Perplexing Expectations

Hájek, Alan; Nover, Harris

doi:10.1093/mind/fzl703

Cited by 17 publications

(9 citation statements)

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“…So the St. Petersburg gamble is worth more to Lin than any finite amount of money. (So far, Lin's reasoning has followed Hájek and Nover 2006, 4. )…”

Section: Don’t Cost a Thingmentioning

confidence: 99%

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Infinite Prospects*

Russell

Isaacs

2020

Philos Phenomenol Research

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Here's the plan. (Section 1) The St. Petersburg gamble cannot be assigned any fair value. (2) Standard arguments for orthodox expected utility theory, involving money pumps or regret, are also arguments against the rationality of preferences that give rise to the St. Petersburg gamble. (3) These arguments are doing something very different from familiar arguments for bounded utilities. (4) The structural requirement the arguments support is compatible with infinite utilities. (5) We describe a representation theorem, and explain how money pumps and regret are connected to the first point about fair values. (6) Some misgivings are expressed.

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“…So the St. Petersburg gamble is worth more to Lin than any finite amount of money. (So far, Lin's reasoning has followed Hájek and Nover 2006, 4. )…”

Section: Don’t Cost a Thingmentioning

confidence: 99%

“…But we’re not in the business of making theorists’ work technically easier—as will be evident. Our goal is to understand the normative limits on preference; this goal is not advanced by artificially truncating the space of possibilities (compare Hájek and Nover 2006, 708–10).…”

Section: Bounded Utilities and Infinite Utilitiesmentioning

confidence: 99%

Infinite Prospects*

Russell

Isaacs

2020

Philos Phenomenol Research

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“…5 See also Hájek and Nover (2006), Hájek andNover (2008), andHájek (2009). 6 See Baker 2007 for appeal to the temporal order in which payoffs are made.…”

Section: Proof Of Limit Agreementmentioning

confidence: 99%

Value Without Absolute Convergence

Lauwers

Vallentyne

2011

SSRN Journal

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We address how the value of risky options should be assessed in the case where the sum of the probability-weighted payoffs is not absolutely convergent and thus dependent on the order in which the terms are summed (e.g., as in the Pasadena Paradox).We develop and partially defend a proposal according to which options should be evaluated on the basis of agreement among admissible (e.g., convex and quasi-symmetric) covering sequences of the constituents of value (i.e., probabilities and payoffs).A finitely additive theory of (e.g., prudential or moral) value holds that, where there are only finitely many parts, the value of a whole is the sum of the value of its parts. We address the problem of how to extend this sum-principle when there are an infinite number of parts.Sometimes the sum of an infinite number of values is a well-defined finite number. For example, the values 1, -½, ¼, -1/8, 1/16, …, (-1/2) n , … add up to 2/3. Indeed, whatever the order with which the terms are added together, the resulting total is equal to 2/3. The series 1 -½ + ¼ -1/8 + 1/16 -…+ (-1/2) n +… is absolutely convergent, which means that it converges (i.e., has a finite limit) and the series of the absolute values of its terms also converges (1 + |-½| + ¼ + |-1/8| + 1/16 +…+ |(-1/2) n | +… = 2). For absolutely convergent series, rearranging the order of the terms does not change to resulting total, and thus the total of the terms is well-defined. Sometimes, however, the order does matter. For example,

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“…(And even if the intuitions were of equal strength, that still may not justify tilting exclusively for decision theory in its current form. It may only justify a pluralism of such theories, one privileging the Archimedean axiom, another privileging dominance-see Colyvan 2006 and the final section of Hájek and Nover 2006 for further discussion of pluralism. )…”

Section: Dominance Vs the Archimedean Axiommentioning

confidence: 99%

“… See Nover and Hájek (2004), andHájek and Nover (2006) for fuller presentations of the payoff and probability tables of both the Pasadena and St. Petersburg games, and further discussion of their expectations.…”

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confidence: 99%

Complex Expectations

Hájek¹,

Nover²

2008

Mind

Self Cite

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In our (2004), we introduced two games in the spirit of the St. Petersburg game, the Pasadena and Altadena games. As these latter games lack an expectation, we argued that they pose a paradox for decision theory. Terrence Fine has shown that any finite valuations for the Pasadena, Altadena, and St. Petersburg games are consistent with the standard decisiontheoretic axioms. In particular, one can value the Pasadena game above the other two, a result that conflicts with both our intuitions and dominance reasoning. We argue that this result, far from resolving the Pasadena paradox, should serve as a reductio of the standard theory, and we consequently make a plea for new axioms for a revised theory. We also discuss a proposal by Kenny Easwaran that a gamble should be valued according to its 'weak expectation', a generalization of the usual notion of expectation.

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Perplexing Expectations

Cited by 17 publications

References 0 publications

Infinite Prospects*

Infinite Prospects*

Value Without Absolute Convergence

Complex Expectations

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