Multi-attribute utility
In decision theory, a multi-attribute utility function is used to represent the preferences of an agent over bundles of goods under uncertainty.
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 ordinal utility 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. This 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> 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 cardinal utility functions. The article Von Neumann–Morgenstern utility theorem 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):(0,0)> and <(2,0):(2,0)>. The preferences here can be represented by cardinal utility functions which take several variables (the attributes). Such functions are the focus of the current article.
The goal is to calculate a utility function 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 is higher under A than under B:
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