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The book treats two approaches to decision theory: (1) the normative, purporting to determine how a 'perfectly rational' actor ought to choose among available alternatives; (2) the descriptive, based on observations of how people actually choose in real life and in laboratory experiments. The mathematical tools used in the normative approach range from elementary algebra to matrix and differential equations. Sections on different levels can be studied independently. Special emphasis is made on 'offshoots' of both theories to cognitive psychology, theoretical biology, and philosophy.

Inhaltsverzeichnis

Frontmatter

Decisions Involving a Single Actor

Frontmatter

Introduction: An Overview of Decision Theory

Decision theory deals with situations in which one or more actors must make choices among given alternatives. These alternatives may be courses of action to be undertaken, objects to possess, amounts of money to pay for them, and so on. They may also be ‘what to believe’. For instance, a jury deliberating on the guilt or innocence of a defendant must eventually choose collectively between ‘believing that the defendant is guilty’ and ‘believing that he is innocent’.
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1. Orders and Scales

A fundamental concept underlying theories of preferences (an integral part of decision theory) is that of a binary relation on a set of elements.
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2. Optimization

In situations represented as decisions under certainty, the actor chooses among a set of alternatives in one-to-one correspondence with a set of outcomes. If the actor knows which of the outcomes he most prefers, clearly the rational choice is of the alternative associated with the most preferred outcome.
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3. Decisions under Uncertainty and Risk

We will now suppose that the outcome of a decision (regarded as a choice of action among a finite number of alternatives) depends not only on the course of action chosen but also on a particular state of nature that obtains at the time. In our model the alternatives will be represented by the rows of a matrix, the states of nature by the columns. The entries in the matrix will represent the utilities of the outcomes determined jointly by the actor’s choices and by the states of nature:
$$\begin{array}{*{20}{c}} {} \\ {{A_1}} \\ {{A_2}} \\ {} \\ {{A_i}} \\ {} \\ {{A_m}} \\ \end{array} \left[ {\begin{array}{*{20}{c}} {{S_1}} & {{S_2}} & \cdots & {{S_i}} & \cdots & {{S_n}} \\ {{u_{11}}} & {{u_{12}}} & \cdots & {{u_{1j}}} & \cdots & {{u_{1n}}} \\ {{u_{21}}} & \cdots & \cdots & \cdots & \cdots & \cdots \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {{u_{il}}} & \cdots & \cdots & {{u_{ij}}} & \cdots & {{u_{in}}} \\ \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\ {{u_{ml}}} & \cdots & \cdots & \cdots & \cdots & {{u_{mn}}} \\ \end{array} } \right]$$
(Matrix 3.1)
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4. Subjective Aspects of Risk

A decisive stimulus to the development of formal decision theory was provided by the formulation of the principle of expected gain. We have seen how this principle was used to settle the question of how the stakes of an interrupted gambling game were to be divided. The term ‘moral expectation’ given to the principle at the time reflected a normative interpretation of expected gain: what the player could ‘justly’ expect.
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5. Modifications and Generalizations of Expected Utility Theory

As has been mentioned, the beginnings of the formal (that is, mathematically rigorous) theory of probability are usually traced to the correspondence between P. Fermat and B. Pascal concerning certain problems arising in gambling. One of them was how to divide the stakes of an unfinished game of chance. The solution involved the concept of expected gain or ‘moral expectation’, as it was once called — the weighted average of the possible gains (positive or negative), where the weights are the probabilities of realizing these gains. The value of a gamble (or a lottery) defined in this way could then be associated with its expected gain in this sense. Thereby a rational decision in the context of gambling or purchasing tickets in a lottery could be defined as one that prescribes taking the gamble or purchasing the ticket if and only if the corresponding expected gain exceeds the stake or the price of the ticket.
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6. Multi-objective Decisions

A sign of maturity is the recognition that one can’t have everything. Compromise and trade-off are almost always unavoidable in real life decision situations, not only when several parties with non-coincident interests are involved but also when one own’s objectives or desires compete with one another for attention or priorities. A person buying a car wants it to be safe, comfortable, economical and low priced. He can’t possibly satisfy all these objectives at once. A person looking for a job wants it to pay well, to offer opportunities for advancement, to be interesting, and so on. Generally, he will have to sacrifice some of these desiderata in order to satisfy others. Thus, the question invariably arises: ‘How much of this is worth how much of that?’ ‘How much more am I willing to pay for an apartment for being nearer to my place of work by how much?’ In considering candidates for a position, how is experience to be weighed against intelligence or against an attractive personality?
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7. Theory of Social Choice

The theory of social choice is concerned with the problem of aggregating the preferences of several persons into a single preference order. The problem has the same structure as that of aggregating the preference orders of a single person with regard to several aspects of alternatives into a single preference order on the set of alternatives. For this reason, the theory of social choice comes logically not under the topic of collective action (as one might suppose) but rather under the topic of multi-objective decisions. This is so because the several participants in a situation defined as a problem of social choice do not really select courses of action among several available ones when they present their preference orders on a set of alternatives. Therefore they are not ‘actors’, as actors are defined in the theory of decision. To be sure, there are situations where voters do choose strategies, that is, choose among different preference orders to present, which may or may not represent their true preferences but which they believe will be more likely to lead to a more desirable aggregated order. These situations can be considered as n-person decision problems, and we will examine them in the context of non-cooperative n-person games (see Chapter 14. At this time, however, we will suppose that each voter acts ‘sincerely’, that is, presents his ‘actual’ preference order. The decision is now up to some agency, which must combine all these submitted preference orders into a ‘social’ preference order.
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8. Individual Psychology of Decision-making

In the preceding chapters we have been concerned for the most part with nor¬mative decision theory, where the central problem is how a decision-maker ought to act in a given situation. Clearly, the introduction of ‘ought’ inadvertently involves a system of values, for it is with respect to values that a given decision is regarded as ‘good’ (one that ought to have been taken) or ‘bad’ (one that ought not to have been taken). In normative decision theory these values are always assumed to be given when a problem is formulated. That is to say, utilities are assumed to have been assigned to the various possible outcomes of the various possible decisions.
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Non-cooperative Games

Frontmatter

9. Two-person Constant Sum Games

In Part I all decision problems were formulated from the point of view of a single actor. He was not necessarily an individual: he could be a corporation, an institution, or a state. His ‘singleness’ was expressed in a set of goals, values, utilities, and the like. To be sure, the outcomes of his decision depended not on these alone. Usually another agency, which could be called Chance, acted like a decision-maker in the sense of choosing among ‘states of the world’ resulting from events over which the actor had no control, for example a spin of the roulette wheel, the market price of a commodity, and the like. But this agency was not a true actor in the sense used here, since it received no payoffs and so was assumed to be indifferent about the states of the world. To be sure, there are people who believe it will rain just because they left their umbrella at home or that the sun will shine just because they took it along. But such beliefs can be credited to lapses of rationality.
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10. Some Topics in Continuous Games

In Chapter 9 we examined two-person constant sum games with finite numbers of pure strategies. These games can be represented by matrices where the rows designate one player’s pure strategies and the columns the other’s. The number of pure strategies available to each player may be superastronomical (as for example, in chess) so that determining strategies by standard algorithms, such as the simplex method, is out of the question. Limits on the games that can be actually solved in this way need not imply limits on the theoretical conclusions valid for all finite games of a given type. The conclusions do not, however, necessarily hold for games with infinite numbers of strategies.
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11. Two-person Non-constant Sum Games

So far the fundamental question posed by the formal theory of decisions could be phrased as follows: given a decision situation, that is, a choice between a set of alternatives, how should a rational actor determine his choice so as to maximize the utility of the outcome? In the context of decisions under certainty, the answer was obvious. The alternative should be the one that leads (with certainty) to the outcome to which the largest utility has been assigned. What problems still remained had to do with methods of finding that alternative (optimization problems).
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12. Social Dilemmas

Social dilemmas (or social traps, as they are sometimes called) are situations in which each participant appears to be acting rationally, and yet the result is to everyone’s disadvantage. For example if a fire breaks out in a crowded theatre and everyone rushes to the exits, everyone may be trapped in the resulting crush.
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13. Applications of Game Theory in Biology

As we have seen, the earliest applications of mathematically rigorous decision theory were in the context of gambling. To be sure, most ‘decisions’ in these situations were made by Chance, but the human actor could at least decide whether to accept or decline a gamble or where to place his bets. These decisions could be guided by the powerful concept of expected utility, first formulated precisely in the seventeenth century.
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14. Theory of Voting

The most familiar type of democratic decision is majority rule. It reflects the rather vague principle of ‘the greatest good for the greatest number’ (Bentham, 1780[1948]) and can be so interpreted if social good is measured by the number of people pleased by the result of an election or a referendum, provided there are exactly two candidates or two ways of voting on an issue.
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Collective Decisions

Frontmatter

15. Two-person Cooperative Games

The dilemma generated by the bifurcation of ‘rationality’ into individual and collective rationality vanishes once collective rationality is regarded as the basis of rational decision-making by two or more actors. Psychologically, this paradigm reflects at least a partial fusion of the players’ consciousnesses. We can surmise that to the extent that it makes sense to speak of consciousness of non-humans, this fusion is complete in social insects, for example, Possibly such fusion takes place also in some mammals or birds, animals known to act ‘altruistically’ toward their young and toward kin, sometimes even toward non-kin, of their species. We humans sometimes experience such fusion and call it ‘empathy’. Empathy manifests itself in the anguish we sometimes feel when we witness the suffering of another.
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16. N-person Cooperative Games

The theory of the two-person cooperative game presupposes the formation of a coalition between the two players enabling them to coordinate their strategies so as to assure a Pareto-optimal outcome. The principal problem posed by the theory of the cooperative game is that of apportioning the payoffs accruing to the players in the Pareto-optimal outcomes of the game.
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17. The Allocation Problem

The simplest allocation problem is that of dividing something between two persons in a way that appears, according to certain criteria, to be ‘fair’. For example, in dividing an apple between two children, it seems fair to give one half to each. Since, however, it may not be easy to divide the apple into two exactly equal parts, one or the other recipients may complain.
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18. Indices of Power

In his book, Politics among Nations, Morgenthau (1973) draws a parallel between economics and political science. He regards the problem of distribution (or allocation) as the common theme of both disciplines. The distinction between them, according to Morgenthau, is that economics is concerned with the distribution of goods, services, resources, and the like, while political science is concerned with the distribution of power.
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19. Theories of Coalition Formation

As we have seen, the main thrust of the theory of cooperative games has been based on the concept of a solution, typically represented by a set of imputations, that is, distributions of some collectively achieved gain among the players. The sum of the payoffs defining an imputation represents the most the players acting as a grand coalition can attain. It follows that the principal concern of this branch of game theory is with conflict resolution. The solution of a game represents, after all, the resolution of the conflicts of interests among the players. In the context of a normative theory, a solution of a game (in the form of an imputation) is rationalized as satisfying certain a priori stated principles of fairness or stability.
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20. Game-theoretical Concepts in Economics

In discussing applications of game theory, Shubik (1982) points out a major misconception. Maximin strategies and the saddlepoint solution are assumed to be central to the game-theoretic analysis of multi-person decision making. The source of this misconception is easy to see. The theory of games grew out of a rigorous analysis of so-called parlour games (e.g., chess, poker). In contrast to games associated with sports, where physical prowess and flexibility are often crucial, the decisive factor in parlour games is strategic acumen. Indeed, these games are often called games of strategy. Moreover ‘winning’ (as opposed to losing) is usually a principal motivation in playing these games. Thus, the two-person constant sum game represents the best known type of game of strategy. It is also the starting point in the development of game theory. Indeed 6 of the 12 chapters of von Neumann and Morgenstern’s fundamental treatise (1947) are devoted to zero sum games. Of these, two chapters deal with the two-person zero sum game, in which the interests of the players are diametrically opposed. The other four deal with zero sum games with more than two players. The players, however, are supposed to split into two coalitions with diametrically opposed interests. In this way, the theory is partly ‘reduced to the preceding case’ a method of theory building to which the mathematician frequently resorts.
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21. Psychology of Collective Decision-making

In discussing the psychology of individual decision-making, we sought to investigate the thought processes of the human individual in an environment impinging upon him and responding to his actions. This individual had available courses of action and envisaged possible outcomes of choices among them, which he evaluated according to a preference order.
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Concluding Remarks

Concluding Remarks

A common pattern in the development of the sciences is the process of generalization. The process is especially clear in the mathematically formalized sciences and is most prominent in mathematics itself. The generalization of the concept of number exemplifies the process. Originally, the extension of the number system from natural integers to fractions and negative numbers was a consequence of practical applications of mathematics to measurement and commercial transactions. But already in ancient Greece with the appearance of mathematics based on strict deduction, extensions of the number system went on independently of practical experience. The concept of the irrational number has no experiential counterpart: one cannot obtain an irrational number as a result of a measurement, for example. These numbers remain ideational concepts.
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Backmatter

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