A mixture preorder is a preorder on a mixture space (such as a convex set) that is compatible with the mixing operation. In decision theoretic terms, it satisfies the central expected utility axiom of strong independence. We consider when a mixture preorder has a multi-representation that consists of real valued, mixture-preserving functions. If it does, it must satisfy the mixture continuity axiom of Herstein and Milnor (1953). Mixture continuity is sufficient for a mixture-preserving multi-representation when the dimension of the mixture space is countable, but not when it is uncountable. Our strongest positive result is that mixture continuity is sufficient in conjunction with a novel axiom we call countable domination, which constrains the order complexity of the mixture preorder in terms of its Archimedean structure. We also consider what happens when the mixture space is given its natural weak topology. Continuity (having closed upper and lower sets) and closedness (having a closed graph) are stronger than mixture continuity. We show that continuity is necessary but not sufficient for a mixture preorder to have a mixture-preserving multi-representation. Closedness is also necessary; we leave it as an open question whether it is sufficient. We end with results concerning the existence of mixture-preserving multi-representations that consist entirely of strictly increasing functions, and a uniqueness result.

We give two social aggregation theorems under conditions of risk, one for constant population cases, the other an extension to variable populations. Intra and interpersonal welfare comparisons are encoded in a single ‘individual preorder’. The theorems give axioms that uniquely determine a social preorder in terms of this individual preorder. The social preorders described by these theorems have features that may be considered characteristic of Harsanyi-style utilitarianism, such as indifference to ex ante and ex post equality. However, the theorems are also consistent with the rejection of all of the expected utility axioms, completeness, continuity, and independence, at both the individual and social levels. In that sense, expected utility is inessential to Harsanyi-style utilitarianism. In fact, the variable population theorem imposes only a mild constraint on the individual preorder, while the constant population theorem imposes no constraint at all. We then derive further results under the assumption of our basic axioms. First, the individual preorder satisfies the main expected utility axiom of strong independence if and only if the social preorder has a vector-valued expected total utility representation, covering Harsanyi’s utilitarian theorem as a special case. Second, stronger utilitarian-friendly assumptions, like Pareto or strong separability, are essentially equivalent to strong independence. Third, if the individual preorder satisfies a ‘local expected utility’ condition popular in non-expected utility theory, then the social preorder has a ‘local expected total utility’ representation. Fourth, a wide range of non-expected utility theories nevertheless lead to social preorders of outcomes that have been seen as canonically egalitarian, such as rank-dependent social preorders. Although our aggregation theorems are stated under conditions of risk, they are valid in more general frameworks for representing uncertainty or ambiguity.

We show that a strongly independent preorder on a possibly infinite dimensional convex set that satisfies two of the following conditions must satisfy the third: (i) the Archimedean continuity condition; (ii) mixture continuity; and (iii) comparability under the preorder is an equivalence relation. In addition, if the preorder is nontrivial (has nonempty asymmetric part) and satisfies two of the following conditions, it must satisfy the third: (i′) a modest strengthening of the Archimedean condition; (ii) mixture continuity; and (iii′) completeness. Applications to decision making under conditions of risk and uncertainty are provided, illustrating the relevance of infinite dimensionality.

According to the priority view, or prioritarianism, it matters more to benefit people the worse off they are. But how exactly should the priority view be defined? This article argues for a highly general characterization which essentially involves risk, but makes no use of evaluative measurements or the expected utility axioms. A representation theorem is provided, and when further assumptions are added, common accounts of the priority view are recovered. A defence of the key idea behind the priority view, the priority principle, is provided. But it is argued that the priority view fails on both ethical and conceptual grounds.

Egalitarians think that equality in the distribution of goods somehow matters. But what exactly is egalitarianism? This article argues for a characterization based on novel principles essentially involving risk. The characterization is then used to resolve disputed questions about egalitarianism. These include: the way egalitarianism is concerned with patterns, in particular its relationship to strong separability; the relationship between egalitarianism and other distributive views, such as concerns with fairness and with giving priority to the worse off; and the relationship between egalitarianism and evaluative measurement. But egalitarianism is subject to a particularly severe form of the levelling-down objection, and is claimed to be false.

Parfit advertised the priority view as a new and fundamental theory in the ethics of distribution. He never discusses risk, and many writers follow suit when discussing the priority view. This article formalizes two popular arguments for a commonly accepted risk-free definition of the priority view. One is based on a direct attempt to define the priority view, the other is based on a contrast with utilitarianism and egalitarianism. But it argues that neither argument succeeds, and more generally, that it is not possible to make sense of the priority view in a risk-free framework. As a diagnosis, the article suggests that the literature on the priority view has paid insufficient attention to axiomatization and has most likely mistaken the proper role of the use of evaluative measurements in theorizing about the ethics of distribution. Reasons are eventually offered for thinking that despite appearances, approaching the priority view in terms of risk from the outset is quite natural.

The priority view has become very popular in moral philosophy, but there is a serious question about how it should be formalized. The most natural formalization leads to ex post prioritarianism, which results from adding expected utility theory to the main ideas of the priority view. But ex post prioritarianism entails a claim which is too implausible for it to be a serious competitor to utilitarianism. In fact, ex post prioritarianism was probably never a genuine alternative to utilitarianism in the first place. By contrast, ex ante prioritarianism is defensible. But its motivation is very different from the usual rationales offered for the priority view. Given the untenability of ex post prioritarianism, it is more natural for most friends of the priority view to revert to utilitarianism.

One major strand of ethical theorizing proceeds by supposing that there exist what I will call individual goodness measures. These are functions from lotteries over entire world histories to the real numbers which somehow purport to measure how good such lotteries are for individuals. Utilitarianism plainly presupposes that such measures exist when it says that one lottery is better than another just in case it contains a greater sum of individual goodness. And so do various well known ways of departing from utilitarianism.

Utilitarianism and prioritarianism make a strong assumption about measures of how good lotteries over histories are for individuals, or for short, individual goodness measures. Given some idealizing assumptions about interpersonal and intrapersonal comparisons, they presuppose that any individual goodness measure can be transformed into any other individual goodness measure by a positive affine transformation. But it is far from obvious that the presupposition is correct, so both theories face the threat of presupposition failure. The usual response to this problem starts by assuming that what implicitly determines the set of individual goodness measures is independent of our discourse about utilitarianism and prioritarianism. I suggest reversing this response. What determines the set of individual goodness measures just is the body of platitudes we accept about utilitarianism and prioritarianism. This approach vindicates the utilitarian and prioritarian presupposition. As a corollary, it shows that individual goodness measures are expectational, and provides an answer to an argument due to Broome that for different reasons to do with measurement, prioritarianism is more or less meaningless.

Theoretical defenses of the principle of double effect (pde) due to Quinn, Nagel and Foot are claimed to face severe difficulties. But this leaves those of us who see something in the case-based support for the pde without a way of accounting for our judgments. This article proposes a novel principle it calls the mismatch principle, and argues that the mismatch principle does better than the pde at accounting for our judgments about cases and is also theoretically defensible. However, where the pde makes claims about the permissibility of actions, the mismatch principle makes claims only about the evaluation of agents; and where the pde explains the cases in terms of intending harm, the mismatch principle explains them in terms of a quite different feature of the agent's will.

The article takes as its starting point the assumption that (a) competing accounts of moral rules should be judged by the distribution of benefits and burdens which would arise from everyone accepting these rules, and that (b) these benefits and burdens are understood in a way which has a substantial resource or freedom-based component. This starting point is compatible with contractualism and various forms of rule consequentialism, and will yield a morality in which people have significant freedoms. The main claim of the article is that as a consequence of these freedoms, and because of phenomena connected with cost internalization, this morality will also impose strong constraints on harming and weak constraints on allowing harm. Thus the starting point ends up vindicating commonsense morality.

On the agent-relativity thesis, what an agent ought to do is a function of the evidence available to her about the consequences of her potential actions. On the objectivity thesis, what an agent ought to do is a function of what the consequences of her potential actions would be, regardless of the evidence available to her. This article argues for the agent-relativity thesis. The main opposing argument, due to Thomson, points to cases where a bystander can see that an agent is about to do something which, unknown to the agent, would have terrible consequences, and says to the agent: "You ought not to do that!" The bystander's utterance seems true, but it is argued that this is consistent with the agent-relativity thesis, which also enjoys support from other directions.

Moral puzzles about actions which bring about very small or what are said to be imperceptible harms or benefits for each of a large number of people are well known. Less well known is an argument by Warren Quinn that standard theories of rationality can lead an agent to end up torturing himself or herself in a completely foreseeable way, and that this shows that standard theories of rationality need to be revised. We show where Quinn’s argument goes wrong, and apply this to the moral puzzles

The two envelope paradox can be dissolved by looking closely at the connection between conditional and unconditional expectation and by being careful when summing an infinite series of positive and negative terms. The two envelope paradox is not another St. Petersburg paradox and one does not need to ban talk of infinite expectation values in order to dissolve it. The article ends by posing a new puzzle to do with infinite expectations.

Theories of rights seem well equipped to explain widely accepted claims about the morality of harming. But can they explain popular claims about the morality of imposing risks of harm? Many think not. But a plausible theory of rights can explain those claims if it says we have the right that others not impose risks of harm upon us. That is a good reason to believe we have that right. There are many objections to the claim that we have that right, but none of them are sound. Among the topics covered are: the permissibility of rights infringements, trivial rights infringements, the permissibility of risk impositions, the aggregation of rights, consent, self-defense, punishment, and compensation.

Standard theories of liability say that X is liable to Y only if Y was harmed, only if X caused Y harm, and (usually) only if X was at fault. This article offers a series of criticisms of each of these claims, and use them to construct an alternative theory of liability in which the nature of X's having imposed a risk of harm on Y is central to the question of when X is liable to Y, and for how much. The article ends with some conjectures on ignorance as an excusing condition.

The article is a plea for ethicists to regard probability as one of their most important concerns. It outlines a series of topics of central importance in ethical theory in which probability is implicated, often in a surprisingly deep way, and lists a number of open problems. Topics covered include: interpretations of probability in ethical contexts; the evaluative and normative significance of risk or uncertainty; uses and abuses of expected utility theory; veils of ignorance; Harsanyi's aggregation theorem; population size problems; equality; fairness; giving priority to the worse off; continuity; incommensurability; nonexpected utility theory; evaluative measurement; aggregation; causal and evidential decision theory; act consequentialism; rule consequentialism; and deontology.

We present an abstract social aggregation theorem. Society, and each individual, has a preorder that may be interpreted as expressing values or beliefs. The preorders are allowed to violate both completeness and continuity, and the population is allowed to be infinite. The preorders are only assumed to be represented by functions with values in partially ordered vector spaces, and whose product has convex range. This includes all preorders that satisfy strong independence. Any Pareto indifferent social preorder is then shown to be represented by a linear transformation of the representations of the individual preorders. Further Pareto conditions on the social preorder correspond to positivity conditions on the transformation. When all the Pareto conditions hold and the population is finite, the social preorder is represented by a sum of individual preorder representations. We provide two applications. The first yields an extremely general version of Harsanyi’s social aggregation theorem. The second generalizes a classic result about linear opinion pooling.

We show that without assuming completeness or continuity, a strongly independent preorder on a possibly infinite dimensional convex set can always be given a vector-valued representation that naturally generalizes the standard expected utility representation. More precisely, it can be represented by a mixture-preserving function to a product of lexicographic function spaces.