Multiattribute utility theory

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Revision as of 10:02, 6 July 2007

The purpose for using utility theory in decision making is to create a mathematical model to aid the process. It gives the decision maker the ability to quantify the desirability of certain alternatives. Utility theory is for design scenarios where uncertainty and risk are considered.

Axioms

Decision theory has a set of axioms, which are meant to eliminate inconsistencies and suboptimal choices when it comes to trade-offs and uncertainty. They provide the basis for good decision making, and structures the problem so determining the utility function is more straightforward. However, it is important to remember that the axioms are just guidelines and are meant to guide decision-making. More likely it is the decision maker who is inconsistent or wrong. The first three axioms are for determining a value function to allow ranking of alternatives. The second three axioms structure a preference function.

• Completeness of complete order: X is preferred/less preferred/equally preferred to Y
• Transitivity: if X is preferred over Y, and Y is preferred over Z, then X is preferred over Z
• Monotonicity: more of an attribute is preferred, or less of an attribute is preferred
• Probabilities exist and can be quantified
• Monotonicity of Probability: decision maker prefers a larger probability of a good outcome than a smaller probability of a good outcome
• Substitution Independence: preferences are linear with respect to probability

Building the Utility Function

To determine the utility value, or the desirability, of the design, there are five steps.

1. Identify significant design attributes and generate alternative designs
2. Verify relevant attribute conditions
3. Evaluate Single Attribute Utility (SAU) function and trade-off preferences
4. Combine SAUs into multi-attribute utility function (MAU)
5. Select alternative with the highest MAU value by ranking the alternatives

Selection of Attributes

Attributes and their bounds are selected so that the designer's preference will be reflected in the attribute features. The range of the attribute must be selected so that it it is useful, manageable, and should indicate the expected performance of the design. When choosing the attributes, they must be:

1. Complete, such that important aspects are reflected in the design formulation
2. Operational, so that design decision analysis can be meaningfully implemented
3. Non-redundant, so there's no double counting
4. Minimal, for simplicity

Lottery

The lottery is used to determine preference. The designer varies the probability of a particular outcome until an indifference point between lottery and certainty is found. Certainty equivalence, which is the value where the designer doesn't care about the lottery between the best and the worst, is used, where the certainty value is the guaranteed result compared to the lottery between two extreme values when there is a probability p of the of the best or worst.

Development of Single Attribute Utility

SAU functions are obtained by set of lottery questions based on certainty equivalence. They are monotonic functions, where the best outcome is set at 1, and the worst at 0. SAU functions are then developed to describe the designer's compromise between the best and worst based on the lottery questions. Certainty equivalence is used, where the certainty value is the guaranteed result compared to the lottery between two extreme values when there is a probability p of the of the best or worst.

An analytical function is typically used for preference description, and exponential functions are usually better suited for it.¹ The general form is u(x) = ae^bx + c, where a and b are parameters which guarantee the utility is normalized between 0 and 1, and c is the risk coefficient which shows degree of risk attitude, reflecting rate at which risk attitude changes with different attribute level. Linear functions are sometimes used as well.

Development of Multi-attribute Utiltiy

• • Mathematical combination of all the SAU functions, with scaling constants results in MAU, which is the overall utility function with all attributes considered
  • Scaling constants reflect designer's preference on the attributes, acquired based on scaling constant lottery questions and preference independence questions
  • Additive and multiplicative formulations. Advantage of an additive formulation is its simplicity, but assumptions can be restrictive. Multiplicative MAU funtion shown in Eq. 4.2