Infinite Previsions and Finitely Additive Expectations

Cambridge Journal of Economics, 2014

The well-known Knightian distinction between quantifiable risk and unquantifiable uncertainty is at odds with the dominant subjectivist conception of probability associated with de Finetti, Ramsey and Savage. Risk and uncertainty are rendered indistinguishable on the subjectivist approach insofar as an individual's subjective estimate of the probability of any event can be elicited from the odds at which she would be prepared to bet for or against that event. The risk/uncertainty distinction has however never quite gone away and is currently under renewed theoretical scrutiny. The purpose of this article is to show that de Finetti's understanding of the distinction is more nuanced than is usually admitted. Relying on usually overlooked excerpts of de Finetti's works commenting on Keynes, Knight and interval valued probabilities, we argue that de Finetti suggested a relevant theoretical case for uncertainty to hold even when individuals are endowed with subjective probabilities. Indeed, de Finetti admitted that the distinction between risk and uncertainty is relevant when different individuals sensibly disagree about the probability of the occurrence of an event. We conclude that the received interpretation of de Finetti's understanding of subjective probability needs to be qualified on this front.

… Probabilities and their Applications. Ithaca, NY, 2001

In this paper coherent risk measures and other currently used risk measures, notably Value-at-Risk (VaR), are studied from the perspective of the theory of coherent imprecise previsions. We introduce the notion of coherent risk measure defined on an arbitrary set of risks, showing that it can be considered a special case of coherent upper prevision. We also prove that our definition generalizes the notion of coherence for risk measures defined on a linear space of random numbers, given in literature. We also show that Value-at-Risk does not necessarily satisfy a weaker notion of coherence called ‘avoiding sure loss’ (ASL), and discuss both sufficient conditions for VaR to avoid sure loss and ways of modifying VaR into a coherent risk measure.

De Finetti introduced the concept of coherent previsions and conditional previsions through a gambling argument and through a parallel argument based on a quadratic scoring rule. He shows that the two arguments lead to the same concept of coherence. When dealing with events only, there is a rich class of scoring rules which might be used in place of the quadratic scoring rule. We give conditions under which a general strictly proper scoring rule can replace the quadratic scoring rule while preserving the equivalence of de Finetti's two arguments. In proving our results, we present a strengthening of the usual minimax theorem. We also present generalizations of de Finetti's fundamental theorem of prevision to deal with conditional previsions. (de Finetti, 1974, vol. 1) provides two criteria of coherence for previsions (or probabilistic forecasts) over a set of bounded random variables. The first criterion is formulated under the assumption that previsions serve as fair prices for buying and/or selling contracts of the form c(X − P (X)). Here, P (X) is the prevision of the bounded random variable X, and X is defined on the set Ω whose generic element is denoted ω. With c positive, the decision maker will pay the price cP (X) and receive cX(ω) units in state ω. With c negative, the decision maker sells this gamble on X and receives a payment of cP (X) units. With c = 0, the decision maker remains at his status-quo fortune. Furthermore, conditional previsions are also defined as fair prices for buying and/or selling contracts of the form cI B [X − P (X|B)]. Here, B is an event and I B is its indicator. The contract has 0 value if B does not occur. It is easy to see that, if B = Ω, P (X|B) has the same operational meaning as P (X). In this way, previsions are special cases of conditional previsions. We refer to conditional previsions for which B = Ω as marginal previsions when the need arises. When we refer to previsions unqualified, we mean both marginal and conditional previsions.

Lecture Notes in Computer Science, 2013

The investigation reported in this paper aims at clarifying an important yet subtle distinction between (i) the logical objects on which measure theoretic probability can be defined, and (ii) the interpretation of the resulting values as rational degrees of belief. Our central result can be stated informally as follows. Whilst all subjective degrees of belief can be expressed in terms of a probability measure, the converse doesn't hold: probability measures can be defined over linguistic objects which do not admit of a meaningful betting interpretation. The logical framework capable of expressing this will allow us to put forward a precise formalisation of de Finetti's notion of event which lies at the heart of the Bayesian approach to uncertain reasoning. 1 Introduction: the epistemic structure of de Finetti's betting interpretation De Finetti's theory of subjective probability is well-known and widely scrutinized in the literature (cf. [2]), so we will review only on those aspects which are directly relevant to our present purposes 1. Let θ 1 ,. .. , θ n be events of interest. De Finetti's betting problem is the choice that an idealised agent called bookmaker must make when publishing a book, i.e. when making an assignment B = {(θ i , β i) : i = 1,. .. , n} in which each event of interest θ i is given value β i ∈ [0, 1]. Once a book has been published, a gambler can place bets S i ∈ R on any event θ i by paying S i β i to the bookmaker. In return for this payment, the gambler will receive S i , if θ i obtains and nothing otherwise. Note that "betting on θ i " effectively amounts, for the gambler, to choosing a real-valued S i which determines the amount payable to the bookmaker 2 .

2011

An account of the first encounter, in 1938, of de Finetti with the notion of martingale is given. The reasons, of an actuarial nature, which led him to deal with the subject are explained, along with a description of the ensuing original contributions to the specific field of martingales. The value of some of his conclusions is, then, discussed in the light of later development of the theory. The note is completed by a few remarks about an interpretative problem concerning the connection between the risk aversion criterion and the (actuarial) criterion based on the riskiness level.

Social Science Research Network, 2000

Statistical Methods and Applications, 2014

Let L be a linear space of real random variables on the measurable space (Ω, A). Conditions for the existence of a probability P on A such that E P |X| < ∞ and E P (X) = 0 for all X ∈ L are provided. Such a P may be finitely additive or σ-additive, depending on the problem at hand, and may also be requested to satisfy P ∼ P 0 or P P 0 where P 0 is a reference measure. As a motivation, we note that a plenty of significant issues reduce to the existence of a probability P as above. Among them, we mention de Finetti's coherence principle, equivalent martingale measures, equivalent measures with given marginals, stationary and reversible Markov chains, and compatibility of conditional distributions.

International Journal of Approximate Reasoning, 2015

Betting methods, of which de Finetti's Dutch Book is by far the most well-known, are uncertainty modelling devices which accomplish a twofold aim. Whilst providing an (operational) interpretation of the relevant measure of uncertainty, they also provide a formal definition of coherence. The main purpose of this paper is to put forward a betting method for belief functions on MV-algebras of many-valued events which allows us to isolate the corresponding coherence criterion, which we term coherence in the aggregate. Our framework generalises the classical Dutch Book method.

2021

In utility theory, certain infinite gambles pose problems, of which the St. Petersberg paradox is the most well-known. Various methods have been used to eliminate the paradoxes, including the supposition that utility is bounded, or that probabilities must decline faster than utility so as to always result in finite expectation values, or simply excluding the divergent gambles by definition. We present a unified resolution of the matter that allows for unbounded utility and accommodates convergent infinite gambles. We also argue that merely excluding the divergent gambles, by whatever method, does not entirely avoid the associated paradoxes, which in practical terms already show up for finite approximations to them.

Journal of Mathematical Economics, 2009

When real-valued utilities for outcomes are bounded, or when all variables are simple, it is consistent with expected utility to have preferences defined over probability distributions or lotteries. That is, under such circumstances two variables with a common probability distribution over outcomes -equivalent variables -occupy the same place in a preference ordering. However, if strict preference respects uniform, strict dominance in outcomes between variables, and if indifference between two variables entails indifference between their difference and the status quo, then preferences over rich sets of unbounded variables, such as variables used in the St. Petersburg paradox, cannot preserve indifference between all pairs of equivalent variables. In such circumstances, preference is not a function only of probability and utility for outcomes. Then the preference ordering is not defined in terms of lotteries. Let T be independent of Y and let it have the uniform distribution on the interval [0,1]. Partition the interval [0,1] into the subintervals I 0 =[0, ], and I 1 ,…, I 2 m 1, where the last 2 m 1 intervals are all of equal length, 2 -m . Define the events B i ={T I i } for i=0,1,…,2 m 1 .

We prove the existence of an expected utility function for preferences over probabilistic prospects satisfying Strict Monotonicity, Indifference, the Common Ration Property, Substitution and Reducibility of Extreme Prospects. The example in Rubinstein (1988) that is inconsistent with the existence of a von Neumann-Morgenstern for preferences over probabilistic prospects, violates the Common Ratio Property. Subsequently, we prove the existence of expected utility functions with piecewise linear Bernoulli utility functions for preferences that are piece-wise linear. For this case a weaker version of the Indifference Assumption that is used in the earlier existence theorems is sufficient. We also state analogous results for probabilistic lotteries. We do not require any compound prospects or mixture spaces to prove any of our results. In the second last section of this paper, we “argue” that the observations related to Allais paradox, do not constitute a violation of expected utility maximization by individuals, but is a likely manifestation of individuals assigning (experiment or menu-dependent?) subjective probabilities to events which disagree with their objective probabilities.

Lecture Notes in Computer Science, 2013

We consider conditional random quantities (c.r.q.'s) in the setting of coherence. Given a numerical r.q. X and a non impossible event H, based on betting scheme we represent the c.r.q. X|H as the unconditional r.q. XH + µH c , where µ is the prevision assessed for X|H. We develop some elements for an algebra of c.r.q.'s, by giving a condition under which two c.r.q.'s X|H and Y |K coincide. We show that X|HK coincides with a suitable c.r.q. Y |K and we apply this representation to Bayesian updating of probabilities, by also deepening some aspects of Bayes' formula. Then, we introduce a notion of iterated c.r.q. (X|H)|K, by analyzing its relationship with X|HK. Our notion of iterated conditional cannot formalize Bayesian updating but has an economic rationale. Finally, we define the coherence for prevision assessments on iterated c.r.q.'s and we give an illustrative example.

Statistical Methods & Applications, 2014

Journal of Mathematical Psychology, 1988

This paper presents an axiomatization of subjective risk judgments that leads to a representation of risk in terms of seven free parameters. This is shown to have considerable predictive ability for risk judgments made by 10 subjects. The risk function retains many of the features of the expectation modeIs--e.g., a constant number of parameters independent of the number of outcomes-but it also allows for asymmetric effects of transformations on positive and negative outcomes. This arises by axiomatizing independently the behavior with respect to gambles having entirely positive outcomes and those with entirely negative outcomes. Complex gambles are decomposed into these, and zero outcomes, using the expected risk property. The resulting risk function compares favorable with other functions previously suggested. It is also demonstrated that preference judgments are distinct from risk judgments, and that the theory does not apply as well to the former.