From risk to opportunity Lecture 5

From risk to opportunityLecture 5 John Hey and Carmen Pasca

First a PS from Lecture 4: the relationship between Rawls and risk/uncertainty • Rawls concentrated only on the worst thing that could happen in any situation: he wanted to make as good as possible the worst thing that could happen. • This means that he ignored all other possible outcomes – however many there were, how probable they were, and what they were. • Concentrating just on the worst meant that he was effectively putting probability 1 on the worst and zero probabilities on all the others. • So he did not need to worry about any true objective or subjective probabilities... • ...cleverly avoiding having to think about probabilities. • Another way of thinking about this is that the decision-maker is infinitely risk averse – only thinking about the worst thing that could happen – if you like a total pessimist.

Lecture 5 Epistemological approaches to risk: Definition • Epistemology: from Greek epistēmē, meaning "knowledge, understanding", and logos, meaning "study of") is the branch of philosophy concerned with the nature and scope (limitations) of knowledge.It addresses mainly the following questions: • What is knowledge? • How is knowledge acquired? • To what extent is it possible for a given subject or entity to be known? • We shall relate it to risk and uncertainty. • Afterwards you could try this quiz or this one. • .

Lecture 5 Epistemological approaches to risk: A question for you • Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car worth €50,000 to you; behind the others, nothing. You get what is behind the door you pick. You pick a door, say No. 1 (but the door is not opened), and the host, who knows what's behind the doors, opens another door, say No. 3, which has nothing. He then says to you, “Would you pay €10 to pick door No. 2 instead of door 1?" Is it to your advantage to switch your choice? • Let us have a show of hands? Who would change? Who would not? • We want you to think about this. We will explain later.

Lecture 5 Epistemological approaches to risk: Overview • This lecture examines the transformation of risk into an abstract concept; this started in the eighteenth century, on the initiative of Buffon, Laplace and Condorcet; and continued into the twentieth century with Keynes, Savage and Allais. • The use of this term is based on a subtle combination of knowledge and uncertainty. • The epistemological understanding and definition of risk will be investigated from a multidisciplinary approach.

Lecture 5 Epistemological approaches to risk: Transversal concept • Risk is an ordered application of knowledge to the unknown. • Each discipline has a particular approach with which it analyses the unknown, in such a way as to systematise its randomness and convert it into risk. • This epistemological approach suggests the concept of risk can act as a mirror, reﬂecting the preoccupations, strengths, and weaknesses of each discipline as they grapple with uncertainty.

Lecture 5 Epistemological approaches to risk: Keynes • Keynes in his Treatise on Probability made the theory of probability synonymous with the theory of logic. • Probability-relations are logical relations between propositions, such relations expressed the degrees of rational belief of individuals. • Epistemology help to explain how individuals come to know these probability-relations and degrees of rational beliefs.

Lecture 5 Epistemological approaches to risk: Keynes • Keynes’ Theory of Knowledge: the distinction between direct and indirect knowledge, between that part of our rational belief which we know directly, and that part which we know by argument. • According to Keynes we have direct acquaintance with three main classes of objects: sensations, meanings and perceptions. • We pass from direct to indirect knowledge by means of argument, this requiring the perception of logical relations between propositions (probability-relations).

Lecture 5 Epistemological approaches to risk: Keynes • Knowledge, truth and rational belief: in Keynes‘ epistemology, knowledge, truth and rational belief are inseparably bound together. • Keynes commences the Theory of Probability with a discussion of “The Meaning of Probability”, the object of which is to reveal the common element in such words as probable, likely, doubtful and uncertain (he thought that probability is undefinable).

Lecture 5 Epistemological approaches to risk: Keynes • Rational choice is the key element or the starting point of Theory of Probability. • Rational Choice: depends on probability comparisons based on the total information available, such information including the success or failure of predictions. • Keynes thought that the theory with the highest probability is always the rationally preferred theory, and while predictive outcomes enter into the determination of probabilities, they do not directly determine theory choice.

Lecture 5 Epistemological approaches to risk: Keynes • The sense of objectivity: two senses of objectivity are used in the Theory of Probability. • The objectivity of probabilities derives from logic, from a timeless world independent of individual psychology. • Keynes links objectivity to reality, probabilistic reasoning being viewed as the foundation of all knowledge about the world, including science. • Probability knowledge is real and objective, both in reference to logic and in reference to the world.

Lecture 5 Epistemological approaches to risk: Keynes • Risk and Uncertainty according to Keynes’s ideas: • Central to Keynes's epistemology is the distinction between risk and uncertainty. • Risk is when probabilities can be measured; uncertainty exists when no such measure is possible. • All the mathematical models underlying our recently-crashed financial system assumed that it was possible to measure risk and therefore insure or hedge against loss. • Individuals could miscalculate the probabilities, but, given the assumption of rationality, their mistakes would cancel out. (But this was obviously wrong.)

Lecture 5 Epistemological approaches to risk: Keynes • Keynes’ point is that economics “deals with introspection and values….with motives, expectations, psychological uncertainties”. • The future cannot always be predicted, because the future is changeable. • It is changeable, in large part, because it is what we choose to make it. • Thinking about the future as largely uncertain is the only rational basis of individual action. • But some parts of the future may be predictable.

Lecture 5 Epistemological approaches to risk: Savage • We now go onto Savage’s epistemological approach to risk. • Risk refers to situations where the probabilities are known. • Uncertainty refers to situations where the probabilities are unknown. • Savage (1954) considers choice under subjective risk (choice under uncertainty) as well as choice under objective risk.

Lecture 5 Epistemological approaches to risk: Savage • In The Foundations of Statistics, Savage (1954) advanced a theory of decision making under uncertainty and used that theory to define choice-based subjective probabilities. • He intended these probabilities to express the decision maker’s beliefs, thereby furnishing Bayesian statistics with behavioural foundations. • This gives an interpretation of probability as a numerical expression of beliefs.

Lecture 5 Epistemological approaches to risk: why Savage is important • The notion of probability emerged in the 1650’s with a dual meaning; the relative frequency of a random outcome in repeated trials and a measure of a decision maker’s degree of belief in the truth of propositions, or the likely realization of events. • Both the “objective” and the “subjective” probabilities, as these interpretations are now called, played important roles in the developments that lead to the formulation of Savage’s subjective expected utility model.

Lecture 5 Epistemological approaches to risk: Savage • In the early stages of their respective evolutions, the notion of utility was predicated on the existence of objective probabilities, and the notion of subjective probabilities presumed the existence of some form of utility. • Savage introduced a new analytical framework and conditions that are necessary and sufficient for the existence and joint uniqueness of utility and probability, and the characterisation of individual choice as expected utility maximizing behaviour. • This led to Subjective Expected Utility Theory.

Lecture 5 Epistemological approaches to risk: Structure • In decision problems the elements of structure that ought to matter to a rational agent are: • A set of possible states of the world; • A set of possible actions; • For each possible action, a set consisting of the consequences of that action in each possible state of the world. • An agent will have preferences among the consequences, and these preferences ultimately will dictate preferences among possible actions.

Lecture 5 Epistemological approaches to risk: an example • Suppose we have already elicited Ms X’s utility function u(.) over money (we will explain how to do this later). • Suppose this person satisfies Savage’s axioms. • Suppose there is some event E to which an objective probability cannot be attached, and Ms X is asked “for what amount of money, R, are you indifferent between being given R and being given the gamble which pays you W if event E happens and nothing otherwise?” • If she answers and gives her number R, then it follows that Ms X is implicitly attaching probability p to the event E where p is given by • p = [u(R) – u(0)]/[u(W) – u(0)] • We will explain later. Note that R must be between 0 and W and hence p must be between 0 and 1.You should try and check this result yourself.

Lecture 5 Epistemological approaches to risk: Allais • With Maurice Allais’ contributions the theory of decision under uncertainty became behavioural. • In 1988, Maurice Allais (1911-2010) was awarded the Nobel Prize in Economics in Memory of Alfred Nobel, but not just for his contributions to epistemology or decision under risk. • The experiments conducted in 1953 at the individual level by Maurice Allais, and his famous paradox, have shown that the model of economic rationality has little descriptive power in relation to the psychology of the real decision makers in real situations. • He was the originator of the now-famous Allais Paradox – which suggests a violation of Expected Utility theory – even with objective probabilities.

Lecture 5 Epistemological approaches to risk: Allais • In 1952, at a symposium held in Paris, Allais presented two studies in which he criticised the descriptive and predictive power of the choice theory (Expected Utility Theory) of the “American School” and especially Friedman’s stance by demonstrating the results of some experiments in which symposium participants were asked to make a series of choices in conditions of risk, which showed that many of them systematically violated the assumptions of Expected Utility Theory. • Apparently Savage was there and he violated his own axioms! He was very upset but changed his choices the next day!

Lecture 5 Epistemological approaches to risk: Allais’ methodology • The experiments proposed by Allais are founded on two points (the first being the assumptions he wants to test, and the second the experimental method): • First, the properties of the choice that characterize the expected utility function must be identified in axiomatic form; the properties are: completeness, transitivity, continuity and independence. • Second, subjects are presented with pairs of binary choices selected in such a way that particular combinations of the answers involves the violation of at least one of the axioms.

Lecture 5 Epistemological approaches to risk: An example • A simple experiment which tests the economists’ idea of rationality in a very strict form. (This slide and the next, with discussion on the third.) • Do you prefer Situation A or Situation B? • Situation A: First being given €1000, then getting €500 more • Situation B: First being given €1000, then a 50% chance of getting €1000 more a 50% chance of getting €0 more Which do you prefer? A or B?

Lecture 5 Epistemological approaches to risk: An example • Do you prefer Situation C or Situation D? • Situation C: First being given €2000, then losing €500 of it • Situation D: First being given €2000, then a 50% chance of losing €1000 of it a 50% chance of losing none of it • Which do you prefer? C or D?

Lecture 5 Epistemological approaches to risk: Are you mad? • If your behaviour respects economists’ rationality. • If you chose A over B you should choose C over D. • If you chose B over A you should choose D over C. • If you do anything else you are … • …mad… • … according to most economists ideas of rationality. • This is simple – it is just a framing problem – the two problems are identical. • Actually this is an example from Kahneman and Tversky (1979).

Lecture 5 Epistemological approaches to risk: An example • An experiment which tests the validity of Savage’s Axiom of Independence. • Do you prefer Situation A or Situation B? • Situation A: Certainty of receiving €300 • Situation B: a 80% chance of winning €400 a 20% chance of winning €0

Lecture 5 Epistemological approaches to risk: Example continued • Do you prefer Situation C to Situation D? • Situation C a 25% chance of winning €300 a 75% chance of winning nothing. • Situation D: a 20% chance of winning €400 an 80% chance of winning nothing

Lecture 5 Epistemological approaches to risk: Are you mad? • If your behaviour respects the Independence Axiom (of Expected Utility Theory)… • If you chose A over B you should choose C over D. • If you chose B over A you should choose D over C. • If you do anything else you are … • …mad… • … according to SEU and Savage! • You should work out why and we will explain later.

Lecture 5 Epistemological approaches to risk: More generally • Our cognitive limitations are so severe that massive reductions to full beliefs (certainty) are indispensable if we wish to be capable of reaching conclusions and making decisions. • It is a crucial drawback of the Bayesian model that it does not take into account the cognitive limitations of actual human beings . • A much more constructive approach is to discuss how a rational being with limited cognitive capabilities can make rational use of these capabilities.

Lecture 5 Epistemological approaches to risk: Conclusions • Keynes saw risk as characterised by objective probabilities based on logic and reason. • Savage constructs a structure in which probabilities, both in conditions of risk and of uncertainty, are subjective (and therefore may have no connection with reality – whatever that is). • Allais added to subjectivism the limits of human cognitive ability and hence the need for behaviourism.

Lecture 5 • Goodbye!

From risk to opportunity Lecture 5