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 (n^th) trader will execute his trade, the n-1^st 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 (n^th) 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:

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:  η_H or η_L.

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 η_H, 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 η_L, 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 η_L when the informed traders bid low demand (X_iL).  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:  MX_i /MP < 0.  However, unlike in IPT, we cannot say that MX_i /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:

where λ is the fraction of the traders who are informed,

X_u is the per

                              Table 7.1.  Bayes Game Notation

____________________________________________________________

Symbol        Definition

____________________________________________________________

ρ_L     (objective) prior probability of a low signal, η_L.

ρ_H               (objective) prior probability of a high signal, η_H.

X_iH              high demand for the risky asset by the informed traders.

X_iL               low demand for the risky asset by the informed traders.

X_uH              high demand for the risky asset by the uninformed traders.

X_uL              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(X_iH#η_L).

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(X_uH#X_iH).

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 ρ_L 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(X_iL#η_H).

                the probability that the uninformed traders will have low demand when they infer the informed traders had low demand, Pr(X_uL#X_iL).

_____________________________________________________________

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 η_L, and the uninformed individuals choosing a high demand in response, which drives the price of the risky asset above its fundamental value (conditioned on η_L).

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 η_H than when they buy low and the signal is η_L:  .  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 S_j = [0,1].  A pure strategy for the informed (i) traders is a scalar α ε S_i, and for the uninformed (u) traders is a scaler β ε S_u.[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 S_i.

    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, S_i, S_u) if

    (a)  and

    (b)  and

    (c)  

Alternatively, the pair (α*,β*) is a Nash Equilibrium if α* ε φ_i(β*) and β* ε φ_u(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

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, η_L.  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 X_uH 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 ρ_L.  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 X_iH, 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 ρ_L, the following set of propositions clarify the Bayesian Nash equilibria under varying prior probability specifications.  First observe that for some value of ρ_L the equilibrium value of α* is set equal to 1.  Let ρ_L = z denote this value, then

Proposition 1:  If ρ_L = 1, the Bayesian Nash Equilibrium of the game is (α*,β*) = (0,0).

Proof:  If ρ_L = 1, then from (7.14),  α* = 0  and

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 < ρ_L < 1, then (α*,β*) are defined in Theorem 1.

Proof:  The condition z < ρ_L < 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 ρ_L = 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 # ρ_L < z, then the Bayesian Nash equilibrium of the game is (α,β) = (1,1).

Proof:  The condition 0 # ρ_L < 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-à-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.