The optimal diet selection algorithm

The optimal diet selection algorithm specifies how to construct a diet of information items which, given certain assumptions, maximises profitability (Stephens & Krebs

1986). It assumes that the forager has access to information about the prevalence and

profitability of these items and can select them in an all or none manner. It also assumes

that the time taken to recognise the items is effectively zero.

In order to explain the rational behind this algorithm it is instructive to start by considering what was discussed above in section 6.1. The user must alternate between spending time, looking for profitable patches, and looking for profitable items in those

patches. Given the fact that in information seeking alternative patches, such as items on

a results list, are of the same prevalence, the expected time to locate the next patch is

assumed to be a constant tb

Within each patch there will be a number of items. For each item, i, the user can

estimate how much information the item is likely to offer, and how much time this

will take to exploit or process, twi So for a patch containing k items the total expected

^ Here t/,has subscript bbecause it is the time taken to do the between-patch foraging. This notation is

used in order to maintain consistency with literature on information foraging.

® Here is given, w, as a subscript because it is the time taken to do the within-patch foraging. This

Chapter 2 section 6.2 103

amount of information is ^ g. , and the total time taken to exploit these items and 1=1

k

obtain the information is . Thus the total rate at which information will be

gained from the patch, R{k) is^:

R(k)= _k

1=1

Equation 1 The rate of information gain for a diet

o / k

items

If the reader is to maximise their rate of information gain they must select items such

that R(k) is maximised. To see how this can be done, suppose the user has evaluated the profitability, „ of each item in the patch and ranked them in terms of that profitability.

i is calculated as:

Now suppose the user has selected k of the top ranked items, and is considering whether k to select the k + f ^ item. The total information gained from those items would be ,

f=l

’ In food foraging theory, varies depending on the type of item the patch contains. This is expressed as

prevalence, , , which is 1/th, so that Equation 1 is formulated as R(k) = 1=1

1 + ^ i . f ^ (=1

and the total time gained would be ^ Then in order to ensure selecting this (=1

next item does not decrease the diet’s profitability:

f k ^

Eg,

V i=i ^ V (=1 J + gt+i f k A —

_f

* ) h ^ w k + l V /=i / V i=i V This gives: t+t W&+] 1=1

Equation 2 The condition that must hold if the

k+1^

item is to be added to a diet

o /k

items.

Essentially in order to maximise the profitability of the diet the user must add the items

to the diet in order of increasing rank until the profitability of the diet of the top k items,

R(k) is less than the profitability of the next item being considered, . Thus when the

k+7* item is less profitable then that of the diet consumed so far, the user should

abandon the item’s patch and look for another, more profitable one.

This algorithm provably maximises the profitability of selections within a patch

(Stephens & Krebs 1986). It can be summarised as follows:

1. Calculate the profitability of, „ of each item i.

2. Rank the items according to their profitability. Where i = 1 is the most profitable

item and i = n is the least profitable one.

If these items have been exploited then the cost and gain should be calculated using actual values, rather then expected values.

Chapter 2 section 6.3 105

3. A dd the item s to the diet in order o f decreasing profitability until the

profitability o f a diet of the top k item s, R{k), is less than the profitability o f the

next item being considered, . In m athem atical term s, item s should be added

to the diet until the condition in Equation 2 is broken.

The way this algorithm w orks is illustrated by Figure 16. As the item s are added to the diet, the profitability o f the diet as a w hole increases at slow ing rate since the profitability of the item s decreases. Once including the next item decreases the diets overall profitability, it is not worth continuing. T herefore, the optim al point at w hich to stop adding item s to the diet occurs ju st before the ‘profitability o f d ie t’ line crosses the ‘item profitability’ one.

180 160 140 120 100 80 60 40 20 0 H --- Diet profitability, R(k) Item profitability, . N CO o m T— N CO o m T - M T - T - o j c o c o T f - ^ - m o t o Rank of item profitability (k)

Figure 16 An example o f the relationship o f between item profitability and the profitability o f diets including items 1,2, ... k.

In document Supporting document use through interactive visualisation of paragraph-level metadata. (Page 103-106)