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Forfatter: Simon Toft Nielsen
Studienummer: SN70199
Vejleder: Stefan Hirth
A game-theoretic analysis of price bubbles in financial markets
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Abstract
Price bubbles have had profound influence on economies in the latest centuries. Studying the
phenomenon has been greatly restricted by the assumption that the efficient market hypothesis
holds. This has incurred a basic lack of understanding investors behavior in price bubbles.
Accepting that the efficient market hypothesis does not hold implies, however, that irrational
investors are directing the market. This raises the issue of how rational investors behavior can be
modeled in an irrational environment.
Game theory is found to be a very efficient tool for handling this issue. First of all, irrational
investors can be modeled quite well. Second, it allows investors to make strategic choices,
depending on their expectations of other investors. The strategic aspect of investing has formerly
been ignored by assuming that rational investors only based decisions on the fundamental value
of stocks.
Furthermore, game theory is in general designed with the purpose of determining players
behavior. Logically, it gives a great framework to analyze investors behavior in price bubbles.
To determine optimal behavior in price bubbles, Abreu & Brunnermeier have presented a
comprehensive model, based on game theory.
This model represents an environment, encompassing both rational and irrational investors.
Additionally, heterogeneity among rational investors leads to a situation, where they become
sequentially aware of the price bubble.
The role of irrational investors is to participate in creating the price bubble, while simultaneously
preventing rational investors from identifying other investors strategies. However, there is a limit
of the amount of selling pressure, irrational investors are able to absorb. When a sufficient
amount of selling pressure is generated, the prices will drop and the bubble burst.
Because of dispersions of opinion, and the absorption capacity of irrational investors, it becomes
rationally optimal to ride the bubble. Until the risk of the bubble bursting exceeds the costs
compared to the benefits of attacking it, investors will stay in the market.
The model additionally presents a broad range of assumptions about the reality.
Violation of these assumptions, logically has major impact on the optimal equilibrium. Discussing
them reveals that they generally are well supported.
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Furthermore, presenting a few investigations of actual behavior of rational investors in the
dot.com bubble also supports the results of the model.
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Content
1) Introduction
5.
1.1) Method
6.
1.2) Limitations
7.
2) Definition of a price bubble
8.
3) Principles of game theory
10.
4) Game theory in price bubbles
15.
5) Rules of the game
18.
6) The Abreu/Brunnermeier model
19.
6.1) Game theoretical background
25.
6.2) Preliminary analysis
26.
6.3) Identifying the optimal behavior of investors
31.
6.3.1) Exogenous crashes
32.
6.3.2) Endogenous crashes
35.
6.4) Optimal behavior
38.
7) Assumptions
40.
8) Supporting results of the model
49.
8.1) “Hedge funds and the technology bubble”
50.
8.2) “Who drove and burst the tech bubble”
50.
9) Conclusion
51.
10) References
53.
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1. Introduction
Analysing modern financial market behavior by game theory is still considered unconventional
compared to traditional asset pricing theories.
However, game theory points out important aspects of investor behavior which contradicts the
fundamental assumptions of traditional theories. Traditional theories are typically founded on the
efficient market hypothesis which constitutes that asset prices are based on every known
information, and that all investors is able to access and benefit from this information1. This implies
that the development of asset prices is unpredictable, since information and news in the future by
nature are unpredictable. As a consequence of unpredictability, the efficient market hypothesis
implies that no investor can consistently outperform the market.
This leads to the idea of the random walk of Wall Street2 where any portfolio of stocks in the long
run will be as good as any other portfolios.
It also leads to the “no-trade theorem”3, where investors cannot benefit from private information.
This means that using private information will have the effect of a signal to other investors. This
will therefore become known information.
Furthermore, if the efficient market hypothesis holds, it implies that a price bubble cannot
develop in the market. The argumentation is, that investors through interpretation of news,
immediately will become aware of the over-valued stocks, and sell them to collect the instant
profit, before other rational investors become aware.
Several papers have proven the weaknesses of the efficient market hypothesis4, and how the
development of asset prices cannot completely be explained by news and information. This shows,
that the asset prices are influenced by other factors, such as historical development, psychology of
investors and other behavioral factors.
1
The Efficient Market Hypothesis: A Survey, 2001
A Random Walk Down Wall Street, Burton Malkiel, 1973
3
Information, trade, and Common Knowledge, Paul Milgrom and Nancy Stokey, 1982
4
On the Possibility of Speculation under Rational Expectations, Jean Tirole, 1982
The limits of price information in market processes, Avraham Beja, 1977
2
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The most important contribution of game theory to the analysis of financial markets is, that it can
be extended to cover a market situation which includes both rational and behavioral traders. In a
price bubble situation, which is the focus of this paper, game theory additionally lays a unique
framework to analyze the optimal behavior of rational investors that hold private information.
This paper describes how game theory can be applied to a price bubble situation. Including a
thorough discussion of the assumptions made especially in relation to the roles of rationality and
information.
Following questions will be answered with a game-theoretic approach:

What is a price bubble

How does a price bubble origin and evolve

What is the optimal behavior of investors in a price bubble
The problems above will be explained using a comprehensive model presented by Abreu and
Brunnermeier in their article “bubbles and Crashes”. Finally, the paper aims to show which
behavior rational investors had during the dot.com bubble, compared to the behavior which the
game theoretic model constitutes as optimal.
1.1 Method
As mentioned, the paper is founded on the model presented by Abreu and Brunnermeier. The
assumptions made, however, will be discussed thoroughly and will draw insights from a broad
range of relevant literature.
The presentation of the model is following the same structure which Abreu & Brunnermeier had
used in their paper Bubbles and Crashes (2003). First presenting their model, then develop some
useful propositions in a preliminary analysis, before finally identifying equilibrium. However, trying
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to grasp the ideas of the model, did not at first seem straight forward. I have felt it necessary to
clarify some of the issues and interpret the mathematical expressions. Therefore I include a few
more steps in the explanation.
The model explains how a price bubble evolves, and determines the optimal behavior. Each step
of the model will be presented, and assumptions will be discussed, so the implications of the
optimal strategy can be analyzed.
In order to compare the behavior of rational investors during the dot.com, with the optimal
behavior identified in Abreu & Brunnermeiers model, I initially wanted to do an investigation on
Hedge funds.
The action ability and the structural design of a hedge fund means, that it is a good representative
of a rational investor in the model.
Gathering data, however, turned out to be a problem that was not possible to overcome. Access
to databases that hold the necessary information was very expensive.
The investigation of rational investors behavior in the dot.com bubble will, therefore, consist of a
short presentation of 2 papers addressing the issue.
(1) “Hedge funds and the technology bubble” by Brunnermeier and Nagel
(2) “Who drove and burst the tech bubble” by Griffin, Harris and Topaloglu
1.2 Limitations:
Abreu & Brunnermeiers model include some further theories especially about synchronizing
events. This part of the model will not be included in this paper.
Additionally the special case where the price equals the fundamental value at the time 𝑡0 , will be
ignored, since this paper focus its attention on broader aspects of the model.
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2. Definition of a price bubble
Price bubbles have had a significant impact in the global economy at numerous times during the
latest centuries. Naming the Tulip mania, the South Sea bubble and the dot.com bubble as a few5.
Kindleberger (1978)6, originally defined price bubbles by the following; “A bubble is an upward
price movement over an extended range that then implodes”. However, this definition seems a
little too simplified, so further explanation is necessary.
A price bubble origins, when a price of an asset rises over its fundamental value. However, it is
complicated to define precisely when this happens.
In the stock market assets are valued as any other goods by the limitations of supply and the
extent of demand. Hence, the price is naturally set by the market, and the price of the asset
reflects its value to the market. The fundamental value of an asset reflects the collected value of
the particular company, including future dividends. Future dividends will naturally depend on an
assessment of how profitable the company will be in the future.
This assessment of the companies profitability in the future is exactly the source of price bubbles
in the financial markets.
Many of the well-known financial bubbles have originated around major technological
innovations7. Companies, that are closely related to or are able to exploit the new technology,
experience exponential increases in stock prices, because investors believe that their future
growth will be higher than the market growth. Under these structural changes unsophisticated
investors tend to believe that the economy has changed as well. They believe that the
technological innovations have created permanently higher growth rates in the particular sector of
the economy.
5
Wikipedia search on; stock market bubble
Maniacs, Panics and Crashes: A History of Financial Crises, 1978
7
Advisors and asset prices: A model of the origins of bubbles
6
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The expectations of permanently higher growth rates will make unsophisticated investors overlyoptimistic in relation to a company’s future dividends, and the intensity of demand will increase
and inflate prices significantly higher than the real fundamental value.
Some literature8 also suggests that the amount of unsophisticated investors compared to the
amount of sophisticated investors will increase during a bubble situation. It is argued that the high
growth rates, attracts private investors who have not yet experienced a downturn in the stock
market. They will for this reason behave over-optimistically and hence, support the development
of the bubble.
As time goes by and the investors’ expectations are revealed to be overly-optimistic, the bubble
crashes and leaves the stock prices at very low levels.
The key for a sophisticated investor is to realize when other investors’ expectations of the
companies future profitability is unrealistic. This will inevitably mean that the stock price will rise
over the fundamental value and create the bubble effect.
This definition of a price bubble implies that it, in the situation virtually, is impossible to be certain
whether or not it is a bubble. Only the future will prove whether or not the expectations to the
companies profitability where realistic. However, it is safe to say that all rational investors, during
a period of rapid price increases, will develop a belief from which they act.
The model of Abreu & Brunnermeier corresponds with this perception of a price bubble. Although,
it is not clearly explained, exactly how they define a price bubble, the overall structure of the
model reveals their perspective9.
From their point of view a mispricing is required to be sustained until a certain amount of
investors are aware before it can be regarded as a price bubble.
As the definition above, a price bubble starts to origin, when the price is separating from the
fundamental value. However, a bubble is not established, until a sufficient number of investors are
aware of the mispricing to burst the bubble, but don’t.
8
John Brooks, The go go years, 1973
Michael Brennan, How did it happen, 2004
9
Bubbles and crashes, s. 180
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It can be argued that as long as investors actually believe in higher future dividends they also
believe that the fundamental value is represented by the price. If enough investors share this
opinion then the mispricing has not established itself as a bubble. In order for a mispricing to
become a bubble situation at least some investors is required to be aware.
The reasoning behind this is to separate minor mispricing from bubble situations.
Summing up, a price bubble is defined as a situation where the price departs from the
fundamental value, because of overly-optimistic expectations to future dividends. When enough
investors have become aware to correct the mispricing, but don’t, the bubble situation is
established.
3. Principles of Game theory
Game theory will, in this paper, outline the framework of the analysis. The following will give the
basic ideas of how game theory works in order to understand how it can be applied to price
bubbles.
Game theory was created and first published by Von Neumann and Morgenstern in 194410.
Recently their ideas have gained increasing success in analyzing economic problems. Experts
believe that the general methodology of economics has changed, and that the principles of game
theory fit in well with the new paradigm11.
Comparing game theory to traditional economic analysis tools, shows a distinctive characteristic of
game theory. The fundamental assumptions are very primitive. This is because game theories set
of point are the actors which essentially are the most basic units in a given economy. Previously
most economic theories started out with higher-level assumptions about actors’ behavior12.
Macroeconomists were typically assuming different behavioral relationships, like the consumption
10
Theory of games and Economic behavior, 1944
Games and information, p. 2
12
Games and information, p. 2
11
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function (the relationship between actors’ income and their consumption). Assuming such
relationships, makes it harder for economists to evaluate results on real-life basis.
The reason is that actors’ behavior is conditioned of these assumptions. This implies, that if the
assumptions do not hold, then results do not hold.
Correspondingly, micro economists often used assumption of sales maximization is problematic.
Sales maximization does not include any cost terms, which indicates that sales maximization can
be viewed as irrational.
Game theory is applying much more simple assumptions. The most important and most general
being that, every actor in a game wishes to maximize his utility function. The actors seek to do
this, given the constraints they are exposed to during the game. Essentially, the assumption
implies, that actors behave rationally13. Traditional economic theories have in recent years
adopted game theories simplicity of assumptions. The paradigms of game theory and traditional
theories have been converging, which is suspected to be the reason of the latest success of game
theory.
The basic idea of game theory is to analyze how players determine their optimal behavior in a
given game.
The first condition that has to be met to use game theory is, that the players strategies must have
an effect on other players strategies. And all players must have an understanding of that relation.
In order to analyze the stock market as a game, this raises a problem, which will be discussed later.
A game consists of players, actions, payoffs and information. All together they form the rules of
the game. The principle of game theory is to take an economic situation and model it, as a game.
The model assigns pay off functions and strategies to each player, and analyzes what happens in
the equilibrium, when every player chooses the strategy, that maximizes his utility pay off.
First of all, the players need to be defined.
13
It can be argued that any goal can behave as an instrument to qualify any behavior as rational. However, it is truly
believed that maximization of utility is the purest form of a goal and accordingly the behavior to reach it is rational.
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A player in a game is an individual, who makes decisions in order to maximize his utility. This,
however, has an exception. In order to structure a game, where payoffs are dependent on the
state of the world, it is appropriate to create a pseudo-player, normally called nature. This player
can given specific probabilities, randomly decide the state of the world. The move of nature can be
either before the players move, or after the players move. If nature moves before the players, the
concept of the Harsanyi transformation and Baye’s rule becomes relevant.
Three types of information exist in games. The first two types are information about natures
move, and information about other players’ moves. Both are highly influential to which strategies
the players will follow. The third type of information is common knowledge which also has an
important role in the fundamental construction of the game. Common knowledge is the term used
to describe information that are known to all players, and that all players know that all players
know, and that all players know that all players knows that all players know... (ad infinitum).
Common knowledge is the information which the players share at the beginning of the game.
Including beliefs concerning the state of the world and the probabilities assigned to natures
moves.
Common knowledge is often named as common priors. This expression will be used further on in
this paper, because the game analyzed involves updating these common prior beliefs.
Information about nature are said to be certain, if nature moves before the players. In these cases
the players are able to adapt their actions to the moves of nature. If the nature moves after the
players, information is said to be uncertain. The uncertainty reflects in the pay off functions.
Normally, players select strategies in order to maximize their utility pay off. When natures moves
are uncertain, the pay off becomes uncertain, which makes it difficult for players to chose the
optimal strategy. This problem can be overcome by rationalizing that, when pay off is uncertain,
players will chose the strategies, which maximizes their expected utility. In these cases, players are
said to have Von Neumann-Morgenstern utility functions14.
14
Games and information, p. 48
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Information about natures move is, however, found to be even more complicated. When players
move after nature, a situation can arise, where natures move is unobserved by at least one of the
players. These cases are called games of incomplete information. A special type of incomplete
information game, is a game with asymmetrical information. In such games, some players hold
private information, which gives them advantages to other players. Private information can
change players probability distributions of natures moves, and hence change his pay off function
and strategy choice.
The pay off function and actions are very closely related. Any strategy by any player has a certain
pay off linked to it. The pay off function is a term used to describe a players pay off, with every
players strategy choices being an endogenous variable of the function.
An equilibrium of a game is the output of the model. A game has an equilibrium for any
combination of strategies by the players.
The most interesting equilibrium is found, where the players are following optimal behavior.
Determining the equilibrium of optimal behavior can be done in a number of different ways:
-
Best worst-payoff: This type of equilibrium is found by letting players choose strategies
that maximize the worst payoff depending on the actions of other players. This is often
found to be pessimistic, and to risk averse to be representative of rational players in reallife games.
-
Dominated strategies: This technique is rarely used, since many games cannot be solved by
this method. Games solved with this method are typically rather simple. A well known
example is the Prisoners Dilemma15. The method can be used if there exists a dominating
strategy. A dominating strategy is better than any other strategy, no matter what strategy
other players choose. A dominant strategy equilibrium is found, where all players choose a
strategy combination that dominates all other possible strategies.
This implies, that the method cannot be used, if the optimal strategy for a player is
dependent on the strategy of other players
15
Games and information, s 20
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-
Iterated dominance: Equilibrium is found by sequentially excluding weakly dominated
strategies, until only one possible strategy is left for each player. Weakly dominant
strategies is characterized by being possible better, but never worse than other possible
strategies. This procedure is also known as backward induction.
-
Nash equilibrium: The idea is, that if all players choose a best response to the choice they
expect other players to take, they will reach a situation, where none of the players can gain
utility from individually altering their strategy. This strategy combination is known as a
Nash equilibrium. This definition implies that any dominant strategy is also a Nash
equilibrium.
The equilibrium found is the solution of the game. Hence, it forecasts the behavior of the players
in the game and the following output.
The certain equilibrium in which all players have chosen the optimal strategy, in order to maximize
their own utility, given the actions of other players, is called a Nash equilibrium.
A Nash equilibrium exists, where none of the players can individually reach higher pay off by
altering their strategy. This implies that a Nash equilibrium can be considered to be quite stable.
Since no players can further maximize their utility by changing strategy, you can expect players to
keep to their strategies. A problem arises, however, if there is more than one Nash Equilibrium in
the game. Therefore, it is important in the analysis of a game’s outcome to evaluate, if the
equilibrium found is unique.
We will later discuss the propositions of a Perfect Bayesian Nash Equilibrium.
A lot of other properties of games, however, need to be explained as well. For instance, it is
important to point out, whether or not coordination is allowed in the game. Logically coordination
will in many cases improve the equilibrium of a game. However, in real life coordination among
players is rare. In many cases, it is the lack of coordination that is the reason why some games
reach such unfortunate equilibriums, even though players have acted rationally with the object of
maximizing utility. Many results of game theoretic analysis are therefore used as a starting point
to solve economic problems by making players in non-cooperative games cooperate.
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Another important aspect is the order of plays. It is essential to determine, if the game should be
modeled with simultaneous or sequential moves. If the game includes sequential moves, it means
that players actions are separated in time. This gives an additional implication of information
which has been briefly mentioned previously. Players acting late in the game are able to benefit
from knowing previous actors strategies. In that sense, they have information presented, that the
first movers did not have. Knowing previous moves from other players in the game has a signaling
effect. This effect can be used both by the first mover and the late mover. First movers can expect
late movers to follow the strategy leader, and late movers can exploit the information given by the
first movers, and thereby reconfigure the probabilities assigned to their pay offs.
The principles of game theory can, by a few configurations, be applied very effectively to a price
bubble situation. This enables economists to analyze players behavior.
The configuration of the basic assumptions of game theory, and the definition of the rules of the
game will be specified in the following chapter.
4. Game theory in price bubbles
Applying game theory to the situation of price bubbles can be done in several ways. One of the
earliest, and maybe most interesting applications, were John Maynard Keynes Beauty Contest
Game16.
Keynes compared the challenges of an investor in the stock market to the challenges facing
ordinary people in the 50’s popular beauty contest quizzes. The beauty contest was a popular quiz
concept from the 50’s, where participants would win a prize, if they were able to pick the model
which would be the most popular, and hence get the most votes.
Thinking about this challenge seems at first very straight forward but given it more thought reveals
the complexity of the game. When first facing the models of which one has to be chosen, an
individual would likely prefer one of the models above the others. Picking this model, the
individual hopes that the general population shares his tastes. However, it is more likely that he
picks a model which he believes would suit the general opinion. You can even argue, that the
individual participant would pick the model, which he believes that other believe will suit the
16
General theory of Employment Interest and Money, 1936
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general opinion. Keynes claims that the same principles apply to investors in the stock market. The
argumentation becomes clearer when using a numeric example.
Given an interval from [0 ; 100], the participant that comes closest to 2/3 of the average number
guessed will win a prize. Thinking of this game, it becomes clear that there exist weakly dominated
strategies. Picking the number 100 for instance, leaves no chance of a win (unless all other
participants pick the number 100, then it will be a draw). This is because 2/3 of 100 are 66,667 and
the winning number will never be higher than this. So picking any number above 66,667 is a
weakly dominated strategy which should be iterated. To a rational participant this consideration
leaves the interval [0 ; 66,667]. The individual participant can, however, assume that his
opponents behave equally rational and intelligent. They would like himself reduce their interval. If
the individual participant, expect this to be the case, he is once again presented with a range of
weakly dominated strategies. Like before he is able to eliminate strategies higher than 2/3 of
66,667 which is 44,444. The game would continue like this in multiple rounds until all weakly
dominated strategies are eliminated, and the optimal behavior of the participant is found. The
optimal behavior, in this case, is picking the number 0, since all other strategies are weakly
dominated.
The example demonstrates the concept of higher order beliefs. It is a term used to explain the
process of how players think about other players beliefs.
This concept is very important to grasp, because the winner of the game will be the player who is
one step ahead. Hence, one order of belief higher than the general participants. Let’s say, that one
of the participants actually goes through all the steps of eliminating the dominated strategies, and
picks the number zero. Even though he has behaved optimal in the game, his winning chances are
not very likely. This is because the general participants, most likely, only go through a few of these
steps.
Experimental research supports this hypothesis17.
Given that, the prize will be handed to the participant who comes closest to 2/3 of the average
number, the optimal-behaving participant probably won’t win.
This game shows an important feature of games. Even though players behave rationally optimal, it
might not benefit them in a game, where opponents do not behave rationally optimal. The same
17
Behavioral Game Theory: Thinking, Learning, and Teaching, Camerer, Ho, Chong, 2001
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goes in an asset price bubble in the stock market. A rational well informed investor might find that
the stock is over-priced and sell. If the general investor is less informed, he will probably stay
invested, and the bubble might even grow further. The rational investor will in this case loose a
profit even though he was right to sell.
The model presented later reveals how the optimal behavior of rational investors, knowing that
the environment includes a considerable amount of irrational investors, is to be one step ahead and win the game.
This feature draws attention to the concept of information which has previously been underscored
as a very important aspect of games. This is because, optimal behavior is dependent on which
information the player has at the time of action. Therefore, asymmetrical information among
players often leads to different actions.
In cases of asymmetrical information, it is possible for all players to act rationally but chose
different strategies. In the case of price bubbles in the stock market, there is asymmetrical
information between well informed investors (knowing of the bubble) and less informed investors
(not knowing of the bubble). This means, that even though everybody behaves rationally
considering their informational level, the well informed player can lose to less informed players.
One of the most important issues of applying game theory to price bubble situations is, how to
handle the number of players. First of all, the number of players in the stock markets are very
high, which makes it impossible to analyze actors behavior with simple tools like game trees.
Second of all, a common idea of the market place, is that there is no monopoly power by any
players. This means, that any individual strategy has no effect in the market, and thereby plays no
role in other players determination of their strategies. This is a major concern, since the
interdependencies of players strategies are the cornerstone of game theory.
Therefore, it must be realized that in a bubble, game players strategies actually will be
interdependent. This is also one of the basic assumptions of the Abreu-Brunnermeier model, that
will be explored later.
It is argued, that the bursting of bubbles is caused by an increasing amount of selling pressure.
Meaning that when the amount of investors, who have sequentially sold the stock, reach a certain
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level, it will take on an effect as a price signal, that the stock is overvalued. The bubble will
immediately burst.
The players strategies are interdependent, since the payoff function to an individual are
dependent on, whether other players buy or sell. As long as the selling pressure is below the
critical level, an informed investor can confidently stay invested until he feels the pressure, and
then he must sell before the burst. This is why the bubble game is also known as a timing game.
The selling strategy must be coordinated with the opponents in the market.
5. The rules of the game:
The initial move is made by nature which decides when and if the bubble situation begins.
Additionally, it decides which types of players become aware.
The players in the game consist of all rational investors in the market. Irrational investors are also
important actors, but not active players. They serve as noise traders which mean that the rational
investors are unable to observe which actions previous players have made.
An important issue for modeling games in the stock market is the large number of players.
Normally, an analysis of stock market behavior is grounded on the assumption that no individual
will have strong enough market power to influence the prices and thereby make investors
strategic decision interdependent.
By setting up an absorption limit for selling pressure, of k investors leaving the market, a
dependency among investors is created. This is because each investors payoff depends on the
bubble have bursted or not, and this is in turn depending on how many investors have sold out.
The information structure is quite complex. Natures move is initially unobserved by all players.
However, immediately after nature’s choice players will start to sequentially receive private
information, revealing natures move until all players are informed. The players do not know how
many other players are already informed.
The players can either take maximum long position or maximum short position, both limited by
their individual financial constraints. Furthermore, the game is dynamic which means that actions
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are taken simultaneously, in each sequence. The impact of the irrational environment is that
players becoming aware in later sequences is not able to detect the actions taken in prior
sequences.
The players strategic choices only influence each other at the critical point, where enough selling
pressure has been generated to burst the bubble. When this point is reached, the payoff to
investors changes from the pre-crash price (including the bubble effect) to the fundamental value.
6. The Abreu-Brunnermeier Model:
With the object of determining the optimal strategy for investors in the stock market, Abreu and
Brunnermeier have constructed a comprehensive model18. During the following presentation of
the model, several assumptions will be made. These assumptions and their implications on the
results will be discussed later.
The model starts of at a random point, in time where (𝑡 = 0). At this time, the price of the stock
equals the fundamental value. Both have previously developed at a rate equaling the risk free
interest rate denoted (𝑟). The risk free interest rate, symbolizes the rate in which a balanced stock
in the market would grow, if the fundamental value equaled the price. At the time 𝑡 = 0, it is
assumed that both the fundamental value and the price is 1$ per stock. From here, the price
begins to follow a new pattern determined by the function:
𝑃𝑡 = 𝑒 𝑔𝑡
,
Pre-crash price
The price does no longer follow the risk free interest rate, but is now determined by a new growth
rate (𝑔). If the new growth is higher than the risk free rate, the price will increase faster than
before. It is assumed that 𝑔 > 𝑟. The development of the price is now exponential and higher
than before.
18
Bubbles and Crashes, 2003
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The higher growth rate is the key to understand the establishment of the bubble. The higher
growth rate is, in the beginning, explained by good news about the new technology, which leads
to higher expectations to the future profit of the company, which leads to a higher stock price.
Until some random point in time, denoted (𝑡0 ), the increasing stock price is explained by an
additional increase in the fundamental value. After 𝑡0 only some part of the stock price is
explained by the fundamental value. This part is given by the function:
𝑃𝑣 = (1 − 𝛽(𝑡 − 𝑡0 ))𝑃𝑡
,
Post-crash price
This means essentially, that the price 𝑃𝑡 includes a fundamental value and a bubble component
which size is described by 𝛽(𝑡 – 𝑡0 ). It is assumed, that the bubble component is a continuous
increasing function of (𝑡 – 𝑡0 ).
It is an important assumption of this model, that the bubble eventually will burst of reasons not
included in the model – exogenous reasons. This means, that the bubble exists in a limited period,
and correspondingly has a maximum size 𝛽̅ . Hence:
𝛽(𝑡 − 𝑡0 ) : [0, 𝜏̅] ↦ [0, 𝛽̅ ]
So,(𝑡 – 𝑡0 ) taking on values between 0 and the maximum life-time of the bubble (𝜏̅) determines
the bubble size 𝛽, which is found in the interval [0, 𝛽]. The maximum size of the bubble is naturally
found, when the bubble reaches its maximum life-time.
The time 𝑡0 is randomly chosen by nature. It follows an exponential distribution with the
cumulative distribution function:
Φ(𝑡0 ) = 1 − 𝑒 −𝜆𝑡0
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So, nature picks randomly the time where the price departs from the fundamental value19. After
this time, the price development can only be explained by irrational behavioral traders, who
expect the growth rate of 𝑔 to continue in infinity.
Rational investors are, however, sophisticated enough to become aware of the bubble situation.
This happens sequentially and from 𝑡0 , an amount of 1⁄𝜂 investors will become aware in any
moment 𝑡 until the time 𝑡0 + 𝜂. At this point, all rational investors are aware of the bubble. In
principle the notation 𝜂, represents the heterogeneity of investors. Because of differences of
opinion, they do not interpret the signals of the stock market the same, which is why they become
aware at different points in time. Depending on when they become aware, they represent a type
of investor.
There are an infinite number of investor types.
Given nature’s choice of 𝑡0 , it is only the players, who become aware in the time interval between
[𝑡0 , 𝑡0 + 𝜂] that are active. The investors, that are not included in that interval, have other
perceptions of the stock market. These investors strategies are in general called irrational.
To protect the bubble from unaware investors attacks, the following is assumed:
𝜆
𝑔−𝑟
<
𝛽(𝜂𝑘)
1 − 𝑒 −𝜆𝜂𝑘
This assumption means, that the relationship between the intensity parameter (determines the
strength of the exponential distribution) and the probability that the critical amount of investors
are aware, must be less than the relationship between growth rate minus risk free interest rate,
and the bubble size at time 𝜂𝑘 20.
In short, the left hand side of the expression is a measure of the probability that the bubble will
burst. The right hand side is a measure of the costs compared to the benefits of attacking the
bubble. Hence, the expression says, that if the probability of a burst is smaller than the cost-
19
This is where the mispricing begins. The bubble situation is, however, not established yet. According to Abreu &
Brunnermeier the bubble starts at time 𝑡0 + 𝜂𝑘. Bubbles and Crashes p. 180, quote: “We label any persistent
mispricing beyond 𝑡0 + 𝜂𝑘 a bubble.”
20
The term will be explained later in more detail. In short it represents the amount of selling pressure needed to burst
the bubble.
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benefit ratio of attacking the bubble, at the time where the critical mass 𝑘 investors become
aware, they will hold onto their stocks.
From this expression, it is required that the difference between 𝑔 − 𝑟 is high enough, so investors
will have costs leaving the bubble. This will make them stay invested. Furthermore the intensity
parameter 𝜆 is low enough to make it sensible to stay invested.
The assumption makes sure that investors do not have interest in selling until after the critical
amount of investors are aware.
Following the fact, that 𝑡0 is randomly chosen by nature, the investor do not know, if he is the first
to know or the last to know. He can only view the market from his individual perspective. We
denote the investor, who becomes aware of the bubble at time 𝑡𝑖 for 𝑡𝑖 .
In order to estimate when the bubble burst, a rational investor will have to determine the time
where the bubble began; 𝑡0 . For investor 𝑡𝑖 , 𝑡0 must be in the interval given by [𝑡𝑖 − 𝜂, 𝑡𝑖 ]. The
interval begins in time 𝑡𝑖 − 𝜂, where he is the last to become aware, and it finishes in time 𝑡𝑖 ,
where he is the first. The distribution of 𝑡0 for investor 𝑡𝑖 is given by:
Φ(t 0 |t i ) =
eλη −eλ(ti−t0 )
eλη − 1
The investor, who is the last possible to become aware from ti’s perspective, is denoted 𝑡𝑘 = 𝑡𝑖 +
𝜂.
The key in this model is, that it is the accumulated selling pressure, which finally burst the bubble.
The point in time, where enough rational investors have become aware to burst the bubble, is in
𝑡0 + 𝜂𝑘. This means, that when 𝑘 amount of investors know of the bubble, a coordinated attack
on the bubble will generate enough pressure to burst it. The non-coordination feature of the game
means that attacking the bubble at this point is not possible.
The action space for investors is described as the continuum interval [0,1]. 0 reflects the
maximum long position and 1 the maximum short position. Correspondingly, the higher you get in
the interval, the higher the selling pressure will be from the individual investor. The selling
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pressure of ti at time t is denoted 𝜎(𝑡, 𝑡𝑖 ). Likewise; [1 − 𝜎(𝑡, 𝑡𝑖 )] is a player’s stock holding. The
strategy profile for an individual investor is:
𝜎 ∶ [0, ∞] × [0, ∞] ↦ [0,1]
This means, that the variables 𝑡 (time) and 𝑡𝑖 (investor type) belong to infinite intervals.
Measurability requirements means, that only strategy profiles, that imply a measureable function
of 𝜎(𝑡,∗) is included in the model21.
The aggregate selling pressure of all investors is given by the following defined integral:
𝑚𝑖𝑛{𝑡,𝑡0 +𝜂}
𝑠(𝑡, 𝑡0 ) = ∫𝑡
0
𝜎(𝑡, 𝑡𝑖 ) 𝑑𝑡𝑖 ,
𝑡 ≥ 𝑡0
The aggregate selling pressure at time 𝑡, is the amount of stocks sold by all individual investors
from t0 until current time (𝑡) or the time 𝑡0 + 𝜂, where all rational investors are informed –
depending on which time comes first. When all rational investors are informed, the selling
pressure is at its maximum.
Knowing that the bubble only will be able to burst, when selling pressure becomes higher than or
equal to 𝑘, the bubbles bursting time can be expressed by:
𝑇 ∗ (𝑡0 ) = inf{𝑡|𝑠(𝑡, 𝑡0 ) ≥ 𝑘
𝑜𝑟
𝑡 = 𝑡0 + 𝜏̅}
This expression says that for a realized 𝑡0 , the bubble will burst either by the aggregate selling
pressure or when the bubble exceeds its maximum lifetime. Whichever comes first. This means
that it is the realization of 𝑡0 and the critical amount of investors 𝑘 that determines the bursting
time. A rational investor, that becomes aware of the bubble at time 𝑡𝑖 , will therefore use his
beliefs about 𝑡0 to determine the bursting time:
Π(𝑡|𝑡𝑖 ) = ∫𝑇 ∗(𝑡
0 )< 𝑡
21
𝑑Φ(t 0 |t i )
𝜎(𝑡,∗) is not always measurable in 𝑡𝑖 .
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As described investor 𝑡𝑖 ’s belief about the distribution of 𝑡0 is given by Φ. The bursting time is the
beginning of the integral where; 𝑡 > 𝑇 ∗ (𝑡0 ). Hence, from 𝑡𝑖 ’s perspective, the bubble will burst
when the time reaches his own belief about the bursting time, determined by his own belief about
𝑡0 .
Investors payoffs depend on the stock prices, minus transaction costs. The stock price is either
given by the pre-crash price or the post-crash price, depending on the time of buying and selling22.
Hence, the expected price is:
(1 − 𝛼)𝑃𝑡 + 𝛼(1 − 𝛽(𝑡 − 𝑡0 ))𝑃𝑡
The variable α reflects the selling pressure. If the selling pressure is higher than 𝑘, then 𝛼 > 0.
This means, that the execution price is determined by the post-crash price. The opposite happens,
if selling pressure is below 𝑘, then 𝛼 = 0.
The transaction costs play a role every time an investor makes a trade. The model seeks to
eliminate the influence of the transaction costs on equilibrium by making the following
assumptions. Transaction costs are high enough to restrict the number of trades to a finite amount
and low enough, not to restrict investors from selling on a bubble. Additionally, it is assumed that
the transaction costs are constant through the time period.
A general payoff function for a random investor 𝑡𝑖 , includes many complex relations. But it all
comes down to the expected value of the traders stock holding, given the strategy chosen.
However, the mathematical interpretation23 of the general function is not necessary to grasp the
idea of the model – or equilibrium behavior. Therefore, it makes more sense to describe the
payoff function in a special case:
22
If an investor trades exactly at the bursting time, the order will be generated at the pre-crash price until the
aggregated selling pressure exceeds k. At this point only the first randomly picked order will go with the pre-crash
price and the rest will be executed at post-crash price.
23
Bubbles and Crashes, 2003, appendix A
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The payoff of investor 𝑡𝑖 who is fully invested in the market until he sells the entire holding at time
𝑡 is given by:
𝑡
∫𝑡 𝑒 −𝑟𝑠 (1 − 𝛽(𝑠 − 𝑇 ∗−1 (𝑠))) 𝑝(𝑠) 𝑑Π(𝑠|𝑡𝑖 ) + 𝑒 −𝑟𝑡 𝑝(𝑡)(1 − Π(𝑡|𝑡𝑖 )) − 𝑐
𝑖
Furthermore, it is assumed that 𝑡𝑖 stays out of the market until the bubble crashes.
Interpreting this mathematical expression, it comes to show that the payoff is given by the area
between time of awareness (𝑡𝑖 ) and the selling time (𝑡). Hence, the payoff is generated by the
increase in stock prices during the period 𝑡𝑖 rides the bubble.
The length of the period 𝑡𝑖 rides the bubble is furthermore depending on his beliefs about 𝑡0 , the
bursting date and the size of the selling pressure.
6.1 Game theoretical background:
Essentially, the game is one of incomplete information. Investors are not able to define their
payoff, because it depends on whether the stock market is in a bubble or not. Traditional game
theoretic approaches of handling this issue, is to use Harsanyis transformation principles, and
make it a game of complete, but imperfect information.
Additionally, the game can be defined as a dynamic game because of the sequential aspects. It
should be stressed, that private information is not signaled by players moves, because the
environment is irrational.
The Harsanyi transformation24 has been used to restructure this game from a game of incomplete
information to a game of imperfect, but complete information25. The principle of the
transformation is to add a move by nature that decides the state of the world.
24
Games and information, s. 51
25
There exists some confusion about these definitions. Originally games of incomplete information could not be
solved which is why they are transformed to games with complete but imperfect information. Basically this
terminology has been neglected and now people just refer to transformed games, as games with incomplete
information.
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The random choice of 𝑡0 , with defined distribution function, should be considered as a move by
nature. It defines the state of the world – if the market is in a bubble or not. This move is initially
unobserved by the players, even though some investors immediately become aware.
The reason why the transformation has to be made is, that players would not be able to define
their payoffs, unless there could be assigned a probability to which state of the world nature has
chosen.
It is essential in the game that the probability distribution of 𝑡0 is shared by all rational investors.
This means, that the probability of being in a bubble is a shared prior belief. This is what is meant
by common priors, and is also called the Harsanyi doctrine, since it is necessary to hold in order to
apply the Harsanyi transformation.
The prior beliefs are, however, allowed to change when investors receive private information by
which they become aware of the price bubble situation. This means, that a situation arises among
investors in which they hold asymmetrical information. The observation of the state of the world is
considered private information, and aware investors correspondingly update their beliefs about
the time 𝑡0 , according to Bayes rule26.
Even though they now know, that they are in a bubble situation, they do not know how many
other investors are aware of this. This means, that their beliefs about how many other investors
know, are included in their payoff function.
The game is viewed from an investor’s individual perspective, assuming that every rational
investor will behave the same way. Therefore, the solution to the game describes the optimal
behavior for any individual investor. The solution is defined as a Perfect Bayesian Nash
Equilibrium. This means, that the optimal behavior is the best response of an individual investor
given that other rational investors follow their optimal behavior.
The definition of the equilibrium is further commented later in the analysis.
6.2 Preliminary analysis:
This section provides some assumptions and propositions that will help define optimal behavior
and thus, equilibrium.
26
Games and information, s. 54
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Definition 1: The equilibrium of this game is defined as a Perfect Bayesian Nash Equilibrium.
As previously mentioned, a Nash equilibrium implies that every player chooses the strategy which
is the best response to what other players choose. The reason why it is Bayesian, is because they
have rationally updated their beliefs, following Baye’s rule during the game.
To have a Nash equilibrium therefore means, that any rational investors will choose the strategy
which leads to the highest expected payoff, given other rational investors will do the same. In
equilibrium all rational investors will actually do the same.
At equilibrium in Abreu & Brunnermeiers bubble game, this implies that; an investor whose stock
holding is less than maximum, can expect that all other bubble-knowing investors who became
aware before him, also hold less than their maximum.
Lemma 1:
𝜎(𝑡, 𝑡𝑖 ) ∈ {0,1} ∀ 𝑡, 𝑡𝑖
This assumption means that the action space per period is only allowed to take on values 0 or 1.
The reason for this is to simplify the model. By only allowing investors to be fully invested (1) or
completely out of the market (0), the aggregate selling pressure can be found as the amount of
investors who have left the market.
The elimination of partial buying and selling, means that a rational investor knowing of the bubble
will not be able to gradually sell the stock to reduce risk. This implies that investors in the model
are risk-neutral. This is clearly not realistic, but as explained by Abreu & Brunnermeier it has not
any impact on the results in the model. If partial trading is allowed, it would only make it much
more complicated to determine the aggregate selling pressure. Hence, the implication that this
assumption has on the model is, that the aggregate selling pressure might exceed the critical limit
(𝑘) at another point in time. The general theory of the optimal behavior of investors would be the
same.
Corollary 1:
𝜎(𝑡, 𝑡𝑖 ) = 1 ↦ 𝜎(𝑡, 𝑡𝑗 ) = 1 ∀ 𝑡𝑗 ≤ 𝑡𝑖
Out of the market
𝜎(𝑡, 𝑡𝑖 ) = 0 ↦ 𝜎(𝑡, 𝑡𝑗 ) = 0 ∀ 𝑡𝑗 ≥ 𝑡𝑖
Fully invested
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This says that if investor 𝑡𝑖 has become aware of the bubble prior to investor 𝑡𝑖 . And if 𝑡𝑖 is
completely out of the market then 𝑡𝑗 is also out of the market (sold out prior to 𝑡𝑖 ). Likewise, if
𝑡𝑖 became aware prior to 𝑡𝑗 , then 𝑡𝑗 will be fully invested when 𝑡𝑖 is fully invested.
This corollary is put together by definition 1 and lemma 1. Definition 1 states that the investor who
becomes aware first will act first, and lemma 1 states that all investors must be either fully
invested or out of the market.
It is also known as the cut-off property and is a direct implication of the sequential awareness.
Definition 2:
𝑇(𝑡𝑖 ) = inf{𝑡|𝜎(𝑡, 𝑡𝑖 ) > 0}
𝑇(𝑡𝑖 ) expresses the first time investor 𝑡𝑖 sells stocks. It is found, at the greatest lower bound of
time 𝑡27, where the strategic decision 𝜎(𝑡, 𝑡𝑖 ) takes on a value higher than zero. With lemma 1 in
mind, the strategic decision will take on the value 1 and 𝑡𝑖 will leave the market.
Corollary 2:
𝑇 ∗ (𝑡0 ) = min{𝑇(𝑡0 + 𝜂𝑘 , 𝑡0 + 𝜏̅)}
By Corollary 1, we know that when investor 𝑡0 + 𝜂𝑘 sells his stocks all rational investors prior to
him have already sold. This means, that at the time where 𝑡0 + 𝜂𝑘 sells, enough selling pressure
has been generated to burst the bubble. So, the bubble will burst at this moment, unless it has
already bursted for exogenuous reasons at the time 𝑡0 + 𝜏̅.
Definition 3:
𝑡0𝑠𝑢𝑝𝑝 (𝑡𝑖 )
The function expresses the lower bound of support for investor 𝑡𝑖 ’s configurated belief about 𝑡0 at
the time of selling (𝑇(𝑡𝑖 )).
Lemma 2:
𝑡0𝑠𝑢𝑝𝑝 (𝑡𝑖 ) ≥ 𝑡𝑖 − 𝜂𝑘
𝑡0 = 𝑡𝑖 − 𝜂𝑘 expresses the situation where 𝑡𝑖 is the investor who generates the critical amount of
selling pressure to burst the bubble. This lemma states, that the lower bound of investor 𝑡𝑖 ’s
beliefs about t0 in time 𝑇(𝑡𝑖 ) always will be higher or equal to 𝑡𝑖 − 𝜂𝑘. This implies, that at the
time 𝑇(𝑡𝑖 ) where 𝑡𝑖 sells out stocks, a maximum of 𝑘 investors have become aware – and sold –
27
The first possible 𝑡, where 𝜎(𝑡, 𝑡𝑖 ) takes on a value higher than 0.
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prior to 𝑡𝑖 . This ensures, that investor 𝑡𝑖 sells out prior to the burst (given his individual belief
about 𝑡0 ). Hence, it is called the preemption lemma.
Lemma 3-4-5:
They are not found to be of significant importance to this paper. However, it should be mentioned
that these lemmas prove that the function for the bursting time 𝑇 ∗ , and the inverse 𝑇 ∗−1 is strictly
increasing and continuous, and that the function of the selling time 𝑇 is also continuous.
This ensures the mathematical liability of the analysis.
Lemma 6:
Pr[𝑇 ∗−1 (𝑇(𝑡𝑖 ))|𝑡𝑖 , 𝐵 𝑐 (𝑇(𝑡𝑖 ))] = 0
,
𝑡𝑖 > 0
This lemma implies, that the probability of the bubble bursting exactly at the time where investor
𝑡𝑖 sells is equal to zero. The probability, that the bubble will burst exactly at the time 𝑡𝑖 sells is
given by 𝑇 ∗−1 (𝑇(𝑡𝑖 )). 𝐵 𝑐 (𝑇(𝑡𝑖 )) expresses the probability that the bubble has not yet bursted at
the time 𝑡𝑖 sells.
Proposition 1:
𝑡 ≥ 𝑇(𝑡𝑖 )
If the previous assumptions hold, rational investors will choose maximum short position at all
times, later than or equal to the first selling time.
This states that in equilibrium investor 𝑡𝑖 , will apply a ‘trigger strategy’ which means that after
leaving the market at time 𝑇(𝑡𝑖 ), he will stay out until the bubble bursts.
The proposition implies that the optimal strategy for investors in the bubble is a trigger strategy.
To prove this, imagine an equilibrium where 𝑡𝑖 re-enters the market after 𝑇(𝑡𝑖 ). Since transaction
costs are high enough to restrict investors from trading constantly, 𝑡𝑖 will stay out of the market in
a certain time period. Lets say, that the time period ends, when investor 𝑡𝑖 + 𝜀 (𝜀 > 0 and
expresses the amount of investors that have left the market, while 𝑡𝑖 was out) leaves the market.
At this point 𝑡𝑖 wishes to re-enter the market. However, given corollary 1, we know that if 𝑡𝑖 is fully
invested all investors that became aware after him will also be fully invested. This means, that 𝑡𝑖
cannot re-enter the market until 𝑡𝑖 + 𝜀 has re-entered. Likewise 𝑡𝑖 + 𝜀 cannot re-enter until 𝑡𝑖 +
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2𝜀 has re-entered and so on. This will essentially postpone 𝑡𝑖 ’s reentering until the bubble bursts
by either selling pressure or exogenous reasons. Hence, 𝑡𝑖 is following a trigger strategy.
As a consequence the analysis of equilibrium can be limited to trigger strategies. And it now
becomes clear why the general payoff function was unimportant. Instead we can focus attention
on the payoff function presented on page 15.
The function has been slightly altered; 𝜋(𝑠|𝑡𝑖 ) expresses the conditional density of Π(𝑡|𝑡𝑖 ), which
is 𝑡𝑖 ’s conditional cumulative distribution function)
𝑡
∫𝑡 𝑒 −𝑟𝑠 (1 − 𝛽(𝑠 − 𝑇 ∗−1 (𝑠))) 𝑝(𝑠) 𝜋(𝑠|𝑡𝑖 ) 𝑑𝑠 + 𝑒 −𝑟𝑡 𝑝(𝑡)(1 − Π(𝑡|𝑡𝑖 )) − 𝑐:
𝑖
Lemma 7:
The hazard rate is given by the expression:
ℎ(𝑡|𝑡𝑖 ) =
𝜋(𝑡|𝑡𝑖 )
1 − Π(𝑡|𝑡𝑖 )
This says that the hazard rate of the bubble bursting at the time t for investor 𝑡𝑖 is given by the
ratio of the conditional density function for 𝑡𝑖 ’s belief about the bursting time, and the conditional
probability that it is not bursting. The hazard rate is a measure of the individual investors risk
assignment to the bubble bursting.
Lemma 7 gives the ‘sell out condition’, which is provided by differentiating the above payoff
function with respect to 𝑡:
ℎ(𝑡|𝑡𝑖 ) <
𝑔−𝑟
𝛽(𝑡 − 𝑇 ∗−1 (𝑡))
If 𝑡𝑖 ’s hazard rate is smaller than the cost-benefit ratio of attacking the bubble at time 𝑡, 𝑡𝑖 will
keep holding the stocks. The benefit of attacking the bubble is determined by the bubble size at
the time of attack. The costs of being out of the market, if the bubble does not burst at the time 𝑡
is given by the difference between the bubble growth rate and the risk free interest rate.
Interpreting this means, that if trader 𝑡𝑖 believes that the risk of the bubble bursting at time 𝑡 is
smaller than the cost-benefit ratio of being out of the market at time 𝑡, then he will stay invested.
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This can be reversed, meaning that if 𝑡𝑖 ’s hazard rate is higher than the cost-benefits of attacking
the bubble, then he will leave the market.
Hence, this is called the ‘sell out condition’. It determines the investors optimal selling time in both
types of crashes.
6.3 Identifying the optimal behavior of investors:
The model includes 2 types of crashes – by selling pressure or exogenous reasons. To give a
wholesome determination of optimal behavior, this paper describes the reasoning and the
relation behind the behavior in both crash types.
In order to define equilibrium it is assumed that investor 𝑡𝑖 expects the bubble to burst at the time
𝑡0 + 𝜁. Given that 𝑡𝑖 ’s belief about 𝑡0 is distributed by:
Φ(t 0|t i ) =
eλη − eλ(ti −t0 )
eλη − 1
We are able to directly derive his belief about the bursting date. From 𝑡𝑖 ’s perspective his
distribution of the bursting date 𝑡𝑖 + 𝜏 becomes:
Π(𝑡𝑖 + 𝜏|𝑡𝑖 ) =
eλη − eλ(ζ−τ)
eλη − 1
The expression ζ represents the period of time between 𝑡0 and the expected bursting date.
Withdrawing τ from this gives 𝑡𝑖 .
Furthermore, by differentiating the payoff function considering these beliefs, the hazard rate
becomes:
𝜆
ℎ(𝑡𝑖 + 𝜏|𝑡𝑖 ) = 1−𝑒 −𝜆(𝜁−𝜏)
This reflects 𝑡𝑖 ′𝑠 hazard rate that the bubble will burst at 𝑡0 + 𝜁, after he have ridden the bubble 𝜏
period after becoming aware.
This expression of the hazard rate is useable in determining equilibrium since the term 𝜁, is easily
replaced with, whatever expectation investor 𝑡𝑖 has.
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6.3.1 Exogenous crashes:
𝜆
1−𝑒 −𝜆𝜂𝑘
≤
(𝑔−𝑟)
̅
𝛽
In this case the bubble will crash before enough selling pressure have been generated.
Interpreting the expression gives the following. 𝑔 – 𝑟 / 𝛽̅ expresses the cost-benefit ratio from
𝜆
attacking the bubble at its maximum size. 1−𝑒 −𝜆𝜂𝑘 represents the hazard rate at the time 𝑡𝑖 beliefs
that the critical amount of investors will sell. 𝑡𝑖 ’s belief about this point in time derived from his
beliefs about 𝑡0 and consequently his beliefs about the bursting date.
If the dispersion of opinion 𝜂 is sufficiently large, the hazard rate at this point in time will be lower
than the cost-benefit ratio. This means that 𝑡𝑖 will delay selling out until the sell-out condition is
met.
This delay effectively means that the bubble will burst for exogenous reasons, since it together
with the large 𝜂 means that 𝑡0 + 𝜏 1 + 𝜂𝑘 > 𝑡0 + 𝜏̅.
Even though enough investors are informed of the bubble to burst it, the condition above makes it
more profitable to ride it further, and let it burst for exogenous reasons.
In equilibrium all investors follow that behavior and endogenous selling pressure will not have
influence.
This means that the optimal behavior is to ride the bubble in a period, and sell prior to the burst.
This explanation is proved by the following section.
Definition 4:
𝜏(𝑡𝑖 ) = 𝑇(𝑡𝑖 ) − 𝑡𝑖
The period of time investor 𝑡𝑖 chooses to ride the bubble (𝜏) is given by the difference between
the time of selling and the point of awareness.
Recapping from earlier, every outcome of 𝑡0 will be exposed of an exogenous crash when the
bubble reaches its maximum size of 𝛽̅ at the time 𝑡0 + 𝜏̅. Every informed investor knows this, and
knows that to sell out before the crash the selling period are limited to the interval [𝑡𝑖 , 𝑡𝑖 + 𝜏̅].
However, there is no reason why the bubble will not crash before 𝑡𝑖 + 𝜏̅. Either by endogenous or
exogenous reasons. It is none the less obvious that the riding time 𝜏(𝑡𝑖 ) < 𝜏̅ in equilibrium.
Proposition 2:
1
𝑔−𝑟
𝜏 1 = 𝜏̅ − 𝜆 ln (𝑔−𝑟−𝜆𝛽̅) < 𝜏̅
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Proposition 2 claims that there is a unique symmetric trading equilibrium at 𝜏 1 which is smaller
than 𝜏̅ (see figure 1).
If investor 𝑡𝑖 believe that the bubble will burst at 𝑡0 + 𝜁. This means that for 𝑡𝑖 the distribution of
the bursting date (𝑡𝑖 + 𝜏) is given by:
Π(𝑡𝑖 + 𝜏|𝑡𝑖 ) =
eλη −eλ(ζ−τ)
eλη − 1
This belief is derived from 𝑡𝑖 ’s belief about 𝑡0 .
Differentiating the payoff function including these beliefs gives an expression of the hazard rate:
𝜆
ℎ(𝑡𝑖 + 𝜏|𝑡𝑖 ) = 1−𝑒 −𝜆(𝜁−𝜏)
Figure 1: Identifying the optimal strategy in exogenous crashes.
Kilde: Econometrica; Vol. 71 No. 1; Page 188; Bubbles and Crashes; Dilip Abreu & Markus K. Brunnermeier; 2003;
ℎ=
𝜆
1 − 𝑒 −𝜆(𝜏̅−𝜏)
𝑔−𝑟
𝛽̅
𝜆
1 − 𝑒 −𝜆𝜂𝑘
𝜏
𝜏̅ − 𝜂
𝜏̅ − 𝜂𝑘
𝜏1
𝜏̅
The hazard rate and cost-benefit ratio is depicted as a function of 𝜏. The cost-benefit ratio is
independent of 𝜏, because the time of the exogenous crash is independent of 𝜏. This makes the
benefits of attacking the bubble a constant equaling the size at the bubbles maximum lifetime.
This is why the cost-benefit ratio is depicted as a constant. Naturally the hazard rate is strictly
increasing in an exponential development. The longer the investor chooses to ride the bubble the
higher is the risk.
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As stated earlier, equilibrium is found where the hazard rate equals the cost-benefit ratio. If 𝑡𝑖
believes that the bubble is bursting for exogenous reasons the term 𝜁 in the hazard rate is
substituted with 𝜏̅.
This gives the final depiction of the hazard rate. It represents the probability from 𝑡𝑖 ’s perspective
that the bubble will burst at each moment in time.
The equilibrium is denoted 𝜏 1 . If 𝜏 is higher than 𝜏 1 the risk of the bubble bursting is higher than
the costs compared to the benefits of attacking – hence it is optimal for investors to be at the
maximal short position. If 𝜏 is lower than 𝜏 1 the reverse is optimal since the risk of the bubble
bursting is lower than the cost-benefit ratio of attacking it. This is given by the sell-out condition.
Therefore the optimal strategy is the trigger strategy of selling at 𝜏 1 . This means, that the bubble
will crash at 𝑡0 + 𝜏̅, since the size of 𝜂 makes 𝑡0 + 𝜏 1 + 𝜂𝑘 > 𝑡0 + 𝜏̅.
This defines the symmetric equilibrium and the optimal behavior of investors. A symmetrical
equilibrium means that all players will apply this strategy.
To prove that the equilibrium 𝜏 1 is unique we look at a possible alternative equilibrium. The
alternative optimal riding time is denoted 𝜏(𝑡𝑗 ). This riding time is smaller than the previous
optimal riding time.
For investor 𝑡𝑗 the smallest possible value of 𝑡0 is still required to be higher than or equal to 𝑡𝑗 −
𝜂𝑘 because of the preemption lemma.
If 𝑡𝑗 ’s belief about the lower support of 𝑡0 is higher than 𝑡𝑗 − 𝜂𝑘, then investor 𝑡𝑗 will not only sell
out because of concerns of an exogenous crash. He will also be concerned of endogenous crashes.
Looking at figure 1, the sell-out condition will in this case be violated unless 𝜏(𝑡𝑗 ) = 𝜏 1 .
If 𝑡𝑗 ’s belief about the lower support of 𝑡0 is equal to 𝑡𝑗 − 𝜂𝑘, the sell-out condition will also be
violated. Since the riding time of 𝑡𝑗 is shorter than 𝑡𝑖 , his hazard rate cannot be higher than 𝑡𝑖 ’s.
Given the initial assumption, we also know that the hazard rate is less than the cost-benefit ratio
at the bubbles maximum lifetime.
This means that selling out prior to 𝜏 1 , is not optimal because the hazard rate is to low compared
to the cost-benefits of attacking it.
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To conclude the optimal strategy of investors is to stay with the bubble and sell out after riding the
bubble in the optimal period of r1 which is just prior to the exogenous burst given their beliefs
about 𝑡0 .
𝜆
6.3.2 Endogenous crashes:
1−𝑒 −𝜆𝜂𝑘
>
(𝑔−𝑟)
̅
𝛽
Given this assumption the bubble-crash will be caused by selling pressure. The equilibrium
behavior in this case is found by iteration of non-best response strategies. The principle is much
like ‘the beauty contest games’. Since the bubble burst at the time 𝑡0 + 𝜏̅ if no investors sell,
investor ti will seek to sell just prior to that moment (𝑡𝑖 + 𝜏 1 ). Given the assumption that selling
pressure will burst the bubble, the bubble will burst when the amount of selling pressure reaches
a critical level; 𝑡0 + 𝜏 1 + 𝜂𝑘. Given the initial assumption, the bursting time is now reduced which
means that investors will sell even earlier at time 𝜏 2 , (𝜏 1 > 𝜏 2 ). The backward induction process
will continue to reduce the time in which investors are able to ‘ride the bubble’ until it reaches 𝜏 ∗ .
Traditional analyzes of this bubble behavior logically eliminates the chance of developing a bubble.
However the process loses its effect gradually because the size of the bubble gets smaller as the
riding time is reduced. When the size of the bubble reduces, the benefits of attacking the bubble is
also reduced which means that investors will keep riding it.
Proposition 3:
𝜆
1−𝑒 −𝜆𝜂𝑘
>
(𝑔−𝑟)
̅
𝛽
Assuming that the hazard rate, when the critical amount of investors becomes aware, is higher
than the costs related to the benefits of attacking the bubble when it reaches its maximum lifetime. There will be a trading equilibrium where investors 𝑡𝑖 ≥ 𝜂𝑘 sell out at 𝜏 ∗ , after becoming
aware, and investors 𝑡𝑖 < 𝜂𝑘 sell out at 𝜂𝑘 + 𝜏 ∗ .
This means that in the symmetric equilibrium the bubble will burst at 𝑡0 + 𝜂𝑘+𝜏 ∗ . At this time the
bubble component will be an optimal fraction (𝛽 ∗ ) of the pre-crash price.
The equilibrium time of riding the bubble, 𝜏 ∗ is found by the following.
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𝜏 ∗ = 𝛽 −1 (
𝑔−𝑟
𝜆
1 − 𝑒−𝜆𝜂𝑘
)
Investor 𝑡𝑖 ≥ 𝜂𝑘 means that the investor has become aware of the bubble after the critical
amount of rational investors has become aware. However, the equilibrium behavior for this
investor (to maximize utility with the restriction of the hazard rate) is to sell 𝜏 ∗ periods after
becoming aware.
Investor 𝑡𝑖 < 𝜂𝑘 means that the investor has become aware of the bubble prior to the critical
mass which means that his optimal behavior is to sell at 𝜂𝑘 + 𝜏 ∗ in order to ride the bubble as well
as possible.
The time period 𝜏 is bounded by the interval; [0 , 𝜏̅ – 𝜂𝑘]. If 𝜏 is larger than 𝜏̅ – 𝜂𝑘 then the
bubble will burst for exogenous reasons before the investor leaves the market. Given these
considerations the bubble will burst at the time 𝑡0 + 𝜂𝑘 + 𝜏 = 𝑡0 + 𝜁 ; where ζ denotes 𝜂𝑘 + 𝜏.
Substituting this expression into the hazard rate we obtain the following:
𝜆/1– 𝑒 −𝜆(𝜁 – 𝑟)
→
𝜆/1– 𝑒 −𝜆(𝜂𝑘)
This gives the optimal hazard rate.
Equilibrium is presented in figure 2. The hazard rate is independent of the time period of riding the
bubble. This means that the hazard rate is found to be a constant.
The cost benefit ratio of, attacking the bubble are on the other hand dependent on 𝜏. Knowing
that the bubble 𝛽(𝜂𝑘 + 𝜏) is an increasing function, the benefits of attacking it, is also increasing
as 𝜏 increases. Equilibrium will be found when the benefits of attacking the bubble have increased
enough compared to the costs of attacking it, so that it equalizes the investors’ hazard rate.
This is the optimal time for leaving the market. As soon as the investors’ hazard rate is higher than
the costs compared to the benefits of attacking, the investor should sell in order to maximize his
utility. Hence, a rational investor will follow this behavior.
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Figure 2: Identifying optimal behavior in endogenous crashes.
Kilde: Econometrica; Vol. 71 No. 1; Page 191; Bubbles and Crashes; Dilip Abreu & Markus K. Brunnermeier; 2003;
𝑔−𝑟
̅
𝛽 (𝜂𝑘 + 𝜏)
ℎ∗ =
𝜆
1 − 𝑒 −𝜆𝜂𝑘
𝜏
𝜏
∗
Shows the hazard rate and the cost benefit ratio of attacking the bubble as a function of the riding
time 𝜏. The hazard rate is constant and the cost-benefit ratio of attack is decreasing as a
consequence of the increasing bubble size. Equilibrium behavior is found where the two
expressions equalize each other.
Since 𝜏 ∗ > 0 when the cost-benefit ratio of attacking is higher than the hazard rate, and 𝜏 ∗ < 𝜏̅ −
𝜂𝑘 when the hazard rate is higher than the cost-benefit ratio, equilibrium can be said to be
symmetric.
To show that equilibrium is also unique is very complicated.
First of all we should remember equilibrium from the exogenous crash 𝑡𝑖 + 𝜏 1 . This equilibrium
will affect the amount of selling pressure if the optimal riding time 𝜏 ∗ (𝑡𝑖 ) > 𝜏 1 for some 𝑡𝑖 .
It is therefore necessary to analyze the behavior of the investor which rides the bubble latest. This
investor is denoted 𝑡𝑗 .
From the preemption lemma we know that at the time 𝑡𝑗 sells ( 𝑇(𝑡𝑗 )), the latest possible time the
bubble originated is 𝑡𝑗 − 𝜂𝑘 where 𝑡𝑗 is the latest investor to become aware. Otherwise the bubble
would already have bursted.
More specifically the preemption lemma says: 𝑡0𝑠𝑢𝑝𝑝 (𝑡𝑗 ) ≥ 𝑡𝑗 − 𝜂𝑘.
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If 𝑡0𝑠𝑢𝑝𝑝 (𝑡𝑗 ) > 𝑡𝑗 − 𝜂𝑘, it would mean that 𝑡𝑗 ’s hazard rate at 𝑇(𝑡𝑗 ), is higher than the cost-benefits
of attack at the maximum bubble size. 𝑡𝑗 would therefore sell prior to 𝑇(𝑡𝑗 ).
If 𝑡0𝑠𝑢𝑝𝑝 (𝑡𝑗 ) = 𝑡𝑗 − 𝜂𝑘, the bubble would burst at 𝑡0 + 𝜂𝑘 + 𝜏(𝑡𝑗 ). Since 𝜏(𝑡𝑗 ) > 𝜏(𝑡𝑖 ) the
minimum hazard rate of 𝑡𝑗 is given by ℎ∗ = 𝜆⁄1 − 𝑒 −𝜆𝜂𝑘 . At the same time the benefits of
attacking the bubble will be higher at 𝜂𝑘 + 𝜏(𝑡𝑗 ) than at 𝜂𝑘 + 𝜏 1 . This means that the cost-benefit
of attack is lower than the hazard rate and 𝑡𝑗 should have sold. Therefore is the sellout condition
violated, and the equilibrium is unique since the bubble can only burst of endogenous selling
pressure. 𝜏 ∗ cannot be higher than 𝜏 1 .
Furthermore. For 𝑡𝑖 > 𝜂𝑘 multiple equilibria can arise since the optimal riding time is only
determined by 𝜏(𝑡𝑖 ). It is therefore proved that the minimum and maximum of 𝜏(𝑡𝑖 ) cannot both
comply with the sell-out condition.
For 𝑡𝑖 < 𝜂𝑘 equilibrium is also unique. Their selling time equals the selling time of the last
investors who got aware. Selling out prior to that is clearly not an advantage since they will not
optimize the payoff from the investment.
6.4 Optimal behavior:
What is important to remember in this game is that any active players (aware investors), is looking
at the price bubble from their own perspective. Because of sequential awareness investors will
also sequentially sell.
Looking at the results should be done separately for endogenous and exogenous crashes. The
reason for this is the fundamental assumption behind them.
Beginning with exogenous crashes, the consequences of the previous assumption is that they
delay selling out at the time in which they expect the critical amount of investors to burst the
bubble. Therefore they commit their focus to ride the bubble until the hazard rate of the bubble
bursting from exogenous reasons becomes higher than the costs-benefits of attacking it.
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This means that investors becoming aware early will be able to ride the bubble effectively while
investors becoming aware to late will ride the bubble through the crash.
An important issue of this model is to identify which factors determine which kind of crash the
bubble will go through. Recapping from earlier we know that the dispersion of opinion is
significant. Likewise the absorption limit of irrational investors also has influence. If 𝑘 is sufficiently
large then it will have the same effect as 𝜂.
In an endogenous crash the optimal behavior is to ride the bubble for 𝜏 ∗ periods. This means, that
given 𝑡𝑖 ’s belief about the time where the bubble began a bursting date is estimated.
Correspondingly a payoff function based on these beliefs and the strategic possibilities is
formulated. Through iteration of non-best response strategies equilibrium is found. In reality an
investor would therefore ride the bubble for a while until costs of attacking the bubble compared
to the benefit equals the hazard rate of the bubble bursting.
In a larger perspective this means that rational investors that have become aware before the
critical mass of investors would be able to effectively ride the bubble and sell out prior to the
burst. Investors becoming aware after the critical mass would also ride the bubble, however, it will
burst before they sell out.
To completely comprehend why it in some cases can be considered optimal behavior of rational
investors to ride the bubble through the burst we have to focus attention on the fundamental
assumption of rational behavior. That is to maximize profit of the investments while still
considering risks. The proofs showing that the equilibriums, both from exogenous and
endogenous crashes are symmetric and unique is necessary to implement results. That an
equilibrium is symmetric means that the behavior is considered optimal if all rational investors
follow it. That equilibrium is unique means that it is the only best-response strategy a rational
investor is able to follow in order to maximize utility.
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7. Assumptions:
The following assumptions are in general very interdependent. This means, that even though they
can be discussed separately, their implications on results might be more complex, because they
support each other.
Common priors: The assumption of common knowledge states all rational investors share the
same beliefs about the structure of the model. This includes the initial probability distribution of
𝑡0 , the role of irrational investors in the market and so on.
The biggest reason to raise questions about this assumption is, that the model at the same time
assumes heterogeneity among the investors. It seems unlikely, that the heterogeneity do not also
affect the beliefs prior to the game.
The assumption is however, necessary in order to use Harsanyi’s transformation principles.
Violation of the assumption of common priors would mean, that the investors would hold
asymmetric information prior to becoming aware of the bubble. In this context the defined
conditions of the Perfect Bayesian Nash Equilibrum will also be violated. Investor 𝑡𝑖 will not be
able to correctly assume, that investors who become aware later than 𝑡𝑖 , will be invested if he is
still invested himself.
Rational and irrational investors: Surely the assumption of rationality plays an important role for
game theory in general. In this context, since the model also includes irrational players, it becomes
even more important.
Discussing the implications of rationality for game theory as a whole, the starting point must
necessarily be the general assumption made by Morgenstern & Von Neumann28. It says, that to be
able to expect which actions players take, they must follow rational behavior with regards to
maximizing their utility function. Hence, the game theoretic definition of acting rationally is to
maximize utility.
28
Theory of games and economic behavior, 1994
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Moving into Abreu & Brunnermeiers model it is specified that maximum utility in this game is
given by the maximum profit of investments conditional on the hazard rate. Investors who follow
this behavior are what they call rational investors.
However, I personally believe that rationality is a rather fluent concept, depending on a specified
goal. In relation to this understanding, all actions can be justified as rational – including the
investment decisions made by so-called irrational investors in Abreu & Brunnermeiers model. This
is why, I find that the discussion of the rationality concept in this model, must be with relation to
the goal of the players.
Abreu & Brunnermeier ignore this perspective which is possibly done in order to simplify. Even so,
I think it is important to stress my belief that rational behavior is dependent on the goal, and that
the goal of investors in the stock market not always is to maximize profit conditioned on risk. For
example investors might also condition their wish to maximize profit on ethical principles. It can
even be argued, though a little naïve, that some investors wish to support a good idea by investing
in their stocks.
Not taking this perspective into account, implies that all investors in Abreu & Brunnermeiers
model take actions in order to maximize utility, given consideration to risk.
This means, that all investors, whose behavior do not seek to maximize their investment profit,
while at the same time keeping their risk aversion in mind is considered to be irrational and hence,
not active players. This makes sense, even though it raises the question of why individuals behave
in ways that do not maximize utility. And equally important, if irrational investors actually exist in
the market.
Fortunately, many investigations have been made on the subject.
Fischoff, Slovic and Liechtenstein’s29 experimental investigations argue that some investors
decisions are biased because of overconfidence, when assigning probabilities to positive events.
This leads to overly optimistic investment decisions, which in general terms leads to trend trading.
Trend trading essentially means, that decisions are based on trends. In a bubble situation with the
trend of stock prices increasing, the overly optimistic investors will choose to follow the trend with
29
Facts and fears, 1977
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the over-confident belief that prices will increase further. Delong, Shleifer, Summers and
Waldmann30 labels this as positive feedback traders.
Furthermore Schiller31 states that the amount of these irrational investors have been increasing.
He argues that a general social movement has made it popular to invest in financial markets. This
has invited a broad range of new and unsophisticated investors. This indicates that the assumption
of irrational investors being in the market is valid.
The central part of Abreu & Brunnermeiers assumption is, that the market power of these
irrational investors is high enough to effectively avoid rational investors to correct mispricing. This
constitutes what is known as the limits of arbitrage.
Barbaris & Thaler32 describes what is meant by that expression. When irrational investors, cause
mispricing of stocks, Friedman33 and Fama34 argue that the efficient market hypothesis will secure
prices returning to normal. The efficient market hypothesis claims that rational investors using
arbitrage will be able to correct the prices.
The limit of arbitrage is reached, when irrational investors become sufficiently powerful to
maintain a mispricing. In that case, even rational investors, might not try to correct it, because of
fear of betting against the market.
Given the previously described insights from especially Schiller, it is believed that the assumption
about having irrational investors in the environment holds.
However, I find that an important clarification is missing from Abreu & Brunnermeiers model. In
their paper, it is hard to identify, what separates irrational investors from rational investors, that
have not yet become aware. Initially, I believed that there were none. Rational investors that have
not become aware yet act irrationally by exactly the same reasons as the irrational investors
described above. Because of differences in opinions and beliefs.
On the other hand, you can argue that the difference is located in the method of investing.
Rational investors make decisions on the basis of the fundamental value of stocks, whereas
30
Positive feedback investment strategies…, 1990
Stockprices and social dynamics, 1984
32
A survey of behavioral finance, 2003
33
1953
34
The behavior of stock market prices, 1965
31
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irrational investors support their decisions with behavioral or psychological factors. As mentioned
earlier, it seems that Abreu & Brunnermeier expect all investors in the market to share the same
goal.
In this light, the difference between rational investors and irrational investors become a question
of basic knowledge – or information. Rational investors have the knowledge which is necessary to
maximize profits. They know, that they have to look at a stocks fundamental value, in order to
evaluate whether or not the price matches. Irrational investors do not have that knowledge and
decide instead on the basis of how they think the stocks will change.
However, it still comes down to a matter of information.
To conclude Abreu & Brunnermeiers definition of what exactly constitutes an irrational investor
seems to be missing. At the same time it is, however, believed that the general assumption of
irrational investors being in the market holds. This means, that it has no influence on the direct
results, but makes it difficult to understand who the players are, in a real-life situation. In my
opinion, investors in the model can be grouped in two ways.
The first option is that there are 2 groups of investors. A group of irrational investors, and a group
of rational investors. The distinction between them is, that rational investors seek to maximize
profit considering risk, by trading mispriced stocks evaluated by their fundamental value. Irrational
investors share the same goals, but seek to accomplish these by trading stocks on the basis of
trends, beliefs and other psychological factors.
All rational investors are surely not alike. They share investment method, but evaluate companies’
fundamental value differently. The way, in which they interpret news, is dependent on the
individual investor’s belief. These are different, which is why they become aware of the bubble, at
different moments in time.
Irrational investors trading behavior take a role as noise-trading. This means that rational investors
that have not yet become aware are not able to read the price signals created by informed
rational investors trading behavior. The absorption capacity of the irrational investors is equal to
the amount of k selling pressure. When the amount of k selling pressure is reached, the price
signals will turn visible, and investors will instantly sell out and burst the bubble.
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The second option is, that there is one collected group of investors. From this point of view they
are labeled as rational investors, whenever they become aware. In this sense, it is their behavior,
caused by awareness, which defines them as rational. There is no distinction between an unaware rational investor and an irrational investor.
Obviously, the first option fits the model best. Additionally, it also relates well to the studies
presented earlier in this section.
Sequential awareness: This assumption is obviously closely related to the above rationality
discussion. It states, that from the random point in time 𝑡0 , 1⁄𝜂 of rational investors will become
aware of the bubble situation.
Taken from the discussion above, there exists an amount of 𝜂 rational investors. From Abreu &
Brunnermeier we also know that 𝜂 relates to an interval stretching from zero to infinity.
The reasoning behind sequential awareness is, that there is heterogeneity among the infinite
group of investors. Abreu & Brunnermeier argue, that heterogeneity among investors reflects
dispersions of opinion and beliefs. In relation to investing, heterogeneity can generally influence
investors in two ways. Through opinions of in which media information should be collected, or
how this information is interpreted.
Rational investors might be biased by personal beliefs in their choice of media and how they rate
the value of different media. Sources of financial news can for instance be brokerage firms,
financial statements, newspapers, internet, and so on.
The interpretation process is even more complicated. A broad range of theories is presented on
human psychics, but this is not found necessary to comment in this paper. Instead it will be
assumed that human individuals are heterogenous.
Whether or not this heterogeneity impacts, the stock market is, however, very interesting.
As mentioned, in this game, the consequence of heterogeneity is, that rational investors become
sequentially aware of the bubble. Basically, because they disagree on the fundamental value of
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the stock. Ghysels & Juergens35 show in their paper that the volatility of stocks can be explained by
heterogeneity of investors. This shows that investors different trading behavior can be explained
by differences in perception. Their results are backed by Boswijk, Hommes and Manzan36. It
should however, be noted that Boswijk, Hommes and Manzans investigation views heterogeneity
as primarily influencing irrational investors.
In my opinion, the assumption of sequential awareness has good support from both empirical
research and logic. Even though rational investors in practice, are often representatives for
institutional investors, the choices are still made by individuals.
Especially, in price bubbles, I believe that the heterogeneity of investors is large. When we look at
the definition of a price bubble, and the numerous examples of previous price bubbles, it becomes
obvious, that it is often closely related to a major technological achievement. When previously
unknown technology is developed, investors are likely to create opinions based on their own
beliefs. Simply, because they do not know how the technology works, and who is able to exploit it.
This also creates incentive to follow the trend to avoid other investors gaining from knowing
something that I don’t.
It is also interesting, to discuss the role which sequential awareness has in the bubble game. If the
heterogeneity is large, the dispersion of opinion among investors is also large. This means, that the
time needed to generate selling pressure increases. During this period, the bubble might burst for
exogenous reasons.
Hence, the amount of heterogeneity directly affects the bursting of a bubble.
Exogenous crashes: The outline of this assumption is, that a price bubble eventually will burst,
despite that there is not generated enough selling pressure. The impact of this assumption is, that
the game is given a finite horizon. A finite horizon would in traditional game theoretic principles,
ironically, exactly generate selling pressure. Furthermore, rational investors would anticipate each
other’s moves, and backward induction would eliminate the bubble.
Interestingly, this will not happen in this model.
35
36
Stock market fundamental and heterogeneity of beliefs, 2003
Behavioral heterogeneity in stock prices, 2006
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In detail, the assumption says, that in cases where the hazard rate of investors is lower than or
equal to the costs-benefits of attacking the bubble, at the time where it reaches its maximum life
time (t0 + ŕ), enough selling pressure to burst the bubble will never be created.
For this reason, the exogenous crash constitutes a limit of time, an investor can ride the bubble
which leads to an equilibrium behavior.
In this model a bubble will crash, when it reaches its maximum size of β.
Interpreting these exogenous crashes is not straight forward. As in any other markets, stock prices
are controlled by supply and demand. Normally the supply of stocks is stable37, which is why stock
prices are highly influential of demand. Since the stock prices are directly determined by demand
the price cannot fall unless demand decreases and selling pressure is generated.
This logically means that all bursts of price bubbles is driven by selling pressure. However, in this
model two types of selling pressure are present. The first type of selling pressure is what
generates an endogenous crash. In this case, the selling pressure is gradually increasing because
informed rational investors follow equilibrium behavior and leaves the market. The second type of
selling pressure is not clearly defined in Abreu & Brunnermeiers paper.
Thinking about what factors that can influence the collapse of a price bubble can, however, give
some good perspectives to describe this second type of selling pressure.
As modeled exogenous crashes become actual, when the heterogeneity of investors are very
large, and the absorption capacity of selling pressure is high (high market power of irrational
investors). This says that investors’ different interpretations of the market, and/or the markets
size, and the noise from irrational investors, makes it capable of handling high amounts of selling.
So, the reason why the selling pressure is not high enough basically, means that too few rational
investors have become aware of the bubble.
Seen in this light, it makes sense to assume that exogenous crashes are caused by a major event or
incident, that sends a clear signal of a bubble. This will lead irrational investors to immediately sell.
37
Stock issuing is considered as special cases.
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Johansen and Sornette38 more clearly define exogenous crashes, as crashes caused by unforeseen
news impacting the market.
Yossefmir, Huberman and Hogg39 have proved, that as price bubbles develop, they will get more
sensitive to exogenous shocks. They claim, that as a bubble persists, it gets more and more
dependent on the optimism of irrational investors. This is in line with how the noise-traders in
Abreu & Brunnermeiers model are able to absorb selling pressure from rational investors.
Naturally, as the selling pressure gets closer to the capacity constraint, more and more pressure is
put on the irrational investors. Even though, they are not aware.
As mentioned in the beginning of this section, a major implication is that the game is given a finite
horizon. The reason why this does not dissolve the bubble is already explained, and will not be
discussed further. However, the introduction of a finite horizon in this game, indirectly give
supporters of the efficient market hypothesis a possibility to refuse the results found.
The argument is, that even though irrational noise traders might affect prices for a while, they will
in a longer perspective loose influence. Biased irrational investors will bit by bit loose money to
arbitraging rational investors, and after a while they will be completely out of the market.
Financial constraints: This assumption is primarily necessary for theoretical reasons. In practice
the amount of actors, and the collected value of trades in the stock market make this assumption
unnecessary in relation to what we have defined as a price bubble.
It simply implies, that no individual investor has the financial strength to burst the bubble,
whenever he wishes.
The game theoretic consequence of this, is that the non-coordination feature of the game, is what
gives the investors the possibility to earn a return higher than the market.
In many other games, prisoners dilemma for instance, equilibrium is found to be a paradox, where
non-coordination leads them to decisions, which on an individual level is worse, than if they had
been able to coordinate. In this game, it is found to be an equally interesting paradox, that noncoordination from an individual investors perspective should be viewed positively.
38
39
Endogenous vs. exogenous crashes in financial markets, 2008
Bubbles and market crashes, 1994
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Furthermore, this assumption does an effective job defining which bubbles the identified optimal
behavior works in.
Any over-pricing of any stock can essentially be called a price bubble. This assumption limits how
large a price bubble is required to be, in order to utilize the optimal behavior found in this game.
No pricesignalling: This assumption implies, that the amount of trade by irrational noise-traders is
high enough to prevent un-aware rational investors from seeing price signals. Price signals will be
caused by rational investors which have become aware early. They will sell out at their optimal
time and thereby cause the stock price to fall.
The assumption of no price signals is also meant to support the mathematical assumption, that 𝛽
is a continuous increasing function of (𝑡 − 𝑡0 ). This means, that the price of the stocks is not able
to decrease, because of rational investors selling pressure.
Initially, it can be argued that the unpredictability (if irrational investors is unbiased) of irrational
investors might not even hide the price signals at all. Assuming, that irrational investors trading
decisions is made on the basis of individual psychological factors, their collective investments
might not point in a uniform direction. Their investments might be unbiased. Following this, price
signals can only be avoided, if the actions of irrational investors are biased.
Following the argumentation of Schiller and Delong + others, and the conclusions from the
previous discussions of rationality and heterogeneity, it seems likely that expectations of irrational
biasedness is true. This supports the assumption of “no price signals”.
If the assumption does not hold the optimal riding time of the bubble will decrease. Optimal riding
time for investors will be the period until the first investor sells. This will give a price signal to all
rational investors that are not aware. Consequently, the bubble will burst by endogenous reasons.
It is interesting to notice, that the bubble would still survive in the period until the first investor
that became aware sell out. However, the riding time will be decreased significantly.
Rational investors do not sell if they are not aware of the mispricing: I find this assumption a bit
problematic, but it is basically necessary in order to investigate behavior.
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It is difficult to say, if it is unrealistic or not. I believe that the assumption would hold. Rational
investors, who is not aware of the bubble believe, that the price increase represents increasing
fundamental value. From this perspective, the stock is viewed as a good investment object, and
there will be no reason to leave the market.
However, that argumentation is difficult to prove.
Importance of transaction costs: This assumption states, that transaction costs are large enough
to limit trades to a finite number. At the same time they may not be so large to avoid selling when
the stock is overpriced.
The point of this is to avoid that transaction costs influence equilibrium.
8. Supporting results of the model:
Doing the discussion of the assumptions made in the model, a few problems have been identified.
In order to conclude, whether or not the model matches the reality, despite the problems
identified, the results from a few investigations of the technology bubble (dot.com bubble) will be
presented.
The technology bubble of the late 90’s is probably the best known bubble in modern financial
markets. It developed upon the commercialization of the Internet, which created a huge amount
of new companies with visionary business ideas. Many investors believing in these visions bought
huge amounts of stocks. As the companies began to put their ideas into the market place, it
became obvious that many of the promising visions could not deliver the expected earnings.
Investors realizing this began to sell out and essentially bursted the bubble.
When comparing the results of these investigations to Abreu & Brunnermeiers model, it is
assumed, that institutional and/or hedge funds are representative for rational investors.
Institutional investors are typically less inclined to follow behavioral reasons, such as personal or
psychological beliefs.
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8.1) Hedge funds and the technology bubble:
The study presented here is structured very closely to the study I originally intended to do.
It analyzes the portfolios of US hedge funds doing the dot.com bubble. The data is gathered from
the 13f filings of institutional holdings. A regulation by the SEC (US Securities and Exchange
Commision) requires hedge funds managing more than $100 million to report their securities
holdings each quarter. Analyzing this data, makes it possible to determine how technology-bubble
stocks were weighted in hedge fund portfolios.
The results show that doing the bubble period (1998-2000), hedge fund portfolios were
significantly overweighed with technology-bubble stocks.
This indicates, that rational investors, indeed did ride the tech-bubble. This supports the optimal
behavior found in Abreu & Brunnermeiers model.
The second interesting result is, that when looking at stocks individually, hedge funds also reduced
their holdings prior to the burst. This also corresponds with the optimal behavior in the model.
8.2) Who drove and burst the tech bubble:
This study has classified stock brokers into nine groups, depending on their primary customer
types. The brokers’ transactions are possible to identify by Nasdaq trading information. By using
the transaction information on brokers typically used by institutional investors, you are able to
estimate the volume of trades made by institutions during the bubble period.
This means, that it can be analyzed which investor types rode the bubble and who succeeded to
sell prior to the crash. The following groups were identified: Individual general, Individual fullservice, Individual discount, Individual daytrading, Institutional, Largest investment banks, Hedge
funds, Derivatives, Mixed.
Concerning the comparison to Abreu & Brunnermeiers, it is assumed that the groups; Institutional,
Large investment banks and Hedge funds are representative as rational investors. In the following
these groups will be collectively named as institutional investors.
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The first conclusion, that is interesting with relation to this model, is that all broker groups
(including institutional) participated in building the bubble. This supports the models optimal
behavior, that rational investors will ride the bubble, even after becoming aware.
The paper even suggests, that institutional trading was the primary reason for the build-up in
prices, and that institutions kept buying overvalued stocks late in the bubble. This indicates, that
institutions were buying stocks even after they became aware, which surely does not seem to be
the optimal behavior. The reason could be that institutions, even late in the bubble, were not
aware of the situation.
However, the results also show us, that institutional investors around the peak of the bubble
began selling out prior to individuals. This also supports, that institutional investors followed
equilibrium behavior in the dot.com bubble.
9. Conclusion:
I believe, that Abreu & Brunnermeiers model of price bubbles reveals some of the techniques by
which game theory can contribute in analyzing a wide range of economic problems. Specifically,
analyzing investors behavior in price bubbles is interesting in order to explain how they develop.
This is fundamentally important, as they have profound influence on the development of financial
markets and economies in general.
Furthermore, the empirical evidence of price bubbles in the past is considered to be the greatest
argument against the efficient market hypothesis. This implies, that the study of behavior in price
bubbles can be part of identifying the reasons why the efficient market hypothesis does not hold.
This paper shows, how heterogeneity among rational investors, in an environment of irrational
investors can explain the development of a bubble.
The bubble will initially start because of some sort of temporal structural change in the economy.
Typically, the reason is a major technological achievement, which leads to expectations of high
future dividends. These expectations lead to high demands of certain stocks, both from indigenous
investors and new investors. New investors are often overly-optimistic irrational investors, who
get drawn in by the high expectations.
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This process makes the stock price rise over the fundamental value and establish a price bubble.
Because of dispersion of opinion, rational investors will become aware of the bubble at different
times during its development. However, because of irrational investors noise-trade they will not
be able to correctly determine at which time the bubble began. They will therefore, rely on their
private beliefs when choosing strategy.
The optimal strategy of investors is to ride the bubble for a period before selling out and
completely leave the market. The optimal time to sell is when the probability of the bubble
bursting equals the costs compared to the benefits of attacking it.
Furthermore, an interesting aspect of analyzing the optimal behavior reveals, that it is the lack of
coordination among investors that makes them able to benefit from the bubble situation. This
happens at the expense of irrational investors.
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