Directed search with privately informed buyers and capacities

Question

My question seems to be a basic one and there should be a rather well-known reference in the (IO or labor) search literature. I will upvote (and comment on) any relevant answer and I will accept an answer pointing to a reference addressing exactly the model below.

In the simplest case, consider the following model:

There are 2 sellers ($j \in \{1,2\}$) and each seller offers a single good. In stage 1, both sellers set an individual price $p_j$.

There are 2 buyers ($i \in \{1,2\}$) who are privately informed about their valuation $v_i$ which is an iid draw from some distribution $F$. In stage 2, buyers observe the prices and decide which seller (if any) they want to visit.

Buyers have single-unit demand. Buyer $i$'s payoff is $v_i - p_j$ if they got a good and zero if they got rationed or did not visit any seller. Sellers do not have opportunity costs (and cannot produce additional goods), so their payoff is $p_j$ if they trade.

A guess for a reasonable equilibrium candidate: In equilibrium, both sellers post the same price $p_1=p_2=p$, buyers with $v_i<p$ abstain from buying and high-value buyers ($v_i\geq p$) randomize equally over both sellers.

To evaluate deviations, it matters what happens when. say, $p_1 > p_2$. Of course this depends on the rationing function, i.e., what happens when two buyers show up at the same seller. Fixing a certain form of rationing (random rationing, efficient rationing, introducing an underlying model in which sellers have additional preferences over customers, etc), one can express the expected utility from visiting each seller in a reduced form and solve. Now I guess my question is: How is this done in the literature (which I find interesting, but don't know much of)? I am happy about references and suggestions and I will provide some myself.

Answer 1

I am aware of classic models such as Kreps & Scheinkman (1983). I am also aware of Burdett, Shi & Wright (2001), but note that their buyers are ex-post symmetric as both of them have value $v_i=v=1$.

I am also aware of the literature on competing auctioneers, e.g., Peters & Severinov (1997) and Virag (2010). The buyers' equilibrium selection strategy in such models is quite cool: If sellers post second-price auctions with reserve prices $r_1>r_2$, buyers of value $v<r_2$ abstain from buying, buyers of value $v \in [r_2,x]$ visit low-price seller 2 with certainty and buyers of value $v>x$ visit each seller with probability $1/2$. One can solve for this cutoff value $x>r_1$. Under a condition on the lower bound of $F$, there is indeed an equilibrium in which both sellers set the same reserve price (which is below the lowest possible buyer value). Knowing this, one can write down an underlying wasteful costly rationing process that essentially replicates the buyers' selection equilibrium above. However, I find this a little far fetched.

Any other approaches? As I said, this is not my literature - I just find the problem too basic to be neglected.