Decision-Making Utility Function

A Decision-Making Utility Function is a numeric-output function that can be optimized to support a decision making task.

• AKA: Cost-Benefit Fitness Function, Objective Function.
• Context:
  • range: a Utility Value.
  • It can range from being a Loss Function/Cost Function to being a Reward Function/Profit Function.
  • It can (often) be a Ranking Function used in decisioning task that assigns a value to each candidate choice.
  • It can range from being a Objective Utility Function (game-centric utility) to being a Subjective Utility Function (e.g. agent-centric preference).
  • It can range from being a Past Utility Function to being an Expected Utility Function.
  • It can range from being a Ordinal Utility Function to being a Continuous Utility Function.
  • It can be used in Cost-Benefit Analysis/Cost-Utility Analysis.
  • …
• Example(s):
  • a Personal Happiness Function.
  • an Expected Company Profitability Function.
  • a von Neumann-Morgenstern Utility Function.
  • a Consequentialist Utility Function.
  • …
• Counter-Example(s):
  • a Cost Function.
  • an Operating System Utility.
  • a Utility Service, such as an electricity service.
  • an Indifference Curve.
  • a Decision-Making Process Pattern.
• See: Function Optimization, Risk Management, Mathematical Optimization, Machine Learning Task, Computational Neuroscience, Optimization Problem, Game Theory, Entropic Risk Measure, Economic Demand, Social Science, Utility.

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References

2017

• (Wikipedia, 2017) ⇒ https://en.wikipedia.org/wiki/Loss_function Retrieved:2017-8-6.
  • In mathematical optimization, statistics, econometrics, decision theory, machine learning and computational neuroscience, a loss function or cost function is a function that maps an event or values of one or more variables onto a real number intuitively representing some "cost" associated with the event. An optimization problem seeks to minimize a loss function. An objective function is either a loss function or its negative (in specific domains, also variously called a reward function, a profit function, a utility function, a fitness function, etc.), in which case it is to be maximized.

    In statistics, typically a loss function is used for parameter estimation, and the event in question is some function of the difference between estimated and true values for an instance of data. The concept, as old as Laplace, was reintroduced in statistics by Abraham Wald in the middle of the 20th century. In the context of economics, for example, this is usually economic cost or regret. In classification, it is the penalty for an incorrect classification of an example. In actuarial science, it is used in an insurance context to model benefits paid over premiums, particularly since the works of Harald Cramér in the 1920s. In optimal control the loss is the penalty for failing to achieve a desired value. In financial risk management the function is mapped to a monetary loss.

2015

• http://en.wiktionary.org/wiki/utility_function
  • A mathematical function that assigns a real number to every element of the outcome space in a way that captures the agent's preferences over both simple and compound lotteries.

2015b

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Preference_(economics) Retrieved:2015-4-4.
  • In economics and other social sciences, preference refers to the set of assumptions related to ordering some alternatives, based on the degree of happiness, satisfaction, gratification, enjoyment, or utility they provide, a process which results in an optimal “choice” (whether real or theoretical). The character of the individual preferences is determined purely by taste factors, independent of considerations of prices, income, or availability of goods.

    With the help of the scientific method many practical decisions of life can be modelled, resulting in testable predictions about human behavior. Although economists are usually not interested in choices or preferences in themselves, they are interested in the theory of choice because it serves as a background for empirical demand analysis.

2015c

• (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Preference_(economics)#Strict_versus_weak Retrieved:2015-4-4.
  • The possibility of defining a strict preference relation [math]\displaystyle{ \succ\! }[/math] from the weaker one [math]\displaystyle{ \succsim\! }[/math] , and vice versa, suggest in principle an alternative approach of starting with the strict relation [math]\displaystyle{ \succ\! }[/math] as the primitive concept and deriving the weaker one and the indifference relation. However, an indifference relation derived this way will generally not be transitive. According to Kreps "beginning with strict preference makes it easier to discuss noncomparability possibilities".

2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Utility#Quantifying_utility Retrieved:2014-9-23.
  • It was recognized that utility could not be measured or observed directly, so instead economists devised a way to infer underlying relative utilities from observed choice. These 'revealed preferences', as they were named by Paul Samuelson, were revealed e.g. in people's willingness to pay:

    Utility is taken to be correlative to Desire or Want. It has been already argued that desires cannot be measured directly, but only indirectly, by the outward phenomena to which they give rise: and that in those cases with which economics is chiefly concerned the measure is found in the price which a person is willing to pay for the fulfilment or satisfaction of his desire.Template:Rp

2014b

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Utility Retrieved:2014-8-31.
  • Utility, or usefulness, is the ability of something to satisfy needs or wants. Utility is an important concept in economics and game theory, because it represents satisfaction experienced by the consumer of a good. Not coincidentally, a good is something that satisfies human wants and provides utility, for example, to a consumer making a purchase. It was recognized that one can not directly measure benefit, satisfaction or happiness from a good or service, so instead economists have devised ways of representing and measuring utility in terms of economic choices that can be counted. Economists have attempted to perfect highly abstract methods of comparing utilities by observing and calculating economic choices. In the simplest sense, economists consider utility to be revealed in people's willingness to pay different amounts for different goods.

2014c

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/utility#Economic_definitions Retrieved:2014-8-31.
  • In economics, utility is a representation of preferences over some set of goods and services. Preferences have a (continuous) utility representation so long as they are transitive, complete, and continuous.

    Utility is usually applied by economists in such constructs as the indifference curve, which plot the combination of commodities that an individual or a society would accept to maintain a given level of satisfaction. Individual utility and social utility can be construed as the value of a utility function and a social welfare function respectively. When coupled with production or commodity constraints, under some assumptions, these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a central concept in welfare economics.

    In finance, utility is applied to generate an individual's price for an asset called the indifference price. Utility functions are also related to risk measures, with the most common example being the entropic risk measure. There has been some controversy over the question whether the utility of a commodity can be measured or not. At one time, it was assumed that the consumer was able to say exactly how much utility he got from the commodity. The economists who made this assumption, belong to the 'Cardinalist School' (of Economics).

2013

• http://plato.stanford.edu/entries/epistemic-utility/
  • QUOTE: Traditional utility theory (also known as decision theory) explores a particular strategy for establishing the norms that govern which actions it is rational for us to perform in a given situation. The framework for the theory includes states of the world, actions, and, for each agent, a utility function, which takes a state of the world and an action and returns a measure of the extent to which the agent values the outcome of performing that action at that world. We call this measure the utility of the outcome at the world. …

    … In epistemic utility theory, the states of the world remain the same, but the possible actions an agent might perform are replaced by the possible epistemic states she might adopt, and the utility function is replaced, for each agent, by an epistemic utility function, which takes a state of the world and a possible epistemic state and returns a measure of the purely epistemic value that the agent would attach to being in that epistemic state at that state of the world. So, in epistemic utility theory, we can appeal to epistemic utility to ask which of a range of possible epistemic states it is rational to adopt, just as in traditional utility theory we appeal to utility to ask which of a range of possible actions it is rational to perform.

2011

• http://plato.stanford.edu/entries/preferences/#ChoFunThePro
  • QUOTE: There is a strong tradition, particularly in economics, to equate preference with choice. Preference is considered to be hypothetical choice, and choice to be revealed preference.
    Given an alternative set A, we can represent (hypothetical) choice as a function C that, for any given subset B of A, delivers those elements of B that a deliberating agent has not ruled out for choice. For brevity's sake we will call them 'chosen elements'. The formal definition of a choice function is as follows:

    C is a choice function for A if and only if it is a function such that for all B ⊆ A: (1) C(B) ⊆ B, and (2) if B ≠ ∅, then C(B) ≠ ∅.

    A large number of rationality properties have been proposed for choice functions. The five most important of these are described here.

    If B ⊆ A then B ∩ C(A) ⊆ C(B) (Property α, “Chernoff”)

    This property states that if some element of subset B of A is chosen from A, then it is also chosen from B. According to property α, removing some of the alternatives that are not chosen does not influence choice. This is a very basic and quite reasonable requirement of choice

2013b

• http://en.wikipedia.org/wiki/Utility
  • … Utility is usually applied by economists in such constructs as the indifference curve, which plot the combination of commodities that an individual or a society would accept to maintain a given level of satisfaction. Individual utility and social utility can be construed as the value of a utility function and a social welfare function respectively. When coupled with production or commodity constraints, under some assumptions, these functions can be used to analyze Pareto efficiency, such as illustrated by Edgeworth boxes in contract curves. Such efficiency is a central concept in welfare economics.

2009

• (Conitzer et al., 2009) ⇒ Vincent Conitzer, Matthew Rognlie, and Lirong Xia. (2009). “Preference Functions That Score Rankings and Maximum Likelihood Estimation.” In: Proceedings of the 21st international jont conference on Artifical intelligence.
  • QUOTE: In social choice, a preference function (PF) takes a set of votes (linear orders over a set of alternatives) as input, and produces one or more rankings (also linear orders over the alternatives) as output. Such functions have many applications, for example, aggregating the preferences of multiple agents, or merging rankings (of, say, webpages) into a single ranking. The key issue is choosing a PF to use.

2007

• (Mishan & Quah, 2007) ⇒ Edward J. Mishan, and Euston Quah. (2007). “Cost-Benefit Analysis." Routledge ISBN:978-0-415-35037-2
  • QUOTE: p.89 All of the variables in each person's utility function … are deemed to be entirely within his control. The parameters within each person's utility function, however, are the set of prices; and these are determined by the system as a whole.

2003

• (Korb & Nicholson, 2003) ⇒ Kevin B. Korb, and Ann E. Nicholson. (2003). “Bayesian Artificial Intelligence." Chapman & Hall/CRC.
  • QUOTE: Given a general ability to order situations, and bets with definite probabilities of yielding particular situations, Frank Ramsey [231] demonstrated that we can identify particular utilities with each possible situation, yielding a utility function. If we have a utility function [math]\displaystyle{ U(O_i \vert A) }[/math] over every possible outcome of a particular action [math]\displaystyle{ A }[/math] we are contemplating, and if we have a probability for each such outcome [math]\displaystyle{ P(O_i \vert A) }[/math], then we can compute the probability-weighted average utility for that action - otherwise known as the expected utility of the action. … It is commonly taken as axiomatic by Bayesians that agents ought to maximize their expected utility.

2003b

• (Freund et al., 2003) ⇒ Yoav Freund, Raj Iyer, Robert E. Schapire, and Yoram Singer. (2003). “An Efficient Boosting Algorithm for Combining Preferences.” In: The Journal of Machine Learning Research, 4.
  • QUOTE: We study the problem of learning to accurately rank a set of objects by combining a given collection of ranking or preference functions. This problem of combining preferences arises in several applications, such as that of combining the results of different search engines, or the "collaborative-filtering" problem of ranking movies for a user based on the movie rankings provided by other users.

1998

• (Kohavi & Provost, 1998) ⇒ Ron Kohavi, and Foster Provost. (1998). “Glossary of Terms.” In: Machine Leanring 30(2-3).
  • Utility: See Cost.
  • Cost (utility/loss/payoff): A measurement of the cost to the performance task (and/or benefit) of making a prediction Y' when the actual label is y. The use of accuracy to evaluate a model assumes uniform costs of errors and uniform benefits of correct classifications.

1995

• (Dawkins, 1995) ⇒ Richard Dawkins. (1995). “River Out of Eden: A Darwinian View of Life." Basic Books. ISBN:0465016065.
  • QUOTE: “Utility function” is a technical term not of engineers but of economists. It means “that which is maximized.” Economic planners and social engineers are rather like architects and real engineers in that they strive to maximize something. … By watching the behavior of individuals throughout their lives, you should be able to reverse-engineer their utility functions. If you reverse-engineer the behavior of a country’s government, you may conclude that what is being maximized is employment and universal welfare. For another country, the utility function may turn out to be the continued power of the president, or the wealth of a particular ruling family, the size of the sultan’s harem, the stability of the Middle East or maintaining the price of oil. The point is that more than one utility function can be imagined. It isn’t always obvious what individuals, or firms, or governments are striving to maximize. But it is probably safe to assume that they are maximizing something. This is because Homo sapiens is a deeply purpose-ridden species. The principle holds good even if the utility function turns out to be a weighted sum or some other complicated function of many inputs. Let us return to living bodies and try to extract their utility function. There could be many but, revealingly, it will eventually turn out that they all reduce to one. A good way to dramatize our task is to imagine that living creatures were made by a Divine Engineer and try to work out, by reverse engineering, what the Engineer was trying to maximize: What was God’s Utility Function? Cheetahs give every indication of being superbly designed for something, and it should be easy enough to reverse-engineer them and work out their utility function.

1987

• (Frank, 1987) ⇒ Robert H. Frank. (1987). “If homo economicus could choose his own utility function, would he want one with a conscience?.” In: The American Economic Review
  • QUOTE: … that it will sometimes be in a selfish person's interest to have a utility function that predisposes … of other readily observable physical symptoms vary systematically with a person's affective condition … We must thus compare the mutant's expected payoff to that of persons who lack …

1985

• (Brans & Vincke, 1985) ⇒ Jean-Pierre Brans, and Ph Vincke. (1985). “Note — A Preference Ranking Organisation Method: (The PROMETHEE Method for Multiple Criteria Decision-Making)." Management science 31, no. 6
  • QUOTE:... 3. Extension of the Notion of Critaion . This extension is based on the introduction of a preference function giving the preference of the decision-maker for an action a with regard to b. This function will be defined separately for each criterion; its value will be between 0 and 1.

1979

• (Kahneman & Tversky, 1979) ⇒ Daniel Kahneman, and Amos Tversky. (1979). “Prospect Theory: An analysis of decision under risk." Econometrica: Journal of the Econometric Society.
  • QUOTE:... (iii) Risk Aversion: u is concave (u" < 0). A person is risk … on the assumption that people often know how they would behave in actual situations of choice, and on … If people are reasonably accurate in predicting their choices, the presence of common and systematic violations of …

1966

• (Lancaster, 1966) ⇒ Kelvin J. Lancaster. (1966). “A New Approach to Consumer Theory.” In: The Journal of Political Economy
  • QUOTE: … Utility or preference orderings are assumed to rank collections of characteristics and only … are implicit in the classical “diet problem” of Stigler (1945), and multidimensioned utilities have been … has been made by Morishima (1959) but in the context of single-dimensioned utility. …

1738

• (Bernoulli, 1738) ⇒ Daniel Bernoulli. (1738). “Specimen Theoriae Novae de Mensura Sortis (Exposition of a New Theory on the Measurement of Risk)." http://www.jstor.org/discover/10.2307/1909829
  • QUOTE: Ever since mathematicians first began to study the measurement of risk, there has been general agreement on the following proposition: Expected values are computed by multiplying each possible gain by the number of ways in which it can occur, and then dividing the sum of these products by the total number of possible cases.

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