It is widely held that Bayesian decision theory is the final word on how a rational person should make decisions. However, Leonard Savage--the inventor of Bayesian decision theory--argued that it would be ridiculous to use his theory outside the kind of small world in which it is always possible to "look before you leap." If taken seriously, this view makes Bayesian decision theory inappropriate for the large worlds of scientific discovery and macroeconomic enterprise. When is it correct to use Bayesian (...) decision theory--and when does it need to be modified? Using a minimum of mathematics, Rational Decisions clearly explains the foundations of Bayesian decision theory and shows why Savage restricted the theory's application to small worlds. The book is a wide-ranging exploration of standard theories of choice and belief under risk and uncertainty. Ken Binmore discusses the various philosophical attitudes related to the nature of probability and offers resolutions to paradoxes believed to hinder further progress. In arguing that the Bayesian approach to knowledge is inadequate in a large world, Binmore proposes an extension to Bayesian decision theory--allowing the idea of a mixed strategy in game theory to be expanded to a larger set of what Binmore refers to as "muddled" strategies. Written by one of the world's leading game theorists, Rational Decisions is the touchstone for anyone needing a concise, accessible, and expert view on Bayesian decision making. (shrink)

Binmore argues that game theory provides a systematic tool for investigating ethical matters.

This article criticises one of Stuart Rachels' and Larry Temkin's arguments against the transitivity of 'better than'. This argument invokes our intuitions about our preferences of different bundles of pleasurable or painful experiences of varying intensity and duration, which, it is argued, will typically be intransitive. This article defends the transitivity of 'better than' by showing that Rachels and Temkin are mistaken to suppose that preferences satisfying their assumptions must be intransitive. It makes cler where the argument goes wrong by (...) showing that it is a version of Zeno's paradox of Achilles and the Tortoise. (shrink)

Natural Justice is a bold attempt to lay the foundations for a genuine science of morals using the theory of games. Since human morality is no less a product of evolution than any other human characteristic, the book takes the view that we need to explore its origins in the food-sharing social contracts of our prehuman ancestors. It is argued that the deep structure of our current fairness norms continues to reflect the logic of these primeval social contracts, but the (...) particular fairness norm a society operates is largely a product of cultural evolution. In pursuing this point, the book proposes a naturalistic reinterpretation of John Rawls' original position that reconciles his egalitarian theory of justice with John Harsanyi's utilitarian theory by identifying the environment appropriate to each. (shrink)

Ken Binmore's previous game theory textbook, Fun and Games, carved out a significant niche in the advanced undergraduate market; it was intellectually serious and more up-to-date than its competitors, but also accessibly written. Its central thesis was that game theory allows us to understand many kinds of interactions between people, a point that Binmore amply demonstrated through a rich range of examples and applications. This replacement for the now out-of-date 1991 textbook retains the entertaining examples, but changes the (...) organization to match how game theory courses are actually taught, making Playing for Real a more versatile text that almost all possible course designs will find easier to use, with less jumping about than before. In addition, the problem sections, already used as a reference by many teachers, have become even more clever and varied, without becoming too technical. Playing for Real will sell into advanced undergraduate courses in game theory, primarily those in economics, but also courses in the social sciences, and serve as a reference for economists. (shrink)

Can people be relied upon to be nice to each other? Thomas Hobbes famously did not think so, but his view that rational cooperation does not require that people be nice has never been popular. The debate has continued to simmer since Joseph Butler took up the Hobbist gauntlet in 1725. This article defends the modern version of Hobbism derived largely from game theory against a new school of Butlerians who call themselves behavioral economists. It is agreed that the experimental (...) evidence supports the claim that most people will often make small sacrifices on behalf of others and that a few will sometimes make big sacrifices, but that the larger claims made by contemporary Butlerians lack genuine support. (shrink)

Game theory has proved a useful tool in the study of simple economic models. However, numerous foundational issues remain unresolved. The situation is particularly confusing in respect of the non-cooperative analysis of games with some dynamic structure in which the choice of one move or another during the play of the game may convey valuable information to the other players. Without pausing for breath, it is easy to name at least 10 rival equilibrium notions for which a serious case can (...) be made that here is the “right” solution concept for such games. (shrink)

Experimental results on the Ellsberg paradox typically reveal behavior that is commonly interpreted as ambiguity aversion. The experiments reported in the current paper find the objective probabilities for drawing a red ball that make subjects indifferent between various risky and uncertain Ellsberg bets. They allow us to examine the predictive power of alternative principles of choice under uncertainty, including the objective maximin and Hurwicz criteria, the sure-thing principle, and the principle of insufficient reason. Contrary to our expectations, the principle of (...) insufficient reason performed substantially better than rival theories in our experiment, with ambiguity aversion appearing only as a secondary phenomenon. (shrink)

Games are played everywhere: from economics and online auctions to social interactions, and game theory is about how to play such games in a rational way, and how to maximize their outcomes. This VSI reveals, without mathematical equations, the insights the theory can bring to everything from how to play poker optimally to the sex ratio among bees.

Do conventions need to be common knowledge in order to work? David Lewis builds this requirement into his definition of a convention. This paper explores the extent to which his approach finds support in the game theory literature. The knowledge formalism developed by Robert Aumann and others militates against Lewis’s approach, because it shows that it is almost impossible for something to become common knowledge in a large society. On the other hand, Ariel Rubinstein’s Email Game suggests that coordinated action (...) is no less hard for rational players without a common knowledge requirement. But an unnecessary simplifying assumption in the Email Game turns out to be doing all the work, and the current paper concludes that common knowledge is better excluded from a definition of the conventions that we use to regulate our daily lives. (shrink)

This short paper begins with a summary of the views of a sympathetic game theorist on the current state of play in what is still called the New Institutional Economics. It continues with a much abbreviated summary of my own attempts to treat justice as a kind of institution in the hope that this will serve as a case study in how game theory can serve as a useful intellectual framework for the study of human institutions.

If some external enforcement agency compels us to honor deals reached in the original position, then Harsanyi has shown that the outcome will be utilitarian. Under the same hypotheses, Rawls claims that the outcome will be egalitarian. This chapter confirms that Harsanyi is correct. It goes on to use the concept of an empathy equilibrium to predict the standard of interpersonal comparison needed to operate a utilitarian norm that will evolve in the medium run.

’Tis vain to talk of adding quantities which after the addition will continue to be as distinct as they were before; one man’s happiness will never be another man’s happiness: a gain to one man is no gain to another: you might as well pretend to add 20 apples to 20 pears.

This is the second part of a two-part paper. It can be read independently of the first part provided that the reader is prepared to go along with the unorthodox views on game theory which were advanced in Part I and are summarized below. The body of the paper is an attempt to study some of the positive implications of such a viewpoint. This requires an exploration of what is involved in modeling “rational players” as computing machines.

This paper argues that we need to look beyond Bayesian decision theory for an answer to the general problem of making rational decisions under uncertainty. The view that Bayesian decision theory is only genuinely valid in a small world was asserted very firmly by Leonard Savage [18] when laying down the principles of the theory in his path-breaking Foundations of Statistics. He makes the distinction between small and large worlds in a folksy way by quoting the proverbs ”Look before you (...) leap” and ”Cross that bridge when you come to it”. You are in a small world if it is feasible always to look before you leap. You are in a large world if there are some bridges that you cannot cross before you come to them. As Savage comments, when proverbs conflict, it is proverbially true that there is some truth in both—that they apply in different contexts. He then argues that some decision situations are best modeled in terms of a small world, but others are not. He explicitly rejects the idea that all worlds can be treated as small as both ”ridiculous” and ”preposterous”. The first half of his book is then devoted to a very successful development of the set of ideas now known as Bayesian decision theory for use in small worlds. The second half of the book is an attempt to develop a quite different set of ideas for use in large worlds, but this part of the book is usually said to be a failure by those who are aware of its existence.2 Frank Knight [15] draws a similar distinction between making decision under risk or uncertainty.3 The pioneering work of Gilboa and Schmeidler [7] on making.. (shrink)

Some behavioral economists argue that the honoring of social norms can be adequately modeled as the optimization of social utility functions in which the welfare of others appears as an explicit argument. This paper suggests that the large experimental claims made for social utility functions are premature at best, and that social norms are better studied as equilibrium selection devices that evolved for use in games that are seldom studied in economics laboratories.

This chapter presents an overview of the book. It argues that the metaphysical approach to ethics is a failure and that the time has come to take a scientific view of morality. A social contract is taken to be the set of common understandings that allow the citizens of a society to coordinate. Such social contracts are seen as the product of biological and cultural evolution. To survive, a social contract must therefore be an equilibrium in the repeated game of (...) life played by a society. Since the folk theorem of repeated game theory says that there are large numbers of such equilibria, fairness norms then become explicable as an equilibrium selection device that selects one of the many efficient equilibria of a society's game of life. It is suggested that the deep structure of such fairness norms is captured by John Rawls' notion of the original position, and is therefore universal in the human species. On the other hand, the standard of interpersonal comparison needed as an input to the original position is culturally determined. (shrink)

The following responses to the scholars who were kind enough to comment on my Natural Justice in this symposium have been kept to a minimum by addressing only issues where I think a misunderstanding may have arisen.

Philosophers used to say that all their endeavours were merely a footnote to Plato. In ethics, this is still largely true.

Robert Aumann argues that common knowledge of rationality implies backward induction in finite games of perfect information. I have argued that it does not. A literature now exists in which various formal arguments are offered in support of both positions. This paper argues that Aumann's claim can be justified if knowledge is suitably reinterpreted.

This paper uses a 4 × 4 expansion of the Hawk–Dove Game to illustrate how sexual drift in a large genotype space can shift a population from one equilibrium in a smaller phenotype space to another. An equilibrium is only safe from being destabilized in this way when implemented by recessive alleles.

A persistent argument against the transitivity assumption of rational choice theory postulates a repeatable action that generates a significant benefit at the expense of a negligible cost. No matter how many times the action has been taken, it therefore seems reasonable for a decision-maker to take the action one more time. However, matters are so fixed that the costs of taking the action some large number of times outweigh the benefits. In taking the action some large number of times on (...) the grounds that the benefits outweigh the costs every time, the decision-maker therefore reveals intransitive preferences, since once she has taken it this large number of times, she would prefer to return to the situation in which she had never taken the action at all. We defend transitivity against two versions of this argument: one in which it is assumed that taking the action one more time never has any perceptible cost, and one in which it is assumed that the cost of taking the action, though (sometimes) perceptible, is so small as to be outweighed at every step by the significant benefit. We argue that the description of the choice situation in the first version involves a contradiction. We also argue that the reasoning used in the second version is a form of similarity-based decision-making. We argue that when the consequences of using similarity-based decision-making are brought to light, rational decision-makers revise their preferences. We also discuss one method that might be used in performing this revision. (shrink)

This article is extracted from a forthcoming book, ‘Natural Justice’. It is a nontechnical introduction to the part of game theory immediately relevant to social contract theory. The latter part of the article reviews how concepts such as trust, responsibility, and authority can be seen as emergent phenomena in models that take formal account only of equilibria in indefinitely repeated games. Key Words: game theory • equilibrium • evolutionary stability • reciprocity • folk theorem • trust • altruism • responsibility (...) • authority. (shrink)

The target article by Henrich et al. describes some economic experiments carried out in fifteen small-scale societies. The results are broadly supportive of an approach to understanding social norms that is commonplace among game theorists. It is therefore perverse that the rhetorical part of the paper should be devoted largely to claiming that “economic man” is an experimental failure that needs to be replaced by an alternative paradigm. This brief commentary contests the paper's caricature of economic theory, and offers a (...) small sample of the enormous volume of experimental data that would need to be overturned before “economic man” could be junked. (shrink)

This paper offers an experimental test of a version of Rubinstein’s bargaining model in which the players’ discount factors are unequal. We find that learning, rationality, and fairness are all significant in determining the outcome. In particular, we find that a model of myopic optimization over time predicts the sign of deviations in the opening proposal from the final undiscounted agreement in the previous period rather well. To explain the amplitude of the deviations, we then successfully fit a perturbed version (...) of the model of myopic adjustment to the data that allows for a bias toward refusing inequitable offers. (shrink)

This chapter reviews the calumnies that are commonly directed at moral naturalists. Philosophical rationalism is compared unfavorably with the empirical tradition by drawing attention to the inadequacies in the reasoning that supposedly leads to Kant's categorical imperative. Moral naturalism is defended on the same basis as biological naturalism. Moral relativism — the big no-no for metaphysical moralists — is defended against a variety of common criticisms; reductionism gets the same treatment. Finally, the life of David Hume is advanced as a (...) counter-example to those who feel that naturalism is necessarily dehumanizing or dispiriting. (shrink)

Knowledge was traditionally held to be justified true belief. This paper examines the implications of maintaining this view if justication is interpreted algorithmically. It is argued that if we move sufficiently far from the small worlds to which Bayesian decision theory properly applies, we can steer between the rock of fallibilism and the whirlpool of skepticism only by explicitly building into our framing of the underlying decision problem the possibility that its attempt to describe the world is inadequate.

When judging what is fair, how do we decide how much weight to assign to the conflicting interests of different classes of people? This subject has received some attention in a utilitarian context, but has been largely neglected in the case of egalitarian societies of the kind studied by John Rawls. My Game Theory and the Social Contract considers the problem for a toy society with only two citizens. This paper examines the theoretical difficulties in extending the discussion to societies (...) with more than two citizens. (shrink)

When do we care for others as we care for ourselves? William Hamilton showed that we should be expected to care for our family members in proportion to our degree of relationship to them. Such reasoning explains why eusociality evolved independently at least twelve times in the order Hymenoptera, which includes ants, bees, and wasps, but only three times elsewhere in the animal kingdom. It also verifies Thomas Hobbes' answer to the question: Why cannot mankind live sociably one with another (...) as bees or ants? (shrink)

This chapter reviews the relevant anthropology, starting with the apparent universality of the golden rule — do as you would be done by — in hunter-gatherer societies. It points out that all pure foraging societies have two properties: they do not tolerate bosses, and they share very fairly. A putative explanation of the first property is offered that appeals to the game theory discipline of mechanism design. The second property is explained as an evolutionary consequence of the implicit insurance contracts (...) that are common in animals as ahedge against hunger and starvation. If so, then we have an argument in support of the contention that Rawls' original position captures the deep structure of fairness norms wired into our brains. The analogy with Chomsky's deep structure of language is close. (shrink)

Playing for Real is a problem-based textbook on game theory that has been widely used at both the undergraduate and graduate levels. This Coursepack Edition will be particularly useful for teachers new to the subject. It contains only the material necessary for a course of ten, two-hour lectures plus problem classes and comes with a disk of teaching aids including pdf files of the author's own lecture presentations together with two series of weekly exercise sets with answers and two sample (...) final exams with answers. There are at least three questions a game theory book might answer: What is game theory about? How is game theory applied? Why is game theory right? Playing for Real is perhaps the only book that attempts to answer all three questions without getting heavily mathematical. Its many problems and examples are an integral part of its approach. Just as athletes take pleasure in training their bodies, there is much satisfaction to be found in training one's mind to think in a way that is simultaneously rational and creative. With all of its puzzles and paradoxes, game theory provides a magnificent mental gymnasium for this purpose. It is the author's hope that exercising on the equipment provided by this Coursepack Edition will bring the reader the same kind of pleasure that it has brought to so many other students. (shrink)

My answer to the question why? is relatively uncontroversial among anthropologists. Sharing food makes good evolutionary sense, because animals who share food thereby insure themselves against hunger. It is for this reason that sharing food is thought to be so common in the natural world. The vampire bat is a particularly exotic example of a food-sharing species. The bats roost in caves in large numbers during the day. At night, they forage for prey, from whom they suck blood if they (...) can, but they aren’t always successful. If they fail to obtain blood for several successive nights, they die. The evolutionary pressure to share blood is therefore strong. The biologist Wilkinson [59] reports that a hungry bat begs for blood from a roostmate, who will sometimes respond by regurgitating some of the blood it is carrying in its own stomach. This isn’t too surprising when the roostmates are related, but the bats also share blood with roostmates who aren’t relatives. The behaviour is nevertheless evolutionarily stable, because the sharing is done on a reciprocal basis, which means that a bat is much more likely to help out a roostmate that has helped it out in the past. Bats that refuse to help out their fellows therefore risk not being helped out themselves in the future. Vampire bats have their own way of sharing, and we have ours. We call our way of sharing “fairness”. If the accidents of our evolutionary history had led to our sharing in some other way, it would not occur to us to attribute some special role to our current fairness norms. Whatever alternative norms we then.. (shrink)

This paper is a comparative analysis of egalitarianism and utilitarianism from a naturalistic perspective that offers some insight into the manner in which we come to make interpersonal comparisons of welfare.

Savage denied that Bayesian decision theory applies in large worlds. This paper proposes a minimal extension of Bayesian decision theory to a large-world context that evaluates an event E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document} by assigning it a number π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} that reduces to an orthodox probability for a class of measurable events. The Hurwicz criterion evaluates π\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\pi $$\end{document} (...) as a weighted arithmetic mean of its upper and lower probabilities, which we derive from the measurable supersets and subsets of E\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E$$\end{document}. The ambiguity aversion reported in experiments on the Ellsberg paradox is then explained by assigning a larger weight to the lower probability of winning than to the upper probability. However, arguments are given here that would make anything but equal weights irrational when using the Hurwicz criterion. The paper continues by embedding the Hurwicz criterion in an extension of expected utility theory that we call expectant utility. (shrink)

Where should experimental economics go next? This paper uses the literature on inequity aversion as a case study in suggesting that we could profit from tightening up our act.

This paper summarizes a theory of fairness that replaces the metaphysical foundations of the egalitarian theory of John Rawls and the utilitarian theory of John Harsanyi with evolutionary arguments. As such, it represents an attempt to realize John Mackie’s call for a theory based on the data provided by anthroplogists and the propositions proved by game theorists. The basic claim is that fairness norms evolved as a device for selecting one of the infinity of efficient equilibria of the repeated game (...) of life played by our prehuman ancestors. (shrink)

Metaphysical moral theories can be loosely be divided into theories of the Good or Right. Naturalistic theories can be said to be theories of the seemly — what is appropriate in a particular place and time. This chapter explains how rights and duties can fit within a theory of the seemly. You have a right to do something if you do not have a duty not to do it. You have a duty to do something if you would otherwise diverge (...) from equilibrium play in the social contract currently operated by your society. (shrink)

Economists have in recent years found the theory of games to be an attractive route for exploring imperfectly competitive markets. In this collection of articles, some of the best minds in contemporary economics on both sides of the Atlantic explore both the potential and the limitations of this theoretical framework.

This chapter looks to the broad consequences of the theory of fairness advanced in this book to questions of social reform. The traditional spectrum of political attitudes that ranges from the utilitarian left to the libertarian right is rejected in favor of a more realistic opposition between the neofeudal societies in which we currently live, and the prospect of fairer and freer societies that could be created by a planned program of decentralization which is identified with the whiggery that inspired (...) the Glorious Revolution of 1688 and the Declaration of Independence of 1776. The contribution that game theory can make to such a program is briefly reviewed along with the deficiencies of the current theory. (shrink)

This chapter surveys the relevant bargaining theory, namely the Nash bargaining solution, the utilitarian bargaining solution, and the egalitarian bargaining solution. The importance of how interpersonal comparisons of utility are made is emphasized.

After demolishing metaphysical notions of moral behavior, John Mackie's Inventing Right and Wrong argues that the way forward is through the study of anthropology and game theory. This chapter begins a review of the basic ideas of game theory by explaining both the rational and the evolutionary interpretation of Nash equilibria in various games, including the Prisoners' Dilemma, the Stag Hunt Game, the Ultimatum Game, the Centipede Game, and the Nash Demand Game.

Sympathy refers to caring about another to some degree as one cares for oneself. Empathy refers to the capacity to put yourself in the position of others to see things from their point of view. Empathetic preferences compare being one person in one situation with being another person in another situation. John Harsanyi showed that mild assumptions imply that to have empathetic preferences is the same thing as having rates at which the utility units of different people are to be (...) traded off against one another. A consequence is that empathetic preferences have the same structure as Hamilton's rule, and so may have originated from strangers adopted into a clan being treated as surrogate kinfolk. But since empathetic preferences are cultural artifacts, they are subject to cultural evolution. The idea of an empathy equilibrium is introduced to capture the medium-run effects of such cultural evolution on empathetic preferences. (shrink)

This chapter explores the consequence of taking Rawls' concerns about the strains of commitment to their logical extreme. If there is no external enforcement at all, so that all agreements must be self-policing, it is shown that deals reached in the original position will generate an egalitarian outcome, as Rawls would wish. The conclusions are broadly consistent with the class of laboratory results that psychologists refer to as “modern equity theory”. The concept of an empathy equilibrium is used to predict (...) the standard of interpersonal comparison needed to operate an egalitarian norm that will evolve in the medium run. The manner in which this standard should be expected to respond to need, effort, ability, and status is then explored. (shrink)