We now turn to the second cornerstone of prospect theory, the weighting function. In the illustration of the certainty effect described above, we saw that subjects did not base their decisions on objec-tive probabilities. The weighting function is the functional form that models the objecobjec-tive probabili-ties into subjective decision weights π(p), each of which are assigned to the value of a given out-come, v(x). This means, that decision weights should not be interpreted as probability measures but rather as an emphasis parameter derived from the probability. The decision weight then measures the impact v(x) has on the decision problem.
22 Kahneman and Tversky (1979), page 279
There exists numerous formulations of the weighting function but in the following the specific tional will be of the rank-dependent or cumulative form (Kahneman & Tversky, 1992). This func-tional is defined upon the probabilities associated with the prospects (x,p). To define the cumulative functional, the prospects are ranked in increasing order by their outcomes. The ranked prospects from i =-m to n, are denoted (xi, pi) where the subscript denotes the rank but also the nature of the prospect, i.e. strictly negative prospects are in the domain –m ≤ i<0 and strictly positive prospects are in the domain 0≤ i<n. A strictly positive prospect is defined as one where all outcomes are non-negative whereas a strictly non-negative prospect has exclusively non-negative outcomes.
To derive the mechanics of the weighting function we introduce the strictly increasing function w defined on the probability distribution of a prospect (xi, pi) where w+ is defined on the domain of strictly positive prospects and w- is defined similarly on strictly negative prospects. At the margin w+ (0)= w- (0)=0 and w+ (1)= w- (1)=1. Thus the weighting function is defined separately at the margins and in the positive and negative domain as
(5.2) πn+ =w+(pn), π−−m =w−(p−m)
(5.3) πi+ =w+(pi +...+ pn)−w+(pi+1 +...+ pn) 0≤i≤n−1 (5.4) )πi− =w−(p−m+...+ pi)−w−(p−m +...+ pi−1 1−m≤i≤0 This formulation states that the decision weight πi+ in the domain of gains is defined as the differ-ence between the emphases put on all outcomes that are at least as good as or better than xi and those that are strictly better than xi. Similarly, the decision weight πi− associated with a negative outcome is the difference between the emphasis put on all outcomes at least as bad as xi and those that are strictly worse. This means that the decision weights can be interpreted as a marginal contri-bution of the ith outcome. At the end points the decision weight and w coincide since there exists no outcomes better than xn or worse than x-m respectively.
Then π is an increasing function of the probabilities. So, when the probability of an event increases, so does the value of π. At the margins, π(0) = 0 and π(1) = 1, which means that an event that sim-ply cannot occur (p=0) is also weighted as 0 and all decision weights are normalized relative to the certain event23.
23 Obviously, this is a matter of choice. Setting the maximum decision weight equal to 1 naturally imposes consequent restrictions downwards on all other decision weights.
5.4.1 Behaviour of the Weighting Function
Having defined the mechanics of the weighting function, I now turn to the implications for the be-haviour of the function. The patterns of bebe-haviour and attitudes towards risk uncovered previously also govern the weighting function; this is important to note – the evidence presented by Kahneman and Tversky as in section 4.2 above is described by the weighting function as well as the value function. As mentioned, it is the interaction between the value attached to outcomes and the weight put on their probability of materializing that describes behaviour. The key findings from section 4.2 that apply to the weighting function as well, are the risk averse preferences for gains and risk seek-ing preferences for losses of moderate or high probability. Moreover, subjects were found to over-weight small probabilities. Elaborating on this point, Kahneman and Tversky discovered, through an extensive empirical study, that the shape of the weighting function favours risk seeking for small probabilities of gains and risk aversion for small probabilities of loss. This is illustrated through an extract of the experimental evidence in the table below:
Table 5.1.
N=25 Gain Loss
P ≤ 0.1 p ≥ 0.5 P ≤ 0.1 p ≥ 0.5
Risk seeking 78 10 20 87
Risk neutral 12 2 0 7
Risk averse 10 88 80 6
Source: Kahneman and Tversky (2000), p. 55.
The table shows, which portion of subjects that made risk seeking, risk averse, and neutral deci-sions. For example, for gains 78% of subjects made risk seeking choices for small probabilities.
From this table we see that a fourfold pattern emerges; for small probabilities, the majority of the subjects were risk seeking in the domain of gains whereas the opposite occurred in the domain of losses. For high probabilities, this pattern is reversed. Here we notice that subjects acted risk aver-sive in the domain of gains and risk seeking in the domain of losses, confirming the previous find-ings.
The figure below depicts a weighting function in which these attributes are illustrated:
Fig. 5.2 Weighting Function
0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0
0,0 0,2 0,4 0,6 0,8 1,0
p
w(p)
w + w
-Kahneman and Tversky label these effects “the fourfold pattern of risk attitudes” and the weighting function is seen to capture this pattern. The figure shows that the patterns of w+ and w- respectively have similar cumulative representation as functions of the probabilities. They are concave at small values of p and they take on values that are larger than the corresponding p’s. Similarly they are convex for moderate and high values of p and take on values that are smaller than the corresponding values of p24. Note that moderate and high probabilities are underweighted due to the convexity of the weighting function. This corresponds to the certainty effect uncovered in section 4.2. All mod-erate and high probabilities are underweighted relative to certainty due to subjects’ risk aversion in the domain of gains and risk seeking in the domain of losses.
In the following section, I represent explicit functional forms of the weighting- and value function in which these attributes are incorporated, following Kahneman and Tversky (1992).