Given an agent with some set X of choices, a utility function u maps from the set X to the real numbers R. The mapping is such that the agent prefers x1 to x2 if and only if u(x1) > u(x2). This completes the definition of an ordinal utility function.

A cardinal utility function satisfies additional conditions which allow easy consideration of probabilities. One way to state these conditions is that probabilities defined on X are required to be linear over u. This means that we can now consider probabilistic mixes of choices from X (with probabilities summing to 1). For example, one valid mix would be 0.25 probability of x1 with 0.75 probability of x2, and a second valid mix would be 0.8 probability of x3 with 0.2 probability of x4. A cardinal utility function must satisfy the condition that the agent prefers the first mix to the second mix if and only if 0.25u(x1) + 0.75u(x2) > 0.8u(x3) + 0.2u(x4).

Cardinal utility functions can also be formalized in other ways. E.g., another way to put it is that the relative differences between utilities must be meaningful. For instance, if u(x1) - u(x2) > u(x3) - u(x4), then the agent prefers x1 to x2 more than it prefers x3 to x4. (This property need not hold for ordinal utility functions.)

Other notes:

  • In my experience, ordinal utility functions are normally found in economics, whereas cardinal utility functions are found in game theory (where they are essential for any discussion of mixed strategies). Most, if not all, discussions on LW use cardinal utility functions.

  • The VNM theorem is an incredibly important result on cardinal utility functions. Basically, it shows that any agent satisfying a few basic axioms of 'rationality' has a cardinal utility function. (However, we know that humans don't satisfy these axioms. To model human behavior, one should instead use the descriptive prospect theory.)

  • Beware of erroneous straw characterizations of utility functions (recent example). Remember the VNM theorem—very frugal assumptions are sufficient to show the existence of a cardinal utility function. In a sense, this means that utility functions can model any set of preferences that are not logically contradictory.

  • Ordinal utility functions are equivalent up to strictly increasing transformations, whereas cardinal utility functions are equivalent only up to positive affine transformations.

  • Utility functions are often called payoff functions in game theory.

Stupid Questions Open Thread Round 3

by OpenThreadGuy 1 min read7th Jul 2012209 comments

8


From the last thread:

From Costanza's original thread (entire text):

"This is for anyone in the LessWrong community who has made at least some effort to read the sequences and follow along, but is still confused on some point, and is perhaps feeling a bit embarrassed. Here, newbies and not-so-newbies are free to ask very basic but still relevant questions with the understanding that the answers are probably somewhere in the sequences. Similarly, LessWrong tends to presume a rather high threshold for understanding science and technology. Relevant questions in those areas are welcome as well.  Anyone who chooses to respond should respectfully guide the questioner to a helpful resource, and questioners should be appropriately grateful. Good faith should be presumed on both sides, unless and until it is shown to be absent.  If a questioner is not sure whether a question is relevant, ask it, and also ask if it's relevant."

Meta:

  • How often should these be made? I think one every three months is the correct frequency.
  • Costanza made the original thread, but I am OpenThreadGuy. I am therefore not only entitled but required to post this in his stead. But I got his permission anyway.

Meta:

 

  • I still haven't figured out a satisfactory answer to the previous meta question, how often these should be made. It was requested that I make a new one, so I did.
  • I promise I won't quote the entire previous threads from now on. Blockquoting in articles only goes one level deep, anyway.