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)

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)

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)

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

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)

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)

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)

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 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.

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)

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)

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.

’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.

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.

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 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.

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 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)

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)

We use probability-matching variations on Ellsberg’s single-urn experiment to assess three questions: (1) How sensitive are ambiguity attitudes to changes from a gain to a loss frame? (2) How sensitive are ambiguity attitudes to making ambiguity easier to recognize? (3) What is the relation between subjects’ consistency of choice and the ambiguity attitudes their choices display? Contrary to most other studies, we find that a switch from a gain to a loss frame does not lead to a switch from ambiguity (...) aversion to ambiguity neutrality and/or ambiguity seeking. We also find that making ambiguity easier to recognize has little effect. Finally, we find that while ambiguity aversion does not depend on consistency, other attitudes do: consistent choosers are much more likely to be ambiguity neutral, while ambiguity seeking is much more frequent among highly inconsistent choosers. (shrink)

This volume explores from multiple perspectives the subtle and interesting relationship between the theory of rational choice and Darwinian evolution. In rational choice theory, agents are assumed to make choices that maximize their utility; in evolution, natural selection 'chooses' between phenotypes according to the criterion of fitness maximization. So there is a parallel between utility in rational choice theory and fitness in Darwinian theory. This conceptual link between fitness and utility is mirrored by the interesting parallels between formal models of (...) evolution and rational choice. The essays in this volume, by leading philosophers, economists, biologists and psychologists, explore the connection between evolution and rational choice in a number of different contexts, including choice under uncertainty, strategic decision making and pro-social behaviour. They will be of interest to students and researchers in philosophy of science, evolutionary biology, economics and psychology. (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)

Our paper “Experimental Economics: Where Next?” contains a case study of Ernst Fehr and Klaus Schmidt’s work in which it is shown that the claims they make for the theory of inequity aversion are not supported by their data. The current issue of JEBO contains two replies, one from Fehr and Schmidt1 themselves, and the other from Catherine Eckel and Herb Gintis. Neither reply challenges any claims we make about matters of fact in our critique of Fehr and Schmidt on (...) inequity aversion, although it is clear that if they could have refuted any single factual sentence then they would have done so. Both replies therefore implicitly concede that the facts quoted in our case study are correct. All the other issues raised in the two replies are just so much dust kicked up to distract attention from the only question that matters: Is it scientific to proceed like Fehr and Schmidt or is it not? Fehr and Schmidt say yes. So do Eckel and Gintis. The implications are quite far-reaching for those like us who think it is obvious that the answer is no. What other claims asserted by the school of Gintis et al can we trust? (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)

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 commentary on Philip Kitcher’s Ethical Project compares his theory of the evolution of morality with my less ambitious theory of the evolution of fairness norms that seeks to flesh out John Mackie’s insight that one should use game theory as a framework within which to assess anthropological data. It lays particular stress on the importance of the folk theorem of repeated game theory, which provides a template for the set of stable social contracts that were available to ancestral hunter-gatherer (...) communities. It continues by drawing attention to the relevance of Harsanyi’s theory of empathetic preferences in structuring the fairness criteria that evolved as one response to the equilibrium selection problem that the folk theorem demonstrates is endemic in our species. (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.

This paper suggests that rights are best seen as being part of the description of a social state rather than as constituents of the mechanism by means of which society selects a social state. A theory of this kind is outlined in which a social state is modeled as an equilibrium in the game of life played by the citizens of a society.

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)

This article is my latest attempt to come up with a minimal version of my evolutionary theory of fairness, previously summarized in my book Natural Justice. The naturalism that I espouse is currently unpopular, but Figure 1 shows that the scientific tradition in moral philosophy nevertheless has a long and distinguished history. John Mackie's Inventing Right and Wrong is the most eloquent expression of the case for naturalism in modern times. Mackie's demolition of the claims made for a priori reasoning (...) in moral philosophy seem unanswerable to me. (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 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)

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.

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

The coverage is based on a research paper Insidious Discrimination? Disentangling the Beauty Premium on a Game Show which examined the strategic behaviour of participants on a Dutch TV game show Shafted.

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.

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)

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.

Among other things, Baumard et al.'s considers the enforcement and establishment of moral norms, the interpersonal comparison of welfare, and the structure of fairness norms. This commentary draws attention to the relevance of the game theory literature to the first and second topic, and the social psychology literature to the third topic.

This article is a contribution to a symposium celebrating the life of Patrick Suppes. It describes the context in which he made contributions relevant to two extremes of the game theory spectrum. At one extreme, he made an experimental study of whether laboratory subjects learn to use Von Neumann’s minimax theory in games of pure conflict. At the other extreme, he invented a theory of empathetic identification that lies at the root of an approach to making interpersonal comparisons needed for (...) the study of games in which cooperation is central rather than conflict. These pieces of work are peripheral to his major interests, but they nevertheless illustrate how it is possible to be an academic success without conceding anything to current academic fashion. (shrink)