The Borel–Kolmogorov Paradox is typically taken to highlight a tension between our intuition that certain conditional probabilities with respect to probability zero conditioning events are well defined and the mathematical definition of conditional probability by Bayes’ formula, which loses its meaning when the conditioning event has probability zero. We argue in this paper that the theory of conditional expectations is the proper mathematical device to conditionalize and that this theory allows conditionalization with respect to probability zero events. (...) The conditional probabilities on probability zero events in the Borel–Kolmogorov Paradox also can be calculated using conditional expectations. The alleged clash arising from the fact that one obtains different values for the conditional probabilities on probability zero events depending on what conditional expectation one uses to calculate them is resolved by showing that the different conditional probabilities obtained using different conditional expectations cannot be interpreted as calculating in different parametrizations of the conditional probabilities of the same event with respect to the same conditioning conditions. We conclude that there is no clash between the correct intuition about what the conditional probabilities with respect to probability zero events are and the technically proper concept of conditionalization via conditional expectations—the Borel–Kolmogorov Paradox is just a pseudo-paradox. (shrink)

An expected utility model of individual choice is formulated which allows the decision maker to specify his available actions in the form of controls (partial contingency plans) and to simultaneously choose goals and controls in end-mean pairs. It is shown that the Savage expected utility model, the Marschak- Radner team model, the Bayesian statistical decision model, and the standard optimal control model can be viewed as special cases of this goal-control expected utility model.

This paper is devoted to fill the gap in studying logics for biprobability structures. We introduce the logic equation image with two conditional expectation operators and prove the completeness theorem. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.

The general question of defining the expectation value of an operator for a fixed value of another noncommuting observable is considered and explicit expressions are derived. Due to the noncommutivity of operators a unique definition is not possible, and we consider different possible expressions. Special cases which have previously been considered in the literature are shown to be derivable from the methods presented.

We investigate the general properties of general Bayesian learning, where “general Bayesian learning” means inferring a state from another that is regarded as evidence, and where the inference is conditionalizing the evidence using the conditional expectation determined by a reference probability measure representing the background subjective degrees of belief of a Bayesian Agent performing the inference. States are linear functionals that encode probability measures by assigning expectation values to random variables via integrating them with respect to the (...) probability measure. If a state can be learned from another this way, then it is said to be Bayes accessible from the evidence. It is shown that the Bayes accessibility relation is reflexive, antisymmetric and non-transitive. If every state is Bayes accessible from some other defined on the same set of random variables, then the set of states is called weakly Bayes connected. It is shown that the set of states is not weakly Bayes connected if the probability space is standard. The set of states is called weakly Bayes connectable if, given any state, the probability space can be extended in such a way that the given state becomes Bayes accessible from some other state in the extended space. It is shown that probability spaces are weakly Bayes connectable. Since conditioning using the theory of conditional expectations includes both Bayes’ rule and Jeffrey conditionalization as special cases, the results presented generalize substantially some results obtained earlier for Jeffrey conditionalization. (shrink)

Reflection and martingale principles are central to models of rational learning. They can be justified in a variety of ways. In what follows we study martingale and reflection principles in the con...

Several fallacies of conditionalization are illustrated, using the two-envelope problem as a case in point.

Davey suggests that increased resistance to extinction of CRs conditioned to fear-relevant stimuli may be due to more persistent expectancies of the UCS following these stimuli. However, this viewpoint is contradicted by existing empirical evidence that fear-relevant CRs survive an extinction trials series producing extinction of expectancies whereas CRs conditioned to non-fear-relevant CSs do not.

Sequel to Armendt 1986, ‘A Foundation for Causal Decision Theory.’ The representation theorem for causal decision theory is slightly revised, with the addition of a new restriction on lotteries and a new axiom (A7). The discussion gives some emphasis to the way in which appropriate K-partitions are characterized by relations found among the agent’s conditional preferences. The intended interpretation of conditional preference is one that embodies a sensitivity to the agent’s causal beliefs.

Engineering and scientific education conditions us to expect everything, including intelligence, to have a simple, compact explanation. Accordingly, when people new to AI ask "What's AI all about," they seem to expect an answer that defines AI in terms of a few basic mathematical laws.

During evaluative conditioning, a neutral conditional stimulus becomes pleasant or unpleasant after pairings with a positive/negative unconditional stimulus. Measures of US expectancy are...

A plausible view in the literature on the epistemic condition of blameworthiness is the Reasonable Expectation View (RE). According to RE, whether ignorance excuses an agent from deserving blame is a matter of whether the agent could have reasonably been expected to have avoided or corrected the ignorance. This paper does not take up the task of defending this view, but instead examines what it implies for an interesting type of ignorance: moral or political ignorance rooted in myside bias. (...) With the prevalence of increasing political polarization, it is worth examining what a plausible view like RE implies about when, if ever, myside bias-based ignorance excuses the agent from blame—from deserving resentment or indignation. To assess this issue, the paper examines the empirical literature on what it takes to mitigate myside bias and then takes the United States as a case study to examine what RE implies for the actual non-ideal circumstances many agents are in. The paper argues that RE has two revisionary implications for our practice of blame. First, the paper argues that RE implies that political or moral ignorance rooted in myside bias is currently an excuse in a surprising number of cases. Second, our epistemic position to know whether a given instance of political or moral ignorance is excusing is often inadequate. (shrink)

As a rapidly growing emerging capital market, China has attracted the attention of both investors and scholars. To alleviate the expectation of external users of listed companies’ financial statements to “discount” items in levels 2 and 3 of the fair value measurement, listed companies will treat these items as conditional conservatism. It refers to the conservatism of companies when confirming bad news of unrealized gains and losses sooner than confirming good news. A sample was selected for empirical analysis (...) to verify the existence of this relationship. The results are as follows: the higher the proportion of levels 2 and 3 fair value measurements, the stronger the conditional conservatism of the company’s profit and loss; the higher the proportion of cash holding of operating activities in a company’s operating profit and the higher the audit quality, the higher the proportion of levels 2 and 3 fair value hierarchy measurements, and the stronger the conditional conservatism of the company’s profit and loss; and the lesser the internal control defects in a company, the higher the proportion of levels 2 and 3 fair value hierarchy measurements, and the stronger the conditional conservatism of its profit and loss. The findings provide empirical evidence to identify listed companies adopting conditional conservatism to alleviate discounting expectations of fair value items of financial statements’ external users and provide a reference for improving the standards and regulation of listed companies. (shrink)

Expected accuracy arguments have been used by several authors (Leitgeb and Pettigrew, and Greaves and Wallace) to support the diachronic principle of conditionalization, in updates where there are only finitely many possible propositions to learn. I show that these arguments can be extended to infinite cases, giving an argument not just for conditionalization but also for principles known as ‘conglomerability’ and ‘reflection’. This shows that the expected accuracy approach is stronger than has been realized. I also argue that we should (...) be careful to distinguish diachronic update principles from related synchronic principles for conditional probability. (shrink)

Many philosophers think that a necessary condition on moral blameworthiness is that the wrongdoer can reasonably be expected to avoid the action for which she is blamed. Those who think so assume as a matter of course that the expectations at issue here are normative expectations that contrast with the non-normative or predictive expectations we form concerning the probable conduct of others, and they believe, or at least assume, that there is a clear-cut distinction between the two. In this paper (...) I put this widespread assumption under scrutiny and argue that it’s mistaken: although predictive and normative expectations are indeed distinct, there is no sharp separation between them. On the contrary, predictive expectations can have a substantial bearing on normative expectations in two related ways: they can recalibrate what is reasonable to expect of agents when responsibility attributions are at stake and they can help to uncover previously undetected excusing conditions. I illustrate my claims with the famous bystander effect from social psychology and show that it yields predictive expectations that affect normative expectations in these two ways. (shrink)

Causal decision theory attends to probabilities used to obtain an option's expected utility but for completeness should also attend to utilities of possible outcomes. A suitable formula for an option's expected utility uses a certain type of conditional utility.

This paper is concerned with how we ought to think about legitimate expectations in the non-ideal, ‘real’ world. In one (dominant) strand of contemporary theories of justice, justice requires not that each gets what she deserves, but that each gets that to which she is entitled in accordance with what Rawls calls ‘the public rules that specify the scheme of cooperation’. However, that is true only if those public rules are part of a fully just scheme and it is plausibly (...) the case that no such scheme obtains in the real world. Given that, and given the centrality of legitimate expectations to theories of justice, it is vital to think about the status of such expectations in non-ideal circumstances. Having explained the sense in which legitimate expectations have come to play a role often previously associated with desert, a brief argument in favour of an ‘expectations’ view of punishment is considered to show that the system to public rules that generates expectations must itself be (in some sense) just. This argument is illustrated by appeal to just punishment and the relevance of thinking about punishment and not merely distributive justice is defended. In the absence of justice, one possibility would be simply to declare that there are no legitimate expectations and so theories of justice provide no guidance as to who should get what in non-ideal conditions. However, this would be to render such theories more-or-less useless in practice. Through discussion of a series of vignettes, the paper offers an account of how we might think about the demands such expectations place on the systems that give rise to them even in cases where the system is unjust. (shrink)

Climate change increases the frequency and intensity of certain kinds of natural hazard events in alpine areas. This interdisciplinary study addresses the hypothetical possibility of relocating the residents of three alpine areas in Austria: the Sölk valleys, the Johnsbach valley, and the St. Lorenzen/Schwarzenbach valleys. Our particular focus is on these residents’ expectations about such relocations. We find that (1) many residents expect that in the next decades the state will provide them with a level of natural hazards protection, aid, (...) and relief that allows them to continue to live in these valleys; (2) this expectation receives some legal protection but only when it is associated with fundamental rights; and (3) the expectation is morally significant, i.e., it ought to be considered in assessing the moral rightness or justness of relocation policies. These results suggest legal changes and likely extend to many other (Austrian) alpine areas as well. (shrink)

We offer a probabilistic model of rational consequence relations (Lehmann and Magidor, 1990) by appealing to the extension of the classical Ramsey-Adams test proposed by Vann McGee in (McGee, 1994). Previous and influential models of nonmonotonic consequence relations have been produced in terms of the dynamics of expectations (Gärdenfors and Makinson, 1994; Gärdenfors, 1993).'Expectation' is a term of art in these models, which should not be confused with the notion of expected utility. The expectations of an agent are some (...) form of belief weaker than absolute certainty. Our model offers a modified and extended version of an account of qualitative belief in terms of conditional probability, first presented in (van Fraassen, 1995). We use this model to relate probabilistic and qualitative models of non-monotonic relations in terms of expectations. In doing so we propose a probabilistic model of the notion of expectation. We provide characterization results both for logically finite languages and for logically infinite, but countable, languages. The latter case shows the relevance of the axiom of countable additivity for our probability functions. We show that a rational logic defined over a logically infinite language can only be fully characterized in terms of finitely additive conditional probability. (shrink)

Utility theories—both Expected Utility (EU) and non-Expected Utility (non-EU) theories—offer numericalized representations of classical principles meant for the regulation of choice under conditions of risk—a type of formal representation that reduces the representation of risk to a single number. I shall refer to these as risk-numericalizing theories of decision. I shall argue that risk-numericalizing theories (referring both to the representations and to the underlying axioms that render numericalization possible) are not satisfactory answers to the question: “How do I take the (...) (best) means to my ends?” In other words, they are inadequate or incomplete as instrumental theories. They are inadequate because they are poor answers to the question of what it is for an option to be instrumental towards an end. To say it another way, they do not offer a sufficiently rich account of what it is for something to be a means (an instrument) toward an end. (shrink)

This essay gives necessary and sufficient conditions for recovering expected utility from choice behavior in several popular models of uncertainty. In particular, these techniques handle a finite state model; a model for which the choice space consists of probability densities and the expected utility representation requires bounded, measurable utility; and a model for which the choice space consists of Borel probability measures and the expected utility representation requires bounded, continuous utility. The key result is the identification of the continuity condition (...) necessary for the revelation of linear utility. (shrink)

The Expected Shortfall or Conditional Value-at-Risk (CVaR) has been playing the role of main risk measure in the recent years and paving the way for an enormous number of applications in risk management due to its very intuitive form and important coherence properties. This work aims to explore this measure as a probability-dependent utility functional, introducing an alternative view point for its Choquet Expected Utility representation. Within this point of view, its main preference properties will be characterized and its (...) utility representation provided through local utilities with an explicit dependence on the assessed revenue’s distribution (quantile) function. Then, an intuitive interpretation for the related probability dependence and the piecewise form of such utility will be introduced on an investment pricing context, in which a CVaR maximizer agent will behave in a relativistic way based on his previous estimates of the probability function. Finally, such functional will be extended to incorporate a larger range of risk-averse attitudes and its main properties and implications will be illustrated through examples, such as the so-called Allais Paradox. (shrink)

Expectations play an important role in how people plan their lives and pursue their projects. People living in highly industrialized countries share a way of life that comes with high levels of emissions. Their expectations to be able to continue their projects imply their holding expectations to similarly high future levels of personal emissions. We argue that the frustration or undermining of these expectations would cause them significant harm. Further, the article investigates under what conditions people can be thought to (...) hold legitimate expectations, in particular about permissible levels of future emissions. We distinguish differing theories of understanding these conditions, namely authority-based and justice-based theories, that each allows us to systematically distinguish between legitimate and illegitimate expectations. Furthermore, with respect to individuals’ future permissible emissions we give several reasons for holding that such theories cannot identify a particular expectation to a specific level of personal emissions as the only legitimate one. Finally, we argue that the set of legitimate expectations that people hold with respect to a just and effective solution to climate change has normative significance in at least two ways: the differing but equally legitimate expectations ought to be taken into account when justifying what could count as such a solution and when determining the just way of arriving at and implementing such a solution. (shrink)

This paper dwells upon formal models of changes of beliefs, or theories, which are expressed in languages containing a binary conditional connective. After defining the basic concept of a (non-trivial) belief revision model. I present a simple proof of Gärdenfors''s (1986) triviality theorem. I claim that on a proper understanding of this theorem we must give up the thesis that consistent revisions (additions) are to be equated with logical expansions. If negated or might conditionals are interpreted on the basis (...) of autoepistemic omniscience, or if autoepistemic modalities (Moore) are admitted, even more severe triviality results ensue. It is argued that additions cannot be philosophically construed as parasitic (Levi) on expansions. In conclusion I outline somed logical consequences of the fact that we must not expect monotonic revisions in languages including conditionals. (shrink)

This note is a generalization and improved interpretation of the main result of Karni and Schmeidler. A decision-maker is supposed to possess a preference relation on acts and another preference relation on state-prize lotteries, both of which are assumed to satisfy the von Neumann–Morgenstern axioms. In addition, the two preference relations restricted to a state of nature are assumed to agree. We show that these axioms are necessary and sufficient for the existence of subjective expected utility over acts with state-dependent (...) utility functions and a subjective probability measure. This subjective probability measure is unique when conditioned on the set of states of nature in which not all the prizes are equally desirable. (shrink)

Expected utility theory often falls silent, even in cases where the correct rankings of options seems obvious. For instance, it fails to compare the Pasadena game to the Altadena game, despite the latter turning out better in every state. Decision theorists have attempted to fill these silences by proposing various extensions to expected utility theory. As I show in this paper, such extensions often fall silent too, even in cases where the correct ranking is intuitively obvious. But we can extend (...) the theory further than has been done before---I offer a new extension, Invariant Value Theory, which deals neatly with those problem cases and also satisfies various desirable conditions. But other prima facie desirable conditions, including Independence, the theory violates. Is this a problem for the proposal? It may not be---in a new impossibility result, I show that no theory can satisfy Independence in full generality without violating several other conditions that together seem just as plausible. (shrink)

The Desire-as-Belief thesis (DAB) states that any rational person desires a proposition exactly to the degree that she believes or expects the proposition to be good. Many people take David Lewis to have shown the thesis to be inconsistent with Bayesian decision theory. However, as we show, Lewis's argument was based on an Invariance condition that itself is inconsistent with the (standard formulation of the) version of Bayesian decision theory that he assumed in his arguments against DAB. The aim of (...) this paper is to explore what impact the rejection of Invariance has on the DAB thesis. Without assuming Invariance, we first refute all versions of DAB that entail that there are only two levels of goodness. We next consider two theses according to which rational desires are intimately connected to expectations of (multi-levelled) goodness, and show that these are consistent with Bayesian decision theory as long as we assume that the contents of 'value propositions' are not fixed. We explain why this conclusion is independently plausible, and show how to construct such propositions. (shrink)

Some obligations are conditional such that act A is morally optional, but if one chooses A, one is required to do act B rather than some other less valuable act C. Such conditional obligations arise frequently in research ethics, in the philosophical literature, and in real life. They are controversial: how does a morally optional act give rise to demanding requirements to do the best? Some think that the fact that a putative obligation has a conditional structure, (...) so defined, is a strike against its being a genuine obligation. I argue that conditional obligations are to be expected in a moral theory that has moral options. (shrink)

Legitimate expectations should be considered in the transition to a low-carbon society. After explaining under what conditions and circumstances expectations are legitimate, this paper shows that those expectations whose frustration undermines the ability to plan, infringes basic moral rights, or is extremely costly for its bearer might justify a deviation in the baseline of justice in favour of the expectation holder. People should be notified about the likely frustration of their expectations so that they can avoid the frustration of (...) their expectations, adapt their life plans and minimise costs. Since the frustration of legitimate expectations seems unavoidable in the transformation to a low-carbon society, priority should be given to the protection of the expectations of those who cannot be materially compensated. Still, if two groups are eligible for material compensation, we should give priority to protecting the expectation of those who cannot continue with the same life plan as before if their expectations are frustrated. (shrink)

Standard theories of expected utility require that preferences are complete, and/or Archimedean. We present in this paper a theory of decision under uncertainty for both incomplete and non-Archimedean preferences. Without continuity assumptions, incomplete preferences on a lottery space reduce to an order-extension problem. It is well known that incomplete preferences can be extended to complete preferences in the full generality, but this result does not necessarily hold for incomplete preferences which satisfy the independence axiom, since it may obviously happen that (...) the extension does not satisfy the independence axiom. We show, for incomplete preferences on a mixture space, that an extension which satisfies the independence axiom exists. We find necessary and sufficient conditions for a preorder on a finite lottery space to be representable by a family of lexicographic von Neumann–Morgenstern Expected Utility functions. (shrink)

It is an unfortunate but unavoidable feature of even well-ordered democratic societies that governmental administrative agencies often create legitimate expectations (procedural or substantive) on the part of non-governmental agents (individual citizens, groups, businesses, organizations, institutions, and instrumentalities) but find themselves unable to fulfil those expectations for reasons of justice, the public interest, severe financial constraints, and sometimes harsh political realities. How governmental administrative agencies, operating on behalf of society, handle the creation and frustration of legitimate expectations implicates a whole host (...) of values that we have reason to care about, including under non-ideal conditions-not least justice, fairness, autonomy, the rule of law, responsible uses of power, credible commitments, reliance interests, security of expectations, stability, democracy, parliamentary supremacy, and legitimate authority. -/- This book develops a new theory of legitimate expectations for public administration drawing on normative arguments from political and legal theory. Brown begins by offering a new account of the legitimacy of legitimate expectations. He argues that it is the very responsibility of governmental administrative agencies for creating expectations that ought to ground legitimacy, as opposed to the justice or the legitimate authority of those agencies and expectations. He also clarifies some of the main ways in which agencies can be responsible for creating expectations. Moreover, he argues that governmental administrative agencies should be held liable for losses they directly cause by creating and then frustrating legitimate expectations on the part of non-governmental agents and, if liable, have an obligation to make adequate compensation payments in respect of those losses. (shrink)

This paper presents a challenge to enactivist approaches to cognition (e.g. Ward, D., Silverman, D. & Villalobos, M. 2017) that is based on the theoretical commitments behind forms of looking time studies that have been extensively used to probe into the cognitive abilities of infants and nonhuman animals. I briefly summarize the Violation of Expectation (VoE) paradigm (Ginnobili & Olmos 2021) to illustrate why such methods might pose a problem for enactivists and their conception of cognition as a largely (...) representation-free dynamic coupling between organism and environment. I argue that despite the lack of clarity regarding how the notion of expectation should be applied to the minds of neonates and nonhuman animals, there is an inherently representational aspect to expectation, given that it embodies satisfaction conditions. The challenge is, then: given that many forms of enactivism seem to reject the notion of representationas it is used in the VoE literature, how can enactivists make sense of data and results obtained using such research methods? (shrink)

Expectation of others’ cooperative behavior plays a core role in economic cooperation. However, the dynamic neural substrates of expectation of cooperation are little understood. To fully understand EOC behavior in more natural social interactions, the present study employed functional near-infrared spectroscopy hyperscanning to simultaneously measure pairs of participants’ brain activations in a modified prisoner’s dilemma game. The data analysis revealed the following results. Firstly, under the high incentive condition, team EOC behavior elicited higher interbrain synchrony in the right (...) inferior frontal gyrus than individual EOC behavior. Meanwhile, the IBS in the IFG could predict the relationship between empathy/agreeableness and EOC behavior, and this prediction role was modulated by social environmental cues. These results indicate the involvement of the human mirror neuron system in the EOC behavior and the different neural substrates between team EOC and individual EOC, which also conform with theory that social behavior was affected by internal and external factors. Secondly, female dyads exhibited a higher IBS value of cooperative expectation than male dyads in the team EOC than the individual EOC in the dorsal medial prefrontal cortex, while in the individual EOC stage, the coherence value of female dyads was significantly higher than that of male dyads under the low incentive reward condition in the rIFG. These sex effects thus provide presumptive evidence that females are more sensitive to environmental cues and also suggest that during economic social interaction, females’ EOC behavior depends on more social cognitive abilities. Overall, these results raise intriguing questions for future research on human cooperative behaviors. (shrink)