Modifications of the Hurwicz’s decision rule
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Abstract
The Hurwicz’s criterion is one of the classical decision rules applied in decision making under uncertainty as a tool enabling to find an optimal pure strategy both for interval and scenarios uncertainty. The interval uncertainty occurs when the decision maker knows the range of payoffs for each alternative and all values belonging to this interval are theoretically probable (the distribution of payoffs is continuous). The scenarios uncertainty takes place when the result of a decision depends on the state of nature that will finally occur and the number of possible states of nature is known and limited (the distribution of payoffs is discrete). In some specific cases the use of the Hurwicz’s criterion in the scenarios uncertainty may lead to quite illogical and unexpected results. Therefore, the author presents two new procedures combining the Hurwicz’s pessimismoptimism index with the Laplace’s approach and using an additional parameter allowing to set an appropriate width for the ranges of relatively good and bad payoffs related to a given decision. The author demonstrates both methods on the basis of an example concerning the choice of an investment project. The methods described may be used in each decision making process within which each alternative (decision, strategy) is characterized by only one criterion (or one synthetic measure).
Keywords
Decision making under uncertainty States of nature Hurwicz’s criterion Laplace’s criterion Optimal pure strategy1 Introduction
The Hurwicz’s rule is a procedure applied within the decision making process under uncertainty (DMUU). This uncertainty is a consequence of the fact that we are not able to anticipate the future effectively. One may just forecast various phenomena and events, but in many cases it is extremely difficult to estimate the exact value of particular parameters (temperature, company profit, size of the mature crops, demand for a product, product prices, production costs etc.). If these data were known, it would be easy to indicate the best alternative (decision), e.g. the best investment strategy. But when many future factors are not deterministic at the time of the decision, the decision maker (DM) has to choose the appropriate alternative on the basis of some scenarios (states of nature, events) predicted by experts, him or herself. Let us add that the probability of these scenarios may be known or not. The situation where parameters can be presented by means of random variables is characteristic for the decision making under risk. When it is impossible to calculate the likelihoods aforementioned the choice of an alternative is made under uncertainty (Groenewald and Pretorius 2011; Render et al. 2006; Chronopoulos et al. 2011; Sikora 2008; Trzaskalik 2008). Knight (1921) first introduced the idea to apply risk and uncertainty in economics, but these two categories were formally integrated in economic theory by von Neumann and Morgenstern (1944). In this contribution we will focus on the second case which seems to be more frequent in realistic decision problems.
The literature offers dozens of procedures applied in DMUU, such as the classical rules, which will be discussed in Sect. 3, and many diverse extensions or hybrids of these methods (see e.g. Basili 2006; Basili and Zappia 2010; Basili et al. 2008; Ghirardato et al. 2004; Ellsberg 2001; Marinacci 2002). One of them is the Hurwicz’s criterion method (Hurwicz’s OptimismPessimism Approach). This procedure usually leads to reasonable answers, but in some specific situations the Hurwicz’s results may be astonishing. Therefore, the target of this paper is to present two modified Hurwicz’s rules which yield more logical results. The first method is designed for passive decision makers who are only asked to declare their level of pessimism and optimism. The second one requires an active attitude in the decision making process, i.e. the ability to analyze the payoffs matrix and to determine some additional parameters.
The remainder of the paper is organized as follows. Section 2 contains a short description of the decision making under uncertainty and the assumptions adopted in the contribution. In Sect. 3, the most wellknown methods for the DMUU are briefly discussed. A deeper analysis of the Hurwicz’s criterion procedure is presented in Sect. 4. The proposed modified Hurwicz’s techniques are demonstrated in Sect. 5. Finally, conclusions are gathered in Sect. 6.
2 Decision making under uncertainty: characteristics and assumptions
Profit matrix/decision table (general case)
Scenarios and alternatives  \(I_{1}\)  \(I_{j}\)  \(I_{n}\) 

\(S_{1}\)  \(a_{11}\)  \(a_{1j}\)  \(a_{1n}\) 
\(S_{i}\)  \(a_{i1}\)  \(a_{ij}\)  \(a_{in}\) 
\(S_{m}\)  \(a_{m1}\)  \(a_{mj}\)  \(a_{mn}\) 
Notice that the decision making under uncertainty may not necessarily signify the occurrence of a finite number of scenarios with a set of \(m\) profits for each decision. In the other concept of DMUU it is assumed that the exact profit connected with the alternative \(I_{j}\) is not known, but belongs to an interval [\(w_{j},\,m_{j}\)] and then we deal with the decision making under interval uncertainty (Huynh et al. 2007). In such a case each value from this interval is probable. In this contribution we will consider the scenarios’ approach for DMMU which is characterized by a lower degree of uncertainty than the interval approach because only several values from this range are probable.
In the uncertainty case the decision maker may search an optimal pure strategy or an optimal mixed strategy. A pure strategy, in contradiction to a mixed strategy, is a solution assuming that the decision maker chooses and completely executes one and only one alternative. Meanwhile the mixed strategy allows the decision maker to select and perform a weighted combination of several accessible alternatives. The whole paper concerns techniques dedicated for optimal pure strategy’s searching.
When talking about the selection of an optimal alternative we must be aware of the fact that each decision may be defined by a vector of values representing the performance of different criteria and then the DMUU is brought to a multicriteria decision making under uncertainty—MDMUU (Dominiak 2006, 2009). Here we will assume that each alternative may be characterized either by one essential criterion’s value or by one synthetic aggregated value denoting the overall realization of all significant criteria. Therefore only onecriterion problems will be discussed in this article.
3 The most wellknown methods for the decision making under uncertainty
This section contains a brief description of possible classical approaches applied in DMUU with scenarios when the decision maker is interested in finding an optimal pure strategy (Ignasiak 1996; Kaufmann and Faure 1974; Pazek and Rozman 2009; Sikora 2008; Trzaskalik 2008). Let us recall that in the scenarios’ approach decision makers have to choose one of a set number of alternatives with complete information about their outcomes but in the absence of any information or data about the probabilities of the various states of nature (Pazek and Rozman 2009). The decision rules presented below prescribe how an individual faced with a decision under uncertainty should go about choosing a course of action consistent with the individual’s basic judgments and preferences (Pazek and Rozman 2009).
At the end of this section it is worth emphasizing that all criteria of choice may suggest a different optimal strategy for the same problem.
Notice that the first four rules find application when the decision maker intends to perform the selected alternative only once. When he or she contemplates to realize this decision many times, it is recommended to use the Laplace’s criterion.
Let us also add that the rules aforementioned were described for problems with a target maximized (profits, effects, sales). When the objective function tends to the minimum (cost, time etc.), i.e. when payoffs are given as negativeflow rewards, one should apply one of the approaches suggested in GasparsWieloch (2012).
4 Hurwicz’s criterion method: a case study analysis
Payoffs matrix—case study
Investment projects  P1  P2  P3 

Scenarios  
S1  10  1  4 
S2  1  9.5  5 
S3  1  9.5  2 
S4  1  9.5  5 
S5  1  9.5  4 
Let us have a look on these data. The profits related to P1 belong to the largest interval \([w_{1},\,m_{1}] = [1, 10]\) and in one out of five scenarios this project gives the highest benefit. The project P2, with an interval \([w_{2},\,m_{2}] = [1, 9.5]\), is the best in four out of five states of nature. The last business plan (P3) yields, independently on the situation, quite congenial and relatively low gains (\([w_{3},\,m_{3}] = [2, 5]\)).
Hurwicz’s measures for projects P1, P2 and P3 (optimist and pessimist type)
Decision maker  DM A (optimist type)  DM B (pessimist type) 

Project  
P1  \(h_{P1}^A =0.3\times 1+0.7\times 10=7.3\)  \(h_{P1}^B =0.7\times 1+0.3\times 10=3.7\) 
P2  \(h_{P2}^A =0.3\times 1+0.7\times 9.5=6.95\)  \(h_{P2}^B =0.7\times 1+0.3\times 9.5=3.55\) 
P3  \(h_{P3}^A =0.3\times 2+0.7\times 5=4.1\)  \(h_{P3}^B =0.7\times 2+0.3\times 5=2.9\) 
 1.
The project P1 yields a very high profit only in the first scenario. In the four remaining states of nature it leads to the worst gains. For an optimist decision maker, prone to risk, this alternative seems to be the most appropriate. But let us focus on the pessimist one. If he or she assumes that the probability of the worst value is equal 0.7 (i.e. quite much), it would be more reasonable to choose P3 for which the worst value \(w_{3}\) is twice as high as \(w_{1}\). That is why we state that the optimal pure strategy set for the pessimist decision maker does not reflect his or her risk aversion. If the scenario S2, S3, S4 or S5 occurs, the loss of the pessimist DM choosing P1 will be in most cases higher than it would be if P2 or P3 were selected.
 2.
The project P2 gives the best results in four out of five states of nature. We intuitively conclude that this strategy should be safer for the pessimist decision maker than the project P1 which is the best merely in one out of five scenarios. Thus, it is fairly difficult to understand why P2 has obtained a lower Hurwicz’s measure than P1, especially in the pessimist case.
 3.
We are surprised not only with the optimal pure strategy selected by Hurwicz’s rule, but also with the ranking of projects. The order is totally the same for both pessimism indices (P1 dominates P2 and P2 is better than P3). This conclusion is astonishing as well since the parameters \(\alpha \) are significantly different.
Hurwicz’s measures for project P1, P2, P3 (for different values of \(\alpha \))
\(\upalpha \)  P1  P2  P3 

0.00  10.00  9.50  5.00 
0.10  9.10  8.65  4.70 
0.20  8.20  7.80  4.40 
0.30  7.30  6.95  4.10 
0.40  6.40  6.10  3.80 
0.50  5.50  5.25  3.50 
0.60  4.60  4.40  3.20 
0.70  3.70  3.55  2.90 
0.80  2.80  2.70  2.60 
0.83  2.50  2.42  2.50 
0.90  1.90  1.85  2.30 
1.00  1.00  1.00  2.00 
 1.
The order of projects changes only when \(\alpha \) is higher than 0.8! P3 is the best only for decision makers whose coefficient of pessimism exceeds 0.8.
 2.
Despite the fact that P1 and P2 are Paretooptimal and that P2 offers the highest payoffs in four out of five scenarios, the project P2 according to the Hurwicz’s rule does never take the first place in the ranking and does never obtain a higher index than the P1 measure.
 3.
When the parameter \(\alpha \) equals 83.3 %, the Hurwicz’s rule treats the project P1 on equal terms with the project P3, though the first one is much more risky than the third one.
 1.The Hurwicz’s rule takes only extreme payoffs into consideration. Transitional values, i.e. \(a_{ij} \in (w_j ,m_j)\), are ignored. This state entails the following consequence. The position of an alternative in the ranking is merely determined by parameters \(w_{j}\) and \(m_{j}.\) This factor explains why, according to the Hurwicz’s rule, the project P2 is always dominated by P1 in the problem discussed:$$\begin{aligned}&\left( {w_1 =w_2} \right) \wedge (m_1 >m_2)\Rightarrow (h_1 >h_2)\end{aligned}$$(12)$$\begin{aligned}&\left( {w_1 >w_2} \right) \wedge (m_1 =m_2)\Rightarrow (h_1 >h_2 ) \end{aligned}$$(13)
 2.The Hurwicz’s criterion does not take into account the frequency of relatively high and small payoffs belonging to the set of all profits assigned to a given alternative. Therefore, two decisions with the same minimal and maximal profits always obtain an identical Hurwicz’s index, even if one of them contains many small payoffs and the second one—many high payoffs:For example, if the decision maker may choose one out of two alternatives: A1 with rewards 5,1,1,1,1 and A2 with rewards 5,5,5,5,1, the Hurwicz’s rule gives the same value for both: \(h_{A1}=h_{A2} =\alpha \cdot 1+(1\alpha )\cdot 5=54\alpha \), which is rather unfair for the second one.$$\begin{aligned} \left( {w_1 =w_2} \right) \wedge (m_1 =m_2)\Rightarrow (h_1 =h_2) \end{aligned}$$(14)
 1.
in problems with an even distribution of payoffs for each alternative, i.e. when the number of rather optimistic scenarios is similar to the number of scenarios with rather bad results or
 2.
in the process of decision making under interval uncertainty, i.e. when merely the parameters \(w_{j}\) and \(m_{j}\) are known and when each value between them may theoretically occur.
5 Two modified Hurwicz’s criteria
The observations aforementioned encourage us to create a new method (or new methods) which takes into consideration both the minimal and the maximal value for each alternative and the frequency of the worst and the best payoffs. The Sects. 5.1 and 5.2 contain a description of both proposed modified Hurwicz’s methods. The first technique (the APO method) is designed for a passive decision maker who is not interested in carrying out a meticulous analysis of the payoffs matrix. The second procedure (the SAPO method) takes better into account the DM’s preferences, but requires a more active and conscious attitude in the decision making process.
5.1 Method I (the averages of good and bad results weighted by the pessimism and optimism index: APO)
 1.
For each alternative \(I_{j}\) present the payoffs as a nonincreasing sequence \(Sq_j =(a_{1j} ,\ldots ,a_{sj},\ldots ,a_{mj})\) containing \(m\) terms (where \(m\) still denotes the number of scenarios and \(s\) is the number of the term).
 2.Fix the value of the parameter \(C\) which signifies the number of good and bad terms in the sequences [Eq. (15)]:$$\begin{aligned} C=\max \{1,\left\lceil {m\cdot \min \{\alpha ,1\alpha \}} \right\rceil \} \end{aligned}$$(15)
 3.For each alternative calculate the average of good results and the average of bad results according to the Eqs. (16)–(17).$$\begin{aligned} A_j^{I,\max }&= \frac{1}{C}\sum _{s=1}^C {a_{sj}} \quad j=1,\ldots ,n\end{aligned}$$(16)$$\begin{aligned} A_j^{I,\min }&= \frac{1}{C}\sum _{s=mC+1}^m {a_{sj} } \quad j=1,\ldots ,n \end{aligned}$$(17)
 4.For each decision calculate the modified Hurwicz’s measure \((H^\mathrm{I})\) using the following expression:$$\begin{aligned} H_j^I =\alpha \cdot A_j^{I,\min } +(1\alpha )\cdot A_j^{I,\max } \end{aligned}$$(18)
 5.Select the strategy fulfilling the condition (19):As mentioned before the technique presented above is similar both to the Hurwicz’s rule and to the Laplace’s rule. On one side, the APO method takes advantage of the pessimism index. On the other side, it consists in calculating the mean value on the basis of all (when \(\alpha \) equals 0.5) or almost all payoffs (when the decision maker is a moderate optimist or pessimist) connected with a given alternative.$$\begin{aligned} H_j^{I^{*}} =\mathop {\max }\limits _j \{H_j^I \} \end{aligned}$$(19)
The condition (15) allows to fit the cardinality (i.e. the number of elements) of the subsequence of good and bad results to the level of optimism and pessimism. Thus, a radical optimist or pessimist calculates the averages only on the basis of the extreme values of particular alternatives, whereas the quantity of terms considered in both averages is pointedly larger for moderate decision makers, which means that in the second case the transitional values \(a_{ij} \in (w_j ,m_j)\) have an influence on the modified Hurwicz’s measure.
5.2 Method II (the shortened averages of good and bad results weighted by the pessimism and optimism index: SAPO)
In the first suggested method the subsequences of good and bad payoffs are determined by the level of the parameter \(\alpha \). It is a quite comfortable technique, nevertheless such an approach entails the risk of inserting rather bad results in the “good” subsequence or rather good results in the “bad” subsequence. For example, if \(\alpha =0.4\) and the rewards of the alternative A1 are 5,1,1,1,1, the average of good results will include two payoffs: 5 and 1, though the second value is not high at all.

Method II.A. Separately for each alternative in a relative way (using deviation degrees).

Method II.B. Separately for each alternative in an absolute way (using bounds).

Method II.C. Together for all alternatives in a relative way (using deviation degrees).

Method II.D. Together for all alternatives in an absolute way (using bounds).
When the initial ranges are calculated in an absolute way one may use parameters \(b_{j}^\mathrm{max}\) and \(b_{j}^\mathrm{min}\) which signify the lower bound of the “good” range and the upper bound of the “bad” range, respectively. If the initial ranges are defined together for all alternatives, then the bounds are the same and equal to \(b^\mathrm{max}\) and \(b^\mathrm{min}.\)
Here, for simplicity’s sake, we will discuss only the case when the “good” and the “bad” range is defined by means of deviation degrees and these relative ranges have the same allowable deviations for all alternatives. In such a situation we need only two parameters: \(d^\mathrm{max}\) and \(d^\mathrm{min}\), since \(d_{1}^\mathrm{max}=d_{2}^\mathrm{max}={\ldots }=d_{j}^\mathrm{max}=d^\mathrm{max}\) and \(d_{1}^\mathrm{min}=d_{2}^\mathrm{min}={\ldots }=d_{j}^\mathrm{min}=d^\mathrm{min}\).
Remark that the final ranges of “good” and “bad” results are usually shorter than the initial ones due to the parameter \(\alpha \) which represents the DM’s risk aversion and has an impact on \(C\) (see the second step of the following method).
 1.
For each alternative \(I_{j}\) present the payoffs as a nonincreasing sequence \(Sq_j =(a_{1j},\ldots ,a_{sj},\ldots ,a_{mj})\) containing \(m\) terms (where \(m\) still denotes the number of scenarios and \(s\) is the number of the term).
 2.For each alternative generate the subsequence of good results \((SSq_{j}^\mathrm{max})\) and the subsequence of bad results \((SSq_{j}^\mathrm{min})\) using Eqs. (25) and (26):$$\begin{aligned} SSq_j^{\max }&= \left\{ a_{sj} \!\in \! Sq_j :(m_j d^{\max }(m_j w_j )\le a_{sj} \!\le \! m_j )\wedge \left( {\left {SSq_j^{\max } } \right \!\le \! C} \right) \right. \nonumber \\&\quad \left. \wedge (a_{sj} \rightarrow \max ) \right\} \quad j=1,\ldots ,n\end{aligned}$$(25)where \({\vert }SSq_{j}^\mathrm{max}{\vert }\) and \({\vert }SSq_{j}^\mathrm{min}{\vert }\) signify the final cardinalities of both subsequences and the parameter \(C\) is computed according to the constraint (15). The Eq. (25) allows the decision maker to include in the subsequence \(SSq_{j}^\mathrm{max}\) only these elements of the whole sequence which belong to the range determined by the deviation degree \(d^\mathrm{max}\). For example, if \(d^\mathrm{max}=0.2,\,m_{j}=20,\,w_{j}=5\), then the elements of \(SSq_{j}^\mathrm{max}\) should satisfy the following constraint \(a_{sj} \in [200.2(205);20]=[17;20]\). Notice that the final cardinality of \(SSq_{j}^\mathrm{max}\) is additionally limited by \(C\) which depends on the pessimism and optimism indices. Closer to 0 and 1 they are, less elements the subsequence \(SSq_{j}^\mathrm{max}\) may contain. Such a relation may be explained by the fact that more radical the decision maker is, more likely, in his or her opinion, one of the extreme values is. If \(m=10\) and \(\alpha =0.2,\,SSq_{j}^\mathrm{max}\) may constist of at least two elements which, due to the last part of the Eq. (25), must be the highest. Thanks to the parameters \(d^\mathrm{max}\) and \(\alpha \) the decision maker is able to set a subsequence \(SSq_{j}^\mathrm{max}\) which, from his or her point of view, is composed of appropriate payoffs, because the formula (25) takes into consideration both the subjective evaluation of “good” values and the DM’s risk aversion. The Eq. (26) has an analogical interpretation.$$\begin{aligned} SSq_j^{\min }&= \left\{ a_{sj} \!\in \! Sq_j :(w_j \le a_{sj} \!\le \! w_j +d^{\min }(m_j w_j ))\wedge \left( {\left {SSq_j^{\min } } \right \!\le \! C} \right) \right. \nonumber \\&\quad \left. \wedge (a_{sj} \rightarrow \min ) \right\} \quad j=1,\ldots ,n \end{aligned}$$(26)
 3.For each alternative calculate the average of good results and the average of bad results, see Eqs. (27)–(28).$$\begin{aligned} A_j^{II,\max }&= \tfrac{1}{\left {SSq_j^{\max }} \right }\sum _{a_{sj} \in SSq_j^{\max }}{a_{sj}}\quad j=1,\ldots ,n\end{aligned}$$(27)$$\begin{aligned} A_j^{II,\min }&= \tfrac{1}{\left {SSq_j^{\min }} \right }\sum _{a_{sj} \in SSq_j^{\min }}{a_{sj}} \quad j=1,\ldots ,n \end{aligned}$$(28)
 4.For each decision compute the modified Hurwicz’s measure \((H^\mathrm{II})\) using the following expression:The parameters \(m,\left {SSq_j^{\min }} \right ,\left {SSq_j^{\max }} \right \) inserted in the condition (29) enable to take into consideration the size of both subsequences, i.e. the frequency of particular payoffs. As one can see the index \(H^{II}_{j}\) is proportional to the number of good payoffs, i.e. the final cardinality of \(SSq_{j}^\mathrm{max}\), and inversely proportional to the number of bad values, i.e. the final cardinality of \(SSq_{j}^\mathrm{min}\), because a given alternative is more attractive when it contains many high profits and few low results. The fractions \(\frac{m+1\left {SSq_j^{\min }} \right }{m}\) and \(\frac{m1+\left {SSq_j^{\max }} \right }{m}\) are equal to 1 when particular subsequences consist of one term. If \({\vert }SSq_{j}^\mathrm{min}{\vert }\) increases, then the first fraction is smaller than 1, but bigger than 0. The weight \(\frac{m+1\left {SSq_j^{\min }} \right }{m}\) is a kind of punishment for the alternative which number of bad results is high because such a distribution of payoffs is not desirable for the decision maker. On the other hand, if \({\vert }SSq_{j}^\mathrm{max}{\vert }\) increases, then the second fraction is bigger than 1, but smaller than 2. The weight \(\frac{m1+\left {SSq_j^{\max }} \right }{m}\) is a kind of bonus for the alternative which number of good results is high because such a distribution of payoffs is muchdesired.$$\begin{aligned} H_j^{II} =\alpha \cdot \tfrac{m+1\left {SSq_j^{\min }} \right }{m}A_j^{II,\min } +(1\alpha )\cdot \tfrac{m1+\left {SSq_j^{\max }} \right }{m}A_j^{II,\max }\nonumber \\ \end{aligned}$$(29)
 5.Select the strategy fulfilling the condition (30):$$\begin{aligned} H_j^{II^{*}} =\mathop {\max }\limits _j \{H_j^{II} \} \end{aligned}$$(30)
5.3 Demonstration and results
Let us analyze the results generated by both modified Hurwicz’s criteria (APO and SAPO) for the case presented in Sect. 4 (Table 2). The first decision maker (A) was an optimist \((\alpha ^{A}=0.3)\). The second one (B) was a moderate pessimist \((\alpha ^{B}=0.7)\).
 1.The nonincreasing sequences of payoffs:$$\begin{aligned} Sq_{P1} =(10,1,1,1,1) \quad Sq_{P2} =(9.5,9.5,9.5,9.5,1) \quad Sq_{P3} =(5,5,4,4,2) \end{aligned}$$
 2.The parameters \(C\) are the same for both decision makers:$$\begin{aligned} C^{A}=\text{ max }\left\{ 1,\lceil 5 \cdot \text{ min }\left\{ 0.3,0.7\right\} \rceil \right\} =2, C^{B}=\text{ max }\left\{ 1,\lceil 5 \cdot \text{ min }\left\{ 0.7,0.3\right\} \rceil \right\} =2, \end{aligned}$$
 3.
 4.
Table 5 presents the modified Hurwicz’s measures \(H_{j}^\mathrm{I}\) for all business plans (see Eq. 18).
 5.
In both cases (optimist and pessimist type) the alternative fulfilling the condition (19) is the project P2. The ranking of projects changes depending on the value of the pessimism index because in this example: \(A_{P2}^{I,\max } >A_{P1}^{I,\max } >A_{P3}^{I,\max }\) and \(A_{P2}^{I,\min } >A_{P3}^{I,\min } >A_{P1}^{I,\min }\).
Modified Hurwicz’s measures \((H^\mathrm{I})\) for projects P1, P2, P3 (optimist and pessimist type)—method I (APO)
Decision maker  DM A (optimist type)  DM B (pessimist type) 

Project  
P1  \(h_{P1}^{I,A} =0.3\times 1+0.7\times 5.5=4.15\)  \(h_{P1}^{I,B} =0.7\times 1+0.3\times 5.5=2.35\) 
P2  \(h_{P2}^{I,A} =0.3\times 5.25+0.7\times 9.5=8.23\)  \(h_{P2}^{I,B} =0.7\times 5.25+0.3\times 9.5=6.53\) 
P3  \(h_{P3}^{I,A} =0.3\times 3+0.7\times 5=4.4\)  \(h_{P3}^{I,B} =0.7\times 3+0.3\times 5=3.6\) 

its highest value had significantly exceeded the highest payoffs of other projects.

its highest value had occurred in more than one state of nature.

the parameter \(\alpha \) had been close to 0 (radical optimist).
Modified Hurwicz’s measures \((H^\mathrm{II})\) for projects P1, P2, P3 (optimist and pessimist type)—method II (SAPO)
Decision maker  DM A (optimist type)  DM B (pessimist type) 

Project  
P1  \(H_{P1}^{II,A} =0.3\times \frac{4}{5}\times 1+0.7\times \frac{5}{5}\times 10=7.24\)  \(H_{P1}^{II,B} =0.7\times \frac{4}{5}\times 1+0.3\times \frac{5}{5}\times 10=3.56\) 
P2  \(H_{P2}^{II,A} =0.3\times \frac{5}{5}\times 1+0.7\times \frac{6}{5}\times 9.5=8.28\)  \(H_{P2}^{II,B} =0.7\times \frac{5}{5}\times 1+0.3\times \frac{6}{5}\times 9.5=4.12\) 
P3  \(H_{P3}^{II,A} =0.3\times \frac{5}{5}\times 2+0.7\times \frac{6}{5}\times 5=4.8\)  \(H_{P3}^{II,A} =0.7\times \frac{5}{5}\times 2+0.3\times \frac{6}{5}\times 5=3.2\) 
As one can see the first modified Hurwicz’s rule is a little bit similar to the Laplace’s approach, since it is based on an average of almost all payoffs treated as equally likely.
Method II.
 1.The nonincreasing sequences of payoffs:$$\begin{aligned} Sq_{P1} =(10,1,1,1,1) \quad Sq_{P2} =(9.5,9.5,9.5,9.5,1) \quad Sq_{P3} =(5,5,4,4,2) \end{aligned}$$
 2.The parameter \(C\) is still equal to 2:The subsequences of good and bad results (see Eqs. 25–26):$$\begin{aligned} C^{A}=\max \{1,\left\lceil {5\cdot \min \{0.3,0.7\}} \right\rceil \}=2, \quad C^{B}=\max \{1,\left\lceil {5\cdot \min \{0.7,0.3\}} \right\rceil \}=2 \end{aligned}$$$$\begin{aligned} SSq_{P1}^{\max }&= \left\{ a_{s,P1} \in Sq_{P1} :(100.35(101)\le a_{s,P1} \le 10)\wedge \left {SSq_{P1}^{\max }} \right \right. \\&\left. \le 2\wedge (a_{s,P1} \rightarrow \max ) \right. \}=(10) \\ SSq_{P1}^{\min }&= \left\{ a_{s,P1} \in Sq_{P1} :(1\le a_{s,P1} \le 1+0.35(101))\wedge \left {SSq_{P1}^{\min } } \right \right. \\&\left. \le 2\wedge (a_{s,P1} \rightarrow \min ) \right. \}=(1,1)\\ SSq_{P2}^{\max }&= (9.5,9.5) \quad SSq_{P2}^{\min } =(1)\\ SSq_{P3}^{\max }&= (5,5) \quad SSq_{P3}^{\min } =(2) \end{aligned}$$
 3.
 4.
Table 6 presents the modified Hurwicz’s measures \(H_{j}^{II}\) for all business plans (see Eq. 29).
 5.
Here again the project P2 wins in both cases, but it is worth underlying that the ranking changes depending on the parameters \(\alpha \) and \(d.\)
6 Conclusions
The contribution concerns the Hurwicz’s criterion and its limited application for the scenarios’ approach in decision making under uncertainty. As it was stated this rule leads to more logical and rational results when instead of several possible scenarios’ payoffs each value from a given interval is theoretically probable or when the discrete distribution of payoffs is rather uniform. Therefore, two new procedures designed for DMMU with states of nature are proposed in the article. Both procedures enable to take into consideration not only the extreme rewards connected with each decision, but also these payoffs which are close to the minimal and maximal values. The first technique proposed (the APO method) is designed for passive decision makers, i.e. for people interested in getting a quick answer about the best decision just on the basis of their level of risk aversion. The second procedure suggested (the SAPO method) allows to better control the width and the contents of the range of good and bad results considered in the modified Hurwicz’s measure. Nevertheless, this time the decision maker must present a more active and conscious attitude in the decision making process. He or she has to carefully analyze the payoffs assigned to each alternative and to determine the width of the “good” and “bad” intervals by means of bounds or deviation degrees. Here, we analyzed a case concerning the choice of an investment project, but actually the methods described may be used in each decision making process within which each alternative is characterized by only one criterion (or one synthetic measure) and the goal of the decision maker is to find the optimal pure strategy.
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