An early Agent-Based stock market:
replication and participation
László Gulyás1 , Adamcsek Balázs2 and Árpád Kiss2,3
1
Computer and Automation Research Institute
Hungarian Academy of Sciences
Kende u. 13-17 – 1111 Budapest, Hungary
[email protected]
http://www.sztaki.hu/~gulyas/
2
AITIA Inc.
Infopark sétány 1 – 1117 Budapest, Hungary
{abalazs, akiss}@aitia.ai
3
Loránd Eötvös University
Pázmány Péter sétány 1/c – 1117 Budapest, Hungary
Abstract. The Santa Fe Artificial Stock Market (SFI-ASM) is one of
the most prominent models of agent-based finance, a computational approach to study the complex system of the financial market. The SFIASM model has two regimes: in one of them the simulated time series
data is consistent with the rational expectations equilibrium, while in the
other, simulation results appear to be in accordance with actual financial
time series data.
The goal of this paper is twofold. First, it reports on the results of porting an early version of the SFI-ASM model onto the RePast simulation
platform. Replications form a very important methodological step in order to scrutinize computational results. Second, the paper describes an
extension to the model that takes it from the realm of agent-based theoretical experiments to that of participatory simulation. In participatory
simulations some agents are artificial, while human subjects control others. This setup offers a great opportunity to test both the assumptions
and the results of the model. The experiences of the first set of participatory experiments are also discussed, demonstrating how technical
trading may lead to market bubbles.
Keywords. Artificial stock market, Agent-Based models, simulation,
artificial and human agents, participatory experiments, technical trading,
market bubbles.
J.E.L. classification: C63, G19.
M.S.C. classification: 81T80, 91B26, 92B20.
1
Introduction
Economics provides many examples of social systems involving complex interactions among many individuals. Traditional models of these systems seek to
48
László Gulyás, Adamcsek Balázs and Árpád Kiss
simplify human behavior and yet to derive easily characterized aggregate macro
features. In some cases, for example, in case of many finance models, however,
the approach has only yielded mixed success. This led to the emergence of the
novel approach of agent-based finance. The most prominent model of which is
the Santa Fe Artificial Stock Market (SFI-ASM), where artificial agents make
investment decisions.
The SFI-ASM is a model with a risk-free financial asset (e.g., Treasury bills)
available in infinite supply that pays a constant risk-free return rate per period,
and with a single risky stock, whose fundamental share value is unknown to
the traders. Traders are identical except that each trader individually forms
his trading rules over time through an inductive learning process. Each trader
chooses his portfolio of financial assets in each period in an attempt to maximize
his wealth.
At the start of the market process, each trader has a set of rules that it evolves
over time in such a way that new rules are continually being introduced. This
setup yields a stable system with two distinct behavioral regimes. [8] In simple
cases, the simulated time series data is consistent with a rational expectations
equilibrium. In contrast, in more complex setups, the market does not appear to
settle down to any recognizable equilibrium. While market price vaguely follows
the fundamental value of the stock, upward and downward deviations exist that
may be called crashes and bubbles. In this regime, simulation results appear to
be in accordance with actual financial time series data. [1] [8]
This paper first reports on the results of porting the SFI-ASM model onto
the RePast simulation platform. Replication is a very important methodological
step in order to scrutinize computational results. Our implementation confirms
the results in [8]. Furthermore, we describe an extension to the model that takes
it from the realm of agent-based theoretical experiments to that of participatory
simulation. Participatory simulation is, in effect, a bridge between the laboratory
experiments performed in experimental economics and the abstract explorations
carried out in agent-based modeling. In a participatory simulation some agents
are artificial, while human subjects control others. This setup offers a great
opportunity to test both the assumptions and the results of the model. For example, participatory experiments may help calibrating certain parameters. On
the other hand, it may help ascertaining the stability of the outcomes, e.g., when
the artificial agents are confronted with agents with utterly different behavioral
patterns. Also, participatory simulation allows testing certain hypotheses about
human behavior. Our first experimental results contribute to the latter two topics. The results show that even a few agents that play a different strategy from
that published in the original paper may significantly alter aggregate market
performance. Furthermore, the experiments also point out differences in human
and computational agent behavior in this particular setting, and demonstrate
the effects of technical trading.
The paper is structured as follows. Section 2 describes the SFI-ASM model.
This is followed by a summary of our replication efforts. Section 4 introduces
the Participatory Santa Fe Artificial Stock Market (PSFI-ASM) model. Section
An early Agent-Based stock market: replication and participation
49
5 describes the settings of our first experiments with the PSFI-ASM model and
reports on the results obtained. Section 6 concludes the paper.
2
The Santa Fe Artificial Stock Market model
In the SFI-ASM model, time is broken up into discrete time periods. There exists
a risk-free financial asset (e.g., Treasury bills) available in infinite supply that
pays a constant risk-free return rate per period. There also exist a risky stock,
whose dividend is generated by an exogenous stochastic process. Computational
agents, the traders, attempt to maximize their wealth by choosing their portfolio in each time period. There are no complex instruments such as options,
nor is direct interaction between the agents. Traders are identical except that
each trader individually forms his trading rules over time through an inductive
learning process. At the start of the market process, each trader has a set of
randomly generated rules. Each rule determines what the agents should do in
a given market situation. The possible actions are buying or selling a stock, or
doing nothing.
Several versions of the SFI-ASM model exist. More modern versions are improved, among others, in their economic realism and in the use of more sophisticated trading rules. Keeping our sight on our final goal of transforming the
model into a participatory agent-based simulation, we trade-off sophistication
for simplicity. Therefore, in the following we will only consider the early version
of the model published in [8].
2.1
The artificial stock market
Let t = 0, 1, 2, . . . identify time periods, t = 0 corresponding to the initial
state of the system. Moreover, let A = {a1 , a2 , . . . , aN } be the set of agents, the
positive integer N denoting the number of agents in the system. The number of
stocks held by agent a in the tth time period is given by hta ∈ <, < standing for
the set of real numbers. Similarly, mta ∈ < denotes the amount of financial asset
(money) the agent owns. Notice, that both holdings and money is represented as
a real value, for reasons of simplicity discussed further below. The price pt ∈ <
per share of the stock depends on the overall buying and selling behavior of
the agents. The stock, however, may also pay a dividend dt ∈ < in money.
The agents’ money is assumed to be invested in a fixed-rate fund, such as a
savings account, that pays an interest rate r in each period. Agents can thus
make profit in three ways: due to the interest derived from their cash, due to the
dividend stream, or through speculation on price changes of their shares. The
wealth wat ∈ < of agent a is defined as
wat = mta + pt hta .
(1)
At the beginning of the tth period, the interest and the dividend is paid, so that
Mat+1 = r · mta + dt · hta ,
(2)
50
László Gulyás, Adamcsek Balázs and Árpád Kiss
for each a ∈ A and t ≥ 0. Then agents have the chance to change their holdings.
They can sell or buy one share, subject to the availability of shares or money,
respectively. The latter constraint can be relaxed by allowing the agents to borrow money subject to a certain limit. At the end of the period, bids and offers
are matched and the market clears. As there must be a buyer for every seller,
agents may not be able to achieve the holdings they desire.
Let bta = 1, if agent a attempts to buy a share in the tth period, and bta = 0
otherwise. Similarly, let ota = 1, if agent a offers to sell, and ota = 0 otherwise.
Furthermore, let B t and Ot stand for the totals of bids and offers, respectively.
X
Bt =
(3)
bta
a∈A
Ot =
X
ota
(4)
a∈A
If B t = Ot , then all bids and all offers are satisfied, giving
hta = ht−1
+ bta − ota
a
(5)
for each a ∈ A and for each t > 0 (where either bta or ota is zero). However, if
B t > Ot then all offers are fully satisfied, while only a fraction Ot /B t of each
bid is filled, giving
Ot
(6)
+ t bta − ota .
hta = ht−1
a
B
Similarly, the case when B t < Ot yields4
hta = ht−1
+ bta −
a
Bt t
o .
Ot a
(7)
The volume of trade V t in the tth period is then defined as V t = min(B t , Ot ).
Notice that this rationing scheme may result in non-integer holdings.
To complete our description of the market, we need to define how the dividends dt and prices pt are set. Dividends are generated by a discrete, stochastic
colored noise process,
¯ + ξt ,
dt+1 = d¯ + ρ(dt − d)
(8)
where d¯ is the theoretical dividend mean, ρ is a ’speed’ parameter, and ξ t is a
Gaussian noise source with mean 0 and a variance δ 2 . This is the discrete version
of the mean-reverting autoregressive Ohrnstein-Uhlenbeck process (see [8] and
[3]), often used as a simple model of stock market time series. [3] [1] In contrast
to the purely stochastic series of dividends, prices depend on the actual bids
and offers. If more agents want to buy than sell, the price should go up, while it
4
The rationing scheme described here is far from being fully satisfactory. In fact, this
is one point where more modern versions of the SFI-ASM had been significantly
improved. However, for the one-stock scenario discussed in this paper it works well,
even despite its lack of realism.
An early Agent-Based stock market: replication and participation
51
should fall, if the supply exceeds the demand. This is achieved by the following
formula.
pt = pt−1 · (1 + η(B t − Ot ))
(9)
The parameter η has a crucial role in determining market behavior. When
it’s small, the price adjusts slowly to different market conditions. On the other
hand, a large η leads to large oscillations of price. In principle, it is assumed to
be small enough to ensure that η(B t − Ot ) 1.
Obviously, agents have to pay for filled bids and they cash in on satisfied
offers. This yields the following formula for agent a’s money at the end of period
t, which completes the specification of the market.
mta = Mat −
2.2
Vt t t Vt t t
b · p + t oa · p
Bt a
O
(10)
The trading agents
The trading agents of the SFI-ASM model try to maximize their wealth by
regularly changing their portfolio. Their behavior is based on rules that specify
what to do when certain market conditions are met. The general form of the
rules is as follows: (condition, action, strength). The third element of the triple
is a real value, whose role will be discussed later. The action is a simple ternary
choice:
(i) bid: bta = 1, ota = 0,
(ii) offer: bta = 0, ota = 1, or
(iii) neither/hold: bta = ota = 0 (default action).
The condition part of a rule is a fixed-length string of symbols drawn form the
alphabet {0, 1, *}, e.g., 11*0*****1*0. This string is matched against a binary
string (with 0s or 1s only) of the same length, representing the current state of
the market. 0s and 1s in the condition string only match to the same symbols in
the market string, while *s match to any value. The symbols represent market
indicators, such as the price is above the fundamental value, or the price is higher
than the 100-period moving average, etc. When the corresponding statement is
true, the appropriate symbol is 1 and 0 otherwise. Therefore, rules specify certain
actions, based on the state of market indicators, allowing for some of them to
be indifferent to the application of the given rule.
Agents have a set of such rules. Each time the agent has to make a decision,
it first lists those rules whose condition is met and whose strength is positive.
Next it selects one of these randomly, with probability proportional to strength.
The action of this selected rule is then executed. If there is no matched rule,
the agent defaults to the neither/hold action. At the end of the period, the
strength of each rule whose condition was met is updated, according to the
following formula.
t
t
st+1
a,k = (1 − c) · sa,k + Aa,k · c · π
(11)
52
László Gulyás, Adamcsek Balázs and Árpád Kiss
where sta,k is the strength of agent a’s k th rule in the tth time period and c is a
small constant. Aa,k is a numerical representation of the rule’s action, so that

 −1, if the action is sell
Aa,k = +1, if the action is bid .
(12)

0,
otherwise
π t is the net profit made by investing in one share of stock in the tth period.
Its value is given by
π t = pt − (1 + r)pt−1 + dt .
(13)
The structure described above is a classifier system. [6] Its rules classify
the states of the market into categories, and then provide probabilities for each
possible action to be taken in each category. The agents in the SFI-ASM model,
however, may also improve upon their rules. This is done by the application
of a genetic algorithm [6] [7], which is executed at random intervals. When the
algorithm is run, the agent replaces 10-20% of its weakest rules by new ones.
The new rules are copies of some of the strongest ones, selecting candidates with
a probability proportional to the rule’s strength. However, the copied rules are
modified by mutation or crossover. Mutation randomly changes bits of the rule,
with probabilities adjusted so that the average number of *s stays constant.
Crossover combines a pair of ‘parent’ rules, getting a part of the new rule from
one parent, and the rest from the other.
3
Replication results
Testing published results is a most important step in the progress of science.
However, computational results don’t yield easily to tests. This is because the
implementation of the simulation may obscure important details, assumptions
or even mistakes. In addition, the code is poorly suited for publication in research papers, the traditional media of science. Therefore, computational results
may go untested for longer periods of time. This emphasizes the importance of
replication, the vehicle of testing for computer experiments. Replication means
the re-implementation of the simulation by independent authors, based on the
published specification of the model. This section reports on our efforts to replicate, in RePast [10], the version of the SFI-ASM model that was described in
the previous section.
The original version of the SFI-ASM model was implemented in C. [6] [7] It
was later replicated in Swarm and in RePast. For a description of these systems
the interested reader is referred to the websites [11] and [10], respectively. The
replication in RePast provides a good example of the significance of replication.
During this replication effort, Norman Ehrentreich has discovered a flaw in the
mutation operator in the agents’ genetic algorithm that corrupted part of the
results. [3] However, no replication of the earlier model published in [8] was made
so far. Another motivation to choose the early version was its better applicability
to participatory simulations.
An early Agent-Based stock market: replication and participation
3.1
53
The early artificial stock market model
The RePast implementation5 of the early SFI-ASM model closely follows the
specification of Section 2. In the following further details are given with respect
to parameter values and other considerations. These are based on information
in [6], [7], [8] and [3].
The stochastic process generating the dividends is slightly different from the
one given in Equation (8). It takes the following form.
¯ + ξ t , 5 · 10−5 ) ,
dt+1 = max(d¯ + ρ(dt − d)
(14)
In effect, this ensures that dt > 0, for each t ≥ 0. The values of the parameters
are ρ = 0.95 and d¯ = 0.75.
Agents are initially endowed with certain money minit and shares hinit . In the
simulation runs discussed here, agents start with minit = 200 units of money and
hinit = 5 shares of stock. Agents are allowed to borrow money (to buy stocks)
to the limit of their initial monetary endowment. That is, they are allowed to
buy until mta ≥ −minit holds. The interest rate on this credit rd is equal to the
interest rate paid for the fixed-rate investments, i.e., rd = r = 0.01.
Agents have 60 rules whose conditions reflect on the 12 market indicators
shown on Table 1. Bits 11 and 12 are zero information bits, providing a way to
check whether the agents’ behavior is actually dependent on market processes.
The table also shows the possibility of rules whose conditions can never get
matched, because they contain contradictory conditions. For example, a string
in which bit #1 is 0 and bit #2 is 1 would imply that 1/2 < pt · r · dt ≤ 1/4,
a clear contradiction. For this reason, agents have a meta-rule that generalizes
rules that haven’t been matched for a long period of time. Generalization is done
by randomly replacing a few bits in the condition string with the * symbol. The
length of the ’maximum sleeping period’ for rules is 200 in our studies.
Table 1. Market Indicator Bits (based on [6])
Bit
1
2
3
4
5
6
7
8
9
10
11
12
5
Market Indicator
pt · r · dt > 1/4
pt · r · dt > 1/2
pt · r · dt > 3/4
pt · r · dt > 7/8
pt · r · dt > 1
pt · r · dt > 9/8
pt > 5-period moving average price
pt > 10-period moving average price
pt > 100-period moving average price
pt > 500-period moving average price
On: 1
Off: 0
Available from the authors upon request.
54
László Gulyás, Adamcsek Balázs and Árpád Kiss
Agents have another meta-rule to help optimizing their strategies. According
to this, the action of rules with negative strength is reversed. The intuition
behind this is that if a condition was particularly weak for buying, it should be
good for selling, and vice versa.
What is left to specify, is the settings for the genetic algorithm (GA). The
number of rules replaced each time the GA runs is 10 in our experiments. The c
parameter from Equation (11) is 0.01. For new rules, the probability of crossover
is 0.3. In this case the condition strings of the two parents are crossed once, at
a random point, while the action is the action of a randomly chosen parent.
The strength of the crossed-over rule is reset to 0. In case of mutation, a few
randomly chosen bits are changed. The probabilities of going from 1 to 0, 0 to
*, or * to 1 have to be adjusted, however, to ensure that the average number
of *s remains constant. As pointed out by Ehrentreich [3], the model’s original
implementation was flawed at this point. Therefore, our implementation follows
the modified algorithm discussed in [3] to calculate probabilities.
3.2
The results of replication
The main findings of the early SFI-ASM model, reported in [8], can be summarized as follows. In simple cases — when the agent population is small, or they
only use a few rules, or facing a low-variance dividend stream —, the agents converge to an equilibrium, in which price follows fundamental value, volume is low,
and there are no significant anomalies, such as bubbles or crashes. The agents
become relatively homogeneous, using only a few simple rules. On the other
hand, in more complex environments, there is no clear equilibrium. Although,
the price usually stays close to fundamental value, it also displays major upward and downward deviations that may be regarded as bubbles or crashes. The
agents become heterogeneous, and thus trading volume also remains relatively
high.6
Our RePast implementation confirms these findings. Fig. 1 shows a typical
run in a complex environment. Deviations from fundamental value are visible and
trading volume keeps oscillating around 10% of the supply. The latter suggests
that agents are heterogeneous. High volume V implies that both demand and
supply is high, both being at least V . Therefore, the agents’ rule sets must differ
significantly, since facing the same market conditions V of them think it best to
buy, while V of them to sell. Indeed, Fig. 2 shows statistics of the fraction of
used (non-*) bits in the agents’ rule sets. Clearly, some agents have very specific
rule sets (the fraction being close to 1), while others apply very general rules
that only depend on a few indicators.
6
This is in contrast to later versions of the SFI-ASM model (e.g., in [1]). Despite that
they also have two regimes: equilibrium is reached when the agents’ learning rate
is slow, while market-like deviations arise when they learn fast. These results were
later questioned in [3]. It was found that after fixing a technical problem, agents
become similar and price follows fundamental value.
55
An early Agent-Based stock market: replication and participation
a)
b)
Fig. 1. Simulated time series of a) price (smooth curve) and fundamental value (crossed
line) and of b) trading volume. The data is typical of a 100-agent market with fast
learning rate (GA-activation in every 3rd period)
a)
b)
Fig. 2. Statistics of the fraction of used (non-*) bits in the agents rule sets. a) The minimum (crossed line), maximum (rectangles) found in agents, and the average (smooth
line) versus time. b) The distribution of the same measure among the agents
56
László Gulyás, Adamcsek Balázs and Árpád Kiss
Another measure of the agents’ homogeneity is the distribution of their
wealth. Initially, every agent is endowed with the same wealth. In contrast, Fig.
3 shows significant heterogeneity after a certain period of time.
Fig. 3. Wealth distribution in the middle of a typical run. The vertical axis shows
the number of agents in the given tenth (shown on the horizontal axis between the
minimum and maximum wealth)
The above results suggest that, in complex environments, the agents of our
implementation grow heterogeneous, both in their wealth and in their rules. At
the same time, as a community, they ’learn’ to manipulate price in such a way
that it follows fundamental value subject to a certain range of error. The level
of error appears to be dependent on the frequency the genetic algorithm is run.
If learning is slow, the deviations are larger, at least in the beginning. This is in
contrast with [1], but not with [8].
Despite the agents apparent ability to ’follow’ the fundamental value, their
heterogeneity, especially that of their wealth, suggests that interesting ’casting’
lies behind this social learning phenomena. Some agents grew smart and become wealthy, while others learn to be dumb, and loose out. In fact, this is an
obvious consequence of the closed system, but interesting nonetheless. In their
conclusions, Parker et. al. emphasize the self-organizing aspects of the SFI-ASM
[8], pointing at the sellers and buyers that mutually adapt their behavior, and
display organized system-level performance. In light of our replication studies,
this co-evolution appears to mean that some agents, in effect, have learned to
’sacrifice’ their wealth.
An early Agent-Based stock market: replication and participation
4
57
A participatory stock market model
The last section summarized our efforts to replicate the results of an early version of the SFI-ASM model. It was shown that our implementation yields results
similar to those reported in [8]. More importantly, the behavior of the artificial
stock market was, in many respects, similar to that of real world markets. In the
simulated time series, price followed fundamental value with occasional deviations. Arguably, this was due to the co-evolution of the agents’ strategies. That
is, the agents managed to adapt to one-another, i.e., they coordinate despite
their heterogeneity. While significant, these findings leave open the question of
whether these agents would be able to adapt to any strategy in the same way
as they did to those of their fellow agents. In other words, it is unclear how well
their meta-level rules would perform in a less sterile, more open environment.
The same time, it would be interesting to know how the artificial stock market
would appear to humans from the inside. Despite its simplicity, the aggregate behavior of the system is quite complex, lending itself to comparison with complex
real world systems. Less obvious, however, is whether humans would perceive
it as such a complex system. In short, we are interested in what effect human
players would have on the system, and conversely in how the system would affect
them.
Conducting theoretical experiments with human participants is not new to
the practice of science. It’s not unheard of in economics either. Experimental
economics has a long tradition, it’s growing literature dates as far back as the
early 20th century. [5] It’s often been applied to test the effect of human cognition to economic behavior, learning and adaptation. [2] Moreover, it has helped
forming theories about coordination and the failure of it, e.g., in ’social traps’.
[9] A prominent application area of experimental economics has always been
trading, with special emphasis on asset markets.
Traditionally, economic experiments are carried out in a laboratory setting,
participants playing a ’game’ according to a strictly defined protocol. Experimenters may use different techniques to record the unfolding of the game and to
extract results from the recorded data. In the past, typical methods were observation and videotaping, and also questionnaires. Quite naturally, computers began
to play an increasing role recently. The advent of the Internet opened up another
path, broadening the scope of some experiments both in space and in time. It is
safe to say that at least a handful of them is carried out somewhere at any particular moment in time. For example, at the time of this writing, a web-based artificial stock market game is going on at http://eco83.econ.unito.it/sumweb/.
[12] [13]
On the other hand, participatory simulation adds an interesting twist to
the general approach of experimental economics. In these experiments a number
artificial agents also participate in the game. Augmenting the studied population
with programmed actors may help generating particular scenarios for the human
participants. Also, it can be applied as a means to generate crowd behavior.
Furthermore, participatory simulation can be used to achieve a different end.
The introduction of human participants may help testing the sensitivity and
58
László Gulyás, Adamcsek Balázs and Árpád Kiss
assumptions of a computer simulation. It is often helpful to be able to compare
computational thought experiments with the behavior of actual people. This is
the motivation behind the Participatory SFI-ASM model described in the next
section.
Agent-based modeling is especially suitable to this task as its inherent representation precisely defines the interfaces of the individual actors, together with
the allowed interactions both among them and with their environment. This
makes it conceptually straightforward to include agents, whose behavior is defined externally by a human participant. It is important to emphasize that due to
the well-defined interface of the actors, artificial and human agents are undistinguishable from the point of view of the model, or from that of the participating
agents.
4.1
The participatory model
This section introduces the Participatory SFI-ASM model (PSFI-ASM), the participatory extension of the early SFI-ASM model described earlier in this paper.
This model and the experiments described later are the first application of our
General-Purpose Participatory Architecture developed for the RePast agentbased simulation platform. The GPPAR package is a collection of Java classes
[4] that helps transforming RePast models to participatory simulation. It’s applicable to a wide range of models, even to those that were written without the
knowledge of GPPAR. A more detailed technical description of GPPAR will be
reported elsewhere.7
In GPPAR, the simulation runs on a central server. Artificial agents inhabit
the server, while human agents connect to the simulation via the network by
running a client application on their own computer. The connections and all
communication are handled by the GPPAR infrastructure. Therefore, artificial
and human agents are undistinguishable from the point of view of the model.
This is illustrated by the fact that when clients log in, technically, they replace
an artificial agent in the model’s pool.
During the design of GPPAR, special emphasis was given to the ability to
replay experimental runs. This was achieved by an extensive logging functionality
that keeps track of the actions of both the artificial and human agents. See Fig. 4
for an illustration. This was augmented by a special execution mode of GPPAR,
in which the simulation is replayed, based on a given log file. This way, the
unfolding of events can repeatedly be observed, and results are easy to analyze.
In the SFI-ASM model, agents interact by indirect means only. Their only
action is attempting to buy or sell a share of stock, or to remain inactive. Consequently, human participants are presented with an interface where they can
press any of three buttons, corresponding to the above choices. To make their
choices, agents have access to various pieces of information. They are obviously
aware of the market indicators (see Table 1). Also, they must be aware of their
7
The GPPAR package is available from the authors upon request.
An early Agent-Based stock market: replication and participation
59
*******************
New Turn
******************
StockPrice=73.80403083485477
StockDividend=1.3714321781941976
Artificial Action, 43:null*** wealth=704.3731902659351
Artificial Action, 73:bid*** wealth=738.508665119635
Artificial Action, 17:null*** wealth=735.2813705615681
Artificial Action, 16:offer*** wealth=736.0779790234244
Participant Action, 0:null*** wealth=575.0
Artificial Action, 38:bid*** wealth=677.3654319907715
Artificial Action, 11:null*** wealth=735.2813705615681
Artificial Action, 85:offer*** wealth=735.2813705615681
Participant Action, 1:offer*** wealth=741.9104103203854
Artificial Action, 64:bid*** wealth=740.0594743932702
Artificial Action, 46:null*** wealth=751.9104103203854
Artificial Action, 28:bid*** wealth=806.1446346640462
Participant Action, 2:bid*** wealth=677.3654319907715
Artificial Action, 20:bid*** wealth=747.6157392703483
Fig. 4. Log file extract (Participatory SFI-ASM simulation)
own assets: their money and their shares. While the latter doesn’t follow immediately from the model description in Section 2, it is implicit in the limitations
given by the budget constraints, i.e., that they cannot sell when they don’t own
any share and that they cannot spend limitlessly.
For reasons of understandability, we have decided to provide human agents
with the base values of the market indicators. That is, they are not confronted
with a string of bits, but rather, they are given the values of measures (i.e.,
pt ·r ·dt and the moving averages). Also, the last two, zero information bits of the
indicator string are kept from human participants. Following a similar argument,
they are presented with their own money, number of shares and wealth. While
artificial agents are not directly aware of these values, it would be unnatural to
keep them from human players.8
There is also a piece of implicit information available to programmed agents.
Artificial agents update the strengths of their matched rules based on Equation
(11). This includes the net profit made by investing in one share of stock in
the given period as defined in Equation (13). Therefore, human players are also
presented with this information. Also, they would hardly feel comfortable in the
environment unless the running price of the stock is displayed to them. The user
interface of the Participatory SFI-ASM client application is shown on Fig. 5.
8
In fact, it would be interesting to see how the behavior of artificial agents would
change if they were able to evolve strategies that take their money and share holdings, and their wealth into account. It seems reasonable to think that, for example,
the level of risk-aversion should be different when being poor versus after growing
wealthy.
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László Gulyás, Adamcsek Balázs and Árpád Kiss
Fig. 5. The Client Application of the Participatory SFI-ASM model
Our goal, in creating the experimental environment for the human participants, is to present them with a game as real and as exciting as possible. Therefore, we want to imitate the fast-changing, on-line nature of stock markets, in
which prices change by the fraction of a second. However, this presents a technical problem as the original SFI-ASM model is organized around rounds, each
agent making a decision in each round. Clearly, human agents cannot be expected to make decisions in the fraction of a second. The GPPAR package offers
several ways to deal with the extended time requirement of human participants.
The simplest of which is a timing-out system, in which participants are given a
certain amount of time to make their move. If they fail to do so, a default action
is executed. However, giving enough time to deliberate would slow down the
simulation significantly, destroying the illusion of a real-time game. Therefore,
we decided to keep the time-out low (at the order of 0.1sec), but regard the participants’ bids and offers as continuous. That is, the last action is resubmitted in
each round, until the player changes it, e.g., to ’do nothing’. Technically, this is
implemented by defining the default action, submitted when timing-out, as the
last action initiated by the user.
However, there is a more fundamental problem with the real-time version of
the SFI-ASM model. According to the specification in Section 2, interest and
dividend is paid in every round. This results in the continuous growth of wealth,
which, given a high frequency of rounds, becomes a very rapid one. Besides being
at odds with reality, this implies that the relative weight of the profit that can
be made on stocks declines very fast. (This is because of the limited supply of
shares, which is the total sum of the agents’ initial endowments.) To overcome
this problem, we stretched out the growth process by rarefying interest and
dividend payments to every 5th period.
An early Agent-Based stock market: replication and participation
a) Price (smooth line) and fundamental
value (crossed line) time series
61
b) Wealth (smooth line) and base
wealth’ (crossed line) time series
Fig. 6. Motivational time series graphs provided for human agents
In order to foster the illusion of a real-world stock market, we also provide
human agents with time series graphs. First, they are able to see the history of
the price and the fundamental value (i.e., dt /r). This is illustrated on the left
panel of Fig. 6. Another reason for graphic presentation is human preference
for visual information. Also, we want participants to be motivated to give their
bests. Naturally, the goal set for them is to make a fortune. However, due to
reasons discussed above, agents in the SFI-ASM increase their wealth, more or
less independent of their actions. Therefore, we introduce the notion of ’base
wealth’, which stands for the current worth of the agents’ initial endowment. In
other words, it is the wealth of an agent that did nothing, but kept its initial
money in the bank, and its endowment of shares in the drawer. The second
time series graph displayed by the PSFI-ASM client, shown on the right panel
of Fig. 6, provides this information to the users. It shows the time series of the
player’s wealth, together with that of the ’base wealth’. The current difference
between these values is also shown numerically among the agent’s statistics. This
way, players can have a notion of their performance, both current and historical,
which may motivate them to make an effort.
Obviously, the additional information provided for human players (e.g., the
history of values, the running price, etc.) creates asymmetry between the human
and the artificial side. However, as argued above, the asymmetry is already
inherent in the agents’ different computational ability. Thus, we consider the
above setup to be a good and balanced compromise.
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László Gulyás, Adamcsek Balázs and Árpád Kiss
Experimental results
This section presents the results of our first experiments with the Participatory
SFI-ASM model. One major goal of these experiments was to demonstrate the
workability of the PSFI-ASM model. Besides this, however, we were seeking
information about a number of exciting questions.
For example, [1] and [8] claims that their classifier agents adapt to the different behaviors of the market, thus making sure price always stays relatively close
to fundamental value. However, our hypothesis was that this is mainly due to
their inherent similarity. That is, despite the significant differences in their parameters and rule sets, the meta-level algorithm of the agents remains the same.
This fact could ease the process of adaptation, and could smooth out market
deviations. Therefore, we expected that human participation would increase the
deviation from the rational expectations equilibrium. If true, this would present
itself in the form of larger ’crashes’ and ’bubbles’, and perhaps in an increased
level of trading volume.
We were also interested in the performance (relative wealth) of the different
types of agents. On one hand, computational agents apply a very simple learning
algorithm. Despite the simplicity of their rules and their information set, the artificial agents are fast, so the best among them, in effect, optimize their behavior
to the current and past states of the market. On the other hand, human participants face a large set of rapidly changing information and are forced to make
quick decisions. Therefore, we hypothesized that computational agents would
outperform human participants, at least initially. However, humans can carry
over their knowledge to subsequent runs. Hence, we were interested in the rules
the players would develop, and in the effects of potential learning in repeated
games.
Finally, more as a measure of our success of motivating our players, than a
question of theoretical importance, we were interested in how human participants
perceive the market. Will they think it is behaving in a complex fashion, or will
they see through it easily? Will they be excited and start theorizing about their
best strategy, or will they fall back to a simple, monotonic pattern of behavior?
The answers to these questions will be discussed next.
5.1
The first participatory experiments
The first set of experiments was carried out at Loránd Eötvös University, and at
AITIA, Inc., both in Budapest, Hungary. The participants were computer science students, and employees of the company, respectively. They were all skilled
computer users, but they lacked any stock market experience. The participants
used personal computers (running a mix of operating systems, including MS
Windows 2000 and Linux), connected through a local area network. Despite
their physical proximity, participants were not allowed to discuss the game or
their performance, before the end of the experiments. The only information the
players received between the runs was the identity of the overall winner. The
An early Agent-Based stock market: replication and participation
63
PSFI-ASM simulation server was also running on a networked personal computer.
The experiments began with an introductory session, where participants were
told the rules and goals of the game, and were the client application’s interface
was explained to them. The description of the game rules was limited to the
abstract interface of the SFI-ASM agents. That is, they were told about their
possible actions, or about the fact that their bids would not necessarily be filled.
However, they didn’t know the exact rules of market clearing, nor did they have
direct knowledge about how the price was determined. Similarly, they had no
knowledge about the dividend stream, except that it was a dynamic process,
following its internal logic.
The introductory session was followed by a short trial run of the system,
when the participants had a chance to see the system in action. Afterwards,
they could ask questions, which were answered by the experimenters, unless
they were about the internal details of the system, or were directly connected to
one of the hypotheses of the experiments.
Then the experiments were run. Each time, the simulation was stopped without warning, at the discretion of experiment leader who was uninformed about
the actual course of the current run. The reason for this ’random’ stopping rule
was to avoid human strategies that could take the extra information of the experiment’s length into account. In the particular experiments, the system was
run for about 5 minutes (1200-1500 rounds).
Finally, after the intended number of experiments, the participants were
asked to fill in a questionnaire, which contained questions about their strategy,
the use of the different market indicators, about their errors or possible improvements, and about their perception of the game. It also contained questions about
the user interface, and about possible technical improvements.
The results presented in the following section are based on runs with 8-10
human participants, facing 20, 100 or 400 artificial agents.
5.2
Results of the experiments
This section summarizes the key findings of our first set of experiments with the
PSFI-ASM model. We found, and this was backed by the questionnaires, that
the players enjoyed the game, so we succeeded in constructing a challenging
environment that motivates users. The players also confirmed our expectations
about the preference for visual information. In fact, they were asking for more
graphs, and, as one would expect, for more information.
More importantly, however, we found that our first hypothesis was true. The
presence of human traders yielded higher deviations. One measure of deviations
is the cumulative difference between stock price and fundamental value.
Dt =
t X
i di p − r
i=1
(15)
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László Gulyás, Adamcsek Balázs and Árpád Kiss
The a)-c) graphs on Fig. 7 plot Dt versus time, comparing participatory
t
experiments (Dparticipatory
) to corresponding simulations with identical paramt
eters (Dcomputational ). It is clear that significant human presence increases the
level of deviation. The strength of the effect is less clear, however, since Equation (9) implies that larger populations (N ) yield larger deviations, if η remains
constant. To overcome this problem, η was set to a higher value for the small
population (20 agents). Graph d) on Fig. 7 shows the normalized difference after
1050 steps as a function of the percentage of human participants. By normalt
t
ized difference we mean the difference between Dparticipatory
and Dcomputational
,
divided by the number of agents. The graph suggests that the more human players, the larger the deviation is. Unfortunately, we don’t have enough data to test
this hypothesis. Especially, as the graphs show the repeated performance of the
same participants, in the order of b), c) and a). Therefore, the differences may
also show the effect of human learning (this issue will be discussed later).
a) 2% (8 players of 400 agents)
b) 8% (8 players of 100 agents),
second run
c) 40 % (8 players of 20 agents)
d) The scaling of the normalized
difference as a function of the
percentage of human participants
Fig. 7. Cumulative market deviations in runs with and without human players. (The
deviation axis shows the values in millions.)
The effect of human players on the trading volume is less obvious. In our
experiments, human participants sometimes increased the volume with respect
to the identical run; sometimes the value stayed the same, while in other cases
it was actually lowered. However, these last cases corresponded with the huge
initial market bubble discussed next, so perhaps they could be ascribed to computational agents adapting to an extreme initial market.
An early Agent-Based stock market: replication and participation
65
Despite the increased level of market deviations, in the long run price always
oscillated around fundamental value. This suggests that the artificial agents are
able to adapt to the behavior of humans, but it is also clear that, quite naturally,
their ability to keep the system close to equilibrium is limited. This latter statement is even clearer in the light of Fig. 8 summarizing the very first run of our
experiments (not included in Fig. 7). What happened here was that inexperienced human participants wanted to buy unanimously, so the price skyrocketed.
Their quest for shares is understandable psychologically, considering that they
had money (200 units) and the price was low (around 80 unit). The figure also
shows that computational agents were able to bring the system back towards
equilibrium. However, a detailed look at the logs reveal that this happened only
after all the human buyers backed down. This happened despite that humans
only represented a mere 8% of the traders.
The results of this very first run also provided a good example of the effects
of learning by humans. In the subsequent runs, the initial price-peak gradually
diminished, as demonstrated by the graphs of Fig. 9. In fact, most players reflected on this episode in their questionnaires, mentioning that they deliberately
avoided buying in the beginning periods of later runs.
a) 8 players of 100 agents
b) 100 computational agents
Fig. 8. Price (smooth line) and fundamental value (crossed line) history of the first
experimental run, and that of the corresponding simulation
The effect of human learning is also obvious in the performance (relative
wealth) of the agents. Before studying these findings in more detail, let’s consider
a few important issues. As discussed in Subsection 4.1, the wealth of the agents
grows rapidly. Moreover, due to the combination of the limited supply of shares
and frequent payment of interests and dividends, the importance of stock market
profit diminishes relative to the gross wealth. Because of this, the performance
is path dependent in the sense that early successes overweight later ones. This
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László Gulyás, Adamcsek Balázs and Árpád Kiss
a) Second experimental run:
8 players of 100 agents
b) Computational run corresponding
to the second experimental run
c) The third experimental run:
8 players of 20 agents
d) Computational run corresponding
to the third experimental run
Fig. 9. The diminishing peak in the runs subsequent to that shown on Fig. 8
An early Agent-Based stock market: replication and participation
67
makes performance comparisons difficult, as the longer reaction time of human
players further shortens the period when decisive profits can be made. On the
other hand, computational agents always start from a random set of rules, and
thus require time to evolve robust strategies. This factor may or may not be
enough to balance the previous disadvantage of humans.
a) The first experimental run, also shown on Fig. 8. (8 players of 100 agents)
b) The second experimental run. (8 players of 100 agents)
Fig. 10. The performance of human and computational agents in the first two experimental runs
Despite all these concerns, Fig. 10 and Fig. 11 show that computational
agents outperformed humans in the initial runs, but later the players improved
significantly. The graphs display the normalized wealth of the worst and the best
computational and human agents, together with the average normalized wealth
of each type. By normalized wealth we mean the ratio of the agents’ wealth
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László Gulyás, Adamcsek Balázs and Árpád Kiss
and the base wealth (defined in Subection 4.1). Note that as agents can be
indebted, this measure can be negative, too. Fig. 10 a) shows the result of the
large market deviation of the first run. Clearly, even the best human participant
performed worse than the worst computational agent. This situation improved
in the second run (see graph b)) with the average human wealth being about
equal to the wealth of the worst performing artificial agent. This demonstrates
the effect of human agents carrying their knowledge over to the next run.
In the two later runs shown on Fig. 11, the best performing agent was a
human. On graph a) the average human wealth was around the maximum computational performance. On graph b), however, average human performance is
only slightly better than that of the artificial agents. This may be due to the
different number of participating agents in the two runs.
There are a number of striking regularities on the graphs of Fig. 10 and Fig.
11. First, the average of computational performances always appears to be very
close to unity. If there weren’t human players present, this would be a natural
consequence of the closed system. With human participants, however, this points
to a potentially interesting phenomenon. Moreover, it was a human participant,
in all four cases, who gave the worst performance. This suggests that while some
participants learned quickly, some others failed to develop a viable strategy.
Finally, the wealth of worst and best human players appears to follow a similar
path on the two graphs of Fig. 10. Most likely, this is the side effect of the market
bubbles shown on Fig. 8 a) and on Fig. 9 a). Human players were caught up in
buying during the initial period, and they paid a serious price when the bubble
burst.
The ultimately most interesting findings, however, may lie beneath the surface of the results above. It is connected to the strategies developed by the participants. According to [1], theorists and market traders have strikingly different
views about markets. Standard theory assumes identical investors who share
their rational expectations about an asset’s future price. Consequently, speculation cannot be profitable, except by luck; trading volume stays low, and market
bubbles and crashes reflect rational changes in the asset’s valuation. In contrast,
it is needless to argue that traders do speculate in practice. Also, market deviations are often ascribed to ’market psychology’. There is also an interpretation
of these differences at the level of practical rules. If speculation works, technical
rules that are based on only price or trade volume information may be useful.
According to rational expectations theory, however, only fundamental strategies
that relate price to fundamental value by using dividend information will yield
success.
Based on their answers to the questionnaire, initially all participants applied
technical trading rules. They bought if the price was low, and sold if it was high.
Then, gradually, a few of them discovered fundamental strategies. Finally, the
winning strategy of Fig. 11 was perhaps the purest of fundamental strategies:
buy if price < f undamental value, sell if it is the other way around. These
early results suggest that technical trading lends itself easily to inexperienced
traders, but fundamental strategies perform better in this artificial stock market.
An early Agent-Based stock market: replication and participation
a) The third experimental run. (8 players of 20 agents)
b) The fifth experimental run. (8 players of 400 agents)
Fig. 11. The performance of human and computational agents in two later runs
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László Gulyás, Adamcsek Balázs and Árpád Kiss
This latter may well be due to the dominance of computational agents, who
might have developed mostly fundamental strategies. To prove this hypothesis,
however, is a topic of future research.
On the other hand, the contribution of technical trading to market deviations
is clearly demonstrated by the price-peak of the first experimental run (see Fig.
8). When all players applied a simple technical rule, a huge bubble occurred.
As in the subsequent runs they gradually learned to refrain from it, at least in
the initial rounds of the game, the peak disappeared. Similarly, the increased
market deviations of Fig. 7, due to human participants, appear to be an effect of
technical trading. Especially, as both of them decrease in later runs, when players
began adapting fundamental strategies. However, the total number of agents has
also varied, which may also have contributed to the effect. These issues motivate
further research, but the findings appear to suggest that technical strategies
drive large market deviations, at least in this artificial system.
6
Conclusions
This paper reported on the replication and participatory extension of the early
Santa Fe Artificial Stock Market model. Our replication efforts confirmed the
main findings reported in [8]. However, we also pointed out that the market’s
self-organization, when agents mutually adapt to one another and induce stock
price to follow fundamental value, may depend on the bad strategies certain
agents evolve.
On the other hand, the participatory extension demonstrated the viability
of the approach of participatory agent-based simulation. It was showed that
blending techniques of experimental economics and agent-based modeling can
help testing hypotheses both about human economic behavior, and about theoretical assumptions embedded in computational models. Among others, our
experiments confirmed that technical trading could lead to market deviations,
i.e., to bubbles and crashes.
7
Acknowledgements
The authors would like to thank the students of the Game Theory in Artificial
Intelligence class of the Loránd Eötvös University for their participation in the
experiments. Similar thanks are due to the employees of AITIA, Inc. We are also
grateful to Norman Ehrentreich for letting us consult with his replication of the
modern SFI-ASM model
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