A novel approach to axiomatic decision theory

lukeprog

In the standard approach to axiomatic Bayesian decision theory, an agent (a decision maker) doesn't prefer Act #1 to Act #2 because the expected utility of Act #1 exceeds that of Act #2. Instead, the agent states its preferences over a set of risky acts, and if these stated preferences are consistent with a certain set of axioms (e.g. the VNM axioms, or the Savage axioms), it can be proven that the agent's decisions can be described as if the agent were assigning probabilities and utilities to outcomes and then maximizing expected utility. (Let's call this the ex post approach.)

Peterson (2004) introduces a different approach, which he calls the ex ante approach. In many ways, this approach is more intuitive. The agent assigns probabilities and utilities directly to outcomes (not acts), and these assignments are used to generate preferences over acts. Using this approach, Peterson claims to have shown that the principle of expected utility maximization can be derived from just four axioms.

As Peterson (2009:75,77) explains:

The aim of the axiomatization [in the ex ante approach] is to show that the utility of an act equals the expected utility of its outcomes.

...The axioms... entail that the utility of an act equals the expected utility of its outcomes. Or, put in slightly different words, the act that has the highest utility (is most attractive) will also have the highest expected utility, and vice versa. This appears to be a strong reason for letting the expected utility principle guide one's choices in decision under risk.

Jensen (2012:428) calls the ex ante approach "controversial," but I can't find any actual published rebuttals to Peterson (2004), so maybe Jensen just means that Peterson's result is "new and not yet percolated to the broad community."

Peterson (2008) explores the ex ante approach in more detail, under the unfortunate title of "non-Bayesian decision theory." (No, Peterson doesn't reject Bayesianism.) Cozic (2011) is a review of Peterson (2008) that may offer the quickest entry point into the subject of ex ante axiomatic decision theory.

Peterson (2009:210) illustrates the controversy nicely:

...even if [this] discussion may appear a bit theoretical... the controversy over [ex post and ex ante approaches] is likely to have important practical implications. For example, a forty-year-old woman seeking advice about whether to, say, divorce her husband, is likely to get very different answers from the [two approaches]. The [ex post approach] will advise the woman to first figure out what her preferences are over a very large set of risky acts, including the one she is thinking about performing, and then just make sure that all preferences are consistent with certain structural requirements. Then, as long as none of the structural requirements is violated, the woman is free to do whatever she likes, no matter what her beliefs and desires actually are... The [ex ante approach] will [instead] advise the woman to first assign numerical utilities and probabilities to her desires and beliefs, and then aggregate them into a decision by applying the principle of maximizing expected utility.

I'm not a decision theory expert, so I'd be very curious to hear what LW's decision theorists think of the axiomatization in Peterson (2004) — whether it works, and how significant it is.

As Peterson (2009:75,77) explains:

The aim of the axiomatization [in the ex ante approach] is to show that the utility of an act equals the expected utility of its outcomes.

...The axioms... entail that the utility of an act equals the expected utility of its outcomes. Or, put in slightly different words, the act that has the highest utility (is most attractive) will also have the highest expected utility, and vice versa. This appears to be a strong reason for letting the expected utility principle guide one's choices in decision under risk.

Peterson (2009:210) illustrates the controversy nicely:

...even if [this] discussion may appear a bit theoretical... the controversy over [ex post and ex ante approaches] is likely to have important practical implications. For example, a forty-year-old woman seeking advice about whether to, say, divorce her husband, is likely to get very different answers from the [two approaches]. The [ex post approach] will advise the woman to first figure out what her preferences are over a very large set of risky acts, including the one she is thinking about performing, and then just make sure that all preferences are consistent with certain structural requirements. Then, as long as none of the structural requirements is violated, the woman is free to do whatever she likes, no matter what her beliefs and desires actually are... The [ex ante approach] will [instead] advise the woman to first assign numerical utilities and probabilities to her desires and beliefs, and then aggregate them into a decision by applying the principle of maximizing expected utility.

I'm not a decision theory expert, so I'd be very curious to hear what LW's decision theorists think of the axiomatization in Peterson (2004) — whether it works, and how significant it is.

Prediction from reading this post before I delved into the paper: the controversy is going to be about psychology, not decision theory. (After delving into the paper: I'm going to go with 'prediction confirmed.')

So, he uses six axioms. How do they map onto Howard's 5 that I used? It looks like his 0 is essentially "you can state the problem," his 1 and 2 are my choice, his 3 and 4 don't seem to have mirrors, and his 5 is my equivalence. I find it a little worrisome that three of my axioms don't appear to show up in his- probability, order, and substitution- except possibly in 0. They're clearly present in his analysis, but they feel like things that should be axioms instead of just taken for granted, and it's not clear to me why he needed to raise 3 and 4 to the level of axioms.

It's also not clear to me why he puts such emphasis on the independence and "sure-thing" principle. The "sure-thing" principle is widely held to only apply to a certain class of utility functions / "sure things", and there's not a good reason to expect people should or do have those utility functions. (Outcomes, properly understood, are the entire future- and so a game in which I flip a coin and you win or lose $100 can be different from a game in which I flip a coin and give you either $0 or $200, because you're $100 richer in the second game.) Similarly, independence only holds for normatively correct probability functions, and so if you allow normatively incorrect probability functions you have to throw out independence.

(The difference between independence and "sure-thing" is that "sure-thing" applies a transformation to all of the outcomes, which may not be the same transformation to all of the utilities of those outcomes, and independence applies a transformation to all of the probabilities, which will be the same transformation to all of the act utilities for normatively correct probability functions.)

For the controversy, I'll quote him directly:

I am aware of two objections to the proposed ex ante axiomatization. The point of departure of the first objection is the reasonable suspicion that people simply do not have numerical utilities in their heads, or at least have no access to them. Therefore, to assume that utility is an extensive structure is unreasonable since it seems practically impossible for decision makers to state the preferences required for obtaining the utility function. The same holds true for the probability function; where does the exogenously defined probability function come from?

The probability complaint is uninteresting. The "how do we measure utilities?" complaint is serious but a little involved to discuss.

Basically, there are three branches of rationality: epistemic, instrumental, and terminal (I'm not sure I like "terminal rationality" as a name; please suggest alternatives): thinking about uncertainties and probabilities, thinking about actions, and thinking about outcomes.

Peterson's root complaint is that traditional decision theory is silent on the valuable parts of terminal rationality- it only ensures that your elicited preferences are consistent and then uses them. If they're consistent but insane, the expected utility maximization won't throw up any error flags (for example, Danzig's diet optimization which prescribed 500 gallons of vinegar a day), because checking for sanity is not its job.

But pointing that complaint at decision theory seems mistaken, because it's a question of how you build the utility function. The traditional approach he describes in lukeprog's post above (the woman considering divorce) uses casuistry, expecting that people can elicit their preferences well about individual cases and then extrapolate. I think the approach he prefers is deconstruction- isolate the different desires that are relevant to the outcomes in question, construct (potentially nonlinear) tradeoffs between them, then order the outcomes, then figure out the optimal action. The first checks primarily for internal consistency; the second checks primarily for reflective equilibrium. But both can do each, and they can be used as complements instead of substitutes.

TL;DR: There does not appear to be meat to the controversy over axioms, and if there is then Peterson's axioms strike me as worse than Howard's and possibly worse than vNM or Savage. There is meat to the controversy over discovering utility functions, but I don't think the 2004 paper you link is a valuable addition to that controversy. Compare it to the chapter on choice under certainty from Thinking and Deciding.

Suggestions for replacing "terminal":

"outcome"; "ultimate" (or "ultima" if you prefer Latin); "intrinsic" (to draw a contrast with "instrumental"); "telikos" (transliteration of the Greek for final, ultimate, terminal, or last); "endpoint"; "consequence"; "effect"; "impact"; "goal."

Do any of those sound better to you?

Edit -- slight change owing to formatting.

6lukeprog14y

I'm confused by several parts of your reply. I'll select just two of them for discussion. Perhaps you'll have the motivation to try to un-confuse me. * I don't understand your points about the axioms he uses. He uses the axioms required to derive the useful results he aimed for, given his approach to formalizing decision problems, and no more than that. Do you reject the plausibility of one of his axioms, or do you disagree that his results follow from his axioms? * I don't think Peterson denies the usefulness of traditional axiomatic decision theory for checking the consistency of one's preferences, he's just saying that it would also be nice to have a decision theory that can tell you what you should choose given what you believe and what you value. Indeed, this is what many/most people actually do when trying to make decisions "rationally," but this norm wasn't justified with an axiomatic approach until Peterson (as far as I can tell).

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A novel approach to axiomatic decision theory

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A novel approach to axiomatic decision theory

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