Savage Axioms Research Articles

Ellsberg 1961: text, context, influence

AbstractIn 1961 Daniel Ellsberg published an article titled “Risk, Ambiguity, and the Savage Axioms” in the Quarterly Journal of Economics, which became a seminal contribution to the theory of decision-making under uncertainty. This paper analyzes Ellsberg’s 1961 classic, situates it within the context of decision-making theory in the 1950s and early 1960s and within the development of Ellsberg’s ideas, and provides an overview of the experimental and theoretical literature to which it gave rise.

Preference for Knowledge

We examine the subjective value of gaining knowledge in a version of Savage's model for decisions under uncertainty in which the received outcome provides information about which event has obtained. Decision makers commonly value such knowledge either because they want to use it in future decisions or because they are personally interested in it. We find that in our model, the sure-thing principle and several other axioms of Savage are inconsistent with this value for knowledge about events. We provide a representation theorem for a subjective value of knowledge consisting of the sum of expected utility and a function of the information partition generated by the outcomes of an act. We characterize when the value of knowledge can be represented by a subjective value of knowledge about an information partition plus a Shannon entropy cost of processing information. Our results also provide a novel critique of the necessity of Savage's axioms for rational decisions under uncertainty.

Policy Note: Risk and Uncertainty in the Murray–Darling Basin

Water Economics and PolicyOnline Ready No AccessPolicy Nook — Policy Note: Risk and Uncertainty in the Murray–Darling BasinJohn QuigginJohn QuigginSchool of Economics, Faculty of Business, Economics and Law, The University of Queensland, St. Lucia, Brisbane, QLD 4072, AustraliaE-mail Address: [email protected] Search for more papers by this author https://doi.org/10.1142/S2382624X22710059Cited by:0 PreviousNext AboutSectionsView articleView Full TextPDF/EPUB ToolsAdd to favoritesDownload CitationsTrack CitationsRecommend to Library ShareShare onFacebookTwitterLinked InRedditEmail View article References Adamson, D, T Mallawaarachchi and J Quiggin [2009] Declining inflows and more frequent droughts in the Murray Darling Basin: climate change, impact and adaption. Australian Journal of Agricultural and Resource Economics, 53, 345–366. Crossref, Google Scholar Bell, S and J Quiggin [2008] The metagovernance of markets: The politics of water management in Australia. Environmental Politics, 17, 712–729. Crossref, Google Scholar Blaug, M [1962] Economic Theory in Retrospect. Cambridge, UK: Cambridge University Press. Google Scholar Borsch, SJ [2004] Environment and population: The collapse of large irrigation systems reconsidered. Comparative Studies in Society and History, 46(3), 451–468. Crossref, Google Scholar Chambers, RG and J Quiggin [2000] Uncertainty, Production, Choice and Agency: The State-Contingent Approach. Cambridge, UK: Cambridge University Press. Google Scholar Council of Australian Governments (2004). Communique 25. June. http://www.pm.gov.au/News/media_releases/media_Release941.html. Google Scholar Ellsberg, D [1961] Risk, ambiguity and the Savage axioms. Quarterly Journal of Economics, 75, 643–669. Crossref, Google Scholar Grant, S, JJ Kline and J Quiggin [2014] A matter of interpretation?: ambiguous contracts and liquidated damages. Games and Economic Behavior, 85, 180–187. Crossref, Google Scholar Grant, S and J Quiggin [2013a] Bounded awareness, heuristics and the Precautionary Principle. Journal of Economic Behavior & Organization, 93, 17–31. Crossref, Google Scholar Grant, S and J Quiggin [2013b] Inductive reasoning about unawareness. Economic Theory, 54, 717–755. Crossref, Google Scholar Jevons, WS [1871] The Theory of Political Economy. London: Macmillan. Google Scholar Knight, F [1921] Risk, Uncertainty and Profit. New York: Houghton Mifflin Company. Google Scholar Quiggin, J [1992] The pioneer’s curse: selection bias and agricultural land degradation. Journal of Risk and Uncertainty, 5, 241–246. Crossref, Google Scholar Randall, A [1981] Property entitlements and pricing policies for a maturing water economy. Australian Journal of Agricultural Economics, 25, 195–220. Crossref, Google Scholar Rumsfeld, D (2002). Transcript of press briefing. February 12. http://www.defense.gov/transcripts/transcript.aspx?transcriptid=2636. Google Scholar Shepherd, T (2022). ‘They’re coming back’: huge flows set to restore life to parched end of Murray-Darling system. The Guardian, November 14. https://www.theguardian.com/australia-news/2022/nov/14/theyre-coming-back-huge-flows-set-to-restore-life-to-parched-end-of-murray-darling-system. Google Scholar Smith, A (1976 [1776]). An Inquiry into the Nature and Causes of the Wealth of Nations, RH CampbellAS SkinnerWB Todd (eds.), Vol. 1. Oxford, UK: Oxford University Press. Google Scholar Willis, E, M Pearce, B Jorgensen and J Martin (2015). Water supply and governance options for outback towns in South Australia. Technical Report Series No. 15/7, Goyder Institute for Water Research, Adelaide, SA. Google Scholar FiguresReferencesRelatedDetails Recommended Online Ready Metrics History Received 10 January 2023 Accepted 11 January 2023 Published: 9 March 2023 PDF download

Savage's P3 Is Redundant

Savage (1954) provided the first axiomatic characterization of expected utility without relying on any given probabilities or utilities. It is the most famous preference axiomatization existing. This note shows that Savage's axiom P3 is implied by the other axioms, which reveals its redundancy. It is remarkable that this was not noticed before as Savage's axiomatization has been studied and taught by hundreds of researchers for more than six decades.

Four Decades of Transformation in Decision Analytic Practice for Societal Risk Management

The formal mathematical structure for decision making under uncertainty was first expressed in Savage's axioms over 60 years ago. But while the underlying normative concepts for decision making under uncertainty remain constant, the practice of applying these concepts in real-world settings, as conducted by decision analysis (DA) specialists working with agencies and interested parties, has seen a major transformation in recent decades. The purpose of this article is to provide perspectives that characterize and interpret how DA practice for societal risk management questions has grown and is being transformed over the last 40 years. It addresses a series of themes for parsing changes in how DA has evolved toward more flexible approaches, moving beyond strict theoretical assumptions and constrained settings, and addresses multiple interested parties to provide insights rather than a single correct answer. The article clarifies the path from the initial DA formulation as a set of normative axioms, through gradual change into what is now the most flexible and least restrictive form of policy analysis. The article shows how the practice of DA for societal risks has become more attuned to a wide array of interests and perspectives, more behaviorally informed, more creative, and more informative for governance process. It addresses the following themes: the evolution in the basic orientation of DA, the increasingly important role of stakeholders in DA practice, the importance and value of key problem-structuring techniques, and evolution in approaches for eliciting values and technical judgments.

Uncertainty and computational complexity.

Modern theories of decision-making typically model uncertainty about decision options using the tools of probability theory. This is exemplified by the Savage framework, the most popular framework in decision-making research. There, decision-makers are assumed to choose from among available decision options as if they maximized subjective expected utility, which is given by the utilities of outcomes in different states weighted with subjective beliefs about the occurrence of those states. Beliefs are captured by probabilities and new information is incorporated using Bayes’ Law. The primary concern of the Savage framework is to ensure that decision-makers’ choices are rational. Here, we use concepts from computational complexity theory to expose two major weaknesses of the framework. Firstly, we argue that in most situations, subjective utility maximization is computationally intractable, which means that the Savage axioms are implausible. We discuss empirical evidence supporting this claim. Secondly, we argue that there exist many decision situations in which the nature of uncertainty is such that (random) sampling in combination with Bayes’ Law is an ineffective strategy to reduce uncertainty. We discuss several implications of these weaknesses from both an empirical and a normative perspective.This article is part of the theme issue ‘Risk taking and impulsive behaviour: fundamental discoveries, theoretical perspectives and clinical implications’.

Savage's theorem under changing awareness

This paper proposes a simple unified framework of choice under changing awareness, addressing both outcome awareness and (nature) state awareness, and both how fine and how exhaustive the awareness is. Six axioms characterize an (essentially unique) expected-utility rationalization of preferences, in which utilities and probabilities are revised according to three revision rules when awareness changes: (R1) utilities of unaffected outcomes are transformed affinely; (R2) probabilities of unaffected events are transformed proportionally; (R3) enough probabilities ‘objectively’ never change (they represent revealed objective risk). Savage's Theorem is a special case of the theorem, namely the special case of fixed awareness, in which our axioms reduce to Savage's axioms while R1 and R2 hold trivially and R3 reduces to Savage's requirement of atomless probabilities. Rule R2 parallels Karni and Viero's (2013) ‘reverse Bayesianism’ and Ahn and Ergin's (2010) ‘partition-dependence’. The theorem draws mathematically on Kopylov (2007), Niiniluoto (1972) and Wakker (1981).

Daniel Ellsberg et la validation des propositions normatives

In the history of decision theory Daniel Ellsberg is known because his seminal paper “Risk, ambiguity and the Savage axioms” presented the counterexample to Bayesian decision-making that got the normative value of the theory into trouble. An assessment of Ellsberg’s contribution in the more encompassing framework of his long unnoticed 1962 doctoral thesis suggests that, although he did not take part to the ensuing debate, Ellsberg provided a thorough philosophical and methodological background for his critique of Savage’s axiomatization of decision theory. By concentrating mainly on Ellsberg’s analysis of decision-making in his thesis, this paper examines the way he conceived of the possibility to test normative propositions and tries to identify the kind of normative value he attributed to his own suggested solution for the Ellsberg Paradox.

A personal history of Bayesian statistics

The history of Bayesian statistics is traced, from a personal perspective, through various strands and via its re‐genesis during the 1960s to the current day. Emphasis is placed on broad‐sense Bayesian methodology that can be used to meaningfully analyze observed datasets. Over 750 people in science, medicine, and socioeconomics, who have influenced the evolution of the Bayesian approach into the powerful paradigm that it is today, are highlighted. The frequentist/Bayesian controversy is addressed, together with the ways in which many Bayesians combine the two ideologies as a Bayes/non‐Bayes compromise, e.g., when drawing inferences about unknown parameters or when investigating the choice of sampling model in relation to its real‐life background. A number of fundamental issues are discussed and critically examined, and some elementary explanations for nontechnical readers and some personal reminiscences are included. Some of the Bayesian contributions of the 21st century are subjected to more detailed critique, so that readers may learn more about the quality and relevance of the ongoing research. A recent resolution of Lindley's paradox by Baskurt and Evans is reported. The axioms of subjective probability are reassessed, some state‐of‐the‐art alternatives to Leonard Savage's axioms of utility are discussed, and Deborah Mayo and Michael Evan's refutation of Allan Birnbaum's 1962 justification of the likelihood principle in terms of the sufficiency and conditionality principles is addressed.WIREs Comput Stat2014, 6:80–115. doi: 10.1002/wics.1293This article is categorized under:Statistical and Graphical Methods of Data Analysis > Bayesian Methods and Theory

Why the Restricted Bayes/Hurwicz Criterion is Savagian: A Game-Theoretic Perspective of Ambiguity

Ellsberg (1961) introduced the Ellsberg urns experiments as evidence against Savage's theory of subjective expected utility. The restrictive Bayes/Hurwicz criterion (Ellsberg, 2001) is proposed to explain the phenomena of subjects shunning the ambiguous urns. Modeling the interaction between the experimenter and the subject as a game, we find sufficient and necessary conditions for the subject to modify beliefs with the reduced-form criterion. We postulate that diverse ambiguity attitudes result from different perceptions about the payoff structure. As a horse race gambler should adjust his belief after the gambling company publishes new odds, the subject can form different beliefs facing different choice problems. The revealed belief discrepancies can be a result of these modifications, and thus does not violate the Savage axioms about the preferences at the moment of each decision. The ambiguity game interpretation is supported by various neuroscientific and behavioral studies.

Inferring Preferences from Choices Under Uncertainty

If a decision maker, in a world of uncertainty a la Anscombe and Aumann (1963), can choose acts according to some objective probability distribution (by throwing dice for instance) from any given set of acts, then there is no set of acts that allows an experimenter to test more than the Axiom of EUOL (that the DM evaluates objective lotteries with an expected utility function). In fact there is no (common) experimental design that allows an experimenter to test more than EUOL. For any decision problem (or set of decision problems), for any preference relation that satisfies the Axiom EUOL, and for any optimal choice (or collection of choices) given this preference relation, there is another preference relation that satisfies EUOL plus the Savage axioms, for which this choice is also optimal.

SAVAGE AXIOMS, ELLSBERG’S PARADOX AND FUZZY OPTIMAL DECISION-CHOICE RATIONALITY

This essay is devoted to epistemics of Savage axioms and Ellsberg’s paradox in decision-choice actions in relation to fuzzy optimal decision-choice rationality in the space of uncertainties. The uncertainty space is partitioned into non-fuzzy stochastic sub-space and fuzzy-stochastic sub-space. The Savage axioms are argued to emerge from the former which is vagueness-free while Ellsberg thought experiment leading to his paradox takes place in the latter that contains vagueness. The explanation of the rise of the paradox is shown to be the result of non-comparable decision-choice sub-spaces in which both of them work. The topologies of the two sub-spaces and the required mathematics and logic are considered in their epistemic forms. Criticisms are offered on some attempts to explain and resolve the paradox. The paradox, it is argued, cannot be logically resolved in the space in which the Savage axioms are created. It can not either be resolved in the fuzzy-stochastic space with classical paradigm under Aristotelian logic where all propositions are either true or false, it is argued. A resolution of the paradox as it is related to probability estimates is offered through fuzzy logic, mathematics and computational methods of optimal fuzzy decision-choice rationality. The essay is concluded with an epistemic reflection.

Inferring Preferences from Choices Under Uncertainty

If a decision maker, in a world of uncertainty a la Anscombe and Aumann (1963), can choose acts according to some objective probability distribution (by throwing dice for instance) from any given set of acts, then there is no set of acts that allows an experimenter to test more than the Axiom of EUOL (that the DM evaluates objective lotteries with an expected utility function). In fact there is no (common) experimental design that allows an experimenter to test more than EUOL. For any decision problem (or set of decision problems), for any preference relation that satisfies the Axiom EUOL, and for any optimal choice (or collection of choices) given this preference relation, there is another preference relation that satisfies EUOL plus the Savage axioms, for which this choice is also optimal.

Probability, Expected Utility, and the Ellsberg Paradox

The Ellsberg paradox is often cited as evidence for unknowable ambiguity versus computable risk, and a refutation of the Savage axioms regarding expected utility maximization and the program for revealing subjective or belief-type probabilities. This note argues that researchers have been too quick to embrace the Ellsberg critique as a refutation of standard expected utility theory. First, Ellsberg performed no actual experiments, and in fact recent empirical evidence on the Ellsberg paradox argues against Second, simple explanations for the paradox deserve as much attention as theories that introduce a new concept such as ambiguity. One such simple explanation is to consider the Ellsberg thought experiment as part of a meta-experiment that includes repeated draws, in which case the choices described by Ellsberg are consistent with expected utility theory.

A simplified axiomatic approach to ambiguity aversion

This paper takes the Anscombe-Aumann framework with horse and roulette lotteries, and applies the Savage axioms to the horse lotteries and the von Neumann-Morgenstern axioms to the roulette lotteries. The resulting representation of preferences yields a subjective probability measure over states and two utility functions, one governing risk attitudes and one governing ambiguity attitudes. The model is able to accommodate the Ellsberg paradox and preferences for reductions in ambiguity.

Subjective Expected Utility Theory with 'Small Worlds'

We model the notion of a “small world” as a context dependent state space embedded into the “grand world”. For each situation the decision maker creates a “small world” reflecting the events perceived to be relevant for the act under consideration. The “grand world” is represented by an event space which is a more general construction than a state space. We retain preference axioms similar in spirit to the Savage axioms and obtain, without abandoning linearity of expectations, a subjective expected utility theory which allows for an intuitive distinction between risk and uncertainty. We also obtain separation of subjective probability and utility as in the state space models.

IS IT ALWAYS RATIONAL TO SATISFY SAVAGE'S AXIOMS?

This note argues that, under some circumstances, it is more rational not to behave in accordance with a Bayesian prior than to do so. The starting point is that in the absence of information, choosing a prior is arbitrary. If the prior is to have meaningful implications, it is more rational to admit that one does not have sufficient information to generate a prior than to pretend that one does. This suggests a view of rationality that requires a compromise between internal coherence and justification, similarly to compromises that appear in moral dilemmas. Finally, it is argued that Savage's axioms are more compelling when applied to a naturally given state space than to an analytically constructed one; in the latter case, it may be more rational to violate the axioms than to be Bayesian.

Probabilistic sophistication and stochastic monotonicity in the Savage framework

Machina and Schmeidler [Machina, M., Schmeidler, D., 1992. A more robust definition of subjective probability. Econometrica 60, 745–780] show that probabilistic sophistication can be obtained in a Savage setting without imposing expected utility by dropping Savage's axiom P2 (sure-thing principle) and strengthening his axiom P4 (weak comparative probability). Their stronger axiom, however, embodies a degree of separability analogous to P2. In this note, we obtain probabilistic sophistication using Savage's original axiom P4 and a weaker analog of Savage's P2.

Revealing Preferences Graphically: An Old Method Gets a New Tool Kit

Because uncertainty is endemic in a wide variety of economic circumstances, models of decision making under uncertainty play a key role in every field of economics. The standard model of decisions under uncertainty is based on von Neumann and Morgenstern Expected Utility Theory (EUT), so it is natural that experimentalists should want to test the empirical validity of the Savage axioms on which EUT is based. Empirical violations of EUT provoke intriguing questions about the rationality of individual behavior and, at the same time, raise criticisms about the status of the Savage axioms as the touchstone of rationality. These criticisms have generated the development of various theoretical alternatives to EUT, and the investigation of these theories has led to new empirical regularities. For the most part, these experimental investigations use several pairwise choices, a la Allais, to test EUT and its various alternatives, such as weighted utility, implicit expected utility, and prospect theory, among others. Each of these theories gives rise to indifference curves with distinctive shapes in some part of the Marschak-Machina probability triangle, so each theory can be tested against the others by choosing alternatives that the various theories rank differently. In these studies, the criterion used to evaluate a theory is the fraction of choices it predicts correctly. Generally speaking, experimental work has, on the one hand, collected only a few decisions from each subject and, on the other, presented subjects with a serious of choices, designed to compare the predictive abilities of competing theories or discover violations of specific axioms (Colin F. Camerer (1995) and Chris Starmer (2000) review the experimental and theoretical work that focuses on evaluating non-EUT theories). Although this practice is understandable given the purposes for which the experiments were designed, critics say that it limits the usefulness of the data for other purposes. First, while these experiments reveal that violations exist, they give us little sense of how important they are or how frequently they

Rationality of Belief or: Why Savage's Axioms are Neither Necessary Nor Sufficient for Rationality, Second Version

Economic theory reduces the concept of rationality to internal consistency. The practice of economics, however, distinguishes between rational and irrational beliefs. There is therefore an interest in a theory of rational beliefs, and of the process by which beliefs are generated and justified. We argue that the Bayesian approach is unsatisfactory for this purpose, for several reasons. First, the Bayesian approach begins with a prior, and models only a very limited form of learning, namely, Bayesian updating. Thus, it is inherently incapable of describing the formation of prior beliefs. Second, there are many situations in which there is not sufficient information for an individual to generate a Bayesian prior. It follows that the Bayesian approach is neither sufficient not necessary for the rationality of beliefs.