In this section, we shall consider two special cases to illustrate the proposed two methods.
3.4.1 Normally distributed targets
In real applications, the uncertain targets may have different probability distributions.
For example, Tsetlin and Winkler [131] used the normal probability distribution, LiCalzi and Sorato [89] used the Pearson system probability distributions to represent the un-certainty of the target. Choosing a suitable probability distribution for uncertain target T is due to specific problems. As the normal distribution is widely used as a model of quantitative phenomena in the natural and behavioral sciences, we shall assume that the uncertain target is normally distributed over the bounded domain D and with mode value Tm. We assume a DM has three types of monotonic preferences: benefit, cost, and equal/range target. According to the two target-oriented decision methods proposed in previous sections, we can obtain the probability of meeting the normal target with respect to these three target preference types. We will discuss these three cases in great detail.
• Firstly, let us consider the benefit case. Fig.3.1 graphically depicts the pdf, induced probability of meeting the normal target. To distinguish these two methods, PrI(x≥ T) is used to denote the value function induced by the cdf based method, whereas
PrII(x ≥ T) is used to denote the value function induced by the level set based method.
Looking at the induced value functions PrI(x≥T) and PrII(x≥T) with respect to benefit target preference, as shown in Fig.3.1. It is clearly seen that no matter which method is chosen, the induced value function (utility function) corresponds to an S-shaped function, which is equivalent to the S-shaped utility function of Prospect Theory by Kahneman and Tversky [70] as well as is consistent with “Goals as reference point” by Heath et al. [54]. The induced value functions have the following two properties:
1. Gain and loss
The target divides the space of outcomes into regions of gain and loss (or success and failure). Thus, the value function assumes that people evaluate outcomes as gains or losses relative to the reference point Tm.
2. Diminishing sensitivity
The value function draws an analogy to psychophysical process and predicts that outcomes have a smaller marginal impact when they are more distant from the reference point Tm.
Xmin0 Tm Xmax
0.5 1
Normal target T
pdf/Probability of meeting target
p(t)
PrI(x≥T) PrII(x≥ T)
Figure 3.1: Induced value functions with a normally distributed target by means of cdf and level set based methods, with respect to benefit target preference
Remark It should be noted that in their Prospect Theory [70] Kahneman and Tversky assume another principle: outcomes that are encoded as losses are more painful than the similar sized gains are pleasurable. In their words, “losses loom larger than gains”. From Fig. 3.1, the induced value function by target-oriented model does not entirely satisfy this principle. The main reasons for this observation are twofold. The first reason is the distribution type of target. The normal target
is symmetrically distributed around the mode value Tm. Another reason is the bounded domain. In fact, when the attribute value has a bounded domain, and the reflection point in the Prospect Theory is the middle value of the domain, the value function induced by prospect theory will also not satisfy this principle.
In addition, from Fig. 3.1 it is clearly that although those two induced value func-tions have anS-shaped value function, the behaviors of value function are different.
The value function PrII(x ≥ T) induced by the level set based method is stepper towards the mode value Tm of the corresponding target T than that PrI(x ≥ T) by the cdf based method. This practically implies that the level set based value function reflects a stronger decision attitude by the DM towards the target T than that defined by the cdf function. A similar result for this phenomenon is given in Huynh et al.[61].
• In case of cost target, the DM will have a monotonically decreasing preference.
According to Eq. (3.8), we can obtain the value function induced by the cdf based method, denoted as PrI(x≤T). By means of Eqs. (3.18) and (3.21) we can obtain the probability of meeting target based on the level set of pdf, denoted as PrII(x≤ T). Fig. 3.2 graphically depicts the pdf of the normal target T, its induced value functions by the cdf based method and level set based method.
Xmin0 Tm Xmax
0.5 1
Normal target T
pdf/Probability of meeting target
p(t)
Pr
I(x ≤ T) Pr
II
(x ≤ T)
Figure 3.2: Induced value functions with a normally distributed target by means of cdf and level set based methods, with respect to cost target preference
From Fig.3.2, it is clear that no matter which method is chosen, these two methods both induce an inverse S-shaped value function. The reference point Tm divides the value function into two parts: gains and losses (the value below Tm can be viewed as a kind of gains; the value upper than Tm can be viewed a kind of losses).
In addition, the value function draws an analogy to psychophysical process and predicts that outcomes have a smaller marginal impact when they are more distant from the reference point Tm. Finally, the behaviors of value function PrII(x ≤ T) induced by the level set based method is also stepper towards the mode value of the corresponding target than that PrI(x≤T) induced by the cdf based method.
• In case of equal/range target preference, according to Eqs. (3.9)-(3.11) we can obtain the value function induced by the cdf based method, denoted as PrI(x ∼= T). By means of Eqs. (3.18) and (3.23)-(3.25), we can induce the value function via the level set based method, denoted as PrII(x ∼= T). Fig. 3.3 graphically depicts the pdf of the normal target T, its induced value functions by the cdf based method and level set based method with respect to equal/range target preference.
Xmin0 Tm Xmax
1
Normal target T
pdf/Probability of meeting target
p(t)
Pr
I(x ≅ T)
Pr
II(x ≅ T)
Figure 3.3: Induced value functions with a normally distributed target by means of cdf and level set based methods, with respect to equal/range target preference
As the DM assumes interval/range target preference, there will be added loss of value for missing the reference point on the low side, or added loss for exceeding the reference point Tm. Thus the reference point Tm is the reflection point. As illustrated in Fig. 3.3, the value functions induced by the cdf based method and level set based method have a convex shaped function, i.e. below or upper than the mode value Tm is viewed as loss. Furthermore, it is clear that the behavior of value function PrII(x∼=T) induced by the level set based method is stepper towards the mode value Tm of the corresponding target than that PrI(x ∼= T) induced by the cdf based method.
3.4.2 Uniformly distributed target
Furthermore, let us consider a special case. Without additional information about the target distribution, we can assume that the random target T has a uniform distribution onD with the pdf p(t) defined by
p(t) =
1
Xmax−Xmin, Xmin≤t ≤Xmax;
0, otherwise. (3.26)
Under the assumption that the random target T is stochastically independent of any alternative, by means of the cdf based method and the level set based method we can obtain the same value function with respect to benefit and cost target preferences as follows
Pr(xT) =
Pr(x≥T) = Xx−Xmin
max−Xmin, for benifit target;
Pr(x≤T) = XXmax−x
max−Xmin, for cost target. (3.27) In this case, the level set and cdf based methods are equivalent. From Eq. (3.27) it is easily seen that, for benefit and cost attribute there is no way to tell whether the DM selects an alternative by traditional normalization method or by target-oriented model.
In other words, in this case the target-based decision model with the decision function is equivalent to the traditional normalization function. Fig.3.4graphically depicts the value function induced by target-oriented decision model under uniformly distributed target.
Xmin0 Xmax
1
Uniform target T
pdf/Probability of meeting target
Pr(x ≤ T) Pr(x ≥ T)
Figure 3.4: Uniformly distributed target with benefit and cost target preference