Methodological Discussion

Figure 15 below summarizes the methodological conclusions that are addressed in this section. As a first step, the discussion of the cognitive model – forecasting, quoting and learning – will be approached from the perspectives of the object (E1, E2 and E3 in the

figure above) and the interactions (F1, F2). The forecasting model is able to generate heterogeneous forecasts through the various outputs created from the parallel association of distinct rules in the different rule-sets. This is an advantage of the forecasting subsystem as novel behaviour may arise when a few rules are changed in a rule-set in a given environment.

Figure 15 - Summary of Methodological Conclusions

In addition, although the system is sensitive to changes in the relationship between rulesets and environment, it still retains the averaging properties of fuzzy-logic weighted aggregation and de-fuzzification which provide an approximately continuous space for the relationships between inputs and outputs at each time step.

An increase in input signals in the forecasting mechanism seems to result in increased exchange rate volatility but it is very expensive in terms of computational power. With the set of 9 inputs/outputs used in most of the simulations (and their respective membership functions), around 750,000 rules can be constructed overall. The addition of just one input with three membership functions would increase the number of possible rules to more than 2 million. When more rules are included in an agent’s rule set, the GA approximates better the correct rule-set – i.e. the set that includes the rules taken from economic models and the appropriate regulatory rules. Marginal improvement, however, decreases when more than around 35% of the rules are given to the trader. In fact, because environmental conditions are continuously changing, the GA tends to find a rule-set that “on average” performs well or that adapts well to the prevailing environment when conditions do not vary rapidly.

When the entire simulation is run, the number of rules that make up a rule-set radically affects computational complexity. For this reason, and bearing in mind that each agent continually evaluates in parallel 10 distinct rule-sets (for GA performance measures), the number of rules is limited to the set {30, 40, 50} for the simulations. This is a form of incomplete or asymmetric information but where favoured or affected traders are not defined ex-ante (informed vs. uninformed) and their status is co-dependent on the environmental constraints prevailing at each time step. This forces the GA algorithm to specialize in order to find the adequate set of rules for the current environment.

Like real traders, artificial agents have to select from amongst the various alternative models. It is not a question of finding the “correct” model but of identifying the appropriate model for a given context; and switching to another model rapidly when the situation changes. It is this search that generates higher volatility when rules are drawn from the correct or accepted set of rules only. It also accounts for the lower volatility that arises when rules are drawn from the whole universe of discourse (including erroneous rules) as it has a similar impact to greatly increasing the mutation rate, which

in turn polarizes the market to the point where half the market is bearish and the other half is bullish¹⁷ if the market is initialised by giving each trader a ruleset chosen randomly from the whole universe of discourse. Allowing traders to draw rules from the correct pool of rules is the best methodological choice because it is clear that an investor who also learns deductively would not use erroneous, illogical rules.

A great advantage of using a fuzzy logic system is that it allows the researcher implicitly to supply the agents with a set of rules that are deductively and rationally chosen and ensures that the heterogeneity generated in the market does not arise from the use of erroneous models (that human traders would never use anyway).

GA convergence improves significantly when a factor that measures how well the ruleset is forecasting compared to the “fundamental” rule set – the distance to the fundamentals – is included in the performance measure. Although its weight is less than 1/3 of the total indicator, it results in an increase in traded volume and an increase in volatility and heteroscedasticity. It seems that a small attractor is needed in the formation of expectations – just as banks modify their expectations (models) when new information arrives – which increases the level of co-adaptation between rule-sets.

Some coherence is needed amongst agents at the microscopic level of the models (rule-sets) for volatility to emerge – this reminds us of Prigogine’s (1998) requirement for coherence at the microscopic level for non-equilibrium to exist.

An adequate quoting process that at the same time manages inventory and risk exposures has been achieved. The inclusion of feedbacks in the quoting system through indicators that refer to best market quotes, average market quotes, bid/ask spreads, market intention and the agent’s own information (his spread, inventory and risk) provide the basis for market dynamics in a setting similar to the way a trader actually works. Again, deductive logic is implicitly (partially) included through the symbolic interface of fuzzy logic systems. Moreover, it yields an approximately continuous set of functions for the risk and expected return of the trader that are constructed from the quoting rule set.

17 The number of erroneous rules is much greater than the number of correct rules.

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