Approximations of One-dimensional Expected Utility Integral of Alternatives Described with Linearly-Interpolated p-Boxes

Abstract

In the process of quantitative decision making, the bounded rationality of real individuals leads to elicitation of interval estimates of probabilities and utilities. This fact is in contrast to some of the axioms of rational choice, hence the decision analysis under bounded rationality is called fuzzy-rational decision analysis. Fuzzy-rationality in probabilities leads to the construction of x-ribbon and p-ribbon distribution functions. This interpretation of uncertainty prohibits the application of expected utility unless ribbon functions were approximated by classical ones. This task is handled using decision criteria Q under strict uncertainty—Wald, maximax, Hurwicz${}_{\alpha }$, Laplace—which are based on the pessimism-optimism attitude of the decision maker. This chapter discusses the case when the ribbon functions are linearly interpolated on the elicited interval nodes. Then the approximation of those functions using a Q criterion is put into algorithms. It is demonstrated how the approximation is linked to the rationale of each Q criterion, which in three of the cases is linked to the utilities of the prizes. The numerical example demonstrates the ideas of each Q criterion in the approximation of ribbon functions and in calculating the Q-expected utility of the lottery.

Appendix 1: Decision Criteria Under Strict Uncertainty

A lottery is a generic model of an uncertain decision alternative. Let’s assume that the DM has to rank n uncertain alternatives according to preference, which can result in r different holistic consequences $x_{j }$ called prizes, indexed according to the preferences of the DM, $x_{1}$ being the most preferred, and $x_{r}$ being the most unwanted one. The consequences of the $i$th alternative will be $x_{j}$ if the event $\theta _{i,j}$occurs, where $\theta _{i,1}$, $\theta _{i,2}$, ..., $\theta _{i,r}$ , $i=1,2,{\ldots },n$, called states, form a full set of disjoint events. An ordinary lottery may be denoted as (A1) with the conditions in (A2):

$$\begin{aligned} l_{i} = \langle \theta _{i,1} , x_{1}; \theta _{i,2}, x_{2}; {\ldots }; \theta _{i,r} , x_{r} \rangle , \end{aligned}$$

(A1)

$$\begin{aligned}&x_{1} \succsim x_{2} \succsim \ldots \succsim x_{r}\\&\theta _{i,j} \succsim _{l} \varnothing ,\\&\theta _{i,1}\cup \theta _{i,2} \cup \ldots \cup \theta _{i,r}=\varTheta ,\\&\theta _{i,j} \cap \theta _{i,k} = \varnothing \; \text {for}\; j \ne k. \end{aligned}$$

(A2)

Here, $\varnothing$ denotes the null event, $\varTheta$ denotes the certain event, $\succsim _{l}$ denotes the binary relation “at least as likely as” defined over events, and $\succsim$ denotes the binary relation “at least as preferred as”, defined over prizes or lotteries.

Assume the DM has constructed a utility function $u(.)$ over the consequences $x_{j}$, such that it measures her relative preferences in the sense [36]:

$$\begin{aligned} u(x_{j})\ge u(x_{k})\; \text {iff}\; \quad x_j \succsim x_k . \end{aligned}$$

(A3)

Unfortunately, the elicitation of utilities from real DMs results in uncertainty intervals, which is another demonstration of fuzzy rationality. For reasons of simplicity, it is assumed here that the values of the utility function are point estimates of some kind of their uncertainty intervals. It is also assumed that those correctly reflect the preferences of the DM.

If the only thing the DM knows about the uncertainty in the problem is which states are possible and which are not, then the decision is said to be under strict uncertainty. Let $b(.)$ be a discrete Boolean function defined over $\theta _{i,j}$ and with range {‘t’ , ‘f’}, where ‘t’ stands for ‘true’ and is assigned when the state is possible, whereas ‘f’ stands for ‘false’ and is assigned when the state is impossible:

$$\begin{aligned} b(\theta _{i,j} )=\left\{ \begin{array}{ll} {^\prime }t{^\prime },&{} \text {for} \quad \theta _{i,j} \succ _l \varnothing \\ {^\prime }f{^\prime },&{} \text {for} \quad \theta _{i,j} \sim _l \varnothing \end{array} \right. . \end{aligned}$$

(A4)

Here $\sim _{l}$ and $\succ _l$ denote respectively the binary relations “equally likely to” and “more likely than” defined over events. In the strict uncertainty case, for each state $\theta _{i,j}$ (for $i=1,2,{\ldots }, n \,\text {and}\, j=1,2,{\ldots }, r)$ the DM can define the value $b(\theta _{i,j} )$ subject to the condition:

$$\begin{aligned} b(\theta _{i,1})\vee b(\theta _{i,2} )\vee \ldots \vee b(\theta _{i,r} )={^\prime }t{^\prime }. \end{aligned}$$

(A5)

In (A5), $\vee$ is the Boolean operator “and”. The lottery (A1) with the known function $b(.)$ can be better represented as

$$\begin{aligned} l_{i}=\langle \langle \theta _{i,1} , b(\theta _{i,1})\rangle , x_{1};\langle \theta _{i,2} , b(\theta _{i,2})\rangle , x_{2};{\ldots }; \langle \theta _{i,r} , b(\theta _{i,r})\rangle , x_{r}\rangle . \end{aligned}$$

(A6)

The lottery (A6) subject to (A2) and (A5) can be called a strictly uncertain lottery. There are criteria to rank strictly uncertain lotteries. Four of the most widespread are the Savage, Laplace, Wald and $\text {Hurwicz}_\alpha$ criteria.

Savage’s minimax criterion [24] constructs the so-called regret table, and recommends the alternative which minimizes the maximal regret. Its idea will not be used in this chapter.

Laplace’s criterion [37] is based on the principle of insufficient reasoning, which says that if no information is available regarding a set of random events, then they might be assumed equally probable. In this way the probabilities are known and the resulting problem under risk must be solved by the expected utility criterion, which here degenerates to the mean utility of all possible states:

$$\begin{aligned} c_i =\frac{\sum \limits _{\begin{array}{c} j=1 \\ b(\theta _{i,j} )={^\prime }t{^\prime } \\ \end{array}}^r {u(x_j )} }{\sum \limits _{\begin{array}{c} j=1 \\ b(\theta _{i,j} )={^\prime }t{^\prime } \\ \end{array}}^r 1 }. \end{aligned}$$

(A7)

Then the recommended choice is $l_{k}$ for which the mean utility is maximal.

Wald’s criterion of maximin return suggests ranking actions in descending order of their worst outcomes [26]. The security level denotes the worst possible outcome from the ith alternative

$$\begin{aligned} s_i =\mathop {min }\limits _{\begin{array}{c} j=1 \\ b(\theta _{i,j} )={^\prime }t{^\prime } \\ \end{array}}^r \{u(x_j )\}. \end{aligned}$$

(A8)

Then the recommended choice is $l_{k}$ for which the security level is maximal. This criterion is suitable for extreme pessimists.

If the DM is an extreme optimist, she can use the maximax criterion, which suggests ranking actions in descending order of their best outcomes. The optimism level $o_{i}$ denotes the best possible outcome from the ith alternative

$$\begin{aligned} o_i =\mathop {max }\limits _{\begin{array}{c} j=1 \\ b(\theta _{i,j} )={^\prime }t{^\prime } \\ \end{array}}^r \{u(x_j )\}. \end{aligned}$$

(A9)

Then the recommended choice is $l_{k}$ for which the optimism level is maximal. Despite the symmetry between (A8) and (A9), the maximax criterion (A9) is not used in practice, whereas Wald’s criterion is often preferred as a decision tool.

Hurwicz argues that people usually do not express such extremes of pessimism or optimism as the previous criteria suggested [25]. He introduced the optimism-pessimism index $\alpha \in [0;1]$ to weight the security level and the optimism index for each alternative in the form

$$\begin{aligned} h_i^\alpha = \alpha s_{i}+(1-\alpha )o_{i}. \end{aligned}$$

(A10)

Hurwicz suggested to rank alternatives in descending order of $h_i^\alpha$, which can be called the $\text {Hurwicz}_\alpha$ strict uncertainty criterion. Then the recommended choice is $l_{k}$ for which $h_i^\alpha$ is maximal.

The optimism-pessimism index $\alpha$ is a measure of people’s pessimism. It is specific to each DM and applies to all decision situations. In order to elicit $\alpha$, the DM can be offered the choice between: a) lottery l $_{1}$, giving consequences with utility 1, v, and 0 respectively at states $\theta _{1,1}$, $\theta _{1,2}$, and $\theta _{1,3}$, where $b(\theta _{1,1} )={^\prime }t{^\prime }$, $b(\theta _{1,2} )={^\prime }f{^\prime }$ and $b(\theta _{1,3})={^\prime }t{^\prime }$; b) lottery $l_{2}$ giving consequences with utility 1, v, and 0 respectively at states $\theta _{2,1}$, $\theta _{2,2}$, and $\theta _{2,3}$, where $b(\theta _{1,1} )={^\prime }f{^\prime }$, $b(\theta _{1,2})={^\prime }t{^\prime }$, and $b(\theta _{1,3} )={^\prime }f{^\prime }$. The value of v varies until the DM becomes indifferent between the lotteries, at which point $\alpha =1-v$.

The rationality of each criterion can be assessed against a set of reasonable properties of the decisions generated by that criterion [36]. However, analysis shows that any decision criterion under strict uncertainty does not (and will not) possess this set of properties of choice and thus is (and will be) irrational. One possible explanation is that problems, where the DM knows nothing about the uncertainty, do not actually exist.