Optimization, OR and risk

Summary

THE PARADOX OF IDEAL EVIDENCE

Two kinds of criticism have been lodged against the classical Bayesian view that agents should capture the empirical uncertainty they face by a single probability function on the set of factual prospects. So far we have concentrated on the objection that this is too demanding and that, for a variety of reasons, agents may be unable to determine precise probabilities for all prospects. We now turn to the second objection; that a probability assignment (whether precise or not) does not suffice to capture all relevant aspects of the agent's uncertainty.

Recall the example I presented in Chapter 11 to illustrate the difference between three grades of factual uncertainty, in which we have before us an urn containing black and white balls and you are asked to say how likely you think it is that a ball drawn at random will be white. In the ‘risk’ variant of the example, you are told that half the balls are black and half white; in the ‘ambiguity’ variant you are not told the proportions of black and white balls but you do know that they exhaust the colour possibilities, while in the ‘severe uncertainty’ variant you are told neither. Objective Bayesians are agreed that the answer of ‘one-half’ is the only reasonable one in the first situation, but disagree over the second and third, with Precisers continuing to advocate the answer of ‘one-half’ and Imprecisers counselling a wide probability interval; perhaps as wide as [0,1].

This example will serve equally well to bring out the second objection to the Bayesian view. Suppose now that you are in a situation of Grade 2 uncertainty, but are given the opportunity to sample the urn with replacement. Suppose that after 1,000 trials you have drawn 498 white balls and 502 black. How would the advice of the two kinds of objective Bayesian change as a result of the evidence? Not much at all, it would seem. For the Preciser, one-half remains the ascription that maximises entropy. Similarly, since the evidence rules out only extreme ball distributions, the set of probability functions consistent with it is the open interval (0,1). So the objective Impreciser's judgemental state should remain almost as imprecise as before.

Summary

Decision problems abound. Consumers have to decide what products to buy, doctors what treatments to prescribe, hiring committees which candidates to appoint, juries whether to convict or acquit a defendant, aid organisations what projects to fund, monetary policy committees what interest rates to set, and legislatures what laws to make. The problem for descriptive decision theory is to explain how such decisions are made. The problem that normative decision theory faces, on the other hand, is what to advise individuals and groups facing choices such as these. How should they evaluate the alternatives before them? What criteria should they employ? What procedures should they follow?

As these examples illustrate, decisions have to be made in a wide variety of contexts and by different kinds of decision makers. A pervasive feature, however, is the uncertainty that decision makers face: uncertainty about the state of the world, about the alternatives available to them, about the possible consequences of making one choice rather than another and, indeed, about how to evaluate these consequences. Dealing with this uncertainty is perhaps the most fundamental challenge we face in making a decision.

In the last 100 years or so an impressive body of work on this issue has emerged. At its core stands Bayesian decision theory, a mathematical and philosophical theory of both rational belief formation and decision making under uncertainty. The influence of Bayesian thinking pervades this book, to which the amount of space devoted to examining and criticising it will attest. Indeed, I regard Bayesian decision theory as essentially the correct theory for certain classes of idealised decision problems. But many real problems fall outside its scope. This is for two main reasons.

Firstly, Bayesian theory assumes that decision makers are ‘unbounded’: rational, logically omniscient and maximally opinionated. Rational in that their attitudes – beliefs, desires and preferences – are consistent both in themselves and with respect to one another; logically omniscient because they believe all logical truths and endorse all the logical consequences of their attitudes; and opinionated because they have determinate beliefs, desires and preferences regarding all relevant prospects.

Summary

The aim of this book is to develop a decision theory that is tailored for ‘real’ agents – i.e. agents who, like us, are uncertain about a great many things and are limited in their capacity to represent, evaluate and deliberate, but who nonetheless want to get things right to the extent that they can. The book is motivated by two broad claims. The first is that Bayesian decision theory provides an account of the rationality requirements on ‘unbounded’ agents that is essentially correct and is applicable in circumstances in which agents are aware of all the options available to them and are able to form precise judgements about all relevant contingencies. The second is that there are many circumstances in which these conditions are not satisfied and hence in which classical Bayesian theory is not applicable. A normative decision theory, adequate to such circumstances, would provide guidance on how bounded agents should represent the uncertainty they face, how they should revise their opinions as a result of experience and how they should make decisions when lacking full awareness or precise opinions (that they have confidence in) on relevant contingencies. The book tries to provide such a theory.

So many people have helped me with this project over the many years it has taken to complete it that I fear that I will forget to mention many of them. Its origins lie in my PhD dissertation completed under the supervision of Richard Jeffrey, David Malament and Dan Garber. Their support and that of Philippe Mongin, Dorothy Edgington and John Broome in the early years after the start of my career was crucial to my sticking with it. The influence of Dick Jeffrey on my thinking is hard to overestimate. The title of this book mirrors that of a paper of his – ‘Bayesianism with a human face’ – in which he espoused the kind of heterodox Bayesianism that pervades my writing. To me, he was the human face of Bayesianism.

Almost as much of an influence has been Jim Joyce, whom I first met at a workshop on Jeffrey's work some 20 years ago. Big chunks of this book can be read as a dialogue with The Foundations of Causal Decision Theory and his subsequent work on Imprecise Bayesianism.

Summary

INTRODUCTION

The second and third parts of the book developed a model of rational decision making based on a number of idealisations. Firstly, that agents’ attitudes – their beliefs, desires and preferences – are consistent both in themselves and with respect to one another. Secondly, that agents are logically omniscient, in that they believe all logical truths and endorse all the logical consequences of their attitudes. Thirdly, that they are fully aware of all relevant prospects, including what options they have, what the possible states of the world are and what the potential consequences of exercising their options are. And, lastly, that agents are maximally opinionated, in that they have determinate belief, desire and preference attitudes to all prospects under consideration.

These assumptions had a purpose: they simplified the discussion enough to allow the presentation of a comprehensive theory of rationality. There is no doubt, however, that they are unrealistic. All of us violate some of them most of the time and most of them some of the time. Anyone interested in a descriptively accurate account of human reasoning and decision making will therefore want to dispense with them. And indeed there has been a great deal of work in this area over the last few decades, broadly under the banners of bounded rationality and behavioural economics, which has emphasised the extent to which our thinking and choosing are affected by factors such as context, framing, emotions and biases.

Normative theories of rationality also need to dispense with these assumptions; not in order to become more descriptively accurate (that is not their goal) but in order to provide prescriptions relevant to real agents. The aim is not to abandon full rationality for a bounded variant that more accurately corresponds to the way human decision makers are, but to characterise (full) rationality for agents who are bounded in the sense that they are not maximally opinionated, logically omniscient, fully aware or even completely consistent. So each of these assumptions will need to be relaxed to some degree, both in order to examine what modifications are required to the theory of rationality developed so far and in order to say something about the rational way to behave when you know yourself to be bounded in these ways.

Summary

MODERATE HUMEANISM

The maximisation of expected utility hypothesis brings together two separate claims. The first concerns what rationality requires of the relation between the agent's preferences between different prospects and her beliefs and desires. Stripped of mathematical baggage, the claim can be expressed as follows:

Rationality Hypothesis Rationality requires of an agent that she prefer one prospect over another if and only if the expectation of benefit conditional on the truth of the former is greater than the expectation of benefit conditional on the truth of the latter, relative to her degrees of belief and desire.

The Rationality Hypothesis is generally taken to express nothing more than a consistency requirement on the agent's preferences, akin to the requirements that logic places on her beliefs. Consistency requirements are purely formal in nature and place no substantial constraints on the content of any preference, belief or desire taken in isolation. Moreover, the constraints that they place on sets of such preferences, beliefs and desires are not such as to rule out many that we might be inclined to regard as defective in some way; for instance, because they are immoral, self-destructive or just plain ill-considered.

In ordinary talk we tend to be more demanding and speak of beliefs as irrational, even if they are consistent, because they fail to meet some or other standard of adequacy. For instance, we might be inclined to criticise someone for not taking into account all available evidence or for failing to give the long-term consequences of their choices sufficient weight. Such talk, it seems to me, runs together two types of requirements that are best kept separate. One is the requirement that we recognise all the available evidence and that we give appropriate weight to all the possible consequences of our actions; the other that our beliefs be consistent with all the evidence that we recognise and that our preferences for actions be consistent with the weight that we give to each of their possible consequences. The former is a requirement that our judgements respond in an adequate way to the world as it is presented to us; the latter that they fit together in a coherent way.

Summary

MAXIMISATION OF EXPECTED BENEFIT

Decision theory begins with decision problems. Decision problems arise for agents: entities with the resources to represent, evaluate and change the world around them in various different ways, typically within the context of ongoing personal and institutional projects, activities or responsibilities. These projects together with the environment, both natural and social, provide the givens for the decision problems agents face: their material resources for acting, their information and often their standards for evaluating outcomes, as well as the source of the problems they must respond to. Social scientists hold very different views about the relative importance of the different aspects of this background and the decisions that are made within it, but few would doubt that choices made by consumers, doctors, policy makers and so on have the power to shape the course of events.

To face a genuine decision problem, agents must have options: actions that they are capable of performing and, equally, of forgoing if they so choose. To get an idea of what sorts of things count as a decision problem, let's look at a few examples.

Summary

Uncertainty is a pervasive feature of human life and almost all our decisions must be made without certainty about what the consequences of our actions will be for ourselves or others. Human attitudes such as hope, fear and even regret depend on it. Despite this, philosophy has given little attention to uncertainty, largely treating it as just lack of certainty (apparently the real interest). In both the mathematical and empirical sciences, on the other hand, the emphasis has been on the development of techniques to manage it. So central has the concept of probability been to this enterprise that it has come to seem as if uncertainty was nothing other than the flipside of probability.

The concept of probability emerged surprisingly late in human history, in the seventeenth-century work of Blaise Pascal and Pierre de Fermat.1 In this work, and much that followed, probability was conceived both in terms of the stochastic properties of chance processes, such as dice rolls and card deals, and the properties of beliefs about events regarding which full knowledge was lacking. In time this hardened into a distinction between two different forms of uncertainty: objective or aleatory uncertainty, which derives from features of the world (indeterminacy, randomness), and subjective or epistemic uncertainty, which derives from lack of information about it. In modern decision theory, there is a dominant theory of how each form of uncertainty should be quantified (in both cases, by a probability measure) and of how, so quantified, it should weigh in the evaluation of actions. For situations of objective uncertainty (or risk, as it typically called) decision theorists look to the version of expected utility theory originally due to von Neumann & Morgenstern (2007/1944), while for those characterised by epistemic uncertainty they look to subjective expected utility theory, whose classic statement is to be found in the work of Leonard Savage (1974/1954).

This distinction between risk and epistemic uncertainty, important and useful though it may be, does not remotely do justice to the variety of forms and degrees of uncertainty relevant to decision making.

Summary

We not only take attitudes to prospects, we also take conditional attitudes to them: attitudes on the hypothesis, supposition or assumption that something is true. On the assumption that I will not be tired tonight, I would rather go out than stay in. On the supposition that interest rates will rise, I believe that housing prices will fall. And so on. Conditional attitudes, like their unconditional counterparts, come in grades, a feature that is crucial to the role they play in decision making. I might find the prospect of going to the beach desirable, conditional on it being a sunny day, but not at all so, conditional on it being rainy. Whether I should go to the beach depends not only on the relative probability of sun and rain but also on the relative strengths of these conditional desires. One aim of this chapter is to identify the different kinds of suppositions involved in forming graded conditional attitudes, to propose rationality conditions that are characteristic of them and then to examine the implications of these conditions. A second is to explore the role that conditional attitudes play in hypothetical reasoning. Throughout I will assume the conceptual framework developed in the previous chapter.

First, two points should be made about what conditional attitudes are not. A common view is that conditional beliefs and desires given some β are dispositions to believe and desire conditional on β being true or on coming to believe that β. But this view is not quite right. While we should expect there to be a close relation between our conditional attitudes and the attitudes we take when we learn that the condition in question holds, sometimes this relation is disturbed by the conditions under which the learning occurs. I would prefer, were I confronted by a bully, that I would act bravely rather than run away. If I were confronted by a bully, however, I would (I predict) prefer to run. I believe that, were I to drink a lot, my driving ability would be impaired. Were I to drink too much, however, I would believe my driving to be better than ever.

Summary

PROSPECTS AND POSSIBLE WORLDS

Any account of mental representation must explain the parallels between the objects or contents of speech acts and the objects or contents of propositional attitudes.

(Stalnaker, 1984, p. 59)

Agency requires a number of abilities: to recognise and represent salient contingencies or prospects, to form consistent attitudes to them and to act in the light of these attitudes. In the second part of the book I gave an account of the requirements that rationality imposes on the beliefs and desires that an agent forms in negotiating her way around her environment. In doing so I refrained from saying anything specific about the contents of these attitudes, simply bundling them under the category of ‘prospects’. This has certain advantages, not least that it allows this account of rationality to be paired with more than one conception of what the objects of belief and desire are. But on the downside it left open the question of whether the proposed rationality constraints are consistent with any account of prospects, a question whose importance is highlighted by the triviality results previously alluded to. So before tackling how decisions should be guided by beliefs and desires we need to address the question of what prospects are.

The philosophical importance of this question lies in the central role that an account of prospects must play in any general theory of thought, reasoning and action. Prospects serve not only as the objects of attitudes such as belief and desire but also as the contents of sentences that agents use to express such beliefs and desires and to reason about the world and their place in it. An agent who says ‘It will rain on Thursday’ is able to express her belief that it will rain on Thursday by uttering this sentence in virtue of the fact that it has as its content or meaning just the prospect that she believes true. When she reasons from this to the thought ‘I will get wet if I don't take an umbrella’ the inference she makes is assisted by the semantic relationships between these sentences and others.

Summary

AGENTS

To make decisions, agents must have a range of capabilities. Firstly, they must have an ability to represent, and discriminate between, different possible features or states of their environment. Secondly, they must be able to perceive or infer the presence or absence of these features, i.e. to form beliefs about the world. Thirdly, they must be able to compare these possible features or states according to some standard of value – i.e. be able to judge whether they are more or less desirable. And, finally, if they are to be capable of not just thinking about the world but also acting on it, agents need to be able to determine what options they have, to evaluate them and to exercise them. In summary, they must be able to identify both the possible states of the world and their options for changing them and to form cognitive and evaluative attitudes towards both.

This list of conditions may seem quite demanding, but it should be borne in mind that there is no requirement that these judgements be at all refined or detailed, or even that they be conscious. So even cognitively quite primitive entities may well have most or all of the required capabilities. Consider a thermostat, for instance. It inarguably represents features of the world – the temperature – and judges which of these states in the actual one. It also has a crude value system that can be summarised as: cold is bad, warm is good. The actions available to it – switching the heating on or off or doing nothing – are chosen in the light of information it receives about the state of the world and implicitly in view of the consequences of selecting the action.

On the other hand, the way the thermostat makes decisions, if one is happy to speak in such terms, lacks a number of the essential characteristics of human decision making. Although we might say that the reason why the thermostat switches the heating on is that the room is cold and that by doing so it will warm up, the reason we refer to is not the cause of the thermostat's switching on. The thermostat does not itself make the judgements about what would happen if it performed one or other of the actions available to it.

Summary

INTRODUCTION

In the first part of the book we developed a theory of rational prospective agency based around a set of claims about what rationality requires of the judgements of agents: consistency between her beliefs, desires and preferences; between her conditional and unconditional attitudes; and between her attitudes and her choices. This theory can reasonably be criticised both for being too demanding and for being insufficiently so. Too demanding because real agents face resource constraints which make it impossible to form opinions about all possible contingencies and unwise that they attempt to do so. Insufficiently demanding, on the other hand, because it requires of agents only that their beliefs and desires be consistent, and not that they be responsive to what is true or valuable in any way. In the next part of the book we will take up the first challenge. Our task in this chapter will be to respond to the second.

There are two directions in which one might seek to constrain the sorts of attitudes that a rational agent can permissibly hold. First, one might put forward conditions that go beyond consistency but which fall short of simply demanding that the agent believe only truths and desire only what is good (which would be to demand what cannot generally be achieved). Second, one might require that the agent improves her opinions over time by responding appropriately to information acquired through interaction with the environment or through deliberation. Many conditions of both kinds are to be found in the epistemological literature, for different conceptions of belief, but I will focus on those deriving from the widely shared thought that an agent's beliefs should be constrained by the evidence she holds about the state of the world and, more generally, that her attitudes should reflect what she has learnt from experience.

Let us restrict attention to belief for the moment and return to the other attitudes later. Evidence-based constraints on belief can take both a synchronic form – that an agent's beliefs at a particular time should appropriately reflect the evidence that she holds at that time – and a diachronic one – that she should revise her beliefs appropriately in response to the acquisition of new evidence.

Optimization, OR and risk

Refine search

Refine search

Actions for selected content:

1189 results in Optimization, OR and risk

Contents

26 - Evaluating Deployed Decision Support Systems for Security: Challenges, Analysis, and Approaches

14 - Confidence

Summary

Introduction

Summary

Preface

Summary

Frontmatter

11 - Imprecise Bayesianism

Summary

List of Figures

List of Tables

2 - Rationality

Summary

PART II - PROSPECTIVE RATIONALITY

Index

1 - Decision Problems

Summary

3 - Uncertainty

Summary

6 - Conditional Attitudes

Summary

Dedication

8 - Multidimensional Possible-World Semantics

Summary

5 - Rational Belief and Desire

Summary

PART III - FACING THE WORLD

10 - The Learning Agent

Summary

Optimization, OR and risk

Refine search

Refine search

Actions for selected content:

Save Search

1189 results in Optimization, OR and risk

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary

Summary