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game theory can/cannot be seen as decision under uncertainty

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Tradeoff method
Games with incomplete information

game theory can/cannot be seen as decision under uncertainty (pp. 52-53 discuss it): they compare natural (my term; meaning: generated by nature) uncertainty with strategic uncertainty. The latter is mostly related to higher-level, say level k, thinking. The stage-hunt game is simple, requiring little strategic thinking, and the entry game requires much. They find that stage-hunt is similar to risk, but entry is different, by neuro measurements (p. 53 4^th para; p. 58 last column middle of 2^nd para) and also behaviorally based on CEs and correlation. P. 57 2^nd column 1^st para: CEs of entry games were even uncorrelated with those of risk and stage-hunt games.
As do HNO, working with SEU, the authors suggest, following some other economists, that, the moment subjective probabilities have been assigned, the case is (like) decision under risk (abstract l. 3; top of p. 53; p. 58 2^nd para l. 3; p. 59 2^nd column 1^st para last line; p. 59 2^nd column lines -4/-6), and any deviation is taken as impossible to involve subjective probabilities. As I write at HNO, in the source method this is not so. Further, in the entry game, subjects can be perfectly Bayesian with subjective probabilities but still have less preference for the safe option x as it increases because they think it increases the probability of the opponent going for safe, so, it improves the risky probability of winning. They would do the same if such probabilities were generated by some natural process rather than a rational opponent, so that it need not necessarily be a difference between natural and strategic uncertainty.
P. 59: “The anterior insula thus reflects risk preferences and guides choice selection both in individual and [in] social settings.”
P. 60 penultimate para precludes that the findings are entirely driven by social preferences. It can still be that social preferences do play a role, alongside with other effects. %}

Nagel, Rosemarie, Andrea Brovelli, Frank Heinemann, & Giorgio Coricelli (2018) “Neural Mechanisms Mediating Degrees of Strategic Uncertainty,” Social Cognitive and Affective Neuroscience 13, 52–62.

{% Considers n-tuples (x₁,…,x_n) in Reⁿ with n variable. I (not the author) interpret it as 1/n probability prospects. Under EU, certainty equivalents, denoted CE, with utility denoted as , is ^¹([(x₁) + … + (x_n)]/n), with  endogenous. This paper axiomatizes functions CE for which there exists a continuous strictly monotonic . The axioms are (reordered but kept author’s numbering):
(i) The function CE is symmetric;
(v) CE(a,…,a) = a;
(ii) Write CE(x₁,…,x_r,x_r+1,…,x_n) = a; then CE(x₁,…,x_r,x_r+1,…,x_n) = CE(a,…, a, x_r+1, …, x_n);
(iii) CE is continuous and a  CE(x₁,…,x_n)  b if each a  x_i  b for all i;
(iv) x₁ < x₂ implies x₁  CE(x₁,x₂)  x₂.
Condition (ii) is called associativity of the mean. It has a remarkable relation with vNM independence. It is a version of vNM independence: if (x₁,…,x_r) ~ a then [r/n: (x₁,…,x_r), (nr)/n: (x_r+1,…,x_n)] ~ [r/n: a, (nr)/n: (x_r+1,…,x_n)].
So: this can be taken as giving the vNM EU axiomatization for equal-probability prospects, which amounts to all rational-probability prospects, under the restriction of continuous utility!
To excite us even more, the theorem on p. 78 shows that constant absolute risk aversion is equivalent to linear-exponential utility!! (The theorem only states sufficiency but the text directly preceding states necessity.) %}

Nagumo, Mitio (1930) “Über eine Klasse der Mittelwerte,” Japanese Journal of Mathematics 7, 71–79.

{% Discuss behavioral theories (social interactions models, self-control models prospect theory in health) in policy applications by three criteria: (1) providing new insights (2) properly applied; (3) corroborated by evidence. Only PT passes the tests. %}

Nakamura, Ryota & Marc Suhrcke (2017) “A Triple Test for Behavioral Economics Models and Public Health Policy,” Theory and Decision 83, 513–533.

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Nakamura, Yutaka (1984) “Nonlinear Utility Analysis,” Ph.D. Thesis, University of California, Davis, 1984.

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Nakamura, Yutaka (1988) “Expected Utility with an Interval Ordered Structure,” Journal of Mathematical Psychology 32, 298–312.

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Nakamura, Yutaka (1990) “Subjective Expected Utility with Non-Additive Probabilities on Finite State Spaces,” Journal of Economic Theory 51, 346–366.

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Nakamura, Yutaka (1990) “Expected Utility with Nonlinear Thresholds,” Annals of Operations Research 23, 201–212.

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Nakamura, Yutaka (1990) “Bilinear Utility and Threshold Structures for Nontransitive Preferences,” Mathematical Social Sciences 19, 1–21.

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Nakamura, Yutaka (1990) “An Axiomatic Characterization of Quiggin’s Anticipated Utility,” Discussion paper, Inst. Socio-Econ. Plann., University of Tsukaba.

{% Theorem 1 modifies the results of Nakamura (1990, JET) by giving the rank-dependent weighted-utility representation on a rank-ordered set, not on the whole product set. %}

Nakamura, Yutaka (1992) “Multisymmetric Structures and Non-Expected Utility,” Journal of Mathematical Psychology 36, 375–395.

{% The published 2009 paper “SSB Preferences: nonseparable Utilities or Nonseparable Beliefs” gives these results but only for additive measures. The nontransitive nonadditive results have never been published (at least not in 2010). %}

Nakamura, Yutaka (1992) “A Generalization of Subjective Expected Utility without Transitivity and Additivity,” paper presented at Sixth FUR conference, Cachan, France.

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Nakamura, Yutaka (1993) “Subjective Utility with Upper and Lower Probabilities on Finite States,” Journal of Risk and Uncertainty 6, 33–48.

{% Marvelous theorems, but written in a difficult, mathematical, manner. He does not only consider sigma-additive probability measures but, more generally, finitely additive measures. Because of that, he has to deal with ultrafilters, and has to write complex definitions in §2 regarding step probability distributions. On p. 108 last two para’s he introduces n-tuples of outcomes and their cumulative probabilities, as Abdellaoui (2002, Econometrica) will do later. Then he, first, considers only three fixed outcomes (so two-dimenstional subspace!) and proves everything there, as he also did in his 1990-JET paper etc. He can, obviously, put his general representations of 1992 for general rank-ordered sets to good use. Axiom 5 is, however, not just multisymmetry but rather it is very similar to act-independence of Gul (1992, Assumption 2), as explained by Köbberling & Wakker (2003 MOR). Here is an explanation. He uses Wakker’s (1993, MOR) truncation continuity to obtain an extension to nonsimple prospects.
P. 104 penultimate para is correct. Nakamura has a rich probability space, and a general consequence space. Wakker (1993) did the extension to nonsimple probability distributions for general consequences, but had no underlying preference foundation of RDU for simple probability distributions for general consequences, but only for continua of outcomes or, at least, solvability for outcomes (Wakker 1991, in Doignon & Falmagne, eds.). %}

Nakamura, Yutaka (1995) “Rank Dependent Utility for Arbitrary Consequence Spaces,” Mathematical Social Sciences 29, 103–129.

{% Adds a weak independence axiom, his Axiom 2, to the probabilistic sophistication axioms of Machina & Schmeidler (1992), that is necessary and sufficient for the M&S model to be RDU. Section 3 considers the case of unbounded utility, using my 1993 truncation continuity. %}

Nakamura, Yutaka (1995) “Probabilistically Sophisticated Rank Dependent Utility,” Economic Letters 48, 441–447.

{% utility families parametric; characterizes utility that is linear combination of exponential functions. %}

Nakamura, Yutaka (1996) “Sumex Utility Functions,” Mathematical Social Sciences 31, 39–47.

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Nakamura, Yutaka (1997) “Lexicographic Additivity for Multi-Attribute Preferences on Finite Sets,” Theory and Decision 42, 1–19.

{% Generalizes Savage (1954) to nontransitive skew-symmetry, thus extending earlier works by Fishburn and Sugden to nonsimple acts. %}

Nakamura, Yutaka (1998) “Skew-Symmetric Additive Representations of Preferences,” Journal of Mathematical Economics 30, 367–387.

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Nakamura, Yutaka (2000) “Finite-Dimensional Utilities,” Economic Theory 16, 209–218.

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Nakamura, Yutaka (2000) “Threshold Models for Comparative Probability on Finite Sets,” Journal of Mathematical Psychology 44, 353–382.

{% Studies convex nontransitive preferences over lotteries. %}

Nakamura, Yutaka (2001) “Totally Convex Preferences for Gambles,” Mathematical Social Sciences 42, 295–305.

{% Extending Herstein & Milnor etc. to lexicographic %}

Nakamura, Yutaka (2002) “Lexicographic Quasilinear Utility,” Journal of Mathematical Economics 37, 157–178.

{% Additive representation without solvability if there is sufficient denseness. %}

Nakamura, Yutaka (2002) “Additive Utilities on Densely Ordered Sets,” Journal of Mathematical Psychology 46, 515–530.

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Nakamura, Yutaka (2004) “Objective Belief Functions as Induced Measures,” Theory and Decision 55, 71–83.

{% Considers set of lotteries preferred to status quo, equivalent to it, and worse than it, and characterizes it à la vNM. %}

Nakamura, Yutaka (2005) “Trichotomous Preferences for Gambles,” Journal of Mathematical Psychology 48, 385–398.

{% Does nontransitive generalizations in Aumann-Anscombe setup, but only for additive representations and not for nonadditive. %}

Nakamura, Yutaka (2009) “SSB Preferences: Nonseparable Utilities or Nonseparable Beliefs.” In Steven J. Brams, William V. Gehrlein, & Fred S. Roberts (eds.) The Mathematics of Preference, Choice and Order: Essays in Honor of Peter Fishburn. Springer, Berlin, 39–55.

{% The authors argue, and I agree, that weighting functions for uncertainty are too general, and introduce a special class after discussing preceding ones. m-separability means that there is a partition A₁,…,A_m such that, for a weighting function (= capacity) W, W(E) = f(W(E  A₁),…,W(E  A_n)) with f strictly increasing in each variable. It is a sort of ordinal additive separability of the elements of the partition. m-separability with respect to every partition will be equivalent to the additivity condition of qualitative probability I guess, and under sufficient richness will be equivalent to being a monotonic transform of an additive probability measure as this is with probabilistic sophistication. %}

Narukawa, Yasuo & Vicenç Torra (2011) “On Distorted Probabilities and m-Separable Fuzzy Measures,” International Journal of Approximate Reasoning 52, 1325–1336.

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Narens, Louis (1980) “On Qualitative Axiomatizations of Probability,” Journal of Philosophical Logic 9, 143–151.

{% Theorems 2.8.2 & 2.8.3 on p. 83 shows that, if the Archimedean axiom is dropped in Hölder’s lemma, then the operation need no more be commutative. So, in the lemma of Hölder the Archimedean axiom has empirical content. The example is as follows: X is the set of affine functions ax + b on the reals with a  1 and b > 0. The operation o is functional composition, the ordering is f  g if f(x)  g(x) for all x sufficiently large (so lexicographic in a,b). The operation is associative and f  g iff f o h  g o h. The operation is not commutative though, with f = 2x + 1 and g = x + 1 we have fog = 2x + 3 > 2x + 2 = gof.
The violation of commutativity is only infinitesimally small, so I’m not sure if this is really empirical content.
cancellation axioms: Theorems 5.2.1 & 5.2.2 give necessary and sufficient conditions for additive representation of finitely many preferences. Does not need weak ordering. %}

Narens, Louis (1985) “Abstract Measurement Theory.” MIT Press, Cambridge, MA.

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Narens, Louis (2002) “Theories of Meaningfulness.” Lawrence Erlbaum, Mahwah, NJ.

{% conditional probability; Qualitative conditional probability, extended to support theory etc. %}

Narens, Louis (2003) “A Theory of Belief,” Journal of Mathematical Psychology 47, 1–31.

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Narens, Louis (2008) “Meaningfulness and Invariance.” In Lawrence Blume & Steven N. Durlauf (eds.) The New Palgrave: A Dictionary of Economics. The MacMillan Press, London.

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Narens, Louis & R. Duncan Luce (1983) “How We May Have Been Misled into Believing in the Interpersonal Comparability of Utility,” Theory and Decision 15, 247–260.

{% L & Narens 1986.1 %}

Narens, Louis & R. Duncan Luce (1986) “Measurement: The Theory of Numerical Assignments,” Psychological Bulletin 99, 166–180.

{% equity-versus-efficiency: seems to be on it %}

Narloch, Ulf, Unai Pascual, & Adam G. Drucker (2011) “Cost-Effectiveness Targeting under Multiple Conservation Goals and Equity Considerations in the Andes,” Environmental Conservation 38, 417–425.

{% Ch. 17, p. 172: Nash made a Dutch book, well, not a Dutch book but arbitrage, against his students for the 1952 election Stevenson-Eisenhower. %}

Nasar, Sylvia (1998) “A Beautiful Mind. The Life of Mathematical Genius and Nobel Laureate John Nash.” Simon & Schuster, New York.

{% Expert aggregation under ambiguity. Adopts Anscombe-Aumann (AA) framework and assumes identical risk attitudes. Two-stage reduction (p. 545) considers replacing the 2^nd-stage lotteries by their CEs, to escapte from violations of RCLS. Cites the advanced Domotor (1979), showing good knowledge of the literature. %}

Nascimento, Leandro (2012) “The Ex-Ante Aggregation of Opinions under Uncertainty,” Theoretical Economics 7, 535–570.

{% Characterize the general functional that satisfies certainty independence, and that is the point of departure of the variational model, multiple priors, and Chateauneuf & Faro’s (2009) appealing variation on variational (not cited here). They do, nicely, cite Chateauneuf on his 91 foundation of multiple priors. %}|

Nascimento, Leandro & Gil Riella (2010) “On the Uses of the Monotonicity and Independence Axioms in Models of Ambiguity Aversion,” Mathematical Social Sciences 59, 326–329.

{% Axiomatizes a common generalization of maxmin multiple priors and incompleteness-via-unanimity multiple priors, by considering a set of sets M of multiple priors, where for each M maxmin is done, and then preference holds if and only if it is unanimous over all sets M considered. Does it also for the variational model. Uses three-stage Anscombe-Aumann. %}

Nascimento, Leandro & Gil Riella (2011) “A Class of Incomplete and Ambiguity Averse Preferences,” Journal of Economic Theory 146, 728–750.

{% Generalizes the recursive utility model by not having one second-order probability, but having multiple priors there. So it is like their 2011 JET paper, but not going for Bewley-type incompleteness but instead for maxmin. Figure 1 in this paper is a very small variation of Figure 1 of the 2011 JET paper. Strangely enough, they do not cite their 2011 JET paper. %}

Nascimento, Leandro & Gil Riella (2013) “Second-Order Ambiguous Beliefs,” Economic Theory 52, 1005–1037.

{% %}

Nash, John F. (1950) “Non-Cooperative Games.} Ph.D. Thesis, Princeton University, Princeton.

{% Axiom 3 is IIA, not in the Arrow-social-choice sense, but in the revealed-preference sense, for multivalued choice functions. So, again, Nash was the first to have written it, preceding Arrow (1959).
Shubik’s 1982 book writes: “This section by John F. Nash, jr., was written as an informal note dated August 8, 1950; it is reproduced here with the permission of the author.” %}

Nash, John F. (1950) “Rational Nonlinear Utility.” In Shubik, Martin (1982) “Game Theory in the Social Sciences,” Appendix A2, The MIT Press, Cambridge, MA.

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Nash, John F. (1950) “The Bargaining Problem,” Econometrica 18, 155–162.

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Nash, John F. (1950) “Equilibrium Points in n-Person Games,” Proceedings of the National Academy of Sciences 36, 48–49.

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Nash, John F. (1951) “Non-Cooperative Games,” Annals of Mathematics 54, 286–295.

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Nataf, André (1948) “Sur la Possibilité de Construction de Certain Macromodèles,” Econometrica 16, 232–244.

{% (NICE): in 2012 one QALY may cost £30,000 in the UK. In Holland, €80,000 has been mentioned informaly. %}

National Institute for Health and Clinical. Excellence

{% Responsible government agency for damage assessments in connection with oil spills (NOAA) appointed panel of economic experts to evaulate use of contingent valuation. Panel was co-chaired by Arrow and Robert Solow. Panel published a report containing a number of recommendations for contingent valuation.
They recommend binary contingent valuation (“referendum approach”) iso open-ended questions.
Discussed by Johannesson, Jönsson, & Karlsson (1995)
Hypothetical WTP exceeds real WTP %}

National Oceanic and Atmospheric Administration (1993) “Report of the NOAA Panel on Contingent Valuation,” Federal Register 58, 4602–4614.

{% paternalism/Humean-view-of-preference, & real incentives/hypothetical choice: seem to write: “The survey instrument of analysis method shall provide a mechanism for calibrating hypothetical WTP to actual WTP. The trustee(s) shall document the rationale for the selected calibration mechanism. If the survey instrument or analysis method fails to provide such a mechanism or the trustee(s) fails to document the rationale for the selected calibration mechanism, actual WTP shall be presumed to be one-half of stated WTP.” %}

National Oceanic and Atmospheric Administration (1994) “Natural Resource Damage Assessments: Proposed Rules,” Federal Register 59, 1062–1191.

{% probability elicitation; Rasmussen rapport %}

National Research Council Governing Board Committee on the Assessment of Risk (1981) “The Handling of Risk in NRC Reports.” Washington, DC: National Research Council.

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Nau, Robert F. (1985) “Should Scoring Rules be “Effective”?,” Management Science 31, 527–535.

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Nau, Robert F. (1992) “Joint Coherence in Games of Incomplete Information,” Management Science 38, 374–387.

{% A very interesting paper. A subject may take 1:2 bets on an event if his subjective probability of the event exceeds 1/3 as long as the stakes are moderate. But if the stakes are large then the subject does not do this anymore, because he starts doubting his own info (especially if the bet is with an opponent who, if setting large stakes, must be self-assured). So the size of stake that is still accepted is an index of the value of info. One of the very rare papers where a behavioral foundation is given to degree of confidence in subjective probability. %}

Nau, Robert F. (1992) “Indeterminate Probabilities on Finite Sets,” Annals of Statistics 20, 1737–1767.

{% state-dependent utility %}

Nau, Robert F. (1995) “Coherent Decision Analysis with Inseparable Probabilities and Utilities,” Journal of Risk and Uncertainty 10, 71–91.

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Nau, Robert F. (1995) “The Incoherene of Agreeing to Disagree,” Theory and Decision 39, 219–239.

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Nau, Robert F. (1995) “Arbitrage-Free Correlated Equilibria,”

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Nau, Robert F. (1999) “Arbitrage, Incomplete Models, and Other People’s Brains.” In Bertrand R. Munier & Mark J. Machina (eds.) Preferences, Beliefs, and Attributes in Decision Making, Kluwer, Dordrecht.

{% criticisms of Savage’s basic model: argues that states and acts are naturally given, consequences not but the consequence set is product set of acts and states. %}

Nau, Robert F. (2001) “De Finetti Was Right: Probability Does not Exist,” Theory and Decision 51, 89–124.

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Nau, Robert F. (2002) “The Aggregation of Imprecise Probabilities,” Journal of Statistical Planning and Inference 105, 265–282.

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Nau, Robert F. (2003) “A Generalization of Pratt-Arrow Measure to Non-Expected-Utility Preferences and Inseparable Probability and Utility,” Management Science 49, 1089–1104.

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Nau, Robert F. (2006) “The Shape of Incomplete Preferences,” Annals of Statistics 34, 2430–2448.

{% Tradeoff method: Axiom 4 on p. 143;
event/utility driven ambiguity model: utility-driven
This paper does not provide proofs but uses the formula "proof available from the author upon request." It was done, as the author explained to me in an email of March 22, 2006, because the proofs were deemed simple, and not merely to save space. He uploaded proofs and explanations on internet in Sept. 06 on his homepage.
source-dependent utility: uses the Kreps-Porteus (1978) two-stage-expectation representation,
EXPT[(EXP_S[U(f(s))d])d],
where EXP_S[…] denotes expectation over S, etc. The model is EU iff  is linear. It reinterprets the model for ambiguity, where T does not reflect uncertainty at a different time as it does for Kreps & Porteus, but uncertainty from a different source of uncertainty for which there can be more ambiguity. Ambiguity aversion then results if  is concave, so that here we find smaller certainty equivalents. This paper generalizes the model to state-dependent utility, and considers local measures of risk/ambiguity aversion being matrix-generalizations of the Pratt-Arrow measure.
biseparable utility violated %}

Nau, Robert F. (2006) “Uncertainty Aversion with Second-Order Utilities and Probabilities,” Management Science 52, 136–145.

{% State-dependent extensions of smooth ambiguity models. %}

Nau, Robert F. (2011) “Risk, Ambiguity, and State-Preference Theory,” Economic Theory 48, 437–467.

{% Games with incomplete information, correlated equilibrium %}

Nau, Robert F. & Kevin F. McCardle (1990) “Coherent Behavior in Noncooperative Games,” Journal of Economic Theory 50, 424–444.

{% %}

Nau, Robert F. & Kevin F. McCardle (1991) “Arbitrage, Rationality, and Equilibrium,” Theory and Decision 31, 199–240.

{% measure of similarity %}

Navarro, Daniel J. (2007) “On the Interaction between Exemplar-Based Concepts and a Response Scaling Process,” Journal of Mathematical Psychology 51, 85–98.

{% measure of similarity; One point of discussion is the pros and cons of fitting individual or group-average data if there is much noise in the data. %}

Navarro, Daniel J., Thomas L. Griffiths, Mark Steyvers, & Michael D. Lee (2006) “Modeling Individual Differences Using Dirichlet Processes,” Journal of Mathematical Psychology 50, 101–122.

{% %}

Navarro, Daniel J. & Michael D. Lee (2003) “Combining Dimensions and Features in Similarity-Based Representations,” Advances in Neural Information Processing Systems 15, 59–66.

Also appeared as book:

Navarro, Daniel J. & Michael D. Lee (2003) “Combining Dimensions and Features in Similarity-Based Representations.” In Suzanne Becker, Sebastian Thrun, & Klaus Obermayer (eds.) Advances in Neural Information Processing Systems 15, 59–66.

{% Discuss that decisions have often ignored the input of patients’ preferences and argue for it. Consider this issue, however, only in the context of planning clinical trials with the emphasis on the sample size that must be incorporated in a clinical test, and only for the probability tradeoff test. %}

Naylor, C. David & Hilary A. Llewellyn-Thomas (1994) “Can There Be a More Patient-Centred Approach to Determining Clinically Important Effect Sizes for Randomized Treatment Trials?,” Journal of Clinical Epidemiology 47, 787–795.

{% utility families parametric: gives 1exp(c(p^s)/t) as family of inverse-S curves, is utility functions of Ron Howard. %}

Nease, Robert F. (1994) “Risk Attitudes in Gambles Involving Length of Life,” Medical Decision Making 14, 201–203.

{% %}

Nease, Robert F. (1996) “Do Violations of the Axioms of Expected Utility Theory Threaten Decision Analysis?,” Medical Decision Making 16, 399–403.

{% dynamic consistency
Several subjects satisfy independence but then violate two or more axioms. %}

Nebout, Antoine & Dimitri Dubois (2014) “When Allais Meets Ulysses: Dynamic Axioms and the Common Ratio Effect,” Journal of Risk and Uncertainty 48, 19–49.

{% %}

Nehring, Klaus D.O. (1992) “Foundations for the Theory of Rational Choice with Vague Priors.” In John F. Geweke (ed.) Decision Making under risk and Uncertainty: New Models and Empirical Findings, Kluwer Academic Publishers, Dordrecht.

{% It will not be surprising that I disagree with the criticism in this note. My 2010 book explains the case in §7.6. In short, the main problem with this note is that under RDU, w cannot just be applied to any probability as the author does, but only to goodnews probabilities. If we transform badnews probabilities, then the dual of w should be taken. All confusions would have been avoided had the field used the more proper term rank-transformation or goodnews-probability transformation rather than probability transformation. %}

Nehring, Klaus D.O. (1994) “On the Interpretation of Sarin and Wakker’s “A Simple Axiomatization of Nonadditive Expected Utility” ,” Econometrica 62, 935–938.

{% preference for flexibility %}

Nehring, Klaus D.O. (1998) “Preference for Flexibility in a Savage Framework,” Econometrica 67, 101–119.

{% Assumes CEU (Choquet expected utility) with linear utility function. Under CEU, unambiguous events are meant to be those for which the capacity is additive. If on a collection of events the capacity satisfies additivity, then it need not be possible to extend it to the algebra generated by the collection while preserving additivity. This point is reminiscent of the definition of additive probability measures in probability theory, where these are first defined on subcollections and then extended to sigma-algebras, and the subcollections must be appropriate. Def. 4 defines unambiguous, essentially, as SEU preferences for two-elements-partitions. %}

Nehring, Klaus D.O. (1999) “Capacities and Probabilistic Beliefs: A Precarious Coexistence,” Mathematical Social Sciences 38, 197–213.

{% %}

Nehring, Klaus D.O. (2000) “A Theory of Rational Choice under Ignorance,” Theory and Decision 48, 205–240.

{% %}

Nehring, Klaus D.O. (2001) “Ambiguity in the Context of Probabilistic Beliefs,” working paper.

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Nehring, Klaus (2002) “Imprecise Probabilistic Beliefs as a Context for Decision-Making under Ambiguity,” working paper.

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Nehring, Klaus D.O. (2003) “Ellsberg without Allais: A Theory of Utility-Sophisticated Preferences under Ambiguity; working paper.”

{% Harsanyi-like aggregation, an existence result iff common prior %}

Nehring, Klaus (2004) “The Veil of Public Ignorance,” Journal of Economic Theory 119, 247–270.

{% ordering of subsets: the author considers a qualitative probability relation that need not be complete and represents it by a set of priors through unanimous representation (known way to get incompleteness). Gives preference axioms for it. The main axiom is the adaptation of the usual additivity. It here claims that for two equally likely events, each can be partitioned into two equally likely smaller events, and then that the four resulting smaller events are equally likely again (it is formulated somewhat differently and less transparently, as splitting in Axiom 8 p. 1062, but the text following states it is only used as I just described). Richness is through equidivisibility: each set can be split up into two equally likely subsets. Further continuity. A 1/k event is such that, in the terminology of Wakker (1981) the vacuous event and the universal event differ by at least k times that event. It is used to define convergence, and then continuity. By equidivisibility, we can divide the universal event into 2ⁿ equally likely events for each n. A very restrictive implication follows: all probability measures in the set of priors must agree on these events and assign the same probability 2^ⁿ to them. Thus, they all agree on a rich set of events, and we in fact have a rich set of events with known probabilities, something like Anscombe & Aumann (1963) but, fortunately, without multistage setup, so more like the hybrid models of Wakker (2010).
The proof is to split the universal event up into always more refined 2ⁿ equally likely partitions, where the probabilities are 2^ⁿ and then all dyadic numbers. All other events can then be calibrated.
In several places the author claims, and I disagree, that the aforementioned restrictive assumption (all priors agreeing on rich set of dyadic events) can hardly be avoided if one wants uniqueness (of convex closure). He only puts forward Example 1 on p. 1065, but this is only one example showing that without equidivisibility and with nonatomicity instead it does not work. There is much between equidivisibility and nonatomicity, and much besides nonatomicity too. (He also puts forward that any structure can be embedded in a larger structure that has equidivisibility, by adding objective-probability events, on p. 1057 penultimate para and p. 1066 last para) but, again, the same kind of argument can be used to defend virtually any richness assumption in any model whatsoever.)
In several places (e.g. p. 1055 next-to-last para, p. 1058 l. 5) the author writes that his model is to be taken as rational.

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