Generalized Expected Utility as a Universal Decision Rule -- A Step Forward
Hélène Fargier, Pierre Pomeret-Coquot
Published: 26 Apr 2024, Last Modified: 26 Apr 2024UAI 2024 oralConference, Publication Chairs, AuthorsRevisionsBibTeXCC BY 4.0
Keywords: Decision Making under Uncertainty, Generalized Expected Utility, non-additive capacities, algebraic mass function, Choquet integral, Sugeno integral

TL;DR: This paper proposes a generalization of the GEU algebraic framework that allows for the representation of non uniform decision rules and in particular of the Choquet and the Sugeno integrals.

Abstract:
In order to capture a larger range of decision rules, this paper extends the seminal work of [Friedman and Halpern, 1995, Chu and Halpern, 2003, 2004] about Generalized Expected Utility. We introduce the notion of algebraic mass function (and of algebraic Moebius transform) and provide a new algebraic expression for expected utility based on such functions. This utility, that we call "XEU", generalizes Chu and Halpern’s GEU to non-decomposable measures and allows for the representation of several rules that could not be captured up to this point, and noticeably, of the Choquet integral. A representation theorem is provided that shows that only a very weak condition is needed for a rule in order to be representable as a XEU.

Submission Number: 464

Paper Decision
DecisionProgram Chairs24 Apr 2024, 15:42 (modified: 26 Apr 2024, 14:37)Program Chairs, AuthorsRevisions
Decision: Accept (oral)
Meta Review of Submission464 by Area Chair ehXU
Meta ReviewArea Chair ehXU16 Apr 2024, 19:00 (modified: 26 Apr 2024, 16:30)Area Chairs, Authors, Program ChairsRevisions
Metareview:

The paper presents an algebraic representation of decision functions, that has a significant expressive power as it encompasses many known decision rules from the literature.

The work is theoretical yet quite interesting, with some potential to allow for the development of new decision rules. All reviewers did go for an accept, except for a reviewer that provided a review without much specific arguments (I therefore decided to disregard the score, given the review quality).





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Official Review of Submission464 by Reviewer cf8v
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Official ReviewReviewer cf8v22 Mar 2024, 19:28 (modified: 08 Apr 2024, 09:25)Program Chairs, Area Chairs, Reviewers Submitted, Reviewer cf8v, AuthorsRevisions
Q1 Summary And Contributions:

The paper introduces the notion of algebraic mass function (and of algebraic Moebius transform) and provides a new algebraic expression for expected utility based on such functions. It shows that this is a generalisation of many other decision rules models.

I acknowledge to have read the authors rebuttal.
Q2-1 Originality-Novelty: 3: Good: The paper makes non-trivial advances over the current state-of-the-art.
Q2-2 Correctness-Technical Quality: 3: Good: The paper appears to be technically sound, but I have not carefully checked the details.
Q2-3 Extent To Which Claims Are Supported By Evidence: 3: Good: the main claims are supported by convincing evidence (in the form of adequate experimental evaluation, proofs, (pseudo-)code, references, assumptions).
Q2-4 Reproducibility: 3: Good: key resources (e.g. proofs, code, data) are available and key details (e.g. proofs, experimental setup) are sufficiently well-described for competent researchers to confidently reproduce the main results.
Q2-5 Clarity Of Writing: 3: Good: The paper is well organized but the presentation could be improved.
Q3 Main Strengths: a nice generalisation of many other decision rules models
Q4 Main Weakness: none

Q5 Detailed Comments To The Authors: The paper provides a convincing generalisation of many other decision rules models. However, there are some issues to be addressed here and there
* I believe that \Omega should be finite since the very beginning and overal. In general, be carefull about the finiteness/non-finitenesses of sets. So, also MS(U) should be set of finite multisets, otherwise e.g., in prop.4 mean(X), for X non finite is ill defined.
* p.7. in E^Pess, E^Pess', what are the first three paprameters exactly?

Q6 Overall Score: 7: Accept: Technically solid paper, with convincing evidence for claims, innovative aspects, good resources and reproducibility, no unaddressed ethical considerations, and the potential for impact in the field.
Q7 Justification For Your Score: A solid and interesting paper, as far as I'm able to judge.
Q8 Confidence In Your Score: 3: Somewhat confident, but there's a chance I missed some aspects. I did not carefully check some of the details, e.g. novelty, proof of a theorem, experimental design, or statistical validity of conclusions. I am somewhat familiar with the topic but may not know all related work.
Q9 Complying With Reviewing Instructions: Yes
Q10 Ethical Concerns: none
Rebuttal by Authors
RebuttalAuthors (Hélène Fargier, Pierre Pomeret-Coquot)03 Apr 2024, 11:31Program Chairs, Area Chairs, Reviewers Submitted, Authors

Rebuttal:
We thanks a lot the reviewer for the review. As to the two questions:

Yes indeed, Omega and X are supposed to be finite - this is introduced in definition 6, and assumed throughout the paper. We will emphasize this point, including in the introduction. Thanks a lot.

The question of a similar model for the infinite case would, at least from a theoretical standpoint, be one of the directions for future research to explore.

The first three parameters in the expressions of E^{pess} et E^{pess} ' are the domains of valuations used to evaluate degrees of belief, degrees of utility of consequences, and those of acts - thus, within the framework of qualitative utilities, the scale \Lambda (that is, in most articles, the scale [0,1]).





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Official Review of Submission464 by Reviewer sreq
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Official ReviewReviewer sreq22 Mar 2024, 03:23 (modified: 01 Apr 2024, 13:55)Program Chairs, Area Chairs, Reviewers Submitted, Reviewer sreq, AuthorsRevisions

Q1 Summary And Contributions: In order to capture a larger range of decision rules, the authors propose a generalization of Chu and Halpern's GEU algebraic framework, which allows the representation of non-consistent decision rules, in particular Choquet and Sugeno integrals.
Q2-1 Originality-Novelty: 3: Good: The paper makes non-trivial advances over the current state-of-the-art.
Q2-2 Correctness-Technical Quality: 3: Good: The paper appears to be technically sound, but I have not carefully checked the details.
Q2-3 Extent To Which Claims Are Supported By Evidence: 2: Fair: the main claims are somewhat supported by evidence (but the experimental evaluation may be weak, or does not match entirely with the claims, important baselines may be missing, proofs contain important ideas but lack rigor, algorithmic details are only discussed superficially, references are imprecise, assumptions are not sufficiently motivated or explicated, etc.).
Q2-4 Reproducibility: 3: Good: key resources (e.g. proofs, code, data) are available and key details (e.g. proofs, experimental setup) are sufficiently well-described for competent researchers to confidently reproduce the main results.
Q2-5 Clarity Of Writing: 3: Good: The paper is well organized but the presentation could be improved.
Q3 Main Strengths: 1)The paper provides related theoretical proof. 2)The topic of this paper belongs to the forefront of this field. 3)The explanation of relevant work is basically fine.
Q4 Main Weakness: 1)The paper does not give experiments or examples. 2)The paper lacks the prospect of future work.

Q5 Detailed Comments To The Authors:
1)The paper does not give experiments or examples. 2)The paper lacks the prospect of future work. 3)How does the writer argue that his work is better.

Q6 Overall Score: 6: Weak Accept: Technically solid paper, with no major concerns with respect to provided evidence, resources, reproducibility, and ethical considerations.
Q7 Justification For Your Score: The innovation of the paper is strong, and the theorem proof is also sufficient. But the paper does not give experiments or examples.
Q8 Confidence In Your Score: 4: Quite confident. I tried to check the important points carefully. It is unlikely, though conceivable, that I missed some aspects that could otherwise have impacted my evaluation. I am familiar with the research topic and most of the related work.
Q9 Complying With Reviewing Instructions: Yes
Q10 Ethical Concerns: No

Rebuttal by Authors
RebuttalAuthors (Hélène Fargier, Pierre Pomeret-Coquot)03 Apr 2024, 11:46 (modified: 07 Apr 2024, 11:53)Program Chairs, Area Chairs, Reviewers Submitted, AuthorsRevisions
Rebuttal:

Thank you very much for the review.

1)The paper does not give experiments or examples.

A: Adding experiments is challenging due to space limitations (and also because the paper, in its current version, is essentially theoretical), but it would be a future development worth considering.

However, we should be able to add to the current paper a running example to illustrate the rules and their formalization (for example, based on the Ellsberg paradox).

2)The paper lacks the prospect of future work.

A: Regarding the perspectives, they would, for the practical part, be primarily algorithmic (defining a generic dynamic programming algorithm). From a theoretical standpoint, the objective is to define a unified axiomatization of the different existing rules, based on the XEU framework: which property guides the XEU towards one rule or another.

The second theoretical development includes the extension to the infinite case (and studying the impact of the assumption of finiteness).

We will try to put more emphasis on these points in the conclusion.

3)How does the writer argue that his work is better.

A: The major argument of our work, compared to Chu and Halpern's previous results, is to be able to capture decision rules based on Choquet and Sugeno integrals, and more generally rules adapted to non-additive measures. It should also be noted that the XEU components simplify the modeling and definition of new decision rules by separating, in particular, the treatment of knowledge (e.g. variability) by the oplus operator and the treatment of ignorance by the f function. We will try to put more emphasis on these points in the concerned paragraphs.



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Official Review of Submission464 by Reviewer qrGA
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Official ReviewReviewer qrGA21 Mar 2024, 15:57 (modified: 01 Apr 2024, 13:55)Program Chairs, Area Chairs, Reviewers Submitted, Reviewer qrGA, AuthorsRevisions
Q1 Summary And Contributions: This paper extends the generalized expected utility rule developed for decomposable capacities to a more general framework where the capacities are not necessarily decomposable. This general framework includes the GEU, and particular cases such as the Choquet expected utility rule or the Sugeno rule, among others.
Q2-1 Originality-Novelty: 3: Good: The paper makes non-trivial advances over the current state-of-the-art.
Q2-2 Correctness-Technical Quality: 3: Good: The paper appears to be technically sound, but I have not carefully checked the details.
Q2-3 Extent To Which Claims Are Supported By Evidence: 3: Good: the main claims are supported by convincing evidence (in the form of adequate experimental evaluation, proofs, (pseudo-)code, references, assumptions).
Q2-4 Reproducibility: 3: Good: key resources (e.g. proofs, code, data) are available and key details (e.g. proofs, experimental setup) are sufficiently well-described for competent researchers to confidently reproduce the main results.
Q2-5 Clarity Of Writing: 2: Fair: The paper is somewhat clear, but some important details are missing or unclear.
Q3 Main Strengths: The contribution deals with an interesting topic, generalizes existing results related to the expected utility representation with non-additive measures and the results seem correct and sound.
Q4 Main Weakness: If I had to nitpick, I miss a deeper discussion about the role of the function f in the XEU. It is true that the authors discuss its relevance at the beginning of page 6-column 1, but I would suggest to expand this discussion because it would allow the reader to better understand its meaning. As well, I would appreciate if the authors add a (non-trivial) example in Section 3 showing the generality of the presented framework. Finally, I would suggest to thoroughly reviewing the writing.

Q5 Detailed Comments To The Authors:s
Main comments: the contribution deals with an interesting topic, generalizes existing results related to the expected utility representation with non-additive measures and the results seem correct and sound. If I had to nitpick, I miss a deeper discussion about the role of the function f in the XEU. It is true that the authors discuss its relevance at the beginning of page 6-column 1, but I would suggest to expand this discussion because it would allow the reader to better understand its meaning. As well, I would appreciate if the authors add a (non-trivial) example in Section 3 showing the generality of the presented framework. Finally, I would suggest to thoroughly reviewing the writing. Evaluation: Anyway, apart from the comments above, the manuscript perfectly fits with the scope of the conference and I the content is interesting, hence I recommend to accept it. Detailed comments (P=page, C=columna, L=line): -Moebius -> M\” obius -monotonic -> monotone -P1,C2,L10: “…work of Halpern and al.”: do you mean Halpern at al? Please, add reference. -Def 2: monotony -> monotonicity. Also: shouldn’t you detailed monotone increasing? -Example 1: you may specify that you are using a probability measure, a belief function and a possibility measure. -Def 10, last equation: shouldn’t \succeq be \succeq^v? -Def 12 and 13: I suggest to write \succeq instead of \preceq to be consistent with the previous notation. -Thm.1: please, add reference. -P4, C1, L24: “monotonic” -> “monotonicity”. -P4, C1, L-8: “regards” -> “regard” -P4, C2, L9: “such as” -> “such that”. Also, add a final dot at the end of the itemize. -P4, C2, sentence before def.18: it is unclear; please, rewrite. -P5, C1, L4-5: unclear sentence; please, rewrite. -Def.21: “such as” -> “such that”. -P6, C1, paragraphs 1, 2 and 3: I would expand and detail this part to emphasize the relevance of the function f. -P6, C1, L22: “show” -> “shows”.

Q6 Overall Score: 8: Strong Accept: Technically strong paper with novel ideas, with convincing evidence for claims, good resources and reproducibility, no unaddressed ethical considerations, and the potential for impact in the field.
Q7 Justification For Your Score: The contribution deals with an interesting topic, generalizes existing results related to the expected utility representation with non-additive measures and the results seem correct and sound.
Q8 Confidence In Your Score: 4: Quite confident. I tried to check the important points carefully. It is unlikely, though conceivable, that I missed some aspects that could otherwise have impacted my evaluation. I am familiar with the research topic and most of the related work.
Q9 Complying With Reviewing Instructions: Yes
Q10 Ethical Concerns: No.


Rebuttal by Authors
RebuttalAuthors (Hélène Fargier, Pierre Pomeret-Coquot)03 Apr 2024, 12:10Program Chairs, Area Chairs, Reviewers Submitted, Authors
Rebuttal:

Thank you very very much for the detailed review and comments

About the function f indeed, its highlighting is one of the most interesting aspects of the formulation we propose based on the mass function. It does not appear at all in Chu and Halpern's formulation.

For example, in the Choquet integral, one can see that the difference between the pignistic approach, the pessimistic integral, or the optimistic integral lies not in the treatment of knowledge (it's always the same knowledge, mm), but in the neutral/optimistic/pessimistic attitude: in the first case, f takes computes the mean value over each focal set, in the second, the decision-maker is cautious/robust (f=min), while in the third case, he/she would be adventurous (f=max).

From this, one could then imagine a series of other rules that would vary f. For example, a rule of minimax regret per focal element, or the use of an OWA (ordered Weighted Average). However, the practical interest of these new rules and their properties would still need to be studied.

Yes, indeed, as you and another reviewer suggest, it is necessary to add a running example - to illustrate the rules and their formalization (we would consider an example based on the Ellsberg paradox, which highlights the differences between the rules).

Thank you also very much for the detailed comments, which will allow us to improve the quality and readability of the article.

Official Comment by Reviewer qrGA
Official CommentReviewer qrGA04 Apr 2024, 16:23Program Chairs, Area Chairs, Authors, Reviewer qrGA
Comment:
I appreciate authors response. The comments regarding the function f are very interesting. This is exactly what I suggested to include in the revised version of the manuscript to highlight its role.



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Official Review of Submission464 by Reviewer tPcs
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Official ReviewReviewer tPcs19 Mar 2024, 09:35 (modified: 16 Apr 2024, 00:50)Program Chairs, Area Chairs, Reviewers Submitted, Reviewer tPcs, AuthorsRevisions
Q1 Summary And Contributions: The paper proposes a uniform algebraic framework to represent generalized expected utility (GEU) for different uncertainty measures. It extends previous GEU framework to cover a larger scope of decision rules, including those based on non-decomposable uncertainty measures such as Dempster-Shafer theory, Choquet integral, and Sugeno integral. The main results include the sufficient condition for the representability of decision rules in the proposed XEU framework, and sufficient and necessary condition for the exact representation of decision rules based on decomposable measures.
Q2-1 Originality-Novelty: 3: Good: The paper makes non-trivial advances over the current state-of-the-art.
Q2-2 Correctness-Technical Quality: 3: Good: The paper appears to be technically sound, but I have not carefully checked the details.
Q2-3 Extent To Which Claims Are Supported By Evidence: 3: Good: the main claims are supported by convincing evidence (in the form of adequate experimental evaluation, proofs, (pseudo-)code, references, assumptions).
Q2-4 Reproducibility: 4: Excellent: key resources (e.g. proofs, code, data) are available and key details (e.g. proof sketches, experimental setup) are comprehensively described for competent researchers to confidently and easily reproduce the main results.
Q2-5 Clarity Of Writing: 3: Good: The paper is well organized but the presentation could be improved.
Q3 Main Strengths: A nice presentation of a theoretical framework based on novel ideas.
Q4 Main Weakness: The significance of the contribution is less clear. It is unclear how such a framework can help decision makers to use the decision rules.

Q5 Detailed Comments To The Authors:

Because of the generality of the algebraic capacity and its corresponding mass function used in the proposed XEU framework, it can cover a lot of existing decision rules. On one hand, this is indeed a nice result and represents theoretical advance on the endeavor of integrating diverse uncertain formalisms. On the other hand, it is less clear how the work can have impact on the practice of decision-making under uncertainty. There is a short remark in the conclusion regarding advantages of the algebraic framework on the computational aspect of the decision rules. However, it need further elaboration on how easily algorithms developed for general algebraic structures can be transferred to more concrete domains. In addition, the fact that the framework can instantiate a diverse of existing decision rules does not provide much insight on the development of new decision theories. A more insightful direction is perhaps to find new instances of algebraic capacity or mass functions with practical sense and show that the framework can help design meaningful decision rules in those cases.

As there is no guarantee about the unique existence of \boxplus-mass function for a given capacity and as a result, it seems that m_{\mu}^\Boxplus is defined only for commutative group operator \boxplus (Def.19), it is not sure if the restriction is also applied to the definition of XEU since m_{\mu}^\Boxplus explicitly appears in its formulation. If it is the case, then obviously, the applicability of XEU become more restrictive as it excludes many qualitative uncertainty measures. If not, does it mean that XEU may not exist or be not unique for any decision problem and expectation domain?

Below, I list some minor technical issues (or typos) to be clarified.
* Table 1, \Pi violates monotony on {\omega_1} and {\omega1,\omega_3}
* Def.10, typo on the expression for GEU: , :\mid
* Def.12: as x_1 and x_2 are acts, the RHS should use u(x_1) and u(x_2)
* Def.17: W or W^*?
* L.2 after Def.18: identity element or neutral element?
* Def.23: It seems that B is a subset of U?
* Def.24: Is it implicitly assumed that W^*=W and U^*=U in the definition?
* L.4, Sec.4.1: do not are not
* Does the \boxplus-decomposability of a capacity imply the uniqueness of its \boxplus-based mass function? Otherwise, the theorem may not hold for all of its \boxplus-based mass functions.
* In rewriting U^{Pess}, it seems that the definition of nmax presupposes the existence of the minus operator in the domain \Lambda. When \Lambda is not [0,1], is the restriction reasonable?
* In the definition of CEU, x_0\chi seems not defined.
* The section entitled Partial Orderings, L.3: they are?
* Proposition 7: it seems that the definition does not match with the XEU framework as U=\mathbb{R}, W=[0,1], and V=\mathbb{R}^2. According to Def.20, it should be \otimes: \mathbb{R}\times[0,1]\rightarrow\mathbb{R}^2 and {\bf f}:{\cal MS}(\mathbb{R})\rightarrow\mathbb{R}, but arguments of these two operators in the definition do not have correct types. It seems that changing U into \mathbb{R}^2 can partially address the issue (only for the \otimes operator). However, because it is said that DSEU considers real-valued utilit functions, I am not sure if the change complies with the original DSEU.

Q6 Overall Score: 7: Accept: Technically solid paper, with convincing evidence for claims, innovative aspects, good resources and reproducibility, no unaddressed ethical considerations, and the potential for impact in the field.
Q7 Justification For Your Score: It's a good theoretical paper.
Q8 Confidence In Your Score: 4: Quite confident. I tried to check the important points carefully. It is unlikely, though conceivable, that I missed some aspects that could otherwise have impacted my evaluation. I am familiar with the research topic and most of the related work.
Q9 Complying With Reviewing Instructions: Yes
Q10 Ethical Concerns: No


Rebuttal by Authors
RebuttalAuthors (Hélène Fargier, Pierre Pomeret-Coquot)03 Apr 2024, 13:55 (modified: 03 Apr 2024, 14:04)Program Chairs, Area Chairs, Reviewers Submitted, AuthorsRevisions
Rebuttal:

Thank you very very much for the detailed review and comments

    Indeed, the work presented is essentially theoretical and is not aimed at an end user who would seek to make a decision in a practical case. Regarding practical perspectives the one we target is primarily algorithmic (defining a generic dynamic programming algorithm).

On the other hand, this work of general formalization can help the decision-makers "parameterize" a rule not clearly defined at the begining of the decision processs. One can imagine, in an elicitation process, an interactive algorithm that, through proposals for decision comparisons, allows the parameterization of the function f, seen as an OWA (e.g. as done in Loïc Adam, Sébastien Destercke: Incremental Elicitation of Preferences: Optimist or Pessimist? ADT 2021: 71-85).

    Yes, a first direction could be to work on the capacity and imagine new ones. Alternatively, we can also work on function 

    . The presence of function f is one of the most interesting aspects of the formulation we propose (it it not appear in Chu and Halpern's GEU).

For example, in the Choquet integral, one can see that the difference between the pignistic approach, the pessimistic integral, or the optimistic integral lies not in the treatment of knowledge (it's always the same knowledge, mm), but in the neutral/optimistic/pessimistic attitude: in the first case, f takes computes the mean value over each focal set, in the second, the decision-maker is cautious/robust (f=min), while in the third case, he/she would be adventurous (f=max).

From this, one could then imagine a series of other rules that would vary f. For example, a rule of minimax regret per focal element, or the use of an OWA (ordered Weighted Average).

However, the practical interest of these new rules and their properties would still need to be studied and axiomatized - the general algebraic framework could be a basis for such characterizations

    Concerning the more technical questions:

    Does the \boxplus-decomposability of a capacity imply the uniqueness of its \boxplus-based mass function? Otherwise, the theorem may not hold for all of its \boxplus-based mass functions. R: Yes, indeed Friedman Halpern show that "decomposable -> existence of a neutral element", and we show that "existence of a neutral element -> unicity".

    Def.24: Is it implicitly assumed that W^*=W and U^*=U in the definition? R: no

    In rewriting U^{Pess}, it seems that the definition of nmax presupposes the existence of the minus operator in the domain \Lambda. When \Lambda is not [0,1], is the restriction reasonable? R: Yes, indeed.

    Table 1, \Pi violates monotony on {\omega_1} and {\omega1,\omega_3} R: There is an error in the example, sorry (and thanks)

    Proposition 7: (...) It seems that changing U into \mathbb{R}^2 can partially address the issue (only for the \otimes operator). R: However, because it is said that DSEU considers real-valued utilit functions, I am not sure if the change complies with the original DSEU. R: Yes, we have that it mind - we thing about changing U into \mathbb{R}^2 and it would comply with DSEU

(the other points are mainly typos that we have to correct - sorry - and thanks again)
