Directed search with privately informed buyers and capacities
I am aware of classic models such as Kreps &amp; Scheinkman (1983). I am also aware of Burdett, Shi &amp; 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 &amp; 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 $r_1&gt;r_2$, buyers of value $v&lt;r_2$ abstain from buying, buyers of value $v \in [r_2,x]$ visit low-price seller 2 with certainty and buyers of value $v&gt;x$ visit each seller with probability $1/2$. One can solve for this cutoff value $x&gt;r_1$. 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.