1. Classical and Modern Utilitarianism

Fundamentally, an economy is judged by the outcomes it provides to those who live in it, whether in the form of consumption, leisure, power, or an enjoyable worklife. If these outcomes are arrived at without any sacrifice, then they are not terribly interesting to study; for example, we can all enjoy a sunny afternoon without giving it much thought. The study gets more interesting when providing an outcome requires sacrifice and social coordination. For example, if there aren't enough cars for every person in the society, who should get a car? Who should get a beautiful penthouse apartment, and who should get a scruffy tenament? Economists and political philosophers have been debating this question for millennia - at least since Plato's time - and have yet to produce a unified, authoritative answer. Nevertheless, a great deal has been argued, discussed, and learned, and covering this discussion is a useful way to begin thinking about how to evaluate the effects of different economic policies. After all, if we can't agree on what the ideal economic outcome is, there is little point in debating what the most appropriate policy is to reach it.

This chapter will introduce the concept of utility and some utilitarian views on the provision of economic and social outcomes. We will begin with 19 $^{\text{th}}$ Century utilitarianism, outlining the philosophies of the two arguably most influential utilitarian economic philosophers, Jeremy Bentham and Wilfredo Pareto. Utilitarianism continued to gain influence throughout the 20 $^{\text{th}}$ century as it proved to offer an extremely tractable set of assumptions for complex modelling of human behavior, and still provides the basis for most economic modeling. A slight variation on utilitarianism, with some interesting philosophical implications, was produced by John Von Neumann and Oscar Morgenstern in 1944 as they developed the field of game theory to analyze group behavior.

Since the chapter will not begin to discuss any economic activities - production, sales, and so on - talking about ideal distributions of goods may seem a bit like building castles in the sky. Economists are not all-powerful monarchs, and the real world is not so flexible that we can simply re-draw property laws and move around people's assets at a whim to suit our needs. However, the distribution of goods changes in small ways every day. Each time something is bought or sold or given away, the goods of our society are redistributed. By starting out with a discussion of ideals, we will be able to talk about whether each little transaction inches us toward or away from those ideals. For now you will simply have to take the author's word that the concepts in this chapter will be readily applicable when we reach the material on production and distribution. In the mean time, there is some fun to be had in building castles.

1.1 The Framework of Material Self-Interest

All of the theories we examine below assume, in various ways, that a society is a group of people who wish to consume available material resources - resources here meaning the stock of a society's goods such as cars and peanut butter, as well as natural resources - and that there are not enough resources available to satisfy everyone's wants. The purpose of a theory of distribution, in this case, is to judge competing claims on those resources in order to determine which claims are most appropriate to satisfy when not all can be. To take some obvious examples, meat should not be allocated to vegetarians and lawnmowers need not be given to apartment-dwellers. Perhaps fancy cars should be given to those who are most obsessed with them, or perhaps they are simply a waste of money.

It is worth pointing out that in viewing distribution this way, we have already brushed aside some interesting ways of looking at the allocation of goods between individuals. Specifically, we are assuming that the distribution of resources is an end in itself; however, in some cases the distribution may in fact be a consequence of goals that a society deems more important. For example, a society might be comprised of ascetics who believe that all should consume the material goods necessary for survival, but that excessive consumption leads to corruption of the soul. Additional resources should therefore be used to promote spiritual enlightenment, and exactly how this should be done is spelled out in the intricately complex laws of that society's religion.

1]0.9

Gross National Happiness
Upon his ascension to the throne in 1972, King Jigme Singye Wangchuck of Bhutan, a small country in the Himalayas, coined the term of ``Gross National Happiness,'' which he has repeatedly declared more important that Gross National Product, and which he made the object of Bhutanese economic policy throughout his reign. ``Happiness takes precedence over prosperity in our development process,'' he explained. Bhutan's culture is heavily Buddhist, and believes in the attainment of spiritual goals and the avoidance of excess consumerism. For many, it seems to work: ``Maybe we are somewhat isolated from the world, but we feel part of a living community that is not just connected by wires,'' explained one Bhutanese professional. ``That's why 95 percent of us exchange students return home. By and large, you would have to say people are happy here." The government held a conference on the study of Gross National Happiness in 2004, which has been followed by other conferences on the subject in other countries. Many criticize the concept of gross national happiness on the grounds that it is subjective and therefore easy for a government to manipulate so as to declare its subjects happy. Nevertheless, the Bhutanese experience illustrates that not every society seeks material wealth, and therefore not every economic problem involves the competition for resources.

Source for quotes: ``Gross National Happiness'' by Orville Schell, published in Red Herring, January 15 $^{\text{th}}$ , 2002.

If we are to take this society's views at face value, we cannot describe its distribution problem as one of a set of competing self-interests. Yes, different individuals might have different views of how to best allocate things to achieve enlightenment. However, these views would be logical or moral arguments, and we could not balance them the way we could, say, one person's taste for turnips and another person's taste for spinach. A moral argument may be provably right or wrong according to a society's shared values, or have disturbing logical consequences. For example, Muslims (at least in theory) do not drink alcohol, and opening a liquor factory, even for export, might cause a philosophical quandry for a government that is trying to promote Islamiv values; on the other hand, decisions about the production of apples vs. oranges are unlikely to have the same dynamics. While an economist could analyze how well such a society's allocation fits what the rules of material self-interest might produce, we cannot judge the appropriateness of the allocation without making a judgment about the society's values. Clearly such an undertaking is beyond the realm of economics.

Although religious societies offer one clear set of examples where competing claims reflect more than just preferences, there are certainly instances when a secular society also faces allocation questions that are driven by values rather than self-interest. For example, in figuring out the optimal allocation or resources to law enforcement, civil liberties must be taken into account; or, in figuring out how much pollution to allow, we must consider not only preferences for clean air (which can be dealt with in a framework of competing self-interest), but also some people's fundamental rights to be free from extreme levels of pollution, and other peoples fundamental beliefs in the sacredness of natural resources.

The theories of Rawls, outlined in the next chapter, offer some means of dealing with these problems, while utilitarianism offers none.^1.1 Nevertheless, the implicit assumption of maximizing individuals' material consumption runs throughout even Rawls, and it is worth pointing out that a society may often have altogether different goals woven into its allocation decisions. With these caveats, we begin the consideration of how to deal with competing material self-interests.

1.2 Classical Utilitarianism

The notion of ``utility'' as a means of describing a person's well being was developed by the early 19 $^{\text{th}}$ century political economist Jeremy Bentham, though utilitarianism has earlier roots in the writings of Pufendorf and of Ferdinando Galiani, and its philosophical basis was further developed by John Stuart Mill and Henry Sidgwick. Bentham's description of the utilitarian goal is to provide ``the greatest good for the greatest number.'' Essentially, evaluating a particular distribution of goods becomes a question of arithmetic: First, we come up with something like a happiness-meter that spits out a number describing each person's well-being under a particular distribution of goods; then we add up the numbers to get the overall well-being of society. The accompanying selection describes the process in Bentham's own words.

In practice, when evaluating the ``good tendency'' or ``evil tendency'' of something, we need to use numbers to describe how good or bad something is (after all, according to Bentham, we have to sum up these tendencies across individuals, and there is no meaningful way to add them together if they are not numbers). We refer to that something we are evaluating as a ``state of the world,'' which could refer to a political outcome (e.g., a certain person is elected president), an economic outcome (e.g., George gets a motorcycle instead of a flat screen TV), or a state of nature (e.g., an earthquake destroys San Francisco). A number describing how well off a particular individual is in a certain state of the world is called her utility, and a function that offers a utility level for each state of the world is called a utility function. A number describing how well off society as a whole is in a certain state has come to be known as its social welfare, and a function that offers a social welfare level for each state of the world has come to be known as a social welfare function. (The more general theory of social welfare functions is a mid 20 $^{\text{th}}$ century development, due initially to Bergson and Samuelson. Bentham does not use the term.)

The accompanying example illustrates how we might use Bentham's method to evaluate a distribution of goods.

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An Example of Using Classical Utility Functions

Suppose an economy consists of just two people, Pavlov and Schroedinger. The economy's resources consist of two dogs and two cats. There are many ways of dividing up these resources among our two citizens (nine, to be precise, assuming we can't divvy up fractions of a dog or cat), but let's consider just two:

Allocation A: Each person gets one dog and one cat. We follow Bentham's process above and somehow arrive at the conclusion that Pavlov's utility is 20 and Schroedinger's utility is 25. The social welfare is therefore 20+25=45.

Allocation B: Pavlov gets both dogs and Schroedinger gets both cats. This time, we determine that Pavlov's utility is 30, while Schroedinger's is 20. The social welfare is therefore 30+20=50.

According to Bentham's criteria, Allocation B is the preferred allocation, because the total welfare is higher. Note that even though Schroedinger is less happy under allocation B, Pavlov's increase in happiness outweights Schroedinger's decrease.

A few properties of the method are worth noting, because they have come under heavy scrutiny.

First, in judging an outcome, Bentham does not take any values into account besides the preferences of the individuals involved. Such a theory of valuation, in which only the end result matters, has become known as a teleological theory of judgement. One difficulty is that it reduced the kinds of value judgments discussed in the previous section to a matter of preference: Joe is a Buddhist and would prefer that people focus on spiritual goals rather than material wealth; Jane is obsessed with visiting the newest lounge every evening and trying the specialty drink; and we have somehow determined that the utility that Jane gains from her specialty drink habit is greater than the disutility that her mindless consumerism causes for Joe. The outcome from our determination may be fine, but utilitarianism does not allow us to address the conflict between Joe and Jane in any terribly deep way.

As long as we are judging one person's need for a new shirt against another person's need for a hedge trimmer, this lack of depth probably does not matter terribly much. However, Bentham's work is meant to describe principles not only for economic allocation, but also for legislation and political decision-making. There is an underlying assumption that no rights are inalienable; freedom of expression, basic comfort, even life itself, may be traded off for the greater good of others. We can put a sock in Pat Robertson's mouth if it causes everyone else's utility to rise more than it causes Robertson's utility to fall. Many people find such a view of politics difficult to digest. Most utilitarians would respond that they are pragmatic utilitarians, and will only use utility theory to judge between outcomes that do not violate people's basic values. Nevertheless, values can enter many economic and political decisions in small and subtle ways (for example when driving your car causes an ever so slight increase in global warming and all of its myriad political consequences), so some caution is in order.

Second, the way that Bentham uses utility is cardinal: We have determined not only that Pavlov receives a higher utility under allocation B, but how much higher. Specifically, his increase in utility going from A to B is 10; if it had been, say, 4 instead, Allocation B no longer would have looked so attractive. The difficulty with cardinal utility functions is that it is not clear what we are measuring; human emotions are complex, and if there is some gauge of contentment that can be broken into measurable units, psychologists have not discovered it yet. Moreover, even if it did exist, we would have to be able to measure it to make use of it. By contrast, an ordinal utility function does not use any information beyond the order of the utility numbers: A utility of 30 is higher than a utility of 24, and a utility of 24 is higher than a utility of 20, but one does not interpret whether one is a little bit or a lot higher than the other. As we will see in a short while, ordinal utility functions can be justified using much weaker assumptions.

Third, related to the notion of cardinal utility is that Bentham uses utility in a way that is interpersonally comparable. Interpersonal comparability means that we can determine which of two people is better off in a given state of the world and, if the utility function is cardinal, we can determine whose change in utility is larger when the world changes its state. That is, going from allocation A to allocation B above, we are comparing Pavlov's utility gain of 10 with Schroedinger's utility loss of 5. Again, like the problems with cardinal utility, the difficulty of comparability is that it assumes a lot about what goes on in people's heads. We have to assume that Pavlov's happiness can be measured against Schroedinger's; if people derive happiness from fundamentally different sources, this may be a very difficult task.

In practice, we use many measures in our daily lives that are intended to be both comparable and cardinal. For example, when we compare economic growth across countries, we are supposing that a 3% growth rate in one country is equally impressive as a 3% growth rate in another country, and that a 6% growth rate will advance a country twice as far as a 3% growth rate. Others may be comparable and ordinal. For example, Body Mass Index^1.2 (BMI) is a measure of obesity whereby anyone with a BMI over 25 is declared overweight, but a jump in BMI from 20 to 25 is not comparable to a jump in BMI from 25 to 30. Other indeces that we use in daily life are cardinal and non-comparable; for example, a ten-degree temperature jump may be comparable in two cities, but the feel of an 80-degree day may be different depending on the humidity level, altitude, etc. Assuming that an index is cardinal and/or comparable often involves some leap of faith, but how big a leap of faith is required depends on what one is measuring. In spite of its shortcomings, cardinal and interpersonally comparable utility has seen a great deal of usage even up to the present day. Stronger assumptions can often allow one to make stronger conclusions, and it turns out that these two assumptions are necessary to justify a number of powerful theorems.

1.2.1 1.3 Pareto's Contribution to Utility Theory

The unrealistic strength of Bentham's assumptions was seen as a problem even in the 19 $^{\text{th}}$ Century. The turn of the century economist Wilfredo Pareto developed a method for comparing allocations and outcomes that can be used on ordinal, non-comparable utility functions. The definition is quite general, and applies to any kind of outcome of a social decision (such as whether or not to pass a certain piece of legislation); the social decisions we are interested in here are those concerning the allocation of goods.

This definition of superiority does not require cardinal utility; all we need to know to evaluate two allocations is whether people are better off or worse off under one outcome than other another outcome. Ordinal utility functions are sufficient for this. Second, utility does not need to be interpersonally comparable. Although we are evaluating several people's utility jointly, we only need to know whether different people's utilities are higher or lower. We do not need to compare one person's utility gain of, say, 25 with another person's utility loss of 2 or 200, nor do we need to know which of two people is better off in a given state of the world. We only need the information that one person gains and the other loses.

Because Pareto superiority uses weaker assumptions than the classical utilitarian method, it often produces fewer conclusions. For example, in the economy of Pavlov and Schroedinger above, neither outcome is Pareto superior to the other; under allocation A, Schroedinger is better off, while Pavlov is better off under allocation B. Thus, simply using Pareto superiority fails to produce a recommendation as to which allocation is more appropriate.

Second, while using Pareto's definition of efficiency allows us to avoid wasteful outcomes, it does not guarantee that our outcomes will in any way be equitable. Consider the following example.

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1.2.1.1       An Example of Pareto Efficiency

Suppose we have an economy of two people, Ram and Rym. Ram is a vegetarian and Rym is not. The economy's resources consist of four hamburgers and two tofu burgers. Now, consider two allocations:

Allocation A: Each person gets two hamburgers and one tofu burger.

Allocation B: Ram gets both tofu burgers and Rym gets all four hamburgers.

We can easily imagine that both are better off under Allocation B. Ram gives up two hamburgers but doesn't care because he won't eat them anyway. In turn, he gains a tofu burger which he will eat. Rym, on the other hand, has given up one tofu burger for two hamburgers, which she considers a better bet. Thus, Allocation B is Pareto superior to Allocation A. The above decision illustrates how the consideration of Pareto superiority avoids waste: By choosing the allocation under which both are better off, we have avoided ``wasted'' utility. If people derive their utility from consumption of material resources, then we have indirectly avoided wasting those resources.

There are many cases when it can be determined that an economic outcome is wasteful. Suppose, for example, that Jorge drinks 10 bottles of imported beer per month. Then, the town he lives in decides to impose a tax on alcohol consumption to pay for a new park. As a result, the beer is more expensive, and Jorge only buys 8 bottles of beer per month. The two bottles per month that went unconsumed after the tax are a waste of utility: The government did not collect any tax on them, and Jorge did not enjoy drinking them. We have not determined whether the tax revenue was well-spent (it very well may have been), so we do not know if the tax was worthwhile. But we do know that the tax created waste, so it could be described as Pareto inferior to a (rather implausible) tax where Jorge is taxed on the first eight bottles of beer he drinks but not on any additional beer he consumes.

However, consider the following allocation:

Clearly, this is a very unfair allocation. However, assuming Rym places some value (even a small one) in having the tofu burgers, she is better off under Allocation C than under Allocations A or B. Ram, though, is worse off. We can therefore not say that Allocation C is Pareto inferior (or superior) to Allocations A or B, even though it is clearly unfair. To take a policy example, a cut in the capital gains tax^1.3 will cause an increase in investment, which will spur economic activity, thereby eliminating some waste. On the other hand, a cut in payroll taxes may not cause people to change their work hours, since most people tend to work full time regardless of the wage, and so it may not eliminate as much waste. Nevertheless, lowering the capital gains tax and raising the payroll tax would create a redistribution from middle-income to wealthy people, which may not be socially desireable.

It is worth noting that while the classical utilitarianism of Bentham also does not rule out unequal outcomes, we can generally avoid extreme cases. If we assume that people's utilities become extremely negative below a certain level of hardship and rise slowly above a certain level of wealth, then just adding up people's utilities will cause us to avoid outcomes in which anyone is extremely poor, and to not place a great deal of value on income gains by wealthy people, thereby causing us to indirectly place value on equality.

1.3 Constructing an Ordinal Utility Function

As we have seen above, an ordinal utility function is all we need to make judgments about Pareto superiority. It will also turn out to be sufficient to build up a coherent theory of demand, and to make several interesting conclusions about the effects of inflation. Unlike cardinal utility, which requires us to delve into people's heads, we can construct an ordinal utility function for an individual based on three reasonable assumptions about people's preferences. Although these assumptions will not be satisfied under all scenarios, they are not too far-fetched and they get us a great deal of mileage.

This assumption simply states that any pair of bundles under consideration can be ranked as better or worse. A violation of this assumption might be given by acts that people consider morally repugnant. For example, someone repulsed by cannibalism might refuse to decide whether she prefers the meat of a young person or an old person. Nevertheless, for more mundane comparisons, such as apples vs. oranges, the assumption is not terribly strenuous, and has not seen a great deal of objection from theorists.

Transitivity imposes some sort of consistency on people's preferences. One violation of transitivity is a phenomenon known as cyclical preferences: The person prefers bundle A to B, B to C, and C to A. For example, in the game Rock-Paper-Scissors, paper beats rock, scissors beats paper, and rock beats scissors. If people's preferences followed this pattern, we would go round and round naming more preferred bundles without any clear sense that the preferred bundle is making the person better off than the less preferred one. On their face, cyclical preferences merely reflect the tastes of someone who is fickle and cannot make up his mind. On the other hand, people's preferences change depending on the day and the state of the world, so that a person might prefer a new t-shirt to a new coat in the summer, but prefer the new coat in the winter. Therefore, if we are imposing this assumption on the behavior of people we observe, it is important that we are not observing them in too many different scenarios; or else, perhaps, we could observe someone many, many times and hope that a person's average preferences will eventually be revealed this way.

A third and final assumption is more technical:

Continuity basically rules out sudden jumps in people's preferences where everything very close to a particular bundle is ranked one way and the bundle itself is ranked completely differently. Although the assumption appears to be about people's preferences being ``smooth,'' in practice, it is most profoundly an assumption about the relative importance of different goods or dimensions of the outcome. Remember that when there are more than two goods (such as apples and oranges) or two dimensions along which a political outcome is being measured (such as school quality and road quality), the set of outcomes that is ``close'' to another outcome includes those with a little bit less of each good (i.e., some outcomes with a little bit fewer apples as well as some outcomes with a little bit fewer oranges; some outcomes with a slightly lower school quality and some outcomes with a slightly lower road quality). Assumption $U3$ ensures that when we have two dimensions of outcomes under consideration (for example, income and leisure time), that one can substitute one for the other. For example, someone with $500 per week of income and 30 hours per week of leisure time would have the same gain in utility from a little bit more money or a suitably small gain in leisure time.

Figure 1.1: Continuous preferences.

A clear example of preferences where assumption $U3$ is violated is that of lexographical preferences. Suppose we have two goods $X$ and $Y$ . Someone with lexographic preferences will always prefer more of either good, but would prefer any (even very small) additional amount of good $X$ to any (even very large) additional amount of good $Y$ . One possible example of lexographical preferences is indicated by the ``hierarchy of needs'' developed by the psychologist Abraham Maslow.^1.4 Maslow postulated that people order their needs in importance, first satisfying physiological needs (food, sleep, etc), then safety needs, then needs for love or belonging, then needs for self-esteem, then needs for self-actualization. If we take this hierarchy literally, then it says that someone who has a physiological need (for example, thirst), will give up any amount of self-esteem for a glass of water. Or, to preview an example form Chapter , the philosopher John Rawls argued that considerations of liberty take precedence over considerations of enjoyment. Therefore, a person should be willing to give up any amount of enjoyment for a small increase in society's liberty. Either of these types of preferences violate the continuity assumption. If assumption $U3'$ is still satisfied, i.e., if there is only a finite or countable number of outcomes under consideration, we can still construct a utility function that describes such preferences; otherwise, we cannot.

With the assumptions $U1$ to $U3$ , we can generate a complete ranking of bundles of goods from best to worst. For example, if there is a finite number of bundles of goods under consideration, we can line them up in order from the least preferred bundle on the bottom to the most preferred bundle on the top. Obviously some bundles may be side-by-side with each other. As long as preferences are transitive, we will be able to build this stack without ever putting a less-preferred bundle above a more-preferred bundle.

Once we have generated this ranking, we can simply number the bundles from top to bottom, using higher numbers as we go up. In this way, we have constructed an ordinal utility function. When there is an infinite number of possible allocations, the math gets a little trickier (and we need the continuity assumption to ensure that the utilities will stack well), but the underlying method for constructing an ordinal utility function is the same. We will state the result formally:

Although the proof of Debreu's Theorem is intricate when there is an infinite number of possible outcomes, a basic version can be demonstrated using just some number theory; it is given in the appendix. Constructing a utility function in this way demonstrates the clear advantage of the framework of ordinal utility: We do not have to delve into people's heads to measure it, but can derive it from a few assumptions about people's preferences. To be sure, people are fickle, and it is sometimes too much to assume that their preferences are consistent. However, the assumption is not far-fetched.

Moreover, not only are the assumptions about preferences weaker, but preferences are not too difficult to observe. We will study this in detail in the next chapter when we study the technique of revealed preference, but a brief preview is worth mentioning here. Basically, if a consumer can afford both bundles A and B, and chooses bundle B, then we conclude that she prefers bundle B to bundle A (or at the very least doesn't strictly prefer A). We will see that if we can generate enough observed preferences for the same individual, we can begin to approximate that person's utility function.

As we will see, the assumptions of ordinal utility, along with a couple of extra assumptions about preferences, will be enough to generate a consistent theory of consumer demand, in which we can consider whether individual preferences lead to Pareto optimal purchasing decisions. Such a theory of demand was finalized by Pareto and has been known for over 100 years. However, theories of distribution in the 20 $^{\text{th}}$ Century have focused on additional properties beyond Pareto optimality that a reasonable allocation should have; the conception of fairness, notably missing from Pareto optimality, has featured especially strongly among them.

1.4 Game Theory as a Theory of Distribution

In 1944, John von Neumann and Oscar Morgenstern introduced their seminal work, The Theory of Games and Economic Behavior. The purpose of game theory is to analyze the outcome of human interactions when each of the individuals involved has different interests and behaves strategically. Their goal was to be able to make formal statements about group behavior. Nevertheless, many theorists have begun to see game theory as a tool for evaluating more general economic outcomes, because it considers not only the effect of a group interaction on people's well-being, but also the context in which group members (e.g., people living in an economy) interact.

Game theory gets its name because it analyzes strategic human interactions as if they were games. Such interactions have players, moves, strategies, and outcomes. The format of the game of chess is that the two players (black and white) alternate moves, and the outcome is that after several moves, one player wins and the other player loses. Abstracting just slightly, war can also be analyzed this way. Generals have various strategic moves that they may make, either alternately or simultaneously. The outcomes of war are generally more complex than simply winning or losing; some outcomes may require more casualties, and some victories may be more complete than others.^1.5 Game theory was built to be sufficiently abstract to accommodate games with heavily interdependent moves and multiple outcomes.

Many types of political and economic behavior can be analyzed in this vein. They require complex interactions of individuals, and their outcomes depend on precisely which sequence of interactions takes place and how. For example, in industries with a handful of dominant firms, those firms may decide to collude to keep prices high, or to undercut each other in a bid for market share, or to stay out of each others' key markets altogether. Their revenues and profits depend on both the firms' strategies and their competitors' responses. Consumers may respond strategically to the choices of their friends (for example, using Friendster or Facebook or Myspace, depending on the choices of people they know), and household members may make choices about cleaning, cooking, and even work based on how they expect other household members to react.

Like the utilitarian models of the 19 $^{\text{th}}$ Century, game theory considers how different individuals with competing interests might end up using resources. In fact, the individuals playing the games are assumed to have utility functions. But while 19 $^{\text{th}}$ -Century utilitarianism evaluates outcomes based purely on how well they satisfy individual preferences, game theory introduced a new concept into the evaluation of outcomes: The idea that the rules people follow to achieve those outcomes are important, and ought to be evaluated in their own right.

Following rules is an important part of the process of economic allocation, both because rules have moral consequences - to take a simple example, stealing will affect the allocation of goods - and because we encounter unfamiliar economic situations every day. People's usual behavior wheen they face an unknown situation is to follow rules that are in someway parallel to those that worked in a previous, familiar situation. For example, I may be trying to sell a house, and I may not know the correct selling price. In an ideal world, I might commission a study of people looking to buy a house, and learn that way what the level of demand is. But that would be prohibitively expensive. Instead, I will probably look at houses in the same neighborhood of a similar size in similar condition.

Based on the idea that people's economic behavior consists of following rules, we can use game theory to analyze an economic scenario and ask: What behavioral rules are available to the players involved? Which rules are fair or unfair? Which rules lead to the best outcome? In this way, we can consider the rules themselves when evaluating an outcome; we can even make recommendations based on the rules alone when we are unable to observe the complete outcome. Solutions of games fall into two major categories: Cooperative games and non-cooperative games. We will examine each in turn.

1.4.1 Cooperative Games

In a cooperative game, the players can agree on a group strategy in advance, and force each other to stick to an agreed-upon strategy. For the purpose of resource allocation, the type of cooperative game we are most interested in is the ``game'' of bargaining. Bargaining takes place in all kinds of economic circumstances - from workers and employers bargaining over wages (whether individually or collectively) to sellers and buyers bargaining over prices. Even when no overt bargaining takes place, nearly every purchasing transaction could be seen as a bargain of sorts - the grocery store posts a price for milk, and then I walk into the grocery store and decide whether or not to accept that price.

The purpose of a theory of bargaining is not just to predict what the actual outcome might be when various players bargain; it is also to analyze what players might reasonably expect as the result of a bargaining process, and also to determine if certain types of solutions have more appealing characteristics than others. What bargaining theory has to offer to the theory of distribution is the notion of an appropriate allocation as a case of mutual advantage - that is, a situation where all parties can benefit from interacting with each other. Social interactions - whether in the marketplace, at work, among family members, or between friends - take place when everyone involved expects to gain something from participating. It is not always obvious how the rewards from such interaction should be split up; bargaining can sometimes determine the answer. The implicit assumption here is that there is a default allocation if the parties refuse to bargain with each other (namely that they get to keep their initial endowments), and that they engage in economic interactions to improve their allocation over the default.

Compared to Bentham's conception of maximizing a sum of people's utilities (and Pareto's slightly weaker version of finding a utility combination that cannot be improved upon), the notion of seeking mutual advantage builds upon a substantially different basis for arguing that an economic outcome is appropriate. Simple utilitarianism allows the economist to consider re-allocating the entirety of a society's resources when evaluating different allocations, allowing her to give people allotments of goods that had nothing to do with their original endowments (if for some reason it were optimal to do so). For example, if she were able to determine that one person derives intense pleasure from material consumption while a second is perfectly content to starve to death, nothing in the rules of utilitarianism would prevent her from giving all of society's resources to the first person and allowing the second person to do just that. By contrast, under a bargaining scheme, individuals can do no worse than their initial endowment. There is therefore an implicit notion of rights in the bargaining game solution: Individuals interact with society in order to improve their lot, and an individual has the right to not participate in exchange if he sees no benefit.^1.6

At this point, it would probably help to present a formal example of a bargaining game.

Example: The gains from employee referrals

In a recent article^1.7, a pair of sociologists examined a large bank that recruits new employees largely via referrals from current employees. They found that prospective employees referred by current workers were more likely to be hired, and that this saved the company $416.43 per person hired in screening costs (costs were $977.95 for referred applicants vs. $1394.37 for non-referred applicants). Let's suppose that it costs $10 of an employee's time for her to refer someone to refer someone, assuming that she knows someone who is interested, regardless whether the person gets hired (this is a simplistic assumption, but it will help to keep this exercise simple).

Figure 1.2: Possible outcomes from bargaining over the division of a hiring bonus.

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The company then needs to contemplate how much to pay the referring employee as a bonus, in order to give people an incentive to make referrals. The company wishes to only give bonuses when n applicant actually gets hired; otherwise, employees would simply fill out bogus referral forms to collect the bonus. The company hires about 10 percent of all applicants referred.

From the perspective of an employee, the actual pay she expects to get will only be ten percent of the size of the bonus, since when she makes a referral, there is only a ten percent chance that the person will get hired and she will collect the bonus. Therefore, she would never make a referral if the bonus were less than $100, because otherwise she would expect to make less than $10, and this would not be worth the $10 cost of her time for making the referral.

From the perspective of the firm, on the other hand, hiring a referred employee saves $416.33 in screening costs. Therefore, the firm would not be interested in paying the bonus if it were any larger than $416.33, because it would be cheaper to hire non-referred employees.

So, any bonus between $100 and $416.33 is feasible; the question is, how much will the firm and the worker get to keep? The possible outcomes of the game are shown in the large gray triangle in Illustration ; any point along the diagonal line on the upper-right edge of the triangle (i. e., the hypotenuse) is Pareto optimal, since there is no way for either party to keep a larger share of the bonus without the other party giving some up. In this particular game, points in the middle of the triangle are a bit of a silly outcome, since they imply that the two parties will throw away money; however, in other games where the bargaining involves people trading goods or services with each other, such outcomes would just reflect poor trading choices.

The classical solution for a bargaining game was developed by John Nash. It may be stated as follows:

1]0.9

Suppose that a game has two players. The players have default allocations $d_{1}$ and $d_{2}$ . The set $S$ describes all of the possible allocations that are available from bargaining, and the two players have utility functions $u_{1}(s)$ and $u_{2}(s)$ for any allocation $s$ in $S$ .

The Nash Bargaining Solution is the outcome that maximizes the product $(u_{1}(s)-d_{1})(u_{2}(s)-d_{2})$ over all possible allocations $s$ in $S$ .

To see how Nash's solution works in the current example, we'll need to make some simplifying assumptions. First, we assume that each player receives utility in dollars - obviously this is not true, but it allows us to assign utility numbers to outcomes, which we have to do one way or another. Second, we will assume that since the worker only has a ten percent chance of receiving a bonus from a referral, she only gets 10 cents in utility from every dollar of the bonus. Moreover, it costs her $10 to make a referral. So if $B$ is the size of the bonus, then the worker's gain from the bonus is $B/10-10$ , and the firm's gain is $416-B$ . Each has a default allocation of zero dollars, where nobody is referred, nobody incurs any costs, and nobody benefits. So, Nash would maximize the product $(B/10-10)(416-B)$ . This product is shown by the area of the gray rectangle inside the triangle. Those who know calculus can verify that this number is maximized at $B=158$ .^1.8 (As it turns out, the actual bonus paid by the firm was $260.)

Nash motivated his solution to the bargaining problem by presenting four properties that he thought a bargaining solution should have:

Nash's Four Properties for a Bargaining Solution

1. A bargaining solution should be Pareto optimal.
2. A bargaining solution should be symmetric: If the only difference between two players $i$ and $j$ is their labels $i$ and $j$ , that is, if their default allocations are the same and, if, for every possible outcome $s$ where $u_{i}(s)=x$ and $u_{j}(s)=y$ , there is a possible outcome $t$ where $u_{i}(t)=y$ and $u_{j}(t)=x$ , then both players should be given the same utility by the bargaining solution.
3. Third, the bargaining solution should be contraction consistent: If we consider a new, restricted bargaining problem, where we take away possible outcomes but do not take away the outcome that is the solution, and we do not change the default allocations, then the bargaining solution for the restricted problem will be the same as that of the original problem. Contraction consistency a form of fairness, since it implies that players cannot change the result of the bargain by threatening each other with bad outcomes.
4. Fourth, the bargaining solution should be scale invariant: If we take player $i$ 's utilities, $u_{i}(s)$ , and generate a new utility function for player $i$ that is just a multiple of the old one, $v_{i}(s)=au_{i}(s)+b$ , e.g., $v_{i}(s)=10u_{i}(s)+50$ , then the bargaining solution will not change if player $i$ 's utility function is replaced with $v$ . Scale invariance is a way of saying that a player cannot declare himself more important than another player by inflating his utility numbers; what matters for the solution is the relative utilities of outcomes for a given player, but not the differences in utilities between players. In other words, the utilities are somewhat cardinal, but are not comparable.

Nash demonstrated that his bargaining solution is the only solution that satisfies all of these four properties. The precise form of Nash's bargaining solution, and the properties it achieves, are still widely used, but they are not the most consequential result of his 1950 paper.^1.9 What has been much more lasting is its effect on the way distributional outcomes are considered. Unlike the utilitarianism of the 19 $^{\text{th}}$ Century, Nash's framework places a great deal of emphasis on the process by which the solution is arrived at. The individuals involved are supposed to have certain inalienable rights (namely, to walk away with the bundles of goods they came to the bargaining table with), and Nash promotes his solution based on the ways in which it is procedurally fair: Avoiding playing favorites between the bargaining parties.

A long literature has followed in the second half of the twentieth century proposing other general solutions to bargaining problems with other desirable properties. No consensus has developed as to exactly which of these properties the result of a bargaining process should have; rather, the lasting legacy of Nash's work has been that distributive outcomes are no longer considered separately from the series of human interactions that leads to them.

A curious property of the Nash solution is that it defines an optimal solution to a problem simply based on information about the possible distributions of people's utilities, rather than information about the goods themselves. This means, for example, that there is no place for certain types of goods, such as perhaps fresh air or clean water or even control over one's own body, to which there may be an inalienable right. However, this shortcoming is also shared by classical utilitarianism, as well as Pareto optimality. This issue will be addressed in the next chapter by the ideas of John Rawls, which both add to some of the innovative conceptions of game theory and also address some of its shortcomings as a theory of resource allocation.

1.4.2 Non-Cooperative Games

A key assumption behind the bargaining games above is that the players are able to meet and collude to generate a solution. This is a decent description of some social and economic interactions - for example, the interaction between worker and employer, customer and client, or fellow legislators. However, there are many cases where collusion is not possible. For example, people may interact anonymously, for example when the commuting decisions of hundreds of drivers lead to traffic patterns. Or, even if players do interact, they may not be able to enforce compliant behavior on each other. For example, a day laborer and a construction contractor may agree on a wage for a one-day job, but the contractor might not observe the quality of the laborer's work until after the day is over, if ever.

Non-cooperative games are interactions where the players decide separately what strategies to follow. A key result of this limitation is that the outcome of the game may not be Pareto optimal. If players can determine the outcome by collusion, then no one will object to an alternative that makes at least one player better off without harming anyone else, and so that alternative will be chosen. If players cannot collude, on the other hand, there may be situations in which each player's pursuit of his or her individual goals will harm his or her fellow players; it would be Pareto superior for every player to agree to drop the pursuit of such goals, but this cannot be done because the players have no way of colluding.

Figure 1.3: The normal form representation of a prisoner's dilemma game.

A classical example of a non-cooperative game is the prisoner's dilemma, in which each person's pursuit of individual self-interest inflicts so much harm on fellow players that it actually makes everyone else worse off. The traditional story behind the prisoner's dilemma goes something like this: Two men are caught in posession of stolen goods. The district attorney knows that the two men actually stole the goods, but is unable to prove it. So, he locks each prisoner in a room, and offers him the following proposition:

• If neither of you confesses to stealing the goods, you will both be convicted of posession of stolen property, and get four years in prison.
• If you confess, but your partner does not, I will let you go free, and use your testimony to convict your partner for burglary. Your partner will get fifteen years in prison.
• If your partner confesses, but you do not, then you will get fifteen years in prison, and your partner will go free.
• If both of you confess, then you will both be convicted of burglary and get ten years in prison.

The game is represented by the table in Figure . One player's strategies are presented by the different rows; that player is referred to as the row player. The other player's strategies are presented by the different columns, and that player is referred to as the column player. Together, a row and a column define a cell of the table, and so each cell represents an outcome that is the result of the row player choosing a particular strategy and the column player choosing another (possibly identical) strategy. The cell normally lists the players' utilities (player 1's utility, followed by player 2's utility), although in this case, we have used the cells to represent the lengths of the prison sentences.

What's a prisoner to do? Certainly, if the prisoners could collude, they could both choose not to confess. However, they are locked in separate rooms. They might choose to not confess out of fear of future retaliation, i.e., that one prisoner will shoot the other prisoner once he gets out of jail; indeed, one theme in game theory is that if a game is played over and over again, then players can use future rounds of the game to retaliate against each other for non-cooperative behavior, and therefore can implicitly collude even when formal collusion is not possible. John Nash proposed a different solution to this game, which is known as the Nash equilibrium of a non-cooperative game, and should not be confused with the Nash bargaining solution of a cooperative game:

The idea behind the Nash equilibrium is that each player should take the other players' strategies as given, because she has no control over them. Then, given these strategies, she should choose the best possible strategy for herself. When every player is doing so, we have arrived at a set of strategies from which no player has an incentive to deviate, and so that set of strategies is an equilibrium.

In the case of the prisoner's dilemma, notice that no matter what the other player does, each player is better off confessing if he takes the other player's actions as given. If player 2 doesn't confess, then player 1 is better off confessing because he will get zero years in prison instead of 4; if player 2 does confess, then player 1 is better off confessing because he will get 10 years in prison instead of 15. Therefore, the only Nash equilibrium is the outcome in which both players confess. Note that the Nash equilibrium is, in fact, the only outcome of the game that is not Pareto optimal. Note also that if the players could cooperate, then the Nash bargaining solution would be the outcome where neither confesses (assuming that each player has a default allocation of 10 years in prison if he does not cooperate).

The prisoner's dilemma is meant to represent many situations in life where the pursuit of individual self-interest is self-defeating. For example, people may drive SUVs instead of cars because when they get into an accident, the driver of the bigger car is more likely to survive, and they do not want to be the driver of the smaller car. Nevertheless, road safety may be worse if everyone drives SUVs than it would be if everyone drives cars. To take another example, if one person litters, then she saves herself the energy of throwing things away. But if everyone litters, then roads will be dirty. Societies have developed behavioral norms (such as indoctrinating their members not to litter) to avoid the worst outcomes of many prisoner's dilemma games, but there are always unforseen games in which anti-social behavior can crop up.

Figure 1.4: A non-cooperative game with one Pareto optimal Nash equilibrium and one Pareto inferior equilibrium

Not all non-cooperative games lead to perverse outcomes; in fact for many, the Nash equilibrium may be Pareto optimal. Take the following game as an example: Two friends use instant messengers. Both prefer MSN messenger to AOL messenger, because it has more features. Nevertheless, the most important aspect of a messenger for these two friends is that they both are using the same one, because if they were using different messengers then they would be unable to communicate. Here, we suppose that each player gets a utility of five if they are both using AOL messenger, a utility of 10 if they are both using MSN messenger, and a utility of zero if they are using different messengers. There are two Nash equilibria of this game: One in which they are both using AOL, and the other in which they are both using MSN. The MSN equilibrium is Pareto optimal, while the AOL equilibrium is not. What this game illustrates is that people may get locked into using inferior technologies or consuming inferior products if it requires an unrealistic amount of coordination to move to a superior one.

1.5 Conclusions

The view we have sketched in this chapter is more or less where utility theory stands in economics today. Economists still use cardinal utilities sometimes, because they are easy to manipulate to make predictions; but most microeconomic outcomes can be demonstrated using ordinal utilities, which are easier to justify. Nevertheless, ordinal utilities can only be used to show whether or not an outcome is Pareto optimal, which is not as powerful as showing that an outcome maximizes social welfare. Group dynamics have been introduced into utility theory through the field of game theory, and to some extent, this allows us to evaluate the process by which an outcome was arrived at (as well as the outcome itself) because we can study the rules that people follow. Every so often a new piece of research manages to construct a utility function with a slightly different set of assumptions in place of completeness, transitivity, and continuity, and every so often a new solution to a bargaining game is proposed; nevertheless, the Nash bargaining solution and the Nash equilibrium stand as the standard solutions to cooperative and non-cooperative games more than 50 years after their creation.

The teleological element of utility theory - that it makes judgments about well being based on nothing more than an outcome ranking - has been criticized heavily in the last forty years by social philosophers. Nevertheless, any attempts to create a quantifiable measure of individual and social well being have ended up more or less with modified, qualified, or restricted versions of the same theory; the more successful attempts to replace utility theory so far have given up on the attempt to make quantitative judgments and predictions. Utility theory is still the dominant means of evaluating outcomes used by economists; nevertheless, we should always keep its shortcomings in mind. We turn to some of them in the next chapter.

Appendix: A Proof of Debreu's Theorem

A precise proof of Debreu's theorem requires real analysis and some topology, which is not assumed of the reader; more general and complex versions of the theorem are available, depending on exactly how far one wants to push. Nevertheless, we can at least sketch the way that the proof works without resorting to much more than intuition.

The proof that Debreu's Theorem holds using Assumptions $U1$ , $U2$ , and $U3'$ is fairly straight-forward: If a set of possible outcomes only contains as many outcomes as there are integers, then one can put the outcomes in a list (albeit one that would take infinitely long to read from top to bottom). One can start constructing the list with any particular outcome, and then insert each additional outcome in the list above outcomes that are preferred to it and below outcomes that it is preferred to. In this way, we have constructed the list in order of preference; we can then number the first outcome on the list $1$ , the second one $2$ , etc. In this way, we have constructed a utility function.

Now, let's consider the continuous set of preferences described by $U3$ . We first consider the rational numbers in that set, which are the numbers that can be written as fractions ($1/2$ , $3659/27811$ , etc.). The set of rational numbers has two important properties: (1) there are exactly as many rational numbers as there are integers;^1.10 (2) the rational numbers are dense, i.e., for any real number, there is a rational number arbitrarily close to it.

We know from the first paragraph above that it is possible to build a utility function $u(q)$ over the outcomes described by the rational numbers in the set. Now, consider two irrational numbers $x$ and $y$ . We define $u(x)$ as the upper bound of the set of utilities $u(q)$ for rational numbers $q$ that are no better than $x$ , and we define $u(y)$ similarly. If $x$ is exactly as good as $y$ , then because preferences are transitive, the rational numbers that are no better than $x$ are exactly those that are no better than $y$ , so $u(x)=u(y)$ .

On the other hand, suppose that one of the two outcomes is strictly preferred; suppose that it is $x$ that is preferred to $y$ . Now, draw a line between $x$ and $y$ . Because of continuity, we know that some of the outcomes close to $x$ are strictly better than $y$ , and some of the outcomes close to $y$ are strictly worse than $x$ . Let $z_{1}$ be the upper bound (toward $x$ ) of the outcomes that are strictly worse than $x$ , and let $z_{2}$ be the lower bound (toward $y$ ) of the outcomes that are strictly better than $y$ . It must be that $z_{1}$ is closer to $y$ than $z_{2}$ (if not, then any points between $z_{1}$ and $z_{2}$ would be at least as good as $x$ and at least as bad as $y$ , which violates transitivity). Therefore, any points between $z_{1}$ and $z_{2}$ are strictly better than $y$ and strictly worse than $x$ . In particular, choose two rational numbers $x'$ and $y'$ between $z_{1}$ and $z_{2}$ such that $x$ is strictly better than $x'$ , which is strictly better than $y'$ , which is strictly better than $y$ . It then follows that $u(x)\geq u(x')>u(y')\geq u(y)$ , i.e., $u(x)>u(y)$ . We have therefore demonstrated that $u$ is a function mapping outcomes to numbers such that $u(x)>u(y)$ whenever $x$ is strictly preferred to $y$ and $u(x)=u(y)$ whenever $x$ is exactly as good as $y$ . But $x$ and $y$ were chosen arbitrarily, so this holds for any two numbers. In other words, $u$ is a legitimate utility function.

Notice that we have relied at some point or other on all three of our assumptions: Completeness, transitivity, and continuity. If any of the assumptions fails, Debreu's Theorem generally does not hold, although there are weaker versions of the theorem that may hold under some conditions.

Key Concepts and Definitions

• A utility function is cardinal if the differences between utility numbers are meaningful; it is only ordinal if better outcomes are given higher numbers, but the differences between the numbers are not meaningful.
• A utility function is comparable if it can be stated which of any two people is better off (or if they are both equally well off) under any outcome.
• A teleological theory of judgement is one that evaluates only the outcome, and not the means by which that outcome was arrived at.
• An outcome $A$ is Pareto superior to an outcome $B$ if everyone is at least as well off under $A$ as under $B$ , and at least someone is strictly better off. $A$ is Pareto inferior to $B$ if $B$ is Pareto superior to $A$ .
• Only an ordinal utility function is required to determine Pareto superiority or inferiority, though cardinal utility functions can determine this too.
• An outcome is Pareto optimal (or Pareto efficient) if no other outcome is Pareto superior to it.
• A bargaining outcome is symmetric if it gives any two people the same outcome whenever their default allocations are the same and their sets of possible utilities mirror each other.
• A bargaining outcome is contraction consistent if it is not altered by the addition or deletion of other outcomes from the set of possible outcomes (aside from itself and the default allocation).
• A bargaining outcome is scale invariant if multiplying a player's utility function by a constant and/or adding a constant to the utility function does not change the outcome.
• The default allocation is the allocation that both parties of a bargaining game receive if they are unable to reach an agreement.
• The Nash bargaining solution to a bargaining game is the outcome that maximizes $(u_{1}-d_{1})\times(u_{2}-d_{2})$ among all possible outcomes, where $u_{1}$ and $u_{2}$ are the players' utilities, and $d_{1}$ and $d_{2}$ are their default allocations. The Nash bargaining solution is the only solution to a bargaining game that obeys Pareto optimality, contraction consistency, symmetry, and scale invariance.
• A Nash equilibrium of a non-cooperative game is an outcome where each player is maximizing his or her utility while taking the actions of the other players as given. Such an equilibrium may be, but need not be, Pareto optimal.

Study Questions

1. A society consists of two people: Uttam and Suchitra. They have to decide between the following four economic outcomes A, B, C, and D:
   

   
   
   

   Outcome

   
   

   1. Which of the outcomes is Pareto optimal?
   2. Which outcome would maximize a classical social welfare function if the utilities are comparable and cardinal?
   3. Suppose that both Uttam and Suchitra have a default allocation of 5 when they cannot come to an agreement. Which of the four outcomes would be chosen as a Nash bargaining solution?
2. It is illegal in most countries to sell body parts. Yet, some organs, such as kidneys, are so desperately needed that their recipients would be willing to pay tens of thousands of dollars for a kidney. There are, in turn, people so poor that they would gladly donate a kidney for ten thousand dollars. Why would a teleological theory of judgement argue in favor of making it legal to trade body parts? Why do you think that trading body parts is not legal?
3. Jeanine likes food that is spicy, aromatic, and fresh. She prefers Mexican food to Lebanese food because it is spicier. She prefers Indian food to Mexican food because it is more aromatic. And, she prefers Lebanese food to Indian food becaues it is fresher. Can you write down a utility function for Jeanine where more preferred cuisines get higher utility numbers? If not, which of the assumptions about utility do her prefereces violate?
4. David has an insatiable need for water. In fact, no matter how much water he consumes, he would prefer another drop of water to anything else in the world. However, holding his water consumption constant, he would always enjoy more champagne. Can you write down a utility function that reflects these preferences? If not, which assumption do they violate?

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Footnotes

...^1.1

This is not completely true: Some have argued that one can first exclude outcomes that are judged immoral, and then apply utilitarian principles to choose from the rest. However, the determination of which outcomes are immoral cannot be guided by utilitarianism, whereas such a determination may be made from within a Rawlsian framework.

... Index^1.2

Body Mass Index is defined as weight, in kilograms, divided by height, in centimeters, squared.

... tax^1.3

The capital gains tax the rate at which investors are taxed when they sell their investments for more than they paid for those investments; it is therefore effectively a tax on investment income.

...^1.4

Abraham Maslow, ``A Theory of Human Motivation.'' Psychological Review 50: 370-396 (1943).

...^1.5

Much of the funding for von Neumann and Morgenstern's work came from the U.S. military, which needed a framework to help make strategic decisions during World War II.

...^1.6

The assumption of a default allocation is not clearly the optimal way to address the idea that people have rights, since it does not address the question of how individuals arrived at their initial endowments in the first place. For example, the endowments of members of a particular generation may be created by the wealth of their parents; the bargaining framework does not answer the question of whether or not those individuals have a moral right to that wealth or even whether there is a net benefit to society as a whole from those individuals keeping their wealth. These problems are addressed in depth by Rawls, whose framework will provide a solution that may not be to everyone's liking.

... article^1.7

See Chapter 5 of Social Capital: Theory and Research, edited by Nan Lin, Karen Cook, and Ronald S. Burt. The article is by Roberto M. Fernandez and Emilio J. Castilla.

... .^1.8

The product $(B/10-10)(416-B)$ is equal to $-0.1B^{2}+31.6B-4160$ . To find the maximum value of the function, we set its derivative with respect to $B$ equal to zero: $\partial[-0.1B^{2}+31.6B-4160]/\partial B=-0.2B+31.6=0$ , or $B=158.$

...^1.9

See John Nash, "The Bargaining Problem" Econometrica 18: 155-162, 1950.

...^1.10

We know this because each rational number is can be defined by a pair of integers, one on the numberator and one on the denominator. We can write every rational number as a cell in a table where the column number is the numerator, and the row number is the denominator. We can then list the elements of the table diagonally (i.e., the first row and first column, followed by the second row, first column, followed by the first row, second column, followed by the first row, third column, followed by the second row, second column, followed by the third row, first column, etc.) and in this way list any given rational number in a finite number of steps.