Newcomb's Problem Research Articles

C270. Newcomb's problem and the foundations of probability

C270. Newcomb's problem and the foundations of probability

Against causal decision theory

Proponents of causal decision theories argue that classical Bayesian decision theory (BDT) gives the wrong advice in certain types of cases, of which the clearest and commonest are the medical Newcomb problems. I defend BDT, invoking a familiar principle of statistical inference to show that in such cases a free agent cannot take the contemplated action to be probabilistically relevant to its causes (so that BDT gives the right answer). I argue that my defence does better than those of Ellery Eells and Richard Jeffrey; and that it applies, where necessary, to other types of Newcomb problem.

Decision Theory in Light of Newcomb's Problem

Should we act only for the sake of what we might bring about (causal decision theory); or is it enough for a decent motive that our action is highly correlated with something desirable (evidential decision theory)? The conflict between these points of view is embodied in Newcomb's problem. It is argued here that intuitive evidence from familiar decision contexts does not enable us to settle the issue, since the two theories dictate the same results in normal circumstances. Nevertheless, there are several reasons to reject the causal approach: (1) its relative complexity; (2) its commitment to the existence of situations in which every possible act would be irrational; (3) its incorporation of an arbitrary time bias; and (4) its implicit distinction between what ought to be done and what ought to be hoped for.

Newcomb's Perfect Predictor

Since Robert Nozick (1970) first published Newcomb's problem, much has been written of it. Most of the literature has dealt primarily with the case in which the predictor is assumed to be highly reliable but not infallible. As the problem is typically formulated, traditional evidential decision theories, such as Jeffrey's (1965) recommend taking one box. Causal decision theories, such as those developed by Gibbard and Harper (1981), Skyrms (1980, 1982), Lewis (1981), and Sobel (1977) favor taking both boxes. Recently, refined evidential decision theories have been offered by Eells (1981) and Jeffrey (1981) which are intended to prescribe a two-box solution. We think it clear that one should take two boxes and prefer the causal decision theorists' justification of this choice. Our concern here is with the case in which the predictor is not merely reliable, but perfect. In this case, causal decision theories are less satisfactory, for they continue to recommend choosing both boxes, and this recommendation is not so intuitive.' The infallibility of the predictor changes intuitions, because it seems that taking one box guarantees that you become a millionaire, while taking both guarantees that you do not. Gibbard and Harper claim that this appearance is based on mistaking truth-functional conditionals for counterfactuals. We shall argue, though, that the counterfactuals leading to the one-box solution have independent support. As a result, the argument for taking one box is cogent in the infallible case. In some formulations of the puzzle, so is the argument for taking two. In these cases, the problem lies not in either of these competing arguments but in the puzzle itself. Perfect predictor versions of Newcomb's problem can be interpreted in various ways, but on plausible interpretations the prob-

The iterated versions of Newcomb's problem and the prisoner's dilemma

The iterated versions of Newcomb's problem and the prisoner's dilemma

The value of information in Newcomb's Problem and the Prisoners' Dilemma

The acts in Newcomb's Problem and the Prisoner's Dilemma are viewed as experiments. The cost of the information yielded by each act-experiment is compared to the value of information in the two problems, which is zero. The non-dominant act-experiments cost more than their information is worth, and are therefore rejected.

Newcomb's many solutions

After a brief presentation and discussion of two versions of Newcomb's problem, I examine the analyses proposed by Levi (1975). Horgan (1981), and Kyburg (1980). I argue that the first two are not genuine solutions to the problem, but that the third, if appropriately elaborated and modified, is correct.

Newcomb's problem: Recalculations for the one-boxer

Newcomb's problem: Recalculations for the one-boxer

Rational Expectations of Government Policy: An Application of Newcomb's Problem

Muth advanced rational expectations hypothesis that agents predict value of endogenous variables on basis of economic of phenomena in question. As long as there is a unique relevant theory, no ambiguity exists as to what is a rational expectation. But what happens when there is more than one relevant economic theory? In this case, unless we know which theory an agent believes, the rational expectation is not defined. This ambiguity takes on increased importance in debate over policy-effectiveness. Sargent and Wallace [10] have argued that if public holds rational expectations concerning monetary authority's actions, systematic monetary policy will be unable to affect value of any real variables in economy. In this paper we demonstrate by presenting a recently discovered decision-theoretic problem called Newcomb's problem [2] that if a decision maker (e.g., monetary authority) must make one of several decisions in such a way that its payoff is affected by whether or not its action was anticipated by another rational agent with good predictive powers, then one uniquely best rational course of action or monetary policy may not exist.

`Why Ain'cha Rich?'

Some think that in (a suitable version of) Newcomb's problem, it is rational to take only one box. These one-boxers think of the situation as a choice between a million and a thousand. They are convinced by indicative conditionals: if I take one box I will be a millionaire, but if I take both boxes I will not. Their conception of rationality may be called V-rationality; they deem it rational to maximize V, that being a kind of expected utility defined in entirely non-causal terms. Their decision ,theory is that of Jeffrey [2]. Others, and I for one, think it rational to take both boxes. We two-boxers think that whether the million already awaits us or not, we have no choice between taking it and leaving it. We are convinced by counterfactual conditionals: If I took only one box, I would be poorer by a thousand than I will be after taking both. (We distinguish nornal from back-tracking counterfactuals, perhaps as in [4], and are persuaded only by the former.) Our conception of rationality is U-rationality; we favor maximizing U, a kind of expected utility defined in terms of causal dependence as well as credence and value. Our decision theory is that of Gibbard and Harper [ 1 ], or something similar. The one-boxers sometimes taunt us: if you're so smart, why ain'cha rich? They have their millions and we have our thousands, and they think this goes to show the error of our ways. They think we are not rich because we have irrationally chosen not to have our millions. We reply that we never were given any choice about whether to have a million. When we made our choices, there were no millions to be had. The reason why we are not rich is that the riches were reserved for the irrational. In the words of Gibbard and Harper [1],

Causal decision theory

Abstract Newcomb's problem and similar cases show the need to incorporate causal distinctions into the theory of rational decision; the usual noncausal decision theory, though simpler, does not always give the right answers. I give my own version of causal decision theory, compare it with versions offered by several other authors, and suggest that the versions have more in common than meets the eye.

Locke on Causation, Compatibilism and Newcomb's Problem

Journal Article Locke on causation, compatibilism and Newcomb's problem Get access Andrè Gallois Andrè Gallois University of Queensland Search for other works by this author on: Oxford Academic Google Scholar Analysis, Volume 41, Issue 1, January 1981, Pages 42–46, https://doi.org/10.1093/analys/41.1.42 Published: 01 January 1981

Locke on causation, compatibilism and Newcomb's problem

Locke on causation, compatibilism and Newcomb's problem

Eulerian numbers, Newcomb's problem and representations of symmetric groups

Consider a set of n positive integers consisting of μ 1 1's, μ 2 2's,…, μ r r's. If the integer in the ith place in an arrangement σ of this set is σ( i), and a non-rise in σ is defined as σ( i+1)⩽ σ( i), a problem that suggests itself is the determination of the number of arrangements σ with k non-rises. When each μ i is unity, the problem is that of finding the number A( n, k) of permutations of distinct integers 1, 2,…, n with k descents, a descent being defined as σ( i+1)< σ( i). The number A( n, k) is known as an Eulerian number. The problem of finding the number of arrangements with k non-rises of the more general set, when not all of μ i are unity, has appeared in the literature as one part of a problem on dealing a pack of cards, this having been proposed by the American astronomer Simon Newcomb (1835–1909). Both the Eulerian numbers and Newcomb's problem have accumulated a substantial literature. The present paper considers these topics from an entirely new stand-point, that of representations of the symmetric group. This approach yields a well-known recurrence for the Eulerian numbers and a known formula for them in terms of Stirling numbers. It also gives the solutions of the Newcomb problem and some recurrences between these solutions, not all of which have been found earlier. A simple connection is found between Stirling numbers and the Kostka numbers of symmetric group representation theory. The Eulerian numbers can also be expressed in terms of the Kostka numbers. The idea which is novel in this treatment and recurs almost as a motif throughout the paper is that of a skew-hook. This occurs in the first place in a very natural way as a picture of the rises and non-rises of σ, with the nodes of the skew-hook labelled successively as σ(1), σ(2),…. As the paper develops, a new form of skew-hook associated with σ emerges. This does not in general depict the rises and non-rises of σ, and it is now the edges, not the nodes, which carry integer labels. A new type of combinatorial number, here called a ψ-function, arises from these edge-labelled skew-hooks. The ψ-functions are intimately related to the Eulerian numbers and the Newcomb solutions and may have further combinatorial applications. The skew-hook treatment casts fresh light on MacMahon's solution of the Newcomb problem and on his “new symmetric functions”, and, if σ( i)− σ( i+1)⩾ s defines an s-descent in σ, on the enumeration of permutations with k s-descents. Also some characters of the symmetric group with interesting properties and recurrences arise in the course of the paper.

Partial difference equations in m1⩾ m2⩾⋯⩾ mn⩾0 and their applications to combinatorics

Various discrete functions encountered in Combinatorics are solutions of Partial Difference Equations in the subset of N n given by m 1⩾ m 2⩾⋯⩾ m n ⩾0. Given a partial difference equation, it is described how to pass from the standard “easy” solution of an equation in N n to a solution of the same equation subject to certain “Dirichlet” or “Neumann” boundary conditions in the domain m 1⩾ m 2⩾⋯⩾ m n ⩾0 and related domains. Applications include a rather quick derivation of MacMahon's generating function for plane partitions, a generalization and q-analog of the Ballot problem, and a joint analog of the Ballot problem and Simon Newcomb's problem.

Foreknowledge: Nelson Pike and Newcomb's problem

The problem of foreknowledge and freedom presents a challenge to the defender of traditional Western theism. Nelson Pike has argued that the existence of an essentially omniscient God who possesses foreknowledge is incompatible with human freedom. Pike's opponents in this matter, among whom is Alvin Plantinga, argue that no incompatibility has yet been shown. I shall develop the view that neither Pike nor his opponents have conclusively settled the question whether foreknowledge and freedom are compatible. Furthermore there is a reason why the issue has not been, and may never be, conclusively settled: it is an inherently paradoxical issue. To aid in demonstrating this point I shall discuss a modified version of a paradox that has come to be known as Newcomb's Problem.

Causation, Compatibilism and Newcomb's Problem

Causation, Compatibilism and Newcomb's Problem

Causation, compatibilism and newcomb's problem

Causation, compatibilism and newcomb's problem

Schlesinger on the Newcomb problem

Schlesinger on the Newcomb problem

Newcomb's Problem and Prisoners' Dilemma

The relationship between Newcomb's problem, which involves an apparent paradox of prediction, and Prisoners' Dilemma is explicated. After describing a resolution to Newcomb's problem, due to John A. Ferejohn, that renders the two contradictory principles of choice in Newcomb's problem (dominance and expected utility) consistent, I show Prisoners' Dilemma to be a “symmetricized” version of Newcomb's problem in its payoff structure. The assumption about predictability of choice made for one player in Newcomb's problem, when applied to both players in Prisoners' Dilemma–one considered as a leader and the other as a follower–offers a resolution to this dilemma that, while consistent with the resolution offered by metagame theory, rationalizes the existence of a metagame solution within a probabilistic, rational-choice framework. The relevance of the mutual-predictability assumption to the solution of arms races, and tragedy-of-commons situations generally, is discussed.