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Abstract

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

In real life, people often need to make decisions in face of uncertainty. The traditional approach to rational decision making under uncertainty is the theory of expected utility. However, the Ellsberg paradox shows that the preference pattern exhibited by ordinary people often violates the expected utility theory. Similar to the Ellsberg paradox, Machina proposes two additional paradoxes, which challenge various important nonexpected utility models developed in the literature. This paper attempts to provide a unified treatment of all these paradoxes by extending the ordered weighted averaging operator based decision model to allow the degree of optimism to take multiple values instead of a single value.

1. INTRODUCTION

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

Over the past decade, there has been an increasing research focus on the study of decision making under uncertainty in the field of intelligent systems. An effective theoretical framework of decision making under uncertainty can provide us a basis for the design and implementation of intelligent systems, especially the intelligent decision making support systems.[1-4]

The classic expected utility theory[5, 6] has been commonly considered as the standard theory of decision making under uncertainty. However, Ellsberg[7] presented an example which has now became known as the Ellsberg paradox, and demonstrated that its commonly observed preferences cannot be accommodated by the classic expected utility theory. Since Ellsberg's seminal work, a lot of effort has been devoted to the development of decision theoretic frameworks that generalize the classic expected utility theory to resolve the Ellsberg paradox. The most widely discussed nonexpected utility models include the Choquet expected utility (CEU) theory,[8] the maxmin expected utility theory,[9] the variational preference model,[10] the α-maxmin model,[11] and the smooth model of ambiguity aversion.[12] We will provide a brief review of the CEU theory in the next section. Roughly speaking, the maxmin expected utility theory employs a set of probability measures rather than a single probability function to represent uncertainty, and then chooses the option that maximizes the lower expected utility with respect to that set of probability measures. On the basis of the maxmin expected utility, the variational preference model introduces an ambiguity index α to better capture the idea of ambiguity aversion. By contrast, the smooth model makes use of the notion called second-order probabilities and a nondecreasing transformation function to accommodate ambiguity-aversion preferences.

In the spirit of the Ellsberg paradox, Machina[13] proposes two paradoxes known as the 50:51 example and the reflection example, which raise a question to the CEU theory. In addition, he argues that the plausible preference patterns in those paradoxes do not conform to the CEU theory. This result has been already confirmed by some empirical experiments,[14] which shows that a large majority of participants actually choose according to the plausible preference, rather than the one predicted by the theory. More recently, it has been shown[15] that not only the CEU theory but also the other decision models mentioned above cannot resolve the Machina paradoxes in a satisfactory manner. So the Machina paradoxes actually pose a serious challenge to most of the existing approaches to decision making under ambiguity. For this reason, the authors agree with the suggestion in Ref. [14] that new approaches to modeling ambiguity in decision making are needed.

One potential candidate is the ordered weighted averaging operator (OWAO)-based model proposed by Yager.[16, 17] However, we show in this paper that this model cannot properly deal with the Machina paradoxes when the degree of optimism is assumed to be unique. Moreover, we attempt to provide a unified account for all the paradoxes by modifying the original OWAO-based model to allow for multiple degrees of optimism. In view of these results, we think that our modified decision model can provide a systematic approach to decision making under ambiguity, and thus merits further investigation.

The remainder of this paper is organized as follows: Section 'DECISION THEORETIC FRAMEWORK FOR RESULTS' briefly reviews some decision theoretic frameworks including the expected utility theory, the CEU theory, and the OWAO-based decision model. Section 'REVIEW OF THE ELLSBERG AND MACHINA PARADOXES' presents the Ellsberg paradox and the Machina paradoxes, together with the commonly observed preference patterns in these decision paradoxes. Section 'THE UNIQUENESS PROBLEM' focuses on the OWAO-based decision model and the Machina paradoxes, and shows that this model cannot properly resolve these paradoxes when a unique degree of optimism is assumed. Section 'PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT' discusses a modified OWAO-based decision model, and demonstrates that this model can accommodate the typical preference patterns in all three paradoxes. Section 'CONCLUDING REMARKS' concludes the paper and suggests some possible future work.

2. DECISION THEORETIC FRAMEWORK FOR RESULTS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

In this section, we shall outline three important models of decision making under uncertainty with a view toward understanding the central representation results. These models differ in the way of describing uncertainty and the specification of preference structures, but they all aim at determining some index that reflects the decision maker's preferences over available options.

2.1. Expected Utility Theory and CEU Theory

Since Savage's seminal work,[5] it has become commonplace to consider the familiar, canonical decision problem in which inline image is a set of states of the world, and inline image is a set of possible outcomes, each of whose elements inline image is the outcome of act inline image in state inline image. In other words, the set A of acts is a set of functions from states to outcomes. This decision problem can be summarized by the following table:

Table I. Canonical decision matrix
 s1⋅⋅⋅inline image⋅⋅⋅inline image
a1c11⋅⋅⋅inline image⋅⋅⋅inline image
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
inline imageinline image⋅⋅⋅inline image⋅⋅⋅inline image
⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅
inline imageinline image⋅⋅⋅inline image⋅⋅⋅inline image

As we mentioned earlier, the theory of expected utility has been vastly employed in different fields including economics and artificial intelligence to account for rational choices under uncertainty. Within this framework, it assumes that a decision maker's preferences over acts should obey certain axioms such as weak ordering, independence, and Archimedean. According to the expected utility theory, a numerical representation can be obtained from the axioms, which quantifies values and beliefs on the basis of the decision maker's preferences. More precisely, there exists a real-valued utility function inline image that reflects the decision maker's desires about the outcomes, and a personal probability measure inline image that represents her belief about the states. On the basis of these functions, one can compute the expected utility of each act, and to be rational, the decision maker should choose the act that maximizes the index of expected utility. Formally, the representation theorem in expected utility theory can be stated as follows:

  • display math

The CEU theory, which generalizes the expected utility theory, has been discussed extensively in the literature of decision making under uncertainty.[8, 18] Roughly speaking, the model of CEU replaces the independence axiom on preferences in the framework of expected utility theory with a weaker requirement called comonotonic independence (see condition (ii) in  Ref. [8]), and it turns out that the decision maker's preferences can be represented through a Choquet integral with respect to a unique capacity v, instead of a unique probability function. More precisely, this theory employs a nonadditive measure v defined over states to represent the decision maker's beliefs about uncertainty. In this framework, the measure inline image is called a capacity which satisfies that inline image, inline image, and inline image if inline image. And it also replaces the standard expected utility formula with an alternative notion of CEU with respect to v. Similarly, we can establish the following representation theorem for the CEU theory:

  • display math

where u is a real-valued utility function defined on S.

In particular, when the option a is a real and finite valued function on S, it is obvious to obtain a ranking of the values that a attains, say inline image. It thus follows that there is a ranking over the utilities namely, inline image. Define inline image. Given this, the CEU, denoted as inline image, is then given by[19]:

  • display math(1)

where inline image. This formula provides a convenient way of computing the CEU without reference to the Choquet integral. We will often use this formula in the following discussion.

2.2. OWAO-Based Decision Theory

Real life decisions are often made in the presence of uncertainty with respect to the states of the world. Dempster–Shafer (D–S) theory[20] provides a powerful mathematical structure for modeling decision makers' uncertainty as it can describe both the cases of risk and ignorance using the same formulation. When using this theory in decision making, however, we need to aggregate the information concerning the collection of possible outcomes. In, Refs. [4, 16, 17, 21] Yager discussed a new aggregation framework based on the OWAOs, which has been shown useful in applying the D–S structure to decision making under uncertainty. This subsection briefly recaps the key ideas of Yager's OWAO-based approach to decision making where D–S belief structures are used for representing uncertainty.

Let us begin with a formal definition of the OWAO as introduced in.[16]

Definition 1.. An OWAO of dimension n is a function inline image with an associated weighting vector inline image such that: (i) inline image, and (ii) inline image, and for any collection of values inline image,

  • display math(2)

where inline image is the ith largest element in the collection inline image.

There are several different ways of defining measures associated with the weights of an operator, which have been suggested in the literature.[22, 16] Here we are not going to review all of these measures. Instead, we shall focus on a particular approach for identifying the appropriate weights, which is suggested by O'Hagan.[22] Notice that the above definition of the operators actually captures a range of ideas concerning a decision maker's risk attitude toward the outcomes in the face of uncertainty. In particular, it can be used to model how optimistic the decision maker is in the cases of ignorance. So it is natural to consider the idea of using a degree of optimism as the starting point for the determination of the appropriate weights. After deciding her subjective degree of optimism β (inline image), the decision maker employs a mathematical programming problem to obtain the appropriate weights, which is given as follows:

Maximize: inline image.

Subject to:

  • display math

In particular, when there are only two values involved in a collection, solving the above mathematical programming problem should give us that inline image and inline image.

As indicated above, another important feature of the OWAO-based decision theory is to use the D–S belief structure to represent the uncertainty that a decision maker faces. To better understand this theory, we provide a brief review of the basic idea of D–S theory.[20] It has been shown that the D–S theory is an important and useful framework for representing imperfect information and knowledge, which has often been applied to construct intelligent systems.[2, 4, 16, 17, 21, 23]

As a generalization of the traditional probability theory, the framework of D–S theory introduces the following basic concept called mass function that allows the possibility of not having all the basic properties of probabilities.

Definition 2.. Let S be a frame of discernment (i.e., a set of the states of the world). The function inline image is called a basic probability assignment or a mass function over S if inline image and inline image. If inline image, then A is said to be a focal element of m.

It is worth discussing some basic differences between the D–S theory and the standard probability theory. Let us consider the following example. Suppose that we draw a ball from the urn randomly, where there are 300 red (r), blue (b), or green (g) balls. We know that at least 100 of them are red. How should we model this case where we only have partial information about the proportion of the balls? According to the D–S theory, it can be modeled by a mass function: inline image, and inline image, where inline image. According to the probability theory, by contrast, we need to specify a probability function p such that inline image, inline image, and inline image, where inline image. In addition, such a probability function p needs to be justified by a great deal of evidence. Another major difference between these two theories is that the mass function given by the D–S theory does not have to be additive, while the probability function in probability theory needs to satisfy the property of additivity. As a matter of fact, a probability function can be regarded as a special case of mass function. It may be shown that if inline image and inline image with A being a singleton (i.e., inline image), then m is indeed a probability function.

In the framework of Bayesian decision theory, the notion of probability measure has been used to reflect a decision maker's information concerning the states of the world. In the cases of insufficient information, however, a decision maker's belief about the states of the nature may not be adequately represented by a single probability function. By contrast, the D–S theory provides a more reasonable way for describing uncertainty with respect to the environment. In essence, a mass function can reflect the decision maker's imprecise beliefs about available evidence and information.

Nevertheless, when the cardinality of the focal element of a mass function m is greater than 1, we are not able to calculate the expected utility of a given act using the standard expectation formula. To address this problem, Yager[16] proposed an OWAO-based approach to decision making uncertainty in terms of a generalized notion of expected utility. The basic procedure of this approach can be summarized as follows:

  1. Represent the agent's beliefs about the states by a mass function m.
  2. Determine a collection of weights to be used in OWAO aggregation. More precisely, for each different cardinality of focal elements, solve the mathematical programming problem using the degree of optimism β.
  3. For each act inline image and each focal element inline image, find the collection inline image of utilities corresponding to that focal element, that is, inline image.
  4. For each inline image, compute the value inline image using formula (2).
  5. Calculate the generalized expected utilities using the following equation:
    • display math(3)
  6. Choose the act inline image that has the maximal value as a decision.

We illustrate the above decision procedure with the following example, taken from Yager.[16]

Example 3.. Consider the decision problem described in Table II. How should a rational decision maker choose?

Table II. A decision situation
 s1s2s3s4s5
a17512136
a212105112

For step (i), assume that the decision maker obtains the following mass function based on available information about this scenario:

  • display math

Regarding step (ii), suppose that the decision maker decides inline image as her subjective degree of optimism. Then by solving the mathematical programming problem described above, one can obtain the following weights:

No. of argumentsω1ω2ω3ω4ω5
20.750.25   
30.620.270.11  
40.520.270.140.07 
50.460.260.150.080.05

Accordingly, one can calculate the collections inline image of utilities given below:

  • display math

For step (iv), the values v11 can be obtained using formula (2):

  • display math

Similarly, one can calculate the other inline image values namely, inline image, inline image, inline image, inline image, and inline image.

In step (v), by formula (3), the decision maker can obtain the generalized expected utilities of a1 and a2:

  • display math

As the final step, this decision procedure recommends that the decision maker should choose the option a1, since inline image.

3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

This section presents the Ellsberg and Machina paradoxes which have been extensively studied in the past. The former one poses a serious challenge to the expected utility theory, while the latter one raises questions about both the expected utility theory and the CEU theory.

3.1. The Ellsberg Paradox

The Ellsberg paradox[7] is a well-known, widely discussed example concerning ambiguity in decision making, which has stimulated development of many alternatives to the classic expected utility theory. We shall present Ellsberg's original decision problems below.

Suppose that there is an urn containing 300 balls, among which 100 balls are known to be red, and the remaining are either blue or green with the exact proportion unknown. More precisely, there are k blue or green balls in the urn, where k can range from 0 to 200. Due to this partial information (ambiguity), the probabilities of the event that a blue ball is drawn and that a green ball is drawn are both indeterminate.

Suppose that an individual is invited to make a decision in each of the choice problems depicted by Table III and Table IV respectively. More specifically, the participant needs to choose between two options in each of the cases. The details of the payoffs associated with the options are given in the tables below. The participant is then asked to answer the following two questions: (i) given the choices a1 and a2 in Table III, which one do you prefer? (ii) given the choices a3 and a4 in Table IV, which one do you prefer?

Table III. First decision problem
 r: redb: blueg: green
a1$100$0$0
a2$0$100$0
Table IV. Second decision problem
 r: redb: blueg: green
a3$100$0$100
a4$0$100$100

As discussed in Ref. [7], Ellsberg invited students of the Harvard Business School to carefully consider how they would choose in each of these decision problems. A commonly observed pattern of preference with regard to these problems is given as follows: inline image and inline image. One can easily verify that such a preference pattern does violate the expected utility theory, which dictates that inline image if and only if inline image.

3.2. The Machina Paradoxes

In the same manner as Ellsberg, Machina[13] presents examples of ambiguous choice problems where the typical ambiguity-averse preferences cannot be explained by both the expected utility theory and the CEU theory. After the description of the example, we shall show in detail how the Machina-styled paradoxical choices violate the CEU theory.

Like in Ellsberg's thought experiment, you as a decision maker are asked to consider an urn with 101 balls, each of which is labeled with a number 1, 2, 3, or 4. You do not know exactly how many balls are marked with one of those numbers, but you are informed that exactly 50 are labeled with either 1 or 2, and the remaining balls are labeled with either 3 or 4. In other words, you do not know the exact probability of the event “a ball marked with number k is drawn” for any inline image; whereas you do know that the probability of the event “a ball marked with either 1 or 2” is exactly inline image, and the probability of the event “a ball marked with either 3 or 4” is exactly inline image.

Now consider the choice between a pair of options as described in Table V, where inline image means that the ball with number k is drawn (inline image). Which one would you prefer?

Table V. First decision scenario
 s1s2s3s4
a1$8000$8000$4000$4000
a2$8000$4000$8000$4000

Then suppose that you are offered another pair of options as depicted in Table VI. Which one would you prefer?

Table VI. Second decision scenario
 s1s2s3s4
a3$12000$8000$4000$0
a4$12000$4000$8000$0

In Ref. [13], Machina has convincingly argued that the plausible preferences over these two pairs of options should be like this: inline image and inline image, which exhibit a similar preference pattern as observed in Ellsberg's example. However, both the expected utility theory and the CEU theory preclude such a preference pattern with regard to Machina's choice problem. It is relatively easy to see how this violates the expected utility theory, and so we omit the detailed explanation. Below we demonstrate the paradoxical aspect of Machina's example using the CEU theory.

Let us start with the calculation of the Choquet expected utilities for the options in the first scenario. Let inline image and inline image be a capacity. By formula (1), we obtain the following equations:

  • display math

With regard to the second scenario, we have the following equations:

  • display math

On one hand, if you strictly prefer a1 to a2, that is, inline image, the representation theorem requires that inline image, which means that:

  • display math

This implies the following result:

  • display math(4)

On the other hand, if you strictly prefer a4 to a3, that is, inline image, the theorem requires that inline image, that is,

  • display math

which implies the following result:

  • display math(5)

The inequalities in (4) and (5) together show that according to the CEU theory, your preferences in these decision situations should satisfy that inline image if and only if inline image. However, Machina has argued that the reasonable choice pattern for these problems typically does not obey such a requirement. More precisely, Machina demonstrates that this example indeed violates the comonotonic independence axiom, which is a weaker assumption in comparison to the independence axiom in the expected utility theory. Since the Machina-styled preference relations, namely inline image and inline image contradict this comonotonic independence principle, it immediately follows that these preferences cannot be accommodated within the framework of the expected utility theory, since the independence axiom is one of its fundamental assumptions. Such a phenomenon is also commonly referred to as the 50:51 example in the literature.

Besides the above example, Machina also presents another choice problem known as the reflection example, which poses difficulties to the CEU theory as well. Unlike the previous setting, assume that you have an urn with 100 balls marked with number 1, 2, 3, or 4. You are informed that half of the balls are marked with 1 or 2, and the other half of them are marked with 3 or 4. Similarly, you do not have any further information concerning the exact proportion of the balls. Now consider the decision problem described in Table VII. Which option would you prefer?

Table VII. Reflection example
 s1s2s3s4
a5$4000$8000$4000$0
a6$4000$4000$8000$0
a7$0$8000$4000$4000
a8$0$4000$8000$4000

Intuitively, according to the available information given above, it seems reasonable to say that the events inline image and inline image are equally likely, and there is not enough information for determining which event with a single state occurring is more likely. In this case, we call the states s1 and s2 (respectively, s3 and s4) informationally symmetric. Observe that the option a5 can be obtained from a8 by exchanging the payoffs in those informationally symmetric states. The same is true for the options a6 and a7. In other words, the payoffs of one option are the “mirrored” images of those associated with the other option. For this reason, we say that a5 (respectively, a6) an (informationally symmetric) reflection of a8 (respectively, a7). In view of this, it seems plausible to expect that a decision maker with the preference inline image should have the “reflected” preference ranking, namely inline image, since being indifferent between reflection options seems quite sensible. However, we show below that such a requirement turns out to be inconsistent with the CEU.

In addition, a number of experimental studies (see, for instance, Ref. [14]) on the reflection example shows that a typical pattern of preference is given by inline image and inline image. It is obvious that this conforms with the reflected preference relations required by the intuitive argument presented above. Furthermore, the CEU theory also precludes this common pattern of preference, since it requires that inline image if and only if inline image. To see this, let us carry out simple calculations using the CEU theory.

Suppose that you strictly prefer a5 to a6. According to the representation theorem, we have that

  • display math

which implies the following inequality:

  • display math(6)

Similarly, if you also have a strict preference of inline image, it follows that the exact same inequality as Equation (6) should hold as well. Thus, the CEU theory requires that inline image if and only if inline image.

As a matter of fact, the Machina paradoxes are quite powerful in the sense that several existing decision theories prohibit the commonly observed patterns of preference. It has been shown in Ref. [15] that the Machina paradoxes raises serious challenges not only to the CEU theory, but also to a number of widely discussed models of decision making under uncertainty.

4. THE UNIQUENESS PROBLEM

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

As pointed out by Yager,[16] the OWAO-based approach to decision making embraces some natural properties of rational decision making, such as symmetry and Pareto optimality. One may hope that such a decision model can provide reasonable theoretical explanations for the Machina paradoxes introduced in the previous section. As we shall show in this section, however, it turns out that the OWAO-based decision model described in Section 'DECISION THEORETIC FRAMEWORK FOR RESULTS' cannot account for the Machina paradoxes, although it is easy to verify that it can resolve the Ellsberg paradox in a reasonable way. In particular, we shall argue that the assumption about the uniqueness of degree of optimism leads to such a problem.

Before entering into the examination, let us first recall that the decision maker is required to have a subjective degree of optimism to determine the weights of an OWAO. An essential assumption made by the OWAO-based decision procedure is that such a degree of optimism β should be unique. In light of this, let us turn to the detailed analysis of the Machina paradoxes using the OWAO-based decision model. Due to its importance, we state the result explicitly as follows.

Result 4.. Under the assumption that the subjective degree of optimism is unique, the OWAO-based decision model cannot accommodate the common pattern of preference in the 50:51 example.

Proof. As discussed before, in the 50:51 example the decision maker only has partial information concerning the proportion of the marked balls. A mass function, instead of a probability measure, is more appropriate for representing uncertainty in this case. Given the available information, the mass function defined over the frame inline image should satisfy the following conditions:

  • display math(7)

According to Definition 2., the sets inline image and inline image are called the focal elements of the mass function m. In the sequel, we write these two sets as B1 and B2, respectively.

Now suppose that the decision maker chooses β as his degree of optimism for both decision problems in the example. As described before, we need to solve the mathematical-programming problems to obtain the proper weights. Since both focal elements of m are of cardinality 2, we have that the weights ω1 and ω2 are given by inline image and inline image.

In this example, the collections inline image of utilities corresponding to options inline image (inline image) and inline image (inline image) are described as follows:

  • display math

Now we can use formula (2) to compute the values inline image given below:

  • display math

Then by formula (3), one can obtain the generalized expected utilities of all the options:

  • display math

Accordingly, the following result follows:

  • display math

which implies that inline image iff inline image. Therefore, the OWAO-based decision model prohibits the common pattern of preference observed in the experiments, namely, inline image and inline image.

It is worth pointing out that the above result shows that no matter what value the degree of optimism β takes, the OWAO-based approach to decision making cannot account for the common pattern of preference in the 50:51 example. In this sense, this result establishes that the presence of such a paradoxical choice is highly robust against all possible values of β in the OWAO-based decision model, as long as that value is unique.

Now let us proceed to the analysis of the reflection example by applying the OWAO-based approach to decision making. Not surprisingly, the OWAO-based decision model precludes the commonly observed preference in the reflection example as well.

Result 5.. Under the assumption that the subjective degree of optimism is unique, the OWAO-based decision model cannot accommodate the common pattern of preference in the reflection example.

Proof. In the reflection example, it seems reasonable to say that the available information gives rises to the following mass function defined over inline image:

  • display math(8)

Similarly, we denote its focal elements inline image and inline image as B1 and B2.

Assume again that the decision maker chooses β as his degree of optimism for this decision problem given in the reflection example. For the same reason as before, we have the weights ω1 and ω2 that are given by inline image and inline image. For each option inline image (inline image), we can then find the collection inline image of utilities corresponding to the focal element inline image (inline image), which are given as follows:

  • display math

Then the values inline image can be obtained using formula (2):

  • display math

By formula (3), we can thus calculate the generalized expected utilities of those options:

  • display math

Simple computation gives us the following equations:

  • display math

This means that inline image, which is inconsistent with the typical preference relations in the reflection example. Therefore, we can conclude that the OWAO-based decision model cannot account for the reflection example.

The same remark concerning robustness made after the proof of Result 1 holds true for this case as well, since the above proof does not depend upon the choice of the β-value.

As indicated before, we think that the major reason why these negative results arise is that the OWAO-based decision theory assumes the uniqueness of the degree of optimism β. First, to the best of our knowledge, there is no discussion in the literature about how to determine the value of β in general, and about the justification for its uniqueness. Recall that the subjective degree of optimism β is needed for the determination of the weights that are eventually used to calculate the generalized expected utilities of all the options. So when β is assumed to be unique, the set of weights for each collection under value of β is unique as well. Intuitively, it seems implausible that a decision maker would in general hold the same degree of optimism and thus the same set of weights with respect to different decision problems. Under different situations, a decision maker may change her attitude toward uncertainty based on all available information including the structures of the decision problems. In view of this, it seems more reasonable to allow that the degree of optimism is not unique in the OWAO-based decision model.

Furthermore, it is worth mentioning that the Machina paradoxes challenge not only the OWAO-based approach to decision making, but also some other approaches similar to it. One of such instances is the so-called Transferable Belief Model (TBM)[24] in artificial intelligence, since it can be recognized as a decision theory constructed based on the D–S theory as well. The basic idea of this approach is to transfer a mass function to a probability function using Laplace's insufficient-reason principle. Its transformation rule can be formally described as follows:

  • display math

The intended interpretation for the pignistic probability inline image is that when the decision maker is uncertain about the outcomes of some consequence set, she will consider all these outcomes as equally probable. This means that the weights for computing the expected utilities are always the same. In view of this, it is fair to say that the TBM also makes the uniqueness assumption similar to the OWAO-based decision model. The difference between the assumptions of these models is that the former one assumes the uniqueness of the weights directly, whereas the latter one requires the degree of optimism to be unique, which results in the uniqueness of the weights. Once the weights are determined, the generalized expected utility of an option can then be calculated in the same way as in the OWAO-based decision model. It is easy to verify that the TBM also prohibits the common pattern of preferences in the Machina paradoxes, since the set of weights given in the TBM is just a special case of that given in the OWAO-based decision model where the weights are evenly distributed. This gives us another reason to believe that the uniqueness assumption is the main reason why the OWAO-based decision model fails to explain the Machina paradoxes.

In the light of the foregoing discussions, we think that the assumption concerning the uniqueness of the degree of optimism should be relaxed when applying the OWAO-based framework to decision making under uncertainty. In the next section, we shall discuss a slightly modified version of the OWAO-based decision model without requiring the degree of optimism to be unique. It may be hoped that such a theory can accommodate the kinds of preference found in the Machina paradoxes, and thus provides a unified account of all these paradoxes.

5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

In the previous section, it became clear that the OWAO-based approach to decision making cannot account for the Machina paradoxes when the degree of optimism is assumed to be unique. To resolve these paradoxes, it seems natural to consider the idea of relaxing the uniqueness assumption by allowing various β-values as the degrees of optimism. In this section, we present a slightly modified model of the OWAO-based approach, where a decision maker is permitted to have multiple subjective degrees of optimism. In addition, we show that such a decision model provides a unified solution to the Ellsberg paradox and the Machina paradoxes.

As briefly mentioned above, there are several possible reasons why we should allow the degree of optimism β to be nonunique. One possible reason is that people may have different attitudes toward uncertainty, given different decision situations. The intended interpretation of the degree of optimism is that it reflects how optimistic a decision maker is when evaluating the outcomes of decisions. It is thus reasonable to admit the possibility that a decision maker may use different degrees of optimism when applying the OWAO-based decision procedure. As we shall see, this is exactly the motivation for considering different β-values in the Ellsberg paradox and the 50:51 example.

Another sensible reason is that some structural information about the decision problems may provide justification for employing different degrees of optimism for the determination of the weights. In certain cases, there may be some recognizable pattern in the structure of the decision problems, which is important for the evaluation of the outcomes. However, such structural information cannot be appropriately modeled using existing concepts in the OWAO-based framework. In our view, when there is structural information available to the decision maker, such as different payoff patterns, the model should reflect such information using different degrees of optimism. In fact, the reflection example serves as a nice example of such a consideration concerning relevant structural information. For the above reasons, we propose to modify the OWAO-based decision model so that it can accommodate cases where multiple degrees of optimism are permitted.

On the basis of this idea, we need to revise some steps of the decision procedure described in Section 'OWAO-Based Decision Theory'. In particular, step (ii) in the original model should be modified as follows:

(ii') For each inline image, determine a collection of weights to be used in OWAO aggregation. More precisely, for each different cardinality of focal elements, solve the mathematical programming problem using that degree of optimism inline image.

In view of this, the remaining steps of that procedure should be carried out with respect to the corresponding degree inline image. Since there may exist multiple inline image values, these typically lead to different sets of weights for a collection that are associated to the OWAOs. A natural question thus arises as to which set of weights should be used for calculating the values inline image corresponding to the option inline image. In general, this depends upon the decision situation in question, since the number of possible β values is also determined by the decision problem itself. In the sequel, we shall illustrate the basic idea of the modified OWAO-based decision model by reexamining the Ellsberg paradox and the Machina paradoxes.

First let us analyze the Ellsberg paradox by applying our modified OWAO-based decision model. It is not surprising that this model can account for the Ellsberg paradox in a sensible way, since the original decision model can accommodate the Ellsberg-styled preference. Nevertheless, it is easier to illustrate how our model works using this problem, and so let us go through the details of the analysis.

Notice first that in the Ellsberg paradox, the decision maker is presented with two decision problems. As we argued above, it seems reasonable to say that the decision maker may evaluate those outcomes under different decision situations based on different degrees of optimism. Thus, we should employ two different β values for the determination of the appropriate weights. As before, we state the result as follows:

Result 6.. The modified OWAO-based decision model with different degrees of optimism can accommodate the Ellsberg-styled preference in the Ellsberg paradox.

Proof. On the basis of the partial information given in the Ellsberg paradox, we have a mass function defined over the frame inline image.

  • display math

We denote the focal elements inline image and inline image as B1 and B2, respectively.

Now assume that the decision maker assigns a degree of optimism β for the first decision situation, and inline image for the second decision situation. Then by solving the mathematical programming problems, we have that the weights ω11 and ω12 corresponding to option a1 are the same as those of option a2, that is, inline image, and inline image; while the weights corresponding to options a3 and a4 are the same, given by inline image, and inline image.

Given the decision problems, we can find the collections inline image of utilities described as follows:

  • display math

We can then calculate the values inline image by formula (2).

  • display math

Further, according to formula (3), we have:

  • display math

Accordingly, we have the following results:

  • display math

which means that our modified model can resolve the Ellsberg paradox if and only if inline image and inline image. In other words, according to our decision model, the decision maker who exhibits the Ellsberg-styled preference pattern may actually hold different degrees of optimism in these two decision problems where those degrees should be less than 0.5. Therefore, it seems reasonable to claim that our modified OWAO-based model can account for the Ellsberg paradox.

It can be seen from the above proof that there is no requirement on the relationship between these two β-values to resolve the Ellsberg paradox. The only condition that both of them are less than 0.5 is sufficient. This is exactly the reason why the original OWAO-based approach with a single degree of optimism can accommodate the Ellsberg-styled preference.

Now let us turn to the 50:51 example. Similarly, there are two decision problems involved in this example, and thus it is natural to analyze the problems in terms of different degrees of optimism. As we shall see, however, this example illustrates a different feature in comparison with the Ellsberg paradox.

Result 7.. The modified OWAO-based decision model with different degrees of optimism can accommodate the commonly observed preference pattern in the 50:51 example.

Proof. According to the available information given in the 50:51 example, the appropriate mass function defined over inline image should satisfy the following conditions:

  • display math

We write the focal elements inline image and inline image as B1 and B2, respectively.

Now suppose that the decision maker decides to use a degree optimism β in the first decision scenario, and a degree of optimism inline image in the second decision scenario. It thus follows by solving the mathematical programming problems that the weights are given as follows: inline image, inline image, inline image, and inline image. It is worth noting that the generalized expected utilities of a1 and a2 should be calculated using the weights obtained from β, while those of a3 and a4 should be computed from the weights determined by inline image.

Since the mass function is exactly the one specified in formula (7), the collection inline image of utilities can be described in the same way as that specified in the proof of Result 1. On the basis of this, we can then calculate the values inline image using formula (2):

  • display math

According to formula (3), the generalized expected utilities of these four options are given as follows:

  • display math

which implies that:

  • display math(9)
  • display math(10)

On one hand, simple computation using formula (9) gives us the following result:

  • display math

This means that a decision maker who prefers a1 to a2 should assign a degree of optimism inline image when evaluating these options in the first decision scenario.

One the other hand, it follows from formula (10) that:

  • display math

The above result says that a decision maker who has the preference of inline image should assign a degree of optimism inline image in the evaluation of these options of the second decision problem.

It is obvious that the above two conditions on β and inline image are not contradicted with each other. Therefore, we can conclude that our modified OWAO-based decision model can account for the paradoxical choice in the 50:51 example.

In light of the above proof, it becomes more clear why the original decision model with a unique degree of optimism cannot resolve the 50:51 example, since there is no single β-value satisfying both conditions specified above. In this sense, the 50:51 example does exhibit some special features that distinguishes it from the Ellsberg paradox.

Finally, let us apply our modified OWAO-based decision model to the reflection example. Unlike the previous two paradoxes, the options available to the decision maker are presented within the same decision problem. In view of this, it seems implausible to say that the decision maker should employ different degrees of optimism to evaluate these options. Nevertheless, there is some structural information in this decision problem that justifies the consideration of multiple degrees of optimism. As argued before, a rational decision maker should recognize that the payoff structures of option a5 and a8 are similar in the sense that their payoffs are mirrored to each other relative to the informationally symmetric states; and the same holds true for the options a6 and a7. For this reason, the decision maker should incorporate this information into her evaluation of these options by associating different pairs of options with different degrees of optimism. More precisely, when calculating the generalized expected utilities, the degree of optimism used for the options a5 and a8 should be different from that used for a6 and a7. On the basis of this idea, we can establish that our approach provides a sensible solution to the reflection example.

Result 8.. The modified OWAO-based decision model with different degrees of optimism can accommodate the commonly observed preference pattern in the reflection example.

Proof. Similar to the previous analysis in Result 2, we employ the same mass function defined over inline image to represent uncertainty in the reflection example (see formula (8)):

  • display math

As usual, we denote its focal elements inline image and inline image as B1 and B2.

As discussed above, we assume that when evaluating those options, the decision maker chooses two different degrees of optimism, say β and inline image, to reflect the informationally symmetric structure of these options. Given this, we can solve the mathematical programming problems for β and inline image respectively, which gives us two sets of weights, namely, inline image and inline image, while inline image and inline image.

Since the mass function is the same as in the previous analysis, we can describe the collections inline image of utilities corresponding to the option inline image (inline image) and the focal element inline image (inline image) as those specified in the proof of Result 2. Then we can calculate the values inline image using formula (2):

  • display math

According to formula (3), the generalized expected utilities of these four options can be obtained as follows:

  • display math

Thus, we obtain the following result:

  • display math

The above result basically means that a decision maker who has the following preference pattern: inline image and inline image should use different degrees of optimism satisfying inline image in the evaluation of these options. It is easy to find a pair of β and inline image that obeys such a condition. This establishes that our modified OWAO-based decision model can accommodate the typical preference pattern of the reflection example.

As we noted earlier, the reflection example poses a challenge for various theories of decision making under uncertainty, since the plausible preference pattern observed in this problem cannot be typically accommodated by these models. In our view, the missing structural information is the major reason that causes the troubling issue for these theories. Here we have successfully incorporated this information into our framework by using different degrees of optimism, and we have shown that our approach can actually account for the Machina-styled preference pattern in the reflection example. This in turn corroborates our previous diagnosis about the problem.

6. CONCLUDING REMARKS

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

The Ellsberg paradox and the Machina paradoxes have become standard thought experiments for testing whether a decision model is adequate for representing ambiguity. Over the years, a lot of work has been done to extend the framework of the classic expected utility to accommodate these paradoxes. However, the Machina paradoxes pose serious difficulties for most of the existing decision models including the OWAO-based decision model. In this paper, we have presented a modified OWAO-based approach to decision making under uncertainty where a decision maker is permitted to employ multiple degrees of optimism to evaluate available options. In Section 'PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT', we outline possible reasons for the idea that we should allow a decision maker to use multiple degrees of optimism. Moreover, we have shown that the modified model can resolve the Ellsberg paradox and the Machina paradoxes. In view of this, it is fair to say that we have provided a unified treatment for all these paradoxes.

In the decision theory, there has been independent interest in the elicitation of personal beliefs about uncertainty. Its main goal is to find appropriate operationalizable methods so that they guarantee to elicit a person's “honest” opinion about some uncertain event of interest. In this paper, we have not addressed a similar issue concerning subjective degrees of optimism yet. Therefore, one interesting project for future work is to investigate the question of how to determine a decision maker's degree of optimism using some operationalizable methods.

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References

The authors would like to thank Professor Ronald R. Yager and the anonymous reviewer for their helpful comments. Research reported in this paper is supported by National Fund of Philosophy and Social Science (No. 11BZX060), Humanity and Social Science Youth Foundation of Ministry of Education of China (No. 11YJC72040001), National Natural Science Foundation of China (No. 61173019), and major projects of the Ministry of Education, People's Republic of China (No. 10JZD0006).

References

  1. Top of page
  2. Abstract
  3. 1. INTRODUCTION
  4. 2. DECISION THEORETIC FRAMEWORK FOR RESULTS
  5. 3. REVIEW OF THE ELLSBERG AND MACHINA PARADOXES
  6. 4. THE UNIQUENESS PROBLEM
  7. 5. PARADOXES REVISITED: TOWARD A UNIFIED ACCOUNT
  8. 6. CONCLUDING REMARKS
  9. Acknowledgments
  10. References
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