towards natural-language reasoning agent-based artificial

Chueh-Yung Tsao 1

Agent-Based Artificial Stock Markets:

Towards Natural-Language Reasoning Artificial Adaptive Agents (4)

• Linn & Tay (2001a). ``Fuzzy Inductive Reasoning, Expectation Formation and the Behavior of Security Prices,’’ JEDC.

• Linn & Tay (2001b). ``Fuzzy Inductive Reasoning and Nonlinear Dependence in Security Returns: Results from Artificial Stock Market Environment,’’ working paper.

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Motivations

• Some might question whether it is reasonable to assume that traders are capable of handling a large number of rules.

• The previous study on artificial stock market have reported that some statistical properties of simulated returns do not match the real returns.

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Assumptions

• Neoclassical Financial Market Models:– Rational Expectation– Deductive Reasoning

• This Model:– Bounded Rationality– Inductive Reasoning Process– Fuzzy Notion

SFASM

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Inductive Reasoning Process

• Two-step Process– Possibility-elaboration

Creating a spectrum of plausible hypotheses based on our experience and the information available.

– Possibility-reductionThese hypotheses are tested to see how well they

connect the existing incomplete premises to explain the data observed. Reliable hypotheses will be retained ; unreliable ones will be dropped and ultimately replaced with new ones.

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Fuzzy Notion

• Literature Supports: – Smithson (1987), Smithson and Oden (1999)

• Some Reasons: – Justifying the assumption that agents are able to

process and compare hundreds of different rules simultaneously when making choices.

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The Model (Market Environment)

• Two Assets:

Payoff Units

Stock d ~ AR(1)* N

Risk-free Bond r ~ Fixed Infinite

*The current dividend, dt, is announced and becomes public information at the start of time period t.

ttt dddd ερ +−+= − )( 1

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The Model (Market Environment)

• N Agents:– Utility Function (CARA):

Ui,t(Wi,t) = -exp(-λWi,t )

(homogeneous, time-independent, time-additive, state-independent, and zero time-preference utility function)

– Expectation: heterogeneously– Decision: share holdings of stock– Object: maximizing subjective expected utility of next

period wealth

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1. At time t, the dividend, dt, realizes.

2. Forecast :– using the recently best performance rule base

3. Submit demand function:

][ 11, ++ + ttti dpE

2,

11,, ˆ

)1(][ˆ)(

ti

ttttitti

rpdpEpx

σλ+−+

= ++

Market Flow

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Market Flow (cont.)

4. The market declares a price pt that will clear the market:– tatonement process

5. Evaluate the forecasting error for each rule base:

6. Update rule bases every k periods: – Using GAs

2,,1

2,,1

2,, )]()[()1( ttjitttjitjit dpEdpee +−++−= −− θθ

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Expectation

• The forecast equation hypothesis used is:

where a and b are forecast parameters.

bdpadpE ttttt ++=+ ++ )()( 11

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Decision Flow

CrispConditions

FuzzyDecisions

CrispDecisions

FuzzyNotions

defuzzify

fuzzifyInside

ThinkingOutside

Environment

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Fuzzy Condition-Action Rule

• The format of a rule is:– ``If specific conditions are satisfied then the

values of the forecast equation parameters are defined in a relative sense’’.

– e.g. ``If {price/fundamental value} is low, then a is low and b is high’’.

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Fuzzy Condition-Action Rule

• Five market descriptors (five information bits) are used for the conditional part of a rule:– p*r/d, p/MA(5), p/MA(10), p/MA(100),

p/MA(500)

• Two forecast parameters (two forecast bits) are used for the conditional part of a rule:– a & b

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Fuzzy Condition-Action Rule

• We present fuzzy information about a variable with the codes: 1 2 3 4 0

low moderately-low moderately-high high absence

• We present fuzzy information about a parameter with the codes: 1 2 3 4

low moderately-low moderately-high high

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Membership Function for Descriptor

lowmoderately-low moderately-high

high

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Membership Function for forecast parameter ‘a’

low

moderately-low moderately-high

high

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Membership Function for forecast parameter ‘b’

low

moderately-low moderately-high

high

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Fuzzy Condition-Action Rule

• In general, we can write a rule as:– [x1, x2, x3, x4, x5| y1, y2], where x1, x2, x3, x4, x5 ∈

{0, 1, 2, 3, 4} and y1, y2 ∈ {1, 2, 3, 4}.

• We would interpret the rule

[x1, x2, x3, x4, x5| y1, y2] as:

– ``If p*r/d is x1 and p/MA(5) is x2 and p/MA(10) is x3 and p/MA(100) is x4 and p/MA(500) is x5, then a is y1 and b is y2’’

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Rule Base• Single fuzzy rule can not specify the

remaining contingencies. Therefore, three additional rules are required to form a complete set of beliefs.

• Fore this reason, each rule base contains four fuzzy rules.

• At any given moment, agents may entertain up to five different market hypothesis rule bases.

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Rule Base (an example)

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Defuzzify of Fuzzy Decisions

• We employ the centroid method , which is sometimes called the center of area

,method to translate the fuzzy decisions into specific values for a a and b.

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Example

Consider a simple fuzzy rule base with the following four rules.

1st rule: If 0.5p/MA(5) is low then a is moderately high and b is moderately high.

2nd rule: If 0.5p/MA(5) is moderately low then a is low and b is high.

3rd rule:If 0.5p/MA(5) is high then a is moderately low and b is moderately low.

4th rule:If 0.5p/MA(5) is moderately high then a is high and b is low.

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Example (cont.)

• Now suppose that the current state in the market is given by p = 100, d = 10, and MA(5) = 100.

• This gives us, 0.5p/MA(5) = 0.5.

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Response of 1st rule (example)

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Response of 2nd rule (example)

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Response of 3rd rule (example)

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Response of 4th rule (example)

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Summary

Rule Membership Decisions

• 1st Rule 0

• 2nd Rule 0.5

• 3rd Rule 0

• 4th Rule 0.5

a is mode rate ly high b is mode rate ly high. a is low b is high. a is mode rate ly low b is mode rate ly low . a is high b is low .

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Defuzzify of Forecast ‘Parameters a’ ‘and b’

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Genetic Algorithms

• GAs are applied to retain the reliable rule bases, drop the unreliable rule bases, and create new rule bases.

• The fitness measure of a rule base is calculated as follows:

where β is constant and s is the specificity of the rule base.

sef jitjit β−−= 2,,,,

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The Market Experiments Linn & Tay (2001a)

• Experiment 1 (slow learning)– k = 1000– Using best rule base with probability 1.

• Experiment 2 (fast learning)– k = 200– Using best rule base with probability 1.

• Experiment 3 (fast learning with doubt)– k = 200– Using best rule base with probability 99.9%.

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Why we introduce ‘a state of doubt’ to catch the actual figure of kurtosis?• Although during the first few hundred of time steps,

kurtosis is always rather large ( because of initialized randomly and trying to figure out how to coordinate), once agents have identified rule bases that seem to work well, excess kurtosis decrease rapidly.

• From that point on, it is extremely difficult to generate further excess kurtosis without exogenous perturbation, because it is difficult to break the coordination among agents.

• We suspect the large kurtosis observed in actual returns series may have originated from such exogenous events as rumors or earnings surprises.

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The Market ExperimentsLinn & Tay (2001b)

• Experiments: – Experiment 1 (slow learning)– Experiment 2 (fast learning)

• Benchmarks:

– Disney and IBM stocks

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Experiments Parameters

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Results (Linn & Tay (2001a))

• The results of this model are similar to those of LeBaron et al. (1999) in which their model is based upon a crisp but numerous rules.

• A modification of the model, i.e., fast learning with ‘doubt’, is shown to produce return kurtosis measures that are more in line with actual data.

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• It is found that the market moves in and out of various states of efficiency. Moreover, when learning occur slowly, the market can approach the efficiency of a REE

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Results (Linn & Tay (2001b))

• Normality: – rejects normality for each series (Jarque-Bera

test)

• Linearity: – exists linear dependent for each series (Ljung-

Box Q test)– does not exist any linear dependent for each

ARMA fitted residual series (Ljung-Box Q test)

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• Non-linearity: – exists nonlinear dependent for each ARMA

fitted residual series (using both correlation dimension and BDS test methods)

• ARCH Effect: – exists ARCH behavior for each ARMA fitted

residual series (Ljung-Box Q test and LM test)– does not exist any ARCH effect for each

ARMA-TARCH fitted residual series (Ljung-Box Q test and LM test)

– exists other nonlinear dependent for each ARMA-TARCH fitted residual series (BDS test)

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• Other Non-linearity– exists other nonlinear dependent for each

ARMA-TARCH fitted residual series (BDS test)

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Conclusions

• These two papers begin by presenting an alternative model of decision-making behavior, genetic-fuzzy classifier system, in capital markets where the environment that investors operate in is ill-defined.

• The results indicate that the model proposed in this paper can account for the presence of nonlinear effects observed in real markets.

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Conclusions (cont.)

• The framework offers an alternative perspective on capital markets that extends beyond the traditional paradigms.