Thomas Whalen
Professor of Decision Sciences
Georgia State University
Atlanta, GA 30303-3083, USA
ABSTRACT There are many practical decision problems and information channels whose most natural probability representation involves ranges or regions in probability space. A major source of such problems is team decisions which require optimal use of limited or costly communication. Probabilities may be communicated in rounded form, or events whose probability exceeds a prespecified threshold may be identified. Another major concern is ordinal information about probability. This ordinal information may come as a summary message from a teammate, or more directly -- e.g. by observing a random walk process after an unknown number of steps. All the abovett cases, and many others, can be expressed by systems of linear constraints on probabilities. Some of them, but not all, can also be represented by the basic probability assignments of Dempster-Shafer's theory. Several methods for decision making with linearly constrained probabilities are introduced and compared in a simulation experiment. On both theoretical and empirical grounds the most promising appears to be the extended Laplace criterion, based on a concept of second order maximum entropy. An alternative approach using an interactive graphic decision support system for problems of this type is also presented using a simple example.
Keywords: Interval probabilities, second order probabilities, decision analysis
1. Risk and Ignorance
The general problem of decision making under uncertainty involves a set of n states of nature, a set of k alternative actions, and a utility function. The utility function assigns a vector of n values to each alternative action; each element of this vector specifies the value of the action under the corresponding state of nature. The k utility vectors typically take the form of row vectors collected into a k by n utility matrix associating a specific value to each (state, action) pair.
Standard treatments of decision making under uncertainty are divided into two separate branches: decisions under risk and decisions under ignorance (Bunn, 1984; Resnik, 1986). Under risk, the numeric probability of each state of nature is also assumed to be known or estimated. This enables us to reduce the utility vector of each alternative action to a scalar expected utility, by weighting each possible utility by the probability of the corresponding state of nature. The action whose expected utility is highest is selected.
Under ignorance, the decision maker has no knowledge at all about the probabilities of the states of nature. Many criteria exist for making a decision without recourse to probability. Implicitly or explicitly, each of these criteria also reduces an action's vector of possible utilities to a scalar figure of merit to facilitate comparisons between alternative actions. This requires replacing the weighting role of the missing probabilities with some other weighting scheme. The Laplace criterion emphasizes all states of nature equally. The Hurwicz criterion (of which maximax and maximin are special cases) emphasizes the most favorable and/or the most unfavorable states of nature. The minimax regret criterion emphasizes the states of nature for which the decision makes the most difference.
2. Intermediate Cases
Decisions must very often rely upon probability information that falls between the well studied extremes of pure risk and pure ignorance. This is especially true in team decision making (Marschak, 1955; Ho and Chu, 1972) when one team member assesses a probability distribution but because of time or other constraints can only communicate a standard, concise description of the distribution to the actual decision maker. Each message that can be sent corresponds to a region within a probability space with (n-1) dimensions, where n is the number of states of nature.
Note that the authors and publishers of handbooks, almanacs, or other sources of potentially useful information can be viewed as generalized "teammates" of everyone who consults their publications.
Sometimes the decision maker has enough information to arrange the possible states of nature in order from most probable to least probable without being able to numerically specify the probabilities of individual states of nature. This ordinal information may come as a summary message from a teammate, or it may come more directly. For example, if a random walk process is observed after a large but unknown number of steps, the most probable direction can be inferred but not how probable it is. This is useful information for implementing a stochastic control system, since it is as important to keep probabilities of opposite discrepancies in balance as it is to keep them small in order to minimize long run overall drift.
Alternatively, the decision maker may be given information about which states of nature, if any, have a probability above a specified threshold. Another important special case of incomplete probability information arises when probabilities are in rounded form; for example, we may be told that P(A) = .2, P(B)=.3, and P(C) =.4 to the nearest tenth, where A, B, and C are a mutually exclusive exhaustive event set whose unrounded probabilities must sum to 1.
3. Linearly Constrained Probabilities
All the above cases, and many others, can be expressed by systems of linear constraints on probabilities. The true probability distribution (or a teammate's true subjective distribution) is not known in detail, but the decision maker knows that it must lie within a particular region in probability space.
If a decision maker receives enough information to determine a precise (objective or subjective) probability assessment, the probability region is reduced to a single point and the recipient faces a problem of decision making under pure risk. On the other hand, if the recipient can derive no information about the sender's subjective probabilities, the probability region is the whole of probability space, constrained only by the ordinary axioms of probability. In this case, the recipient's problem is equivalent to decision making under pure ignorance.
4. Partial Second Order Ignorance
In the general case, the decision maker is ignorant about what probability distribution holds within a region of possible probability distributions. Each point in the region specifies an ordinary probability distribution over the original states of nature. This probability distribution together with the utility matrix for (state-action) pairs in turn specifies an expected value for each action. Thus each point in the region of possible probability distributions specifies an expected utility for each action. The decision maker knows that the true probability distribution over states of nature corresponds to one of the points, but has no information about the relative likelihood of the points within the region.
This is equivalent to a second order problem of decision making under ignorance in which the role of "states of nature" is filled by probability distributions over the original states of nature. In the absence of an appropriate second order probability distribution, the decision maker must rely upon some other considerations to weight the expected return or regret of each probability distribution, just as in ordinary decision making under ignorance.
It is relatively straightforward to find the corner points of a region in probability space defined by a system of linear constraints and to calculate the expected utility arising from each alternative action at each corner point. For any possible probability distribution, the expected utility for an action is a linear combination of the expected utilities of that action at these corner points. Therefore the maximum and minimum expected utility for each alternative action can be found by examining only these corner points.
5. Graphical Analysis When n=3
Suppose that the uncertainty of a decision problem can be expressed using just three possible states of nature. The space of possible probability distributions
with respect to these three events forms a plane triangle bisecting the unit cube, as shown in Figure 1. This fact enables us to graph any trinomial probability as a point on a set of triangular coordinates. The three corners of the triangle represent respectively the three degeneraye probability distributions which assign a probability of 1 to the corresponding states of nature.
Figure 2 shows the partitioning of trinomial probability space corresponding to information that completely orders the three probabilities. The three vertices, labeled (1,0,0), (0,1,0), and (0,0,1) represent degenerate probability distributions in which one of the three states (respectively A, B, or C) is certain. The point labeled (.1,.6,.3) is an example of a probability distribution compatible with the ordinal information that P(B) _ P(C) _ P(A) corresponding to the lower left hand region of the probability space. Similarly, figures 3, 4, and 5 show the partitionings for information systems that inform the decision maker which, if any, of the states has a probability greater than a particular threshold probability, denoted L. In Figure 3 the threshold is L = 1/2, in Figure 4 L = 1/3, and in Figure 5 L = 1/4.
Figure 6 shows the 166 different regions of probability space that arise from rounding each of the probabilities of three exclusive exhaustive events to the nearest tenth. The hexagonal and trapezoidal regions represent cases where the three rounded probabilities add up to 1.0. The upwards pointing triangles at the three corners represent the cases when one probability is rounded to 1.0 and the other two are rounded to zero. The upwards pointing triangles in the interior of Figure 6 contain probability distributions such as (.86,.06,.08) which when rounded add up to more than 1.0. Finally, the downwards pointing triangles contain probability distributions such as (.84.,03,.13) or (.94,.03,.03) which when rounded add up to less than 1.0.
6. When Do Linear Constraints Define Dempster-Shafer Evidence?
The Dempster-Shafer theory of evidence (Shafer, 1976; Zadeh, 1986) concerns one particular type of incomplete probability knowledge, represented by basic probability assignments. However, this model does not account for some kinds of probability knowledge that are of great practical importance.
An "evidence" (Shafer, 1976) is equivalent to a collection of lower bounds on the probabilities of all the event sets in a universe of discourse. Given a system of linear constraints on probabilities, these lower bounds can be found by linear programming. For example, when there are three possible states of nature, we can find lower bounds on the probability of the six nontrivial event sets using six linear programs all with the same constraints. Let L1 be the minimum possible value of P(s1) subject to the constraints (including P(s1)+P(s2)+P(s3) = 1 and P(s1), P(s2), P(s3) > 0). Similarly, let L2 and L3 be the minima of P(s2) and P(s3), L12 = min{P(s1)+P(s2)}, L13 = min{P(s1)+P(s3)}, L23 = min{P(s2)+P(s3)}.
Basic probability assignments, if they exist, can then be determined from the lower probabilities:
m1 = L1, m2 = L2, m3 = L3, m12 = L12-L1-L2, m13 = L13-L1-L3, m23 = L23-L2-L3, and m123 = 1-(m1+m2+m3+m12+m13+m23) = 1-L12-L13-L23+L1+L2+L3.
It is convenient to compute the final quantity first; if it is nonnegative then a valid basic probability assignment exists and the system of linear constraints constitutes an evidence in the sense of Shafer. If the calculated value for m123 is negative this immediately shows that the system of linear constraints does not correspond to any Dempster-Shafer evidence.
When there are only two possible states of nature, the ordinal information that state 1 is more probable than state 2 corresponds to the probability threshold information that P(s1)>.5. This can be represented by the basic probability assignment m(s1)=.5, m(s2)=0, m(s1Us2)=.5. However, when there are more than two possible states of nature, as in Figure 2, ordinal information about probabilities can never be expressed by basic probability assignments.
Probability threshold information does not always correspond to basic probability assignments. In Figure 3 (probability threshold = .5), each of the three corner regions corresponds to a bpa; the central region is the intersection of three regions all of which have a bpa, but it cannot be represented by any bpa. The information systems shown in Figures 4 and 5, on the other hand, are entirely describable in Dempster-Shafer terms.
Rounded probabilities, as in Figure 6, can sometimes be represented by basic probability assignments but not when the rounded probabilities add up to less than 1.0. For example, probabilities of .33, .33, and .34 would be rounded to .3, .3, and .3. The latter would be a useful approximation to the true probabilities but it cannot be expressed as a basic probability assignment. The rounded information system (Figure 6) contains five types of regions. Let us examine each type in turn to see whether they correspond to a Dempster-Shafer evidence.
The 36 hexagonal regions in Figure 6 correspond to the simplest sort of rounded probabilities, in which none of the three rounded values are zero or 1 and the three sum to exactly one. One such rounded distribution is (.6,.3,.1). This defines the following system of linear constraints: .55>P(s1)<.65, .25<P(s2)<.35, .05<P(s3)<.15.
Six linear programs give us L1=.55, L2=.25, L3=.05, L12=.85, L13=.65, and L23=.35.
1-L12-L13-L23+L1+L2+L3) = 1-1.85+.85 = 0
Since 0 > 0, the region is an evidence with the following bpa:
m1=.55, m2=.25, m3=.05, m12=.05, m13=.05, m23=.05, m123=0.
Table 1 summarizes these calculations for all five types of regions. Note that since 1-L12-L13-L23+L1+L2+L3 is negative in the last case (downward pointing triangles), no corresponding basic probability assignment exists
Converting from a Dempster-Shafer evidence to a system of linear constraints is very straightforward. The lower bound for the probability of each state of nature is equal to its basic probability assignment, and the upper bound is the sum of all the basic probability assignments of sets containing that state of nature.
7. Decision Criteria
A logical first step in making a decision under uncertainty is dominance screening. Potter and Anderson (1980) discuss dominance screening in the context of linearly constrained Bayesian priors. Ordinary linear programming can find the maximum and minimum values of the difference between the expected utility (EU) of one alternative and that of another. One alternative decision dominates another if the maximum and the minimum difference have the same sign. (A common error is to assume that the maximum EU of the dominated act must be less than the minimum EU of the act that dominates it. In fact two utility ranges can overlap even if one action always has higher EU than the other for every particular feasible probability distribution.)
Table 1:
Are Rounded Probabilities Dempster-Shafer Evidences?
Table 1:
Are Rounded Probabilities Dempster-Shafer Evidences?
+---------------------------------------------------------------------+
| | | | Lower |Calcu-|Basic |
|Shape |Example |Probability | Proba- |lated |Probability|
| | |Bounds | bilities |m123 |Assignment |
+---------+----------+-----------------+-----------+------+-----------+
|36 |(.6,.3,.1)|0.55 _P(s1)_ 0.65| L1 = 0.55| 0 |m1 = 0.55 |
|Hexagons | |0.25 _P(s1)_ 0.35| L2 = 0.25| |m2 = 0.25 |
| | |0.05 _P(s1)_ 0.15| L3 = 0.05| |m3 = 0.05 |
| | | | L12 = 0.85| |m12 = 0.05 |
| | | | L13 = 0.65| |m13 = 0.05 |
| | | | L23 = 0.35| |m23 = 0.05 |
| | | | | |m123= 0 |
+---------+----------+-----------------+-----------+------+-----------+
|27 |(.6,.4,0.)|0.55 _P(s1)_ 0.65| L1 = 0.55| 0.05 |m1 = 0.55 |
|Trape- | |0.35 _P(s1)_ 0.45| L2 = 0.35| |m2 = 0.35 |
|zoids | | 0 _P(s1)_ 0.05| L3 = 0| |m3 = 0 |
|At Edges | | | L12 = 0.95| |m12 = 0.05 |
| | | | L13 = 0.55| |m13 = 0 |
| | | | L23 = 0.35| |m23 = 0 |
| | | | | |m123= 0.05 |
+---------+----------+-----------------+-----------+------+-----------+
|3 |(1.,0.,0.)|0.95 _P(s1)_ 1| L1 = 0.95| 0.05 |m1 = 0.95 |
|Upward | | 0 _P(s1)_ 0.05| L2 = 0| |m2 = 0 |
|Pointing | | 0 _P(s1)_ 0.05| L3 = 0| |m3 = 0 |
|Triangles| | | L12 = 0.95| |m12 = 0 |
|At | | | L13 = 0.95| |m13 = 0 |
|Corners | | | L23 = 0| |m23 = 0 |
| | | | | |m123= 0.05 |
+---------+----------+-----------------+-----------+------+-----------+
|45 |(.6,.4,.1)|0.55 _P(s1)_ 0.65| L1 = 0.55| 0.05 |m1 = 0.55 |
|Upward | note |0.35 _P(s1)_ 0.45| L2 = 0.35| |m2 = 0.35 |
|Pointing | sum |0.05 _P(s1)_ 0.15| L3 = 0.05| |m3 = 0.05 |
|Triangles| = 1.1 | | L12 = 0.9| |m12 = 0 |
|In | | | L13 = 0.6| |m13 = 0 |
|Interior | | | L23 = 0.4| |m23 = 0 |
| | | | | |m123= 0.05 |
+---------+----------+-----------------+-----------+------+-----------+
|55 |(.5,.3,.1)|0.45 _P(s1)_ 0.55| L1 = 0.5|-0.05 |m1 = ERR |
|Downward | note |0.25 _P(s1)_ 0.35| L2 = 0.3| |m2 = ERR |
|Pointing | sum |0.05 _P(s1)_ 0.15| L3 = 0.1| |m3 = ERR |
|Triangles| = 0.9 | | L12 = 0.85| |m12 = ERR |
| | | | L13 = 0.65| |m13 = ERR |
| | | | L23 = 0.45| |m23 = ERR |
| | | | | |m123= ERR |
+---------+----------+-----------------+-----------+------+-----------+
Typically, more than one nondominated alternative will remain.
To reach a final decision, it is helpful to calculate a figure of merit to
represent the attractiveness of each action by a single number. When each
state's probability is fully determined, expected utility is the figure of
merit. When the probability is underdetermined, there are two approaches
to calculating a figure of merit. One approach first evaluates the range
of expected utilities possible for an action and then reduces this range
to a single representative expected utility. The other approach first reduces
the range of probability distributions to a single distribution and then
calculates just one expected utility based on this representative probability
distribution.
8. Representative Utility Approaches
The two most common ways to reduce a range of utilities to a single figure of merit are the maximin criterion and the midpoint criterion. Both are special cases of the Hurwicz family of criteria, which use a general weighted average of the minimum and maximum possible utility. The maximin criterion expresses conservatism in decision making, while the midpoint criterion seeks to optimize average performance.
The extended Hurwicz criterion selects the action with the greatest value of
a*[max{E(utility)}] + (1-a)*[min{E(utility)}]
taking max and min over the set of admissible probability distributions and taking expectation over states of nature according to each particular distribution. In particular, when the optimism coefficient a equals zero the extended Hurwicz criterion becomes extended maximin. If the constraints on probability are correct and remain constant for many iterations of the decision maker's action, the long-run average utility of the extended maximin criterion's selected action cannot possibly fall below the indicated value, while that of other actions might be below this value for some possible probability distribution. Similarly, when a=.5 the extended Hurwicz criterion becomes the extended midpoint criterion, while when a=1 it reduces to the extended maximax criterion.
9. Representative Probability Approaches
On the other hand, many authors (Jaynes, 1968; Gottinger, 1990) argue that uncertainties about probabilities ought to be resolved as objectively as possible; in other words, without reference to utilities. Under this assumption, Gottinger has shown that the only reasonable choice for a representative probability distribution from a range is the distribution whose entropy is highest (the Laplace criterion). These arguments are convincing, but their direct application to the probabilities of states of nature can lead to discarding most or all of the available information. For example, the standard maximum entropy (Laplace) form for a complete order over probabilities is equivalent to the maximum entropy form for total ignorance!
This dilemma can be resolved using a second order maximum entropy concept that preserves more real information while satisfying the requirements that motivate the original maximum entropy concept. (Whalen and Brönn, 1990) Rather than considering the probability distribution over the original set of states, we consider a second-order probability distribution over points in probability space (See Figures 2-6). Applying the maximum entropy principle to this distribution implies that all points in probability space should be considered equally likely. Thus the representative point for a region of probability space is the mean point of that region.
Geometrically, the ordinary maximum entropy distribution for a region in probability space (as in Figures 1 - 6) is the point in the region closest to the center of the entire probability space. The second-order maximum entropy distribution for a region is the center of that region itself. Under total ignorance, the region in question is the entire probability space, and both versions of maximum entropy select the same representative point; i.e. the center of the space.
10. Simulation Experiments
A series of simulation experiments compared four methods of determining a figure of merit (maximin, midpoint, standard Laplace, and extended Laplace) using seven different information systems:
(1) the null information system in which the decision maker has no information about probability,
(2) the ordinal information system of Figure 2, which ranks the three probabilities from lowest to highest (six possible messages),
(3) the information system of Figure 3, which identifies any state of nature whose probability is greater than .5 (four possible messages),
(4) the information system of Figure 4, which identifies any state(s) of nature whose probability is greater than 1/3 (six possible messages),
(5) the information system of Figure 5, which identifies any state(s) of nature whose probability is greater than .25 (seven possible messages),
(6) the information system of Figure 6, which gives the approximate probability of each of the three events, rounded to the nearest tenth, and
(7) the perfect information system in which the decision maker knows the exact probabilities of the three states.
Ten thousand trinomial distributions were generated using a uniform second-order distribution: p1 = 1 - _R, p2 = S*(1-p1), p3 = 1-p1-p2 where R and S are uniformly distributed random fractions. (This procedure makes the proportion of generated probability distributions falling in a region proportional to the area of the region in the graph of probability space.) Ten thousand 3X3 utility matrices were randomly generated; the highest utility in each matrix was 100 and the lowest zero, with other utilities uniformly distributed. Each pairing of a criterion with an information system selected an action, and the expected utility of that action was recorded for a total of ten thousand iterations. The lowest mean expected value was 64.255 (maximin criterion, null information system), and the highest mean expected value was 71.748 (perfect information system).
For the "rounded" information system, a fifth decision criterion is also shown. In this criterion, the expected value is simply calculated using the three rounded probabilities, regardless of whether they sum to 0.9, 1.0, or 1.1.
Table 2 summarizes the experimental findings. The table shows the mean expected utility of each combination of one of the seven information systems with one of the four decision criterion, expressed as a percentage of the range of mean expected utility from the lowest (64.255) to the highest (71.745). Thus, the percentages represent the proportion of the maximum benefit that can be derived from probability information.
TABLE 2: Results of Simulation Experiments
TABLE 2: Results of Simulation Experiments
+-------------+--------+--------+--------+---------+--------+-------+
| | | | | Stan- | Exten- | |
| | # of | Maxi- | Mid- | dard | ded | As |
| |Messages| min | point | Laplace | Laplace|Rounded|
+-------------+--------+--------+--------+---------+--------+-------+
| None| 1 | 0.0% | 33.9% | 48.0% | 48.0 | |
|Threshold=1/2| 4 | 78.0% | 86.4% | 80.9% | 88.57%| |
| Ordinal| 6 | 81.1% | 89.7% | 48.0% | 88.55%| |
|Threshold=1/3| 6 | 84.7% | 92.4% | 48.0% | 92.2% | |
|Threshold=1/4| 7 | 85.2% | 91.6% | 79.0% | 92.3% | |
| Rounded| 166 | 98.8% | 99.1% | 98.6% | 99.5% | 99.4% |
| Perfect| 10000 | 100.0% | 100.0% | 100.0% | 100.0% | |
+-------------+--------+--------+--------+---------+--------+-------+
Since the results in Table 2 are average results, the values
for the midpoint criterion are automatically better than the corresponding
values for maximin. Similarly, the second-order Laplace criterion is automatically
higher than the conservative standard Laplace.
Not surprisingly, there is a general tendency for the performance of the various techniques to increase with increasing richness of information as measured by the number of alternative messages. But there are some noteworthy exceptions.
One very striking finding is the poor performance of the ordinal information system. By the standard Laplace criterion, ordinal information is no better than no information at all. Under the other three criteria, knowing which one of the six ordinal regions contains the true probability is always less useful than knowing which one of the six regions derived from a probability threshold of 1/3 contains it. In the two representative probability approaches (standard Laplace and extended Laplace), the six-message ordinal information system is actually inferior to the four-message information system with probability threshold .5! Furthermore, learning which of the four regions for L=1/2 contains the true probability and applying the second-order Laplace criterion obtains 88.57% of the benefit of the perfect information system. This is better than that obtainable by any of the four criteria applied to the six-region ordinal information system.
The additional bandwidth of the seven-message information system in Figure 5 versus the six-message system in Figure 4 does not guarantee improved performance. Under the midpoint criterion, the seven-message information system with threshold .25 is inferior to the six-message information system with threshold 1/3, while under the standard Laplace criterion the four-message information system with probability threshold .25 outperforms both six-message information systems and the seven-message information system. The only decision criterion which comes close to consistently rewarding richer information with better performance is the extended Laplace, although even here the performance with ordinal is very slightly poorer than the performance with threshold .5.
Comparing decision criteria under a given information system, the extended Laplace consistently outperforms the others except in the case of the ordinal information system, in which it is not quite as good as the midpoint criterion. The standard Laplace is consistently the worst except with the information system with probability threshold = .5, in which it is better than the maximin criterion but worse than the midpoint and extended Laplace.
Figure 7 compares the decision criteria for the three probability threshold information systems and the rounded probability information system. (The horizontal axis, labeled "bandwidth," is the logarithm to the base 2 of the number of messages in the information system, ranging from 2 bits for the four-message system to 7.375 bits for the 166-message system.)
11. Application to Decision Support Systems
We are currently designing and implementing a variety of decision support systems (DSS) based on second order representative utility approaches. These systems will be used to carry out a program of experimentation comparing the actual effectiveness of these competing concepts of decision making under partial ignorance. These decision support systems include a maximin DSS, a maximax DSS, a Hurwicz DSS which combines the latter two, and a minimax regret DSS.
In the maximin DSS, the user picks an alternative action Ai to serve as standard. The first step is to find the minimum and maximum possible expected utilities for Ai subject to the system of linear constraints which define the region of possible probability distributions. The DSS graphs this information in the form of a 45-degree line segment from the point (min(euAi),min(euAi)) to the point (max(euAi),max(euAi)).
For each possible expected utility of the standard alternative, min(euAi) < x < max(euAi), the DSS then finds the minimum possible expected utility for each of the other alternative actions when euAi = x, denoted min(euAj |euAi=x). This is done by adding an additional constraint, euAi = x, to the system of linear constraints which define the region of possible probability distributions. The sensitivity analysis
features of linear programming make it possible to completely identify min(euAj |euAi=x) as a function of x by evaluating a relatively small number of LP problems.
The maximin DSS then graphs the minimum possible expected utility of each alternative action given each possible expected utility (euAi) of the standard alternative. Suppose that action Ai is the standard and x is one possible expected utility for action Ai. Min(euAj | euAi=x) = x, so the DSS will plot a point at (x,x) in the 45-degree line corresponding to action Ai. Each of the other alternative actions Aj, j =/ i, generates a piecewise linear curve composed of points of the form (x,min(euAj | euAi=x)) showing the minimum possible expected utility of action Aj over all possible probability distributions for which the expected utility of action Ai is x.
The maximax DSS differs from the maximin DSS only in plotting points (x,max(euAj | euAi=x)) instead of points (x,min(euAj | euAi=x)) for each action Aj other than the standard Ai. The "Hurwicz" DSS plots a linear combination (typically a simple average) of the expected utilities used by the maximin DSS and the maximax DSS.
The minimax regret DSS uses the range of possible expected regrets (erg) for the standard alternative instead of range of possible expected utilities, and plots max(ergAj | ergAi=x) to show the maximum possible expected regret of action Aj for all possible probability distributions for which the expected regret of action Ai is x. (This DSS is based on the extended minimax regret criterion, which selects the act whose max(E(regret)) is least. Thus, the criterion emphasizes those probability distributions where the decision makes the greatest difference to the consequences, measured by the difference between the expected utility of a given action and the expected utility of the optimal action for that probability distribution.)
12. DSS Example
Table 3 shows a utility matrix that will be used to illustrate one of the projected decision support systems. Suppose the decision maker's knowledge of the probabilities of the three states of nature is as indicated by the basic probability assignment m1=1/3, m2=0, m3=0, m12=1/6, m13=1/6, m23=0, m123=1/3. (This is the most restrictive bpa that contains all probability distributions for which s1 is the most likely state.) Table 3 gives the corner points of the corresponding probability region along with the expected utilities of the three alternative actions at these points.
Figure 8 shows the maximin DSS display for this situation
when action A1 is used as the standard. Both axes range from 0 to 4 because
that is the full range of utilities in Table 2. With the given information,
the expected utility of action A1 (euA1) ranges from 1 to 2, so the minimum
expected utility function for action A2 is a straight line running from (1,1)
to (2,2). The dark lines labeled euA2 and euA3 are found by examining the
points in the probability region for which euA1 has a given value and taking
the minimum value of euA2 (resp. euA3) over those points. (The lighter lines
show the maximum possible expected utilities of euA2 and euA3 for each possible
value of euA1. Note that there is only one possible probability distribution
for which euA1 = 2, so the minimum and maximum values of each action coincide
when euA1 =2.)
TABLE 3: Utility Matrix TABLE 4:
+-------+--------------------+ Corner points of probability region
| | States | +-----+-----+-----+------+------+------+
+-------+------+------+------+ | p1 | p2 | p3 | euA1 | euA2 | euA3 |
|Actions| s1 | s2 | s3 | +-----+-----+-----+------+------+------+
+-------+------+------+------+ |0.33 |0.33 |0.33 | 1 | 1.83 | 2.83 |
| A1 | 2 | 1 | 0 | |0.5 |0 |0.5 | 1 | 1.25 | 3.25 |
| A2 | 1.5 | 3 | 1 | |0.5 |0.5 |0 | 1.5 | 2.25 | 1.25 |
| A3 | 0.5 | 2 | 6 | |1 |0 |0 | 2 | 1.5 | 0.5 |
+-------+------+------+------+ +-----+-----+-----+------+------+------+
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Figure 6
