Experiential Learning of
The Efficient Market Hypothesis:
Two Trading Games
Andreas Park
Assistant Professor
University of Toronto
Department of Economics
150 St. George Street
Toronto, ON, M5S 3G7, Canada
Phone: 416-978-4189
Fax: 416-978-6713
Email: andreas.park@utoronto.ca
Experiential Learning of
The Efficient Market Hypothesis:
Two Trading Games
Abstract
In goods markets an equilibrium price balances demand and supply; in a financial
market an equilibrium price also aggregates people’s information to reveal the true value
of a financial security. Although the underlying idea of informationally efficient markets
is one of the centerpieces of capital market theory, students often have difficulties
grasping and accepting that asset prices fulfill this dual role of information revelation and
demand-supply aggregation. The author presents two simple classroom games that
illustrate the workings of information transmission and aggregation through prices. The
games are easy to comprehend, simple to implement and short. Each game takes about 30
minutes, including classroom discussions, and by the end students will have an intuitive
feel for informational efficiency.
Keywords: efficient capital markets, information aggregation, trading
JEL codes: A22, C91, D82, G14
1
One of the hallmarks of finance theory is that financial markets are informationally
efficient, that is, that asset prices correctly reflect market participants’ information. Yet
this notion is difficult to convey in the classroom: to many students it seems rather
miraculous that prices should aggregate information. In this article I propose two
classroom games that help students get a feel for the process of information aggregation.
Both games have a common theme: students are presented an item that represents
the fundamental value of a company’s share. Its value is represented by a glass jar of
nickels or small, identically sized glass objects and it is thus tangible, yet uncertain. With
some effort and patience, students can count the objects and thus get a good estimate of
the asset’s true value. Students record their first estimate before trading starts so that in
the aftermath they can determine if indeed prices reveal the average opinion. Asking
them to also record their estimate after trading has concluded allows them to document
whether there is learning from prices.
Students’ opinions about the share’s value will differ, but if prices aggregate
information correctly, then, loosely, the market price that equilibrates demand and supply
should reveal the average opinion. When there are also sufficiently many opinions, this
average should be close to the true value of the underlying asset. And indeed I usually
observe both outcomes.
The first game represents an idealized Walrasian market in which students submit
demand-supply-schedules for an asset. Demands and supplies are aggregated centrally
and a market clearing price is determined in the spirit of a Walrasian auction (henceforth:
the Walrasian game). The second game mimics the dynamic trading environment of most
2
traditional stock exchanges. Students trade assets “face-to-face” and all transactions are
publicly listed (henceforth: the face-to-face trading game). The details are outlined below.
For the Walrasian game I observed that the market price is close to and usually
slightly below the average opinion.1 Moreover, provided that there are sufficiently many
students present, prices also reflect the true value. There are several possible explanations
for why the price is slightly below the average opinion, for instance, people may be
avoiding the winner’s curse, there may be too much noise in the small sample, or people
may be simply risk averse. The classroom discussion will help students get an intuitive
grasp of these concepts. Aggregate demand and supplies are usually monotonic, and it
never fails to impress students how smoothly the market aggregates their different
opinions.
With face-to-face trading, prices typically fluctuate widely in the first few minutes.
The simplest explanation for this fluctuation is that the first few trades are arranged oneon-one without public guidance as to what the correct price might be. Eventually, prices
settle on a tentative upward trajectory. If the group is not too large and trading is
transparent, then after about two-thirds of the trading time, activity comes to a stop for
several minutes. Towards the end trading activity resumes and often prices fall a little.
This is consistent with learning: early trades are only observed with a lag, whereas later
trades are more transparent because there are fewer transactions. There are two types of
buyers: those with high valuations and those who speculate on rising prices. Together
these two types create a tentative upward trajectory of prices. As prices rise, it becomes
harder to find someone with a high enough valuation to sell shares to. Eventually, as the
market closing time approaches, speculators try to unload their positions, causing price
3
drops. After the game, the spread of opinions tightens, which illustrates learning from
prices.
There are three simple messages that students can derive from playing these games:
(1) the wisdom of masses holds, that is, the average prior opinion is close to the truth, (2)
markets and prices aggregate opinions, and (3) people often act according to the efficient
market hypothesis even if they have not yet been taught the concept. Although these
learning objectives obtain independently of the trading mechanism, the trading outcomes
differ in the two games, illustrating the importance of the organization of financial market
trading. In a nutshell, in the Walrasian game prices are generally closer to the true value
and the average opinion than in the face-to-face trading game.
There are several possible reasons for the differences. For instance, there may be
behavioral biases such as the unwillingness to realize losses. Another explanation is that
endowments may be too limited and short selling is not allowed. The latter hinders the
revelation of low valuations compared to the Walrasian game, which implicity allows
short-selling and where endowments play no role.
The Walrasian game is thus the cleanest way to show how well prices aggregate the
average opinion. The face-to-face trading game, on the other hand, can illustrate the
relevance of limited endowments or short-selling constraints. The latter game also
provides an intuitive background for discussions of no-trading results, which state that
two parties cannot agree to be both benefiting from a trade. Moreover, it also illustrates
other trading motives such as the greater-fool-fallacy that is based on the idea that while
one may be a fool to buy, one expects to find an even greater fool who one can sell to at a
higher price.
4
Although prices usually converge to the true underlying value, one must be careful
when setting up the game. It matters how many shares are available for trading: the more
shares there are, the less trading activity is observed. Even if prices ultimately do not
reflect the true value, the classroom discussion helps students greatly to understand how
prices aggregate information and why market frictions may impede this mechanism.
The ultimate goal of the games is for students to get a feel for trading and the
informational role of prices, and to establish credibility for the idea that financial market
prices aggregate information. In contrast to most games that are used in the financial
markets experimental literature (discussed in the next section), the games proposed here
do not require students to compute the value of a stock with a mathematical model. The
simple setup and its easy implementation make it more likely that the learning objectives
are achieved. The game is suitable for most standard undergraduate finance courses, and
may be useful in graduate financial economics courses or speciality courses on financial
market microstructure.2
In the next section I review the literature on market-efficiency games and their
adaptations to the classroom. I then explain the setups of my two trading games, outline
some common findings of the games and provide guidelines for the classroom discussion.
The appendices contain the games’ instructions.
LITERATURE ON FINANCIAL MARKET (CLASSROOM) GAMES
The games that I propose are a blend of classroom market games in the tradition of Smith
(1962) and winner’s curse games as described, among others, in Thaler (1988). In
contrast to all other games in the literature that mimic financial market trading, students
are not expected to compute the value of a financial asset. This simplifies the
5
implementation of the games in the classroom greatly and avoids distorted outcomes
caused by the students’ limited mathematical capabilities.
Several financial market game authors studied the emergence of bubbles in market
experiments, starting with Smith, Suchanek, and Williams (1988), and, more recently,
Lei, Noussair, and Plott (2001) and Haruvy and Noussair (2006). The frameworks
employed in these articles are very similar, and Ball and Holt (1998) provided an
adaptation of these setups for a classroom environment. In all these games, there is an
underlying fundamental value that is constant (or decreasing) over time.3 All information
is public, and students can effectively compute the (risk-neutral) fundamental value by
backward induction. In these games one frequently observes large positive deviations
from this value, and this is usually interpreted as the emergence of a bubble.
In a second strand of literature Plott and Sunder (1988) and Forsythe and Lundholm
(1990) examined how information is aggregated through prices. In these games people
receive private signals, and the pooled information reveals the true value of the
underlying asset. These studies usually find that the existence of informational efficiency
crucially depends on the market structure.
All of these experiments require that participants understand a problem on a
relatively high level of abstraction. At the very least, participants have to compute a
(discounted) sum of expected future payments and understand backward induction. It
cannot be overemphasized that problems and settings that seem simple to academic
economists may be hard for our student audience.4
Information aggregation is also taught in the context of auctions. Bazerman and
Samuelson (1983) were the first to auction glasses of pennies, nickels, or paper clips.5
6
The penny-auction is usually designed to illustrate the winner’s curse, and so when
setting up the game, one tries to ensure that winner’s-curse-type overbidding occurs. A
first step towards this goal is making it difficult to estimate the true value. For instance, in
Bazerman and Samuelson (1983) the value of the object is $8, but the average opinion is
only about $5. The purpose of the games that I propose is different: people are supposed
to be able to estimate the value correctly (and they do on average), and prices are meant
to reveal information (which they do).
The first definition of the efficient markets by Fama (1965) is still the one found in
most textbooks, yet it gives very little guidance as to exactly how information
aggregation is achieved, and under which conditions we would accept or reject that a
price or market is informationally efficient. The seminal micro foundation of the efficient
market hypothesis (EMH) was provided by Grossman (1976, 1977) and Grossman and
Stiglitz (1976, 1980) and the games proposed here are best described as an experimental
implementation of Grossman’s work. However, for teaching purposes, especially on the
undergraduate level, Grossman’s models are usually considered to be too advanced.
Moreover, the underlying theoretical models are very specific, usually combining
constant absolute risk aversion (CARA) utility functions with normally distributed asset
returns. This specific structure makes an experimental implementation difficult. The
games that I outline here can thus be seen as filling a gap, allowing instructors to discuss
the information aggregation role of prices as well as, tentatively, allowing a simple
illustration of Grossman’s results.
7
THE SETUP OF THE TRADING GAMES
I ran the games with several groups of undergraduate students: about 2/3 were third or
fourth year students in economics or commerce, who had one term training in basic
finance; the remaining third were second year students of commerce and economics with
two years of standard training in economics (but no training in finance). I also ran the
games with a class of Financial Economics graduate students, who usually proceed to
become analysts or traders in major Canadian financial institutions. The group sizes
varied between 13 and 36 people; larger classes were split in half. One needs about two
instructors per 20 students to organize trading efficiently.
Each game requires four steps: first, explaining the rules, second, playing the game,
third, analyzing and summarizing the information collected during the game and
Insert Figure 1 here.
fourth, discussing the outcomes in class. Ideally one would perform Steps 1 and 2 in the
last 20 minutes of the class or tutorial/discussion section that precedes the lecture where
the efficient markets are to be taught, so that one can analyze the data between classes
(Step 3) and then discuss the outcomes in the following lecture (Step 4).6
Common Setup for Both Games
The asset. The security is represented by a glass filled with a number of identical objects.
The objects and the jars should be chosen in such a manner that it is possible to get a
good estimate of the number of objects. I find that small glasses (spice jars) filled with
8
$2-$3 worth of U.S. or Canadian 5 cent coins or glass-pebbles worth 5 cents each work
well. Photos of sample shares are in Figure 1. Note that a bank-roll of nickels is worth $2.
The number of jars. I suggest using one jar per three participants. Market activity and the
number of shares are negatively related. If there are too few jars, on the other hand, then
the game no longer resembles a market but becomes an auction.
Ex ante information extraction. Before trading starts, participants are allowed to examine
the jar for a limited amount of time. The time constraint ensures some residual
uncertainty, leading to differences in perception. One usually observes that some, though
not all, participants try to count the number of coins (some will be taking notes).
Assessing the average opinion. Each student will be assigned a trader ID. Before the start
of trading/the submission of orders, participants are asked to record their initial
assessment of the value of the asset. This information is to be collected right away. In the
face-to-face setting students will be asked for their valuation again after the games
concludes. If they see their original opinion they may feel inclined to give the same
number to be consistent.
Suitable class size. The Walrasian game can be played with classes of arbitrary size. Face
to face trading requires a manageable group size of about 15-20 people. If the group is
too large, trade reporting becomes logistically challenging.7
Timing relative to the teaching of informationally efficient markets. I suggest letting
students play the game prior to the lecture on informationally efficient markets. This
ensures that students approach the games with an open mind instead of wondering what is
required of them given the material that was just covered in class. One can then combine
9
the formal introduction of informational efficiency with the discussion of the game(s)
(see Section 5).
Organization of Trading
In addition to standard finance classes, I use both games in a course on financial market
microstructure to illustrate the institutional differences in market trading arrangements.
These two games allow an illustration of two major mechanisms: continuous trading as
practiced on most stock exchanges during the day and simultaneous order submission as
practiced during the opening sessions of exchanges. 8
Organization of the demand-supply-schedule trading process. Each student receives a
sheet of paper with the printout of a table. The first column of this table lists prices (in
cents), and in the second and third column students are to enter the number of units they
would buy and sell if the price in the first column were the equilibrium price.
The maximum quantity that students can buy or sell at any price is restricted (for
instance, 10 units) and there is a fixed number x of assets in supply.
The instructions (see Appendix A) state that all individual participants’ demands
and supplies will be added up for each price. The price at which the aggregate demand
minus supply is closest to x is the market price. Students are also informed how their
payoffs are computed. These payoffs depend on the price that they pay and the true value
of the asset, and are computed as
(quantity demanded at the market price − quantity supplied at the market price)
× (true value − market price).
10
Students record the number of units that they wish to buy or sell in the table that is given
to them, their orders are collected, entered in a spread sheet and the market clearing price
is computed.
Organization of the face-to-face trading process. The face-to-face trading occurs in a
simple, lightly regulated double auction market, in structure akin to floor-based open
outcry markets (for instance, the New York Board of Trade).9 Deals are arranged face-toface, trading tickets are filled out, and prices plus transactions are reported.
A fixed number of shares is distributed at random to students. Students are told that
those who receive a share have to pay the instructor the true value of the stock after
trading concludes and that those who own the stock after trading concludes will be paid
the true value by the instructor.10 Therefore there is a firm terminal value and full
information revelation after trading concludes. All other payments are self-explanatory
buy-sell transactions.
Students are asked to leave their seats to roam the classroom in search of trading
partners. The trading process itself is unregulated. For instance, they can negotiate with
several people at a time or they can auction their share. Once an agreement is reached,
both trading parties record their transactions on trading tickets, and the seller reports the
trade to the central trading desk. Transaction prices are then centrally listed on the
blackboard. Instructions are in Appendix B.
The trading session lasts for roughly 8-12minutes, depending on the class size. The
rule of thumb is that the larger the class size, the longer the trading time should be. With
groups of 15-20 students 7 to 8 minutes are sufficient. Although it may be beneficial to
have students trade for longer (so that there is more time for price adjustment and
11
learning), the time pressure is important to ensure that speculators feel the need to unload
their positions.
A word about payments. It may be controversial to ask students for payments in a
classroom environment; in my sessions no money changed hands. Despite this, students
usually take this exercise very seriously and treat their trading as if it were for real
money—they cherish gains and loathe losses. After all, there is pride involved when
making a good transaction or when beating their classmates. As a control I also ran the
games twice during paid experimental sessions and found no difference in behavior. For a
more detailed discussion on how to properly motivate students to take classroom games
seriously see Marks, Lehr, and Brastow (2006) (their footnote 12).
COMMON OUTCOMES
Common outcomes of the Walrasian game.
Figure 2 displays an example of the aggregate demand and supply schedules from the
Walrasian game trading session. The column chart is the transaction volume that could be
realized for the given prices. The increasing curve is the aggregate supply as a function of
the price, the decreasing curve is the aggregate demand. The underlying true value was
$2.35, the average opinion was $2.41, the median opinion was $2.25, the market price
would be between $2.20 and $2.30. Allocational efficiency was at 82 percent. An aside:
Stock exchanges often describe their opening procedure as a process that tries to
determine the price that maximizes volume. The graph illustrates this notion because
volume is indeed maximal ‘around’ the transaction price.
12
In general, most students submit monotonic demand and supply schedules, with demands
that decline with prices, and supplies that increase. There are always some who submit
erratic schedules, but the effect of such orders on the equilibrium price is typically
negligible. Prices can also deviate from the average opinion because of some residual
noise that cannot be eliminated in a small sample.
Insert Figure 2 here.
That prices are slightly below the average opinion is consistent with two theoretical
concepts. First, there may be a small degree of risk aversion. Second, if the price were set
at the average opinion, then on average people should be indifferent between buying and
selling. This is, however, not market-clearing because there is also a fixed supply of x
shares. Therefore, if the average person would obtain a share at his/her own assessment
then s/he would be a victim of the winner’s curse. By the same token, if prices are above
the average opinion, then this may indicate that students were subject to the winner’s
curse.
In three out of seven sessions, the price was below or just marginally above the average
opinion. In four sessions, the price was not below the average opinion; notably this
occurred for the groups that had no training in finance. These groups also paid on average
more than (a) their assessment, (b) the average opinion and (c) the k-th highest opinion,
where k is the number of issued shares.
Although the students in these sessions violated the theoretical prediction, there is a
point to be made that at least on average students acted rationally, that is, on average they
13
submitted demand-supply schedules that avoided the winner’s curse. To see this I
computed the students’ virtual gains: this number is obtained by multiplying each
student’s net demand with the difference of their prior valuation and the market price. For
instance, if a student has valuation 310 and would buy 5 shares at price p = 300, then his
or her virtual profit is 5 × (310 − 300) = 50. Table 1 summarizes the above observations.
As can be seen in column labelled average virtual profits, the average virtual profit was
positive. Also, the column labelled number virtual losses indicates the number of people
who made virtual losses. Overall, less than 42 percent of students made virtual losses;
with finance-trained students this number is, somewhat comfortingly, even smaller.
Finally, one can ask if those with the highest valuations also ended up buying the
shares; I dub this the allocation efficiency of the trading process. As can be see from
Table 1, allocation efficiency is highest for the finance-trained groups (82-92 percent).
Insert Table 1 here.
Common Outcomes with face-to-face trading.
Figure 3 plots the price development for an example of a face-to-face trading session. The
figure plots trading prices against the time when the transaction occurred in a face to face
example. The data is from one of the control sessions where people were paid; these were
the only sessions presented here for which I had recorded the times of trades. There were
13 people in the session, the true value was $2.85, and there were 4 assets that could be
traded. The average opinion was $3.23, the median $2.90, the 4-th highest opinion was
$3.00; allocation efficiency at the end was 75 percent. Prices rise along a linear trend.
14
Although it is not visible from graph, in the last two minutes of trading there almost no
activity; only towards the very end did people engage in negotiations again.
Generally, in the smaller sized groups (15-20 people), students first negotiate trades
bilaterally. After one or two minutes, when the first prices are posted, trading becomes
more organized: people often start an open auction by lifting their shares and shouting
prices.11 In the middle of the trading session volume declines, and instead the auctioneers
try to sell their assets at very high prices (higher than the last transaction price). Towards
the end, transaction activity picks up again, and is often accompanied by a slight
downswing in prices (which is observed but not statistically significant). Overall with
face-to-face trading, one observes a tentative price increase during the first 2/3 of the
session, followed by low activity, followed by a slight downswing in prices. Incidentally,
this U-shaped volume pattern is also observed on most stock markets.
Students buy for two reasons: first, they think that the security is undervalued. Second,
they buy because they speculate that there may be somebody else who has a higher
valuation. For a given class size, roughly 2/3 of students are actively transacting. A
sizeable number of the non-transacting students try to trade but cannot agree on a price.
This is worthy of discussion because people who have a low opinion of the asset cannot
easily convey this belief. If short-sales were allowed or if there were larger endowments
of assets, then those who believe that the asset is overvalued could potentially bring the
price down by selling.
Because the shares are allocated randomly and there are fewer shares than students,
before trading starts, early sellers are not necessarily those who have a low opinion;
15
instead, all participants are in search for the highest opinion. Once they realize that these
people are rare, prices slowly fall as speculators unload their positions.
Insert Figure 3 here.
The number of transactions is negatively related to the number of assets,12 that is,
the more shares there are the lower is the number of transactions. Moreover, there is also
a negative correlation between the average trading price (scaled by the true value) and the
number of assets, that is, the more shares are available the lower the price.
The range of opinions shrinks/tightens from before to after trading and the variance
of opinions also shrinks. The average price is above the average opinion by about 11
percent, above the true value by 9 percent, and above the k-th highest opinion by 4.5
percent, where k is the number of issued shares.
There is also a substantial amount of speculative trading: on average more than 50
percent of the buyers trade above their valuation, whereas only 25 percent of the sellers
transact at prices below their valuation. On average buyers lose in expectation, whereas
sellers gain. Allocational efficiency on average is at 64 percent which is lower than in the
Walrasian game. All of the above findings are summarized in Table 2.
Speculation may account for the (virtual) losses that buyers are willing to incur in
the process, but one is left wondering why people hold on to their shares beyond the end
of the game when they could have unloaded them earlier. Looking only at the inefficient
allocations and excluding a session with poor data recording, 71 percent of the buyers
who held on to their shares until the end of trading made an average virtual loss. Half of
16
the last owners could have made a profit had they sold at (one of) the last trading prices.
This point is worth mentioning to students for it documents the well-known phenomenon
that people tend to hang on to their losses.
SUGGESTIONS FOR THE CLASSROOM DISCUSSION
The classroom discussion following the game is a vital component of the learning
exercise and should clarify key concepts. The discussion should center around the
question of whether or not prices aggregate information, and the discussion should
include the following components.
First, students should discuss how they came up with their valuation. This way they
learn that there were some people with better and some with worse assessment
techniques.
Second, the instructor should lead the discussion to let individuals describe how
they learned from market behavior. Most likely, those with the most extreme initial
valuations revised their opinion after observing trading behavior.
Third, informational efficiency of prices can be assessed relative to two numbers:
the true value and the average opinion. Usually these two measures are close, the true
value can only be revealed if the average opinion is sufficiently close to it. If the two are
not close, then prices can still be close to the average opinion and thus aggregate
information. Irrespective of whether the true value is revealed, the game will help
students understand how actions reveal information and how prices then aggregate
information.
As stated before, I propose to introduce the standard teaching material on
17
efficient markets only after the game has been played. This material can then be
intertwined with the discussion of the game outcomes. In what follows I will outline a
few core concepts from the theory of informationally efficient markets.
The literature on market efficiency originated in Fama (1965). He introduced three
notions of efficient markets as follows: first, markets are weak-form efficient if all
information from past market trading is reflected by the current price. Next there is the
semi-strong form when all publicly available information regarding the prospects of a
firm is reflected in the current price. Finally, under the strong-form efficiency, all
information relevant to the firm is reflected by the current price, including insider
information; Campbell, Lo, and MacKinlay (1997) used the term private information.
These three types of market efficiency have their origin in empirical work, and early
work lacked a micro foundation. Grossman (1976) filled the gap by developing a
framework in which people’s private information is incorporated in a rational
expectations market clearing price. In the games presented here, arguably, people obtain a
private signal from examining the jar of nickels,13 and thus the games test to what extent
prices incorporate information in the sense of Grossman.14 Because Grossman provided a
micro foundation for the EMH, by association the games illustrate the efficient market
hypothesis in Fama’s sense. To wit, because the private signals are based on public
information it is reasonable to assert that the relevant EMH studied here is the semistrong form.
The games also help students to get a better understanding of the standard empirical
tests of the efficient market hypothesis. One of the implications of the weak-form EMH is
that prices are a sub-martingale, or, more loosely, they are a random-walk. Consequently,
18
a so-called technical analysis, which is the extraction of information about the future
movement of prices from past prices, should have no merit. The face-to-face setting can
actually illustrate why this may or may not be accurate. For instance, it is not true that
every trading-action is fully informative, because people speculate so that a buy-trade
may originate either from someone who has a favorable opinion or from a speculator.
Students understand this idea well, and many apply it.
The classroom discussion can then establish that speculative behavior, to some
extent, can be understood as trying to exploit alleged short-term inefficiencies, that is,
situations in which the price reflects information inaccurately. At the same time,
speculators often lose money and although this is not directly caused by the weak-form
efficient nature of prices, it illustrates the greater-fool fallacy. It also illustrates that the
common initial upward trend is not a useful predictor for future price movements.
Next, an implication of the semi-strong form of the EMH is that a fundamental
analysis (a careful analysis of the value of the jar) should have no or little merit as prices
will incorporate the information that analysts have obtained. When acting on the basis of
an accurate analysis, one can avoid making the wrong decision: in the demand-supplyschedule game, most people made virtual profits, that is, given their assessment they took
positions that would be profitable from their subjective perspectives. Moreover, people
who were more careful in their counting usually did better. In part this thus illustrates
what a fundamental analysis is and why it may have merit.
At the same time, in the face-to-face trading game, a careful analysis may not pay
out: prices are usually high, often exceeding the average opinion. Invariably there will be
some students who merely eye-balled the value and thus arrived at a poor quality
19
assessment and others, who came up with a careful assessment method that allowed a
precise estimate. These latter people are often the ones who believe that trading prices are
too high. Yet because endowments are limited and short-selling is not allowed, they
cannot utilize their knowledge, and their opinion does not enter prices adequately.
This observation can lead to inspiring classroom discussion of real-world
institutional and regulatory restrictions: for instance, mutual funds and most pension
funds are banned from taking short positions whereas hedge-funds are not.15
Another issue that merits discussion is that in the face-to-face setting, all profits and
losses add to zero. As a consequence, on every trade, one party gains, one loses. Of
course, in reality trading can be Pareto-improving because of diversification or tax
benefits; the face-to-face trading game can thus inspire a discussion of these trading
motives.
It should be stressed that the games do not provide a perfect test of the validity of
any level of efficient market hypothesis. Indeed, as Forsythe and Lundholm (1990)
pointed out, it is generally impossible to devise a test that definitively determines whether
or not market efficiency prevails. That being said, it is usually plainly visible in the
games that some form of information aggregation occurs; for instance, prices never
grossly deviate from the opinions in the room. In summary, for teaching purposes the
games provide ample evidence in favor of Grossman’s information aggregation models.
20
CONCLUSION
I presented two simple trading games that allow students to develop an intuitive grasp of
how financial market prices aggregate people’s opinions.
The first game is a setting that is close to an idealized Walrasian market: people
submit a demand-supply schedule that lists precisely how much they are willing to buy or
sell at each price. The Walrasian market clearing price usually reflects the average
opinion and is also often very close to the true value.
The second game mimics intra-day trading on financial markets, with all its frantic
activity and standard shortcomings. Although prices never grossly deviate from the
opinions in the room, they tend to be higher than in the Walrasian game. This allows the
instructor to lead students into meaningful, well-founded discussions on, for instance, the
impact of short-selling constraints and limited endowments on the inclusion of negative
opinions.
The major innovation of the games is their simplicity: in contrast to all other
financial market trading games in the literature, students do not have to compute a value
that is based on a mathematical construction. Instead, students only have to estimate the
number of coins in a jar. Thus although I provided a teaching module, there are potential
implications for experimental research that may use the games’ simplicity to check the
robustness of existing findings.
21
Appendix A
INSTRUCTIONS FOR STUDENTS: DEMAND-SUPPLY SCHEDULE
You will participate in a simple trading simulation that follows these steps
1. I will show you the asset. Your job is to estimate its true value.
2.
You then have to submit your demand-supply “schedule”.
3.
A market price is established (demand=supply).
4.
The true cash value of the asset is announced.
5.
Trading gains/losses are computed.
Profits and losses. To illustrate gains and losses, suppose the market clearing price
is 150 whereas the true value is 140. Example 1: You bought 5 assets. Your payoff:
5×(140−150) = −50. Example 2: You sold 10 assets. Your payoff: 10×(150−140) = 100.
Bottom line: If you buy when the market price is above the true value, then you lose
money; if you buy when the market price is below the true value, then you win. Likewise,
if you sell when the market price is above the true value, then you win and if you sell if
the market price is above the true value then you lose.
The Asset. You will be trading assets. These assets are jars filled with blue or green
pebbles; I will them show to you in a moment. Each pebble is worth 5 cents. You may
look at the jar, but you are not allowed to open it. Please note that the jar itself has no
value (it is only wrapping) and that all jars are identical and contain the same number of
pebbles.
22
Apart from the jars that are bought and sold by your fellow classmates there is a
fixed supply of 8 jars.
Before we start trading, we will ask you to assess the value of a share. You will
find a form in front of you. Please enter your trader ID (a number I give you), and your
assessment. Please also express how confident you are about your own assessment by
marking the respective box.
Order Submission: attached to these instructions is a yellow sheet of paper that
lists prices from 200-500. For each price, please record how many jars you would buy or
sell at this price if this were the market price. (This list is your demand-supply schedule.)
Note that the price is only established after all demands and supplies have been
collected and aggregated.
Rules: At any price, you may buy or sell at most 10 jars. If you do not want to
participate, then simply do not submit the yellow sheet.
23
Appendix B
INSTRUCTIONS FOR STUDENTS: FACE-TO-FACE TRADING
You will now participate in a simple face-to-face trading simulation. Before you is a glass
filled with nickels (5 cent coins). You may look at the jar, but you are not allowed to
open it. The content of these jars represents the fundamental value of one share of
company xyz. Please note that the jar itself has no value. All jars are identical and contain
the same number of nickels.
There are 10 identical jars that can be traded on the market.
Just before we start trading I give these jars to some students in the room at random.
The value of holding a share is as follows:
1. After the game ends, each initial owner of the jars pays the true value of the jar
(i.e., the sum of the nickels).
2. Likewise, those who own the jar at the end of trading will be paid the true value
(in return for the content of the jar).
Before we start trading, I will ask you to assess the value of a jar. You will find a form
in front of you. Please enter your trader ID (the number on the blue or green cover sheet
of the handout), and your assessment.
The organization of trading is simple: markets will be open for 10minutes. 1 minute
before the end I will announce that the market is about to close. All trading must stop
after I announce the end of trading. You may record last minute trades, but you cannot
negotiate further.
24
During these 10 minutes you may buy or sell (if you own it) the jar; you can trade
with any person in the room, at any price that you agree upon. You may also abstain from
trading.
Record keeping. The front of the cover sheet (blue or green) that has been given to you
indicates your trader ID, the back is your trading ticket. On it you must record your
trades. More specifically, recording works as follows:
1. Both parties record their trades on their trading tickets.
2. On your trading ticket you record
•
Your role (buyer or seller),
•
the ID of the other trader (a number)
•
and the price that you agreed upon.
3. The SELLER reports the trade to the market organizer who will record the
IDs of both traders and the transaction price. The market organizer will then
publicize the transaction price by writing it on the blackboard.
25
1
This result is true for the groups of finance students that I played this game with;
behavior of groups that were not trained in finance was slightly different.
2
For the latter two, the games are particularly useful in illustrating the different
trading mechanisms that are commonly used in capital markets.
3
For instance, in the simple game proposed by Ball and Holt (1998), the
fundamental value is $1+$6×(5/6) = $6 and computed as follows: At the end of each
round, each share pays a $1 dividend. After the dividend, each share is destroyed with
probability 1/6, the outcome being determined by the roll of a die. At the end of all
rounds, after the last roll of the die, the surviving asset pays $6. The fundamental value
can then be determined by backward induction. Before the last period, the fundamental
value is $1+(5/6)×$6 = $6. Iterating this argument, the fundamental value is a flat $6 in
every period. The other cited papers additionally allow decreasing or stochastic
dividends. (Comment: $1+$6(5/6)=$6 to clarify)
4
Having run the Ball and Holt game in several classes I have yet to encounter a
student who would compute the fundamental value correctly. I find this quite striking (in
a sample of about 80). All students in my sample are trained in basic finance, all are able
to compute expectations and know backward induction arguments, and most know how
to compute fundamental values, at least for this simple form.
26
5
An alternative is Feinstein (2000), who suggested auctioning an envelope that
contains either a high or a low value and the class organizer has given students some
information about the value. The purpose of his game is to illustrate the strong form of
the efficient market hypothesis.
6
Alternatively, a teaching assistant can compute average opinions and Walrasian
market prices during the lecture or the games.
7
For instance, the lag between arranging a trade, reporting it, and the price
publication becomes too large. Of course large classes can be broken up into subgroups
or teams.
8
Market microstructure matters: markets can be centralized (for instance, stock
exchanges) or decentralized (such as currency markets), they can be order- or quotedriven, there can be a monopolistic specialist (e.g. NYSE or Deutsche Börse) or many
market makers (NASDAQ), there can be a central, onetime clearing system (as in the
opening sessions at TSX, NYSE or for infrequently traded stocks on Paris Bourse) or
trading can occur continuously. The institutional market microstructure differences lead
to differences in trading volume, bid-ask-spreads, intra-day trading patterns and so on.
9
A nice illustration of this trading mechanism can be found in the movie Trading
Places: in one of the last scenes, the two heros engage in floor trading in the trading pit
for concentrated frozen orange juice (CFOJ). One should point out to students, however,
that the two main characters engage in illegal insider trading.
27
10
In contrast to the auctioning of a jar of pennies, instructors won’t be able to earn
their lunch money (see Thaler 1988): they make zero profits.
11
Again this is a noteworthy point in the classroom discussion because it illustrates
that people may have a preference for an open process. Of course, it is also possible that
students merely copy what they see on television or in the movies.
12
This confirms the recent theoretical result by Hong, Scheinkman, and Xiong
(2006), although the specifications that I looked at always had fewer shares than students.
It would probably be interesting to consider the case where all students get share
allocations so that the number of shares exceeds the number of students.
13
Although the jar is public information, when examining it with their ‘information
processing technology’ students create a piece of private information. The jar is thus a
metaphor of a financial statement.
Analysts process the information contained in these statements, and they have
different capabilities of processing quantities of information, of seeing linkages between
different pieces of information, etc.
14
The trading process in the Walrasian game is actually quite similar to Grossman’s
(1977) model where people submit complete demand-supply schedules.
28
15
Brunnermeier and Nagel (2004) show, however, that hedge-funds’ strategies are
more subtle, even if a fund has determined that a stock is overpriced.
29
REFERENCES
BALL S. B. AND C. A. HOLT. 1998. Classroom games: Speculation and bubbles in
asset markets. Journal of Economic Perspectives 12 (1): 207–18.
BAZERMAN M. H. AND W. F. SAMUELSON. 1983. I won the auction but don’t want
the prize.. The Journal of Conflict Resolution 27 (4): 618–34.
BRUNNERMEIER M. AND S. NAGEL. 2004. Hedge funds and the technology bubble.
Journal of Finance 59 (5): 2013–40.
CAMPBELL J. Y. A. W. LO AND A. C. MACKINLAY. 1997 The econometrics of
financial markets. Princeton University Press.
FAMA E. 1965. The behavior of stock market prices. Journal of Business 38 34–105.
FEINSTEIN S. P. 2000. Teaching the strong-form efficient market hypothesis and
making the case for insider trading – A classroom experiment. Journal of Financial
Education 26 (2): 40–44.
FORSYTHE R. AND R. LUNDHOLM. 1990. Information aggregation in an
experimental market. Econometrica 58 (2): 309–47.
GROSSMAN S. 1976. On the efficiency of competitive stock markets where trades have
diverse information. The Journal of Finance 31 (2): 573–85.
--------------------- 1977. The existence of futures markets noisy rational expectations and
informational externalities. The Review of Economic Studies 44 (3): 431–49.
GROSSMAN S. J. AND J. E. STIGLITZ. 1976. Information and competitive price
systems. American Economic Review 66 (2): 246–53.
--------------------- 1980. On the impossibility of informationally efficient markets. The
American Economic Review 70 (3): 393–408.
HARUVY E. AND C. N. NOUSSAIR. 2006. The effect of short selling on bubbles and
crashes in experimental spot asset markets. Journal of Finance 61 (3): 1119–58.
HONG H. J. SCHEINKMAN AND W. XIONG. 2006. Asset float and speculative
bubbles. Journal of Finance 61 (3): 1073–117.
LEI V. C. N. NOUSSAIR AND C. R. PLOTT. 2001. Nonspeculative bubbles in
experimental asset markets lack of common knowledge of rationality vs. actual
irrationality. Econometrica 69 (4): 831–59.
MARKS M. D. LEHR AND R. BRASTOW. 2006. Cooperation versus free riding in a
threshold public goods classroom experiment. The Journal of Economic Education 37
(2): 156–70.
PLOTT C. R. AND S. SUNDER. 1988. rational expectiations and the aggregation of
diverse information in laboratory security markets. Econometrica 56 (5): 1085–118.
SMITH V. L. 1962. An experimental study of competitive market behavior. The Journal
of Political Economy 70 (2):111-137.
SMITH V. L. G. SUCHANEK AND A. W. WILLIAMS. 1988. Bubbles, crashes and
endogenous expectations in experimental spot asset markets. Econometrica 56: 1119–
51.
THALER R. H. 1988. Anomalies the winner’s curse. The Journal of Economic
Perspectives 2 (1): 191–202.
31
TABLE 1: Outcomes of the Walrasian Games.
True Finance Average Standard k-th highest Number Price Allocation efficiency Average Number
Value trained
deviation
of traders
virtual
virtual
profits
losses 290
no
247
60
275
28
315 64%
40
14
340
no
248
93
320
36
330 64%
103
16
285
no
239
108
255
27
300 67%
43
7
205
no
212
55
250
26
290 52%
7
12
290
yes
280
42
320
17
285 88%
195
2
235
yes
241
54
250
17
225 82%
216
1
340
yes
339
106
250
11
280 92%
398
2
32
TABLE 2: Summary Statistics from Face-to-Face Trading
True Value
allocation efficiency
finance course
number of Assets
number of transactions
number of people
number of active people
ø price
ø opinion before
ø opinion after
σ opinion before
σ opinion after
kth opinion before
kth opinion after
max ante opinion
min ante opinion
max post opinion
min post opinion
buyer ante V − p
buyer post V − p
seller ante p − V
seller post p − V
ø of ante & post opinion
ø price/( ø post & ante)
(buyer ante V − p)/(ø post & ante)
(buyer post V − p)/(ø post & ante)
(seller ante p − V)/(ø post & ante)
(seller post p − V)/(ø post & ante)
buyer neg % ante
buyer neg % post
seller neg % ante
seller neg % post
205
29%
yes
7
12
21
13
233
254
228
137
31
250
240
800
150
300
175
44(172)
8(28)
-13(53)
1(35)
241
97%
18%
3%
-5%
0%
33%
42%
58%
42%
285
85%
yes
13
16
26
20
313
282
283
67
31
275
300
500
200
310
200
-20(82)
-19(56)
23(68)
28(52)
282.5
111%
-7%
-7%
8%
10%
69%
50%
19%
13%
205
67%
yes
6
12
16
12
289
231
269
48
40
200
220
250
105
250
120
-45(51)
-39(45)
54(41)
50(41)
250
116%
-18%
-16%
22%
20%
67%
67%
0%
0%
285
75%
yes
4
16
16
12
317
300
305
73
49
300
300
350
100
315
150
-53(132)
-38(58)
57(91)
50(52)
302.5
105%
-17%
-13%
19%
17%
56%
69%
25%
13%
205
50%
no
8
36
33
24
295
243
246
54
53
285
300
300
140
350
150
-52.5(91)
22(180)
49(126)
-9(153)
244.5
121%
-22%
9%
20%
-4%
59%
50%
35%
41%
285
43%
no
13
38
36
31
243
193
198
95
65
300
300
700
180
500
200
-33(87)
-8(92)
37(103)
26(83)
195.5
124%
-17%
-4%
19%
13%
53%
34%
24%
34%
290
80%
no
10
28
31
26
290
228
272
63
57
250
300
360
125
360
150
-43(64)
-4(60)
59(71)
11(50)
250
116%
-17%
-2%
24%
4%
67%
55%
22%
39%
340
82%
no
11
31
35
22
258
255
268
78
67
290
250
400
100
400
120
5(66)
24(72)
4(42)
-4(67)
261.5
99%
2%
9%
2%
-2%
35%
32%
35%
30%
285
75%
yes
4
18
13
13
328
323
308
103
64
300
300
500
225
500
250
-37(47)
-17(26)
33(36)
23(29)
315.5
104%
-12%
-5%
10%
7%
73%
55%
18%
18%
290
60%
yes
5
10
16
13
269
226
243
38
29
250
260
300
150
275
150
-1.9(100)
-19(57)
-17(127)
11(93)
234.5
115%
-1%
-8%
-7%
5%
61%
61%
33%
17%
Ø
65%
111%
-9%
-3%
11%
7%
57%
52%
27%
25%
Note: Symbol ø signifies an average and σ a standard deviation. The k in kth highest opinion corresponds to the number
of assets available in that setup. ‘neg’ stands for ‘negative’. ‘Buyer ante V-p’ is the buyer’s ex ante valuation minus the price
at which s/he traded, averaged over all transactions (similarly for ‘Buyer post’ and ‘Seller ante/post’)
33
Figure 1: Photos of the Jars. Both jars contain glass pebbles, where each pebbles is worth 5c/. The left
jar contains 68 pebbles ($3.40), the right jar contains 47 pebbles (= $2.35).
34
160
140
120
80
60
40
20
340
320
300
280
260
240
220
200
180
160
140
120
0
100
Quantity
100
Price in Cents
Volume
Demand
Supply
Figure 2: An example outcome for the Walrasian Game.
35
400
380
360
Price in cents
340
320
300
280
260
240
220
200
1:00
2:00
3:00
4:00
5:00
6:00
7:00
Time
Figure 3: An example from face-to-face trading.
36
8:00