Rival formalizations of a decision problem

DT is not my strength either, but it seems like Peterson is just doing a simple slight-of-hand trick. The trickery is concealed in this line:

Now, since you know nothing about the probabilities of the two states, you decide to regard them as equally probable

This conflicts with the earlier claim that our priors for each location are 1/3. Changing the way the table looks does not mean that we are allowed to change that prior information. P(P) is still 1/3, and P(LA or NY) is still 2/3. Calling these "two states" conceals the fact that you have manipulated the prior information, which is what's creating the "paradox."

The trick is that until you add in the prior, you don't actually have a decision theory problem, only part of one; making the states equally probable is adding information, and shuffling states around and then making the states equally probable is adding different information.

A fully-specified decision theory problem is one that could be written as a function which takes as input a strategy and a random number generator, and outputs a utility score. If you have to add any information - priors, structure, expected opponent-strategy - then you have an underspecified problem, which puts you back in the realm of science.

This problem is similar to the bead jar guess problem. Essentially the problem is where priors come from and it doesn't have a general solution within the context of Bayesianism. Bayes can tell you how to update your priors, but not what your initial priors should be.

The best thing to do in this problem, when you're not sure what priors you should assign, is to work backwards and figure out what priors you need to arrive at one solution or the other. In this case:

Let P = Pr(Julia Roberts goes to Paris). Then E(Stay) = 10(1-P) = 10 - 10P and E(Go) = 5P + 6(1-P) = 6 + P. So E(Stay) > E(Go) if 10 - 10 P > 6 + P or 4 > 11 P or P < 4/11.

Now, instead of trying to decide "what does the Holy Doctrine of Indifference direct us to do in this situation" we can think about the real question: is the probability that Julia Roberts goes to Paris less than 4/11?

The question being asked here is really "what priors should I assign?". I don't think I have the answer, but allow me to restate the problem:

"The subject is known to be headed for one of three places, A, B and C. What is the probability they turn up in each place?" To which the answer is 1/3 each by indifference.

Now why did it look ok to us to merge B and C into one option, (B or C)? Because (in the original problem) B and C were cities located in the same country, the U.S., and that prior geographical information had been incorporated into the problem. When we condition on the knowledge that A is in one country (France) and B and C are in another (the U.S.) the problem is, well, no longer symmetrical. And I confess I'm now actually unsure how or if indifference or maximum entropy can be applied to this now asymmetric problem.

It has been super interesting to read all your contributions to lukeprog's post; this 'paradox' is no doubt interesting, because there seemed to be a shared gut reaction as to something wrong about the above formulation. I have stumbled across this page with the exact dilemma as lukeprog while reading Peterson's Introduction to Decision Theory book (2nd edition). As you have all pointed out, it would seem that there is something inherently fishy about his formulation of this particular example of 'Rival Formalisations'.

I think if you follow the logic of the initial axioms he uses in the book, this example that he provides does not follow. To give you some context, he formulates this 'paradox' by invoking two 'axioms'; the principle of insufficient reason (ir) and merger of states (ms). These principles are as follows (Peterson 2009 page 35):

• The Principle of Insufficient Reason (IR): If (Pi) is a formal decision problem in which the probabilities of the states are unknown, then it may be transformed into a formal decision problem (Pi)' in which equal probabilities are assigned to all states.
• Merger of states (MS): If two or more states yield identical outcomes under all acts, then these repetitious states should be collapsed into one, and if the probabilities of the two states are known, then they should be added.

From these two principles he first applies the IR rule and then the MS rule to formulate the 'paradox' above (Peterson 2009 page 35).

As per the above post by lukeprog, Peterson, in his (1/3, 1/3, 1/3) (P, LA, NY) example, insists that the probabilities of LA and NY can be added together to make 2/3rds and that this is a correct application of IR and MS.

The principle of IR would instead contradict this application, as he adds the 1/3 probabilities from NY and LA as if these probabilities are apriori known. Contrary to Peterson, they are not apriori known. That's why IR was invoked in the first place. When evoking MS, it requires that the probabilities of the states to be known in order for the probabilities to be added. From IR, we know that these probabilities have, rather, been arbitrarily assigned into equal proportions, precisely because the probabilities of these states (P, NY, LA) in question are apriori unknown. It should not follow from this that the probabilities of NY can be added to LA. To do so is to suggests that the probabilities are apriori known and unknown at the same time, which is a contradiction.

A good question one could ask is what the difference between 'collapsing' states and adding probabilities, and whether it affects the above analysis. Much like the 'Sure-Thing Principle', states with identical outcomes are collapsable into one because the probability component is irrelevant; regardless of the likelihood of each state is, the outcomes are the same. I think that is why, in this example, collapsing NY/LA into (NY or LA) is permissible but adding probabilities without an apriori known origin is not. This would suggest that LA and NY should first collapse into one state because of its identical outcomes, and then as a result of the apriori unknown (and unknowable) probabilities of states P and (LA or NY), should 1/2 and 1/2 then be assigned to these states.

I believe this is where Peterson's application of these two principles fall short, and contradicts his own application. With this is mind, this would suggest the correct application would be to first use MS (premise 1 of MS) on (LA or NY), then treat that as one "state", then assign 1/2, 1/2 probabilities to P and (LA or NY). To do so otherwise would be to contradict IR and MS.

I think this would be the my way of explaining the seemingly 'paradoxical' outcome of Peterson's example; careful reading would suggest that his application is no way compatible with the initial axioms.

Please refer to (Peterson 2009) page 33-35 on An Introduction to Decision Theory (Second Edition) for more reading.

Kind Regards,

Derek

As far as I can tell, this is just the standard complaint about the (naive?) Principle of Indifference and doesn't have much to do with decision theory per se. E.g., here's Keynes talking about a similar case. The most plausible solutions I know of are to either 1. insist that there simply are no rational constraints besides the axioms of probability on how we should weight the various possibilities in the absence of evidence and hence the problem is underdetermined (it depends on our "arbitrary" priors), or 2. accept that this is a real problem with Bayesian epistemology and hope something better comes along that doesn't model all doxastic attitudes as probabilities.