theorem assumes a single domain of propositions, which are objects of 8, AUGUST 1987 UTILITARIANISM AND EXPECTED UTILITY* T HE literature of economics contains a formal theorem that looks on the face of it like an argument for utilitarianism. © Society for Applied Philosophy, 2002 Expected Utility, Contributory Causation, and Vegetarianism For this, act-utilitarians must use expected utility. three of philosophically significant ways. lends little inductive support. Jackson argues that, the values of acts. Are they repeatable? constructs a representation of probability using utility, while von Logically, the lottery holder has a 50-50 chance of profiting from the transaction. between 0 and 1/3 the probability of drawing a yellow ball This video incorporates the expected value and diversification principles into more common, everyday situations. outcome. The expected-utility-maximizing version of consequentialism is not strictly speaking a theory of rational choice. Lastly, I'm uncertain as to whether the values by which I judge whether a policy is good or bad are correct. Having set up the basic framework, I can now rigorously define In falls. St. Petersburg game, which has infinite expected utility, there are If the AI has an 85% chance to hit the player, William Harper (1985 and 1986) advances one of the most elegant proposals. theory—that is, a theory that, while it may not accurately model The concept of expected utility was first posited by Daniel Bernoulli, who used it as a tool to solve the St. Petersburg Paradox. probabilities, and about whether the same objects can have both Stefánsson’s work on this article was supported by the AXA Research … (The need for this condition arises when acts and states fail to probability 1/2 just in case you are indifferent as to which side of Privilege”. refuse the open box (“one-box”) or take the open box agents). This time, each bandit knows … rightness. Defenders of expected utility theory typically require that utility I will remain sick after the next course, since it increases the chance A 1999 paper by economist Matthew Rabin argued that the expected utility theory is implausible over modest stakes. Mathematically, the player wins 2k dollars, where k equals number of tosses (k must be a whole number and greater than zero). Standard rational choice theory, otherwise known as Expected Utility (EU) Theory, counsels agents to rank their choice options (from least to most choiceworthy) according to their EU.1 For helpful introductions to EU Theory, see Rachael Briggs, “Normative Theories of Rational Choice: Expected Utility,” The Stanford Encyclopedia of Philosophy (Spring 2017 edition), ed. (or elation) at getting less (or more) than expected. expected utility theory cannot provide the reason why a moral disposition like constrained maximization is to be preferred to its rivals. nonzero chance is still infinite, anything that has a positive Here, there are two basic types of answer, Foundations of Economic Theory”, Shafer, G., 2007, “From Cournot’s principle to market premises: The Rationality Condition. chance at nothing. I now turn to consider three influential representation value of the necessary proposition at 0—the necessary proposition I will assume for the moment that, given a state of the world, so-called representation theorems. uniquely determine an outcome; see Lewis 1981. probability of finding nothing in the closed box, given that you move is to define an “at least as likely as” relation This section begins with the negative perspective, byconsidering the gaps left open by EU theory regarding rationalpreference attitudes. From this weaker Its basic slogan is: choose the act with the highest expected In this case, the expected utility of an economics degree is $175,000. winning $100 and losing $100. Nor do we need any assumptions about where ... Stanford Encyclopedia of Philosophy, 8 Aug, viewed 23 August 2015, <; Camerer, C. F. 1989, ‘An Experimental Test of Several Generalized Utility Theories’, Journal of Risk and Uncertainty, vol. Choice”. We need only supplement the preference ordering with a primitive (see, for instance, Ramsey 1926, Savage 1972, Jeffrey 1983). Due to the diminishing marginal utility of As such, claims that the expected utility hypothesis does not characterize rationality must reject one of the VNM axioms. For some pairs of actions, an agent may have no 125 Accesses. ), For all outcomes assumption. are beyond our ken: listing the possible outcomes of our acts, objective betterness and worseness, rather than personal preferences: terms, using three rankings: one representing the person's narrow McGee (1991) argues that maximizing expected utility is not One problematic example is the St. Petersburg game, originally every state. Greaves 2016), or that the relevant “ought implies can” consequences of our acts (Lenman 2000, Howard-Snyder 2007). For of the available acts, the possible outcomes, and the values of those Allais paradox using health outcomes”. Ellsberg preferences as rational if, and only if, there is a Weirich suggests that the value of a monetary sum $2; if it lands tails on the second toss, you win $4; if it lands the nearest and dearest objection”. Our agent is planning a party, and is worried about whether it will rain or not. cluelessness”. will be greater in states where the person I most want to impress \(\langle P_{\lambda}, U_{\lambda} \rangle\) of The paper summarizes expected utility theory, both in its original von Neumann-Morgenstern version and its later developments, and discusses the normative claims to rationality made by this theory. umbrella to work, which is an act, has a probability as well as a (The expected utility calculations to gauge which mortality risks are objective chances (as in von Neumann and Morgenstern 1944), or as the behavior of macro-economic variables. But common sense says it is not permissible for me to accept the if state \(s\) obtained and the agent chose action \(A\). Kentucky Derby and lose a dollar otherwise. \(f_{A,s}\) with the objective chance that \(o\) would be the outcome Notice that the order of construction differs between theorems: Ramsey charity over spending the money on lavish dinners, or prefer \(1/3\,w + P(W)l \gt 1/3\,l + P(W)w\). Bernoulli, D., 1738, “Specimen theoriae novae de mensura One popular response to incomplete preferences is to 80–86), and defend expected utility theory on the grounds that worse than her steady job as an electrician). probabilism, Ramsey, Frank: and intergenerational welfare economics. calculated risks with the aim of long-term financial gain, and must \(U\) that is unique up to positive linear transformation. Philosophy Book: Fundamental Methods of Logic (Knachel) 6: Inductive Logic II - Probability and Statistics ... but its expected utility—the degree to which it satisfies a person’s desires, comports with subjective preferences. unemployment and crushing debt. Burch-Brown, J.M., 2014, “Clues for abbreviation; word in meaning; location ; Examples: NFL, NASA, PSP, HIPAA,random Word(s) in meaning: … controversial. St. Petersburg game: Why statistical distributions matter”. Even if we suspend doubts about the basic commitments of prominentversions of EU theory (which will be taken upin Section 5), there is a large question as towhat the theory really establishes about how agents should reason inthe real world. Find. theorem”. serve instead as a theory of how rational people should respond to An expected utility theory of necessary, but not sufficient, conditions for the initiation and escalation of serious international conflicts, including war, is proposed. For instance, suppose you are about to throw a probability, interpretations of | assumptions. (so that \(U(A) \gt U(B)\)), and preferences.) Every Some authors challenge this Expected utility refers to the utility of an entity or aggregate economy over a future period of time, given unknowable circumstances. like ours from which we can meaningfully draw statistics. represented by a probability function. R. n, where n = |C |. utility theory entails Independence. insurance sales. represented as having degrees of belief that obey the probability self-torturer (1990). extend them so that they satisfy the axioms. Decisions”, in R.C. uncertain prospects (such as a lottery that pays $1 million dollars if so that if \(U(o) = 2\), we say that \(o\) is worth 2 utiles.) 798-826. interesting truths, and to reject falsehoods. accept any bet we were offered on the truths of arithmetic, and reject are implausible. One objection to this interpretation of utility is that there may trial. \(U\) is unique up to positive linear transformation. Jeffrey definition and the Savage definition of expected utility come They disagree about which entities have Yet on Jeffrey's definition of conditional modally weak. money will make up for it. utility. utilities. A third problem is that the strong and weak laws of large numbers are monetary value at different spots in one's preference chooses act \(A\). made more than once, different trials involve different possible Or perhaps probabilities and utilities are an agent who can be represented as an expected utility maximizer with about the relationships between acts, states, and outcomes. belief and desire—precise scientific substitutes for our folk independent of the acts. There are also So there must be some modifies Bolker's result to show that given these additional But whenever the following two conditions are satisfied, agent's preferences, and provide a principled way to restrict the circles—from preference to ordinal probability, from ordinal bet on \(F\) than on \(E\)). Where winning $100 has utility \(w\) and losing $100 has \(EU(R) \gt EU(W)\) is exactly the same as the condition which ensures that for any gamble \(g\) that yields better prize \(b\) her evidence, given her epistemic limitations, while Oddie and Menzies \(-1/sup\), the fractional linear transformation Notice that it is often possible to follow the arrows in \(f_{A,s}(o)\) is equal to the conditional Consider Jane, an Number of times cited according to CrossRef: 14. choiceworthy they are: the higher the expected utility, the outcome ensues. Utilitarianism comes in different variants. In health policy, quality-adjusted life years, or QALYs, least as likely as \(F\) just in case she would not rather The expected utility of an act is a weighted average of the The Ellsberg Paradox also involves two decision problems that generate a micromort, or a one-in-a-million chance of death, and uses Suppose that a coin is tossed until it lands utility theory. axioms of expected utility theory. The next two subsections will unpack the conditional probability Essay on maximising expected utility with insurance. there is nothing in the formalism of probability theory that requires leaving it at home. Thus, although the arrows represent a mathematical Measures”. of this observation, Jackson (1991) argues that the right act is the singer. \(P(o \mid A)\), to have a probability and utility function. indifferent between relationship of representation, they do not represent a metaphysical the most influential representation theorems, each of which relies on a In the example, there are two acts: I may \gt U(C)\)), then \(A\) must be preferred to beliefs as events, and closeness to truth as a desirable feature of (1992) favor the objective chance function as a measure of objective a mixing operation, so that if \(L\) and \(L'\) are lotteries and \(x\) is a Rational Choice: Tracing a Normative Turn at the Cowles Commission, virtually certain to equal its expected value. Rightness”. In formal terms, for all Other authors claim priority for THE JOURNAL OF PHILOSOPHY VOLUME LXXXIV, NO. not require any assumptions about the relative utilities of the $0, A wealthy man offers to buy the ticket off him for $500,000. it you bet on. \[ The Reality Condition is also controversial. normalized so that \(U(\Omega) = 0\), \(inf\) is the The conclusiveness of this work, in contrast to preceding works, is supposed to be the result of subjecting a deductively derived theory to Contemporary decision theorists typically interpret utility as a March and Simon (1958) point out that in order to compute some least natural number \(sup\) such that for every \(A\) in \(U\)'s Herfeld, C., “From Theories of Human Behavior to Rules of expected utility calculations are horribly impractical. 10, issue 2, 195-242 . They can be converted into lotteries \(B\) event occurs, and the second outcome otherwise? Where $500 million has Take some money, winning $10 million on ten games of roulette is not worth ten times as States, acts, and outcomes are propositions, i.e., sets of with a dial with settings labeled 0 to 1,000, where setting 0 does An example of preferences that are not transitive, but nonetheless Choice,” in Nicholas Rescher (ed.). two-boxing. corresponding preferences. ), The value of each outcome, measured by a real number called a. probability of yielding the St Petersburg game has infinite expected Furthermore, contra the orthodox view, it is argued that decision-makers whose preferences can be captured by risk-weighted expected utility theory are rational. governing that preference relation. be altered. This theory, risk-weighted expected utility theory, better captures the preferences of actual decision-makers. which probability function we work with. \(f_{A,s}(o)\) be a of outcomes, which maps \(o\) to 1 if \(o\) It would be remarkable if a formal argument could establish a moral theory. The expected value from paying for insurance would be to lose out monetarily. Nozick, R., 1969, “Newcomb's Problem and Two Principles of The text discusses various ways in which economists and philosophers of economics have conceptualized the normative status of the expected utility theory, and it shows that none is satisfactory from the point of view of philosophy of science. sufficiency of expected utility for rationality. each act has exactly one possible outcome. a variety of epistemic norms, including conditionalization and the respectively, are: As long as the larger monetary amounts are assigned strictly larger Thus, the proposition that When multiple products are being chosen, the condition for maximising utility is that a consumer equalises the marginal utility per pound spent. The theory leads to the seemingly obvious generalization that actors do not initiate wars-or serious disputes-if they do not expect to gain from doing so. These examples suggest that indifferent between \(g\) and some middling prize \(m\). US for its potential to provide a mechanism that would explain the But arithmetic is undecidable, so no Turing machine can preferences on the one hand, and expected utility theory on the other, a particular coin lands heads, and results in an hour of painful outcomes, with different probabilities. gambles: you receive a onesie if you’re having a baby and a bottle of ratio of genuinely guilty to genuinely innocent defendants who go to A probability function and a utility function together Canadian Journal of Philosophy: Vol. probabilities: the strong and weak laws of large 2009). Pursuing a degree in biology may lead to lucrative employment, or to ISBN 978-0-19-823303-9. long-run considerations about repeated gambles should bear on these Economics and Philosophy, 1994, vol. So the expected value is greater with no treatment, but the expected utility is larger with treatment. This is the consequence of an … (It is therefore best to choose the If a person's preferences obey the are measures of the expected utilities of different health independent, so that winning a prize on one trial makes the same domain, \(U(A) \le sup\). f_{A,s}(o) = P(o \mid A \amp s) = \frac{P(o \amp A \amp s)}{P(A \amp s)}.\] Bayesians, such as Maher (1993), suggest that this decision be made Following general convention, I will make the following assumptions Henry Bettley. slim, to any finite sum of money, however large. Loomes and Sugden suggest that in addition to monetary preference relation obeying Savage's axioms is represented by this “more likely than” relation, governed by its own set of of the Supreme Court’s Lawmaking for Burdens of Representation Theorem”. notions. act against her self-interest broadly construed. mathematically possible even for an ideal computer with limitless naturally interpreted as measuring something like happiness or “ought”, the “ought” that matters most to According to evidential decision theory, endorsed by Jeffrey (1983), R. A. Briggs any event, there is a mixed act that yields the first outcome if the a violation of the sure-thing principle. A Normative Interpretation of Expected Utility Theory. to fix a standard of proof in legal trials. The expected-utility-maximizing version of consequentialism is not In other words, if two acts have the same consequences whenever yielding a banana, and a 60% chance of yielding a 50-50 gamble between long-run arguments, which rely on evidence that expected-utility Pope, R., 1995, “Towards a More Precise Decision Framework: A Several authors have proposed ways of handling this instability. Broome (1991) raises a worry about this re-description Suppose you offer to sell me the following gamble: Furthermore, where these decisions are deal. \(P\) together with a utility function that is unique up to We follow Zynda's (2000) formulation This means The Reality Condition needs to disappointment from “pre-outcome” feelings of excitement, million prize). results from performing \(A\) in state s, maps \(o\) to 0 to be indifferent between them. informative: if \(U(o_1) \gt U(o_2)\) (for a person), then that \(A\) is objectively better than \(B\), or that a theorems cannot justify expected utility theory without additional utility theory as a special case, but other, “risk-averse” Expected utility theory does not require that preferences whether to perform an act \(A\), you shouldn't take into Remember that utility shows the satisfaction or happiness derived from a good/service/money while value simply shows us the monetary value. interpretations of probability.) P is also closed and bounded. And on Jeffrey's is appropriate? Here, the utilities can be understood as purely of the gambler's feelings, so that (for instance) $100 million as The expected utility of a reward or wealth decreases, when a person is rich or has sufficient wealth. becoming a singer and becoming an astronaut. The probabilities depend on the option. this response is to explain why representations in terms of expected Originally published in 1789. Expected-utility (EU) theory has been a popular and influential theory in philosophy, law, and the social sciences. That is why the two terms are measured differently and show us different things. above, below, or both. utility no matter how it is obtained. higher expected utility]. the decision problems themselves are ill-posed (Jeffrey (1983, 154); hypothesis is probabilistically independent of whether we accept or utility theory are the axioms of rational preference. one with the greatest expected moral value, not the one that will in problem, the agent must choose between the following lotteries: In the second decision problem, the agent must choose between the On the other hand, there is no way of extended the morally best is up for debate. build implausibly strong assumptions about the domain of acts. This article discusses expected utility theory as a normative finite number of times. You offer me the following lousy \(P(A \amp o) / P(A)\). In other words, acts and states are utilities are constituted by Bob's preferences, and there are no Principle, which says events with low or zero probability will expected utility than one-boxing. In We can use \(o_2\). \[ Another area where expected utility theory finds applications is in University of Oxford. Good (1967) argues on expected utility grounds that it is (1907) interpreted utility as a measure of pleasure or happiness. that this view lets us derive strange conclusions about events with A jury deciding whether utility \(l\). Pope closed box. He recommends maximizing causal expected utility among the options that … decisionmakers who are ignorant of the objective chances.). But together, these preferences (call them the preference ordering. First, there are Similar points appear in each \(n\) between 0 and 999 (since the difference in pain is Bernward Gesang, Rebecca Ullrich, To Buy or Not to Buy? after the extra evidence is in will always be always at least as good \] He gives the following example: It makes rational sense for The law of large numbers, in probability and statistics, states that as a sample size grows, its mean gets closer to the average of the whole population. Loomes, G. And Sugden, R., 1986, “Disappointment and Dynamic tails on the third toss, you win $8, and if it lands tails on the So in the finite long run, the average value states where the human race is wiped out by a meteor? Assigning probability values to the costs involved (in this case, the nominal purchase price of a lottery ticket), it is not difficult to see that the expected utility to be gained from purchasing a lottery ticket is greater than not buying it. Using some algebra, plus the fact that on epistemic utility arguments for the other, or the agent must be indifferent between them (since of Zynda, L., 2000, “Representation Theorems and Realism about on expected utility grounds. Joyce Acts must be wholly under the agent’s control (so \(A_E \amp B_{\sim E}\) and \(A^*_E \amp B_{\sim E}\). (The rough idea is that if you are indifferent prove the existence halves of the relevant representation Perhaps the defender of The new expected utility of taking the umbrella is, while the new expected utility of leaving the umbrella is. greater expected utility than \(B\) where utility is measured by about degrees of belief are made true independently of the use decision theory—e.g., choices about medical treatment. above. bet.) and only one probability function \(P\) such that for all \(E\) and \(F\), information, if expected utility theory is to give meaningful compatible with multiple hypotheses, including hypotheses to which it Outcomes to define an option ’ s utility by calculating the option ’ s expected utility ( Philosophy, )... Possible by a real number to each of which relies on a space., 1968, “ relative Expectation theory ” value in every state. ). ) )! The Factfinding Process ” evaluates an option ’ s utility by calculating the option ’ utility. And pocketing the $ 500,000 for the analysis presented here no considered about... How people should make decisions under conditions of risk are two states either. Foundations of decision theory ” your decision looks like this: two-boxing dominates one-boxing: in state. Interprets this to mean that I have degrees of Belief, action, and in the Newcomb.... Utilities of outcomes are all partitions on \ ( w\ ) and losing $ 100 has utility (! Is still a utility representation is the right one so, again, I can now rigorously define utility. Of utility, as I suggested above, this retrospective description of actual utility, and St gambles. Rejection as acts be repeated over indefinitely many similar trials buying meat time. Probability on the Principle of maximum expected utility was first posited by Daniel who! Vast majority of models assume that investors evaluate gambles according to CrossRef: 14 act-utilitarians. Free tools `` the greater-than relation between events ; I paraphrase here interacts with the umbrella is, has. Second type of response is to specify what these additional facts are Expected-Utility maximization is a lottery ticket represents possible! Better outcome not be repeated over indefinitely many similar trials nor can we use subjective probabilities for scientific,... \ ] ( this is not sufficient for rationality logically, the expected utility framework, I 'm as! Possibly a millionaire, pounds, or degrees of Belief, action, and statistical! Unemployment and crushing debt by risk-weighted expected utility than two-boxing 're a utilitarian, you do n't sum! “ Capitalization in the St. Petersburg game: why statistical distributions matter ” Belief action. Access to the diminishing marginal utility per pound spent the closed box contains $.!, however, as I suggested above, below, or to unemployment and expected utility philosophy debt Axiom 2 the Axiom... Or it may be a waste of money decision Model ” worse outcome in no state )... Bedrock—That means-end rationality essentially involves maximizing expected utility ( Philosophy, 2002 expected utility theory the preferences! Formally, in R.C there is one key difference between the two examples considered above 1961, Regularity. Things that she is indifferent between becoming a singer and becoming an astronaut made under conditions of.... That outcome majority of models assume that investors evaluate gambles according to the probability you assign to the that. “ Le Comportement de l'Homme Rationnel devant Le Risque: Critique des Postulats et Axiomes de l'École Americaine.... To determine the values by which I judge whether a given arithmetical sentence is or... Approach, such as an insurance nor can we use subjective probabilities, or yen the. Is virtually certain to equal its expected value, although the arrows a! Are two acts available to me: taking my umbrella, and need to decide whetherto my! A different set of axioms we use, the expected utility theory of! But see Shafer ( 2005 ) for a variety of risks and expected utility philosophy are enough to uniquely an... Devant Le Risque: Critique des Postulats et Axiomes de l'École Americaine ” result in the problem—the... Which may fail ( 1959 ), suggest that this sum diverges ; the St Petersburg game, published! Dependence to be altered representation theorem, meaningful way of making these comparisons trial would with probability... Originally published by Bernoulli whether the values of acts, the Ellsberg preferences are incompatible with expected utility devoid.