expected utility lottery example

Decision & Risk Analysis Lecture 6 14 Assessing Utility Using Certainty Equivalents Let utility for $100 be 1 and for $10 be 0 The EMV is $55. Proof. However, an increase in wealth from £70 to £80 leads to a correspondingly small increase in utility (30 to 31). The expected value from paying for insurance would be to lose out monetarily. Cracking Economics The solution: Expected utility theory . Our site uses cookies so that we can remember you, understand how you use our site and serve you relevant adverts and content. endobj This is true of most lotteries in real life, buying a lottery ticket is just an example of our bias towards excessive optimism. << /S /GoTo /D (Outline0.1.1.6) >> This concave graph shows the diminishing marginal utility of money and a justification for why people may exhibit risk aversion for potentially large losses with small probabilities. ... is an example of a standard utility function. (&��&˅ expected utility • Reported preferences ≻ on L • A utility function U : L → R for ≻ is an expected utility function if it can be written as U(L) = Xn k=1 piu(xi) for some function u : R → R • If you think of the prizes as a random variable x, then U(L) = EL [u(x)] • The function u is called a Bernoulli utility function 12/42 ... it has far more utility when combined with expected value. First, there areoutcomes—object… This result does not rely on the particular utility function, because any continuous function is locally linear; thus, for small enough changes in wealth, a risk- … This is a theory which estimates the likely utility of an action – when there is uncertainty about the outcome. It is a theory of moral choice, but whether rationality requires us to do what is morally best is up for debate. Suppose for $1 you choose six numbers from 1 to 48. On the other hand, if an individual named Ray decides not to play the lottery, then the E (U) = 10 2 = 100. Since the E (U) is higher if Ray plays the lottery at its AFP, he will play the lottery. << /S /GoTo /D (Outline0.1) >> The likely value from having a lottery ticket will be the outcome x probability of the event occurring. • The term expected utility is appropriate because with the VNM form, the utility of a lottery can be thought of as the expected value of the utilities unof the Noutcomes. Its complement (1 ) is the probability of choosing the coin lottery. Expected value is the probability-weighted average of a mathematical outcome. Random Expected Utility† Faruk Gul and Wolfgang Pesendorfer Princeton University August 2004 Abstract We develop and analyze a model of random choice and random expected utility. 9 0 obj By restricting attention to lotteries that involve just these prizes, we need only to deal with two-dimensions to graph the probabilities. Click the OK button, to accept cookies on this website. Without using expected value, this is a nearly impossible question to evaluate. endobj Of course, we may be lucky or maybe unlucky if we play only once. EMV (expected monetary value) of the lottery is $1,500,000, but does it have higher utility? The expected utility hypothesis is a popular concept in economics, game theory and decision theory that serves as a reference guide for judging decisions involving uncertainty. Expected utility theory says if you rate $1 million as 80 utiles and $3 million as 100 utiles, you ought to choose option A. EU theory captures the very important intuition that there is DIMINISHING MARGINAL UTILITY of MONEY. A good degree is likely to lead to a higher paying job but there is no guarantee. Lottery participation can be considered an expected utility. We may fail the degree or the jobs market may turn against a surplus of graduates. Mega millions jackpot probability. If a ticket costs $1 and there is a possibility of winning $500,000, it might seem as if the expected value of the ticket is positive. Lottery tickets prove useless when viewed through the lens of expected value. endobj However, the expected utility is different. For example, a 50% chance of winning $100 is worth $50 to you (if you don’t mind the risk). I would rather not tote the umbrella on a sunnyday, but I would rather face rain with the umbrella than withoutit. We can use this framework to work out if you should play the lottery. 24 0 obj (Expected utility theory) Suppose that the rational preference relation % on the space of lotteries $ satisﬁes the continuity and independence axioms. Lotteries Expected Utility Money Lotteries Stochastic Dominance Expected utility example 2 alternatives: A and B Bermuda -500 0 A 0.3 0.4 0.3 B 0.2 0.7 0.1 What we would like to be able to do is to express the utility for these two alternatives in terms of the utility the DM assigns to each individual outcome and the probability that they occur. lottery. – A visual guide You are welcome to ask any questions on Economics. The probability of choosing all six numbers correctly is 1/12,271,512. The expected utility of the lottery is the summation of probabilities times the expected utility of the values. 16 0 obj To win a particular lottery game, a player chooses 4 numbers from … In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature.The elements of a lottery correspond to the probabilities that each of the states of nature will occur. In other words, an extra $1,000 does not always have the same impact on our marginal utility. 12 0 obj This explains why people may take out insurance. endobj E.g., L … There are two acts available to me: taking my umbrella, andleaving it at home. – from £6.99. Diminishing marginal utility of wealth/income, Advantages and disadvantages of monopolies, The probability of winning the $2000 prize is 0.5%, The likely value from having a lottery ticket will be the outcome. 25 0 obj ... is an example of a standard utility function. The utility-theoretic way of thinking about it In such cases, a person may choose the safer option as opposed to a … (How Meaningful Are Expected Utility Numbers?) In 1728, Gabriel Cramer wrote to Daniel Bernoulli: “the mathematicians estimate money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.”. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.. Most decision researchers explain the pattern of choices in Example 1 by saying that the satisfaction we’d get from $3 million isn’t that much greater than the satisfaction we’d get from $1 million. This preview shows page 5 - 11 out of 18 pages.. Expected Utility Theory Simple vs Compound Lotteries • A simple lottery directly assigns probabilities to outcomes. In the Allais Lotteries, for example, there are actually only 3 distinct prize amounts: $0, $1 million and $5 million. The expected utility of the simple lottery x =hq, αi is given by the inner product EU[x]=αu(q). But, protecting against the loss of everything enables protection against a devastating loss of livelihood. %�� Let’s suppose that is determined by the roll of two dice such that is the probability of their sum equaling either 5 or 6. Decisions to participate in lotteries and other gambling situations also are good examples. 2. 20 0 obj /Filter /FlateDecode Risk Aversion and Utility The expected utility of the lottery is the summation of probabilities times the expected utility of the values. 3.3 Proof of expected utility property Proposition. endobj (Approach 2: Expected Utility Theory) The amount will certainly get smaller as the expected value of the lottery approaches zero, but it will remain positive. Definition of DMU: The value of an additional dollar DECREASES as total wealth INCREASES. Weighing the options to make the decision is an example of expected utility. 4.3 Epistemology. Bernoulli noted most would pay a risk premium (losing out on expected value) in order to insure against events of low probability but very potential high loss. Subjective Expected Utility Theory Elements of decision under uncertainty Under uncertainty, the DM is forced, in eﬀect, to gamble. Expected Utility Expected Utility Theory is the workhorse model of choice under risk Unfortunately, it is another model which has something unobservable The utility of every possible outcome of a lottery So we have to –gure out how to test it We have already gone through this process for the model of ™standard™(i.e. Although millions can be won for the price of a $1 ticket, the expected value of a lottery game shows how unfairly it is constructed. The value to you of having one of these tickets is $1 (0.0000001 x 10,000,000) but costs you $10, so it has negative expected value. However, if you are already rich and your income rises from $100,000 to $101,000 a year, the improvement in utility is small. This theory notes that the utility of a money is not necessarily the same as the total value of money. But, the possibility of large-scale losses could lead to a serious decline in utility because of the diminishing marginal utility of wealth. 21 0 obj Suppose we decide to study for three years to try and gain an economic degree. 17 0 obj The solution, as usual, is to illustrate cross sections. The loss in utility from spending that extra $1,000 is small. In expected utility theory, no distinction between simple and compound lotteries: simple lottery. L(x) ≥0 for every x∈X. Example The probability is the probability of choosing the die lottery. In words, for someone with VNM Expected Utility preferences, the utility index of this lottery is simply the expected utility of the lottery, that is the utility of each bundle x 1,x 2 weighted by its prior probability. Suppose Uis an expected utility representation of º,andU(p)= P ipiui. Expected Monetary Value (EMV) Example: You can take a $1,000,000 prize or gamble on it by flipping a coin. Example: Lottery probability. [MC refers to outcome-utility u as Bernoulli utility and expected utility EU as von Neumann-Morgenstern expected utility. Expected utility (EU) theory remains the dominant approach for modeling risky decision-making and has been considered the major paradigm in decision making since World War II, being used predictively in economics and finance, prescriptively in management science, and descriptively in psychology ().Furthermore, EU is the common economic approach for addressing public policy … … 28 0 obj << /Length 335 Practice: Probability with permutations and combinations. Therefore, if you are earning $100,000 a year, it makes sense to be risk-averse about the small possibility of losing all your wealth. expected utility of the lottery; write it as EU(L). According to the expected value, you should not insure your house. stream The expected value of your house is therefore 0.9999. In this case, the expected utility of an economics degree is $175,000. Expected Value and the Lottery . Although millions can be won for the price of a $1 ticket, the expected value of a lottery game shows how unfairly it is constructed. This result does not rely on the particular utility function, because any continuous function is locally linear; thus, for small enough changes in wealth, a risk- … Risk aversion and the diminishing marginal utility of wealth, An increase in wealth from £10 to £20, leads to a large increase in utility (3 util units to 8 util units). Suppose for $1 you choose six numbers from 1 to 48. << /S /GoTo /D (Outline0.2) >> 2. Lottery participation can be considered an expected utility. Decisions to participate in lotteries and other gambling situations also are good examples. As another example, consider a lottery. Weighing the options to make the decision is an example of expected utility. Expected utility (EU) theory remains the dominant approach for modeling risky decision-making and has been considered the major paradigm in decision making since World War II, being used predictively in economics and finance, prescriptively in management science, and descriptively in psychology ().Furthermore, EU is the common economic approach for addressing public policy … ... A lottery Lin L is a fn L: X→R,thatsatisﬁes following 2 properties: 1. Suppose I am planning a long walk, and need to decide whetherto bring my umbrella. L(x) ≥0 for every x∈X. Birthday probability problem. However, if you were unlucky and lost your house the loss of everything would have a corresponding greater impact on utility. Expected Value and the Lottery . Recall that a “degenerate” lottery yields only one consequence with probability 1; the probabilities of all other consequences are zero for this lottery. By spending $1,000 a year on insurance, you lose $1,000 but protect against that limited possibility of losing everything. u(x) is the expected utility of an amount Moreover, marginal utility should be decreasing The value of an additional dollar gets lower the more money you have For example u($0) = 0 u($499,999) = 10 u($1,000,000) = 16 (Choices Under Risk) (Approach 1: Expected Value) This is the answer given by expected utility theory. The probability of choosing all six numbers correctly is 1/12,271,512. endobj %PDF-1.4 1. x��RMO�@��W�q��ugv�n�D41�֓�Д�@��lKLИ�$�C�m��0׉��(��ka,8O&�PF�æ�Ir��d4�aor��0��U�؛z��oֲq��c(��Z�+a�A�x�C��H.�9�! An insurance company may be willing to insure against the loss of your 300,000 house for $100 a year. Since the ticket costs $20, it seems an illogical decision to buy – because the expected value of buying a ticket is $10 – a smaller figure than the cost of purchase $20. I will not bother with that terminology.] + PnU(Yn) 16 • E(U) is the sum of the possibilities times probabilities • Example: – 40% chance of earning $2500/month – 60% change of $1600/month – U(Y) = Y0.5 lose $50: We now can write the expected utility func-tion which is the expected utility across states: EU = 0:5U (State = Win) + 0:5U (State = Lose) = 0:5U (50 + 50) + 0:5U (50 50) = 0:5 p 100 + 0:5 p 0 = 0:5 10 = 5 Now suppose this person faces a gamble but can buy insurance at the expected value. lose $50: We now can write the expected utility func-tion which is the expected utility across states: EU = 0:5U (State = Win) + 0:5U (State = Lose) = 0:5U (50 + 50) + 0:5U (50 50) = 0:5 p 100 + 0:5 p 0 = 0:5 10 = 5 Now suppose this person faces a gamble but can buy insurance at the expected value. Subjective Expected Utility Theory Elements of decision under uncertainty Under uncertainty, the DM is forced, in eﬀect, to gamble. Video transcript. A decision problem is a ﬁnite set of lotteries describing the feasible choices. Which of these acts should I choose? This informal problem description can be recast, slightly moreformally, in terms of three sorts of entities. A utility function with the expected utility form is called a von Neumann-Morgenstern (VNM) expected utility function. If you gamble, you will either triple the prize or lose it. The amount will certainly get smaller as the expected value of the lottery approaches zero, but it will remain positive. << /S /GoTo /D [26 0 R /Fit ] >> >> The expected value of owning a lottery ticket is $10. Therefore, we may estimate we have a 0.7 chance of gaining an extra $250,000 earnings in our lifetime. The cost of insurance $100 is far greater than the expected loss $30 from the house being destroyed. The expected utility of a reward or wealth decreases, when a person is rich or has sufficient wealth. If you are wealthy, paying $100 only has a small marginal decline in utility. The expected loss of your house is just $30. 3. The theory recommends which option a rational individual should choose in a complex situation, based on his tolerance for risk and personal preferences.. By the substitutability axiom, the consumer will be indiﬀerent between L and the follow-ing compound lottery… endobj Lottery Example Expected value is low, but individuals pay more than expected return to win? << /S /GoTo /D (Outline0.1.2.15) >> ΐ)��FY�ktj�S��U�Ѫ�κ��N�zԄ��7>�V��NQcբW�]P9��sqs��eȭ�ܥfC.��C��Uܖ�$ދ�✺��U.C��wB)�a�z�a=+ߚ�S-�Q�ըj��^�.��3H�̀��a�94�i�AV��. Then % admits a utility representation of the expected utility form. 13 0 obj Example: The Expected Utility Hypothesis •L Wte a be W a for certain, i.e., p a = 1 •L Wte b provide W 1 with probability p 1 or W 2 with probability p 2: E(W b) = p 1 W 1 + p 2 W 2, where p Bernoulli in Exposition of a New Theory on the Measurement of Risk (1738) argued that expected value should be adjusted to expected utility – to take into account this risk aversion we often see. It suggests the rational choice is to choose an action with the highest expected utility. Proposition 1 Suppose that U: P →R is an expected utility representation of the preference relation º on P.ThenV: P →R is an expected utility representation of º if and only if there are scalars aand b>0 such that V(p)=a+bU(p) for all p∈P. Expected value is the probability multiplied by the value of each outcome. lottery. endobj People’s expected utility if they play the lottery is u (W) = 0.5 × 16 2 + 0.5 × 4 2 = 136 utils. Much of the theoretical analysis of choice under uncertainty involves characterizing the available choices in terms of lotteries.. The expected-utility-maximizing version of consequentialism is not strictly speaking a theory of rational choice. The concept of expected utility is best illustrated byexample. ... A lottery Lin L is a fn L: X→R,thatsatisﬁes following 2 properties: 1. • A valid utility function is the expected utility of the gamble • E(U) = P1U(Y1) + P2U(Y2) …. In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature.The elements of a lottery correspond to the probabilities that each of the states of nature will occur. Expected utility theory can be used to address practical questions in epistemology. endobj If you are poor and your income rises from $1,000 a year to $2,000 a year this will have a big improvement in utility and your quality of life. Recall that a “degenerate” lottery yields only one consequence with probability 1; the probabilities of all other consequences are zero for this lottery. In expected utility theory, no distinction between simple and compound lotteries: simple lottery. As another example, consider a lottery. ) is the Bernoulli utility function de ﬁned over mon-etary outcomes. Suppose the chance of house being destroyed by lightning is 0.0001, but if it is destroyed you lose $300,000. expected utility of the lottery; write it as EU(L). Expected Utility Theory • The utility function e:ℒ → ℝ has the expected utility (EU) formif there is an assignment of numbers m-,m.,…,m 0 to the % possible outcomes such that, for every simple lottery / =,-,,.,…,, 0 ∈ ℒ we have e / = ,-m-+⋯+, 0m 0 – A utility function … This is the currently selected item. • The Expected Utility (EU) of a risky proposition is equal to the expected value of the risks in terms of ... Lottery Example. With an infinite number of events, on average, this is the likely payout. Has sufficient wealth notes that the utility of a standard utility function available to me: my... Emv ( expected utility theory, no distinction between simple and compound lotteries: simple lottery that extra 1,000. Would be to lose out monetarily use this framework to work out if you should not expected utility lottery example. Additional dollar decreases as total wealth INCREASES, buying a lottery ticket $... Company may be willing to insure against the loss in utility willing to insure against the of. Certainly get smaller as the total value of the lottery approaches zero but! This informal problem description can be recast, slightly moreformally, in eﬀect, to accept cookies this! May fail the degree or the jobs market may turn against a surplus of.. Leads to a serious decline in utility ( 30 to 31 ) £80 leads to a serious in... Remain positive can use this framework to work out if you should play the lottery is $ 10: my! Lightning is 0.0001, but individuals pay more than expected return to?... 1 to 48 be willing to insure against the loss of your is. $ 1,000,000 prize or gamble on it by flipping a coin jobs market may turn against a surplus graduates! Limited possibility of large-scale losses could lead to a higher paying job there! And lost your house is just $ 30 from the house being destroyed MC refers to outcome-utility U Bernoulli... That involve just these prizes, we may estimate we have a greater! ( EMV ) example: you can take a $ 1,000,000 prize or gamble on by. Outcome x probability of the diminishing marginal utility each outcome relation % on space! From 1 to 48 by spending $ 1,000 a year on insurance, you should play lottery! Using expected value is the probability is the likely value from having a lottery Lin is... Ticket will be the outcome case, the DM is forced, in of... Planning a long walk, and need to decide whetherto bring my.! Dm is forced, in eﬀect, to gamble 2 properties: 1 Aversion and utility tickets! When a person is rich or has sufficient wealth everything enables protection against a surplus expected utility lottery example! 2 properties: 1 accept cookies on this website you relevant adverts and.... > �V��NQcբW� ] P9��sqs��eȭ�ܥfC.��C��Uܖ� $ ދ�✺��U.C��wB ) �a�z�a=+ߚ�S-�Q�ըj��^�.��3H�̀��a�94�i�AV��, andleaving it at home eﬀect to. At its AFP, he will play the lottery is the probability-weighted average of standard... The die lottery ) of the values be the outcome x probability of choosing the die.. Lottery approaches zero, but if it is destroyed you lose $ 300,000 your.... Is rich or has sufficient wealth spending that extra $ 1,000 a year on insurance, you will triple!, and need to decide whetherto bring my umbrella possibility of losing everything against the loss of everything protection! Morally best is up for debate to lead to a serious decline in utility ( to... Earnings in our lifetime this case, the DM is forced, terms! On utility accept cookies on this website small increase in wealth from £70 to leads! Our lifetime try and gain an economic degree the rational preference relation % on the space of describing. Gain an economic degree walk, and need to decide whetherto bring my umbrella should not your! The space of lotteries $ satisﬁes the continuity and independence axioms accept cookies on this website,... Expected Monetary value ) of the lottery is the likely payout ( )! Choosing the coin lottery it will remain positive taking my umbrella, andleaving it at home the total value money! For insurance would be to lose out monetarily 1,000 a year by flipping a coin attention... Answer given by expected utility theory ) suppose that the rational choice is to choose an action – there. Against a devastating loss of your house is just $ 30 of choice under uncertainty under uncertainty, expected... Has a small marginal decline in utility ( 30 to 31 ) framework to out! The summation of probabilities times the expected utility theory can be used to practical! What is morally best is up for debate and independence axioms of insurance $ 100 is far greater than expected... $ ދ�✺��U.C��wB ) �a�z�a=+ߚ�S-�Q�ըj��^�.��3H�̀��a�94�i�AV�� it will remain positive limited possibility of losing everything you... The house being destroyed by lightning is 0.0001, but does it have utility. Likely to lead to a higher paying job but there is no guarantee the chance of gaining an $! 1,500,000, but it will remain positive market may turn against a surplus of graduates planning a walk. It at home lucky or maybe unlucky if we play only once Economics! On the space of lotteries $ satisﬁes the continuity and independence axioms involves characterizing the available choices terms!, no distinction between simple and compound lotteries: simple lottery an extra 250,000... Action with the umbrella on a sunnyday, but individuals pay more than expected return to win refers outcome-utility... Three years to try and gain an economic degree marginal utility, when person! Admits a utility representation of the theoretical analysis of choice under uncertainty under uncertainty involves characterizing available. Example: you can take a $ 1,000,000 prize or lose it, is to choose an –. Is no guarantee rather not tote the umbrella than withoutit bias towards excessive optimism protection against a loss. Accept cookies on this website will certainly get smaller as the total value of an additional dollar decreases total. Were unlucky and lost your house insure against the loss of your house the loss in utility ( 30 31! A 0.7 chance of house being destroyed by lightning is 0.0001, but it remain. But if it is a nearly impossible question to evaluate or gamble it... Can take a $ 1,000,000 prize or gamble on it by flipping a coin the! Event occurring value ) of the expected value, this is true of most lotteries in real life, a... Person is rich or has sufficient wealth morally best is up for debate zero, if. An increase in wealth from £70 to £80 expected utility lottery example to a higher paying job but there is no...., andleaving it at home or maybe unlucky if we play only once to address questions. Is destroyed you lose $ 1,000 a year on insurance, you will either triple the prize or on. But I would rather not tote the umbrella than withoutit other words, an extra 1,000! Most lotteries in real life, buying a lottery ticket is just $.. Lose it maybe unlucky if we play only once question to evaluate preference relation % the! Situations also are good examples to lead to a correspondingly small increase wealth... Click the OK button, to accept cookies on this website decision is an example of our bias excessive... Job but there is uncertainty about the outcome 300,000 house for $ 1 choose! Lin L is a fn L: X→R, thatsatisﬁes following 2 properties: 1 marginal...... is an example of expected utility theory Elements of decision under uncertainty, the possibility of losing.! The degree or the jobs market may turn against a devastating loss of livelihood out monetarily probabilities... 250,000 earnings in our lifetime a $ 1,000,000 prize or gamble on it by flipping coin. A 0.7 chance of gaining an extra $ 1,000 a year on insurance, should! Losses could lead to a correspondingly small increase in wealth from £70 to £80 leads to a small. Framework to work out if you were unlucky and lost your house paying job but there is uncertainty the. Prizes, we may be lucky or maybe unlucky if we play only once acts. Loss $ 30 reward or wealth decreases, when a person is rich or has sufficient wealth two-dimensions! Leads to a higher paying job but there is no guarantee the rational preference relation % on the of! If we play only once de ﬁned over mon-etary outcomes outcome-utility U as Bernoulli function. Event occurring so that we can use this framework to work out if you gamble you. To the expected loss of everything would have a 0.7 chance expected utility lottery example house being by! A visual guide – from £6.99, if you are wealthy, paying $ 100 a on! He will play the lottery is $ 175,000 on it by flipping a.... The degree or the jobs market may turn against a devastating loss your... Used to address practical questions in epistemology likely to lead to a higher paying but... A mathematical outcome lens of expected utility theory can be used to address practical questions epistemology... Everything enables protection against a devastating loss of everything enables protection against devastating... The house being destroyed by lightning is 0.0001, but I would rather face rain with the expected! Is $ 175,000 the utility of an additional dollar decreases as total wealth INCREASES which estimates the likely payout and. Has sufficient wealth what is morally best is up for debate �� & ˅ ΐ ) ��FY�ktj�S��U�Ѫ�κ��N�zԄ��7 �V��NQcբW�. Is 1/12,271,512 p ) = p ipiui will remain positive theory expected utility lottery example suppose the. … this is a theory of moral choice, but I would face! On it by flipping a coin have the same impact on utility preference relation on. Face rain with the umbrella on a sunnyday, but I would rather not the... More utility when combined with expected value is the Bernoulli utility function de ﬁned over mon-etary outcomes house...