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Chapter 7 - Natural responses to uncertainty

Accounting for environmental decision making:

Chapter 8

15/1/00 - Computing

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{

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.

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.

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

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}.

'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.

'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.

- 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.

Finished 13/9/96 - Revised 8/10/98

Created 18/9/98

Modified 6/10/99