Antique Dealer Problem I have a problem related to the industry I work in which I think can be addressed with a probabilistic graph model. Unfortunately, this is something I know nothing about.
I'm paraphrasing the problem to make it simple:
Imagine we have a huge number of antique dealers (say 10,000) and each year, there are 1,000,000 new antiques coming into the market. People bring these new antiques to the dealers and the dealer chooses to

*

*buy the antiques (so they can sell them in their store), or

*they pass

At this point, the seller has 2 options:

*

*take the antique to another dealer

*decide not to sell

Some dealers get offered lots of antiques and some get very low numbers. Some dealers get offered the best antiques and others tend to get offered lower-quality ones. So, the pass rate might be high for a particular dealer for different reasons. They might have very high standards, or they might simply get offered a lot of very low-quality stuff.
Importantly, there are a lot of cases where multiple dealers see the same antique. This may mean that we can compare how well the dealers discriminate between the items. E.g.

*

*if a dealer buys an item which 4 other dealers have passed-on, it suggests that dealer is not very good at discriminating good antiques from bad ones.

*If 2 dealers pass hundreds of items to each other, then we can see how much they agree on their decisions.

In framing the problem, I'm imagining dealers as nodes in a network and the edges representing a 'pass' where sellers take items from one dealer (who passed) to another one. We know which dealers got which items, and which items they bought/passed-on.
One important factor might be the number of times the item has been passed-on or whether the seller took it out of the market. These might be indications that the item was low-quality.
A key source of bias is the sellers' prejudice about the dealer. They might choose one dealer over another because they are well-known or likely to give a better price. We can infer that this prejudice exists from the data, but I don't think it's something we can quantify.
Assuming we have complete data, we would like to estimate:

*

*what percentage of ALL of the antiques in the world would a dealer pass-on (including those that they never got to see)?

This would be a proxy to measure which dealers are best at discriminating good quality antiques from bad ones.
 A: You are correct; this can be described as a graphical model.  I have done a limited amount of work on the topic of antiques in the past.  However, you are missing part of the background theory, so your question is “ill-posed.”
The antiques business is more complex than what you are describing.  At least as important, you are ignoring market segmentation and the rules of competition.
There are a variety of types of competition.  Economics tends to oversimplify discussions of that unless we are talking with each other.  If you are modeling behaviors, then you end up needing a more in-depth discussion.
One of the general forms of competition that exists is called monopolistic competition.  It is generally presented in contrast to monopoly or pure competition.  In pure competition, such as in the market for red hard winter wheat, sellers are price takers.  They cannot do anything at all to impact their revenue, so they only control their costs.
On the opposite side of the coin is a monopoly.  Revenue is totally under the control of the monopolist, and they can maximize their profits.
In monopolistic competition, other people offer similar but slightly different things.  The seller wants people to pay a premium for the object they have and not treat another object as so close that they want the same price for a similar item at another dealer.
That is the first complexity.  A dealer offered item $B$ may reject it because they already own item $A$, and having both on a shelf may reduce the existing premium on $A$.  No antique dealer wants to compete against themselves.
The second thing that you are missing is price discrimination.  The antique industry, as a whole, uses all three types of price discrimination.  Third degree price discrimination, the most common type, segments the market so that each dealer can earn higher profits than if it did not exist.
Imagine a town with two antique dealers; one only sells low quality antiques and the other only high quality antiques.  Both dealers will make a higher profit than they would if they both sold all types of antiques.  Market segmentation is a highly valuable tool to produce higher profits selling the same goods.
Indeed, some dealers own both kinds of stores.  There is a major retailer that owns two store brands.  There is a high degree of overlap in terms of the items sold.  They package their goods centrally.  If sold in store A, good X will sell for \$5 per unit and in store B will sell for \$50 per unit.
Walmart, if offered the chance to sell Lamborghini’s, would decline.  It is outside their product line and target audience.  Not accepting the object for sale increases their profits.
There is a lot more going on as well.  Dealers often get their product from auctions.  There is an entire theory around auctions, and it would be too challenging to rewrite all of auction theory as it relates to antiques.  Nonetheless, auctions provide dealers with both local information about local pricing and with low cost inventory because of how they behave in auctions when compared with the general public.  There is a profit-making strategy in auctions that differs from other types of forums.
A discriminating dealer may reject an item, not because it isn’t valuable, but because other items, even if of less value, may be sold at so low a price that their margin will be better.
Now to your specific question:

Assuming we have complete data, we would like to estimate what percentage of ALL of the antiques in the world would a dealer pass-on (including those that they never got to see)?

Yes, this can be done, but it isn’t meaningful to ask without further constraining the question.
With information on local income and wealth, market segmentation, and so forth, you could do something like a Poisson regression to give you a count of items or a logistic regression to predict the probability for an individual object. Still, for your question to be realistic, it needs to be far better defined.
The difficulty is that if you didn’t collect the correct data, you would end up with an “endogeneity” problem.  In other words, the object you are predicting alters the items you are using to predict the outcome.  The more challenging issue with a complete set of data is removing endogeneity.
For example, imagine I sell only type A objects, and everybody knows that.  Another seller sells type B objects.  The very fact that people know that I sell type A objects may change their behavior.  That may change the predictors, so I need to control for who does and does not have knowledge.  Possibly, people that know I only sell type A objects may go out of town, figuring as the only seller of A, I will give them a worse price than if they went to a major city.  People who are ignorant will go to both sellers and just take my price as a fair price, feeling lucky to sell it to anybody.
The buyer and the seller both influence each other.  You have to control for that.  That is the hard part of this problem over anything else.
