Utility

10 Key excerpts on "Utility"

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  eBook - ePub

  Game Theory

  A Nontechnical Introduction

  • Morton D. Davis(Author)
  • 2012(Publication Date)
  • Dover Publications

    (Publisher)

  The problem is to find a way for players to convey their attitudes in a form that is useful to the decision maker. Statements such as “I detest getting caught in the rain” or “I love picnics” do not help someone decide whether to call off a picnic when the weather prediction is an even chance of rain. There is no hope of completely describing subjective feelings quantitatively, of course, but, using Utility theory, it is possible to convey enough of these feelings (under certain conditions) to satisfy my present purpose.

  Utility Functions: What They Are, How They Work

  A Utility function is simply a “quantification” of a person’s preferences with respect to certain objects. Suppose I am concerned with three pieces of fruit: an orange, an apple, and a pear. The Utility function first associates with each piece of fruit a number that reflects its attractiveness. If the pear was desired most and the apple least, the Utility of the pear would be greatest and the apple’s Utility would be least.

  The Utility function not only assigns numbers to fruit; it assigns numbers to lotteries that have fruit as their prizes. A lottery in which there is a 50 percent chance of winning an apple and a 50 percent chance of winning a pear might be assigned a Utility of 6. If the utilities of an apple, an orange, and a pear were 4, 6, and 8 respectively, the utilities would reflect the fact that the person was indifferent (had no preference) between a lottery ticket and an orange, that he or she preferred a pear to any other piece of fruit or to a lottery ticket, and that he or she preferred a lottery ticket or any other piece of fruit to an apple.

  Also, Utility functions assign numbers to all lotteries that have as prizes tickets to other lotteries; and each of the new lotteries may have as its prizes tickets to still other lotteries, so long as the ultimate prizes are pieces of fruit.

  This is still not enough, however. Von Neumann and Morgenstern demand one more thing of their Utility functions that make them ideally suited for their theory. The Utility functions must be so arranged that the Utility of any lottery is always equal to the weighted average Utility of its prizes. If in a lottery there is a 50 percent chance of winning an apple (which has a Utility of 4) and a 25 percent chance of winning either an orange or a pear (with utilities of 6 and 8, respectively), the Utility of the lottery would necessarily be 5½ = (.5)(4) + (.25)(6) + (.25)(8).

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  Game Theory for Political Scientists

  • James D. Morrow(Author)
  • 2020(Publication Date)
  • Princeton University Press

    (Publisher)

  Chapter Two Utility Theory Game theory is based on Utility theory, a simple mathematical theory for repre-senting decisions. In Utility theory, we assume that actors are faced with choices from a set of available actions. Each action provides a probability of producing each possible outcome. Utility is a measure of an actor's preferences over the outcomes that reflects his or her willingness to take risks to achieve desired out-comes and avoid undesirable outcomes. The probabilities of obtaining each outcome after taking an action represent uncertainty about the exact conse-quences of that action. We calculate an expected Utility for an action by multiplying the Utility of each possible outcome by the probability that it will occur if the action is cho-sen, and then summing across all possible outcomes. Utilities for outcomes are chosen so that the magnitude of expected utilities concur with preferences over actions. Actions with larger expected utilities are preferred. Given the proba-bilities that actions produce outcomes and preferences over actions, we can calculate utilities over outcomes so that actions with larger expected utilities are preferred. Utility theory is closely tied to probability theory and is almost as old. As in the case of probability theory, the rigorous analysis of gambling problems drove the early development of Utility theory. Daniel Bernoulli 1 first worked on Utility theory to explain why the attractiveness of gambles did not neces-sarily equal the gambler's monetary expectation. After this initial observation, Utility theory lay dormant until Jeremy Bentham advanced utilitarianism as a philosophy in the 1800s. Bentham's Utility theory was, mathematically speak-ing, quite sloppy and is not useful for developing a rigorous theory of decision. Consequently, Utility was rejected as a useful concept until the middle of the twentieth century.

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  The Economics of Resource Allocation in Health Care

  Cost-utility, social value, and fairness

  • Andrea Klonschinski(Author)
  • 2016(Publication Date)
  • Taylor & Francis

    (Publisher)

  In summary, the basic ingredients of Jevons’ economics are Bentham’s utilitarianism, psychophysiology, and the use of the differential calculus. Jevons strictly rejected the labor theory of value and made the individuals’ feelings of pleasures and pains the building blocks of his theory. Psychophysiology’s mechanistic conception of the human mind allowed him to apply the methods of the natural sciences and, in particular, mathematics to the investigation of individual decision making as a calculus of pleasure and pain. The application of the calculus, in turn, permitted Jevons to focus on marginal changes in the amounts of Utility or, what comes to the same thing in a psychophysiological framework, on the marginal units of pleasure the individual economic agent derives from commodities. These changes are not proportional to the increase in the amount of a commodity but are in fact decreasing. This principle of diminishing marginal Utility enabled Jevons to derive the equimarginal principle according to which the individual allocates his resources so as to maximize his own pleasure or, as it were, Utility. Methodologically, the economic theory of Utility-maximizing behavior can hence be considered the “child of the marriage of Utility with the technique of marginal increments and decrements, which itself led directly to the consideration of extremal problems” (Dobb 1973: 172).

  As to the adoption of the Benthamite Utility concept it deserves emphasis again that while Bentham’s Principles first and foremost addressed the legislator who should build institutions to the advantage of all, Jevons’ individualistic account of Utility maximization was totally detached from any societal concerns. Here, the Utility concept serves an explanatory function within demand theory. To accomplish that task, it referred to “subjective scales of valuation which were supposed to reside in the consumer’s mind” (Endres 1999: 602). Thereby, pleasure or Utility are considered as being quantities that provide “agents with a monotonic criterion by which to carry out the ordering of the alternative outcomes they face” (Warke 2000a: 20). Put differently, Utility provided for an ordering principle, explaining how subjects generate their preference rankings (see Mandler 2001: 374). The maximization of pleasure, then, was regarded as the subjects’ aim and motive for action. Henceforth, Utility maximization in economics became more and more associated with the idea of individual rationality (see Cudd 1993: 106) and the problem an economic agent faces became framed as the problem of allocating his resources “in such a way that his well-being is enhanced to the greatest degree possible” (Colvin 1985: 9). The publication of TPE can thus be conceived as the hour of birth of the economic man, i.e., of the “discrete, self-contained, self-interested” individual of modern microeconomics (Colvin 1985: 5), aiming at the maximization of pleasure (see Little 1957: 10). Put differently, the TPE gave rise to the fundamental principle of modern economics that “economic behaviour is maximising behaviour subject to constraints” (Blaug 1997: 280).42

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  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2020(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Utility theory Chapter summary The objective of this chapter is to provide a quantitative representation of players’ preference relations over the possible outcomes of the game, by what is called a Utility function. This is a fundamental element of game theory, economic theory, and decision theory in general, since it facilitates the application of mathematical tools in analyzing game situations whose outcomes may vary in their nature, and often be uncertain. The Utility function representation of preference relations over uncertain outcomes was developed and named after John von Neumann and Oskar Morgenstern. The main feature of the von Neumann–Morgenstern Utility is that it is linear in the probabilities of the outcomes. This implies that a player evaluates an uncertain outcome by its expected Utility. We present some properties (also known as axioms) that players’ preference relations can satisfy. We then prove that any preference relation having these properties can be represented by a von Neumann–Morgenstern Utility and that this representation is determined up to a positive affine transformation. Finally we note how a player’s attitude towards risk is expressed in his von Neumann–Morgenstern Utility function. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 2.1 Preference relations and their representation A game is a mathematical model of a situation of interactive decision making, in which every decision maker (or player) strives to attain his “best possible” outcome, knowing that each of the other players is striving to do the same thing.

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  Social Choice and Individual Values

  • Kenneth J. Arrow, Eric S. Maskin(Authors)
  • 2012(Publication Date)
  • Yale University Press

    (Publisher)

  'Op. tit., pp. 15-31, 617-632. See also W. S. Vickrey, Measuring Marginal Utility by Reactions to Risk, Economelrica, Vol. 13, October, 1945, pp. 319-333. 9 10 THE NATURE OF PREFERENCE AND CHOICE [CHAP. H find that there is a Utility indicator (unique up to a linear transforma-tion) which has the property that the value of the Utility function for any probability distribution of certain alternatives is the mathematical expectation of the Utility. Put otherwise, there is one way (unique up to a linear transformation) of assigning utilities to probability distribu-tions such that behavior is described by saying that the individual seeks to maximize his expected Utility. This theorem does not, as far as I can see, give any special ethical significance to the particular Utility scale found. For instead of using the Utility scale found by von Neumann and Morgenstern, we could use the square of that scale; then behavior is described by saying that the individual seeks to maximize the expected value of the square root of his Utility. This is not to deny the usefulness of the von Neumann-Morgenstern theorem; what it does say is that among the many different ways of assigning a Utility indicator to the preferences among alternative probability distributions, there is one method (more precisely, a whole set of methods which are linear transforms of each other) which has the property of stating the laws of rational behavior in a particularly con-venient way. This is a very useful matter from the point of view of developing the descriptive economic theory of behavior in the presence of random events, but it has nothing to do with welfare considerations, particularly if we are interested primarily in making a social choice among alternative policies in which no random elements enter. To say otherwise would be to assert that the distribution of the social income is to be governed by the tastes of individuals for gambling. The problem of measuring Utility has frequently been compared with the problem of measuring temperature. This

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  Mathematical Modeling in the Social and Life Sciences

  • Michael Olinick(Author)
  • 2014(Publication Date)
  • Wiley

    (Publisher)

  CHAPTER 8 Introduction to Utility Theory Some reckon time by stars, And some by hours; Some measure days by Dreams, And some by flowers; My heart alone records My days and hours. —Madison Cawein I. Introduction This chapter continues the axiomatic discussion, begun in Chapter 7, of certain aspects of measurement theory. We consider again the problem that motivated the development of the material in the preceding chapter from a new point of view. The problem is to construct a numerical measurement of “happiness”; in particular, to assign numbers that measure how happy a particular student would be if she were assigned various different courses by the college’s registrar. The point of view of this chapter is called Utility theory. The theory dates back at least 200 years to a time when nobles of the French court asked mathematicians for advice on how to gamble. Quite a rich theory has been developed, and various aspects of it have been tested experimentally in situations requiring decision making with incomplete knowledge. Consider the set S of possible choices of courses to which the student might be assigned. Using the mechanisms of Chapter 7, or some other scheme, it is determined that the student prefers course x over course y and course y over course z. Utility theory aims to assign numerical weights to these preferences. Suppose we offer the student a choice: she may have course y, her intermediate choice, or she may flip a coin. If the coin comes up heads, she gets course x, while if it comes up tails, she gets course z. Which option does she prefer: the certainty of y or the gamble between x and z? If the coin is weighted so that it always comes up heads, then she will certainly always prefer the gamble: there is a certainty that she will receive her first choice. If the coin is weighted so that it always lands with tails showing, then she will forego the gamble and take course y.

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  Game Theory

  • Michael Maschler, Eilon Solan, Shmuel Zamir(Authors)
  • 2013(Publication Date)
  • Cambridge University Press

    (Publisher)

  2 Utility theory Chapter summary The objective of this chapter is to provide a quantitative representation of players’ preference relations over the possible outcomes of the game, by what is called a Utility function. This is a fundamental element of game theory, economic theory, and decision theory in general, since it facilitates the application of mathematical tools in analyzing game situations whose outcomes may vary in their nature, and often be uncertain. The Utility function representation of preference relations over uncertain outcomes was developed and named after John von Neumann and Oskar Morgenstern. The main feature of the von Neumann–Morgenstern Utility is that it is linear in the probabilities of the outcomes. This implies that a player evaluates an uncertain outcome by its expected Utility. We present some properties (also known as axioms) that players’ preference relations can satisfy. We then prove that any preference relation having these properties can be represented by a von Neumann–Morgenstern Utility and that this representation is determined up to a positive affine transformation. Finally we note how a player’s attitude toward risk is expressed in his von Neumann–Morgenstern Utility function. 2.1 Preference relations and their representation • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • A game is a mathematical model of a situation of interactive decision making, in which every decision maker (or player) strives to attain his “best possible” outcome, knowing that each of the other players is striving to do the same thing.

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  Calculus and Techniques of Optimization with Microeconomic Applications

  • John Hoag(Author)
  • 2007(Publication Date)
  • WSPC

    (Publisher)

  Section IX Preferences, Utility, and Demand In this section, we will take up the problem of the consumer. Here, we assume that the consumer has well-defined preferences. It turns out that preferences together with a budget set are sufficient to determine demand. However, as we want to use the elegant calculus techniques we have developed, we will generate the tools necessary for this endeavor. We start, then, with a definition of preferences. From this definition, we build up a Utility function and then proceed to the maximization of Utility subject to a budget constraint. A mechanism for determining the testable hypotheses for demand, comparative statics, is then provided. Once the case of choice under conditions of certainty is examined, we then turn to the problem of choice under uncertainty. A brief discussion of the conditions under which maximizing expected Utility makes sense is then provided followed by some of the problems generated by that assumption. This material will be important for the section on the firm where, after some initial discussion, game theory is also examined. ELEMENTS OF A MAXIMUM Economics is often characterized as the social science that studies choice. These choices arise out of the scarcity problem that is at the root of all of eco-nomics. To solve a choice problem, we need two basic pieces of information: a set of alternatives among which to choose and a criteria or objective to some-how rank the alternatives. For the consumer, the alternatives are defined by the budget. The criterion is to maximize Utility. Thus we assume that the con-sumer acts to maximize Utility subject to the budget. We start with a definition of maximum. IX.1. Definition (also V.13.) Let S be a set of real numbers. We say that A is a maximum for S if there is no a ∈ S so that a > A . 217

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  Game AI Pro

  Collected Wisdom of Game AI Professionals

  • Steven Rabin(Author)
  • 2013(Publication Date)
  • A K Peters/CRC Press

    (Publisher)

  113 An Introduction to Utility Theory David “Rez” Graham 9 9.1 Introduction Decision making forms the core of any AI system. There are many different approaches to decision making, several of which are discussed in other chapters in this book. One of the most robust and powerful systems we’ve encountered is a Utility-based system. The general concept of a Utility-based system is that every possible action is scored at once and one of the top scoring actions is chosen. By itself, this is a very simple and straightforward approach. In this article, we’ll talk about common techniques, best practices, pitfalls to avoid, and how you can best apply Utility theory to your AI. 9.2 Utility Utility theory is a concept that’s been around long before games or even computers. It has been used in game theory, economics, and numerous other fields. The core idea behind Utility theory is that every possible action or state within a given model can be described with a single, uniform value. This value, usually referred to as Utility , describes the use-fulness of that action within the given context. For example, let’s say you need a new toy for your cat; so you go online and find the perfect one. One website has it for $4.99 while another website sells the exact same toy for $2.99. Assuming delivery times are the same, you will likely choose the toy for $2.99. That option typically has a higher Utility than the toy for $4.99 because, in the end, you are left with more money. This process gets more difficult when you need to compare the value of two things that aren’t directly comparable. For instance, in the previous example let’s assume that the two 9.1 Introduction 9.2 Utility 9.3 Principle of Maximum Expected Utility 9.4 Decision Factors 9.5 Calculating Utility 9.6 Picking an Action 9.7 Inertia 9.8 Demo 9.9 Conclusion 114 Part II. Architecture websites have different delivery times.

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  Stochastic Dominance and Applications to Finance, Risk and Economics

  • Songsak Sriboonchita, Wing-Keung Wong, Sompong Dhompongsa, Hung T. Nguyen(Authors)
  • 2009(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  For each decision-maker with her own  , there exists a Utility function (of her own, unique up to a positive linear transformation) u : Z ! R such that X; Y 2 X ; X  Y () Eu ( X ) > Eu ( Y ) ; provided that the expectations exist. This is known as the von Neumann-Morgenstern expected Utility rep-resentation of preference relations. When this is the case, decision-makers try to maximize their expected utilities, so that their behavior (in making choices under uncertainty) follows the so-called “expected Utility maxi-mization theorem.” The existence of Utility functions (and hence of a numerical represen-tation of  ) is restricted to “rational”people. We will detail below what that means. Needless to say whether everybody is rational, in a sense to be speci…ed, is open to debate! In fact, economists (and psychologists) have already raised concerns about the applicability of von Neumann-Morgenstern’s model to all real-world problems. This is exempli…ed by: (i) Allais paradox (1953): a choice problem showing the inconsistency of expected Utility theory. (ii) Ellsberg paradox (1961): people’s choices in experimental economics violate the expected Utility hypotheses, leading to a generalized expected Utility theory based on Choquet integral (see Chapter 5 for background). (iii) Non-expected Utility theory (1979): from the work of two psycholo-gists Daniel Kahneman and Amos Tversky leading to the so-called “Prospect Theory.” 42 CHAPTER 1. Utility IN DECISION THEORY Interested students should read Fishburn [44] or Kreps [81] for more details. Now we need to consider the case of arbitrary probability measures , e.g., on the real line R , which are probability laws of real-valued random variables (e.g., investment returns or losses).

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