# Used goods: private or common value auctions?

Consider used goods sold in typical market places ranging from your old computer to a house.

I am interested in identifying the information and valuation structure if this were sold in an auction. For a moment, consider a set of typical assumptions used in a Bayesian Nash equilibrium of a sealed bid first price auction.

An example would be putting an used iPad on the Facebook marketplace and waiting people to contact, but unambiguously, interested buyers must submit the bids by a certain due date.

How does the valuation and information structure look like here? I would assume it is a hybrid between private and common value auction. For a moment, let's exclude any signal avaialble for the seller.

Typically, each bidder $$i$$ will obtain a signal from a distribution. All bidders have access to the distribution's information, but each bidder observes only the signal it receives. So, in the pure private value auction, $$V_i($$X$$)=x_i$$ where $$X=\{X_1,...,X_N\}$$ is the set of signal each bidder receives, but $$i$$ values based on its signal only (e.g., a piece of painting).

Clearly, this is not the case for the iPad. This object has consumption value (e.g., actual usage). So, while bidder $$i$$ might obtain some information from the scratch mark of the used iPad ($$X_i=x_i$$), there is the minimum utility the used iPad would provide to anyone.

The tricky part to me is even this has many layers. Let $$Y_i$$ be the random variable representing the productivity bidder $$i$$ expects to yield by using the iPad. For example, if a bidder tries to give the used iPad to its toddler to watch the Paw Control ($$Y_i$$), then the utility it yields would be significantly different from the person who tries to use it for work with a certain amount of annual salary ($$Y_j$$).

I was thinking that the bidder-independent value of the used iPad relates to it being able to connect to WiFi, show the time and function as alarm, and the like: the basis functions. Let's denote this value $$Z$$ without any subscript.

Then, what is the used iPad's value? What would be a sensible and simple example of how $$V$$ looks like (e.g., additively separable, multiplicative)?

Is it $$V(iPad)= f(X_1,...,X_N;Y_1,...,Y_N;Z)$$?

• Are you aware of models of auctions with interdependent (or affiliated) values? Commented Jul 31 at 17:21
• @HerrK. Hey, it has been a while! hope things are well with you. Yes! do you have any reference of an exercise using interdependent values set up? Commented Jul 31 at 20:30
• Thanks. Hope you're doing well too! For reference, take a look at Chapter 6 of Krishna's (2010) Auction Theory. Some of the end-of-chapter problems may be helpful as well. Commented Aug 1 at 15:26
• @HerrK. Excellent suggestion - thank you! Commented Aug 4 at 20:00