The exposition of meta-probability is well done, and shows an interesting way of examining and evaluating scenarios. However, I would take issue with the first section of this article in which you establish single probability (expected utility) calculations as insufficient for the problem, and present meta-probability as the solution.

In particular, you say

What’s interesting is that, when you have to decide whether or not to gamble your first coin, the probability is exactly the same in the two cases (p=0.45 of a $2 payout). However, the rational course of action is different. What’s up with that?

Here, a single probability value fails to capture everything you know about an uncertain event. And, it’s a case in which that failure matters.

I do not believe that this is a failure of applying a single probability to the situation, but merely calculating the probability wrongly, by ignoring future effects of your choice. I think this is most clearly illustrated by scaling the problem down to the case where you are handed a green box, and only two coins. In this simplified problem, we can clearly examine all possible strategies.

• Strategy 1 would be to hold on to your two dollar coins. There is a 100% chance of a $2.00 payout
• Strategy 2 would be to insert both of your coins into the box. There is a 50.5% chance of a $0.00 payout, 40.5% chance of a $4.00 payout and a 9% chance of a $2.00 payout.
• Strategy 3 would be to insert one coin, and then insert the second only if the first pays out. There is a 55% chance of $1.00 payout, a 4.5% chance of a $2.00 payout, and a 40.5% chance of a $4.00 payout.
• Strategy 4 would be to insert one coin, and then insert the second only if the first doesn't pay out. There is a 50.5% chance of a 0.00$ payout, a 4.5% chance of a $2.00 payout, and a 45% chance of a $3.00 payout.

When put in these terms, it seems quite obvious that your choice to open the box would depend on more than the expected payoff from only the first box, because quite clearly your choice to open the first box pays off (or doesn't pay off) when opening (or not opening) the other boxes as well. This seems like an error in calculating the payoff matrix rather than a flaw with the technique of single probability values itself. It ignores the fact that opening the first box not only pays you off immediately, but also pays you off in the future by giving you information about the other boxes.

This problem easily succumbs to standard expected value calculations if all actions are considered. The steps remain the same as always:

1. Assign a utility to each dollar amount outcome
2. Calculate the expected utility of all possible strategies
3. Choose the strategy with the highest expected utility

In the case of two coins, we were able to trivially calculate the outcomes of all possible strategies, but in larger instances of the problem, it might be advisable to use shortcuts in the calculations. However, it still remains true that the best choice will still be the one you would have gotten if you had done out the full expected value calculation.

I think the confusion arises because a lot of the time problems are presented in a way that screens them off from the rest of the world. For example, you are given a box, and it either has $10.00 or $100.00. Once you open the box, the only effect it has on you is the amount of money you got. After you get the money, the box does not matter to the rest of the world. Problems are presented this way so that it is easy to factor out the decisions and calculations you have to make from every other decision you have to make. However, decision are not necessarily this way (in fact in real life, very few decisions are). In the choice of inserting the first coin or not, this is simply not the case, despite having superficial similarities to standard "box" problems.

Although you clearly understand that the payoffs from the boxes are entangled, you only apply this knowledge in your informal approach to the problem. The failure to consider the full effects of your actions in opening the first box may be psychologically encouraged by the technique of "single probability calculations", but it is certainly not a failure of the technique itself to capture such situations.