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Surreal Decisions
Philosophy and Phenomenological Research 100 (1):5474 (2020)
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Games such as the St. Petersburg game present serious problems for decision theory.1 The St. Petersburg game invokes an unbounded utility function to produce an infinite expectation for playing the game. The problem is usually presented as a clash between decision theory and intuition: most people are not prepared to pay a large finite sum to buy into this game, yet this is precisely what decision theory suggests we ought to do. But there is another problem associated with the St. (...) 

This paper is a response to Paul Bartha’s ‘Making Do Without Expectations’. We provide an assessment of the strengths and limitations of two notable extensions of standard decision theory: relative expectation theory and Paul Bartha’s relative utility theory. These extensions are designed to provide intuitive answers to some wellknown problems in decision theory involving gaps in expectations. We argue that both RET and RUT go some way towards providing solutions to the problems in question but neither extension solves all the (...) 

A decade ago, Harris Nover and I introduced the Pasadena game, which we argued gives rise to a new paradox in decision theory even more troubling than the St Petersburg paradox. Gwiazda's and Smith's articles in this volume both offer revisionist solutions. I critically engage with both articles. They invite reflections on a number of deep issues in the foundations of decision theory, which I hope to bring out. These issues include: some ways in which orthodox decision theory might be (...) 

Representation theorems are often taken to provide the foundations for decision theory. First, they are taken to characterize degrees of belief and utilities. Second, they are taken to justify two fundamental rules of rationality: that we should have probabilistic degrees of belief and that we should act as expected utility maximizers. We argue that representation theorems cannot serve either of these foundational purposes, and that recent attempts to defend the foundational importance of representation theorems are unsuccessful. As a result, we (...) 

Lara Buchak sets out a new account of rational decisionmaking in the face of risk. She argues that the orthodox view is too narrow, and suggests an alternative, more permissive theory: one that allows individuals to pay attention to the worstcase or bestcase scenario, and vindicates the ordinary decisionmaker. 

This paper revisits the Pasadena game (Nover and Háyek 2004), a St Petersburglike game whose expectation is undefined. We discuss serveral respects in which the Pasadena game is even more troublesome for decision theory than the St Petersburg game. Colyvan (2006) argues that the decision problem of whether or not to play the Pasadena game is ‘illposed’. He goes on to advocate a ‘pluralism’ regarding decision rules, which embraces dominance reasoning as well as maximizing expected utility. We rebut Colyvan’s argument, (...) 

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 (...) 

The Pasadena paradox presents a serious challenge for decision theory. The paradox arises from a game that has welldefined probabilities and utilities for each outcome, yet, apparently, does not have a welldefined expectation. In this paper, I argue that this paradox highlights a limitation of standard decision theory. This limitation can be (largely) overcome by embracing dominance reasoning and, in particular, by recognising that dominance reasoning can deliver the correct results in situations where standard decision theory fails. This, in turn, (...) 

NonArchimedean probability functions allow us to combine regularity with perfect additivity. We discuss the philosophical motivation for a particular choice of axioms for a nonArchimedean probability theory and answer some philosophical objections that have been raised against infinitesimal probabilities in general. _1_ Introduction _2_ The Limits of Classical Probability Theory _2.1_ Classical probability functions _2.2_ Limitations _2.3_ Infinitesimals to the rescue? _3_ NAP Theory _3.1_ First four axioms of NAP _3.2_ Continuity and conditional probability _3.3_ The final axiom of NAP (...) 

This article discusses how the concept of a fair finite lottery can best be extended to denumerably infinite lotteries. Techniques and ideas from nonstandard analysis are brought to bear on the problem. 



We examine the notions of negative, infinite and hotter than infinite temperatures and show how these unusual concepts gain legitimacy in quantum statistical mechanics. We ask if the existence of an infinite temperature implies the existence of an actual infinity and argue that it does not. Since one can sensibly talk about hotter than infinite temperatures, we ask if one could legitimately speak of other physical quantities, such as length and duration, in analogous terms. That is, could there be longer (...) 

Probabilism is committed to two theses: 1) Opinion comes in degrees—call them degrees of belief, or credences. 2) The degrees of belief of a rational agent obey the probability calculus. Correspondingly, a natural way to argue for probabilism is: i) to give an account of what degrees of belief are, and then ii) to show that those things should be probabilities, on pain of irrationality. Most of the action in the literature concerns stage ii). Assuming that stage i) has been (...) 

We propose an alternative approach to probability theory closely related to the framework of numerosity theory: nonArchimedean probability (NAP). In our approach, unlike in classical probability theory, all subsets of an infinite sample space are measurable and only the empty set gets assigned probability zero (in other words: the probability functions are regular). We use a nonArchimedean field as the range of the probability function. As a result, the property of countable additivity in Kolmogorov’s axiomatization of probability is replaced by (...) 

Among recent objections to Pascal's Wager, two are especially compelling. The first is that decision theory, and specifically the requirement of maximizing expected utility, is incompatible with infinite utility values. The second is that even if infinite utility values are admitted, the argument of the Wager is invalid provided that we allow mixed strategies. Furthermore, Hájek has shown that reformulations of Pascal's Wager that address these criticisms inevitably lead to arguments that are philosophically unsatisfying and historically unfaithful. Both the objections (...) 



A. Pascal's statement of his wager argument is couched in terms of the theory of probability and the theory of games, and the exposition is unclear and unnecessarily complicated. The following is a ‘creative’ reformulation of the argument designed to avoid some of the objections which have been or might be raised against the original. 

Only one traditional objection to Pascal's wager is telling: Pascal assumes a particular theology, but without justification. We produce two new objections that go deeper. We show that even if Pascal's theology is assumed to be probable, Pascal's argument does not go through. In addition, we describe a wager that Pascal never considered, which leads away from Pascal's conclusion. We then consider the impact of these considerations on other prudential arguments concerning what one should believe, and on the more general (...) 

In his monograph On Numbers and Games, J. H. Conway introduced a realclosed field containing the reals and the ordinals as well as a great many less familiar numbers including $\omega, \,\omega/2, \,1/\omega, \sqrt{\omega}$ and $\omega\pi$ to name only a few. Indeed, this particular realclosed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG, it may be said to contain (...) 

By recourse to the fundamentals of preference orderings and their numerical representations through linear utility, we address certain questions raised in Nover and Hájek 2004, Hájek and Nover 2006, and Colyvan 2006. In brief, the Pasadena and Altadena games are welldefined and can be assigned any finite utility values while remaining consistent with preferences between those games having welldefined finite expected value. This is also true for the St Petersburg game. Furthermore, the dominance claimed for the Altadena game over the (...) 

Many philosophers have argued that "degree of belief" or "credence" is a more fundamental state grounding belief. Many other philosophers have been skeptical about the notion of "degree of belief", and take belief to be the only meaningful notion in the vicinity. This paper shows that one can take belief to be fundamental, and ground a notion of "degree of belief" in the patterns of belief, assuming that an agent has a collection of beliefs that isn't dominated by some other (...) 

The Pasadena game invented by Nover and Hájek raises a number of challenges for decision theory. The basic problem is how the game should be evaluated: it has no expectation and hence no welldefined value. Easwaran has shown that the Pasadena game does have a weak expectation, raising the possibility that we can eliminate the value gap by requiring agents to value gambles at their weak expectations. In this paper, I first prove a negative result: there are gambles like the (...) 

We introduce a St. Petersburglike game, which we call the ‘Pasadena game’, in which we toss a coin until it lands heads for the first time. Your payoffs grow without bound, and alternate in sign (rewards alternate with penalties). The expectation of the game is a conditionally convergent series. As such, its terms can be rearranged to yield any sum whatsoever, including positive infinity and negative infinity. Thus, we can apparently make the game seem as desirable or undesirable as we (...) 









A. Pascal's statement of his wager argument is couched in terms of the theory of probability and the theory of games, and the exposition is unclear and unnecessarily complicated. The following is a ‘creative’ reformulation of the argument designed to avoid some of the objections which have been or might be raised against the original. 