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.

What sense does it make that we can derive a utility function if the probability of taking $30 over $20 is either 1 or 3/4, but not anything else?

It seems that he should account for the fact that this subjective probability will update. For example, you quoted him as saying

This means that if the probability that you choose salmon is 2/3, and the probability that you choose tuna is 1/3, then your utility of salmon is twice as high as that of tuna.

But once I know that u(salmon)/u(tuna) = 2, I know that I will choose salmon over tuna. I therefore no longer assign the prior subjective probabilities that led me to this utility-ratio. I assign a new posterior subject probability — namely, certainty that I will choose tuna. This new subjective probability can no longer be used to derive the utility-ratio u(salmon)/u(tuna) = 2. I have learned the utility-ratio, but, in doing so, I have destroyed the state of affairs that allowed me to learn it. I might remember how I derived the utility-ratio, but I can no longer re-derive it in the same way. I have, as it were, "burned up" some of my prior subjective uncertainty, so I can't use it any more.

Now suppose that I am so unfortunate as to forget the value of the utility-ratio u(salmon)/u(tuna). However, I still retain the posterior subjective certainty that I choose salmon over tuna. Now how am I going to get that utility-ratio back? I'm going to have to find some other piece of prior subjective uncertainty to "burn". For example, I might notice some prior uncertainty about whether I would choose salmon over $5 and about whether I would choose tuna over $5. Then I could proceed as Peterson describes in the photo example.

So maybe Peterson's proposal can be saved by distinguishing between prior and posterior subjective probabilities for my choices in this way. Prior probabilities would be required to be consistent in the following sense: If

my prior odds of choosing A over B are 1:1, and
my prior odds of choosing B over C are 3:1,

then

my prior odds of choosing A over C have to be 3:1.

Thus, in the photo example, the prior probability of taking $30 over $20 has to be 3/4, given the other probabilities. It's not allowed to be 0.99. But the posterior probability is allowed to be 1, provided that I've already "burned up" some piece of prior subjective uncertainty to arrive at that certainty. In this way, perhaps, it makes sense to say that "the probability of taking $30 over $20 is either 1 or 3/4, but not anything else".

I wrote,

Now suppose that I am so unfortunate as to forget the value of the utility-ratio u(salmon)/u(tuna).

But, on reflection, the possibility of forgetting knowledge is probably a can of worms best left unopened. For, one could ask what would happen if I remembered that I was highly confident that I would choose salmon over tuna, but I forgot that I was absolutely certain about this. It would then be hard to see how to avoid inconsistent utility functions, as you describe.

Perhaps it's better to suppose that you've shown, by some unspecified means, t... (read more)

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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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