Rival formalizations of a decision problem

Decision theory is not one of my strengths, and I have a question about it.

Is there a consensus view on how to deal with the problem of “rival formalizations”? Peterson (2009) illustrates the problem like this:

Imagine that you are a paparazzi photographer and that rumour has it that actress Julia Roberts will show up in either New York (NY), Los Angeles (LA) or Paris (P). Nothing is known about the probability of these states of the world. You have to decide if you should stay in America or catch a plane to Paris. If you stay and [she] shows up in Paris you get $0; otherwise you get your photos, which you will be able to sell for $10,000. If you catch a plane to Paris and Julia Roberts shows up in Paris your net gain after having paid for the ticket is $5,000, and if she shows up in America you for some reason, never mind why, get $6,000. Your initial representation of the decision problem is visualized in Table 2.13.

Table 2.13

	P	LA	NY
Stay	$0	$10k	$10k
Go to Paris	$5k	$6k	$6k

Since nothing is known about the probabilities of the states in Table 2.13, you decide it makes sense to regard them as equally probable [see Table 2.14].

Table 2.14

	P (1/3)	LA (1/3)	NY (1/3)
Stay	$0	$10k	$10k
Go to Paris	$5k	$6k	$6k

The rightmost columns are exactly parallel. Therefore, they can be merged into a single (disjuntive) column, by adding the probabilities of the two rightmost columns together (Table 2.15).

Table 2.15

	P (1/3)	LA or NY (2/3)
Stay	$0	$10k
Go to Paris	$5k	$6k

However, now suppose that you instead start with Table 2.13 and first merge the two repetitious states into a single state. You would then obtain the decision matrix in Table 2.16.

Table 2.16

	P	LA or NY
Stay	$0	$10k
Go to Paris	$5k	$6k

Now, since you know nothing about the probabilities of the two states, you decide to regard them as equally probable… This yields the formal representation in Table 2.17, which is clearly different from the one suggested above in Table 2.15.

Table 2.17

	P (1/2)	LA or NY (1/2)
Stay	$0	$10k
Go to Paris	$5k	$6k

Which formalisation is best, 2.15 or 2.17? It seems question begging to claim that one of them must be better than the other — so perhaps they are equally reasonable? If they are, we have an example of rival formalisations.

Note that the principle of maximising expected value recommends different acts in the two matrices. According to Table 2.15 you should stay, but 2.17 suggests you should go to Paris.

Does anyone know how to solve this problem? If one is not convinced by the illustration above, Peterson (2009) offers a proof that rival representations are possible on pages 33–35.