Conclusion

Recognizing that an act’s risk is a consequence of the act yields a version of expected-utility maximization that does not need adjustments for risk in addition to the probabilities and utilities of possible outcomes. This treatment of an act’s risk justifies the expected-utility principle, and the mean-risk principle, for evaluation of an act. Rational attitudes to risks explain the rationality of acting in accord with the principles. They ground the separability relations that support the principles. The expected-utility principle justifies a substantive, and not just a representational, version of the decision principle of expected-utility maximization. Consequently, the principle governs a single choice and not just sets of choices. It demands more than consistency of the choices in a set. It demands that each choice follow the agent’s preferences, and these preferences explain the rationality of a choice that complies with the principle.

Regulation of Risks

Governments regulate risks on behalf of the people they serve. Given that regulatory agencies aim for regulatory measures that the public would endorse if rational and informed, the mean-risk method of evaluating acts provides valuable guidance. It offers a way of constructing for a citizen informed probability and utility assignments for a regulation’s possible outcomes, and using these assignments to obtain for the citizen an informed utility assignment for the regulation. The theory of cooperative games combines the utility assignments of multiple agents to support a collective act, and under simplifying assumptions, supports an act that maximizes collective utility, defined as a sum of the act’s utilities for the agents, in the tradition of utilitarianism. This approach to regulation accommodates acts targeting information-sensitive, evidential risks as well as acts targeting physical risks. Verification of a reduction in an evidential risk can meet the standards of objectivity that the law adopts.

Return-Risk Evaluation of Investments

In finance, a common way of evaluating an investment uses the investment’s expected return and the investment’s risk, in the sense of the investment’s volatility, or exposure to chance. A version of this method derives from a general mean-risk evaluation of acts, under the assumption that only money, risk, and their sources matter. Although the method does not require a measure of risk, finance investigates measures of risks to assist evaluations of risks. An investment creates possible returns, and the variance of the probability distribution of their utilities is a measure of the investment’s risk. This measure neglects some factors affecting an investment’s risk, and so is satisfactory only in special cases. Another measure of risk is known as value-at-risk, or VAR. It also neglects some factors affecting an investment’s risk, and so should be restricted to special cases.

Rolling Back Idealizations

Common principles of rationality adopt idealizations that a normative model makes explicit. A standard method of generalizing the model is to relax an idealization and revise principles of rationality to accommodate the change. Three idealizations to relax are: (1) cost-free reflection (2) certainty of the contents of propositions entertained, and (3) utility assignments that are stable throughout a choice. This chapter sketches ways of relaxing these idealizations. It shows how deliberations using beliefs and desires may approximate results of deliberations using probabilities and utilities. It generalizes probability and utility assignments for uncertainty about the content of a proposition by relativizing the assignments to a way of understanding the proposition. Finally, it presents principles of choice for decision problems with unstable utility assignments to acts.

Rational Management of Risks

The literature on expected utility formulates in two ways the principle to maximize expected utility. One version of the principle requires choices that literally maximize expected utility. The other version requires choices that are “as if” maximizing expected utility. The first principle is substantive and applies to a single choice. The second principle is representational and applies to a set of choices; it requires only a type of consistency among choices and is weaker than the first, substantive principle. The justification of the substantive version of the principle is that following it amounts to following preferences among the options in a decision problem. The justification takes an option’s risk as a consequence of the option’s realization. Making this move yields a simple decision principle, using evaluations of options that do not add weights to the probabilities or the utilities of an option’s possible outcomes.

Attitudes

Probabilities and utilities of possible outcomes yield the expected utilities of the acts an agent considers in a decision problem. This chapter introduces probability and utility as the book’s decision principles understand these functions. It has them attach to propositions that declarative sentences express, and it takes their values to represent the strengths of attitudes—strengths of doxastic attitudes in the case of probabilities and strengths of conative attitudes in the case of utilities. Desires and aversions, typical conative attitudes, may have narrow or wide evaluative scope. Intrinsic desires have narrow scope, and extrinsic desires have wide scope. Utility assignments may, correspondingly, have narrow or wide scope. The intrinsic utility of a risk evaluates the risk taken by itself, whereas the extrinsic or comprehensive utility of the risk evaluates all that accompanies the risk. Methods of measurement apply to these types of probability and utility, as the appendix demonstrates.

Types of Risk

A chance of a bad event is one type of risk. The volatility of an act’s possible outcomes is another type of risk. Distinguishing these two types of risk is normatively significant because different general principles of rationality govern attitudes to them. A measure of the risk of a bad event may use the probability-utility product for the event. A measure of the volatility of an act’s possible outcomes may, in some cases, use the variance of the probability distribution of the utilities of the act’s possible outcomes. Given that an act’s risk belongs to every possible outcome, it arises from an equilibrium between the risk’s effect on an outcome’s utility and the utilities of outcomes’ effect on the risk.

Combinations of Acts

Combining acts combines their risks. Their risks are not additive, either using their sizes or their intrinsic utilities. This creates the possibility of hedging, that is, adopting a risk to lower overall risk. Evaluation of combinations of acts that produce risks depends on whether the acts are simultaneous or in a sequence. Rationality evaluates a set of simultaneous acts as one act, given an ideal agent’s awareness of performing all together and at will. It evaluates a sequence of acts by evaluating its components one by one; their rationality suffices for the sequence’s rationality. This method of evaluation contrasts with evaluation of a sequence by comparing its utility to the utilities of rival sequences. Rationality does not evaluate sequences using utility maximization, even when the sequences bring neither unanticipated information nor changes in goals, because sequences are not options in the sense of being performable at will.

Evaluation of an Act

The expected-utility principle asserts that an act’s utility equals its expected utility, that is, a probability-weighted average of the utilities of the act’s possible outcomes. The mean-risk principle asserts that an act’s utility equals the sum of (1) the act’s expected utility ignoring the act’s risk and (2) the intrinsic utility of the act’s risk. The justification of both principles uses the independence of evaluations of risks and prospects, taking them in isolation. The scope of intrinsic evaluations of risks and prospects makes the evaluations independent, and their independence grounds the additivity of evaluations of an act’s risks and prospects, and also the additivity of an evaluation of the act’s risk, in the sense of its exposure to chance, and the act’s evaluation ignoring its risk.

Rational Attitudes toward Risks

Rationality requires aversions to risks, considering them in isolation, but permits attractions to risks, taken as means to attainment of goals. It specifies the strength of an intrinsic aversion to a risk in the sense of a chance of a bad event, but leaves open the strength of an intrinsic aversion to a risk in the sense of an act’s exposure to chance. Principles of consistency and proportionality govern attitudes to risks and attitudes to prospects, taken as chances for good events. Rationality prohibits being more intrinsically averse to losses than attracted to gains of equal magnitude, provided that losses and gains are assessed using their intrinsic utility assignments. However, it permits attitudes to risks to change as risks move into the past. Requirements concerning attitudes to risks and prospects ground evaluation of acts using their expected utilities.