Utility Functions in Artificial Intelligence

The agents use the utility theory for making decisions. It is the mapping from lotteries to the real numbers. An agent is supposed to have various preferences and can choose the one which best fits his necessity.

Utility scales and Utility assessments

To help an agent in making decisions and behave accordingly, we need to build a decision-theoretic system. For this, we need to understand the utility function. This process is known as preference elicitation In this, the agents are provided with some choices and using the observed preferences, the respected utility function is chosen. Generally, there is no scale for the utility function. But, a scale can be established by fixing the boiling and freezing point of water. Thus, the utility is fixed as:

 U(S)=uT for best possible cases

 U(S)= u⊥  for worst possible cases.

A normalized utility function uses a utility scale with value uT=1, and u⊥ =0. For example, a utility scale between uT and u⊥  is given. Thereby an agent can choose a utility value between any prize Z and the standard lottery [p, u_; (1−p), u⊥]. Here, p denotes the probability which is adjusted until the agent is adequate between Z and the standard lottery.

Like in medical, transportation, and environmental decision problems, we use two measurement units: micromort or QUALY(quality-adjusted life year) to measure the chances of death of a person.

Money Utility

Economics is the root of utility theory. It is the most demanding thing in human life. Therefore, an agent prefers more money to less, where all other things remain equal. The agent exhibits a monotonic preference(more is preferred over less) for getting more money. In order to evaluate the more utility value, the agent calculates the Expected Monetary Value(EMV) of that particular thing. But this does not mean that choosing a monotonic value is the right decision always.

Multi-attribute utility functions

Multi-attribute utility functions include those problems whose outcomes are categorized by two or more attributes. Such problems are handled by multi-attribute utility theory.

Terminology used

  • Dominance:  If there are two choices say A and B, where A is more effective than B. It means that A will be chosen. Thus, A will dominate B. Therefore, multi-attribute utility function offers two types of dominance:
  • Strict Dominance: If there are two websites T and D, where the cost of T is less and provides better service than D. Obviously, the customer will prefer T rather than D. Therefore, T strictly dominates D. Here, the attribute values are known.
  • Stochastic Dominance:  It is a generalized approach where the attribute value is unknown. It frequently occurs in real problems. Here, a uniform distribution is given, where that choice is picked, which stochastically dominates the other choices. The exact relationship can be viewed by examing the cumulative distribution of the attributes.
  • Preference Structure: Representation theorems are used to show that an agent with a preference structure has a utility function as:

U(x1, . . . , xn) = F[f1(x1), . . . , fn(xn)],

            where F indicates any arithmetic function such as an addition function.

Therefore, preference can be done in two ways :

  • Preference without uncertainty: The preference where two attributes are preferentially independent of the third attribute. It is because the preference between the outcomes of the first two attributes does not depend on the third one.
    • Preference with uncertainty: This refers to the concept of preference structure with uncertainty. Here, the utility independence extends the preference independence where a set of attributes X is utility independent of another Y set of attributes, only if the value of attribute in X set is independent of Y set attribute value. A set is said to be mutually utility independent (MUI) if each subset is utility-independent of the remaining attribute.
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