# A novel approach to axiomatic decision theory

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)](http://commonsenseatheism.com/wp-content/uploads/2012/05/Peterson-From-Outcomes-to-Acts-a-non-standard-axiomatization-of-the-expected-utility-principle.pdf) 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)](http://www.amazon.com/Introduction-Decision-Cambridge-Introductions-Philosophy/dp/0521888379/) 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)](http://books.google.com/books?id=PyJ_4KYwxNwC&lpg=PA29&dq=A%20Philosophical%20Assessment%20of%20Decision%20Theory&pg=PA405#v=onepage&q&f=false) 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)](http://www.amazon.com/Non-Bayesian-Decision-Theory-Beliefs-Desires/dp/9048179572/) 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)](http://commonsenseatheism.com/wp-content/uploads/2012/05/Cozic-Review-of-Non-Bayesian-Decision-Theory.pdf) 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.