The effect of relative decision making in the heterogeneous
expectation model for forecasting
Wouter Hendriksen∗
Evelien van der Hurk†
Jos. H. T. Rohling‡
February 16, 2012
Abstract
How to explain and predict price development in markets has long been an important
research topic. The traditional assumption in economics is that people behave rationally.
However, the current observed market behavior with periods of high volatility and strong
trends is not easily explained by rational acting agents.
Behavioral studies have shown that people in general do not act rationally. Recent literature
on forecasting market prices focusses on including bounded rational behavior. Comparing lab
experiments with the results of agent based models, it is shown that heterogeneous expectations
may partly explain the behavior of markets.
However, the bounded rationality does not only limit itself to the way people form expectations, it also extends itself to how decisions are evaluated. Behavioral studies show that
generally people make relative decisions. For example, a company could compare the success of
its strategy to the full set of strategies, but it is also possible that it measures it success only in
comparison to its competitors. This translates to market forecast models as agents evaluating
their strategy with respect to the performance of their direct neighbors only, compared to an
evaluation to the whole set of strategies.
In this study we incorporate this aspect of relative decision making by introducing a network
to the heterogeneous expectation model of Hommes (2011). We investigate whether the micro
patterns of strategy distribution over the population and the macro patterns of market price
development change when assuming relative decision making instead of fully informed decision
making.
We find that both the average degree as well as the network topology has a strong influence on the strategy distribution of the agent population. Moreover, this distribution changes
dynamically and is generally instable.
”In the real world people are heterogeneous: they do different things and change their strategies over
time. They learn and adapt their behavior, switch between strategies and that creates nonlinearities.”
Cars Hommes, Tinbergen magazine, fall 2010.
1
Introduction
People base their current actions on their expectations about the future. Since the work of Muth
(1961) and Lucas (1971) most research in Economics has assumed people to act rationally, and build
models for economic and market analysis based on the rational expectations hypothesis. Models of
∗ Technical
University Delft, Delft, The Netherlands, email:[email protected]
of Decision and Information Sciences, Rotterdam School of Management, Erasmus University Rotterdam, Rotterdam, The Netherlands, e-mail: [email protected]
‡ Leiden University Medical Center, Leiden, The Netherlands, email: [email protected]
† Department
1
market price development assuming rational behavior fail to forecast the high volatility and strong
trends observed. Work of Tversky and Kahneman (1981) shows that people generally do not make
rational decisions. Current research focusses on how new behavioral models including bounded
rationality may explain market development.
An overview of research on bounded rationality that focussed on applications in finance is presented
in LeBaron (2006) and Hommes (2006), more recently in Hommes and Wagener (2009) and Chiarella
et al. (2009).
A recent study on this topic is Hommes (2011). He focusses on Learning to forecast models where
the agents have heterogeneous expectations. Comparing the results of lab experiments with that of
the simulation model, he shows that heterogeneous expectations explain part of the patterns in the
lab experiment. Therefore, real life stock market development may be explained by heterogeneous
expectations.
In this heterogeneous expectation model, agents have full knowledge of all available forecasting
strategies and the performance of those strategies. In practice, complete knowledge about everyones
strategies is unlikely. Also, the behavioral study of Tversky and Simonson (1993) shows that people
make relative decisions. Intuitively, it seems more important for a company to perform better than
its competitors, than to invest in obtaining all information on all strategies of all actors in the
market. A model including relative decisions combined with limited knowledge therefore might
explain market behavior better.
In this study we extend the heterogeneous expectation model of Hommes (2011) to include relative
decision making. We introduce a network through which agents gather information about different
forecasting strategies. Decision of keeping or switching his strategy is based on the agents’ own
performance relative to the performance of the neighbors of the agent.
We compare our results to the results of both the real life experiment and the results of the agent
based models in Hommes (2011). Two of the questions presented in Hommes (2011), relevant for
this research are:
• How do individual forecasting rules interact at the micro level and which aggregate outcome
do they co-create at the macro-level?
• Will coordination occur, even when there is limited information, or will heterogeneity persist?
At the micro level, we focus on the distribution of forecasting strategies of the agents in the network.
The macro-level is in our setting identical to the one in Hommes (2011): price development. We
compare our model results to the results of both the lab experiment and the agent model in Hommes
(2011).
Comparing different networks with several average degrees, we see that relative decision making
causes a highly dynamic behavior of the distribution of forecasting strategies over the agents. The
topology and degree distribution both influence the development of strategies in the network. Still
all networks generate unstable results for the distribution in terms of the best strategy and the
distribution of strategies over the agents in the network.
2
Methodology
We investigate the effect of relative decisions on forecasting market price by introducing a network
structure to the agent model with heterogeneous expectations from Hommes (2011). We implement
2
different network structures with several levels of average degree and compare our results to the
results of Hommes (2011). We implement our model using Python1 and the Networkx2 package.
2.1
Heterogeneous expectations
We use the set of four heuristics as defined in Hommes (2011) that were obtained from fitting results
from the lab experiments: adaptive expectations (ADA), a strong and weak trend (WTR, STR),
and an anchor and adjustment rule (LAA). The heuristics are:
ADA pe1,t+1 = 0.65pt−1 + 0.35pe1,t
WTR
STR
pe2,t+1
pe3,t+1
LAA pe4,t+1
(1)
= pt−1 + 0.4(pt−1 − pt−2 )
(2)
= pt−1 + 1.3(pt−1 − pt−2 )
pav + pt−1
+ (pt−1 − pt−2 )
= t−1
2
(3)
(4)
Pt−1
where pav
t−1 = (
j=0 pj )/t is the sample average of past prices.
2.2
Networks and the choice of forecast strategy
We consider several networks to look at the effect of different topologies. We consider a regular
graph, where only direct neighbors are connected and no long distance connections are present.
All nodes will have the same degree. We create this network in Python by creating a WattsStrogatz network (Watts and Strogatz, 1998) with probability p = 0. The second architecture is
the small-world architecture, that contains few long-range random connections. Here we use the
Watts-Strogatz network with probability p = 0.2. Furthermore, we run experiments for the random
network, which is created by a Watts-Strogatz network with probability p = 1. Finally, we look at a
scale-free random network specified as a Barabási-Albert graph (Barabási and Albert, 1999). This
last network contains so-called hub-nodes. These nodes contain a large number of connections, but
are few in number, while a large number of nodes only contain a small amount of connections.
We compare these networks by keeping the average degree of each node approximately the same
over all networks. We are interested in information propagation of strategies through the network,
the effect of limited information and relative decision making. We surely create a different situation
from the full information model of Hommes (2011) for all nodes (agents) that have a degree of less
than 3. Because there are four forecasting strategies, agents with a degree less than 3 cannot have
full information. We will also test the networks with a larger degree to investigate if the results are
more in accordance to the full information model.
In our model, choice is relative. Also, a strategy can only be chosen when one of the current
neighbors of the agent, or the agent itself, follows that strategy. Finally, the probability of choosing
a different strategy now depends on the attractiveness of the known alternatives. Therefore the
decision rule, slightly adopted from Hommes (2011), is now dependent on the agent a and the set
of different strategies of it’s neighbors Ia :
na,i,t = δna,i,t−1 + (1 − δ)
exp(βUi,t−1 )
P
(5)
i⊂Ia
with
Ui,t = −(pt−1 − pei,t−1 )2 + ηUi,t−2
1 www.python.org
2 networkx.lanl.gov
3
(6)
2.3
Experiments
We set our parameters equal to the values in Hommes (2011): β = 0.4, η = 0.7, δ = 0.9. For each of
the graphs mentioned in 2.2 we take the number of edges per node to be (re)wired equal to 2, 4 and
49. For 49 edges per node, the average degree is about 24, hence the chance that all agents will have
full information is high, and therefore this setting is comparable to the model of Hommes (2011).
The other two settings lead to an average degree between 1 and 4. Consequently the number of
agents having limited information is significant.
We use a population of 100 nodes and run 5 experiments of 50 periods for each network and degree.
For each experiment, we set the price development equal to the price development with negative
feedback used in the lab experiment. Notice that this is a slightly different approach from Hommes
(2011). He fits the strategies to the data of the lab experiments, while we mimic the lab experiment
with our agent based model. In this study we restricted ourselves to negative feedback in the price
development.
We focus on the development of the distribution of forecasting method over the agent population
(the micro level) as well as the development of price (the macro level). We compare the results
of each network to the results of both the lab experiment and the agent based model of Hommes
(2011).
3
Results
The price development of the process with negative feedback is rather stable. This is similar in
all runs, and all network settings of our experiment, and may be due to the choice for negative
feedback.
For every run we generated three graphs showing price development, expectations, and the distribution of the strategy over agents. Furthermore for each time step we made a graph of the network
where the color of the nodes represent the strategy. In this way we can see the change in forecast
strategy over time. Typical results are shown in Figure 1 and Figure 2.
When focussing on the distribution of strategies over the agents in the network, we see a dynamic
pattern. In many cases, one strategy rises, but just before it can take over the full network, it’s
decreasing. In some situations though, the strategies are more equally distributed over the full
population during the 50 time period.
Focussing on the networks with the lowest average degree (between 1 and 2) we see that the most
prevalent strategies are ADA and LAA in all network topologies. Some difference between the
networks remain: in regular networks more often both strategies take a significant share of the
agent population, while in random networks it is either of the two, dependent on the simulation
run, that is significantly the most used strategy in the population. Scale free networks show a
similar pattern, but almost always the ADA strategy is the most common strategy in the network.
In general the distribution seems to go to some equilibrium at the end of the 50 periods.
For the networks with the highest degree, the most used strategies still include ADA, but exchanged
LAA for WTR in most cases. Here however we see a strong rise and fall of these strategies, where the
turning point is usually around period 20. Here the pattern seems to be more dynamic and possibly
distribution could drastically change over a longer period of time. There is not much difference in
this pattern over the different networks. The rise and fall of strategies is also something we can
recognize from the results of Hommes (2011).
Around an average degree of 4 we could very well be in a phase transition. In some runs, distribution
mimics patterns of low average degree networks. In other runs, the behavior is more similar to
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Figure 1: The price, expectations and ratio’s of strategies are displayed in this figure. The graphs are
typical examples of the model outcomes made by using different networks. The ratio’s of strategies in the
network more dynamic behavior at higher average degrees, indicating the strong influence of number of
connections between agents. All networks contain a 100 agents (n=100) and the time span is a 50 steps.
The random network (a), the regular network (b) and the Small-World network (d) are created with a
Watts-Strogatz graph, having a p equal to 1, 0, 0.2, respectively. The Scale-free network (c) is created
based on a Barábasi-Albert graph having an average degree of 3.8, similar to the Random network which
has a average degree of 4. The Small-world and Regular network have an average degree of 24.
networks with a high degree. In these networks, often agents are in between the two types as well:
some agents will have full information, similar to a high degree, other will have few neighbors, similar
to the average low degree distribution. Which group of agents wins depends on the precise network
topology of the run, and the outcome of the stochastic forecasting process. Because the behavior
of average degree 4 is so volatile, which could possibly indicate a phase transition, emphasizes the
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Figure 2: The strategies of agents is displayed with a graphical representation, where the colors of
the nodes indicated the strategy. White - ADA, orange - WTR, red - STR, dark red - LAA.
importance of limited information in the heterogeneous forecasting process.
4
Conclusions and future research
Our results show that there is a clear difference between relative decision making and full information. When agents have less neighbors, and hence limited information, the strategy distribution
of the network and the development of that over time is quiet different from a model with many
connections and that is therefore close to full information.
In the low average degree networks, the network structure plays a significant role. The different
networks were specifically chosen because information will spread differently in a network with
short distances as a scale free network in comparison to a regular or random network. The fact
that results change with different networks shows that the relative decision making and the level of
information available to agents has a significant effect on the choice of forecasting heuristic.
At an average degree of about 4, which is the borderline between full information and relative
choice, we see a very unstable pattern. This could very well indicate a phase transition between
the relative decision making process and decision making with full information. Consequently it
underlines the difference between a network structure with relative decision making and the full
information design of Hommes (2011).
Suggestions for future research
In this paper we focussed on the effect of relative decision making on forecasts. We have just
investigated a few of the settings for decision making and networks. Our results give an indication
that there is a significant difference between relative decision making and decision making with full
information. However, many questions remain for future research, among which:
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• Is there a difference between positive and negative feedback of price?
• Is there an actual phase transition around an average degree of 4? Are there more phase
transitions?
• What is the effect of more heuristics for forecasting?
• What network structures cause the dynamics, especially the rise and fall of a strategy?
• How well do relative decision models fit practice?
5
Acknowledgements
This research was enabled through a grand of The Netherlands Organization for Scientific Research
(NWO).
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