In this section we, first, present two different forecasting heuristics and study the dynamics of the corresponding models with homogeneous expectations. Then, we combine the heuristics in the switching model.
We evaluate the explanatory power of different models in two ways. In this section we apply a first test. We compare a 50 −periods ahead model simulation (so-called simulated path) with the experimental outcomes. The price dynamics, generated by a model, must have some of the characteristics of the price developments observed in the experiments. In the negative feedback market the price must fluctuate heavily in the first few periods and then converge quickly to the equilibrium value, see Fig. 1a. In the positive feedback market the price must oscillate slowly and either gradually converge towards the constant level or not converge at all, see Fig. 1b. After identifying a model that satisfies these requirements for the simulated path of 50 periods, we apply a second test in Section 4. There the model performance in one-period-ahead forecasting is investigated numerically, the models are optimized over parameters and tested for robustness.
First-order heuristics
Before defining the switching model a small number of forecasting heuristics has to be chosen. A prediction heuristic must fulfill a certain number of conditions to be relevant. It should be informationally accessible to the experiment’s participants, and it should be simple and intuitive for them to use. We define the following two heuristics. The first heuristic is an adaptive heuristic given by
$$ p_{t+1}^e=wp_t+(1-w) p_t^e, $$
(6)
where an agent’s expectation of the price in the next period, \(p_{t+1}^e\), depends on the price of today and agents’ expectation about the price of today, with weights w and (1 − w), respectively.
The second prediction heuristic is a trend heuristic of the form
$$ p_{t+1}^e=p_t+\gamma (p_t-p_{t-1}), $$
(7)
where the expectation for the price in the next period depends on the last price plus γ times the last price change. This heuristic is interpreted as people expecting a constant trend in price developments.
Dynamics generated by single heuristics
What kind of aggregate behavior does each of these heuristics imply for the negative and positive feedback environments? Figure 2 shows the deterministic simulated paths with the adaptive and trend heuristics, i.e., 50-periods ahead simulations by the model with homogeneous expectations. The first forecasts generated by the heuristics are set to 50 in this simulation.
The adaptive heuristic (Eq. 6) with w = 0.75 generates oscillatory converging price dynamics in the negative and monotonically converging price dynamics in the positive feedback markets, see Fig. 2a. This outcome is similar to the dynamics observed in the negative feedback sessions, but is very different from the oscillations observed in the positive feedback sessions. Notice that a decrease of the weight w in Eq. 6 leads to less oscillating and, eventually, to monotonic convergence under negative feedback and very slow convergence under positive feedback, as illustrated by example with w = 0.25.
Straightforward stability analysis reveals that the trend heuristic (Eq. 7) produces converging prices in the negative feedback market when − 21/20 < γ < 1/40, and converging prices in the positive feedback market when − 41/40 < γ < 21/20. For γ outside of the interval, the price dynamics under corresponding feedback diverge. Interestingly, all the values of γ estimated in the positive feedback sessions of the HHST experiment fall within the interval of convergence under positive feedback. But the dynamics with these γ’s would diverge under negative feedback. The dynamics of the model where expectations are generated by the trend heuristic are illustrated in Fig. 2b. For γ = 1 the dynamics under negative feedback (left panel) does not converge to the equilibrium but remain bounded because the forecasts in the simulations are limited to the interval [0, 100]. When γ is changing to 0.5 the initial oscillations are less wild but eventually converge to the same 2-cycle. The right panel of Fig. 2b shows that for γ = 1 the price oscillates on the positive feedback market, resembling the experimental outcome. When γ decreases the oscillations are less pronounced and convergence eventually become monotonic, as shown for γ = 0.5.
Heuristic switching model
In the model of heterogeneous expectations, different heuristics can be used. Consequently, the average price expectations in Eqs. 1 and 2 is given by
$$\overline{p_t^e} = \sum\limits_{h=1}^H n_{h,t}p^e_{h,t}\,, $$
(8)
where H is the number of heuristics and n
h,t
is the impact of heuristic h at time t, which depends on the past performance of the heuristic.
The performance of the heuristics is measured by squared forecasting errors, consistently with incentives given to the participants in the experiment. The performance of heuristic h at time t is given by
$$ U_{h,t}=-(p_{t}-p_{h,t}^e)^2 +\eta U_{h,t-1}\,, $$
(9)
where p
t
is the realized price obtained by applying Eqs. 1 or 2, respectively. The parameter 0 ≤ η ≤ 1 represents the weight that agents attribute to past forecasting errors. The impact of heuristic h changes according to a discrete choice model with asynchronous updating (Hommes et al. 2005; Diks and van der Weide 2005)
$$ n_{h,t+1}=\delta n_{h,t} + (1-\delta) \frac{\exp(\beta U_{h,t})}{Z_{t}}\,, $$
(10)
where \(Z_{t}=\sum_{h=1}^H \exp(\beta U_{h,t})\) is a normalization factor.
Two parameters are important in Eq. 10. The parameter δ is inversely related to the frequency with which every agent updates “active” forecasting heuristic. Positive values of δ capture the tendency of people to stick to their previously chosen rule despite the evidence that an alternative rule performs better. Such inertia is widely reported in experiments (Kahneman 2003). In a large population, δ is also the average percentage of agents who do not update their heuristic in every period. The parameter β ≥ 0 determines how strongly those agents who update their heuristic react to a difference in performance between heuristics. If β = 0 agents will not consider the differences in the performance of the heuristics at all; all heuristics will be given equal impacts. If, on the contrary, the value of β is very large, agents who update their forecasting heuristic will all switch to the best performing heuristic.
In order to simulate the model, one should
-
choose H different forecasting heuristics;
-
fix three learning parameters, β, η and δ;
-
initialize prices in order for the heuristics to yield the initial forecasts;
-
initialize the impacts for all heuristics so that the initial forecasts are combined to determine the average price forecast.
Given these initializations, the model works as follows. For every time t, first, the forecasts \(p^e_{h,t}\) of H heuristics are computed on the basis of past prices. Second, they are combined using Eq. 8 to provide the average price forecast. Third, the price predicted by the model at time t is computed using Eq. 1 for the negative feedback market or Eq. 2 for the positive feedback market. This price is denoted simply as p
t
. Fourth, the performance of every heuristic U
h,t
is calculated using Eq. 9 on the basis of the realized price p
t
. Finally, the relative impacts of heuristic for the next period are computed using Eq. 10. The same steps are then repeated for time t + 1, and so on.
Dynamics of the heuristics switching model
Let us apply the heuristic switching model (HSM) given by Eqs. 9 and 10 to the experimental results of HHST. The parameters of the heuristics are chosen as w = 0.75 for the adaptive heuristic (Eq. 6), and γ = 1 for the trend heuristic (Eq. 7). Recall that the heuristics with these parameters describe the two markets relatively well: the negative feedback market is well described by the adaptive heuristic, while the positive feedback market is well described by the trend heuristic. After some trial and error simulations, we set the parameters of the HSM to β = 1.5, δ = 0.1 and η = 0.1. We also choose equal initial impacts of both heuristics and p
0 = 50 as the initial price.
Figure 3a and b show the outcome of the model’s simulations in, respectively, the negative and positive feedback environments. In the left panels the two types of dynamics of the simulated path are shown. The lines show the simulation without noise in the laws of motion (1) and (2), while the circles correspond to the simulation with the same noise realization ε
t
as in the experiments. The right panels show the evolution of the heuristics’ impacts for the simulated path without noise. We observe striking difference in the dynamics between the negative and positive feedback environments. Indeed, the price dynamics of the heuristic switching model do adhere to the characteristics of the experimental outcomes in both treatments. In the negative feedback market the price oscillates heavily in the first periods and then quickly converges. In the positive feedback market the price slowly and (when augmented by the experimental noise) persistently oscillates around the equilibrium.
It is particularly informative to analyze the evolution of the heuristics’ impacts. When the feedback is negative, the impact of the trend heuristic immediately falls to almost 0 and increases later on only at the stage when the price has already converged to the equilibrium level, i.e., when the predictions of both heuristics are similar. When the feedback is positive, the opposite phenomenon takes place with the trend heuristic dominating from the beginning of the simulations. The intuition of this result is as follows. The trend heuristic performs well during the long phases of the trends and performs poorly during the periods with frequent fluctuations around the constant price. At the same time, an extensive use of the trend heuristic results in the trends under the positive feedback and in oscillations under the negative feedback, see Fig. 2b. Thus, under the positive feedback, the success of the trend heuristic reinforces its use, which makes the trend in prices sustainable. The adaptive heuristic performs relatively poorly during the trend phases and loses its impact. On the other hand, under the negative feedback, the trend heuristic generates oscillatory dynamics on which it performs very poorly, much worse than adaptive heuristic. Coordination on the adaptive heuristic leads to fast convergence through initial oscillations, as shown in Fig. 2a.
The initial 10 periods of the simulations are explained now, but what happens next? In the positive feedback market agents attach a higher impact to the trend heuristic at the trend phases and decrease their impacts when the price development changes direction. Even when all subjects use the trend-following heuristic, the trend cannot be sustained forever and, at a certain moment, the trend will be reverted and the impact of the adaptive heuristic will grow. This occurs in the periods 15–17 of the simulations. Afterwards the downwarding trend reinforces the use of the trend heuristic, but since the price is already close to the equilibrium, the relative impacts of the heuristics are similar. Notice that the model generates such convergence to the equilibrium only in the absence of noise, see the left panel of Fig. 3b. In the negative feedback market, the price is stabilized at the level close to the equilibrium during the periods 10–15. However, the steady-state dynamics with price at the equilibrium level is not stable in the model with switching. Indeed, when the price stabilizes both heuristics give the same predictions and their impacts are the same. But the trend heuristic reinforces a trend and leads to the overshooting of the equilibrium level. As a result, dynamics converge to the 2-period cycle with price being very close to the equilibrium level but jumping around it.Footnote 5 At the cycle, the forecast of the trend heuristic is worse than the forecast of the adaptive heuristic, which results in their different impacts: around 80% of the adaptive heuristic and around 20% of the trend heuristic.
The heuristic switching model is also able to reproduce the same pattern of dependence on the initial condition, which was observed in the HHST experiment. Figure 4 shows that the different aggregate outcomes (convergence and oscillations) within the same environment of the positive feedback can be attributed to the path-dependent property of the HSM. Depending on the initial price level, the model produces qualitatively different outcomes during 50 periods. Both without (the left panel) and with noise (the right panel), the price generated by the model stays closer to the fundamental level during all simulation, when the initial price, p
0, is closer to the fundamental level.
Discussion
The first simulations of the heuristic switching model point to a behavioral explanation of the difference in the experimental outcomes between positive and negative feedback markets. When people, as subjects in the experiment or agents in the model, cannot make strategic decisions due to the absence of full knowledge of the environment they are operating in, they rely on behavioral rules of thumb. In the learning to forecast experiment different rules are possible, and some of them provide better forecasts than others. The learning of agents then takes a form of evaluating different forecasting possibilities and switching to those which performed better in the past. Agents in the HSM learn individually (not socially, through the interaction with others) by applying a counterfactual analysis of alternative forecasting rules on the basis of past data. As a result, the population of agents switch to more successful rules and the aggregate dynamics may change its properties (e.g., the trend in prices may revert). Then, via the re-evaluation of performances, dynamics feed back to the distribution of the rules’ impacts. Three parameters of the model allow to capture the behavioral characteristics such as imperfect switching behavior and, consequently, heterogeneity (especially when β is small), inertia in switching (when δ is close to 1), and short memory of past performances (when η is close to 0).
While the behavioral assumptions underlying our model are known from the behavioral literature,Footnote 6 our model does not aim to provide a precise description of the behavior of subjects in the experiment.Footnote 7 Instead, the aim of the model is to outline a mechanism explaining both negative and positive feedback markets at the same time. Indeed, in the simulations discussed in Section 3.4 the same heuristics and values of the learning parameters have been used. Our model is, essentially, a parsimonious version of the numerous computational learning models based on genetic algorithms (see, e.g., Arifovic 1996 and Hommes and Lux 2011) or its modifications such as Individual Evolutionary Learning (see Arifovic and Ledyard 2007 and Anufriev et al. 2011).
Our model also stresses importance of heterogeneity in the explanation of the experiments. According to the results of Section 3.2, the homogeneous expectations model with simple first-order heuristics we considered can not explain negative and positive feedback experimental data simultaneously. The results of the experiments can be explained, however, by assuming that agents learn to change their forecasting methods. The heuristic switching model can be simulated with many heuristics. For example, the model analyzed in Anufriev and Hommes (2012) had 4 different heuristics. We found, however, that the two heuristics are sufficient to reproduce the result of the HHST experiment qualitatively, and preferred such parsimonious version of the model over other possible specifications.Footnote 8