretation is instrumental in that it allows us to use the weighted
average probability q in our computation. For concreteness, suppose n = 2 and χ = .5. If this is interpreted as 0.5
fraction of rivals are informed (meaning there is 1 informed rival and 1 uninformed rival), the expected payoff from A
is µ{pI pU π(2) + [pI (1 − pU ) + pU (1 −
pI )]π(1) + (1 − pU )(1 − pI )π(0)}. If χ is interpreted as prior probability instead,
m
P2
2
the expected payoff is µχ2 m=0 m
pI (1 − pI )2−m π(m) + 2µχ(1 − χ)[pI pU π(2) + [pI (1 − pU ) + pU (1 − pI )]π(1) + (1 −
m
P2
2
pU (1 − pU )2−m π(m) = µ{[χpI + (1 − χ)pU ]2 π(2) + 2[χpI + (1 − χ)pU ][χ(1 − pI ) +
pU )(1 − pI )π(0)] + µ(1 − χ)2 m=0 m
m
P2
2
(1 − χ)(1 − pU )]π(1) + [χ(1 − pI ) + (1 − χ)(1 − pU )]2 π(0)} = µ m=0 m
q (1 − q)2−m π(m), the form we use. This
‘discrepancy’ appears in this paper because we cannot resort to the Strong Law of Large Numbers, as often used in the
literature by assuming a continuum of agents.
14
Note that even though an informed fund takes q as a given, our focus is on equilibria in which each informed fund
chooses the same pI .
13
17
• If ΠI (0, pU ) < 0, then pI = 0.
This observation leads us to the characterization of the best response pI in the following lemma.
Lemma 3. The best response function of the informed funds to the uninformed funds’ strategy
pU is given by (for χ 6= 0)
pI (pU , χ, n) =




1




if n < n1 ,
pdn −(1−χ)pU
χ






0
if n1 ≤ n ≤ n2 ,
if n > n2
where n1 and n2 are (implicitly) defined as
pdn1 = (1 − χ)pU + χ
pdn2 = (1 − χ)pU
6
pdn
pdn1
q(pU , χ, n)
pdn2
-
n1
n2
n
Figure 1: The Unconditional Probability of Choosing A, q(pU , χ, n), as a function of n for
given pU and χ. Between n1 and n2 , q and pdn coincide.
The informed funds’ response to the uninformed funds’ probability pU of choosing A depends
on whether the informed funds find pU aggressive or conservative. This, in turn, depends on the
realized number of funds. If the realized n is small, the informed funds find pU conservative in
18
the sense that there are too few funds investing in A. If this condition is true even when all the
informed funds choose the pure strategy A, the optimal symmetric strategy of the informed funds
is to attack A with probability one. To see this, notice that when the informed funds choose
strategy A with probability one, the unconditional probability of choosing A is q = (1 − χ)pU + χ.
And for sufficiently small n (n < n1 ), pdn > q meaning the expected excess payoff of choosing A
over F , Gdn (q) is positive.
Analogously, when the realized n is large, pU results in excessive concentration in A, and the
informed funds avoid A altogether. When the informed funds completely avoid choosing A, the
unconditional probability of choosing A is q = (1 − χ)pU . And for sufficiently large n (n > n2 ),
pdn < q meaning the expected excess payoff of choosing A over F , Gdn (q), is negative. Thus,
choosing F with probability one is the optimal strategy for the informed funds.
For the intermediate values of n (n1 ≤ n ≤ n2 ), the uninformed funds’ strategy pU results
in only a moderate number of funds in A. In this situation, if all the informed funds choose A
with probability one, there are too many funds in A, making A less profitable than F . Likewise,
if all the informed funds choose F with probability one, there are too few funds in A making A
more profitable than F . The informed funds fail to coordinate on either of the pure strategies A
or F and end up randomizing between them. Their mixing probability pI is such that the two
strategies A and F are equally profitable – q = (1 − χ)pU + χpI equals pdn implying Gdn (q) = 0.
Now let us characterize the optimal strategy pU (λ, χ) of the uninformed funds. If the uninformed funds choose strategy pU and the informed funds act optimally, the (unconditional)
probability q(pU , χ, n) that a given fund chooses strategy A is (see Figure 1)
q(pU , χ, n) = (1 − χ)pU + χpI (pU , χ, n)
The expected excess payoff ΠU of an uninformed fund from choosing strategy A over F is
∞ −λ n X
n X
e λ
n
Π (pU ) = µ
q(pU , χ, n)m (1 − q(pU , χ, n))n−m π(m) − 1
n!
m
U
=
n=0
∞
X e−λ λn
n=0
n!
m=0
Gdn (q(pU , χ, n))
19
For pU to be the optimal strategy of the uninformed funds, we must have15
• If ΠU (1) > 0, then pU = 1.
• If pU < 1, then ΠU (pU ) ≤ 0. If pU > 0, then ΠU (pU ) ≥ 0.
• If ΠU (0) < 0, then pU = 0.
The following proposition confirms the existence of an equilibrium (we abuse the notation to write
pI (pU (λ, χ), χ, n) as pI (λ, χ, n)).
Proposition 5. The equilibrium {pU (λ, χ), pI (λ, χ, n)} under information asymmetry exists and
is unique for all λ, χ and n.
When the informed funds act optimally, the excess payoff of investing in A over F , Gdn , is
positive when the realized population size is small (this is why the informed funds set pI = 1 for
small n) and negative when the realized population is large (in this case pI is set to zero). When
the uninformed funds’ probability of choosing A, pU , increases, the range of values of n over which
A is more profitable than F shrinks and the range over which A is less profitable than F expands.
As a result, the excess payoff of choosing A over F for the uninformed funds averaged over all
realizations of n, ΠU , falls as pU increases. The uniqueness of the equilibrium is a consequence
of this fact.
B
Properties of the Equilibrium
One natural question under asymmetric information is how much value of information the informed funds are able to enjoy compared to the uninformed funds due to the information advantage. We define the value of information as the difference between the expected payoffs
15
Note that even though an uninformed fund takes pU as a given, our focus is on equilibria in which each uninformed
fund chooses the same pU .
20
(suppressing λ):
R(χ, n) ≡ [pI (χ, n) − pU (χ)]Gdn (q(χ, n))




[1 − pU (χ)]Gdn (pdn1 ) if n < n1 ,




= 0
if n1 ≤ n ≤ n2 ,






−pU (χ)Gd (pd )
if n > n2
n n2
As expected, R is non-negative for each value of n and is positive for some values of n (recall
Gdn (pdn1 ) > 0 for n < n1 and Gdn (pdn2 ) < 0 for n > n2 ). The expected value of information is given
by
R(χ) ≡
∞ −λ n
X
e λ
n=0
n!
R(χ, n),
This value tells us how much an investor’s payoff would increase by becoming informed, holding
χ fixed for her competitors. As a result, this function naturally appears in a two-period extension
in Section VI.A, where investing in A gives a chance to be informed in the next period. The next
proposition tells us how the likelihood of a fund being informed, χ, impacts this expected value
of information.
Proposition 6. The expected value of information is decreasing in the expected number of informed funds: R0 (χ) ≤ 0.
This result is intuitively appealing – the more the number of informed funds, the less is their
informational advantage. From the discussion above and the discussion of Lemma 3, we know that
the informed funds have strictly higher expected payoff in case that they are able to coordinate in
choosing one of the strategies – A or F . The advantage disappears when the informed funds fail to
coordinate on the strategies (this happens for the intermediate values of the realized population
size n). Obviously, the more the number of informed funds, the harder it is to achieve coordination
among them. Therefore, when the number of informed funds is larger, the value of information
is zero for a larger range of n, driving down the average value.
21
The next proposition summarizes the various comparative statics results under asymmetric
information.
Proposition 7.
∂pU (λ, χ)
≤0
∂λ
pU (λ, 0) = psλ , pU (λ, 1) = 1
∂pU (λ, χ)
pU (λ, χ)
1 − pU (λ, χ)
−
≤
≤
1−χ
∂χ
1−χ
When the informed funds act optimally, they avoid investing in A when the realized rival
population size n is large because the number of funds in A is likely to be large, making A less
profitable than F . When λ is large, n is likely to be large. This implies the uninformed funds, in
turn, should optimally choose A less often – pU falls with λ.
IV
Capital Disclosure Policy
We are now ready to analyze regulation policies regarding disclosure of capital available for
investment in a strategy to the investors. Suppose there is a regulator who has better information
than each individual investor about the amount of capital available. A real world example that
comes close to such a regulator is Securities Exchange Commission. In this case, even though each
individual hedge fund itself may not observe its competitors’ capital base, SEC with which each
hedge fund is required to register, observes this information albeit with noise. In this section, we
ask how the regulator should disclose her information about the capital base to the investors in
order to maximize the social welfare.
To keep the analysis simple, we assume that the asymmetry of information between the funds
and the regulator is extreme – no fund observes the total number of funds n + 1 (they only know
that n is drawn from Pois(λ)) but the regulator observes n perfectly. The full disclosure policy –
perfectly disclosing n to each fund – results in the decentralized equilibrium under no population
22
uncertainty p = pdn . We know that this equilibrium is inefficient (Corollary 1).16 This observation
naturally leads us to consider less-than-perfect disclosure policies. The set of such policies is
potentially huge. We restrict our discussion to disclosure policies that treat the funds equally
(no favoritism) and take the form of providing the funds signals about the capital size, n. In our
setting, noisy public signals are detrimental for the social welfare (see below). Therefore, in the
following, we explore a disclosure policy that allows the regulator to push the equilibrium under
the decentralized economy closer to the social optimum p∗n using private signals.
We consider a two-tiered (disclosure) policy tool:
(i) The regulator can decide whether to intervene or not.
(ii) If the regulator intervenes, she discloses n privately to the individual funds with a chosen
probability.
We denote the probability of intervention by τ and the probability with which to disclose n
privately to the individual funds by χ(n) (a function of n).17 So a disclosure policy is the pair
(τ, χ(·)). When the regulator does not intervene, all the funds stay uninformed (of n) and when
she intervenes, some funds become informed and the others continue to stay uninformed after the
disclosure. The optimal policy is the pair (τ ∗ , χ∗ (·)) that maximizes the social welfare.
To solve for the optimal policy, we first find the optimal disclosure probability schedule χ(·)
given the intervention probability τ . In the next step, we find the optimal intervention probability
τ ∗ given the optimal schedule χ(·).
A
Optimal Disclosure Probability Schedule
For this subsection, fix the intervention probability τ . Let χ(·) be the corresponding optimal
schedule. The figure below shows the sequence of events that unfold when the regulator intervenes.
16
At the other extreme, if no information is disclosed to the funds, the social welfare in the resulting equilibrium
p = psλ is even lower.
17
More generally, we can also allow the probability of intervention to depend on n: τ (n). While this generalization
could produce interesting results, we limit ourselves here to a constant probability τ . The main goal of this section is
to show that the regulator needs a commitment device to be able to improve welfare over the decentralized economy
under full information. This will be shown to be true even in the restricted space of constant τ .
23
n is realized
No fund knows n
All funds know (τ, χ(·))
Planner reveals n
to each fund with
probability χ(n)
Uninformed funds
update their
prior on n
Game is played: uninformed
funds choose A with
probability p̂U and
informed funds choose
A with probability p̂I (n)
Figure 2: Sequence of Events at t = 0 when Regulator Intervenes
A Feasible Implementation of χ(·): The regulator provides each fund a private signal that
reveals the true n with probability χ(n) and reveals nothing with probability 1 − χ(n). The
signals are statistically independent, and χ(n) can be interpreted as a measure of precision or
informativeness of the signals. What we require is that the regulator has a device to control this
precision.18 In this implementation, the regulator treats each fund equally ex-ante. After the
signal realization, some funds would be completely informed and others completely uninformed.
The important thing to note is that the regulator does not know (and has no need to know) which
funds are informed and which are not; the regulator does not favor any fund over others ex-ante
and ex-post.
Ineffectiveness of Public Signals: A natural alternative to the disclosure policy described
above is that the regulator generates a noisy public signal that confines n within some region. A
18
It may appear uncommon that the regulator is able to control the precision of private signals, not public. However,
it should be noted that it is very common in the literature to assume that the precision of private signals are exogenously
given from the individual fund’s perspective. We implicitly assume that the precision is affected by the state of the
overall economy which the regulator can influence. If the precision of private signals is considered to be a choice variable
in the spirit of information acquisition, it is not hard to imagine that the regulator can influence the cost of acquiring
information, which plays a qualitatively similar role to the precision control in this paper.
24
public signal leaves all the funds equally informed ex-post. However, this policy is detrimental to
the welfare. Proposition 4 tells us that under uncertainty, funds invest more aggressively in the
negative-externality asset. This means that upon observing the public signal, the funds choose
strategy A with probability larger than in the deterministic case, pdn (assuming n is the mean
value of the number of rivals conditional on the signal). This reduces social welfare since it peaks
at p∗n and p∗n < pdn . Therefore, generating a public signal about n has lower social welfare than
simply disclosing n to all the funds.
Let p̂U denote the optimal strategy of the uninformed funds given the policy (τ, χ(·)). The
corresponding optimal strategy of the informed funds is p̂I (n) = pI (p̂U , χ(n), n). Then, the
(unconditional) probability with which a given fund chooses strategy A is
q̂(n) = (1 − χ(n))p̂U + χ(n)p̂I (n)
The expected payoff of the two types of funds when the regulator intervenes are 1 + p̂U Gdn (q̂(n))
and 1 + p̂I (n)Gdn (q̂(n)) respectively. Thus, the payoff averaged over the two types is
h
i
h
i
(1 − χ(n)) 1 + p̂U Gdn (q̂(n)) + χ(n) 1 + p̂I (n)Gdn (q̂(n))
= 1 + q̂(n)Gdn (q̂(n))
Therefore the regulator’s problem boils down to
h
i
max (n + 1) 1 + q̂(n)Gdn (q̂(n))
χ(·)
This program compares to the so-called ‘unconstrained’ program of the planner in Section II:
maxp Sn (p). The unconstrained program has the solution p = p∗n . Therefore, the above stated
constrained program is equivalent to choosing χ(·) such that q̂(n) is as close to p∗n as possible.
Figure 3 depicts the constrained optimal q̂(n) for the various values of n.
The corresponding optimal χ(·) is described in the following lemma.
25
Lemma 4. When the uninformed funds’ strategy is p̂U , the optimal disclosure probability schedule
is given by
χ(n) =


p∗n −p̂U



1−p̂U



0






1 −
if n < N1 ,
if N1 ≤ n ≤ N2 ,
pdn
p̂U
if n > N2
where N1 and N2 are (implicitly) defined as
p∗N1 = pdN2 = p̂U .
pd
6
6
pd
p∗
q̂(n) = p∗n
p̂U
q̂(n) = p̂U
p̂U
p∗
-
n
-
N1
N2
(a) Case: n < N1
6
N1 n N2
(b) Case: N1 ≤ n ≤ N2
pd
p̂U
p∗
q̂(n) = pdn
-
N1
N2
n
(c) Case: n > N2
Figure 3: Optimal Information Revelation for three different ranges of n
26
The above lemma states that, when the regulator intervenes, it is not optimal to disclose the
population size perfectly to the funds. In fact, for the moderate values of n, the optimal policy
is to keep all the funds uninformed. This non-trivial form of the disclosure policy is due to the
wedge between the social optimum (p∗n ) and the decentralized equilibrium under full information
(pdn ). The optimal policy enhances the welfare by making the negative externality interact with
the population uncertainty.
B
Optimal Intervention Probability
We arrive at the optimal intervention probability τ ∗ of the regulator by dividing our analysis
into three steps. First, we analyze the special case of full intervention τ = 1 and show why it is
suboptimal. Second, we determine the optimal ex-ante strategy of the uninformed funds p̂U for
a given probability of intervention τ . Lastly, given the optimal strategy p̂U , we determine τ ∗ .
B.1
Full Intervention
Suppose the regulator intervenes for sure and implements the optimal disclosure probability schedule χ(·) obtained in the previous subsection. Let q̂n be the corresponding (unconditional) probability with which a given fund chooses strategy A. As we can see from Figure 3, when n < N2 ,
we have q̂n < pdn implying strategy A has higher payoff than that of F (recall Gdn (p) > 0 for
p < pdn ). When n ≥ N2 , we have q̂n = pdn implying the two strategies have the same payoff. This
means, the payoff of A averaged over all realizations of n is higher than that of F . Therefore, the
optimal strategy for the uninformed funds is to set p̂U = 1. This, in turn, yields q̂(n) = pdn for
all n. The social welfare in this case is same as in the decentralized economy with no population
uncertainty.
In this sense, the full intervention by the regulator suffers a ‘policy trap’ (to borrow the term
from Angeletos, Hellwig, and Pavan (2006)).19 The uninformed funds internalize the implicit
insurance provided by the regulator against the large capital size, and thus invest aggressively in
19
Angeletos, Hellwig, and Pavan (2006) study a global-game setup in which policy effectiveness is undermined due to
the signaling effect of policy announcement.
27
the negative-externality strategy. This limits the welfare enhancements the regulator can achieve.
One plausible way for the regulator to get around this trap is to not intervene (and thus disclose
her information) sometimes, the case we discuss next.
B.2
Partial Intervention
The above discussion suggests that, for there to be any room for welfare improvement beyond
the decentralized economy under no population uncertainty, we need the uninformed funds to
be not fully aggressive: p̂U < 1. This can be achieved when the regulator intervenes partially:
τ < 1.20 Under partial intervention, when she intervenes, the regulator discloses n privately to
the individual funds with probability χ(n). When she does not intervene, all the funds remain
uninformed and play the strategy p̂U .
Commitment Problem: Once n is realized and if it does not belong to the interval [N1 , N2 ],
the regulator wants to intervene for sure as it is welfare improving at that value of n (see Lemma
4). That is, the regulator faces a commitment problem. Therefore, partial intervention is feasible
only if the regulator can commit to implement χ(n) with probability τ which is strictly less than
one. Henceforth, we assume the regulator has access to such a commitment device.
The excess payoffs of playing strategy A over F when the regulator intervenes is Gdn (q̂(n)),
and it is Gdn (p̂U ) when she does not intervene. In order to compute the expected excess payoff
from the perspective of the uninformed fund, we also need to determine the posterior probability
they attach to each possible n. This posterior is different from Pois(λ) because the probability
that a fund stays uninformed, 1 − χ(n), is different for different n.
Event I denotes whether the fund is informed (I = 1) or not (I = 0). Event J denotes
whether the regulator intervenes (J = 1) or not (J = 0). The joint probability that a fund stays
uninformed, n is realized, and the regulator intervenes – the regulator observes n – is given by
Pr(I = 0, n, J = 1) = τ (1 − χ(n))
20
e−λ λn
,
n!
At the other extreme, τ = 0, p̂U = psλ , which reduces to the population uncertainty case of Section II with no
regulator.
28
while the joint probability that a fund stays uninformed, n is realized, and the regulator does not
intervene is
Pr(I = 0, n, J = 0) = (1 − τ )
e−λ λn
.
n!
The marginal probability of being uninformed is, therefore,
Pr(I = 0) =
∞ −λ n
X
e λ
n=0
n!
[τ (1 − χ(n)) + (1 − τ )] .
Thus, the expected excess payoff from the perspective of an uninformed fund is
∞
ΠU
τ (p̂U )
X
1
Pr(I = 0, n, J = 1)Gdn (q̂(n)) + Pr(I = 0, n, J = 0)Gdn (p̂U )
=
Pr(I = 0)
n=0
∞
X
i
e−λ λn h
τ (1 − χ(n))Gdn (q̂(n)) + (1 − τ )Gdn (p̂U )
n!
n=0
#
" ∞
X e−λ λn
1
τ
(1 − χ(n))Gdn (q̂(n)) + (1 − τ )Gsλ (p̂U )
=
Pr(I = 0)
n!
=
1
Pr(I = 0)
n=0
U
Then, if ΠU
τ (1) ≥ 0, p̂U = 1, otherwise p̂U is the unique probability that sets Πτ (p̂U ) = 0. The
first term in the bracket is positive (and finite). So, if Gsλ (1) > 0, p̂U = 1. If Gsλ (1) < 0, ΠU
τ is
negative for sufficiently small τ . In this case p̂U ∈ (psλ , 1). The uniqueness stems from the fact
that both Gdn (q̂(n)) and Gsλ (p̂U ) are strictly decreasing in p̂U ; see the proof of Proposition 5.
B.3
Optimal Intervention
We are now ready to determine the optimal probability of intervention τ ∗ of the regulator. The
expected social welfare per participant is given by
∞ −λ n h
i
X
e λ
d
τ d
Ŝ(τ ) ≡ 1 +
τ q̂(n)Gn (q̂(n)) + (1 − τ )p̂U Gn (p̂U )
n!
n=0
∞
X
=1+τ
n=0
e−λ λn
q̂(n)Gdn (q̂(n)) + (1 − τ )p̂U Gsλ (p̂U )
n!
Since the intervention frequency τ is a committed value decided before the regulator observed n,
the welfare need be averaged over different values of n. Therefore, the expected welfare should
29
be computed ‘per capita,’ denoted by Ŝ(·). Otherwise, the social welfare can be big just because
of a large n.21
Now τ ∗ ≡ argmaxτ Ŝ(τ ) is the optimal intervention probability. And p̂U is determined as in
the case of random intervention with τ = τ ∗ . The second term in Ŝ(·) with τ corresponds to
the upside of the commitment to τ less than 1. With τ < 1, p̂U becomes less than one and the
economy moves toward the social optimum for small n away from p̂U = 1. The cost of τ < 1 is
reflected in the last term with 1 − τ . This term represents the welfare when the planner fails to
intervene. Since p̂U is always bigger than the social optimum at λ, this term is always negative,
expressing the social welfare loss from playing p̂U by all (uninformed) participants. Moreover, it
can be shown that τ = 1 is a local maximum. However, the following proposition shows that τ = 1
is not necessarily globally optimal and the social welfare improves when the planner commits to
a randomized intervention rather than takes ex-post optimal strategies. As can be seen in the
proof, the region of τ in which Ŝ(τ ) > Ŝ(1) expands as λ becomes bigger, with the supremum of
τ such that Ŝ(τ ) > Ŝ(1) converging to 1.
Proposition 8. For sufficiently large λ, there exists τ < 1 such that Ŝ(τ ) > Ŝ(1).
When the average population size is small, a large probability of choosing strategy A is desirable, which is what happens when τ = 1. However, for a large average population size, in order
to make the uninformed funds invest less aggressively in strategy A, it is optimal to allow for the
possibility of no intervention by the planner (τ < 1).
We have illustrated three different ways of determining p̂U – full, partial, and optimal intervention. Lemma 4 is a general result that applies to all possibilities because the only difference
under intervention is the value of p̂U .
V
An Example
In our plots below we set µ = 5.
21
In previous sections, since n was known, S and Ŝ can be used interchangeably. Since Ŝ has a simpler expression,
we use it mostly from here on, where Ŝn (p) = 1 + pGdn (p), the benchmark.
30
Figure 4 summarizes the results of section II: p∗n < pdn < psn and each value is decreasing
in n. Figure 5 displays pU in terms of λ under asymmetric information, given χ = 0.5. This
downward sloping graph inherits the same intuition as Figure 4: all other things being equal, the
equilibrium strategy (of uninformed) becomes less aggressive when competition is expected to be
severe. Figure 6 shows that ∂pU /∂χ is increasing in χ. We presented a range for this derivative
in Section III.B. One implication of the inequalities is that the slope can be very steep when
χ is close to 1, which is clearly visible in Figure 6. Figure 7 and 8 illustrate what we learned
from Proposition 6 and Lemma 4, respectively. Figure 7 shows that the value of information is
decreasing in χ. Figure 8 exhibits the fact that the regulator does not have to fully reveal her
information to maximize welfare.
Figure 4: Equilibria Under No Information Asymmetry, with population uncertainty (solid
line), without population uncertainty (dash-dotted line), and under social optimum (dashed line)
31
Figure 5: Equilibrium Strategy of Uninformed, pU , as a function of the expected population size,
λ, for χ = 0.5
Figure 6: Equilibrium Strategy of Uninformed, pU , as a function of the degree of asymmetry, χ,
for λ = 10 (solid line) and λ = 35 (dashed line)
32
Figure 7: Value of Information, R, as a function of the degree of asymmetry, χ, for λ = 20
Figure 8: Welfare-Maximizing Partial Information Revelation as a function of the realization
of n observed by the regulator, for λ = 20 and τ = 0.5
33
VI
A
Extension
Two Periods
Some investments are undertaken simply to learn about their return prospect. We can introduce
this ‘learning by doing’ attribute in our setup by extending the one-shot investment game to
more than one period. Uninformed funds have an incentive to make an initial investment in the
negative-externality strategy due to the possibility of learning the number of rivals by observing
the realized return on the intermediate dates. Therefore, we expect investment in the negativeexternality strategy to be even more aggressive when there is a possibility of learning. We illustrate
this intuition by using a simple two period model.
There are three dates, t = 0, 1, 2. At t = 0, n + 1 uninformed funds enter the game. In each
period, they play the one-period game as in the previous sections and earn realized payoff which
depends on the number of funds playing A. In order to avoid additional complexity without
changing qualitative results, we assume that, if a fund chooses A, she learns n perfectly.22
Let p2 denote the symmetric strategy at t = 0, where the superscript means the total number
of periods. If it is a mixed strategy (0 < p2 < 1), the expected payoff from A and F should be
the same. The expected payoffs at t = 1 from A and F are simply 1 + Gsλ (p2 ) and 1, respectively.
We will drop the subscript λ which is fixed in this section. Since we assume that a fund playing
A at t = 0 perfectly knows n at t = 1, p2 is the probability that a fund is informed. That is, p2
plays the role of χ in the case of asymmetric information. Hence, the profits at t = 2 from A and
F are 1 + pI (p2 , n)Gdn (q(p2 , n)) and 1 + pU (p2 )Gdn (q(p2 )), respectively. With a discount rate δ,
therefore, a necessary condition for a mixed equilibrium strategy is given by
1 + Gs (p2 ) + δ
∞ −λ n
X
e λ
n=0
n!
[1 + pI (p2 , n)Gdn (q(p2 , n))] = 1 + δ
∞ −λ n
X
e λ
n=0
n!
[1 + pU (p2 )Gdn (q(p2 ))].
This condition can be expressed in a simpler way by using the value of information:
Gs (p2 ) + δR(p2 ) = 0.
22
(3)
When a fund playing A observes π(m), she updates her prior on n by imposing a condition n ≥ m. This imperfect
learning weakens the results with perfect learning only quantitatively, not qualitatively.
34
Since R(·) > 0, we conclude that
p2 > p1 = ps .
If there is no solution to (3) in [0, 1], then p2 = 1, which obviously satisfies the inequality above.
Equation (3) clearly shows the benefit of learning in the form of the discounted value of information. This benefit gives the originally uninformed funds more incentive to play strategy A,
through which they learn about the population size.
VII
Conclusion
We consider an investment game in which funds’ investment decisions are strategic substitutes.
This happens, for example, when there is a possibility of fire sales. In this situation, the average
return of an investment strategy is decreasing in the number of investors employing that strategy.
We ask how the investment decision is influenced when the investors do not know the total
number of the other investors in their strategy. We find that the investors invest more aggressively
in the strategy compared to the case when there is no population uncertainty. This happens
because, when the investment payoff is convex in the number of investors, an investor’s profit
from the possibility of fewer rivals is more in magnitude than her loss from the possibility of
many rivals.
If there is a planner who observes the realized number of investors in the market, she can
strategically reveal her information to the investors in order to mitigate the negative externality.
We find that under the optimal revelation policy, the planner does not reveal her information to
all the funds. This non-trivial revelation policy is a result of the externality that creates a wedge
between the first best case and the decentralized equilibrium under full information.
We model the strategic substitutability in a reduced form by assuming that the investment
payoffs are some exogenously specified decreasing functions of the number of investors in respective
investment. This lets us focus exclusively on the effect of population uncertainty on the investment
decisions. Even though we consider only two investment strategies in our model, the case of more
than two strategies is not qualitatively different. Convexity of the investment payoff function is
natural and has been assumed elsewhere in the literature. For the most part, we assume that one
35
of the two investment strategies has no externality – it pays unity irrespective of the number of
investors investing in it. In the case of both the strategies having the negative externality, the
investment decision is more complicated and a definitive prediction is elusive.
To our knowledge, this paper is the only paper along with Stein (2009) in the finance literature
that takes population uncertainty as one of the central ingredients of a model.
VIII
A
A.1
Appendix
Portfolio Approach
No Population Uncertainty
The number of rivals is n, known to all the n + 1 funds. If each of the n rivals invest p ∈ [0, 1] in
A, the (n + 1)th fund solves
max αµπ(np + α) + 1 − α
α
The first and second order conditions are
αµπ 0 (np + α) + µπ(np + α) − 1 = 0
2π 0 (np + α) + απ 00 (np + α) < 0
In the symmetric equilibrium, α = p and the above conditions become
Gd (p, n) ≡ pµπ 0 ((n + 1)p) + µπ ((n + 1)p) − 1 = 0
2π 0 ((n + 1)p) + pπ 00 ((n + 1)p) < 0
Also,
∂Gd (p, n)
= µπ 0 ((n + 1)p) + pµπ 00 ((n + 1)p) (n + 1) + µπ 0 ((n + 1)p) (n + 1)
∂p
yπ 00 (y)
= µπ 0 (y) n + 2 + 0
π (y)
where y ≡ (n + 1)p. The planner’s objective
Ŝn (p) ≡ pµπ ((n + 1)p) + 1 − p
Ŝn0 (p) = pµπ 0 ((n + 1)p) (n + 1) + µπ ((n + 1)p) − 1 < Gd (p, n)
yπ 00 (y)
00
0
Ŝn (p) = µ(n + 1)π (y) 2 + 0
π (y)
From here on, we assume a simple form of π
π(x) =
1
1+x
36
Then −xπ 00 (x)/π 0 (x) = 2x/(1 + x) < 2 for x > 0. This means ∂Gd (p, n)/∂p < 0 and Ŝn00 (p) < 0
for all p and n. Note, Gd (0, n) = µ − 1 > 0. If Gd (1, n) > 0, then pdn = 1. Otherwise, pdn is given
by Gd (pdn , n) = 0. p∗n is given by Ŝn0 (p∗n ) = 0. Since Ŝn0 (p) < Gd (p, n), p∗n < pdn . Now define
f (p, n) ≡ (n + 1)p + 1
∂f
∂f
therefore,
= n + 1,
=p
∂p
∂n
Then,
µ
µp
−
−1
f (p, n) f (p, n)2
µ(np + 1)
=
−1
[(n + 1)p + 1]2
∂Gd (p, n)
µp
= − 3 [f − 2p]
∂n
f
µp
= − 3 [1 + (n − 1)p] < 0
f
Gd (p, n) =
This implies pdn is decreasing in n. Therefore, so far, we have verified that Propositions 1 and 2
hold for the portfolio approach.
A.2
Population Uncertainty
Note that
∂ 2 Gd (p, n)
2µp2
=
[f − 3p]
∂n2
f4
2µp2
= 4 [1 + (n − 2)p]
f
Since ∂ 2 Gd /∂n2 is non-negative for all p when n ≥ 1, Gd is convex in n for all n ≥ 2. To be precise,
since n is a nonnegative integer, convexity is defined as Gd (p, n − 1) + Gd (p, n + 1) > 2Gd (p, n)
for n ≥ 1. However, it is ambiguous at n = 1 because the second derivative can be negative at
n = 0. We check whether Gd is convex at n = 1 by direct computation.
i
1h d
1
2p + 1
2(p + 1)
G (p, 0) + Gd (p, 2) − 2Gd (p, 1) =
+
−
µ
(p + 1)2 (3p + 1)2 (2p + 1)2
2p2 (5p3 + p2 − 3p − 1)
=−
.
(p + 1)2 (2p + 1)2 (3p + 1)2
This expression is positive when p < 0.8235, a fairly high value of probability of doing A. In this
case, Gd is globally convex in n and
E[Gd (p, n)] > Gd (p, λ),
(where λ ≡ E[n]) due to Jensen’s Inequality.
Eg(pnd , n) > g(pnd , n) = 0.
37
The equilibrium under population uncertainty is given by the probability psλ which satisfies
E[Gd (psλ , n)] = 0.
But
0 = E[Gd (psλ , n)] > Gd (psλ , λ)
Since Gd is decreasing in p for all n and Gd (psλ , λ) < 0 = Gd (pdλ , λ), it follows that
psλ > pdλ
(4)
when pdλ < 0.8235, determined by λ and µ. This implies that population uncertainty increases
crowding if λ is large enough. It should be noted that this condition on p is only sufficient. Since
Gd is convex except at most one value of n, (4) possibly holds in a significantly bigger region of
(λ, µ) than p < 0.8235. This verifies Proposition 3.
B
Proofs
Proof of Proposition 1 We have
Gdn (p) = µE d [π(m); n, p] − 1
n X
n m
d
p (1 − p)n−m π(m)
where E [π(m); n, p] ≡
m
m=0
Some useful identities about E d [·] (for any function u(·)):
E d [mu(m); n, p] = npE d [u(m + 1); n − 1, p]
E d [(n − m)u(m); n, p] = n(1 − p)E d [u(m); n − 1, p]
E d [u(m); n, p] = pE d [u(m + 1); n − 1, p] + (1 − p)E d [u(m); n − 1, p]
∂E d [u(m); n, p]
= nE d [u(m + 1) − u(m); n − 1, p]
∂p
The symmetric equilibrium is given by the probability pdn that sets Gdn (pdn ) = 0. In order to
prove the existence and uniqueness, note (i) Gdn (0) = µ − 1 > 0, (ii) Gdn (1) = µπ(n) − 1 < 0,
and (iii)
∂Gdn (p)
= µnE d [π(m + 1) − π(m); n − 1, p] < 0
∂p
Then by the intermediate value theorem, there exists a unique solution to Gdn (p) = 0 in the
range (0, 1). The previous three observations along with the relation
Gdn+1 (p) − Gdn (p) = µE d [π(m); n + 1, p] − µE d [π(m); n, p]
= µpE d [π(m + 1) − π(m); n, p] < 0
deliver the second part of the proposition.
38
Proof of Proposition 2 Express
Ŝn (p) =
n+1
X
m=0
d
n+1 m
p (1 − p)n+1−m [µmπ(m − 1) + n + 1 − m] + H(p)
m
= µE [mπ(m − 1); n + 1, p] + (n + 1)(1 − p) + H(p)
Using the identities listed in the proof of the Proposition 1, we can write
Ŝn (p) = µ(n + 1)pE d [π(m); n, p] + (n + 1)(1 − p) + H(p)
= (n + 1)[1 + pGdn (p)] + H(p)
∂Gdn (p)
0
d
+ H 0 (p)
Ŝn (p) = (n + 1) Gn (p) + p
∂p
< (n + 1)Gdn (p)
If pdn < 1, then ∀p ≥ pdn , Ŝn0 (p) < (n + 1)Gdn (p) ≤ 0. Therefore, p∗n < pdn . If pdn = 1, then
obviously p∗n ≤ pdn .
Proof of Proposition 3 We have
Gsλ (p) = µE s [π(m); λ, p] − 1
∞
X
e−λp (λp)m
s
π(m)
where E [π(m); λ, p] =
m!
m=0
Note
E s [π(m); 0, p] = 1
∂E s [π(m); λ, 1]
= E s [π(m + 1) − π(m); λ, 1] < 0
∂λ
lim E s [π(m); λ, 1] = 0
λ→∞
This implies that the solution of E s [π(m); λ, 1] = 1/µ exists and is unique. Denote this
by γ(µ) (note γ(µ) is increasing in µ). The equilibrium is given by the probability psλ that
sets Gsλ (psλ ) = 0. If Gsλ (1) ≥ 0, psλ = 1. If Gsλ (1) < 0, λ > γ(µ) and we have psλ = γ(µ)/λ < 1.
Proof of Lemma 2 The claim is equivalent to showing that for any positive convex function
π(m),
E s [π(m); λ, 1] > E d [π(m); n, λ/n]
We show this by making two observations about E d [π(m); n, p]:
1. E d [π(m); n, p] is convex in n.
39
Using the identities listed in the proof of the Proposition 1,
(E d [π(m); n + 2, p] − E d [π(m); n + 1, p]) − (E d [π(m); n + 1, p] − E d [π(m); n, p])
= pE d [π(m + 1) − π(m); n + 1, p] − pE d [π(m + 1) − π(m); n, p]
= p2 E d [π(m + 2) − 2π(m + 1) + π(m); n, p] > 0
2. We have the identity
s
E [π(m); λ, p] =
∞ −λ n
X
e λ
n!
n=0
E d [π(m); n, p]
To see this
∞ −λ n
X
e λ
n=0
n!
n ∞ −λ n X
X
n m
e λ
p (1 − p)n−m π(m)
E [π(m); n, p] =
n!
m
m=0
n=0
m
∞ ∞
X
X
n [(1 − p)λ]n
p
−λ
π(m)
=
e
1−p
m
n!
n=m
m=0
∞
∞
m
X
X [(1 − p)λ]n
p
π(m)
=
e−λ
[(1 − p)λ]m
1−p
m!
n!
d
=
=
m=0
∞
X
m=0
∞
X
m=0
s
n=0
e−λ (λp)m
π(m) (1−p)λ
e
m!
e−λp (λp)m
π(m)
m!
= E [π(m); λ, p]
Applying Jensen’s inequality, we get
∞ −λ n
X
e λ
n=0
n!
"
d
E [π(m); n, p] > E
d
π(m);
∞ −λ n
X
e λ
n=0
n!
#
n, p
So, E s [π(m); λ, p] > E d [π(m); λ, p]
Setting λ = n and p = λ/n in this relation yields the desired claim.
Proof of Lemma 3 Given the definitions of pdn1 and pdn2 , the bullet points just above Lemma 3
can be reexpressed as
• Gdn (pdn1 ) > 0 =⇒ pI = 1
• pI < 1 =⇒ Gdn (q) ≤ 0. pI > 0 =⇒ Gdn (q) ≥ 0.
• Gdn (pdn2 ) < 0 =⇒ pI = 0
Let us first characterize pdn . Define n01 such that Gdn0 (1) = 0. For n ≤ n01 , Gdn (1) ≥ Gdn0 (1) =
1
1
0, implying pdn = 1. For n ≥ n01 , Gdn (1) ≤ Gdn0 (1) = 0 and since Gdn (p) is strictly decreasing
1
in p, pdn is the unique solution of Gdn (pdn ) = 0. Moreover, since Gdn+1 (p) < Gdn (p) for n and
p, pdn+1 < pdn for n ≥ n01 .
For n < n1 , we will show Gdn (pdn1 ) > 0 which implies pI (n) = 1. If pdn1 = 1, n1 = n01 and
40
for n < n1 , Gdn (pdn1 ) = Gdn (1) > Gdn1 (1) = Gdn0 (1) = 0 as required. But if pdn1 < 1, n1 > n01 .
1
For n < n01 , Gdn (pdn1 ) > Gdn (1) > Gdn0 (1) = 0. And for n01 ≤ n < n1 , pdn > pdn1 implying
1
Gdn (pdn1 ) > Gdn (pdn ) = 0.
For n > n2 , we will show Gdn (pdn2 ) < 0 which implies pI (n) = 0. If pdn2 = 0, n2 = ∞ and the
claim is trivially true. If pdn2 > 0, pdn < pdn2 which implies Gdn (pdn2 ) < Gdn (pdn ) = 0.
We claim for n1 ≤ n ≤ n2 , q(n) ≡ (1 − χ)pU + χpI (n) = pdn . If q(n) > pdn , Gdn (q(n)) <
Gdn (pdn ) = 0 =⇒ pI (n) = 0 =⇒ q(n) = pdn2 . If pdn2 = 0, q(n) = 0 < pdn , a contradiction. If
pdn2 > 0, Gdn (pdn2 ) < 0 =⇒ pdn < pdn2 =⇒ n > n2 , a contradiction. Similarly, if q(n) < pdn ,
Gdn (q(n)) > Gdn (pdn ) = 0 =⇒ pI (n) = 1 =⇒ q(n) = pdn1 . If pdn1 = 1, q(n) = 1 ≥ pdn , a
contradiction. If pdn1 < 1, Gdn (pdn1 ) > 0 =⇒ pdn > pdn1 =⇒ n < n1 , again a contradiction.
Proof of Proposition 5 We will show (i) ΠU (0) > 0, and (ii) ΠU (pU ) is strictly decreasing in
pU . Then, if ΠU (1) > 0, pU (λ, χ) = 1, otherwise pU (λ, χ) is the unique probability that sets
ΠU (pU (λ, χ)) = 0. Given the form of pI (pU , χ, n) in Lemma 3, we have

d

pn1 if n < n1 ,
q(pU , χ, n) = pdn for n1 ≤ n ≤ n2 ,

 d
pn2 if n > n2
Therefore,
U
Π (pU ) =
=
∞ −λ n
X
e λ
n=0
nX
1 −1
n=0
n!
Gdn (q(pU , χ, n))
n2
∞
X
X
e−λ λn d d
e−λ λn d d
Gn (pn1 ) +
Gn (pn ) +
n!
n!
n=n
n=n2 +1
1
e−λ λn d d
Gn (pn2 )
n!
Definitions of pdn1 , pdn and pdn2 imply respectively that the first term is positive, the second
term is zero and the third term is negative.
We handle the boundary case χ = {0, 1} separately. At χ = 0, q(pU , 0, n) = pU and
U
Π (pU ) =
=
∞ −λ n
X
e λ
n!
Gdn (q(pU , 0, n))
n=0
Gsλ (pU )
which implies pU (λ, 0) = psλ . pI (λ, 0, n) is obtained from Lemma 3. Define n∗ such that
pdn∗ = psλ . Then


1
pI (λ, 0, n) = psλ


0
if n < n∗ ,
if n = n∗ ,
if n > n∗
(Note pI (λ, 0, n∗ ) = limχ→0 (pdn∗ − (1 − χ)psλ )/χ = psλ ). So the equilibrium is unique for
χ = 0.
At χ = 1, pdn1 = 1 implying n1 = π −1 (1/µ) and pdn2 = 0 implying n2 = ∞ (see Corollary
2) and so the third term in the expression of ΠU (pU ) goes to zero. Therefore, ΠU (1) > 0
41
implying pU (λ, 1) = 1. Again from Lemma 3, pI (λ, 1, n) = 1 for n < n1 and pI (λ, 1, n) = pdn
for n ≥ n1 yielding uniqueness of the equilibrium.
Now assume χ ∈ (0, 1). When pU = 0, pdn2 ≡ (1 − χ)pU = 0, implying n2 = ∞. Therefore,
ΠU (0) > 0. Now differentiate ΠU wrt pU 23
∞
∂ΠU (pU ) X e−λ λn ∂Gdn (q(pU , χ, n)) ∂q(pU , χ, n)
=
∂pU
n!
∂p
∂pU
n=0
= (1 − χ)
nX
1 −1
n=0
∞
X
e−λ λn ∂Gdn (pdn1 )
+ (1 − χ)
n!
∂p
n=n2 +1
e−λ λn ∂Gdn (pdn2 )
<0
n!
∂p
The inequality follows from the fact that Gdn (p) is strictly decreasing in p ∀p.
Proof of Proposition 7 We have already computed pU (λ, 0) and pU (λ, 1) in the proof of Proposition 5.
For the purpose of this proof, reexpress the excess payoff function of the uninformed fund
as
U
Π (pU , λ, χ) =
∞ −λ n
X
e λ
n=0
n!
Gdn (q(pU , χ, n))
In the proof of Proposition 5, we saw ∂ΠU (pU , λ, χ)/∂pU < 0 when χ < 1. Using the second
identity listed in the proof of proposition 3, we obtain the partial derivative
∞
∂ΠU (pU , λ, χ) X e−λ λn d
=
[Gn+1 (q(pU , χ, n + 1)) − Gdn (q(pU , χ, n))]
∂λ
n!
=
+
+
n=0
nX
1 −1
n=0
nX
2 −1
n=n1
∞
X
n=n2
e−λ λn d
[Gn+1 (pdn1 ) − Gdn (pdn1 )]
n!
e−λ λn d
[Gn+1 (pdn+1 ) − Gdn (pdn )]
n!
e−λ λn d
[Gn+1 (pdn2 ) − Gdn (pdn2 )] < 0
n!
The first term and the third term are negative because of the fact Gdn+1 (p) < Gdn (p) ∀n, p.
The second term is zero by the definition of pdn .
When pU (λ, χ) < 1, the condition ΠU (pU (λ, χ), λ, χ) = 0 holds. Differentiating this condition wrt λ, we get
∂pU (λ, χ)
∂ΠU (pU (λ, χ), λ, χ)/∂λ
=− U
∂λ
∂Π (pU (λ, χ), λ, χ)/∂pU
23
Strictly speaking, q(pU , n) is not differentiable wrt pU when n is n1 or n2 – the left derivative at n = n1 is 1 − χ and
the right derivative is zero and vice versa at n = n2 . Nevertheless, the conclusion that ΠU (pU ) is strictly decreasing in
pU is still true.
42
Since the two partial derivatives are negative for all pU , we conclude
∂pU (λ, χ)
≤0
∂λ
∀λ, χ
The relation
−
1 − pU (λ, χ)
1−χ
≤
∂pU (λ, χ)
pU (λ, χ)
≤
∂χ
1−χ
is a simple rearrangement of the relation 0 ≤ h(χ) ≤ 1 derived in the proof of proposition 6.
Proof of Proposition 6 The first inequality in the last relation of Proposition 7 implies that if
for some χ = χ, pU (λ, χ) = 1, then pU (λ, χ) = 1 for all χ > χ (we continue to suppress λ as
an argument of the various functions for brevity). We show R0 (χ) ≤ 0 separately for χ < χ
and χ ≥ χ.
For χ ≥ χ, we have
R(χ) ≡
∞ −λ n
X
e λ
n!
n=0
∞
X
=−
n=n2 +1
R(χ, n)
e−λ λn d d
Gn (pn2 )
n!
As χ increases, pdn2 (=1 − χ) decreases which implies both n2 and Gdn (pdn2 ) increase. Since
Gdn (pdn2 ) < 0, this implies R is decreasing in χ.
For χ < χ, pU (λ, χ) < 1 and the following equilibrium condition holds.
∞ −λ n
X
e λ
n=0
n!
Gdn (q(χ, n)) = 0
where q(χ, n) has the form

d

pn1
q(χ, n) = pdn

 d
pn2
if n < n1 ,
for n1 ≤ n ≤ n2 ,
if n > n2
where n1 and n2 are (implicitly) defined as
pdn1 = (1 − χ)pU (χ) + χ
pdn2 = (1 − χ)pU (χ)
Now define h(χ) ≡ ∂pdn1 /∂χ = 1 − pU (χ) + (1 − χ)p0U (χ) and differentiate the equilibrium
43
condition wrt χ24
∞ −λ n
X
e λ ∂Gd (q(χ, n)) ∂q(χ, n)
n
n=0
⇔ h(χ)
nX
1 −1
n=0
n!
∂p
∂χ
∞
X
e−λ λn ∂Gdn (pdn1 )
+ (h(χ) − 1)
n!
∂p
n=n2 +1
P∞
⇔ h(χ)
=0
e−λ λn ∂Gdn (pdn2 )
=0
n!
∂p
n=n2 +1
=P
d (pd )
∂G
−λ
n
n n1
n1 −1 e λ
n=0
n!
∂p
d d
e−λ λn ∂Gn (pn2 )
n!
∂p
+
P∞
n=n2 +1
d d
e−λ λn ∂Gn (pn2 )
n!
∂p
Since Gdn (p) is decreasing in p ∀p, h(χ) ∈ [0, 1]. Now turn to R
R(χ) =
=
∞ −λ n
X
e λ
n=0
∞
X
n=0
n!
[pI (χ, n) − pU (χ)]Gdn (q(χ, n))
e−λ λn
pI (χ, n)Gdn (q(χ, n))
n!
The last equality follows from the equilibrium condition stated above. Differentiating wrt χ
∞ −λ n X
e λ ∂pI (χ, n) d
∂Gdn (q(χ, n)) ∂q(χ, n)
R (χ) =
Gn (q(χ, n)) + pI (n, χ)
n!
∂χ
∂p
∂χ
0
n=0
Gdn (q(χ, n)) = 0 for n ∈ [n1 , n2 ] and pI (n, χ) = 1 for n < n1 and pI (n, χ) = 0 for n > n2 .
So, the first term in the above summation can be dropped and we are left with
R0 (χ) = h(χ)
nX
1 −1
n=0
e−λ λn ∂Gdn (pdn1 )
<0
n!
∂p
Proof of Lemma 4 As a first step in our analysis, we note that if, for some n, χ(n) = 0 then
q̂(n) = p̂U and if χ(n) = 1 then q̂(n) = p̂I (n) = pdn since this reduces to the deterministic
case of section II. We plot p̂U , pd and p∗ in the Figure 3. Our goal is to minimize, for each
n, the vertical distance between q̂(n) and p∗n by choosing an appropriate χ(n).
The optimal strategy of the social planner is split into three regions based on n. In the
region n < N1 (see Figure 3(a)), as χ(n) increases from zero to one, q̂ increases from p̂U to
pdn , the range that includes p∗n . Therefore, by continuity of q̂, we can find a value of χ(n)
such that the social optimum is achieved: q̂(n) = p∗n . In this case, all the informed funds
choose strategy A with probability (p̂I (n) = 1), since q̂(n) < pdn . Moreover, due to the fact
that p∗n is decreasing in n, the optimal χ(n) is also decreasing in n – the smaller is q̂, the
smaller is χ(n).
In the intermediate region (N1 ≤ n ≤ N2 , Figure 3(b)), we have the ordering p∗n < p̂U < pdn .
As χ(n) increases from zero to one, q̂(n) increases from p̂U to pdn for all n < N2 drifting
away from p∗n . Note for q̂(n) < pdn , p̂I (n) = 1 implying q̂(n) ≥ p̂U . Therefore, the optimal
24
The qualification of the type in the footnote 23 applies to differentiating q(χ, n) and pI (χ, n) wrt χ at n = n1 , n2 in
the following.
44
policy is to leave all the funds uninformed, χ(n) = 0. In this case q̂(n) = p̂U (which implies
p̂I (n) = 1) and the social optimum of the centralized economy is not achieved.
When n is large (n > N2 , Figure 3(c)), q̂(n) becomes smaller as the planner shares her
information with more funds, because those informed funds, observing the large value of n,
will refrain from investing in strategy A. In other words, for a large n, more information
revelation induces more participants to tilt their strategy toward F , drawing the economy
closer to the social optimum. However, the social planner cannot push q̂(n) down to an
arbitrary level. When n > N2 , pdn is the lower limit of q̂(n), keeping q̂(n) from reaching
p∗n < pdn for any value of χ(n). To see this, note if q̂(n) < pdn , p̂I (n) = 1 implying q̂(n) ≥
p̂U > pdn which is a contradiction to the initial assumption (also note that q̂(n) > pdn implies
p̂I (n) = 0). Moreover, if χ(n) passes a threshold, q̂(n) does not change and stays at pdn
(at this threshold p̂I (n) starts increasing from zero). It should be noted that pdn is the
equilibrium strategy when there is no population uncertainty (χ(n) = 1), as discussed in
Section II. A new finding here is that the planner does not have to reveal her information
to all the funds to achieve the same level of social welfare. It appears reasonable to assume
that the social planner prefers the lowest value of χ(n) when multiple values of χ(n) deliver
the same social welfare, for instance, because it may be costly to convey information. Under
this assumption, the optimal χ(n) chosen by the planner is the value such that the informed
funds start choosing A with positive probability. Since pdn is downward sloping, the planner
should increase the optimal χ(n) for larger n.
The previous discussion is summarized as follows

∗

pn if n < N1 ,
q̂(n) = p̂U if N1 ≤ n ≤ N2 ,

 d
pn if n > N2
Proof of Proposition 8 For an expositional purpose, we assume pdn < 1 for all n > 0.
" ∞
#
X e−λ λn
1
ΠU
τ
(1 − χ(n))Gdn (q̂(n)) + (1 − τ )Gsλ (p̂U )
τ (p̂U ) =
Pr(I = 0)
n!
n=0
Ŝ(τ ) = 1 + τ
∞ −λ n
X
e λ
n=0
n!
q̂(n)Gdn (q̂(n)) + (1 − τ )p̂U Gsλ (p̂U ).
τ
∗
First, evaluate ΠU
τ when p̂U = p1 . Recall that N1 = 1 in Lemma 4 when the planner
intervenes. Thus the expected excess payoff is written as
"
#
N2 −λ n
X
1
e λ d ∗
U ∗
−λ
d
s ∗
Πτ (p1 ) =
τ e (1 − χ(0))G0 (1) + τ
Gn (p1 ) + (1 − τ )Gλ (p1 ) ,
Pr(I = 0)
n!
n=1
where p∗1 = pdN2 . Note that q̂(n) = pdn and, therefore, Gdn (q̂(n)) = 0 beyond N2 . Also,
χ(n) = 0 when N1 = 1 ≤ n ≤ N2 . Since p∗1 > psλ for sufficiently large λ, Gsλ (p∗1 ) < 0 for
large values of λ. As will be clear later, we are only interested in large values of λ.
45
∗
So ΠU
τ (p1 ) is negative when
τ <−
Gsλ (p∗1 )
P 2
e−λ [(1 − χ(0))Gd0 (1) + N
n=1
λn d ∗
n! Gn (p1 )]
− Gsλ (p∗1 )
≡ τc (λ)
∗
Here we assume that the denominator is positive, otherwise ΠU
τ (p1 ) is negative for all τ , a
simpler case.
Since χ(n), p∗1 and N2 are independent of λ and Gsλ (p∗1 ) is negative and decreasing in λ, it
can be easily seen limλ→∞ τc (λ) = 1. So for any τ < 1, we can find λc such that τ < τc (λ)
for all λ > λc . From here on, fix τ < 1 and suppose λ > λc .
There are simple but useful implications from this observation. For all λ > λc , since ΠU
τ is
∗ ) < 0, we have p̂ < p∗ at this τ , whereby ΠU (p̂ ) = 0. In addition,
decreasing and ΠU
(p
U
U
τ
τ
1
1
it follows N1 > 1 in defining q̂(n) for this p̂U .
Turning our attention to Ŝ(·), first note that
Ŝ(1) = 1 + e−λ Gd0 (1),
since q̂(n) = pdn and Gdn (pdn ) = 0 for all n > 0. On the other hand,
Ŝ(τ ) = 1 + τ
=1+τ
=1+τ
∞ −λ n
X
e λ
n=0
∞
X
n=0
∞
X
n!
q̂ τ (n)Gdn (q̂ τ (n)) + (1 − τ )p̂τU Gsλ (p̂τU )
∞
X e−λ λn
e−λ λn τ
q̂ (n)Gdn (q̂ τ (n)) − τ
p̂τU (1 − χ(n))Gdn (q̂ τ (n))
n!
n!
n=0
e−λ λn
n=0
n!
[q̂ τ (n) − p̂τU (1 − χ(n))]Gdn (q̂ τ (n)),
τ
where we use the relation given by ΠU
τ (p̂U ) = 0. We add a subscript τ to emphasize that
the values are computed for this specific τ .
The difference of welfare is
Ŝ(τ ) − Ŝ(1) = τ
∞ −λ n
X
e λ
n=0
NX
1 −1
n!
[q̂ τ (n) − p̂τU (1 − χ(n))]Gdn (q̂ τ (n)) − e−λ Gd0 (1)
e−λ λn ∗
[pn − p̂τU (1 − χ(n))]Gdn (p∗n ) − e−λ Gd0 (1)
n!
n=0
n
o
−λ
≥e
λτ [p∗1 − p̂τU (1 − χ(1))]Gd1 (p∗1 ) − Gd0 (1) .
=τ
(5)
The second equality comes from the facts that q̂ τ (n) = p̂τU and χ(n) = 0 in [N1 , N2 ], and
Gdn (q̂ τ (n)) = Gdn (pdn ) = 0 beyond N2 . The inequality is derived from (i) N1 > 1, (ii) p∗1 ≥ p̂τU
for all n < N1 , (iii) 0 ≤ χ(n) ≤ 1, and (iv) Gdn (p∗n ) > 0 for all n. Combining the conditions,
we observe that each term in the sum is positive and the sum spans at least up to n = 1,
which is (5).
46
Lastly, suppose we increase λ a little in the equilibrium condition for the uninformed:
τ
Pr(I = 0)ΠU
τ (p̂U )
=τ
NX
1 −1
n=0
N2
X
e−λ λn
e−λ λn d τ
d ∗
(1 − χ(n))Gn (pn ) + τ
Gn (p̂U ) + (1 − τ )Gsλ (p̂τU ) = 0.
n!
n!
n=N1
s
τ
d
s
First, note that p̂τU > psλ , because ΠU
τ (pλ ) > 0 from above. Hence, p̂U = pN2 > pλ , implying
λ > N2 . It follows that each positive term in the first two sums become smaller by the
increase in λ, since all n lie on the left of the hump in Pois(λ). Other pieces in the sum
than the Poisson probability do not depend on λ. The last negative term also decreases in
U
λ. Therefore, Pr(I = 0)ΠU
τ and, therefore, Πτ becomes negative for an increase in λ. In
τ
∗
τ
turn, p̂U is decreasing in λ, and p1 − p̂U is bounded away from zero for large λ.
In the curly bracket in (5), all values are independent of λ other than itself and p̂τU . Thus,
for sufficiently large λ, the term in the bracket is bigger than zero and
Ŝ(τ ) > Ŝ(1).
We conclude that τ = 1 cannot be the global maximum for large λ.
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