Multi-attribute utility

In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods either under conditions of certainty about the results of any potential choice, or under conditions of uncertainty.

In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods either under conditions of certainty about the results of any potential choice, or under conditions of uncertainty. u(x_1,...,x_n) = r(v(x_1,...,x_n)) The problem with this approach is that it is not easy to assess the function r. When assessing a single-attribute cardinal utility function using VNM, we ask questions such as: "What probability to win $2 is equivalent to $1?". So to assess the function r, we have to ask a question such as: "What probability to win 2 units of value is equivalent to 1 value?". The latter question is much harder to answer than the former, since it involves "value", which is an abstract quantity. A possible solution is to calculate n one-dimensional cardinal utility functions - one for each attribute. For example, suppose there are two attributes: apples ( x_1 ) and bananas ( x_2 ), both range between 0 and 99. Using VNM, we can calculate the following 1-dimensional utility functions: u(x_1,0) - a cardinal utility on apples when there are no bananas (the southern boundary of the domain); u(99,x_2) - a cardinal utility on bananas when apples are at their maximum (the eastern boundary of the domain). Using linear transformations, scale the functions such that they have the same value on (99,0). Then, for every bundle (x_1',x_2') , find an equivalent bundle (a bundle with the same v) which is either of the form (x_1,0) or of the form (99,x_2) , and set its utility to the same number. Often, certain independence properties between attributes can be used to make the construction of a utility function easier. Some such independence properties are described below. u(x_1,x_2)=u_1(x_1) + u_2(x_2) PROOF: \longrightarrow If the attributes are additive-independent, then the lotteries L and M , defined above, are equivalent. This means that their expected utility is the same, i.e.: E_L[u]=E_M[u] . Multiplying by 2 gives: u(x_1,x_2)+u(y_1,y_2)=u(x_1,y_2)+u(y_1,x_2) This is true for any selection of the x_i and y_i . Assume now that y_1 and y_2 are fixed. Arbitrarily set u(y_1,y_2)=0 . Write: u_1(x_1) = u(x_1,y_2) and u_2(x_2) = u(y_1,x_2) . The above equation becomes: u(x_1,x_2) = u_1(x_1)+u_2(x_2) \longleftarrow If the function u is additive, then by the rules of expectation, for every lottery L : E_L[u(x_1,x_2)] = E_L[u_1(x_1)] + E_L[u_2(x_2)] This expression depends only on the marginal probability distributions of L on the two attributes. This result generalizes to any number of attributes: if preferences over lotteries on attributes 1,...,n depend only on their marginal probability distributions, then the n-attribute utility function is additive: u(x_1,\dots,x_n) = \sum_{i=1}^n{k_i u_i(x_i)} where u and the u_i are normalized to the range [0,1] , and the k_i are normalization constants. Much of the work in additive utility theory has been done by Peter C. Fishburn. u(x_1,x_2)=c_1(x_2)+c_2(x_2)\cdot u(x_1,x_2^0) when x_2^0 is a constant value for attribute 2. Similarly, If attribute 2 is utility-independent of attribute 1: u(x_1,x_2)=d_1(x_1)+d_2(x_1)\cdot u(x_1^0,x_2) If the attributes are mutually utility independent, then the utility function u has the following multi-linear form: u(x_1,x_2)=u_1(x_1)+u_2(x_2)+k\cdot u_1(x_1)\cdot u_2(x_2) Where k is a constant which can be positive, negative or 0. When k=0 , the function u is additive and the attributes are additive-independent. When k\neq 0 , the utility function is multiplicative, since it can be written as: [k u(x_1,x_2)+1]=[k u_1(x_1)+1] \cdot [k u_2(x_2)+1] where each term is a linear transformation k\cdot+1 of a utility function. These results can be generalized to any number of attributes. Given attributes 1,...,n, if any subset of the attributes is utility-independent of its complement, then the n-attribute utility function is multi-linear and has one of the following forms: Additive, or - Multiplicative: 1 + k u(x_1,\dots,x_n) = \prod_{i=1}^n{1+k k_i u_i(x_i)} where: The u and the u_i are normalized to the range [0,1] ; The k_i are constants in [0,1] ; k is a constant which is either in (-1,0) or in (0,\infty) (note that the limit when k\to 0 is the additive form). AI implies mutual UI. The opposite is, in general, not true; it is true only if k=0 in the multi-linear formula for UI attributes. But if, in addition to mutual UI, there exist x_1,x_2,y_1,y_2 for which the two lotteries L and M , defined above, are equivalent - then k must be 0, and this means that the preference relation must be AI. UI implies PI. The opposite is, in general, not true. But if: there are at least 3 essential attributes, and: all pairs of attributes {1,i} are PI of their complement, and: attribute 1 is UI of its complement, then all attributes are mutually UI. Moreover, in that case there is a simple relation between the cardinal utility function u representing the preferences on lotteries, and the ordinal utility function v representing the preferences on sure bundles. The function u must have one of the following forms: Additive: u(x_1,...,x_n) = v(x_1,...,x_n) Multiplicative: u(x_1,...,x_n) = [exp(R\cdot v(x_1,...,x_n))-1]/[exp(R)-1] where R\neq 0 . PROOF: It is sufficient to prove that u has constant absolute risk aversion with respect to the value v. The PI assumption with n\geq 3 imply that the value function is additive, i.e.: v(x_1,\dots,x_n)=\sum_{i=1}^{n}{\lambda_i v_i(x_i)} Let x_1, z_1 be two different values for attribute 1. Let y_1 be the certainty-equivalent of the lottery . The UI assumption implies that, for every combination (w_2,\dots,w_n) of values of the other attributes, the following equivalence holds: (y_1,w)\sim The two previous statements imply that for every w, the following equivalence holds in the value space: \lambda_1 v_1(y_1) + \sum_{i=2}^n{\lambda_i v_i(w_i)} \sim This implies that, adding any quantity to both sides of a lottery (through the term \sum_{i=2}^n{\lambda_i v_i(w_i)} ), increases the certainty-equivalent of the lottery by the same quantity. The latter fact implies constant risk aversion.

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