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What is the Difference Between Cheerful Price and Shadow Price?

by Zachary Robertson1 min read28th Mar 20217 comments

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I read Yudkowsky's post on cheerful price and came away confused as to why there was no reference to shadow price. It seems the two are effectively the same modulo a perspective change. Am I missing something, are Yudkowsky (and others) somehow unaware of shadow price, or is it something else entirely?

Argument

The shadow price for a constrained optimization problem is the change in the optimal value of the primal utility function with respect to a change in constraints. More technically it's the dual variables associated with the dual optimization problem.

Yudkowsky's Tl;dr is, well comparable to the length of this post so to compare I'll just list the first bullet,

The cheerful price for doing something is the price that gives you a cheerful feeling about the transaction.

People do what they do to achieve their ends (utility) while obeying constraints, physical or otherwise. If they already were going to do the 'something' to be proposed the cheerful price would be zero. This is also true for the shadow price. If the optima lies within the set after perturbing the constraints there is no shadow price.

Say we ask someone to do something they would not naturally do. This means we are going to propose a change in constraints. The shadow price will compensate for the decrease in the person's utility, compared to no intervention.

This is premised on the idea that we can substitute utility with some other quantity. If it's money then the shadow price for doing something (to constraints) is the price that leaves the person's optimality unchanged. It seems clear that if the person's objective value is their utility then the change in value is their cheerfulness to the request. Hence, the shadow price is the cheerful price.

Addressing Reductionism

The most immediate reaction I expect is that somehow the cheerful price has to be 'more' than the fair price or 'catered' towards a specific part of the inner utility optimizer. However, these all seem to be straight-forward modifications of the concept of shadow price. The sorts of things that immediately come up when you apply the concept. It doesn't make sense to treat this as a novel concept.

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I'd say that the cheerful price is a primarily psychological concept, while shadow price is a more analytical one, and that is the whole point - when what you feel and what you think you ought to feel disagrees, the concept of cheerful price is explicitly telling you to not worry about the mismatch, and go with the former.

I suppose this is the most correct answer. I'm not really updating very much though. From my perspective I'll continue to see cheerful price as a psychological/subjective reinvention of shadow price.

Edit: It seems clear in this context, shadow price isn't exactly measurable. Cheerful price is just the upper estimate on the shadow price.

The Wikipedia page is not exactly easy to read for someone not already familiar with the term. My impression (possibly wrong) is that "shadow price" = estimate of a hypothetical market price, if the thing is actually not on the market (therefore it is impossible to look at the actual price).

So, first, there is a difference in usage. "Shadow price" is used in context of buying something for money, or producing it (which would cost you money and other resources). "Cheerful price" is used in context of selling, typically your work and time. In other words, you estimate "shadow price" of things you want, and "cheerful price" of things someone else wants you to do.

Second, there is a difference in precision. "Shadow price" tries to be as accurate as possible (in a situation where it is difficult to be accurate). "Cheerful price" tries to be safe. That is, if your probability distribution is like: most likely $200, but there is a small chance it is $100 or $300, you would choose $200 for the "shadow price", but $300 for the "cheerful price".

Suppose I have a state consisting of : number of apples, number of bananas. My utility is  - i.e. I have no terminal desire for money, and my utility for apples and bananas is separable (aka a sum, with each term dependent on only one of the two). I also have a budget constraint: , i.e. price of apples times number of apples plus price of bananas times number of bananas is at most , the amount of money I start with.

A useful technique for this sort of problem is to separate it into two optimization problems (thus the term "separable" for the utility). Here's how that works:

  • First, pick some amount of money  to budget to apples, and budget the rest  to bananas.
  • Then, maximize  subject to , and separately maximize  subject to .

(Note: if  and  were higher-dimensional, i.e. each involved a whole bunch of variables, then this would still work, and indeed that's the case in which it's typically interesting/useful to do this.)

Assuming that each sub-problem can be solved efficiently, all we have to do is adjust the budget-split  to be optimal. That's where the shadow price comes in: the shadow prices of each sub-optimization problem tells us how much utility we would have to trade off in the -problem to get some amount of utility in the -problem, or vice-versa. To put it differently: the shadow price summarizes all the information about the -decisions which is relevant to the -decisions. To put it yet another way: the shadow price is the opportunity cost of spending a marginal dollar on  rather than .

And this exactly the way people use shadow prices in practice, all the time. When I'm trying to avoid spending a lot of money on a new gadget, it's not because I value money as an end in itself, it's because I can use money saved to gain utility in other problems in my life. When trying to decide how much to spend on the gadget, I don't need to directly think about all the other problems in my life - I don't have to think about it all as one giant optimization problem. Instead, I just have a rough intuition for how much utility I can get in the other problems of my life from $X extra, and then I decide whether I can get more utility than that by spending $X extra on the new gadget. That rough intuition for how-much-utility-$X-is-can-buy is the shadow price.

Now we're ready to answer the question. The shadow price quantifies the opportunity cost, so if I'm paid my shadow price, then that's just barely enough to cover my opportunity cost. It's reimbursing me for the opportunity cost of doing-the-thing rather than spending my time and resources on other problems in my life, but it's just reimbursing me, nothing extra. I'm not actually eager to take the deal, I'm indifferent. 

The idea of a cheerful price is that it does give me something extra. It's not just paying my opportunity cost, I'm not just indifferent between taking the deal or not; it's giving me more utility than I'd get from spending my time and resources on other problems in my life.

Your example is interesting and clarifies exchange rates. However,

The shadow price quantifies the opportunity cost, so if I'm paid my shadow price, then that's just barely enough to cover my opportunity cost.

This is an interpretive point I'd like to focus on. When you move a constraint, in this case with price, the underlying equilibrium of the optimization shifts. From this perspective your usage of the word 'barely' stops making sense to me. If you were to 'overshoot' you wouldn't be optimal in the new optimization problem.

At this point I understand ... (read more)

The other answers are pretty great but I just want to point out that it very well could be the case that 'cheerful prices' are just a specific subset of 'shadow prices'.

The concept of cheerful price also seems aimed at a very different audience!

But translation is hard – I'm not sure the two prices are different 'fundamentally'.

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I really like this question!