8 0 obj �G֘ More gen­er­ally, for a lot­tery with many p… After you’ve repeated this process enough, we can deduce what your favorite good of all the listed goods is. stream To do so, he had to make use of VNM theory. stream 3.3. vNM vNM expected utility theoryexpected utility theory a)a) Intuition Intuition [L4] b) A i ti f d tiAxiomatic foundations [DD3] 4.4. Receive 1.00e+0 Banana lottery ticket(s)or 1.00e+0 Carrot lottery ticket(s) Indifferent. x�uP=O1eί�狝O���8$$�Nb��*]�J���s��P$��q����v�y�3�y�~��9@!��c����HhW���� ������1�#��oZ��_k�pe,ʹ �:�� ��z��mf BI�a|5'J���Qvfe�]ɧj��+���R�"v�e�K�G�A������>��>yI��E�T�\��xk�Y6���D�C�����c�8�����1%_�d��2D%@᯼�1GP>��Y_p�N�l����J&� T��4?l]endstream What is a von Neumann-Morgenstern expected utility function? 3 The theorem then proved that if an agent is VNM-rational, then there exists some utility function (commonly called the VNM utility function) such that the agents decisions coincide with the decisions that maximize that utility function… endobj In the the­o­rem, an in­di­vid­ual agent is faced with op­tions called lot­ter­ies. An individual’s von Neumann-Morgenstern (vNM) utility function is given by U(M) = √? 282 Presenting them with a series of lotteries is at least a different task and it may turn out to be an easier or more accurate one. ) is the Bernoulli utility function de fined over mon-etary outcomes. (a) Calculate the Arrow-Pratt coefficients of absolute and relative risk aversion at the level of wealth w. (b) Calculate the risk premium for a gamble (0.5 16,0.5 4). The expected utility of any gamble may be expressed as a linear combination of the utilities of the outcomes, with the weights being the respective probabilities. <> A \(\frac{100}{n}\) chance of a carrot is better than a \(\frac{1}{n}\) chance of a banana (\(n \geq 101\)). I prefer an apple to a banana but can’t or won’t quantify the magnitude of that preference. 3. Getting back to our earlier examples, … <> x�uPMKA��_���a���ε�� But the somewhat sloppy way I like to think of it is this: If a person has merely ordinal preferences (e.g. Expected utility function U : P → R. represents preferences t on P just like in Lectures 1—2. Here the utility attached to any combination of consumption bundles de- pends on the pattern of consumption in a nonlinear way. endobj To do so, he had to make use of VNM theory. Proposition 1 Assume that % is consistent with expected utility. The extra information is useful, for example, in sidestepping Arrow’s impossibility theorem (which says that it’s impossible to have a good voting system if you only ask people for their ordering of candidates). In decision theory, the von Neumann–Morgenstern utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future. 283 42 0 obj expectation of their utility values, where the expectation is taken with respect to some well-defined pair of probability and utility function. 9 0 obj ... represented by an agent's utility function. De nition:A function f : Rk!R isconcavei f(x;y) 2Rk+1: y f(x)gis convex. <> x��TMo1�k~E���dmlj�붕X$$Jq@ж�J-�_�=��v'U�Zi����=�̍���o��\��;~�П��j�H۳�je?Z(֚�o���,Wn�z��o���G�x���o�:�/���;K�����m_�{l��r�z�'���~��MC�i,+E*~}�>��a��%��ƔS��ݜ5fJ��9d ��fIV3���b�\Jq:��9px?��8�]h�.�΄��r2�J���`��_�al�O�� {�Xs�'�� 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 Given some mu­tu­ally ex­clu­sive out­comes, a lot­tery is a sce­nario where each out­come will hap­pen with a given prob­a­bil­ity, all prob­a­bil­i­ties sum­ming to one. (b) Derive the Hicksian Demand functions for good X and Y given the following utility function: U(X, Y) = √? X = {apple, banana}. The standard descriptions of the mechanism of the VNM utility theorem may be a little opaque. For ex­am­ple, for two out­comes A and B, 1. The v.NM function maps from the space of lotteries to real number as it represents the preference defined on the lottery space while the Bernoulli is defined over sure amounts of money. (2) Interactive VNM. Figure 2 shows a strongly compatible vNM utility function (left panel), and a vNM utility function von Neumann-Morgenstern utility function u : C → R. is not a standard utility function. .� �:x����ll�=2���q|��c`��їDQ;X�w�&v�����\��j�T��ʲH�%��uT��`���RsHl�m ��$�f#e.�\��x��M�q�uz��kP?��W!�|���Rr��L�O\ƨ�9�W��F]=��cщ>�����%��T�e��X�\�endstream Moreover, u ... (VNM) utility function by multiplying it with a positive number, or adding a constant to it; but they do change when we transform it through a Risk Attitude and Shape of the vNM utility function I Our definition of risk attitude applies to any type of preference relation over L. I Now, we investigate the implications of the different risk attitudes when preferences are consistent with expected utility. If you haven't already, check out the Von Neumann-Morgenstern utility theorem, a mathematical result which makes their claim rigorous, and true. L=0.25A+0.75B{\displaystyle L=0.25A+0.75B} de­notes a sce­nario where P(A) = 25% is the prob­a­bil­ity of A oc­cur­ring and P(B) = 75% (and ex­actly one of them will occur). Theorem (Expected Utility Theorem): If % satis es continuity and independence, then it is represented by a vNM utility function. It starts with a few sample goods, but you’re free to add, remove or otherwise alter these. And their description of "a certain way" is very compelling: a list of four, reasonable-seeming axioms. 26 0 obj In their definition, a lottery or gamble is simply a probability distribution over a known, finite set of outcomes. Its popularity stems from the fact that, under the assumption of quadratic utility, mean-variance analysis is optimal. For example, in Figure 1, Lottery 1 depicts a situation ... utility function u : Z →R as in (1) if and only if º satisfies Axioms 1-3. %PDF-1.4 stream Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4] 5. Suppose that an individual has a VNM utility function u(x) = x1/2. Conclusion 1 (1) For every nonempty group T, v T (r ⁎) = v T (r ⁎) = 0. The utility of a lottery follows the standard expected utility formula. 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 ��Ԡ,���J�5�B+������mo]۔Y#���9)�� �Cti�(�d���7�ӮP��Zq7c�� n)s;��Fc�� , �2��d�6j���Tm��j��� ;���L�bi�AU(إ]L��~XU }��TknugT�|]��)7���]v�u�v&�甦=��$7MW��$���X�ucTm#���R�%�M�$T�ק���"�~�I��c.rW�ߩ#.Q��}2@�l2f������q4+��I�FE ����b��/���3��� ��)&�$�}ao�˾�4a�fX��}L�ɶ�"��{��~*�endstream In the rest of the paper, we show that these two 1 Which things would you like to make a utility function out of? There are two important things to note here. The following conclusion is implied by what was written thus far. Modifications made through either of these will give rise to a non-expected utility function, which is supposed to improve the model's descriptive accuracy of people's decision under risk. Such utility functions are also referred to as von Neumann–Morgenstern (vNM) utility functions. 3. But because the theorem is constructive, we can actually give people a feel for it by putting them ‘inside’ the mechanism and showing them the result. • A valid utility function is the expected utility of the gamble • E(U) = P1U(Y1) + P2U(Y2) …. endobj 3.3. vNM vNM expected utility theoryexpected utility theory a)a) Intuition Intuition [L4] b) A i ti f d tiAxiomatic foundations [DD3] 4.4. 3 Risk aversion coefficients and Risk aversion coefficients and pportfolio choice ortfolio choice [DD5,L4] 5. Concavity and Risk Aversion De nition:A set C ˆRk isconvexif it contains the line segment connecting any two of its members. Another example of a utility function that might be used to examine choice under uncertainty is the Cobb-Douglas utility function: ( 1 )" 1-.. UC1,C2,7I", -71" =clC2 . Another example of a utility function that might be used to examine choice under uncertainty is the Cobb-Douglas utility function: ( 1 )" 1-.. UC1,C2,7I", -71" =clC2 . The von Neumann–Morgenstern utility theorem lets us turn an ordinal utility function into a cardinal utility function. endobj 59 0 obj Chooses to maximize a utility function u. u specifies how much utility DM gets from each alternative: u : X → R. Example: DM chooses whether to eat an apple or a banana. 33 0 obj 3 This function is known as the von Neumann–Morgenstern utility function. function: If x;y 2C and 0 1, x + (1 )y 2C. (a) Calculate the Arrow-Pratt coefficients of absolute and relative risk aversion at the level of wealth w. (b) Calculate the risk premium for a gamble (0.5 16,0.5 4). Over time, by answering more questions, we can refine these intervals until they’re arbitrarily small. This preview shows page 6 - 8 out of 8 pages., since different increasing utility functions express different risk pref-erences.But some distributions are better than others for anyone with an increasing vNM utility function. von Neumann and Morgenstern weren't exactly referring to Powerball when they spoke of lotteries (although Powerball is one of many kinds of gambles that the theory describes). Example of 1: Rank-Dependent Utility 3. vNM expected utility theory a) Intuition [L4] b) Axiomatic foundations [DD3] 4. In financial economics, the utility function most frequently used to describe investor behaviour is the quadratic utility function. <> <> Once you’ve decided upon the goods you’re interested in, you can proceed to the next step. the agent’s vNM utility function. %�쏢 endobj stream I like apples exactly twice as much as bananas and would be indifferent between an apple and two bananas (ignoring diminishing marginal utility for the same of exposition).). Hence, we see that dominance by pure strategies coincides with dominance by mixed strategies if the agent is sufficiently risk-averse, and there exists a sufficiently risk-averse utility function which is compatible with the given ordinal preferences. Here the utility attached to any combination of consumption bundles de- pends on the pattern of consumption in a nonlinear way. Based on the questions you answer, we know upper and lower bounds for your value (a carrot is better than \(\frac{1}{100}\) banana but worse than \(\frac{1}{1}\) banana). Figure 2 vNM utility functions for Example 1 with X = {1,2}. The extra information is useful, for example, in sidestepping Arrow’s impossibility theorem (which says that it’s impossible to have a good voting system if you only ask people for their ordering of candidates). endobj This is what makes vNM theory consistent with a wide range of non-standard preferences. A VNM-rational agent satisfies 4 axioms, stated in the article. The extra information is useful, for example, in sidestepping Arrow’s impossibility theorem (which says that it’s impossible to have a good voting system if you only ask people for their ordering of candidates). Going from L 1 to L 2 L 1 306 I can also imagine the basic setup of VNM as useful for preference elicitation. Assume this individual has Rs 4 with him. 50 0 obj Suppose that an individual has a VNM utility function u(x) = x1/2. The utility of a decision problem follows the standard expected utility for-mula weighted by the actual choice probability of each option, added (subtracted) by a bene t (cost) term that depends on the size of the decision problem. A \(\frac{1}{n}\) chance of a banana is better than a \(\frac{1}{n}\) chance of a carrot, by your lights (\(n \geq 2\)). VNM utility is a decision utility, in that it aims to characterize the decision-making of … 10 11 Assumptions about utility with uncertainty • Utility is a function of one element (income or wealth), For example, a firm might, in one year, undertake a project that has particular probabilities for three possible payoffs of $10, $20, or $30; those probabilities are 20 percent, 50 percent, and 30 percent, respectively. Utility function might say u (apple) = 7, u (banana) = 12. Moreover, u ... (VNM) utility function by multiplying it with a positive number, or adding a constant to it; but they do change when we transform it through a We abbreviate v {i} to v i, for every referent individual i ∈ I. The resulting function over lotteries v T is a vNM utility function. U : P → R. is an example of a standard utility function. Here, we have an interactive widget that actually constructs a utility function from a series of questions using the theorem. (c) Calculate the risk premium for a … x��XMkGM�{�9��r�!�VwUL����A���m�r��cI��ϫ����Ѭ�%�xdz�Uկ^�����V���W>_���0�;9_��d��㔌��ݚR��KMJ�:���Q��?\��]�}x�:��3��������������ݣU�ԝ��ʌ����iw�H. This transformation is often useful because a cardinal utility function is much richer and more informative than an ordinal utility function. 284 (c) Calculate the risk premium for a … Prudence coefficient and precautionary savingsPrudence coefficient and precautionary savings [DD5] 6.6. Exercise: Show that if is represented by a vNM utility function, then % is continuous and satis es the independence axiom. In the first text area, enter a list of goods (each on a separate line) for which you’d like to generate a utility function. stream ) is the Bernoulli utility function de fined over mon-etary outcomes. 6. L !R is another vNM utility representing % if and only if 9 >0 and such that U~ = U+ . A great deal of time is spent distinguishing the big U (von-Neumann-Morgenstern)v. small u (Bernoulli Utility Function). 51 0 obj The standard descriptions of the mechanism of the VNM utility theorem may be a little opaque. For example, if you mildly prefer bananas to carrots, you’d click on the banana box when presented with one lottery ticket for each. and reasons well under uncertainty, we can transform those ordinal preferences into a cardinal utility function (e.g. Conversely, by letting the lottery axioms “do the work” in securing a utility function, vNM theory doesn’t imply extra restrictions on bundle preferences—that, is restrictions above and beyond what is required for a utility representation. With this as a numéraire, we can start to visualize your utility function and do so with a chart that appears at the bottom. Preference structures to guarantee the existence of a von Neumann–Morgenstern (vNM) utility function are well known under various structural assumptions, since they have been extensively studied and generalized in the last half century (see, for example, Fishburn, 1970, Fishburn, 1982, Hammond, 1999). stream ��Ń�ڋ��*�}3�b� �7I&y���k��;�����p� ��O�΋D촕E�`{����l�~������Gd�o�5���0�� 25 0 obj To relate impose any restrictions on the differences u(a,x)−u(b,y) when x 6= y. x�e��N1EY�+��,&�c'�lU)��X �*�������"!Kq���\g����}�u0�f���B)�}��ա��Z�)ؗ���N`0�������08��թ����h�SP_��_&��c���Rd-���x�]��`CT _���\^�!�!r 94�S:�vKD�lC oG�}�u8l�1��%ƀ�#�s�Nќ �ܹ���g��ke#��MUR�*��#���j1.SqU�W9�����O������(I>Jts;,u���R�x�!��_���_W|�^�����=(drendstream Utility functions are also normally continuous functions. x�uP=O1eί�狝O���8$$�Nb��*]�J���s��P���v����v�q�3�y�~��9@!�ֱH�N[I$�'�����w�y�ژ���7��_k�pe,ʹ �:�� ��z��mf BI�a|5'J���Q;�����S�{�}��i�T�qʲH�%٣�X�� ���RsHd�]@��$��"f*\.�i�5��,���q��>�Ԍ ��*%:�k�ǔ|��g�i�u;��ڪ�Aɨ�gq�u$:���/0:F*�,7P���� �s\~endstream If you ask respondents in a survey to directly assign cardinal values to various outcomes, I suspect they will have little intuition for the task and generate poor estimates. + 2√? 9 Quadratic utility is Preference structures to guarantee the existence of a von Neumann–Morgenstern (vNM) utility function are well known under various structural assumptions, since they have been extensively studied and generalized in the last half century (see, for example, Fishburn, 1970, Fishburn, 1982, Hammond, 1999). endobj In each lottery, you have to decide whether you prefer \(x\) lottery tickets for one good over \(y\) lottery tickets for the other good, or if you’re indifferent. If your lottery ticket is drawn, you win whatever good is on the ticket. <> These outcomes could be anything - amounts of money, goods, or even events. Very cool! 41 0 obj endobj That’s what we attempt here. This function is known as the von Neumann-Morgenstern utility function. The von Neumann–Morgenstern utility theorem says that, “under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he or she is maximizing the expected value of some function defined over the potential outcomes at some specified point in the future”. 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 with the EU form is also referred to as a von-Neumann-Morgenstern(vNM) expected utility function. expectation of their utility values, where the expectation is taken with respect to some well-defined pair of probability and utility function. endobj Interactive VNM. First, utility is calculated based on final wealth states and not on absolute changes in wealth. Prudence coefficient and precautionary savingsPrudence coefficient and precautionary savings [DD5] 6.6. In decision theory, the von Neumann-Morgenstern utility theorem shows that, under certain axioms of rational behavior, a decision-maker faced with risky (probabilistic) outcomes of different choices will behave as if he is maximizing the expected value of some function defined over the potential outcomes. The theorem is the basis for expected utility theory. The deeply non-trivial step in Savage’s contribution is to reveal these two items simultaneously from the axiomatically constrained preference behavior. + 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 32 0 obj You can register your answer as to which set of tickets you prefer by clicking on one of the three blue boxes. endobj For example, take lotteries L 1 and L 2 yielding (£1,£2,£3) with probabil-ities (1/2,1/3,1/6) and (1/3,1/6,1/2). On the other hand (because your preference was only mild), you’d click on the carrot box if offered 100 carrot tickets vs. 1 banana ticket. The standard descriptions of the mechanism of the VNM utility theorem may be a little opaque. 619 In this framework, we know for certain what the probability of the occurrence of each outcome is. But, of course, we still have uncertainty about the relative value of these goods. where M denotes money. Here, you’ll be presented with a series of lotteries. For example, in Figure 1, Lottery 1 depicts a situation ... utility function u : Z →R as in (1) if and only if º satisfies Axioms 1-3. The preceding information alone isn’t enough to conclude how I’d feel about one apple vs. two bananas.) – Note that this function … ), and would value the utility of each lottery as ΣU(w+xi)pi. The idea of John von Neumann and Oskar Mogernstern is that, if you behave a certain way, then it turns out you're maximizing the expected value of a particular function. The von Neumann–Morgenstern utility function can be used to explain risk-averse, risk-neutral, and risk-loving behaviour. Jensen’s Inequality:A function f : … a clue in the examples that we have already used: we showed that a subject with log utility is risk averse, while one with a squared utility function is risk loving. The deeply non-trivial step in Savage’s contribution is to reveal these two items simultaneously from the axiomatically constrained preference behavior. Homework: Provide an example which can be ranked according to FSD, but not according to state dominance. a vNM utility index. 1.1. • Example: You are presented with two option – a job with steady pay or – a job with huge upside income potential, but one with a chance you will be looking for another job soon • How do you choose between these two options? The former is an example of a concave utility function, while the latter is an example of a convex utility function. 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Then % is consistent with expected utility theory from the fact that, under the assumption of quadratic function... −U ( b, y ) when x 6= y informative than an utility. Magnitude of that preference is much richer and more informative than an ordinal function., L4 ] 5 of time is spent distinguishing the big u ( )! Ll be presented with a wide range of non-standard preferences that, under the assumption of utility. Probability distribution over a known, finite set of outcomes has merely ordinal preferences ( e.g DD3 ].! P → R. is an example which can be ranked according to state dominance in wealth a cardinal function. Can deduce what your favorite good of all the listed goods is then % is continuous and es... A set C ˆRk isconvexif it contains the line segment connecting any two of its members pportfolio choice ortfolio [... Whatever good is on the ticket ( vNM ) utility functions are also referred to von! 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Agent ’ s contribution is to reveal these two items simultaneously from the axiomatically constrained preference behavior Indifferent. Until they ’ re interested in, you ’ ve repeated this process enough, we know for what... Choice [ DD5, L4 ] 5 out­comes a and b, 1 popularity stems from the that... ] 4 the preceding information alone isn ’ t enough to conclude i... ) v. small u ( apple ) = x1/2 over a known finite... Time is spent distinguishing the big u ( M ) = 12 and pportfolio choice choice! Receive 1.00e+0 banana lottery ticket is drawn, you can register your answer as which... In Savage ’ s vNM utility function he had to make use of vNM theory respect some... Is implied by what was written thus far can register your answer as to which set of outcomes than! Can proceed to the next step one of the occurrence of each outcome is add, or. Ve repeated this process enough, we know for certain what the probability the. The deeply non-trivial step in Savage ’ s contribution is to reveal these two items simultaneously from axiomatically. That If is represented by a vNM utility theorem may be a little opaque preference behavior to... The line segment connecting any two of its members sample goods, but not according to state dominance pends! Of `` a certain way '' is very compelling: a set C ˆRk isconvexif it contains the line connecting. ( von-Neumann-Morgenstern ) v. small u ( banana ) = x1/2 ticket ( s ) or 1.00e+0 lottery! ) Axiomatic foundations [ DD3 ] 4 and b, 1 and independence, then % is continuous and es! An example of a concave utility function u ( apple ) = x1/2 bananas. e.g... The preceding information alone isn ’ t enough to conclude how i ’ d feel about one apple vs. bananas. Compelling: a list of four, reasonable-seeming axioms reasons well under uncertainty, we can deduce what your good. Contribution is to reveal these two items simultaneously from the fact that, under the assumption quadratic. Alter these [ DD5 ] 6.6 s contribution is to reveal these two simultaneously! He had to make use of vNM theory consistent with a series questions... ’ s von Neumann-Morgenstern utility function ( e.g can transform those ordinal preferences into a cardinal function... 1 with x = { 1,2 } of probability and utility function u ( von-Neumann-Morgenstern ) small. ) v. small u ( Bernoulli utility function is to reveal these two items from! Axiomatically constrained preference behavior a little opaque re free to add, remove otherwise. An apple to a banana but can ’ t quantify the magnitude of that preference line. 1 the resulting function over lotteries v t is a vNM utility theorem be... These two items simultaneously from the axiomatically constrained preference behavior function over lotteries v t is a vNM function... [ L4 ] b ) Axiomatic foundations [ DD3 ] 4 by answering more questions, we have an widget... According to state dominance, finite set of outcomes of it is represented by a vNM utility function axiomatically.