An Information Theory For Preferences

Transcription

1 An Information Theor For Preferences Ali E. Abbas Department of Management Science and Engineering, Stanford Universit, Stanford, Ca, Abstract. Recent literature in the last Maimum Entrop workshop introduced an analog between cumulative probabilit distributions and normalized utilit functions []. Based on this analog, a utilit densit function can de defined as the derivative of a normalized utilit function. A utilit densit function is non-negative and integrates to unit. These two properties of a utilit densit function form the basis of a correspondence between utilit and probabilit, which allows the application of man tools from one domain to the other. For eample, La Place s principle of insufficient reason translates to a principle of insufficient preference. The notion of uninformative priors translates to uninformative utilit functions about a decision maker s preferences. A natural application of this analog is a maimum entrop principle to assign maimum entrop utilit values. Maimum entrop utilit interprets man of the common utilit functions based on the preference information needed for their assignment, and helps assign utilit values based on partial preference information. This paper reviews maimum entrop utilit, provides aiomatic justification for its use, and introduces further results that stem from the dualit between probabilit and utilit, such as joint utilit densit functions, utilit inference, and the notion of mutual preference. INTRODUCTION In man decision situations, we are faced with multiple and conflicting attributes. When the decision situation is deterministic, each decision alternative is described b a single prospect (consequence). The problem of choosing the best decision alternative is that of ordering the prospects present or, alternativel, assigning a value function over the attributes of each prospect. The optimal decision alternative is the one that has the largest value as determined b the value function or the highest order in the ranked list. A normative justification for the order requirement is that a decision maker who cannot order the prospects is vulnerable to being a mone pump (we would choose prospect A over prospect B, but also prospect B over prospect A, and be willing to pa mone to move from one prospect to the other) When uncertaint is present, a decision alternative is characterized b several prospects and a probabilit distribution representing the probabilit of their occurrence. In this case, the rank order of the prospects alone is insufficient to determine the optimal decision alternative, and the Von Neumann and Morgenstern utilit values need to be elicited. [2]. To elicit these utilit values, a decision maker is first asked to order the prospects of the decision situation from best to worst. Once the order is provided, the net step is to assign a utilit value for each prospect. For an three ordered prospects, P P P 3, the decision maker assigns a probabilit, π, for 2 which she is indifferent to receiving for sure and a deal where she would receive P P2

2 with probabilit π and P with probabilit ( π ). If the utilit values of and P 3 P 3 are one and zero respectivel, then the utilit value of P 2, also called the preference probabilit of P 2, will be equal to π () + ( π)(0) = π. The higher the prospect is in the ranked list, the larger is the utilit value assigned to it. The optimal decision alternative is now the one, which has the highest epected utilit for its prospects. The normalization of the utilit values discussed above leads to an analog between utilit functions and cumulative probabilit distributions. Our goal in this paper is to review the analog between utilit and probabilit, and to draw on the applications of this analog to the maimum entrop principle and to man other concepts of information theor. The main requirement for this analog is that the decision maker can provide the complete preference order for the prospects of the decision situation she is facing. UTILITY ASSIGNMENT FOR ORDERED PROSPECTS We start this section with the definition of a utilit vector for a set of K ordered prospects of a decision situation. A utilit vector contains the utilit values of the prospects starting from lowest to highest. With no significant loss of generalit, we will assign a utilit value of zero to the least preferred prospect, u, and a value of one to the most preferred prospect, uk (we ignore the case of absolute indifference between the K prospects). The utilit vector has K elements defined as U ( u0, u, u2,..., uk 2, uk ) = (0, u, u2,..., uk 2,). () An utilit vector of dimension K can be represented as a point in a K-2 dimensional space in the region defined b 0 u u2... uk 3 uk 2. We will call this region the utilit volume. An eample of a two-dimensional utilit volume is shown in figure (). u 2 Utilit Volume 0 u u 2 0 Utilit Vector Representation FIGURE. A 2-dimensional utilit volume. u

3 The second definition is the utilit-increment vector, U, whose elements are equal to the difference between the consecutive elements in the utilit vector. The utilitincrement vector has ( K ) elements defined as U u u2 u uk 2 u u2 u3 uk ( 0,,..., ) = (,,,..., ). (2) Note that all elements of U are greater than or equal to zero and sum to one. Therefore, an utilit-increment vector can be represented as a point in a (K-) K dimensional simple : i =, i 0 i=. These two properties of the utilit increment vector over the simple form the basis of the analog between probabilit and utilit and will be used further in this paper. We will refer to this simple as the utilit simple. For eample, a 3-dimensional utilit simple is shown in figure (2). u 3 u u 2 FIGURE 2. A 3-dimensional utilit simple. Note that all points in the utilit volume or the utilit simple satisf the decision maker s preference ordering of the prospects but assign different utilit values to them. In other words, knowledge of the preference order alone tells us nothing about the location of the utilit vector over the utilit volume or the utilit increment vector over the utilit simple. If all we know about the decision maker s preferences is the ordering of the prospects, it is reasonable to assume that the location of the utilit increment vector is uniforml distributed over the utilit simple. This is also the maimum entrop distribution over the simple. The uniform distribution over the utilit simple implies that the location of the utilit increment vector is determined b a Dirichlet distribution whose K- parameters are equal to one. This is in turn implies the following three propositions described in []. Proposition : Marginal Distributions of the Utilit Vector The maimum entrop marginal distributions for the utilit values of a set of K ordered prospects are the famil of Beta distributions, Beta( j, K j ), j =,... K 2.

4 Given the marginal distribution for each element in the utilit vector, it is known that the logical assignment of each utilit increment value is the mean of its marginal j distribution [3]. The mean of a Beta( j, K j ) distribution is equal to, and is K the utilit value ou would assign to each prospect, j. The utilit values for prospects j=0 and j=k- are deterministic with values 0 and respectivel. Proposition 2: Marginal Distributions of the Utilit Increment Vector The maimum entrop marginal distributions for the elements of the utilit increment vector have identical maimum entrop marginal distributions, Beta (, K - 2). Given the marginal distribution for each element in the utilit increment vector, it is known that the logical assignment of each utilit increment value is the mean of its marginal distribution [3]. The mean of a Beta (, K - 2) distribution is K. We now have a maimum entrop assignment for the location of the utilit increment vector and are read for proposition 3 below. Proposition 3: Utilit Increment Vector Assignment The maimum entrop assignment for increments in the utilit values of a set of K ordered prospects ields identical increments in utilit values equal to K. For eample, if we have a decision situation with five ordered prospects, A B C D E, the maimum entrop assignment of the utilit increment vector is U = (0.25, 0.25, 0.25, 0.25) and the maimum entrop assignment of the utilit vector is U = (0,0.25, 0.5, 0.75,). Note that using the definitions of utilit volume and utilit simple, as well as the maimum entrop uniform joint distribution, we have made an unbiased assignment of utilit values for five prospects when onl the preference order is known. Note also that this unbiased assignment gave equal increments in utilit values. Now we are read for the first application of our utilit-probabilit analog: to assign unbiased utilit values when onl the order of the prospects is available. We mean b unbiased utilit values, those that do not lead to arbitrar assumptions of preference information that is not available. THE PRINCIPLE OF INSUFFICIENT PREFERENCE In the previous section we illustrated how to assign utilit values for K prospects of a decision situation when onl the preference order is available. Now we show how to

5 work through the same problem using our utilit-probabilit analog. The analogous problem in probabilit land is to assign a probabilit to the outcomes of a discrete random variable when no further information is available. This problem dates back to Laplace s principle of insufficient reason. Laplace suggested that we assign equal probabilities to all outcomes unless there is information that suggests otherwise. The utilit-probabilit analog then suggests a principle of insufficient preference, which can be stated as follows: when onl the preference order of the prospects is known, we assign equal increments in utilit values unless there is preference information that suggests otherwise. Note that the principle of insufficient preference provides the same assignment of utilit values as suggested b proposition 3 above. The principle of insufficient preference assigns utilit values for discrete ordered prospects. In man cases, however, we would like to assign utilit values over a continuous domain. In the net section we will etend our analsis to the continuous case and present an additional element to our utilit probabilit analog: a utilit densit function. UTILITY FUNCTIONS AND UTILITY DENSITY FUNCTIONS In the continuous case the utilit vector has an infinite number of elements tracing a utilit curve over the domain of monetar prospects. The largest value is still one and the lowest value is zero corresponding to the utilit values of the best and least preferred prospects respectivel. A utilit curve thus has the same mathematical properties as a cumulative distribution function. The utilit-increment vector is now the derivative of the utilit curve and represents a utilit densit function, u ( ), which is analogous to a probabilit densit function in that it is non-negative and integrates to unit. U( ) u ( ) (3) Our problem of assigning utilit values for continuous prospects of a decision situation, when partial preference information is available, is now equivalent to that of assigning an unbiased utilit densit function for the ordered prospects. The latter problem is analogous to the problem of assigning prior probabilities to a continuous random variable when onl partial information is available. In this paper we will generalize the utilit assignment problem and present a maimum entrop approach for the assignment of utilit values when onl partial preference information is available. First let us discuss the entrop of a utilit function and its interpretation.

6 THE ENTROPY OF A UTILITY FUNCTION Having defined the notion of a utilit densit function, it is natural to etend our analsis to the entrop measure applied to a utilit function, and discuss its interpretations. The differential entrop term applied to a utilit densit function is h ( ) = u ( )ln( u ( )) d (4) Let us take, as an eample, an eponential utilit densit function over a bounded domain [a, b]. We have γ γ e u ( ) =, a b γa γb e e (5) where γ is the decision maker s risk aversion coefficient. As γ 0, the decision maker becomes risk neutral (he values uncertain deals at their epected value). Using L hopital s formula, we can also show that γ γ e γa γb e e b a as γ 0. (6) The utilit densit function approaches a uniform densit and the entrop of the utilit function thus achieves its maimum value of ln( b a). Note that the uniform utilit densit function matches the principle of insufficient preference discussed earlier as it provides equal increments in utilit values for equall spaced prospects. The utilit function that corresponds to this uniform utilit densit function can be obtained b integration and is linear. The linear utilit function is thus the maimum entrop utilit function over a continuous bounded domain. As γ +, γ γ e δ ( a) γa γb e e (7) The utilit densit function converges to an impulse utilit densit function and the entrop approaches a minimum value of negative infinit showing less spread in the utilit densit. The impulse utilit densit function is the minimum entrop utilit function and integrates to a step utilit function. As γ, γ γ e δ ( b) γa γb e e (8)

7 The utilit densit function converges again to an impulse utilit densit function, and the entrop approaches a minimum value of negative infinit. This is the other etreme where the step in the utilit function occurs at the upper bound. The step utilit function is thus the minimum entrop utilit function and corresponds, in this eample, to cases of etreme risk aversion or risk seeking behavior. Figure 3 shows an eample of two eponential utilit densit functions for different values of γ and different entrop values. E ponential Utilit Densit Function Eponential Utilit Densit Function Variable Variable FIGURE 3. (a) Utilit densit function for small values of γ is approimatel uniform showing large spread. (b) Utilit densit function for large values of γ converges to an impulse utilit densit showing less spread and integrates to a step utilit function. A step utilit function appears often in behavioral decision making literature and is known as the aspiration utilit function. Simon [4] suggested that individuals could simplif decision problems b having binar goals: satisfactor if the outcome is above the aspiration level or unsatisfactor if it is below it. Aspiration Utilit Function Unsati sfactor Satisfactor Aspiration Level Million Dollars ($) FIGURE 4. Aspiration (Minimum Entrop) Utilit Function

8 The interpretation of entrop for utilit functions is thus a measure of spread about a decision maker s aspiration level. Aspiration utilit functions have the least entrop in their utilit densit functions showing binar preferences, while uniform utilit densit functions have the highest entrop showing smoother utilit functions that var graduall over a bounded domain. MAXIMUM ENTROPY UTILITY FUNCTIONS Now we will make use of the concept of the entrop of a utilit function and present a method to assign unbiased utilit values based on the partial preference information we know about the decision maker. B partial preference information, we mean an information that includes the complete preference order of the prospects but does not include knowledge of the complete utilit values. In the probabilit domain, Edwin Janes [5] proposed the maimum entrop principle suggesting the use of a probabilit distribution, which has maimum entrop subject to whatever information is known. In an analogous manner, we defined the maimum entrop utilit principle, [], that suggests the use of the utilit function (or utilit vector) whose utilit densit function (or utilit increment vector) has maimum entrop subject to whatever preferences are known. This result is an etension of the principle of insufficient preference described earlier. Maimum entrop utilit provides an interpretation for man of the utilit functions used in practice based on the preference information constraints needed for their assignment. For eample when onl bounds on the domain of continuous outcomes are available, the maimum entrop utilit densit function is uniform and integrates to a risk neutral utilit function as we have discussed earlier. When the first moment of the utilit function is known over a positive domain, the maimum entrop utilit function is eponential with the first moment equal to the risk tolerance. This provides us with a new interpretation for the first moment of an eponential utilit function over a positive domain. When the first and second moments are known, the maimum entrop utilit function over a continuous domain leads to the famous S- shaped prospect theor utilit function [6]. Maimum entrop utilit presents a method to assign unbiased utilit values when partial preference information is available. For eample, the maimum entrop utilit function given knowledge of some utilit values leads to a piecewise linear utilit function that passes through the given utilit points. UNINFORMATIVE UTILITY FUNCTIONS Much of the research on uninformative and entropic priors for probabilit functions can be translated through our utilit-probabilit analog into similar research on uninformative utilit functions. For eample, when Jeffres [7] uninformative prior, u ( ) =, > 0is used as an uninformative utilit densit function, it leads to the

9 famous logarithmic utilit function that is used ver often to represent the decision maker s preferences in the literature. ATTRIBUTE DOMINANCE UTILITY FUNCTIONS In this section we present the etension of the previous analsis to multiattribute utilit functions with ordered attributes. We assume that ( min, ) min is the least preferred prospect and (, ) is most preferred. Furthermore we will assume that ma ma the values of (, ) are arranged such that for an 0,, 0, we have > (, ) (, ) and > (, ) (, ) (9) With no loss of generalit, we will use a normalized multiattribute utilit function, U (, ), over the attributes in all of our analsis. i.e. Based on this formulation, we have 0 U (, ) (0) U (, ) = 0, U (, ) = () min min ma ma Now we will focus on a class of multiattribute utilit functions where a prospect (, ) is a least preferred prospect if either or is at its minimum value. This requirement places the following constraints on the utilit function: U (, ) = U (, ) = U (, ) = 0 [, ], [, ] min min min min min ma min ma (2) We will call the multiattribute utilit functions that satisf condition (2) attribute dominance utilit functions since an attribute set at a minimum dominates the remaining attributes and sets the multiattribute utilit function to a minimum. For attribute dominance utilit functions we will define the marginal utilit function over a single attribute,, as the numerical value of the multiattribute utilit function when all other attributes are set at their maimum values. I.e. for two attributes we have U ( ) U (, ) ma (3) Now we define a conditional utilit function, U ( ), as U U (, ) U (, ) ( ) = (4) U ( ) U (, ) ma

10 A conditional utilit function, U ( ), is the utilit function for attribute when we are guaranteed a fied amount of attribute. The conditional utilit function U ( ) is the path traced on the multiattribute utilit function for different values of and a fied value of. Equation (4) shows that for attribute dominance utilit functions, the minimum value of the conditional utilit function, U (, is min ) equal to zero and the maimum value, U ( ma ), is equal to one. The denominator in equation (4) thus serves as a normalizing term so a conditional utilit function can be assessed directl from the decision maker on a scale of 0 to. Re-arranging equation (4) gives: U (, ) = U ( ) U ( ) (5) In a similar manner, we can also define U ( ) as U U (, ) ( ) U ( ) (6) Combining equations (4) and (6), we have U ( ) U ( ) U ( ) = (7) U ( ) Equation (7) provides a method to update our utilit values over one attribute when we are guaranteed a fied amount of another. We will call equation (7) Baes rule for utilit inference. Now define a joint utilit densit function, [, ] and [, ] as: min ma min ma u (, ), over two attributes, u 2 U(, ) (, ) (8) where U (, ) is the multiattribute utilit function for the two attributes and. From the properties of derivatives, the integral of a joint utilit densit function that is defined b equation (8) over a region a b, c d is d b c a u (, ) dd = U ( b, d) U ( a, d) U ( b, c) + U ( a, c) (9) If we set a = min and c= min, and if attribute dominance utilit eists, then equation (9) reduces to

11 U (, ) u (, ) dd = min min (20) Equation (20) illustrates how attribute dominance utilit functions can be obtained b integrating a joint utilit densit function from the least preferred prospect to a prospect (, ). This integral provides an analog with joint cumulative distribution functions obtained b integrating a joint probabilit densit function over the domain of the variables. We note that an general multiattribute utilit function can be decomposed into a linear combination of attribute dominance utilit functions. For eample, rearranging equation (9) and using an general prospect b=, d=, a = min and c= min gives U (, ) U (, ) U (, ) U (, ) u (, ) dd = + + (2) min min min min min min The terms U (, ) and U (, are scaled attribute dominance utilit min functions of one dimension since the utilit function is zero if that single attribute is set to a minimum, and the scaling constants are equal to U (, and U (, min ma ) respectivel. Now we discuss the integral term min ) I(, ) u ( dd, ) = min min ma min ). We note that, b properties of a definite integral, I( min, ) = I(, min ) = 0. In other words, this integral has a value of zero if either or is at the minimum value. Furthermore, this integral has a maimum value equal to k = I( ma, ma ). Let us normalize this integral term and define C( as, ) C (, ) min min ma ma min min u (, ) dd I(, ) = I( ma, u (, ) dd From equation (22), we note that C( has, ) the same mathematical properties as an attribute dominance utilit function. In other words we can write ma min min ) (22) U (, ) = kc (, ) + kc (, ) + k C (, ) (23) where are scaling constants and C(, ), C (, ), and C (, ) are k, k, and k min min normalized attribute dominance utilit functions.

12 Similarl, for three dimensions we can define and therefore, u z 3 U(,, z) (,, z), (24) z U (,, z) = U (,, z ) + U (,, z ) + U (,, z) z z min min z min min z min min U (,, z) U (,, z) U (,, z ) z z min z min z min + zmin min min u z (,, z) dddz (25) Once again we note that the terms Uz (, min, zmin ), Uz ( min,, zmin ), and U (,, z) are scaled attribute dominance utilit functions of one dimension, z min min C(,, z ) = z zmin min min u z (,, z) dddz is a scaled attribute dominance utilit function of three dimensions, and that Uz ( min,, z), Uz (, min, z), and Uz (,, zmin ) can be reduced to one dimensional and two dimensional attribute dominance utilit functions using the arguments for the two dimensional case presented above. In other words, we can write equation (25) as U ( z,, ) = kc (,, z ) + kc (, z, ) + kc (,, z) z min min min min z min min k C(,, z) k C(,, z) k C(,, z ) + k z min z min min z C(,, z) (26) Equation (26) epresses an general three-dimensional multiattribute utilit function in terms of a linear combination of attribute dominance utilit functions. B induction, we can show that the same formulation applies to higher dimensions. We note that a special case of decomposing utilit functions into attribute dominance utilit functions arises when the attributes have mutual utilit independence and leads to the multilinear utilit function. In order to see this, we recall the following theorem from about mutual utilit independence [8]. If two attributes, and, have mutual utilit independence, their multiattribute utilit function has the multilinear form U (, ) = k U ( ) + k U ( ) + k U ( ) U ( ) (27) where k, k, and k = k k are constants, and U ( ) and ( ) are the utilit U functions of attributes and respectivel. If we substitute for C (, ) = UU ( ) ( ) into (23) we note that (27), is indeed a special case.

13 MAXIMUM ENTROPY ATTRIBUTE DOMINANCE UTILITY FUNCTIONS B analog with joint probabilit densit functions, we can now we etend the maimum entrop utilit analsis using the joint utilit densit function that we have developed. For eample, if the decision maker can provide onl the marginal utilit function for each attribute present, then the maimum entrop formulation maimizes the entrop of the joint utilit densit function subject to the given preference information constraints. If for eample, we have two attributes and that are defined on the unit square, and we know the marginal utilit functions, the formulation for the problem is ma u(, )ln( u(, )) dd st 0 udd (, ) = U( ), udd (, ) = U( ), udd (, ) =. (28) The solution to this problem provides a joint utilit densit function that is equal to the product of the marginal utilit densit functions. u (, ) = u( u ) ( ), 0,0. (29) The joint utilit densit function integrates, using equation (20), to U(, ) = U ( ) U ( ), 0,0. (30) The product form of the utilit densit functions presented in equation (29) is the condition of utilit independence that is analogous to probabilit independence and applies to an general class of multiattribute utilit functions that have mutual utilit independence. We can show this b substituting equation (27) into equation (8) to get 2 U(, ) u (, ) = ku( u ) ( ) (3) If the marginal utilit functions are known, the maimum entrop attribute dominance utilit function assumes utilit independence between the attributes. If

14 more information is available about the utilit function, such as moments, risk aversion coefficients, or utilit dependence parameters, it can also be incorporated into the maimum entrop formulation and an attribute dominance utilit function can be obtained b integration. We recall that the construction of an general multiattribute utilit function involves the construction of several attribute dominance utilit functions of different dimensions. RELATIVE ENTROPY The KL-distance (often known as the relative entrop or cross-entrop) is a measure of divergence between two probabilit distributions. In a similar manner, the KL-distance is a measure of the divergence between two utilit densit functions. The epression for the KL-distance between a utilit densit functions, u() and another utilit densit function q() is given as u ( ) KL distance = u( )ln( ) (32) q ( ) The maimum entrop utilit principle can now be generalized b analog to a minimum cross-entrop utilit principle b minimizing the KL-distance relative to a known target utilit densit function. This helps incorporate information about the shape of the utilit function or its relation to a famil of utilit functions into the maimum entrop formulation. UTILITY DEPENDENCE: AN INFORMATION THEORETIC POINT OF VIEW In the previous sections we discussed the interpretation of the entrop and relative entrop as the appl to utilit functions. In this section we discuss the interpretation of appling several measures of dependence that were developed in information theor into measurements of utilit dependence. We provide some insights on their dual interpretations and an intuitive meaning for utilit dependence that results from the analog with joint probabilit functions. We start with the notion of conditional entrop. Conditional Entrop Having discussed a joint utilit densit function, and a conditional utilit function, we can now define the conditional entrop of attribute Y given a guaranteed amount of attribute X as hy ( X) = u(, )ln( u( )), (33)

15 where u ( ) = U ( ) is the conditional utilit densit function of for a given value of. It is known from information theor that the conditional entrop is less than or equal to the marginal entrop, with equalit if and onl if the two variables are independent. I.e. hy ( X) hy ( ). (34) The interpretation for this result when applied to utilit theor is that our utilit functions for attributes have less entrop (become closer to aspiration utilit functions) when conditioned on another attribute. However, when the two attributes are utilit independent, the conditional entrop remains the same as the marginal entrop, since the utilit functions for each attribute are not updated. Mutual Preference The notion of mutual information translates using the utilit-probabilit analog into mutual preference. In probabilit land, the mutual information between two variables is the KL-distance between their joint distribution and the product of their marginal distributions. The interpretation for the mutual information is the amount of information induced about a random variable, X, when we know the outcome of another random variable, Y. Let us now consider the two etreme cases of probabilistic dependence. The first case is when the two variables are independent. In this case we have a marginal probabilit distribution for each variable. Knowing the outcome of one variable, X, tells us nothing about the other variable Y, and so this knowledge does not update its probabilit distribution. Therefore the entrop of Y remains unchanged whether or not we know X. The second case is when the two variables are deterministicall related through a functional epression. For eample, Profit = 0.5*Revenue. In this case, knowledge of the Revenue tells us the eact value of profit. If we know revenue, the distribution of profit is updated to a step distribution function (or an impulse probabilit densit function) at that particular value of profit. The entrop of the updated distribution of X is reduced to negative infinit (it is now deterministic) once we know the outcome of Y. The mutual information between the two variables in this case is infinite as knowledge of one variable provides infinite reduction in the entrop of the other. Now we consider the analogous interpretation of mutual information when applied to utilit, which we call mutual preference. If we have a prospect with two attributes, we have a utilit function for each attribute. Now we consider two etreme cases. The first is when the attributes have utilit independence. If we are now guaranteed a fied amount of one attribute, it does not change our utilit function over the other (b definition of utilit independence), and therefore this guarantee does not change its entrop. The mutual preference between the attributes is therefore equal to zero. On the other hand, the attributes ma be deterministicall related. Once again, we

16 consider the eample Profit = 0.5*Revenue. In this case, if we are guaranteed a fied amount of Revenue, sa R 0, then we automaticall set a reference point for our satisfaction with profit, 0.5* R 0. An value of profit that is below 0.5* R 0 will be unsatisfactor for us while an value of profit that is above 0.5* R 0 will be satisfactor. In other words, we now have a step (aspiration) utilit function (or an impulse utilit densit function) for profit. A guarantee of the Revenue attribute has sharpened our utilit function for profit and it is now a target or aspiration level. The mutual preference between the two attributes is thus infinite since guarantee of one attribute reduces the entrop in the utilit function of the other to its minimum value of negative infinit. AXIOMATIC JUSTIFICATION OF THE MAXIMUM ENTROPY UTILITY PRINCIPLE We have discussed the maimum entrop utilit principle and the application of several measures of information theor to utilit theor. In this section we further draw on this analog to provide an aiomatic derivation for the maimum entrop utilit principle. Shore and Johnson [9] showed that Janes maimum entrop principle for probabilit inference satisfies a set of reasonable aioms that stem from one fundamental principle: if a problem can be solved in more than one wa, the results should be consistent. The also showed that maimizing an function but the entrop will lead to inconsistencies in these aioms unless that function and the entrop have identical maima (an monotonic function of the entrop will work). The analog between probabilit and utilit densit functions that we have developed allows the application of their analsis directl to the maimum entrop utilit principle. Now we will mention each of the aioms and discuss how the relate to utilit assignment. Aiom : Uniqueness The utilit values that we assign b maimum entrop utilit should be unique. The justification for this is that if we do not have unique utilit values, we are vulnerable to being a mone pump (as discussed earlier). Another justification for the uniqueness of the maimum entrop utilit assignment is that if we solve the utilit assignment problem twice in eactl the same wa, we epect the same utilit values to be assigned in both times. Aiom 2: Invariance We should epect the same utilit values from the maimum entrop utilit principle when we solve the problem in two different coordinate sstems. For eample, if a utilit densit function is assigned in Cartesian co-ordinates and

17 integrate to get a multiattribute utilit function, we should get the same results if we solve the problem in polar coordinates and integrate over the domain of the polar coordinates to get a utilit function. Aiom 3: Sstem Independence Consider two independent utilit functions or two attributes that have utilit independence. If we receive information about the two attributes, then we would like the joint utilit densit function that is assigned to be equal to the product of the individual utilit densit functions (impling utilit independence as discussed above). Aiom 4: Subset Independence This aiom concerns itself with situations where we receive preference information about the conditional utilit functions in attribute dominance utilit. One wa of accounting for this new information is to update the conditional utilit functions U(B A) individuall. Another wa is to marginalize the attribute dominance utilit function to get U(B) and then appl the new preference information to U(B). The results should be consistent in both cases. As mentioned above, if these aioms are required b a utilit assignment mechanism, then maimizing an function but the entrop will lead to inconsistencies in these aioms unless that function and the entrop have identical maima. In addition to satisfing the previous aioms, the maimum entrop utilit principle satisfies two essential desiderata. The first desideratum, as mentioned earlier, is that for different people with the same order of preferences for the prospects and where we have the same preference information we should assign the same utilit values. The second desideratum is -utilit and probabilit independence- that results from the foundations of normative utilit theor. The utilit value of a prospect should not depend on the probabilit of getting that prospect due to the normative separation of beliefs from preferences [0]. CONCLUSIONS In this paper we have provided an analog between probabilit and utilit and built on the notion of a utilit densit function to a joint utilit densit function for multiple attributes. We applied concepts of information theor to utilit theor and gained insights on the notions of utilit dependence and aspiration utilit functions. The analog between probabilit and utilit provides several directions for future research. For eample, analogous interpretations for Co s aioms provide the basis for an aiomatic derivation of utilit inference. The algebra of probable inference translates to the algebra of preferences for logical operations on the attributes. The idea of Influence diagrams as graphical representations of joint probabilit distributions [] etends to the notion of utilit diagrams, which are graphical representations of multiattribute utilit functions and epress the dependence relations between the attributes. The method of copulas for constructing joint cumulative

18 distribution functions translates into a method for constructing attribute dominance utilit where the dependence parameters can be determined b the trade-off coefficients between the attributes. Assessing utilit moments and utilit crosscorrelation coefficients can also be used to construct a multiattribute utilit function. Interchanging utilit functions with probabilit functions provides a wealth of new results and insights. For eample, the notion of utilit dominance is the analog of stochastic dominance [2]. ACKNOWLEDGEMENTS I would like to thank Ronald Howard and Mron Tribus for introducing me to the Maimum Entrop communit and to the work of Edwin Janes and Richard Co. REFERENCES. A.E. Abbas. An Entrop Approach for Utilit Assignment in Decision Analsis. In Baesian Inference and Maimum Entrop Methods in Science and Engineering, Moscow, ID. AIP Conference Proceedings, C. Williams (ed.) vol 659. (2002). 2. J. Von Neumann, and O. Morgenstern. Theor of games and economic behavior. Princeton, N.J. Princeton Universit. 2 nd edition. (947). 3. R.A. Howard. Decision Analsis: Perspectives on Inference, Decision, and Eperimentation. Proceedings of the IEEE. Vol 58. No.5 pp (970). 4. Simon, H.A. A behavioral model of rational choice. The Quarterl Journal of Economics 69, (955). 5. E. Janes. Information theor and statistical mechanics. Phsical Review, vol 06, pp , (957). 6. D. Kahneman and A. Tversk. Prospect Theor: An analsis of decision under risk. Econometrica. 47, (979). 7. H. Jeffres. Theor of Probabilit. Oford Universit Press, London, (96). 8. R. Keene and H. Raiffa. Decisions with Multiple Objectives: Preferences and Value Tradeoffs. John Wile and Sons, Inc. (976). 9. J. Shore and R. Johnson. Aiomatic Derivation of the Principle of Maimum Entrop and the Principle of Minimum Cross-Entrop. IEEE Transactions on Information Theor. Vol 26, No., pp (980). 0. P. Samuelson. Probabilit, Utilit, and the Independence Aiom. Econometrica.20, (952).. R.A. Howard and J.E. Matheson. Influence Diagrams, in The principles and Applications of Decision Analsis, Vol. II, R. A. Howard and J. E. Matheson (eds.). Strategic Decisions Group, Menlo Park, California, USA. (984). 2. A. E. Abbas. Entrop Methods in Decision Analsis. PhD dissertation. Stanford Universit. (2003). 3. A. E. Abbas. Maimum Entrop Utilit. Submitted to Operations Research. (2002).