= Shapley limit in a buyer-heavy exchange market
{c}
{title2=$s_k\to1,\quad b_k\to0$}
For $k$ unit sellers and $3k$ unit buyers with unit surplus per trade, a coalition's value is the smaller of its seller and buyer counts. Give each player an independent uniform arrival time. Conditional on a specified seller arriving at time $t$, the previous buyers and other sellers have independent binomial counts $B\sim\operatorname{Bin}(3k,t)$ and $S\sim\operatorname{Bin}(k-1,t)$. The seller contributes one exactly when $B>S$. For fixed positive $t$ this probability tends to one; a Chebyshev bound and a split near $t=0$ also give an explicit uniform integral estimate. Averaging gives seller <Shapley value> tending to one. Efficiency and role symmetry give $s_k+3b_k=1$, hence the buyer value tends to zero.
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