Peter Norvig on Utility Theory - Dictionary of Arguments
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Utility theory/AI research/Norvig/Russell: Intuitively, the principle of Maximum Expected Utility (MEU) seems like a reasonable way to make decisions, but it is by no means obvious that it is the only rational way. After all, why should maximizing the average utility be so special? What’s wrong with an agent that
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maximizes the weighted sum of the cubes of the possible utilities, or tries to minimize the worst possible loss? Could an agent act rationally just by expressing preferences between states, without giving them numeric values? Finally, why should a utility function with the required properties exist at all?
Solution: constraints on rational preferences.
Possible preferences: a) the agent prefers A over B, b) he is indifferent between A and B, c) he prefers A over B or is indifferent between them.
The primary issue for utility theory is to understand how preferences between complex lotteries are related to preferences between the underlying states in those lotteries. To address this issue we list six constraints that we require any reasonable preference relation to obey:
1. Orderability: Given any two lotteries, a rational agent must either prefer one to the other or else rate the two as equally preferable. That is, the agent cannot avoid deciding.
2. Transitivity: Given any three lotteries, if an agent prefers A to B and prefers B to C, then the agent must prefer A to C.
3. Continuity: If some lottery B is between A and C in preference, then there is some probability p for which the rational agent will be indifferent between getting B for sure and the lottery that yields A with probability p and C with probability 1 − p.
4. Substitutability: If an agent is indifferent between two lotteries A and B, then the agent is indifferent between two more complex lotteries that are the same except that B
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is substituted for A in one of them.
5. Monotonicity: Suppose two lotteries have the same two possible outcomes, A and B. If an agent prefers A to B, then the agent must prefer the lottery that has a higher probability for A (and vice versa).
6. Decomposability: Compound lotteries can be reduced to simpler ones using the laws of probability. This has been called the “no fun in gambling” rule because it says that two consecutive lotteries can be compressed into a single equivalent lottery (…).
These constraints are known as the axioms of utility theory. >Preferences/Norvig, >Rationality/AI research, >Certainty effect/Kahneman/Tversky, >Ambiguity/Kahneman/Tversky._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Author1]Vs[Author2] or [Author]Vs[term] is an addition from the Dictionary of Arguments. If a German edition is specified, the page numbers refer to this edition.
Stuart J. Russell
Artificial Intelligence: A Modern Approach Upper Saddle River, NJ 2010