Axiom: An axiom is a fundamental premise of an argument for which no further justification is given. Example: According to the asymmetry axiom, no rational agent strictly prefers x to y and y to x.

Bargaining problem: The bargaining problem is a cooperative game with infinitely many Nash equilibria, which serves as a model for a type of situation that arises in many areas of society: A pair of players are offered to split some amount of money between the two of them. Each player has to write down his or her demand and place it in a sealed envelope. If the amounts they demand sum to more than the total amount available the players will get nothing; otherwise each player will get the amount he or she demanded. The players are allowed to communicate and form whatever binding agreements they wish. A general solution to this problem was offered by Nash, who based his proposal on a small set of intuitively plausible axioms.

In the final sections of Chapter 11 we showed that all (two-person) zero-sum games can be solved by determining a set of equilibrium strategies. From a purely theoretical perspective, there is little more to say about zero-sum games. However, nonzero-sum games are more interesting, and require further attention by game theorists. This chapter gives a brief overview of what we currently know about nonzero-sum games. First, we shall give a proper introduction to the equilibrium concept tacitly taken for granted in the previous chapter. We will then go on to analyse a couple of well-known nonzero-sum games. The chapter ends with a discussion of whether game theory has any implications for ethics, biology and other subjects, which some scholars believe is the case.

The Nash equilibrium

The prisoner's dilemma is an example of a nonzero-sum game. This is because the loss or gain made by each player is not the exact opposite of that made by the other player. If we sum up the utilities in each box they do not always equal zero. In the prisoner's dilemma each prisoner will have to spend some time in prison no matter which strategy he chooses, so the total sum for both of them is always negative. To see this point more clearly, it is helpful to take a second look at the game matrix of the prisoner's dilemma (Table 12.1).

The focus of this chapter is on the role of causal processes in decision making. In some decision problems, beliefs about causal processes play a significant role for what we intuitively think it is rational to do. However, it has turned out to be very hard to give a convincing account of what role beliefs about causal process should be allowed to play. Much of the discussion has focused on a famous example known as Newcomb's problem. We shall begin by taking a look at this surprisingly deep problem.

Newcomb's problem

Imagine a being who is very good at predicting other people's choices. Ninety-nine per cent of all predictions made by the being so far have been correct. You are offered a choice between two boxes, B1 and B2. Box B1 contains $1,000 and you know this, because it is transparent and you can actually see the money inside. Box B2 contains either a million dollars or nothing. This box is not transparent, so you cannot see its content. You are now invited to make a choice between the following pair of alternatives: You either take what is in both boxes, or take only what is in the second box. You are told that the predictor will put $1M in box B2if and only if she predicts that you will take just box B2, and nothing in it otherwise. The predictor knows that you know this. Thus, in summary, the situation is as follows.

The von Neumann–Morgenstern theorem (Theorem 5.2) on page 101 is an if-and-only-if claim, so we have to prove both directions of the biconditional. We first show that the axioms entail the existence part of the theorem, saying that there exists a utility function satisfying (1) and (2). In the second part of the proof, the uniqueness part, we prove that the utility function is unique in the sense articulated in (3). Finally, in the third part, we prove that if we have a utility function with properties (1)–(3) then the four axioms all hold true.

Part One

Let us start by constructing the utility function u mentioned in the theorem. Since Z is a finite set of basic prizes the completeness axiom entails that Z will contain some optimal element O and some worst element W. This means that O is preferred to or equally as good as every other element in Z, and every element in Z is preferred to or equally as good as W. Furthermore, O and W will also be the optimal and worst elements in all probabilistic mixtures of Z, i.e. in the set L. This follows from the independence axiom.

This chapter gives an overview of how people do actually make decisions. The headline news is that people frequently act in ways deemed to be irrational by decision theorists. This shows that people should either behave differently, or that there is something wrong with the normative theories discussed in the preceding chapters of this book. After having reviewed the empirical findings, both conclusions will be further considered.

The interest in descriptive decision theory arose in parallel with the development of normative theories. Given the enormous influence axiomatic theories had in the academic community in the latter half of the twentieth century, it became natural to test the axioms in empirical studies. Since many decision theorists advocate (some version of) the expected utility principle, it is hardly surprising that the axioms of expected utility theory are the most researched ones. Early studies cast substantial doubt on the expected utility principle as an accurate description of how people actually choose. However, it was not until 1979 and the publication of a famous paper by Kahneman and Tversky that it finally became widely accepted that expected utility theory is a false descriptive hypothesis. Kahneman and Tversky's paper has become one of the most frequently quoted academic publications of all times. (Kahneman was awarded the Nobel Prize in economics in 2002, but Tversky died a few years earlier.) In what follows, we shall summarise their findings, as well as some later observations.

Game theory studies decisions in which the outcome depends partly on what other people do, and in which this is known to be the case by each decision maker. Chess is a paradigmatic example. Before I make a move, I always carefully consider what my opponent's best response will be, and if the opponent can respond by doing something that will force a checkmate, she can be fairly certain that I will do my best to avoid that move. Both I and my opponent know all this, and this assumption of common knowledge of rationality (CKR) determines which move I will eventually choose, as well as how my opponent will respond. Thus, I do not consider the move to be made by my opponent to be a state of nature that occurs with a fixed probability independently of what I do. On the contrary, the move I make effectively decides my opponent's next move.

Chess is, however, not the best game to study for newcomers to game theory. This is because it is such a complex game with many possible moves. Like other parlour games, such as bridge, monopoly and poker, chess is also of limited practical significance. In this chapter we shall focus on other games, which are easier to analyse but nevertheless of significant practical importance. Consider, for example, two hypothetical supermarket chains, Row and Col.

The term Bayesianism appears frequently in books on decision theory. However, it is surprisingly difficult to give a precise definition of what Bayesianism is. The term has several different but interconnected meanings, and decision theorists use it in many different ways. To some extent, Bayesianism is for decision theorists (but not all academics) what democracy is for politicians: Nearly everyone agrees that it is something good, although there is little agreement on what exactly it means, and why it is good. This chapter aims at demystifying the debate over Bayesianism. Briefly put, we shall do two things. First, we shall give a rough characterisation of Bayesian decision theory. Second, we shall ask whether one can give any rational argument for or against Bayesianism.

What is Bayesianism?

There are almost as many definitions of Bayesianism as there are decision theorists. To start with, consider the following broad definition suggested by Bradley, who is himself a Bayesian.

According to this definition, Bayesianism has two distinct components. The first tells us what your state of mind ought to be like, whilst the second tells us how you ought to act given that state of mind.

This book is an introduction to decision theory. My ambition is to present the subject in a way that is accessible to readers with a background in a wide range of disciplines, such as philosophy, economics, psychology, political science and computer science. That said, I am myself a philosopher, so it is hardly surprising that I have chosen to discuss philosophical and foundational aspects of decision theory in some detail. In my experience, readers interested in specific applications of the subject may find it helpful to start with a thorough discussion of the basic principles before moving on to their chosen field of specialisation.

My ambition is to explain everything in a way that is accessible to everyone, including readers with limited knowledge of mathematics. I therefore do my best to emphasise the intuitive ideas underlying the technical concepts and results before I state them in a more formal vocabulary. This means that some points are made twice, first in a non-technical manner and thereafter in more rigorous ways. I think it is important that students of decision theory learn quite a bit about the technical results of the subject, but most of those results can no doubt be explained much better than what is usually offered in textbooks. I have tried to include only theorems and proofs that are absolutely essential, and I have made an effort to prove the theorems in ways I believe are accessible for beginners.

On 6 September 1492 Christopher Columbus set off from the Canary Islands and sailed westward in an attempt to find a new trade route between Europe and the Far East. On 12 October, after five weeks of sailing across the Atlantic, land was sighted. Columbus had never been to the Far East, so when he landed in Middle America (‘the West Indies’) he believed that he had indeed discovered a new route to the Far East. Not until twenty-nine years later did Magellan finally discover the westward route to the Far East by sailing south around South America.

Columbus' decision to sail west from the Canary Islands was arguably one of the bravest decisions ever made by an explorer. But was it rational? Unlike some of his contemporaries, Columbus believed that the Earth is a rather small sphere. Based on his geographical assumptions, he estimated the distance from Europe to East India to total 2,300 miles. The actual distance is about 12,200 miles, which is more than five times farther than Columbus thought. In the fifteenth century no ship would have been able to carry provisions for such a long journey. Had America not existed, or had the Earth been flat, Columbus would certainly have faced a painful death. Was it really worth risking everything for the sake of finding a new trade route?

Here is a simple test of how carefully you have read the preceding chapters: Did you notice that some axioms have occurred more than once, in discussions of different issues? At least three preference axioms have been mentioned in several sections, viz. the transitivity, completeness and independence axioms. Arguably, this is all good news for decision theory. The fact that the same, or almost the same, axioms occur several times indicates that the basic principles of rationality are closely interconnected. However, this also raises a very fundamental concern: Do we really have any good reasons for accepting all these axioms in the first instance? Perhaps they are all false!?

In Table 8.1 we summarise the relevant axioms by using a slightly different notation than in previous chapters. Recall that the axioms are supposed to hold for all options x, y, z, and all probabilities p such that 1 > p > 0.

A very simple standpoint – perhaps too simple – is that the preference axioms need no further justification, because we know the axioms to be true because we somehow grasp their truth immediately. However, many decision theorists deny they can immediately adjudicate whether a preference axiom is true, so it seems that some further justification is needed.

What more could one say in support of the axioms? According to an influential view, one has sufficient reason to accept a preference axiom if and only if it can somehow be pragmatically justified.

The three towers

Mr Claus, mandarin of the College of Li-Sou-Stan reported that, during his travels,

Exercise 4.1.1Quickly guess the time to the end of the world.

In order to find how much time remains before the end of the world, we consider the more general case in which the priests have n discs on one needle A and must transfer them to the second needle B making use of the third needle C.

Marrying

So far in this book, we have considered the question of finding the best outcome for a single person under a fixed set of rules. What is the best way to bet, given the appropriate odds and probabilities? Should I take out an annuity? What is the shortest route from A to B?

Life becomes much more complicated when there are many people with different goals and the action of one person changes the rules for the others. In this chapter we shall see that, even in these circumstances, mathematics can sometimes provide insight. We shall also see that problems arise which lie outside the province of the mathematician.

We start by looking at problems of the following type. Suppose we wish to form 2n children into pairs. If we match Amber with Bertha and Caroline with Delia but Amber prefers Caroline to Bertha while Caroline prefers Amber to Delia, then the pairing is unstable since Amber and Caroline would both prefer to break up with their present partners and form a pair together. If, however, Amber prefers Caroline to Bertha but Caroline prefers Delia to Amber this particular event will not happen (though there may be other ways in which the pairing is unstable).

Our problem is the following.

The Kindergarten ProblemIs it always possible to arrange 2n children in stable pairs (i.e. so there are not two children in different pairs who would prefer each other to their present partner)?

Bernard Shaw

The title for this chapter is taken from an essay in which Shaw explains the principle of insurance and its relation to the welfare state. Shaw begins, as we have done, on the race-track and continues as follows.

Scissors, Paper, Stone

In the previous chapter we recalled the game of Scissors, Paper, Stone. In it, the two players simultaneously put out their right hands, two extended fingers represent scissors, an open hand paper, and a clenched first stone. Scissors cut paper, paper wraps stone and stone blunts scissors.

How should we play this game? The answer depends on our opponent. If our opponent is a small child we may observe that it never uses stone or that it never repeats the weapon it used in the previous round. You can use this information to ensure that you win more times than you lose.

Exercise 7.1.1Explain why, if you know that your opponent will choose from two specified weapons, you can arrange so that you never lose and sometimes win.

It is more interesting to consider what we should do if faced by an opponent cleverer than ourselves. Whatever plans we make, we must expect them to be anticipated. Under these circumstances, it makes sense to play at random, choosing each weapon with probability 1/3 independent of what has gone before. (We could, for example, throw a die and play scissors if the die shows 1 or 2, paper if the die shows 3 or 4 and stone if the die shows 5 or 6.) If our opponent plays stone, then with probability 1/3 we play scissors and lose, with probability 1/3 we play paper and win and with probability 1/3 we play stone and draw.

A well known newspaper columnist once wrote:

This book requires the knowledge and skill which the columnist believes he once had, together with a rather more open mind. Roughly speaking, it requires the tools available after two years of school or one year of university calculus. I assume the reader can use those tools readily and without too much effort.

The level aimed for is a year or so higher than that I envisaged for The Pleasures of Counting. In that book, I tried to offer something to those readers who skipped the mathematical details but, in this book, the mathematics is the message.

Students who have been trained to think of mathematics as being about finding right answers often find it hard to adjust to subjects like statistics which are about making decisions that may turn out to be right or wrong. The object of this book is to help readers think about how decisions are made and how mathematics can help the process.

Find the largest

Suppose that we are helping some very cultured friends move house. They have already installed several walls of bookcases but left us to unpack the many crates of books which go on the bookcases. We decide to place the books in alphabetical order. This is not as easy as it seems, but we will be helped by various pieces of knowledge and guesswork. However badly jumbled the books have been in packing, there will probably be runs in near perfect alphabetical order. We expect that about a third of the authors will have initial letters A to G, about one third G to N and about one third N to Z. We know that most authors with initial letter W will have second letter A, E, H, I, O, R or Y and so on.

If we seek to mechanise the sorting process involved, then we can either attempt to identify and incorporate all these random and not very precise pieces of information or we must produce methods which make no use of them at all.

To make sure that we do not use extraneous information, let us consider the following model for sorting n cards bearing the numbers 1 to n. The cards are placed face down in front of you. You have an assistant who will look at any two cards that you indicate and tell you which of the two cards bears the largest number.

Coin tossing

In The Napoleon of Notting Hill, Chesterton wrote:

There is no headline journalists more enjoy writing or their readers more enjoy reading than ‘Experts get it wrong again’. But, although the future is covered in mist, we may be able to glimpse vague shapes through that mist and our actions should take account of those glimpses.

Consider, for example, the question of who will be president of the United States of America in 9 years' time. Someone who believes that the future is totally unknowable might offer odds of a thousand million to one against us being able to name the future president.