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Multi-attribute utility

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr">In <a href="Decision_theory" title="Decision theory">decision theory</a>, 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.

<div class="mw-heading mw-heading2"><h2 id="Preliminaries">Preliminaries</h2></div>

A person has to decide between two or more options. The decision is based on the attributes of the options.

The simplest case is when there is only one attribute, e.g.: money. It is usually assumed that all people prefer more money to less money; hence, the problem in this case is trivial: select the option that gives you more money.

In reality, there are two or more attributes. For example, a person has to select between two employment options: option A gives him $12K per month and 20 days of vacation, while option B gives him $15K per month and only 10 days of vacation. The person has to decide between (12K,20) and (15K,10). Different people may have different preferences. Under certain conditions, a person's preferences can be represented by a numeric function. The article <a href="Ordinal_utility" title="Ordinal utility">ordinal utility</a> describes some properties of such functions and some ways by which they can be calculated.

Another consideration that might complicate the decision problem is uncertainty. Although there are at least four sources of uncertainty - the attribute outcomes, and a decisionmaker's fuzziness about: a) the specific shapes of the individual attribute utility functions, b) the aggregating constants' values, and c) whether the attribute utility functions are additive, these terms being addressed presently - uncertainty henceforth means only randomness in attribute levels. This uncertainty complication exists even when there is a single attribute, e.g.: money. For example, option A might be a lottery with 50% chance to win $2, while option B is to win $1 for sure. The person has to decide between the lottery <2:0.5> and the lottery <1:1>. Again, different people may have different preferences. Again, under certain conditions the preferences can be represented by a numeric function. Such functions are called <a href="Cardinal_utility" title="Cardinal utility">cardinal utility</a> functions. The article <a href="Von_Neumann%E2%80%93Morgenstern_utility_theorem" title="Von Neumann–Morgenstern utility theorem">Von Neumann–Morgenstern utility theorem</a> describes some ways by which they can be calculated.

The most general situation is that there are both multiple attributes and uncertainty. For example, option A may be a lottery with a 50% chance to win two apples and two bananas, while option B is to win two bananas for sure. The decision is between <(2,2):(0.5,0.5)> and <(2,0):(1,0)>. The preferences here can be represented by <a href="Cardinal_utility" title="Cardinal utility">cardinal utility</a> functions which take several variables (the attributes).<a href="#cite_note-KR-1">[1]</a>: 26–27 Such functions are the focus of the current article.

The goal is to calculate a utility function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},...,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},...,x_{n})}</annotation>

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</math><img src="./d0f5e63fdf3e4d7733538784e5bc6f77107059e9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.241ex; height:2.843ex;" alt="{\displaystyle u(x_{1},...,x_{n})}" loading="lazy"> which represents the person's preferences on lotteries of bundles. I.e, lottery A is preferred over lottery B if and only if the expectation of the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> is higher under A than under B:

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<annotation encoding="application/x-tex">{\displaystyle E_{A}[u(x_{1},...,x_{n})]>E_{B}[u(x_{1},...,x_{n})]}</annotation>

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</math><img src="./5b97f3015385360fba97bcbcb126ac1ee0100930.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.542ex; height:2.843ex;" alt="{\displaystyle E_{A}[u(x_{1},...,x_{n})]>E_{B}[u(x_{1},...,x_{n})]}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Assessing_a_multi-attribute_cardinal_utility_function">Assessing a multi-attribute cardinal utility function</h2></div>

If the number of possible bundles is finite, u can be constructed directly as explained by <a href="Von_Neumann%E2%80%93Morgenstern_utility_theorem#construction" title="Von Neumann–Morgenstern utility theorem">von Neumann and Morgenstern</a> (VNM): order the bundles from least preferred to most preferred, assign utility 0 to the former and utility 1 to the latter, and assign to each bundle in between a utility equal to the probability of an equivalent lottery.<a href="#cite_note-KR-1">[1]</a>: 222–223

If the number of bundles is infinite, one option is to start by ignoring the randomness, and assess an <a href="Ordinal_utility" title="Ordinal utility">ordinal utility</a> function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v(x_{1},...,x_{n})}">

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<annotation encoding="application/x-tex">{\displaystyle v(x_{1},...,x_{n})}</annotation>

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</math><img src="./10a72ab2554d2281e0231e1f0a2b0c3421d83cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.039ex; height:2.843ex;" alt="{\displaystyle v(x_{1},...,x_{n})}" loading="lazy"> which represents the person's utility on sure bundles. I.e, a bundle x is preferred over a bundle y if and only if the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> is higher for x than for y:

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<annotation encoding="application/x-tex">{\displaystyle v(x_{1},...,x_{n})>v(y_{1},...,y_{n})}</annotation>

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</math><img src="./7713d771f577483f7cde415d346a05bb2a1a1280.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.795ex; height:2.843ex;" alt="{\displaystyle v(x_{1},...,x_{n})>v(y_{1},...,y_{n})}" loading="lazy"></dd></dl>

This function, in effect, converts the multi-attribute problem to a single-attribute problem: the attribute is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

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<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

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</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy">. Then, VNM can be used to construct the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy">.<a href="#cite_note-KR-1">[1]</a>: 219–220

Note that u must be a positive monotone transformation of v. This means that there is a monotonically increasing function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r:\mathbb {R} \to \mathbb {R} }">

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<annotation encoding="application/x-tex">{\displaystyle r:\mathbb {R} \to \mathbb {R} }</annotation>

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</math><img src="./e11c92a8ccbf303f955c2c2eae7dabc56cdee2b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.956ex; height:2.176ex;" alt="{\displaystyle r:\mathbb {R} \to \mathbb {R} }" loading="lazy">, such that:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},...,x_{n})=r(v(x_{1},...,x_{n}))}</annotation>

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</math><img src="./66ee66795b7790f4fab44348c50f8ea945a8d606.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.236ex; height:2.843ex;" alt="{\displaystyle u(x_{1},...,x_{n})=r(v(x_{1},...,x_{n}))}" loading="lazy"></dd></dl>

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 (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

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</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy">) and bananas (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

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</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy">), both range between 0 and 99. Using VNM, we can calculate the following 1-dimensional utility functions:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},0)}</annotation>

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</math><img src="./f00ab4dbec5432fd302dde4da34cca6658f8a24a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.719ex; height:2.843ex;" alt="{\displaystyle u(x_{1},0)}" loading="lazy"> - a cardinal utility on apples when there are no bananas (the southern boundary of the domain);</li>

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<annotation encoding="application/x-tex">{\displaystyle u(99,x_{2})}</annotation>

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</math><img src="./a5309b753e6d54f361e4372278b7bccff1360551.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.882ex; height:2.843ex;" alt="{\displaystyle u(99,x_{2})}" loading="lazy"> - a cardinal utility on bananas when apples are at their maximum (the eastern boundary of the domain).</li></ul>

Using linear transformations, scale the functions such that they have the same value on (99,0).

Then, for every bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1}',x_{2}')}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1}',x_{2}')}</annotation>

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</math><img src="./fde003cf55482b74c8150fb6cb735d9c1c7b867a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.611ex; height:3.009ex;" alt="{\displaystyle (x_{1}',x_{2}')}" loading="lazy">, find an equivalent bundle (a bundle with the same v) which is either of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},0)}">

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</math><img src="./4651c8e5721d3fac22aa26b690a08a039aa322e1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.39ex; height:2.843ex;" alt="{\displaystyle (x_{1},0)}" loading="lazy"> or of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (99,x_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (99,x_{2})}</annotation>

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</math><img src="./52297072a2948e072b1a6a3358129b1d89c78865.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.552ex; height:2.843ex;" alt="{\displaystyle (99,x_{2})}" loading="lazy">, and set its utility to the same number.<a href="#cite_note-KR-1">[1]</a>: 221–222

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.

<div class="mw-heading mw-heading2"><h2 id="Additive_independence">Additive independence</h2></div>

The strongest independence property is called additive independence. Two attributes, 1 and 2, are called additive independent, if the preference between two lotteries (defined as joint probability distributions on the two attributes) depends only on their <a href="Marginal_probability_distribution" class="mw-redirect" title="Marginal probability distribution">marginal probability distributions</a> (the marginal PD on attribute 1 and the marginal PD on attribute 2).

This means, for example, that the following two lotteries are equivalent:

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">: An equal-chance lottery between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},x_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle (x_{1},x_{2})}</annotation>

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</math><img src="./b9bb1427314b1719774232fe0d156a0edacf1ced.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.611ex; height:2.843ex;" alt="{\displaystyle (x_{1},x_{2})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (y_{1},y_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">: An equal-chance lottery between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x_{1},y_{2})}">

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In both these lotteries, the marginal PD on attribute 1 is 50% for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1}}">

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</math><img src="./a8788bf85d532fa88d1fb25eff6ae382a601c308.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{1}}" loading="lazy"> and 50% for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation>

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</math><img src="./eef4db76d658a98219aca14df06d9869d2b43c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}" loading="lazy">. Similarly, the marginal PD on attribute 2 is 50% for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

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</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy"> and 50% for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{2}}</annotation>

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</math><img src="./7377c7399e662562cd420fa5c7ce49cfba574998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{2}}" loading="lazy">. Hence, if an agent has additive-independent utilities, he must be indifferent between these two lotteries.<a href="#cite_note-KR-1">[1]</a>: 229–232

A fundamental result in utility theory is that, two attributes are additive-independent, if and only if their two-attribute utility function is additive and has the form:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=u_{1}(x_{1})+u_{2}(x_{2})}</annotation>

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PROOF:

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If the attributes are additive-independent, then the lotteries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

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</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, defined above, are equivalent. This means that their expected utility is the same, i.e.: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{L}[u]=E_{M}[u]}">

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<annotation encoding="application/x-tex">{\displaystyle E_{L}[u]=E_{M}[u]}</annotation>

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</math><img src="./861e78ba98d8ad811c8db81d12d411a74e480f1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.086ex; height:2.843ex;" alt="{\displaystyle E_{L}[u]=E_{M}[u]}" loading="lazy">.

Multiplying by 2 gives:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})+u(y_{1},y_{2})=u(x_{1},y_{2})+u(y_{1},x_{2})}</annotation>

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</math><img src="./b324f3db342134bf8828ef37a1b14afc5345b985.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.78ex; height:2.843ex;" alt="{\displaystyle u(x_{1},x_{2})+u(y_{1},y_{2})=u(x_{1},y_{2})+u(y_{1},x_{2})}" loading="lazy"></dd></dl>

This is true for any selection of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation>

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</math><img src="./e87000dd6142b81d041896a30fe58f0c3acb2158.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{i}}</annotation>

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</math><img src="./67d30d30b6c2dbe4d6f150d699de040937ecc95f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.939ex; height:2.009ex;" alt="{\displaystyle y_{i}}" loading="lazy">. Assume now that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation>

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</math><img src="./eef4db76d658a98219aca14df06d9869d2b43c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle y_{2}}</annotation>

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</math><img src="./7377c7399e662562cd420fa5c7ce49cfba574998.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{2}}" loading="lazy"> are fixed. Arbitrarily set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(y_{1},y_{2})=0}">

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<annotation encoding="application/x-tex">{\displaystyle u(y_{1},y_{2})=0}</annotation>

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</math><img src="./c869c6a51dfebfcba501d397f4894e071836c255.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.821ex; height:2.843ex;" alt="{\displaystyle u(y_{1},y_{2})=0}" loading="lazy">. Write: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{1}(x_{1})=u(x_{1},y_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle u_{1}(x_{1})=u(x_{1},y_{2})}</annotation>

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</math><img src="./cf7c395b726323eb38f31f9c7d65f1ace0ca5151.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.426ex; height:2.843ex;" alt="{\displaystyle u_{1}(x_{1})=u(x_{1},y_{2})}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{2}(x_{2})=u(y_{1},x_{2})}">

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<annotation encoding="application/x-tex">{\displaystyle u_{2}(x_{2})=u(y_{1},x_{2})}</annotation>

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</math><img src="./298cacd1f9c00797f553ceae68e4c9bbe95f74d5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.426ex; height:2.843ex;" alt="{\displaystyle u_{2}(x_{2})=u(y_{1},x_{2})}" loading="lazy">.

The above equation becomes:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=u_{1}(x_{1})+u_{2}(x_{2})}</annotation>

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</math><img src="./b333fed759ccf80ffb740785b1be04843a1bcac2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.034ex; height:2.843ex;" alt="{\displaystyle u(x_{1},x_{2})=u_{1}(x_{1})+u_{2}(x_{2})}" loading="lazy"></dd></dl>

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<annotation encoding="application/x-tex">{\displaystyle \longleftarrow }</annotation>

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</math><img src="./2bfecf2e8509d078cd7cf56f2ca8dca90598f251.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.738ex; height:1.843ex;" alt="{\displaystyle \longleftarrow }" loading="lazy">

If the function u is additive, then by the rules of expectation, for every lottery <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy">:

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<annotation encoding="application/x-tex">{\displaystyle E_{L}[u(x_{1},x_{2})]=E_{L}[u_{1}(x_{1})]+E_{L}[u_{2}(x_{2})]}</annotation>

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</math><img src="./07fd5bc4980547332c3551ede706346f9c589e2b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:41.115ex; height:2.843ex;" alt="{\displaystyle E_{L}[u(x_{1},x_{2})]=E_{L}[u_{1}(x_{1})]+E_{L}[u_{2}(x_{2})]}" loading="lazy"></dd></dl>

This expression depends only on the marginal probability distributions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

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<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

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</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> 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:<a href="#cite_note-KR-1">[1]</a>: 295

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},\dots ,x_{n})=\sum _{i=1}^{n}{k_{i}u_{i}(x_{i})}}</annotation>

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</math><img src="./2ad6a01f7f7908b2bf37339fd2f135784393fec2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:28.169ex; height:6.843ex;" alt="{\displaystyle u(x_{1},\dots ,x_{n})=\sum _{i=1}^{n}{k_{i}u_{i}(x_{i})}}" loading="lazy"></dd></dl></dd></dl></dd></dl></dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation>

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</math><img src="./14f13cb025ff2e136dcbd2fc81ddf965b728e3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}" loading="lazy"> are normalized to the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

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<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

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</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation>

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</math><img src="./f29138ed3ad54ffce527daccadc49c520459b0b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}" loading="lazy"> are normalization constants.

Much of the work in additive utility theory has been done by <a href="Peter_C._Fishburn" title="Peter C. Fishburn">Peter C. Fishburn</a>.

<div class="mw-heading mw-heading2"><h2 id="Utility_independence">Utility independence</h2></div>

A slightly weaker independence property is utility independence. Attribute 1 is utility-independent of attribute 2, if the conditional preferences on lotteries on attribute 1 given a constant value of attribute 2, do not depend on that constant value.

This means, for example, that the preference between a lottery <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <(x_{1},x_{2}):(y_{1},x_{2})>}">

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<annotation encoding="application/x-tex">{\displaystyle <(x_{1},x_{2}):(y_{1},x_{2})>}</annotation>

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</math><img src="./4e5f44bb272ff866a6a8e9019c9df98052899538.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.875ex; height:2.843ex;" alt="{\displaystyle <(x_{1},x_{2}):(y_{1},x_{2})>}" loading="lazy"> and a lottery <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <(x'_{1},x_{2}):(y'_{1},x_{2})>}">

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<annotation encoding="application/x-tex">{\displaystyle <(x'_{1},x_{2}):(y'_{1},x_{2})>}</annotation>

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</math><img src="./e9c68d26192483053050b2a2d0a4458aca68b56c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.875ex; height:3.009ex;" alt="{\displaystyle <(x'_{1},x_{2}):(y'_{1},x_{2})>}" loading="lazy"> is the same, regardless of the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}}</annotation>

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</math><img src="./d7af1b928f06e4c7e3e8ebfd60704656719bd766.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.009ex;" alt="{\displaystyle x_{2}}" loading="lazy">.

Note that utility independence (in contrast to additive independence) is not symmetric: it is possible that attribute 1 is utility-independent of attribute 2 and not vice versa.<a href="#cite_note-KR-1">[1]</a>: 224–229

If attribute 1 is utility-independent of attribute 2, then the utility function for every value of attribute 2 is a linear transformation of the utility function for every other value of attribute 2. Hence it can be written as:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=c_{1}(x_{2})+c_{2}(x_{2})\cdot u(x_{1},x_{2}^{0})}</annotation>

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</math><img src="./a628163e2b764229b4cb22f555c95d20c3a0880e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.008ex; height:3.176ex;" alt="{\displaystyle u(x_{1},x_{2})=c_{1}(x_{2})+c_{2}(x_{2})\cdot u(x_{1},x_{2}^{0})}" loading="lazy"></dd></dl>

when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{2}^{0}}">

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<annotation encoding="application/x-tex">{\displaystyle x_{2}^{0}}</annotation>

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</math><img src="./ea2922c0126a0cff70d7c82975127e3744f62e7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.384ex; height:3.176ex;" alt="{\displaystyle x_{2}^{0}}" loading="lazy"> is a constant value for attribute 2. Similarly, If attribute 2 is utility-independent of attribute 1:

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=d_{1}(x_{1})+d_{2}(x_{1})\cdot u(x_{1}^{0},x_{2})}</annotation>

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</math><img src="./96f295e2ddc68cb76df292a9ec8c4c6c0f0e1df0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:38.412ex; height:3.176ex;" alt="{\displaystyle u(x_{1},x_{2})=d_{1}(x_{1})+d_{2}(x_{1})\cdot u(x_{1}^{0},x_{2})}" loading="lazy"></dd></dl>

If the attributes are mutually utility independent, then the utility function u has the following multi-linear form:<a href="#cite_note-KR-1">[1]</a>: 233–235

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<annotation encoding="application/x-tex">{\displaystyle u(x_{1},x_{2})=u_{1}(x_{1})+u_{2}(x_{2})+k\cdot u_{1}(x_{1})\cdot u_{2}(x_{2})}</annotation>

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</math><img src="./1abd26186fc8713be1a7e396b8f59dafdf6face7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:48.598ex; height:2.843ex;" alt="{\displaystyle u(x_{1},x_{2})=u_{1}(x_{1})+u_{2}(x_{2})+k\cdot u_{1}(x_{1})\cdot u_{2}(x_{2})}" loading="lazy"></dd></dl>

Where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is a constant which can be positive, negative or 0.

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<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

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</math><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy">, the function u is additive and the attributes are additive-independent.</li>

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<annotation encoding="application/x-tex">{\displaystyle k\neq 0}</annotation>

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</math><img src="./3e4367cc52bf0bebde550f396dde7bf07fa67bdd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.472ex; height:2.676ex;" alt="{\displaystyle k\neq 0}" loading="lazy">, the utility function is multiplicative, since it can be written as:</li></ul>

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<annotation encoding="application/x-tex">{\displaystyle [ku(x_{1},x_{2})+1]=[ku_{1}(x_{1})+1]\cdot [ku_{2}(x_{2})+1]}</annotation>

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</math><img src="./7e417735437fbc8d654e1500f068099c4f6b20a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:46.396ex; height:2.843ex;" alt="{\displaystyle [ku(x_{1},x_{2})+1]=[ku_{1}(x_{1})+1]\cdot [ku_{2}(x_{2})+1]}" loading="lazy"></dd>

<dd>where each term is a linear transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\cdot +1}">

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<annotation encoding="application/x-tex">{\displaystyle k\cdot +1}</annotation>

</semantics>

</math><img src="./36cc7e5cb05aa3494ebb1bb6c16f47290f182b5d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.861ex; height:2.343ex;" alt="{\displaystyle k\cdot +1}" loading="lazy"> of a utility function.</dd></dl>

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:

<ul><li><a href="#Additive_independence">Additive</a>, or -</li>

<li>Multiplicative:<a href="#cite_note-KR-1">[1]</a>: 289–290 </li></ul>

<msub>

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</msub>

<msub>

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</msub>

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</munderover>

<msub>

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<msub>

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</msub>

<msub>

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</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1+ku(x_{1},\dots ,x_{n})=\prod _{i=1}^{n}{1+kk_{i}u_{i}(x_{i})}}</annotation>

</semantics>

</math><img src="./b8a8d270d71cdb35b62863dd6862cc00fd37ddf8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:38.211ex; height:6.843ex;" alt="{\displaystyle 1+ku(x_{1},\dots ,x_{n})=\prod _{i=1}^{n}{1+kk_{i}u_{i}(x_{i})}}" loading="lazy"></dd></dl></dd></dl></dd></dl></dd></dl>

where:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u_{i}}">

<msub>

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</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u_{i}}</annotation>

</semantics>

</math><img src="./14f13cb025ff2e136dcbd2fc81ddf965b728e3d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle u_{i}}" loading="lazy"> are normalized to the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">;</li>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle k_{i}}</annotation>

</semantics>

</math><img src="./f29138ed3ad54ffce527daccadc49c520459b0b0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.011ex; height:2.509ex;" alt="{\displaystyle k_{i}}" loading="lazy"> are constants in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

</semantics>

</math><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy">;</li>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> is a constant which is either in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (-1,0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (-1,0)}</annotation>

</semantics>

</math><img src="./a8f5f5f03e9f4380e41314d8c5d0129861c8ecf6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.976ex; height:2.843ex;" alt="{\displaystyle (-1,0)}" loading="lazy"> or in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0,\infty )}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (0,\infty )}</annotation>

</semantics>

</math><img src="./da17102e4ed0886686094ee531df040d2e86352a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.329ex; height:2.843ex;" alt="{\displaystyle (0,\infty )}" loading="lazy"> (note that the limit when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\to 0}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k\to 0}</annotation>

</semantics>

</math><img src="./3fcc764df099ea8bc091518756ccc1221f6e37ba.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.988ex; height:2.176ex;" alt="{\displaystyle k\to 0}" loading="lazy"> is the additive form).</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Comparison_of_independence_concepts">Comparison of independence concepts</h2></div>

It is useful to compare three different concepts related to independence of attributes: Additive-independence (AI), Utility-independence (UI) and Preferential-independence (PI).<a href="#cite_note-KR-1">[1]</a>: 344

AI and UI both concern preferences on lotteries and are explained above. PI concerns preferences on sure outcomes and is explained in the article on <a href="Ordinal_utility#PI" title="Ordinal utility">ordinal utility</a>.

Their implication order is as follows:

AI is a symmetric relation (if attribute 1 is AI of attribute 2 then attribute 2 is AI of attribute 1), while UI and PI are not.

AI implies mutual UI. The opposite is, in general, not true; it is true only if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle k=0}</annotation>

</semantics>

</math><img src="./6307c8a99dad7d0bcb712352ae0a748bd99a038b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.472ex; height:2.176ex;" alt="{\displaystyle k=0}" loading="lazy"> in the multi-linear formula for UI attributes. But if, in addition to mutual UI, there exist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{1},x_{2},y_{1},y_{2}}">

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1},x_{2},y_{1},y_{2}}</annotation>

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</math><img src="./bf42f264bfd1a66cf0b20d3450cc47541569503b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.257ex; height:2.009ex;" alt="{\displaystyle x_{1},x_{2},y_{1},y_{2}}" loading="lazy"> for which the two lotteries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle L}</annotation>

</semantics>

</math><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, defined above, are equivalent - then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> must be 0, and this means that the preference relation must be AI.<a href="#cite_note-KR-1">[1]</a>: 238–239

UI implies PI. The opposite is, in general, not true. But if:

<ul><li>there are at least 3 essential attributes, and:</li>

<li>all pairs of attributes {1,i} are PI of their complement, and:</li>

<li>attribute 1 is UI of its complement,</li></ul>

then all attributes are mutually UI. Moreover, in that case there is a simple relation between the cardinal utility function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> representing the preferences on lotteries, and the ordinal utility function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v}</annotation>

</semantics>

</math><img src="./e07b00e7fc0847fbd16391c778d65bc25c452597.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" loading="lazy"> representing the preferences on sure bundles. The function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

</semantics>

</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> must have one of the following forms:<a href="#cite_note-KR-1">[1]</a>: 330–332 <a href="#cite_note-2">[2]</a>

<ul><li>Additive: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},...,x_{n})=v(x_{1},...,x_{n})}">

<msub>

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<msub>

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<msub>

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<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x_{1},...,x_{n})=v(x_{1},...,x_{n})}</annotation>

</semantics>

</math><img src="./8550eb6092bf9f8cf7c2142ba38416ee35901bad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.378ex; height:2.843ex;" alt="{\displaystyle u(x_{1},...,x_{n})=v(x_{1},...,x_{n})}" loading="lazy"></li>

<li>Multiplicative: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(x_{1},...,x_{n})=[exp(R\cdot v(x_{1},...,x_{n}))-1]/[exp(R)-1]}">

<msub>

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</msub>

<msub>

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<msub>

</mrow>

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<msub>

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</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(x_{1},...,x_{n})=[exp(R\cdot v(x_{1},...,x_{n}))-1]/[exp(R)-1]}</annotation>

</semantics>

</math><img src="./aeec2e1a9059af96b434d05197a4f1d5a007f672.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:57.124ex; height:2.843ex;" alt="{\displaystyle u(x_{1},...,x_{n})=[exp(R\cdot v(x_{1},...,x_{n}))-1]/[exp(R)-1]}" loading="lazy"></li></ul>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\neq 0}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle R\neq 0}</annotation>

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</math><img src="./6668d74ef5db38b2f2b6202160649ac2cd67e4e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.025ex; height:2.676ex;" alt="{\displaystyle R\neq 0}" loading="lazy">.

PROOF: It is sufficient to prove that u has <a href="Risk_aversion#Absolute_risk_aversion" title="Risk aversion">constant absolute risk aversion</a> with respect to the value v.

<ul><li>The PI assumption with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\geq 3}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle n\geq 3}</annotation>

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</math><img src="./73136e4a27fe39c123d16a7808e76d3162ce42bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.656ex; height:2.343ex;" alt="{\displaystyle n\geq 3}" loading="lazy"> imply that the value function is additive, i.e.:</li></ul>

<msub>

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<msub>

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</munderover>

<msub>

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<msub>

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<msub>

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</msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle v(x_{1},\dots ,x_{n})=\sum _{i=1}^{n}{\lambda _{i}v_{i}(x_{i})}}</annotation>

</semantics>

</math><img src="./0e0dc127bae796303c1c5eb2480bbdf5c3a0b647.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.909ex; height:6.843ex;" alt="{\displaystyle v(x_{1},\dots ,x_{n})=\sum _{i=1}^{n}{\lambda _{i}v_{i}(x_{i})}}" loading="lazy"></dd></dl></dd></dl></dd></dl></dd></dl>

<msub>

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</msub>

<msub>

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<annotation encoding="application/x-tex">{\displaystyle x_{1},z_{1}}</annotation>

</semantics>

</math><img src="./9b9e85f874b09080b49ee4f63a9d8a0e138443a3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.553ex; height:2.009ex;" alt="{\displaystyle x_{1},z_{1}}" loading="lazy"> be two different values for attribute 1. Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{1}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{1}}</annotation>

</semantics>

</math><img src="./eef4db76d658a98219aca14df06d9869d2b43c42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.193ex; height:2.009ex;" alt="{\displaystyle y_{1}}" loading="lazy"> be the certainty-equivalent of the lottery <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle <x_{1}:z_{1}>}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle <x_{1}:z_{1}>}</annotation>

</semantics>

</math><img src="./2e462388f347607361478d750df26747ec155339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.363ex; height:2.176ex;" alt="{\displaystyle <x_{1}:z_{1}>}" loading="lazy">. The UI assumption implies that, for every combination <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (w_{2},\dots ,w_{n})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (w_{2},\dots ,w_{n})}</annotation>

</semantics>

</math><img src="./20892050438d0a4903b942bc9ccfa1f55ff1260a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.588ex; height:2.843ex;" alt="{\displaystyle (w_{2},\dots ,w_{n})}" loading="lazy"> of values of the other attributes, the following equivalence holds:</li></ul>

<msub>

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<msub>

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle (y_{1},w)\sim <(x_{1},w):(z_{1},w)>}</annotation>

</semantics>

</math><img src="./0a6ede92fd0c406e159769b06953a55a1950fa84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.532ex; height:2.843ex;" alt="{\displaystyle (y_{1},w)\sim <(x_{1},w):(z_{1},w)>}" loading="lazy"></dd></dl></dd></dl></dd></dl></dd></dl>

<ul><li>The two previous statements imply that for every w, the following equivalence holds in the value space:</li></ul>

<msub>

</mrow>

</msub>

<msub>

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<msub>

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</munderover>

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</munderover>

<msub>

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<msub>

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<msub>

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<msub>

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<msub>

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</munderover>

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<msub>

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</msub>

<msub>

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda _{1}v_{1}(y_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}\sim <\lambda _{1}v_{1}(x_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}:\lambda _{1}v_{1}(z_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}>}</annotation>

</semantics>

</math><img src="./3b87a66aed8be19362aa3cfa785fd731714ce258.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:80.025ex; height:6.843ex;" alt="{\displaystyle \lambda _{1}v_{1}(y_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}\sim <\lambda _{1}v_{1}(x_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}:\lambda _{1}v_{1}(z_{1})+\sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}>}" loading="lazy"></dd></dl></dd></dl></dd></dl></dd></dl>

<ul><li>This implies that, adding any quantity to both sides of a lottery (through the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}}">

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}}</annotation>

</semantics>

</math><img src="./2d6cc6a8ccf7fcbfd020d049c40963adbc2c7f79.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:12.097ex; height:6.843ex;" alt="{\displaystyle \sum _{i=2}^{n}{\lambda _{i}v_{i}(w_{i})}}" loading="lazy">), increases the certainty-equivalent of the lottery by the same quantity.</li>

<li>The latter fact implies constant risk aversion.</li></ul>

<ul><li><a href="Multi-attribute_auction" title="Multi-attribute auction">Multi-attribute auction</a></li>

<li><a href="Multi-objective_optimization" title="Multi-objective optimization">Multi-objective optimization</a></li>

<li><a href="Multi-issue_voting" title="Multi-issue voting">Multi-issue voting</a></li>

<li><a href="Decision-making_software" title="Decision-making software">Decision-making software</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFKeeneyRaiffa1993" class="citation book cs1">Keeney, Ralph L.; Raiffa, Howard (1993). Decisions with Multiple Objectives. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-521-44185-4</bdi>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> This idea is attributed to Richard F. Meyer and <a href="John_W._Pratt" title="John W. Pratt">John W. Pratt</a>.

</li>

</ol></div></div>

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</style><div id="Decision_theory250" style="font-size:114%;margin:0 4em"><a href="Decision_theory" title="Decision theory">Decision theory</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Core concepts</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Ambiguity_aversion" title="Ambiguity aversion">Ambiguity aversion</a></li>

<li><a href="Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li>

<li><a href="Choice_architecture" title="Choice architecture">Choice architecture</a></li>

<li><a href="Expected_utility_hypothesis" title="Expected utility hypothesis">Expected utility</a></li>

<li><a href="Expected_value" title="Expected value">Expected value</a></li>

<li><a href="Hyperbolic_discounting" title="Hyperbolic discounting">Hyperbolic discounting</a></li>

<li><a href="Leximin" class="mw-redirect" title="Leximin">Leximin</a></li>

<li><a href="Loss_aversion" title="Loss aversion">Loss aversion</a></li>

<li><a href="Path_dependence" title="Path dependence">Path dependence</a></li>

<li><a href="Principle_of_indifference" title="Principle of indifference">Principle of indifference</a></li>

<li><a href="Prospect_theory" title="Prospect theory">Prospect theory</a></li>

<li><a href="Rational_choice_theory" class="mw-redirect" title="Rational choice theory">Rational choice theory</a></li>

<li><a href="Risk_aversion" title="Risk aversion">Risk aversion</a></li>

<li><a href="Risk-seeking" title="Risk-seeking">Risk-seeking</a></li>

<li><a href="Satisficing" title="Satisficing">Satisficing</a></li>

<li><a href="Strategic_dominance" title="Strategic dominance">Strategic dominance</a></li>

<li><a href="Subjective_expected_utility" title="Subjective expected utility">Subjective expected utility</a></li>

<li><a href="Sure-thing_principle" title="Sure-thing principle">Sure-thing</a></li>

<li><a href="Von_Neumann%E2%80%93Morgenstern_utility_theorem" title="Von Neumann–Morgenstern utility theorem">Utility theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision models</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Anscombe-Aumann_subjective_expected_utility_model" title="Anscombe-Aumann subjective expected utility model"> Anscombe-Aumann framework</a></li>

<li><a href="Causal_decision_theory" title="Causal decision theory">Causal decision</a></li>

<li><a href="Decision_field_theory" title="Decision field theory">Decision field theory</a></li>

<li><a href="Emotional_choice_theory" title="Emotional choice theory">Emotional choice</a></li>

<li><a href="Evidential_decision_theory" title="Evidential decision theory">Evidential decision</a></li>

<li><a href="Fuzzy-trace_theory" title="Fuzzy-trace theory">Fuzzy-trace theory</a></li>

<li><a href="Intertemporal_choice" title="Intertemporal choice">Intertemporal choice</a></li>

<li><a href="Naturalistic_decision-making" title="Naturalistic decision-making">Naturalistic decision</a></li>

<li><a href="Normative_model_of_decision-making" title="Normative model of decision-making">Normative model</a></li>

<li><a href="Quantum_cognition" title="Quantum cognition">Quantum cognition</a></li>

<li><a href="Recognition-primed_decision" title="Recognition-primed decision">Recognition-primed decision</a></li>

<li><a href="Rubicon_model" title="Rubicon model">Rubicon model</a></li>

<li><a href="Savage's_subjective_expected_utility_model" title="Savage's subjective expected utility model">Savage's subjective expected utility model</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Decision analysis tools</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Analytic_hierarchy_process" title="Analytic hierarchy process">Analytic hierarchy process</a></li>

<li><a href="Analytic_network_process" title="Analytic network process">Analytic network process</a></li>

<li><a href="Cost%E2%80%93benefit_analysis" title="Cost–benefit analysis">Cost–benefit analysis</a></li>

<li><a href="Cost-effectiveness_analysis" title="Cost-effectiveness analysis">Cost-effectiveness analysis</a></li>

<li><a href="Cost%E2%80%93utility_analysis" title="Cost–utility analysis">Cost–utility analysis</a></li>

<li><a href="Decision_conferencing" title="Decision conferencing">Decision conferencing</a></li>

<li><a href="Decision_curve_analysis" title="Decision curve analysis">Decision curve analysis</a></li>

<li><a href="Decision_rule" title="Decision rule">Decision rule</a></li>

<li><a href="Decision_support_system" title="Decision support system">Decision support system</a></li>

<li><a href="Decision_table" title="Decision table">Decision table</a></li>

<li><a href="Decision_tree" title="Decision tree">Decision tree</a></li>

<li><a href="Decision-matrix_method" title="Decision-matrix method">Decision matrix</a></li>

<li><a href="Decisional_balance_sheet" title="Decisional balance sheet">Decisional balance sheet</a></li>

<li><a href="Gittins_index" title="Gittins index">Gittins index</a></li>

<li><a href="Influence_diagram" title="Influence diagram">Influence diagram</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Scoring_rule" title="Scoring rule">Scoring rule</a></li>

<li><a href="Value_of_information" title="Value of information">Value of information</a>

<ul><li><a href="Expected_value_of_perfect_information" title="Expected value of perfect information">perfect</a></li>

<li><a href="Expected_value_of_sample_information" title="Expected value of sample information">sample</a></li>

<li><a href="Expected_value_of_including_uncertainty" title="Expected value of including uncertainty">uncertainty</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Paradoxes and biases</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Allais_paradox" title="Allais paradox">Allais paradox</a></li>

<li><a href="Certainty_effect" title="Certainty effect">Certainty effect</a></li>

<li><a href="Cognitive_bias" title="Cognitive bias">Cognitive bias</a></li>

<li><a href="Decoy_effect" title="Decoy effect">Decoy effect</a></li>

<li><a href="Disposition_effect" title="Disposition effect">Disposition effect</a></li>

<li><a href="Ellsberg_paradox" title="Ellsberg paradox">Ellsberg paradox</a></li>

<li><a href="Endowment_effect" title="Endowment effect">Endowment effect</a></li>

<li><a href="Framing_effect_(psychology)" title="Framing effect (psychology)">Framing effect</a></li>

<li><a href="Heuristics_in_judgment_and_decision-making" class="mw-redirect" title="Heuristics in judgment and decision-making">Heuristics</a></li>

<li><a href="Newcomb's_paradox" title="Newcomb's paradox">Newcomb's paradox</a></li>

<li><a href="Pseudocertainty_effect" title="Pseudocertainty effect">Pseudocertainty effect</a></li>

<li><a href="Reference_dependence" title="Reference dependence">Reference dependence</a></li>

<li><a href="Regret_(decision_theory)" title="Regret (decision theory)">Regret</a></li>

<li><a href="St._Petersburg_paradox" title="St. Petersburg paradox">St. Petersburg paradox</a></li>

<li><a href="Status_quo_bias" title="Status quo bias">Status quo bias</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Uncertainty and risk</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Decisionmaking_under_deep_uncertainty" class="mw-redirect" title="Decisionmaking under deep uncertainty">Deep uncertainty</a></li>

<li><a href="Exploration%E2%80%93exploitation_dilemma" title="Exploration–exploitation dilemma">Exploration–exploitation</a></li>

<li><a href="Pignistic_probability" title="Pignistic probability">Pignistic probability</a></li>

<li><a href="Robust_decision-making" title="Robust decision-making">Robust decision-making</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Behavioral_economics" title="Behavioral economics">Behavioral economics</a></li>

<li><a href="Game_theory" title="Game theory">Game theory</a></li>

<li><a href="Operations_research" title="Operations research">Operations research</a></li>

<li><a href="Social_choice_theory" title="Social choice theory">Social choice theory</a></li>

<li><a href="Utility_theory" class="mw-redirect" title="Utility theory">Utility theory</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key people</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="David_Blackwell" title="David Blackwell">David Blackwell</a></li>

<li><a href="Bruno_de_Finetti" title="Bruno de Finetti">Bruno de Finetti</a></li>

<li><a href="Morris_H._DeGroot" title="Morris H. DeGroot">Morris H. DeGroot</a></li>

<li><a href="Peter_C._Fishburn" title="Peter C. Fishburn">Peter C. Fishburn</a></li>

<li><a href="Gerd_Gigerenzer" title="Gerd Gigerenzer">Gerd Gigerenzer</a></li>

<li><a href="Itzhak_Gilboa" title="Itzhak Gilboa">Itzhak Gilboa</a></li>

<li><a href="Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li>

<li><a href="R._Duncan_Luce" title="R. Duncan Luce">R. Duncan Luce</a></li>

<li><a href="Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li>

<li><a href="Howard_Raiffa" title="Howard Raiffa">Howard Raiffa</a></li>

<li><a href="Leonard_J._Savage" class="mw-redirect" title="Leonard J. Savage">Leonard J. Savage</a></li>

<li><a href="David_Schmeidler" title="David Schmeidler">David Schmeidler</a></li>

<li><a href="Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li>

<li><a href="Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li></ul>

</div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> Category</div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-05-25" href="https://en.wikipedia.org/wiki/?title=Multi-attribute_utility&oldid=1292180815">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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