AEDM: Modelling responses to uncertainty

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Chapter 7 - Natural responses to uncertainty

Accounting for environmental decision making:
Chapter 8
Modelling responses to uncertainty

New! - More stuff on the logistic map

New! - I've now written a c++ version of this model. The original model used MS Excel. (GKS 6/10/99)

This section looks at the ability of simple systems to respond to external uncertainty. This uncertainty is presented to the system in the form of a "reality" function. The systems are split into sub-populations each with a fixed response. Growth of the sub-populations is governed by a benefit function. The benefit function measures the similar{?? strength? / similarity! 8/10/98 } of the sub-population's response to the reality function. Maximum growth occurs when the response matches the reality function exactly.

In this model different systems to aggregate benefit between the sub-populations are compared against each other. The systems that match the external uncertainty more precisely show faster growth.

Figure 8.1. Simple model for responding to external uncertainty.

Reality Function

If we give ourselves a "reality" function, a function that gives the state of the environment at time t., we can look at how systems adapt to this function to try to gain maximum benefit over time. This is a test signal that looks at the response of the system to a particular function. A interesting function is the "logistic map" as this has areas of stability, periodicity and 'deterministic chaos', depending on the values of {lambda}. This means that systems can be tested against a variety of situations by only changing one parameter.

State of reality at time, t , R(t) is given by:

R(t+1) = {lambda} R(t) (1 - R(t)),

where {lambda} is the feedback parameter that affects the stability of the function.

Figure 8.2. The logistic map for {lambda} =3.8.

Figure 8.3. Relative frequencies of events of value R, for {lambda} = 3.8.

Initial populations:

For this reality function we know that R(t) varies between zero and one, so we can divide the population into sub-populations each with a given response.

Let n(i,t) be the size of sub-population, i, at time, t,

At time t = 0

n(i,0) = 1 , i = 0, ... , 9,.

Response of sub-population, i, r(i) is given by:

r(i) = 0.1i + 0.05

Benefit coefficient:

In order for the populations to adapt to the set of reality, there need to be a feedback between the state of reality and the benefit received by the sub-populations. We can define a benefit coefficient dependent on the state of reality and the response of the sub-population.

The benefit coefficient for sub-population, i, at time, t, b(i,t) is given by:

b(i,t) = {alpha} exp ( -{beta} (R(t) - r(i))2),

where {alpha} is the maximum increase in population size between generations,
and {beta} increases the selectivity between a response and reality.
It is important to remember that the form of this function severely affects the outcome of the model.

Figure 8.4. Form of the benefit function for different values of {alpha} and {beta}.

Population systems.

The final stage of the model is the population systems themselves, a variety of systems are used so that they can be compared. In all of the systems analysed in this model, there is initial diversity of response, given by the fact that there are different sub-populations each with a different response. However, there is no facility for sub-population to produce mutated offspring and regenerate diversity.

'Selfish' system:
This system assumes that the benefit coefficient for each sub-population is completely independent.

n(i,t+1) = b(i,t) n(i,t)

'Altruistic' system:
This system assumes that benefit is aggregated and divided equally between each sub-population. (This is similar to Marx's idea of a general rate of profit, but that assumes that the probabilities of each outcome are known and invested in accordingly)

B(t) = sum over i (b(i,t) n(i,t)) / N(t), where N(t) = sum over i ( n(i,t) )

n(i,t+1) = B(t) n(i,t)

Hybrid system:
From these two systems it is possible to make a hybrid system where a percentage, a, of benefit is aggregated and the rest is kept by the sub-population.

{This reverses the notation I originally used in my thesis. This reversal was suggested by E.J. but I didn't have time to change all the diagrams. GKS 18/9/98}

B (t) = a . sum over i (b(i,t) n(i,t)) / N(t)

n(i,t+1) = ((1-a) . b(i,t) + B(t)) n(i,t)

These equations create a model where systems that gain the most benefit from their responses to the reality function, increase grow{th} fastest.

Results

The model was run for 150 iterations comparing five different systems.

'Selfish' system
'Altruistic' system
Hybrid system with a = 0.05
Hybrid system with a = 0.50
Hybrid system with a = 0.95

The systems were compared with four values of {lambda}:

{lambda} = 1.6, 'stable'.
{lambda} = 3.5, 'period 4'
{lambda} = 3.8, 'chaotic'
{lambda} = 3.95, 'chaotic'

and for two survival functions:

{alpha} = 2, {beta} = 2.5, 'weak selection'
{alpha} = 5, {beta} = 30, 'strong selection'

(The different values of a used here were to limit the final populations sizes into values that could be calculated by a computer. They do not affect the relative sizes of the different systems.)

Figure 8.5, Logistic map {lambda} = 1.6, population over time.

Figure 8.6, Logistic map {lambda} = 3.5, population over time.

Figure 8.7, Logistic map {lambda} = 3.95, population over time.

The results are shown in Figures A.1 - A.16.

These results show that some systems are more responsive to different forms of the reality function than others. Specifically that the "selfish" system does well when there is weak selection or there is a stable reality, as it is the system with the best average response. The hybrid (a = 0.95) system (95% altruistic aggregation) is better when there is strong selection and a periodic or chaotic reality function. This is because the system creates a "memory" of the past probabilities of events and subsidises less likely events to retain this memory. The altruistic system has no ability to change the relative proportions of its response so it cannot adapt to the reality function.

Limitations of this model

This is a very simplistic model and a number of modifications could be made to investigate the response patterns of systems more thoroughly.

Really only one system of aggregation has been tested, the hybrid system with values a = 1.00 (altruistic), 0.95, 0.50, 0.05, 0.00 (selfish). More interesting systems could be designed or abstracted from real social and biological systems.
Different reality functions could be used, i.e. ones with longer periods, that include stochasticity, or pulsed functions.
The model assumes an infinite environment and no competition between the systems.
There was no mutation of response allowed, therefore no defecting mutants could be generated.
There was no feedback between the responses and the environment. The environment affected the benefits accrued by the systems, but the systems did not affect the environment. This makes the uncertainty external.
The range of variation in the environment was bound and the initial diversity of response of the systems was enough to cover all possible values of the reality function.
Only two forms of the benefit function were used. What would happen if the selection was stronger or the function was not uni-modal?
The system and reality function were only one-dimensional.

Extensions to the model

Apart from the above considerations, there is an additional source of information within this reality function. Chaotic functions have attractors that can be mapped in order to predict the value of R(t+1) gives {given} the value of R(t). These predictions usually can't be extended too far into the future, but the may be useful. Do the populations receive enough information to make these predictions, they don't know R(t), but only b(i,t).

Conclusions from this model

Different systems perform better at responding to different types of variation in their environment. The "picture" of their environment that these systems "see", is dependent on both the form of the reality function and on the form of the benefit function. Can this be useful in helping use make better decisions concerning the environment? It can make us look at the decision making structures that we use to create our responses to environmental uncertainty and the biases inherent in these structures to different types of information input. Creating accounting structures that include environmental values is a fine ideal, but if the structures that use those values are unresponsive to their natural time scales and variabilities, then it is wasted information.

[Conclusions]

Finished 13/9/96 - Revised 8/10/98
Created 18/9/98
Modified 6/10/99