The bounds of reason pdf

Methods Citations. Results Citations. Figures from this paper. Citation Type. Has PDF. Publication Type. More Filters. The future of behavioral game theory. Behavioral economics has rejuvenated economic theory and deepened the bonds between economic theory and the other social sciences.

Neoclassical economics does not depend on individual preferences … Expand. Game Theory as Psychological Investigation. Over the course of history mathematics and science have become increasingly entangled with one another.

This has been especially true in the physical science wherein mathematical derivations have … Expand. Highly Influenced.

View 3 excerpts, cites background. Accounting for constitutive rules in game theory. Game theory and rules are deeply intertwined for at least two reasons: first, in many cases rules are necessary to break the indeterminacy that surrounds most of the games; second, in the past 30 … Expand. View 8 excerpts, cites background. This paper argues that most game-theoretic accounts of institutions in economics are eliminativist: they reduce institutions to behavioral patterns the players are incentivized to implement.

However, … Expand. Game theory and rules are deeply intertwined for at least two reasons: first, in many cases rules are necessary to break the indeterminacy that surrounds most of the games; second, in the past thirty … Expand.

View 7 excerpts, cites background. Economics and Philosophy. Game theory is central to understanding human behavior and relevant to all of the behavioral sciences—from biology and economics, to anthropology and political science.

However, as The Bounds of Reason demonstrates, game theory alone cannot fully explain human behavior and should instead complement other key concepts championed by the behavioral disciplines.

Herbert Gintis shows that just as game theory without broader social theory is merely technical bravado, so social theory without game theory is a handicapped enterprise. This edition has been thoroughly revised and updated.

Reinvigorating game theory, The Bounds of Reason offers innovative thinking for the behavioral sciences. This splendid book makes skillful use of figures and algebra, and reads like a charm. He is ever ready to pose unusual questions and to defend unorthodox proposals. The Bounds of Reason is Gintis' most ambitious project to date, one that draws upon all of his extraordinary originality and learning.

One part develops an epistemic theory of the rational actor as an alternative to what is provided by classical game theory, and the other part is a spirited plea to use behavioral game theory as a unifying tool in all behavioral sciences. Both objectives are highly valuable, but combing them both creates friction.

Friction creates heat, and Gintis, who thrives gleefully on controversial issues, may be enjoying the prospect of heated discussions. In fact, game theory is complementary to ideas developed and championed in all the behavioral disciplines.

Behavioral scientists who have rejected game theory in reac- tion to the extravagant claims of some of its adherents may thus want to reconsider their positions, recognizing the fact that, just as game theory without broader social theory is merely technical bravado, so social theory without game theory is a handicapped enterprise.

The reigning culture in game theory asserts the sufficiency of game the- ory, allowing game theorists to do social theory without regard for either the facts or the theoretical contributions of the other social sciences. Game theory is a wonderful hammer, indeed a magical hammer. The most fundamental failure of game theory is its lack of a theory of when and how rational agents share mental constructs.

The assumption that humans are rational is an excellent first approximation. But, the Bayesian rational actors favored by contemporary game theory live in a universe of subjectivity and instead of constructing a truly social epistemology, game theorists have developed a variety of subterfuges that make it appear that ra- tional agents may enjoy a commonality of belief common priors, common knowledge , but all are failures.

This social epistemology characterizes our species. The bounds of reason are thus not the irrational, but the social. That game theory does not stand alone entails denying methodological individualism, a philosophical position asserting that all social phenomena can be explained purely in terms of the characteristics of rational agents, the actions available to them, and the constraints that they face.

This position is incorrect because, as we shall see, human society is a system with emergent properties, including social norms, that can no more be analytically derived from a model of interacting rational agents than the chemical and biological properties of matter can be analytically derived from our knowledge of the properties of fundamental particles. Evolutionary game theory often succeeds where classical game theory fails Gintis The evolutionary approach to strategic interaction helps us understand the emergence, transformation, and stabilization of behav- iors.

In evolutionary game theory, successful strategies diffuse across pop- ulations of players rather than being learned inductively by disembodied ra- tional agents. Evolutionary game theory allows us to investigate the interaction of learning, mutation, and imitation in the spread of strategies when information processing is costly. But evolutionary game theory cannot deal with unique events, such as strangers interacting in a novel environment or Middle East peace negoti- ations.

Moreover, by assuming that agents have very low-level cognitive capacities, evolutionary game theory ignores one of the most important of human capacities, that of being able to reason. Human society is an evolved system, but human reason is one of the key evolutionary forces involved.

This book champions a unified approach based on modal logic, epistemic game theory, and social epistemology as an alternative to classical and a supplement to evolutionary game theory. This approach holds that human behavior is most fruitfully modeled as the interaction of rational agents with a social epistemology, in the con- text of social norms that act as correlating devices that choreograph social interaction.

This approach challenges contemporary sociology, which re- jects the rational actor model. My response to the sociologists is that this rejection is the reason sociological theory has atrophied since the death of Talcott Parsons in The self-conceptions and dividing lines among the behavioral disciplines make no scientific sense. How can there be three separate fields, sociology, anthropology, and social psychology, for instance, studying social behav- ior and organization?

How can the basic conceptual frameworks for the three fields, as outlined by their respective Great Masters and as taught to Ph. In the name of sci- ence, these arbitrarities must be abolished. Game theory is a tool for investigating the world. By allowing us to specify carefully the conditions of social interaction player characteristics, rules, informational assumptions, payoffs , its predictions can be tested and the results can be replicated in different laboratory settings.

For this rea- son, behavioral game theory has become increasingly influential in setting research priorities. This aspect of game theory cannot be overstressed be- cause the behavioral sciences currently consist of some fields where theory has evolved virtually without regard for the facts and others where facts abound and theory is absent.

Economic theory has been particularly compromised by its neglect of the facts concerning human behavior. This situation became clear to me in the summer of , when I happened to be reading a popular introductory graduate text on quantum mechanics, as well as a leading graduate text on microeconomics. The physics text began with the anomaly of blackbody radiation, which could not be explained using the standard tools of electro- magnetic theory.

In , Max Planck derived a formula that fit the data perfectly, assuming that radiation was discrete rather than continuous. The text contin- ued, page after page, with new anomalies Compton scattering, the spectral lines of elements of low atomic number, etc. By contrast, the microeconomics text, despite its beauty, did not contain a single fact in the whole thousand-page volume.

A bounty of excellent economic theory was developed in the twentieth century in this manner. But, the well has run dry. We will see that empirical evidence chal- lenges the very foundations of both classical game theory and neoclassical economics. Future advances in economics will require that model-building dialogue with empirical testing, behavioral data-gathering, and agent-based models.

Preface xvii A simple generalization can be made: decision theory has developed valid algorithms by which people can best attain their objectives. Given these objectives, when people have the informational prerequisites of decision theory, yet fail to act as predicted, the theory is generally correct and the observed behavior faulty. Indeed, when deviations from theoretical pre- dictions are pointed out to intelligent individuals, they generally agree that they have erred. By contrast, the extension of decision theory to the strate- gic interaction of Bayesian decision makers has led to a limited array of useful principles and when behavior differs from prediction, people gener- ally stand by their behavior.

Most users of game theory remain unaware of this fact. Rather, the con- temporary culture of game theory as measured by what is accepted without complaint in a journal article is to act as if epistemic game theory, which has flourished in the past two decades, did not exist.

Thus, it is virtually uni- versal to assume that rational agents play mixed strategies, use backward induction and more generally, play a Nash equilibrium.

When people do not conform to these expectations, their rationality is called into question, whereas in fact, none of these assumptions can be successfully defended. Rational agents just do not behave the way classical game theory predicts, except in certain settings such as anonymous market interactions. The reason for the inability of decision theory to extend to strategic in- teraction is quite simple.

In strategic interaction, nothing guaran- tees that all interacting parties have mutually consistent beliefs. Yet, as we shall see, a high degree of intersubjective belief consistency is required to ensure that agents play appropriately coordinated strategies. The behavioral sciences have yet to adopt a serious commitment to link- ing basic theory and empirical research. Indeed, the various behavioral disciplines hold distinct and incompatible models of human behavior, yet their leading theoreticians make no attempt to adjudicate these differences see chapter This bizarre state of affairs must end.

Physicists generally believe that rigor is the enemy of creative physical insight and they leave rigorous formulations to the mathe- maticians. The truth is its own justification and needs no help from rigor. Game theory can be used very profitably by researchers who do not know or care about mathematical intricacies but rather treat mathematics as but one of several tools deployed in the search for truth.

I assert then that my arguments are correct and logically argued. I will leave rigor to the mathematicians. In a companion volume, Game Theory Evolving , I stress that un- derstanding game theory requires solving lots of problems.

I also stress therein that many of the weaknesses of classical game theory have beauti- ful remedies in evolutionary game theory. Neither of these considerations is dealt with in The Bounds of Reason, so I invite the reader to treat Game Theory Evolving as a complementary treatise. The intellectual environments of the Santa Fe Institute, the Central Eu- ropean University Budapest , and the University of Siena afforded me the time, resources, and research atmosphere to complete The Bounds of Rea- son.

Somanathan, Lones Smith, Roy A. Thanks especially to Sean Brocklebank and Yusuke Narita, who read and corrected the entire manuscript. They are psychological. Anonymous People often make mistakes in their maths.

This does not mean that we should abandon arithmetic. Decision theory depends on probability theory, which was developed in the seventeenth and eighteenth centuries by such notables as Blaise Pascal, Daniel Bernoulli, and Thomas Bayes. A rational actor need not be selfish. Indeed, if rationality implied selfishness, the only rational individuals would be sociopaths.

Beliefs, called subjective priors in decision theory, logically stand between choices and payoffs. Be- liefs are primitive data for the rational actor model. In fact, beliefs are the product of social processes and are shared among individuals. To stress the importance of beliefs in modeling choice, I often describe the rational actor model as the beliefs, preferences and constraints model, or the BPC model.

While there are eminent critics of preference consistency, their claims are valid in only a few narrow areas. Behavioral decision theorists have argued that there are important areas in which individuals appear to have inconsistent preferences.

We show in this chapter that, assuming individuals know their preferences, adding in- formation concerning the current state of the individual to the choice space eliminates preference inconsistency. When we are hungry, scared, sleepy, or sexually deprived, our preference ordering ad- justs accordingly.

The idea that we should have a utility function that does not depend on our current wealth, the current time, or our current strate- gic circumstances is also not plausible. More- over, territoriality in many species is an indication of loss aversion Chap- ter Decision Theory and Human Behavior 3 leased into elaborate spatial models of flowerbeds.

Humans, by contrast, are tested using imperfect analytical models of real-life lotteries. While it is important to know how humans choose in such situations, there is cer- tainly no guarantee they will make the same choices in the real-life situa- tion and in the situation analytically generated to represent it. Evolutionary game theory is based on the observation that individuals are more likely to adopt behaviors that appear to be successful for others.

Neuroscientists increas- ingly find that an aggregate decision making process in the brain synthe- sizes all available information into a single unitary value Parker and New- some ; Schall and Thompson This error prediction mechanism has the drawback of seeking only local optima Sugrue, Corrado, and Newsome Mon- tague and Berns address this problem, showing that the orbitofrontal cortex and striatum contain a mechanism for more global predictions that in- clude risk assessment and discounting of future rewards.

Their data suggest a decision-making model that is analogous to the famous Black-Scholes options-pricing equation Black and Scholes The existence of an integrated decision-making apparatus in the human brain itself is predicted by evolutionary theory. The fitness of an organism depends on how effectively it make choices in an uncertain and varying en- vironment.

But in three separate groups of animals, craniates vertebrates and related creatures , arthropods including insects, spiders, and crustaceans , and cephalopods squid, oc- topuses, and other mollusks , a central nervous system with a brain a cen- trally located decision-making and control apparatus evolved.

The phylo- genetic tree of vertebrates exhibits increasing complexity through time and increasing metabolic and morphological costs of maintaining brain activity. Thus, the brain evolved because larger and more complex brains, despite their costs, enhanced the fitness of their carriers.

Brains therefore are in- eluctably structured to make consistent choices in the face of the various constellations of sensory inputs their bearers commonly experience. Before the contributions of Bernoulli, Savage, von Neumann, and other experts, no creature on Earth knew how to value a lottery. The fact that people do not know how to evaluate abstract lotteries does not mean that they lack consistent preferences over the lotteries that they face in their daily lives.

We usually write the proposition. Decision Theory and Human Behavior 5 3. Thus, completeness implies reflexivity. It is hard to see how this condition could fail for anything we might like to call a preference ordering.

The third condition, independence of irrelevant alternatives IIA means that the relative attractiveness of two choices does not depend upon the other choices available to the individual. For instance, suppose an individual generally prefers meat to fish when eating out, but if the restaurant serves lobster, the individual believes the restaurant serves superior fish, and hence prefers fish to meat, even though he never chooses lobster; thus, IIA fails.

When IIA fails, it can be restored by suitably refining the choice set. For instance, we can specify two qualities of fish instead of one, in the preceding example. In this new choice space, IIA is trivially satisfied. Their analysis applies, however, only to a narrow range of choice situations. Formally, we say that a preference function u W A! We have the following theorem. It is clear that u. Con- versely, if both u. The first half of the theorem is true because if f is strictly increasing, then u.

For the second half, suppose u. Then f. In his Foundations of Economic Analysis , economist Paul Samuelson removed the hedonistic as- sumptions of utility maximization by arguing, as we have in the previous section, that utility maximization presupposes nothing more than transitiv- ity and some harmless technical conditions akin to those specified above.

Rational does not imply self-interested. There is nothing irrational about caring for others, believing in fairness, or sacrificing for a social ideal. Nor do such preferences contradict decision theory. We can then treat p as the price of a unit contribution to charity and model the individual as maximizing his utility for personal consumption x and contributions to charity y, say u.

Indeed, Andreoni and Miller have shown that in making choices of this type, consumers behave in the same way as they do when choosing among personal consumption goods; i. Decision theory does not presuppose that the choices people make are welfare-improving.

In fact, people are often slaves to such passions as smoking cigarettes, eating junk food, and engaging in unsafe sex.

These behaviors in no way violate preference consistency. If humans fail to behave as prescribed by decision theory, we need not conclude that they are irrational. In fact, they may simply be ignorant or misinformed.

However, if human subjects consistently make intransitive choices over lotteries e. The latter is often called performance error. Performance error can be reduced or eliminated by formal instruction, so that the experts that society relies upon to make efficient decisions may behave quite rationally even in cases where the average individual violates preference consistency.

Preference consistency flows from evolutionary biology Robson Decision theory often applies extremely well to nonhuman species, includ- ing insects and plants Real ; Alcock ; Kagel, Battalio, and Green Biologists define the fitness of an organism as its expected number of offspring. Assume, for simplicity, asexual reproduction. A maximally fit individual will then produce the maximal expected number of offspring, each of which will inherit the genes for maximal fitness.

Thus, fitness max- imization is a precondition for evolutionary survival. For instance, moths fly into flames and humans voluntarily limit family size. Rather, organisms have preference orderings that are themselves subject to selection according to their ability to promote fitness Darwin We can expect preferences to satisfy the completeness condition because an organism must be able to make a consistent choice in any situation it habitually faces or it will be outcompeted by another whose preference ordering can make such a choice.

This biological explanation also suggests how preference consistency might fail in an imperfectly integrated organism. Suppose the organism has three decision centers in its brain, and for any pair of choices, majority rule determines which the organism prefers.

Of course, if an objective fitness is associated with each of these choices, Darwinian selection will favor a mutant who suppresses two of the three decision centers or, better yet, integrates them. For instance, smok- ers may know that their habit will harm them in the long run, but cannot bear to sacrifice the present urge to indulge in favor of the far-off reward of a healthy future.

Similarly, a couple in the midst of sexual passion may appreciate that they may well regret their inadequate safety precautions at some point in the future, but they cannot control their present urges.

We call this behavior time-inconsistent. Decision Theory and Human Behavior 9 Are people time-consistent? Take, for instance, impulsive behavior. Ec- onomists are wont to argue that what appears to be impulsive—cigarette smoking, drug use, unsafe sex, overeating, dropping out of school, punching out your boss, and the like—may in fact be welfare-maximizing for people who have high time discount rates or who prefer acts that happen to have high future costs.

Controlled experiments in the laboratory cast doubt on this explanation, indicating that people exhibit a systematic tendency to discount the near future at a higher rate than the distant future Chung and Herrnstein ; Loewenstein and Prelec ; Herrnstein and Prelec ; Fehr and Zych ; Kirby and Herrnstein ; McClure et al. For instance, consider the following experiment conducted by Ainslie and Haslam It is instructive to see exactly where the consistency conditions are vio- lated in this example.

However, time inconsistency disappears if we model the individuals as choosing over a slightly more complicated choice space in which the dis- tance between the time of choice and the time of delivery of the object cho- sen is explicitly included in the object of choice.

Of course, if you are not time-consistent and if you know this, you should not expect that your will carry out your plans for the future when the time comes. Thus, you may be willing to precommit yourself to making these future choices, even at a cost. This is called exponential discounting and is widely assumed in economic models.

For in- stance, suppose an individual can choose between two consumption streams x D x0 ; x1 ; : : : or y D y0 ; y1 ; : : :.

According to exponential discounting, he has a utility function u. The indi- vidual strictly prefers consumption stream x over stream y if and only if U. For instance, continuing the previous example, let z t mean 5 Throughout this text, we write x 2. The value of x0 is thus u. But u. There is also evidence that people have different rates of discount for dif- ferent types of outcomes Loewenstein ; Loewenstein and Sicherman This would be irrational for outcomes that could be bought and sold in perfect markets, because all such outcomes should be discounted at the market interest rate in equilibrium.

But, of course, there are many things that people care about that cannot be bought and sold in perfect markets. Neurological research suggests that balancing current and future payoffs involves adjudication among structurally distinct and spatially separated modules that arose in different stages in the evolution of H. The long-term decision-making capacity is localized in specific neural structures in the prefrontal lobes and functions improperly when these areas are damaged, despite the fact that subjects with such damage appear to be otherwise com- pletely normal in brain functioning Damasio In sum, time inconsistency doubtless exists and is important in model- ing human behavior, but this does not imply that people are irrational in the weak sense of preference consistency.

For axiomatic treatment of time-dependent prefer- ences, see Ahlbrecht and Weber and Ok and Masatlioglu In fact, humans are much closer to time consistency and have much longer time horizons than any other species, probably by several orders of mag- nitude Stephens, McLinn, and Stevens ; Hammerstein We do not know why biological evolution so little values time consistency and long time horizons even in long-lived creatures. Let X be a set of prizes.

We interpret p. If X D fx1 ; : : : ; xn g for some finite number n, we write p. The expected value of a lottery is the sum of the payoffs, where each payoff is weighted by the probability that the payoff will occur. Consider the lottery l1 in figure 1. Note that we model a lottery a lot like an extensive form game—except that there is only one player.

Consider the lottery l2 with the three payoffs shown in figure1. Here p is the probability of winning amount a, q is the probability of winning amount b, and 1 p q is the probability of winning amount c.

A lottery with n payoffs is given in figure 1. The prizes are now a1 ; : : : ; an with probabilities p1 ; : : : ; pn , respectively. C pn an. Lotteries with two, three, and n potential outcomes. Von Neumann and Morgenstern , Friedman and Savage , Savage , and Anscombe and Aumann showed that the expected utility principle can be derived from the assumption that individuals have consistent preferences over an appropriate set of lotteries. X that associates with each state of nature! Note that this concept of a lottery does not include a probability distribution over the states of nature.

Rather, the Savage axioms allow us to associate a subjective prior over each state of nature! We suppose that the individual chooses among lotteries without knowing the state of nature, after which Nature chooses the state! We state this more formally as follows. Suppose we also have A D f!

This axiom says, reasonably enough, that the relative desirability of two lotteries does not depend on the payoffs where the two lotteries agree. Because of A1, this is well defined i. Our third condition asserts that the probability that a state of nature occurs is independent of the outcome one receives when the state occurs.

The diffi- culty in stating this axiom is that the individual cannot choose probabilities but only lotteries. The fourth condition is a weak version of first-order stochastic domi- nance, which says that if one lottery has a higher payoff than another for any event, then the first is preferred to the second. Finally, we need a technical property to show that a preference relation can be represented by a utility function. The proof of this theorem is somewhat tedious; it is sketched in Kreps There is always uncertainty as to the degree of success of the various options in X , which means essentially that each x 2 X determines a lottery that pays i offspring with probability pi.

Then the expected number of offspring from this lottery is. Let L be a lottery on X that delivers xi 2 X with probability qi for i D 1; : : : ; k. The probability of j offspring given L is then kiD1 qi pj. See also Cooper Machina re- views this body of evidence and presents models to deal with them. We sketch here the most famous of these anomalies, the Allais paradox and the Ellsberg paradox.

They are, of course, not paradoxes at all but simply empirical regularities that do not fit the expected utility principle. Maurice Allais offered the following scenario. Which would you choose? This pair of choices is not consistent with the expected utility principle. To see this, let us write uh D u. Why do people make this mistake? Perhaps because of regret, which does not mesh well with the expected utility principle Loomes ; Sugden The Allais paradox is an excellent illustration of problems that can arise when a lottery is consciously chosen by an act of will and one knows that one has made such a choice.

The regret in the first case arises because if one chose the risky lottery and the payoff was zero, one knows for certain that one made a poor choice, at least ex post. Hence, there is no regret in the second case.

But in the real world, most of the lotteries we experience are chosen by default, not by acts of will. Thus, if the outcome of such a lottery is poor, we feel bad because of the poor outcome but not because we made a poor choice.

Another classic violation of the expected utility principle was suggested by Daniel Ellsberg Consider two urns. Urn A has 51 red balls and 49 white balls. Urn B also has red and white balls, but the fraction of red balls is unknown. One ball is chosen from each urn but remains hidden from sight.

Subjects are asked to choose in two situations. Many subjects choose the ball from urn A in both cases. This violates the expected utility principle no matter what probability the subject places on the probability p that the ball from urn B is white. For in the first situation, the payoff from choosing urn A is u. In the second situation, the payoff from choosing urn A is u. This shows that the expected utility principle does not hold.

Whereas the other proposed anomalies of classical decision theory can be interpreted as the failure of linearity in probabilities, regret, loss aversion, and epistemological ambiguities, the Ellsberg paradox strikes even more deeply because it implies that humans systematically violate the following principle of first-order stochastic dominance FOSD.

Let p. The usual explanation of this behavior is that the subject knows the prob- abilities associated with the first urn, while the probabilities associated with the second urn are unknown, and hence there appears to be an added degree of risk associated with choosing from the second urn rather than the first.

If decision makers are risk-averse and if they perceive that the second urn is considerably riskier than the first, they will prefer the first urn. Of course, with some relatively sophisticated probability theory, we are assured that there is in fact no such additional risk, it is hardly a failure of rationality for subjects to come to the opposite conclusion.

The Ellsberg paradox is thus a case of performance error on the part of subjects rather than a failure of rationality. We thus say utilities over outcomes are ordinal, meaning we can say that one bundle is preferred to another, but we cannot say by how much. By contrast, the next theorem shows that utilities over lotteries are cardinal, in the sense that, up to an arbitrary constant and an arbitrary positive choice of units, utility is numerically uniquely defined.

Decision Theory and Human Behavior 19 u. For a proof of this theorem, see Mas-Collel, Whinston, and Green , p. If X D R, so the payoffs can be considered to be money, and utility satisfies the expected utility principle, what shape do such utility functions have? It would be nice if they were linear in money, in which case expected utility and expected value would be the same thing why?

But generally utility is strictly concave, as illustrated in figure 1. We say a function u W X! R is strictly concave if, for any x; y 2 X and any p 2. We say u. To see this, note that the expected value of the lottery is E D px C. But by definition, the utility of the expected value of the lottery is at D, which lies above H.

This proves that the utility of the expected value is greater than the expected value of the lottery for a strictly concave utility function. What are good candidates for u. It is easy to see that strict concav- ity means u But there are lots of functions with this property.

According to the famous Weber-Fechner law of psychophysics, for a wide range of sen- sory stimuli and over a wide range of levels of stimulation, a just noticeable change in a stimulus is a constant fraction of the original stimulus.

If this holds for money, then the utility function is logarithmic. We say an individual is risk-averse if the individual prefers the expected value of a lottery to the lottery itself provided, of course, the lottery does not offer a single payoff with probability 1, which we call a sure thing. We know, then, that an individual with utility function u. Clearly, an individual is risk-neutral if and only if he has linear utility. We may define individual A to be more risk-averse than individual B if whenever A prefers a lottery to an amount of money x, B will also prefer the lottery to x.

We say A is strictly more risk-averse than B if he is more risk-averse and there is some lottery that B prefers to an amount of money x but such that A prefers x to the lottery. Clearly, the degree of risk aversion depends on the curvature of the utility function by definition the curvature of u.

The most plausible explanation is that people enjoy the act of gambling. The same woman who will have insurance on her home and car, both of which presume risk aversion, will gamble small amounts of money for recreation. An excessive love for gambling, of course, leads an individual either to personal destruction or to wealth and fame usually the former. Decision Theory and Human Behavior 21 ticated. This is called the Arrow-Pratt coefficient of ab- solute risk aversion, and it is exactly the measure that we need.

For example, the logarithmic utility function u. Studies show that this property, called decreasing absolute risk aversion, holds rather widely Rosenzweig and Wolpin ; Saha, Shumway, and Talpaz ; Nerlove and Soedjiana Another increasing concave function is u.

Similarly, u. This utility has the additional attractive property that utility is bounded: no matter how rich you are, u. Note that for any of the utility functions u. For u. This is related to the notion that individuals adjust to an accustomed level of income, so that subjective well-being is associated more with changes in income rather than with the level of income. See, for instance, Helson , Easterlin , , Lane , , and 8 If utility is unbounded, it is easy to show that there is a lottery that you would be willing to give all your wealth to play no matter how rich you are.

This is not plausible behavior. Indeed, people appear to be about twice as averse to tak- ing losses as to enjoying an equal level of gains Kahneman, Knetsch, and Thaler ; Tversky and Kahneman b.

More formally, suppose an individual has utility function v. In other words, individuals are two to three times more sensitive to small losses than they are to small gains, they exhibit declining marginal utility over gains and declining absolute marginal utility over losses, and they are very insensitive to change when all alternatives involve either large gains or large losses.

This utility function is exhibited in figure 1. The problem is that for small gambles the utility function should be almost flat. This issue has been formalized by Rabin Most subjects in the laboratory reject this lottery.