Feedback Effects and Speculative Bubbles
in Informational Price Theory
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MICHAEL A. S. GUTH, Ph.D., J.D. |
Feedback Effects and Speculative Bubbles
in Informational Price Theory
Informational price theory (IPT) has become the
main paradigm to explain how prices reveal information. In the older efficient markets literature,
prevailing market prices were thought to impound all available information. Thus a price change could indicate that an
asset was previously under or overpriced but not that it is currently
mispriced.[1] In contrast, IPT implies that prices not
only clear markets but signal future movements. Prices convey information about expectations for future earnings,
so that a rise today may be confirmed by subsequent rises tomorrow - as more
and more people learn the private information.
If
prices fully reflect information, then no one has any incentive to buy
information. Everyone can simply infer
information from prices for free. But
if no one purchases information, then market prices will reflect no
information. Hence everyone will have
an incentive to buy information. The
obvious solution to this conundrum is that prices only partially reflect
information. Grossman and Stiglitz
(1976) developed a model that formalizes this logic.
In
equilibrium, some fraction of the market pays to become informed and the
remaining fraction chooses to infer information from noisy prices.[2]
Prices reflect some, but not all, of
the private information which the informed traders have purchased. The competitive advantage of knowing the
private information compensates the informed for their acquisition cost. At the margin, prices aggregate just enough
information such that the last individual to buy information is indifferent
between paying for information or attempting to infer it from prices.
The
sequence of events in the Grossman and Stiglitz model looks something like
this. Informed traders receive good
news about some asset, buy the asset in mass, and thereby raise its price. Uninformed traders witness the increase in
price, infer that the informed must have received favorable news, and
subsequently buy as well. A similar
logic applies when the informed receive unfavorable news and sell some
asset. The uninformed follow suit. In both instances, purchases or sales by the
informed cause a feedback effect by the uninformed.
One
example where the uninformed might benefit from mimicking the actions of the
informed, without introducing a feedback effect, would be in selecting a
restaurant in an unfamiliar city.
Visitors (the uninformed) might patronize a restaurant that seemed
crowded. They would reason that local
residents (the informed) know the prices and quality of food at that restaurant
and have chosen to eat there. In this
restaurant selection example, the informed derive no benefit from the actions
by the uninformed. But in financial
markets, capital gains will usually depend on the purchasing activities of
other market participants.
The
literature on information and prices has identified three basic speculative
problems with the IPT approach.[3] First, in judging the quality of goods or
services by price, consumers face uncertainty over whether high prices actually
indicate superior quality or merely an attempt by the seller to attract
consumers. If a consultant charges a
high hourly fee, is it because he is superior to other professionals with a
lower fee, or merely a speculative attempt to persuade clients into thinking he
must be good if he charges that high fee?
Only in strictly competitive supply markets would price-taking behavior
by sellers limit the opportunities of low quality vendors to fool new consumers
with high prices.
Second,
attempts at market timing may limit the extent to which the informed act on
their information. If someone receives
a favorable signal on a stock, should he rush out and buy it? How can he know whether the current price
already impounds his private signal? If
other traders are planning to sell the stock and realize short term capital
gains, he might gain from waiting until the price has fallen with their
sales. Hence, no feature of IPT
guarantees the informed will immediately act upon their signal.
Third,
feedback effects have been thought to unravel the foundations of IPT, as
investors then have incentives
to
acquire (and in some cases disseminate) information on the purchasing
proclivities of the market as well as on the `real' factors of earnings,
dividends and so forth. What we end up
with is a model of a market in which even when favorable information on
earnings is available to insiders or informed traders, there is uncertainty as
to whether this will be acted upon; in which `tulip bulb mania' factors can be
important so that rises in prices might reflect only a belief that the market
will value the stock higher next period, independent of earnings; and in which
the chances of actually ferreting out any information on earnings prospects
from observed prices is miniscule (sic).
(Burness, Cummings, and Quirk (1980), p.75)
At first
glance, this line of criticism would seem to eradicate any application of IPT
to financial markets. However, in
moving from words to formal models, we find that generating speculative profits
off the feedback effect is actually more difficult then it sounds. Uninformed traders are watching prices. In order to realize their speculative
profits, informed traders must buy and resell, yet their subsequent sales
must not dissuade the uninformed from buying as a feedback.
This
criticism of IPT also involves questions about strategic interactions usually
applied in noncompetitive environments.
Grossman and Stiglitz developed their IPT model with perfect competition: traders take prices as given. If purchases by informed traders cause
prices to rise, then prices are indeed a strategic variable under the control
of the informed agents - even in the original theory. In financial markets, informed traders could then profit from
manipulating the market price in order to fool the uninformed into buying or
selling as feedback effects. The
question then arises as to what modeling tool best captures the essence of IPT
and the possibility for this strategic interaction.
One
might envision casting the informed agents as Stackelberg leaders with the
uninformed agents acting as followers in a game theory model. However, a Stackelberg equilibrium approach
would entail some lack of intelligence or rationality on the part of one or
more players.[4] The usual IPT dichotomy of intelligence
holds that the uninformed traders `are smart enough'[5]
to know the correlation between market price levels and news. Yet the informed traders are not
smart enough to realize their purchases (sales) induce a feedback effect.[6] In a Stackelberg equilibrium approach, the
leader (the informed) would be smart enough to realize his purchases trigger a
feedback effect, but the follower (the uninformed fraction) does not consider
the leader's strategy in deciding whether to buy. We need a model in which all players act intelligently.
The
first solution developed in this chapter is a Bayesian Nash Equilibrium in a
game of incomplete information.
Informed traders play opposite uninformed traders, each taking the
other's strategy as given. The informed
group, having incurred the cost of acquiring private information, moves
first. The informed group may try to
fool the uninformed into believing that asset returns are better than they
actually are. Sometimes the deception
succeeds. A bubble results, and after
the crash is over and the dust has cleared, the informed group turns out to
have gained at the expense of the uninformed.
Sometimes the deception fails, and the informed traders wind up losing
more money than if they had followed a conventional, fundamental-based
strategy.
The
equilibria of this game, contained in Sections 7.3 through 7.6, indicate the
probability that the informed traders acting as a group will collude (i.e., buy
when they have received a low signal) as well as the probability that the
uninformed, acting as a group, will buy as a feedback effect. To assure that some informed traders do not
deviate from the group strategy, and thereby tip off the uninformed, we employ
a self-enforcing[7] Nash
equilibrium approach. By restricting
individual behavior this way, we are limited to a two-player model, which
cannot adequately describe the operation of densely populated, competitive
markets.
To
generalize the model beyond the two-group, two-player model of informational
price theory, we examine subjective correlated equilibria in the final sections
of this chapter. In this framework, the
informed traders receive correlated - though not identical - signals, and all
players act as individuals, rather than as members of an orchestrated
group. The equilibrium characterizes
the quantity of the risky asset that each individual selects, rather
than the probability of purchasing.
Game
theoretic analysis rarely applies to competitive markets. `Anonymous games' are one class of
`competitive' models where the outcome depends on the actions of each agent
only through the proportion of agents that acts in a certain way. Like anonymous games, the pure strategy equilibria
in this chapter might be interpreted as the frequency of each population that
chooses to engage in this strategic play.
Under this interpretation, we must consider each informed trader as
having, e.g., firm-specific information.
Thus one informed trader's private information and strategic use of that
information will not affect another informed trader's strategy, because their
strategies are focused upon different stocks.
However, since no one individual's purchases should affect prices in a competitive
market, this interpretation seems unrealistic at best. Thus we have reason to question the
competitive market foundation of IPT regardless of the equilibrium concept
chosen.
The
significance of the results in this chapter are (1) the derivation of
speculative bubbles from fully rational, payoff-maximizing behavior on the part
of all market participants;[8] (2) the highlighting of an important feature
of IPT: the two-player model; (3) a demonstration of the stringent
conditions required for the informed traders to derive speculative profits off
the feedback effect, thereby attenuating this criticism of IPT; and (4)
utilization of some mathematical results by Nikaidô and Isodo that can be used
in proofs of the existence of (subjective correlated) equilibria.
Section
7.2 compares our work to related articles in the speculative bubble and
asymmetric information literatures. The
players and extensive form game are presented in Section 7.3, the payoffs in
Section 7.4, and the pure and mixed strategy equilibria in Section 7.5. A negative bubble is illustrated in Section
7.6. Section 7.7 offers an intelligent
critique of the Bayesian Nash Equilibrium speculative bubble models. Section 7.8 generalizes the two-player,
high-low demand model of the first six sections to allow for multiple strategies
and multiple signals like Kyle (1989), and each player acting as an
individual. Section 7.9 presents the
conclusions and a discussion of the noncompetitive assumptions of the model. Table 7.1 in Section 7.3 conveniently lists
the notation used in the Bayesian Nash Equilibria sections of this chapter.
7.2. The
Speculative Bubble and Asymmetric Information
Literatures
The traditional view of speculative bubbles
maintains that they result from irrational behavior by some or all of the
market participants [Kindleberger
(1989)]. Our model is distinguished
from many previous derivations of speculative bubbles in that all the
market participants are fully intelligent and rational. Blanchard and Watson (1982) and Leach (1991)
illustrate bubble formation in which traders will refrain from participating
unless they can rationally determine that the last (nth)
trader will execute his trade, the n-1st trader will execute
his trade, and so on. In these two
articles, players may be induced to buy into the speculative bubble by the fact
that the bubble may continue to grow indefinitely. Every `bubble' in our work, by contrast, bursts with probability
1 as the state of the world is revealed.
DeLong
et al. (1990) have demonstrated convincingly that perfectly rational
speculators can jump on the feedback effect bandwagon and not buck the
trend. Speculators in the DeLong et al.
model do not worry that some (nth) trader may get stuck
holding the overvalued asset. From an
expected profit-maximizing viewpoint, they would be foolish to be overly
concerned with some distant collapse point, as long as they expect to realize
their capital gains while the bubble still exists. Wang (1993) looks at the same basic model as DeLong et al.;
however, `noise trading' is replaced by `supply shocks.' He finds the existence of investors with
imperfect information increases the risk premia on stocks, and investors with
different information will adopt different investment strategies.
Friedman
and Aoki (1986, 1989) provide parametric examples of bubble formation when
investors are oriented towards long-term gains and expectations; traders in
their model ignore short-term capital gains and the potential ability to
manipulate prices. Friedman and Aoki
(1989) have illustrated a negative bubble arising from momentum in price
trajectories over time. The Friedman
and Aoki bubbles can result from overshooting, which tends to be
self-correcting as the game is played over time and the players become more
familiar with price escalations.
O'Flaherty
(1987) looks at bubbles as nonconstant correlated equilibria in coordination
games with asymmetrically informed agents.
He finds a bias towards myopic investment and against public revelation
of private signals. Similarly, Fishman
and Hagerty (1992) show full disclosure of privately held information would
lead insider traders to have zero profits. In the absence of a mandatory
disclosure rule, security prices will generally not be efficient in
transmitting information between informed and uninformed investors.
Fishman
and Hagerty's work extends Kyle's (1985) noisy rational expectations model to
study the impact of insider trading on price efficiency. In Kyle's model, informed traders
strategically choose their transactions knowing their strategies will affect
prices. The resulting Nash equilibrium
quantity orders show that informed traders optimally choose to withhold some of
their information. Kyle never
satisfactorily explained why `noise traders' come into the market and routinely
lose their wealth. Nevertheless,
numerous authors have adopted Kyle's Walrasian auctioneer and noise trader
framework to examine market issues such as intraday patterns in volume and
price variability (Admati and Pleiderer 1988), diversification by the
uninformed (Bhushan 1991), transaction costs to the uninformed (Subrahmanyam
1991b), and market liquidity and price efficiency (Subrahmanyam 1991a).
Smith,
Suchanek, and Williams (1988) and King, Smith, Williams, and Van Boening (1993)
have found evidence for bubble creation in experimental markets from `homegrown
capital gains expectations,' which collapse when dividend and fundamental value
information becomes common knowledge.
Bubble creation in these experimental markets arise more from myopia than
from an attempted manipulation of the uninformed based on feedback effects.
Allen
and Gorton (1993) consider bubble creation resulting from incentive contracts
paid to portfolio managers. The
managers in their model receive no income if their portfolios have non-positive
returns. However, they get to keep a
portion of any positive return they achieve.
This principal-agent contract leads portfolio managers to prefer taking risks
and buying assets above their fundamental value on the chance that they will
continue to rise.
Portfolio
managers and institutional investors are often said to behave like herds. Banerjee (1992) has studied decision rules
that depend on the investment strategies of previous decision makers. He shows that the resulting equilibrium is
inefficient. People tend to be guided
too much by other people's investments, even to the extent of ignoring their
own private information. In Banerjee's
Bayesian Nash Equilibrium, the order of play is important. The first few players' decisions have a much
greater impact on subsequent investment patterns than those of players entering
later in the game.
Our
paper is most similar to work simultaneously derived by Gorton and Pennacchi
(1989), who also develop a game theoretic model of strategic play between
informed and uninformed traders. In the
Gorton and Pennacchi model, the informed always try to manipulate and the
uninformed always buy, whereas in our model this behavior would tend to drive
the uninformed traders out of the market.[9] Gorton and Pennacchi focus on the rise of
new securities that split the cash flows of underlying assets and on government
intervention to protect the uninformed.
Our paper focuses more on the differences between this strategic game
and the predictions of informational price theory, as well an analysis of the
competitive market assumptions.
7.3.
Principals and Extensive Form Game
Consider
a market containing two assets, one safe and one risky. Let the return, r, to the risky asset
depend on a random variable, η, which can be observed at some cost, and another random
variable, ε, that cannot
be observed:
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|
where η and ε are
independent, normally distributed random variables. The informed individuals pay to learn the value of η. Some exogenous risk remains from ε, but the
informed traders have purchased private information that reduces the risk on
the asset's return.
Figure
7.1 shows the extensive form game tree.
Chance moves first by selecting the value of η. To simplify the game tree, we consider only
high or low signals: ηH or ηL.
Figure 7.1.
Bayes Game of Information Inference Through Prices
The
informed traders move second.
They know the value of η and must select a purchasing strategy for the risky
asset. If they receive the signal ηH, we assume the informed traders will have high demand (
) for the risky asset to collect the real earnings, just as
in IPT. Section 7.6 relaxes this
assumption and shows how `negative bubbles' can be formed. When the earnings signal is ηL, the informed traders must choose whether to exploit the feedback
effect. Their expected profits depend
on the purchasing proclivities of the uninformed.
The
uninformed traders move third by choosing a high or low quantity
purchase. They observe the price level
after the informed traders have moved but not the private signal selected by
chance. The uninformed traders can
always infer ηL when the informed traders bid low demand (XiL). Consequently, the uninformed will respond
with low demand in this scenario.
Typically, the uninformed will infer the informed traders have chosen
high demand and wonder if they should do likewise.
In
Figure 7.1, an information set surrounds two nodes that are indistinguishable to the uninformed.[10] The uninformed traders use Bayes' Rule to
infer the real (correct) versus purely speculative (incorrect) prospects for
the asset's earnings. Informed traders
who have purchased solely to exploit capital gains off the feedback effect will
sell off their speculative holdings at the same time that the uninformed are
buying. The uninformed cannot observe
the informed traders' sales, because traders can only see market clearing
prices not the limit orders submitted in advance to the market. Following these moves the game ends. Traders realize payoffs according to the
strategies they adopt.
Per
capita demand,
, for the risky asset by the informed will vary inversely
with the price, P, of the risky asset:
MXi /MP
< 0. However, unlike in IPT, we
cannot say that MXi /Mη > 0. The informed traders may submit either a
high or a low demand schedule upon receiving a low signal. Informed trader demand will also depend on
, which is the probability that the uninformed will buy as a
feedback effect. Let α denote the
probability that the informed will buy the risky asset even though they have
learned
. For the reader's
convenience, Table 7.1 lists the notation used in the next three sections of
this chapter.
Demand
must equal supply each period in equilibrium:
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where λ is the fraction of the traders who are informed,
Xu is the per
Table
7.1. Bayes Game Notation
____________________________________________________________
Symbol Definition
____________________________________________________________
ρL (objective) prior probability
of a low signal, ηL.
ρH (objective) prior
probability of a high signal, ηH.
XiH high demand for the
risky asset by the informed traders.
XiL low demand for the
risky asset by the informed traders.
XuH high demand for the
risky asset by the uninformed traders.
XuL low demand for the
risky asset by the uninformed traders.
α the probability that the
informed traders will buy when they have received a low signal, Pr(XiH#ηL).
α* the value of α that leaves
the uninformed traders indifferent between their strategies of purchasing the
risky asset upon witnessing the informed traders have purchased (and have high
demand).
β the probability that the
uninformed traders will purchase when they infer the informed traders have bought the risky asset, Pr(XuH#XiH).
β* the value of β that leaves
the informed individuals indifferent between their strategies (of various
probabilities α) of trying to manipulate the uninformed traders.
z the
value of ρL which sets α* = 1.
Section 7.6:
Negative Bubble
γ the probability the informed
traders will show low demand when they have received a high signal, Pr(XiL#ηH).
the
probability that the uninformed traders will have low demand when they infer
the informed traders had low demand, Pr(XuL#XiL).
_____________________________________________________________
capita demand by the uninformed, and X s
is the per capita supply of the asset.
Before
deriving the Bayesian Nash Equilibrium of this game of incomplete information,
we can intuitively conclude that if β were sufficiently large, it would always pay for the
informed traders to try to take advantage of the uninformed. If β were sufficiently small, so that the feedback effect
was small, then the informed traders would seldom gain from attempting to
manipulate the uninformed.
Consequently, if β were known to be small, then α would be
small.
By
specifying a range for the exogenous parameters in the model, we will initially
focus on the interior solution where both 0 < α < 1 and 0 < β < 1. The latter condition implies that the
uninformed still expect to profit from sometimes buying when they infer the
informed have bought.
A
speculative bubble is defined as an increase in an asset's price above its
fundamental value, based on the information available in the market. The interior solution illustrates one of
many speculative bubble equilibria in our model. Speculative bubbles arise here from fully rational,
payoff-maximizing behavior by all players.
These price bubbles stem from informed traders buying the risky asset
when they know the earnings signal is ηL, and the uninformed individuals choosing a high
demand in response, which drives the price of the risky asset above its
fundamental value (conditioned on ηL).
7.4.
Payoffs from Alternative Strategies
The payoffs are shown in parentheses at the bottom
of the extensive form game tree, Figure 7.1.
The first element in the parentheses gives the informed traders' payoff,
and the second element is the uninformed traders' payoff. The following assumptions state the relative
magnitudes of the payoffs.
Assumption 1:
> 0 and
> 0 for all j
{1,2,3,4}, k
{1,2,3,4,5}.
Assumption 2: ![]()
Assumption 3: ![]()
Assumption
1 states that, in any event, both informed and uninformed traders earn positive
profits. This assumption explains why
uninformed traders are willing to play the game. The key part of Assumption 2 states informed traders with low
signals receive a higher payoff (
) if they can successfully fool the uninformed than if they
follow the IPT strategy of honestly revealing their signal with low demand
purchases (
). Without this
payoff structure, the informed traders would have no reason to try to
capitalize on the feedback effect.
The key
to Assumption 3 is that the uninformed receive a higher payoff when they buy
high (
) and the signal is ηH than when they buy low and the signal is ηL:
. If this relationship
did not hold, the uninformed traders would always be better off choosing low
demand. The fact that they sometimes
receive higher payoffs from choosing high demand and collecting real returns
actually induces the uninformed to play this speculative feedback game.
The
informed receive their second highest payoff (
) when the signal is
, and they show high demand for the risky asset
and succeed in
fooling the uninformed. This payoff
derives from the capital gains realized when the uninformed traders drive up
the risky asset price with their high demand.
Obviously,
for the informed traders to realize a capital gain, the aggregate demand for
the risky asset - both informed selling and uninformed buying - must move
outward. Either the uninformed must
significantly outnumber the informed, or the informed hold a smaller percentage
of the purchasing power in the market.
The
informed traders receive the third highest payoff
when they respond
with low demand to the signal
. The risky asset
then has a low return, but the informed have cut their exposure. If they attempt to take advantage of the
feedback effect and fail, then the informed receive
, which is their lowest payoff in the game. In this worst case scenario for the
informed, they get a low price for the risky asset which they hold in large
quantities.
The
uninformed traders receive their two highest payoffs when the risky asset has
high earnings
The worst payoff for
the uninformed,
, comes when the informed successfully fool them into bidding
high
even though the
signal is
.
Definition 1: The expected payoff to
the informed traders when they receive a low signal,
, is
The expected payoff
to uninformed traders who witness high prices is
+ ![]()
Notice
that both
and
are continuous in
their respective choice arguments,
and
, over the interval [0,1].
We can therefore invoke Brower's Fixed Point Theorem to prove the
following preliminary lemma on the existence of an equilibrium.
Lemma 1: There exists unique
and
which maximize
and
, respectively.
From the
Kuhn-Tucker conditions for maximums occurring at the limits of a range,
is defined by
and
(7.5)
or
and
(7.6)
or
and
(7.7)
Similarly, the Kuhn-Tucker conditions for
are given by
and
(7.8)
or
and
(7.9)
or
and
(7.10)
Conditions
(7.5) through (7.7) state that informed traders who receive low signals will
always bid low demand for the risky asset (
= 0), always try to exploit the feedback effect (
= 1), or be indifferent between these strategies (
) depending on whether the profit from bidding low,
, exceeds the expected profit from attempting to exploit the
feedback effect, ![]()
Conditions
(7.8) through (7.10) state that uninformed traders inferring high demand by the
informed traders from prices will always mimic their purchases (
= 1), always bid low demand (
= 0), or be indifferent between these strategies (
) depending on the expected profits from bidding high.
7.5.
Bayesian Nash Equilibrium in Pure Strategies
The problem set out in Section 7.3 (that of
determining an optimal investment strategy in light of the feedback effect
generated by informed traders' purchases), with payoffs defined in Section 7.4,
can be modeled and solved using game theory.
The solution for games with incomplete information is the Bayesian Nash
Equilibrium introduced by Harsanyi (1967).
Definition 2: The strategy space for
trading group j is Sj = [0,1]. A pure strategy for the informed (i)
traders is a scalar α ε Si, and for the
uninformed (u) traders is a scaler β ε Su.[11]
Definition 3: The set of best responses
for the informed traders to β is
The set of best
responses for the uninformed traders to α is
{
18
![]()
The functions
and
are the best response
correspondence for the informed and uninformed traders, respectively.
Assumption 4: The uninformed traders use
Bayes' Rule to compute
and
.
Definition 4: The informed traders behave
as a self-enforcing Nash coalition if in either state
or
, any subset of the informed traders chooses to abide by the
per capita demand assigned by the whole informed group.
Assumption 5: The informed traders act as a
self-enforcing Nash coalition in choosing their strategy α ε Si.
Assumption
5 guarantees that no individual informed trader, or any subset of them, will
deviate from the group's strategy and thereby tip off the uninformed
traders. For the case of
, no informed individual would benefit from choosing low
demand instead, because he would end up with a smaller quantity of the risky
asset when its earnings will be high.
The uninformed traders know what the risky asset's price will be when
all the informed traders exhibit high demand.
If any individual informed trader deviates from the high bid collusion
when the signal is
, he would spoil the potential feedback profits not only for
the rest of the informed group but for himself. In Section 7.7, we will return to the limitations imposed by this
assumption.
Definition 5: A strategy pair (α*,β*) is a Nash
Equilibrium for the game G = (V i,
V u, Si, Su) if
(a)
and ![]()
(b)
and
(c)
Alternatively, the pair (α*,β*) is a Nash
Equilibrium if α* ε φi(β*) and β* ε φu(α*): each
strategy is a best response to the other.
Let
denote the prior ![]()
Theorem 1: Under Assumptions 1 through 4,
there exists a Bayesian Nash equilibrium in pure strategies given by
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Proof: From condition (7.7) we know
that for an interior solution for
,
(7.11)
Define
as that probability
that leaves the informed traders who receive a low signal indifferent between
attempting to exploit the feedback effect or not. Then solving (7.11) for
yields
. (7.12)
To obtain an interior solution for
, it is necessary but not sufficient from condition (7.10)
that
and
+
Given
and Assumption 4,
Substituting this latter expression for
in equation (7.10),
and recognizing
+
yields
|
|
Solving (7.13) for that value of
that leaves
uninformed traders facing a high demand by the informed traders indifferent
between choosing high demand or not, we obtain
|
|
By definition of the Kuhn-Tucker conditions (7.7)
and (7.10),
and ![]()
Therefore,
and ![]()
so that
is a Nash equilibrium
with
computed using
Bayesian inference. ![]()
Equation
(7.12) is the informed traders' reaction function when they have received a low
signal, ηL. If β > β*, informed
traders would always try to take advantage of the feedback effect. The informed traders do not behave this way
because they are inherently bad people.
Rather, this strategy simply maximizes their objective function. If the uninformed traders chose XuH with probability β < β*, the informed traders would never expect to gain from
attempting to fool the uninformed; consequently, in these circumstances α = 0.
The
value of equation (7.14), the uninformed traders' reaction function, depends on
the magnitude of ρL. In
particular, the right-hand side of (7.14) is greater than or equal to 0. However, it may also exceed unity. In this case α # 1 < α*, and
optimality condition (7.9) would apply so that β = 1, which is a boundary solution. Figure 7.2 illustrates the interior solution
for α* < 1. The informed and uninformed traders'
reaction functions intersect at the Bayesian Nash equilibrium, (α*,β*).
In
Figure 7.2, when α < α*, the uninformed traders will always choose high demand
(β = 1) after
inferring XiH, because α is sufficiently small.
When α = α*, all of the
uninformed traders' strategies, or mixtures of them, result in the same
expected payoff. Thus the optimal
uninformed traders' reaction function at α = α* is the entire interval β ε [0,1]. Finally,
when α > α* the
uninformed traders never choose high demand (β = 0).
These facts give the uninformed traders' reaction function a stepwise
appearance.
Since
the right-hand side of (7.14) depends on ρL, the following set of propositions clarify the
Bayesian Nash equilibria under varying prior probability specifications. First observe that for some value of ρL
the equilibrium value of α* is set equal to 1.
Let ρL
= z denote this value, then
![]()
Proposition 1: If ρL
= 1, the Bayesian Nash Equilibrium of the game is (α*,β*) = (0,0).
Proof: If ρL
= 1, then from (7.14), α* = 0 and
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|
Accordingly,
=
therefore
for all
which implies
and
Thus
and
so that each strategy
is a best response to the other.
The
equilibrium of Proposition 1 is shown in Figure 7.3. Proposition 1 states that when
65
the uninformed traders know with certainty that high demand by the informed (
) is merely an attempt to speculate off the feedback
effect. Consequently, the uninformed
traders would never choose high demand (
), which in turn leads to
Note
that in both Figure 7.2 and Figure 7.3, the shape of the informed traders'
reaction function remains the same; only the uninformed traders'
Figure 7.2.
Bayesian Nash Equilibrium When
(and
)
Figure 7.3.
Bayesian Nash Equilibrium When ![]()
reaction function has changed. It now lies along the ordinate axis in the
unit quadrant. This position signifies
that
for all values
of
. The following
proposition merely restates the condition for
, which yields the equilibrium previously shown in Figure
7.2.
Proposition 2: If z < ρL
< 1, then (α*,β*) are defined in Theorem 1.
Proof: The condition z < ρL
< 1 is necessary and sufficient to ensure 0 < α* < 1 from
(7.14). It then follows that 0 < β* < 1 by
equation (7.12) and Assumption 2.
Proposition 3: If ρL
= z, the Bayesian Nash
equilibria are given by
α = 1 and β ε [β*,1].
Proof: By definition when
This value for
implies
77
and
for all
from the uninformed
traders' perspective. For the informed
traders taking
as given and for
all
for all
and in particular
for
For all
and
= 0. Therefore, the mixed strategy Bayesian Nash
equilibria will be given by
= 1 together with any
value of
in the interval [
,1].
Proposition 3 shows that when
the uninformed
traders will be indifferent between any of their strategies
95
at the value of
For any value of
the response
will be the
uninformed traders' dominant strategy.
Thus the uninformed traders' optimal reaction function comprises the
upper border
and the right border
of the unit
quadrant. The informed traders
optimally react with
to values of
on the interval
Again, the informed
traders' stepwise reaction function remains the same shape in Figure 7.4 as in
the two previous figures. The multiple
equilibria, at the intersection of the informed and uninformed traders' reaction
functions, are shown in Figure 7.4 by a darkened line connecting endpoints
(1,1) and ![]()
Proposition 4: If 0 # ρL < z, then the Bayesian Nash equilibrium of
the game is (α,β) = (1,1).
Proof: The condition 0 # ρL < z
implies α* > 1 from
(7.14). Thus the uninformed traders
will always choose β = 1, which in turn induces the informed traders to set α = 1 as a best
response. Since each strategy in the
pair (
) = (1,1) is a best response to the other strategy, (1,1) is
a Bayesian Nash equilibrium.
Figure
7.5 shows the equilibrium of Proposition 4.
When the (objective) prior probability of a low signal is sufficiently
small
the uninformed
traders always choose high demand:
, which is why the uninformed traders' reaction function
appears as a straight line on the upper border of the unit quadrant. The informed traders, taking
as given, will always
profit from exploiting the feedback effect ![]()
The
informational asymmetry and the feedback effect permit the informed to exploit
speculative bubble profits off uninformed traders in every scenario
except
Propositions 1
through 4 developed
Figure 7.4.
Bayesian Nash Equilibria When ![]()
Figure 7.5.
Bayesian Nash Equilibria When ![]()
comparative statics for changes in
. Corollary 1
summarizes what happens when payoffs change.
Corollary 1: (a)
; (b)
;
(c)
; and (d) ![]()
Corollary
1 shows that the spread in potential profits to one group influences the other
group's decision to purchase. For
example, part (a) indicates that as the profit margin widens for uninformed
traders to gain from feedback buying; the informed traders, who are aware of
this incentive, respond by increasing
.
We will
conclude this section by commenting on two special kinds of Nash
equilibria. Since the information set
surrounding the informed traders' high demand nodes occurs with positive
probability, the Nash equilibrium of this Bayes game will also be
Trembling-Hand Perfect [Selten (1975) and Simon (1987)]. Furthermore, because the information set
connects all the tree branches, this Bayes game has no subgame short of the
entire tree itself. Therefore, the
concept of a Subgame Perfect Equilibrium [Hellwig and Leininger (1987)] does
not apply to our model.
7.6.
Negative Bubbles: The Symmetric
Game
Feedback effects can also induce speculative
selling. When the signal comes up
favorable,
119,
the informed traders might try to fool the uninformed into thinking earnings
are low. If sales by the informed
traders trigger subsequent sales by the uninformed, the informed might be able
to repurchase these assets at a lower price than they sold them.
Figure
7.6 illustrates an extensive form game, which is symmetrical to the game form
in Figure 7.1. Whereas previously the
feedback effects created a positive bubble, the symmetric game depicts a
negative bubble: a deviation of the
price of the risky asset below its fundamental value conditioned on the
information available to the market.
A
closed-end investment trust illustrates an asset with a negative bubble. Despite numerous attempts to explain the
mispricing, it remains an unsolved puzzle in finance theory even today. These trusts almost always have a market
value less than their liquidation value.
See Thompson (1978) and Weiss, Lehn, and Malmquist (1989). As a general rule, closed-end funds with
unrealized capital gains as well as those with capital appreciation objectives
tend to have higher mispricing discounts.
Closed-end funds that seek to maximize current income sell for
comparatively smaller discounts over their liquidation value. Finally, the generally lackluster
performance of the New York stock market in the 1970s may represent a
macroeconomic negative bubble on stocks vis-ŕ-vis the bond and money markets.
Figure
7.6 shows that informed traders who speculate after receiving a high signal can
obtain a higher profit
120
than if they followed the safer strategy of choosing high demand, which leads
to profits of
. Of course, the
higher profits come at the expense of greater risk of failure. If the uninformed knew the earnings signal
was low
they would show low
demand. Thus the uninformed in this
game do not have a dominant strategy in always choosing high demand, ![]()
Together,
these two payoff structures pose the following decision problem to informed
traders who have received a high signal:
should they pretend the signal is low by exhibiting low demand? Let
denote the
probability
. Uninformed traders
drawing Bayesian inferences from prices face a related question of whether to
choose low demand simply because the informed have. Let
denote this
probability of choosing low demand as a feedback effect.
Theorem 2: There exists a Bayesian Nash
Equilibrium in pure strategies given by

Proof: Follows analogous steps to the
proof of Theorem 1 by maximizing with respect to
the expected payoff
![]()
maximizing with respect to
the expected payoff

Figure 7.6.
Extensive Form Game for Negative Bubbles
and solving for an interior solution. The applicable form of Bayes' Rule for this
game will be
![]()
Theorem
2 shows the creation of a negative bubble from general equilibrium,
payoff-maximizing behavior. With some
equilibrium probability
> 0, the informed
traders will attempt to exploit capital gains off the uninformed traders'
feedback effect when the informed signal is high. With equilibrium probability
> 0, the
uninformed are induced into choosing low demand as a strategy upon witnessing
low demand by the informed. The
combined low demand drives the price of the risky asset below its market
fundamental conditioned on
.
7.7
Critique of the Bayesian Nash Equilibrium Approach[12]
Some critics of IPT have characterized it as
having an `air of unreality.' However,
the orchestrated manipulations by the informed traders, which were covered in
Sections 7.3 through 7.6, strike us as even more unrealistic. In trying to use this analysis to understand
securities markets, it may be helpful to list some of the obstacles we have
confronted.
Payoffs. We
specified the payoffs a priori, so that each player could determine not
only his optimal strategy but also his opponent's optimal strategy under the
varying states of the world. These
payoffs are the
and
vectors. In a numerical example, we could fix the
various quantities required for market clearing as well as an initial
equilibrium price. High demand by the
informed traders would lead to excess demand at the current price. The market would clear only with an increase
in price. We could then record the cost
basis and, eventually, the capital gains for all traders' transactions.
In
principle, these payoffs should be independent of the pure and mixed strategies
chosen by the two groups. Although the
payoff vectors could be transformed into real numbers once we specify the other
model parameters, we have left them as ranked variables. One of the rankings provides that the
informed traders with a low earnings signal receive a higher payoff if they
successfully fool the uninformed than if they simply follow the IPT strategy of
exhibiting low demand:
As previously
mentioned, this relationship could only hold if the informed traders realize
their capital gains. To do this, the
informed must be able to sell their speculative purchases without being
detected. However, the uninformed must infer
(detect) when the informed have bought.
We rely
on the sequence of play to mask the sales order: uninformed traders do not realize the informed have sold until
they see the next market clearing price.
Then it is too late. Thus for
completeness, the two extensive form game trees (Figures 7.1 and 7.6) should
contain an additional, simultaneous move by the informed (at the same time as
the uninformed move) on the one branch of the tree where the informed try to
fool the uninformed.
The need
for specified payoffs at the bottom of the tree severely limits the Nash
equilibrium approach. In financial
markets, payoffs from investments can rarely be written in advance. For that reason, I remain disillusioned with
game theory, in general, as a useful tool for explaining how financial markets
operate. Consider, for example, what
happens if we combine Figures 7.1 and 7.6, so that the informed traders can
attempt to create a speculative bubble regardless of the signal. Figure 7.7 shows the generalized game tree
with two information sets. Note the
tree contains no payoffs at the bottom.
Depending on how we choose to specify payoffs, we can either replicate
the positive and negative bubble equilibria or we can eliminate the bubble
paths (branches of the tree) from consideration. Thus the bubble results are very sensitive to how we specify the
payoffs.
Two-Player,
High-Low Limitations. The mathematics of the formal model as well
as Figures 7.1 through 7.6 all apply to two-player games. However, the informal comments in this
chapter describe many traders who belong to one of two groups: the informed and the uninformed. This two-player environment corresponds
precisely with Informational Price Theory.
Instead of criticizing the relevance of the two-player framework for
populated markets, we can consider ways to generalize the model.
Each of
the traders should have a continuum of strategies, not just high or low
demand. Individual informed traders
should receive correlated but not uniform information signals. The solution for such an economy could then
be either a separating or a pooling equilibrium. The separating equilibrium would be a victory for the efficient
markets hypothesis. The pooling
equilibrium, in which one or more informed traders sells his information to one
or more of the uninformed, would not be efficient. It would also not lead to speculative bubbles, since the informed
traders share news about the low signal.
Figure 7.7.
Generalized Bayes Game in Extensive Form With Unspecified Payoffs
Our
bubble results thus appear to rely on the underlying discrete (high or low)
purchase decisions. We initially viewed
the high-low quantity choice as a minor simplification; now it turns out to be
more crucial than expected. Favorable
news does not equate with a unique price level in either (1) a model in which
informed traders can choose from a continuum of quantities to purchase, or (2)
one with discreet quantity purchases but where the informed traders mix their
strategies independently. Hence, even
if informed traders wanted to send a false signal, they have no price target to
aim for with their purchases.
Computational problems can also restrict the traders' knowledge about a
price target: traders may not have
access to all the information they need to compute future market-clearing
prices.
By
allowing each trader to act as an individual, in the next section, we can
expand the decision range to include no action. Some of the informed may feel prices already impound their
private signals; some may feel they should wait for a better price before
purchasing. This generalized model
should accommodate the market timing criticism of IPT.
Noncompetitive
Behavior. If no individual informed trader can move
prices with his purchases, but all the informed traders as a group can influence
prices, do we still have a competitive market?
Grossman and Stiglitz clearly feel they do; the critics of IPT would
disagree. To the extent either trading
group can form a self-enforcing coalition, then clearly they exert monopolistic
influences on market prices. However,
our analysis has shown just how difficult it may be for such a coalition to
exist and succeed in extracting speculative bubble profits.
Gorton
and Pennacchi (1989) describe `imperfectly competitive rational expectations
equilibrium prices' that are `fully revealing in only two of the four
states.' In contrast, no prices in the
Grossman and Stiglitz (1976) model are fully revealing; the difference between
noisy prices and fully revealing prices compensates the informed for their cost
of acquiring information. Yet Grossman
and Stiglitz state they are modeling competitive markets with asymmetric
information. Gorton and Pennacchi trace
the source of their price inefficiency to insider collusion.
In the
model of Sections 7.3 through 7.5, the informed traders receive payoff
if they choose low
demand upon receiving a low signal.
They receive payoff
if they take part in
the attempted manipulation and fail.
Since
, each informed trader faces an incentive to deviate from the
coalition. The size of the incentive
varies inversely with the probability that the uninformed will buy as a
feedback effect. If the uninformed are
likely to mimic purchases by the informed, then individual informed traders
would prefer to adhere to the coalition's strategy. When they expect the coalition will fail, because the probability
of feedback buying is too low, individual informed investors would rationally
choose to abandon the coalition's strategy.
But if
, then Kuhn-Tucker condition (7.5) applies. Thus the informed traders as a group choose
to bid low demand without any collusion.
If we
changed the payoff structure so that
, then individual informed traders would have no reason to
depart from the group's manipulation strategy.
Assumption 5 under the condition
would be
superfluous. Individual informed
investors would adhere to the coalition strategy by construction. As the game now stands, the speculative
bubble strategy might crumble without Assumption 5.
7.8.
Existence of Subjective Correlated Equilibria
If the informed and uninformed traders decide
what quantity of the risky asset to purchase, rather than the probability of
purchasing a fixed quantity, then Brower's Fixed Point Theorem can no longer be
used to establish the existence of an equilibrium. This section shows that by imposing appropriate concavity and
continuity assumptions on the payoffs in a generalized game form, we can prove the
existence of a correlated equilibrium in pure strategies. Assumption 6 converts our n-player
game of incomplete information into a convex game.
Assumption 6: The noncooperative n-person
game satisfies the following conditions:
(a) The i
th player's strategy space is a compact convex set Si
of a topological linear space Ei.
(b) The i
th player's payoff Vi(s1,... si,....sn)
is concave with respect to his own strategy variable s1
Si.
(c) The
sum of payoffs
is continuous over
the Cartesian product space
(d) For
each fixed
is a continuous
function of the
(n-1)-tuple
respectively.
For
constant sum games, 6(c) is automatically fulfilled. If all the payoffs are continuous over
then 6(c) and 6(d)
are also fulfilled. Kakutani's
fixed-point theorem cannot be applied to convex games since conditions 6(a)
through 6(d) do not guarantee the required upper semi-continuity.
Definition 5: A Nash equilibrium point
is an n-tuple
that maximizes the si-function
, at
for i = 1, 2,
... n.
This
generalized game contains n players constituting both informed and
uninformed traders: each individual is
a player. If the informed traders mix
their strategies independently, then the economy will attain a correlated
equilibrium not a Nash equilibrium.
Each player i has a subjective prior
, which is a probability measure on the space
with
, and an information signal
bout the risky
asset's return r =
. The
are correlated with
, though not the actual value.
Definition 6: A subjective correlated
equilibrium is an n-tuple
such that for every
individual i
(7.15)
for all
.
Note
that the correlated equilibrium is only a condition on individual rationality
rather than market clearing. The
difference between an objective and a subjective correlated equilibrium is
based on assumptions about the
. If the traders'
priors are equal and common knowledge, as with the value of
in Sections 7.3
through 7.6, then we would have an objective correlated equilibrium. Definition 6 generalizes the agreed common
knowledge prior concept to include possible heterogeneous priors.
Recall
from Chapter 2 the distinction between ex ante optimal and ex post
optimal allocations. The subjective correlated
equilibrium only requires the traders' strategies be optimal ex ante to
receiving their private signals. For
the strategies to be optimal even after the traders have received their
signals, Brandenberger and Dekel-Tabak (1987) define an a posteriori
equilibrium, which looks like (7.15) but with the substitution
for the prior, and
presumably
and
for the
strategies. `Uninformed' traders
receive a signal that gives them no information: actual earnings belong to the state space
.
Before
proving the existence of a subjective correlated equilibrium for our
generalized game, we must first introduce a summed expected payoff function:
![]()
which is also concave with respect to the n-tuple ![]()
Theorem 3: Under Assumptions 6(a) - 6(d),
there exists at least one subjective correlated equilibrium point.
Proof: [Adapted from Nikaidô and
Isodo (1955), p.809-10] The concavity
of Vi (by Assumption 6(b)) with respect to the choice
variable
implies
is concave with
respect to
. From Definition 6
and the concavity of
, a point s = (s1,.....sn)
is a subjective correlated equilibrium point if
is maximized at
We must now prove
the existence of a point
such that
for any
Suppose to the
contrary that for each
there exists some
such that
|
|
Define
Then
is open by
Assumptions 6(c) and 6(d) and
from (7.16). The compactness of S by assumption
6(a) enables us to define a set
such that
This condition
implies
for any
Let
with i = 1,
2,.....n. These n
functions are continuous under Assumptions 6(c) and 6(d) and satisfy
and
for any
The continuous
mapping
![]()
maps S into the convex hull of A, C(A),
and therefore maps C(A) into itself.
Since C(A) is homeomorphic to a compact convex set in an
Euclidean space, Brower's Fixed Point Theorem establishes the existence of a
fixed point. Let
denote one such fixed
point. Then
![]()
For such an i that fi(s)
> 0, then
. Since
is concave in
, it follows that
, which is a contradiction.
Q.E.D.
The
subjective correlated equilibria of this game of incomplete information depend
on how we transform the general model into particular situations. Suppose the informed traders receive
(correlated) low signals along with a separate signal that says `fleece the
uninformed'
percent of the
time. Essentially, we give chance three
signals from which to choose: high, low
and fleece, and low and don't fleece.
We do not need to specify whether the uninformed correlate their random
strategies.
The
obvious question is whether such a correlated equilibrium exists: is it optimal for individual informed
traders to obey the signal? The answer
is yes; such a model could fit within the parameters of Theorem 3. Once the `fleece the uninformed' signal is
given, every informed trader is rational to obey it under particular
specifications of the model's
parameters. Public goods theory
may have free rider problems, but apparently public pillaging models have no
free raiders. Of course, we still face
the knotty problem of how the pillaging informed traders realize their capital
gains without tipping off the uninformed.
We will
conclude this section by seeing how far we can go to model a (separating)
correlated equilibrium without invoking the same pitfalls of the Bayesian Nash
equilibrium approach: specifying the
payoffs a priori, two-player model, discrete strategies, etc. Grossman and Stiglitz let
denote the fraction
of the population who choose to become informed. With n players in the model, then
is the number of
informed traders; players
to
are uninformed.
Assume
that the informed players cooperate in attempts to speculate off the feedback
effect. Suppose all the traders play m
original Bayes games of incomplete information simultaneously according to some
special rules. Denote these m
games as
, respectively. In
game
, the informed and uninformed traders select respective
strategies of
and
.
The individual informed traders may choose
different quantities to purchase, i.e., the
and
notation is not meant
to indicate the individuals grouped in a matrix adopt the same strategy. As outside theorists, we can fix different
quantities across the individuals and label a matrix of these strategies as
or
or whatever.
The
special rules[13] then for
this game are that when the informed traders choose the
strategy and the
uninformed traders choose
, the uninformed traders pay each of the
informed traders
their respective expected payoff
, where
and
as the outcome of
. The informed
traders pay the uninformed traders the amount
(7.17)
as the premium for participating in these
speculative feedback games. Each
uninformed trader receives that portion of the amount in (7.17) that represents
his own individual summed payoffs across the m games. Condensing notation, the informed traders
receive a gain of
, while the uninformed receive
.
In our
Bayesian Nash equilibrium model, we assumed
> 0, so that the
uninformed have a reason to play the game.
Expression (7.17) gives a statement of the premium the uninformed
traders expect for agreeing to play the game.
The uninformed know the informed cooperate (share their correlated
signals) to maximize their own payoff.
Although we have not proved it here, we suspect that the amount in
(7.17) and the
strategies are tied
to competitive (fair) play in informational price theory: informed buying only when the signal is good
and selling only when the signal is bad.
If the given n-person game is constant sum, a criterion for the
existence of an equilibrium would be
,
where C is the constant sum.
7.8.
Conclusions and Issues for Future Research
This chapter identified three areas of concern
with informational price theory (IPT):
low quality vendors using high prices to lure customers, market timing
decisions, and the feedback effect. Our
Bayesian Nash equilibrium model of speculative behavior focused on the
feedback effect generated by uninformed traders purchasing when they learn
informed traders have bought.
In the
process of analyzing these feedback effects, we have introduced a game
theoretic analysis of speculative bubbles, which are created from fully
rational, payoff-maximizing behavior on the part of all market
participants. Various assumptions about
the external prior probability for low earnings can eliminate the bubble
equilibria, show bubbles will form with probability equal to one, or lead to
occasional bubble formation. Further
study of feedback effects, particularly in speculative markets, could provide
rich new models that can explain factors ranging from stock market overshooting
and excess volatility to dividend signals sent by firm managers.
The
traditional IPT literature has not addressed individual departures from group
behavior. Individual informed traders
in those markets could set up side deals to sell news directly to one or more
of the uninformed. The stability of the
Grossman-Stiglitz equilibrium, for example, depends on achieving the optimal
percentage of informed traders so that the marginal informed trader is just
compensated for the cost of his information.
If other traders convey this same information to the uninformed at a
lower cost, then the entire information gathering mechanism will unravel or
continually be in a state of disequilibrium.
In
addition to deviations from group behavior, we have identified three other
broad objections to the Bayesian Nash equilibrium approach to speculative
bubbles. First, payoffs for investment
strategies must be specified at the bottom of the extensive form game tree. Without these payoffs, players could not
compute their optimal strategies, which are best responses to the other
player's strategy. We also assumed that
when the informed traders succeed in manipulating the uninformed, they are able
to realize their capital gains (sell the asset) without their sales affecting
the uninformed traders' strategy.
The
two-player, high-low demand framework severely limits a realistic portrayal of
financial markets. Our work has
highlighted the two-player focus of the mathematics of even traditional IPT,
which might go unrecognized due to the informal comments about large numbers of
informed and uninformed traders.
Speculative bubbles arise in the Bayesian Nash equilibrium model due to
the high-low quantity constraint.
Without this constraint, no unique price target exists to signal high
earnings. Uninformed traders may
observe a price increase and wonder if it really reflects high earnings news.
Finally,
the informed traders - acting as a collusive group - influence prices. This fact tends to negate claims that IPT or
the Bayes game model portrays how competitive markets operate under
uncertainty. In neither model will
prices fully reflect the information available in the market. The more competitive the market, the more
individual informed traders face incentives to deviate from group behavior and
sell their private information to the uninformed.
These
combined factors mean that the feedback effect does not create significant
problems for the validity of IPT. The
informed traders face more difficultly in capitalizing on this feedback effect
than the critics of IPT have admitted.
Market timing decisions and attempts to lure customers with high prices
still limit the amount of information reflected in prices.
In the
generalized model of Section 7.8 with many players of each type, the resulting
individual rationality condition is correlated equilibria rather than a Nash
equilibrium. The multiple correlated
equilibria, under specified exogenous conditions, will either be separating or
pooling equilibria. The separating
equilibria are in the Grossman-Stiglitz tradition of homogeneous groups. The pooling equilibrium eliminates the kind
of speculative bubbles studied in this chapter.
The
presence of witless noise traders in many asymmetric information models
represent a principal shortcoming in the literature. We have quantified in Section 7.8 a premium that uninformed
traders expect to receive for playing a game in which others collude and
occasionally take advantage of them.
The means by which the uninformed collect this premium comes from buying
when they infer the informed have bought and earnings are actually high: no manipulation occurs.
The
separating correlated equilibrium, which leads to speculative bubbles, is not
unique. In fact, the equilibrium
appears suspect due to the informational gymnastics required by the informed
traders: they must participate in a
giant, carefully-orchestrated conspiracy.
It is good to know that such equilibria exist as polar cases, but other
more relevant and interesting ones deserve our attention. Financial theorists should continue to look
for them.
Notes
1. The weak
form of the efficient markets hypothesis (EMH) states all information on
past price movements is fully reflected in current market prices. The semi-strong form of the EMH
states all publicly available information is impounded in prices, and the strong
form of the EMH states all information - both publicly and privately held -
is reflected in market prices.
2. Additional
works in the informational price theory literature include Grossman (1976) and
(1978), Kyle (1985), Grinblatt and Ross (1987), and the writings of Shiller
(1990), among others.
3. See
Burness, Cummings, and Quirk (1980) and Shleifer and Summers (1990).
4. Shleifer
and Summers (1990), Kyle (1985), Kindleberger (1989) and others have no
reluctance to introduce irrational investors into their models. Economists have traditionally objected to
irrationality arguments, because they abandon the discipline's expected utility
maximizing hypothesis and permit the explanation of just about any phenomenon.
5. Burness,
Cummings, and Quirk (1980), p.74.
6. How
ironic that the `informed' traders are the stupid players in informational
price theory. In most feedback models,
uninformed traders act as witless noise traders, who continually let the
informed traders exploit them.
7. Bernheim,
Peleg, and Whinston (1987).
8. This
chapter comes from my draft dissertation that I wrote and first presented at a
workshop at the University of California, Los Angeles, back in 1984. At the time, it was the first game
theoretic analysis of speculative bubbles and the first derivation of
bubbles from purely rational behavior.
9. For
specific values of the external parameters of the model, we can derive as a
special case an equilibrium in which the informed traders always try to exploit
the uninformed traders, and the uninformed may be willing to buy if the
objective prior probability of unfavorable news is sufficiently small. See Proposition 4.
10. Simply
from observing an increase in the risky asset's price, uninformed traders can
no longer on average infer η. A high price
can now mean either a high signal (ηH), or a low signal (ηL) together with an attempted manipulation.
11. The
Bayesian Nash Equilibrium in this section yields a probability that the
informed will choose high demand. In
one sense, this probability can be considered as a mixed strategy between high
and low demand. However, since we have
defined the strategy space in this section over probabilities and not over
quantities demanded, a single value for the probability represents a pure
strategy.
12. This
section benefitted greatly from discussions I have had with Brendan O'Flaherty. He also identified the work by Nikaidô and
Isodo to me. However, he bears no responsibility
for any errors contained in my analysis.
13. The
special rules and alternative definition of the minimax theorem are adapted
from an example by Nikaidô and Isodo (1955), p.811.
References
Admati, Anat R., and Paul Pfleiderer, `A Theory of
Intraday Patterns: Volume and Price
Variability,' Review of Financial Studies, Vol. 1:1, pp. 3-40.
Allen, Franklin, and Gary Gorton, (1993),
`Churning Bubbles,' Review of Economic Studies, Vol. 60, pp. 813-836.
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Comment
This
chapter explores one of the many different ways of arguing that security prices
are based on something other than fundamentals. The chief conclusion is that this particular attempt doesn't work
very well, but that doesn't imply that no method is capable of showing that
serious deviations from fundamentals can occur.
The
question is important. Adam Smith's
claim about the capitalist system was that it allocated capital well: the claim had nothing to do with the static
Pareto optimality that dominates modern welfare theorems. Even today, claims of the superiority of
capitalism over socialism are, for 99% of the world, claims about growth, not
claims about static efficiency. Growth
depend on allocating capital. The
system isn't called capitalism incorrectly.
Investigations of bubbles and speculative behavior challenge the most
fundamental and deep-rooted claims for any kind of superiority of the
capitalist system.
One of
the most pervasive difficulties in developing models where speculation matters
are multiple equilibria. Typically,
game theoretic models of security markets end up with several equilibria, one
of which is the fundamental equilibrium and one (or more) of which is a
speculative equilibrium. My 1987 paper
is a good example of this sort of result, as are the subjective correlated
equilibria in the latter part of the chapter.
One of the great strengths of the first model in this chapter is that
the speculative equilibrium is unique.
In the study of speculative equilibria, too much attention has been paid
to existence, and not enough to uniqueness.
What's
so important about uniqueness? As
Jagdish Bhagwati argues in a slightly different context, multiple equilibria
say something about modelling not the world; they are signs of incomplete
modelling, not of an undetermined world.
In reality only one thing happens.
From the existence of multiple equilibria one can draw inferences about
the logical structure of a model; one cannot draw inferences about the reality
the model is trying to capture. Having
a calculator that doesn't take square roots does not entitle you to assert that
the square root of seven is five or that it is indeterminate.
An
example with a model not usually thought of as having multiple equilibria can
make the point clearer. The theory of
human capital is compatible with the president of General Motors wearing a red
tie and also compatible with his wearing a blue tie; so there are (at least)
two equilibria. We conclude from the
existence of these multiple equilibria that the theory of human capital is not
very useful if we are interested in tie color C but no one ever claimed it was.
We do not conclude that the president's tie color is indeterminate or
that it is purple because nothing in the theory of human capital, a very good
theory, rules that out. We conclude
only that to answer tie color questions another model should be used. The same conclusion should be reached
whenever multiple equilibria are encountered.
Brendan O'Flaherty
Department of Economics
Columbia University