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For a transferable utility game on player set $N$, its characteristic function of a coalitional game is $v:2^N\to \mathbb{R}$, where $v(S)$ is the maximum transferable surplus attainable by coalition $S$ and $v(\varnothing )=0$. An imputation is a payoff vector satisfying efficiency ${\sum }_{i\in N}{x}_i=v(N)$ and individual rationality $x_i\ge v(\{i\})$. The core of a cooperative game consists of the efficient payoff vectors satisfyingThese conditions prevent any coalition from blocking the allocation; singleton constraints include individual rationality.

The Shapley value payoff isEquivalently, it is the expected marginal contribution of player $i$ when the players arrive in uniformly random order. The factorial weight counts the orders with exactly $S$ preceding $i$. Summing marginal contributions telescopes along each order, proving Shapley efficiency.

In the four-player market a coalition creates one unit of surplus exactly when it contains the seller and at least one buyer. ThusAll singleton values are zero and $v(N)=1$. Core constraints for the pairs $\{1,j\}$ require $x_1+x_j\ge 1$ for every buyer $j$. Since all payoffs are nonnegative and sum to one, the other two buyers' payoffs must be zero for each such pair. Therefore every buyer receives zero and the seller receives one. This vector satisfies all coalition inequalities, soThe seller's marginal contribution is one unless he arrives before all three buyers, which happens with probability $1/4$. A given buyer contributes one exactly when the seller is first and that buyer is second, with probability $(1/4)(1/3)=1/12$. HenceIn particular the Shapley allocation need not lie in this market's core.

For the market with $k$ sellers and $3k$ buyers, a coalition containing $s$ sellers and $b$ buyers can make precisely $\min (s,b)$ trades, each producing unit surplus. Thus $v(S)=\min (s(S),b(S))$ and $v(N)=k$. Let $s_k$ and $b_k$ denote the common Shapley payoffs within the two roles. Role symmetry and the telescoping efficiency identity giveIt remains to prove $s_k\to 1$, rather than infer it only from competitive intuition.

Generate a uniformly random arrival order by giving every player an independent uniform time in $[0,1]$. Conditional on a specified seller's arrival time $t$, the number $B$ of preceding buyers and number $S$ of preceding other sellers are independent, withThe seller adds a trade precisely when $B>S$. ConsequentlyFor $t>0$, the difference $D=B-S$ has mean $(2k+1)t$ and variance $(4k-1)t(1-t)\le 4kt$. By Chebyshev's inequality,Split the integral at $\delta =k^{-1/2}$. Using the trivial bound one on the first piece givesThis proves the Shapley limit in a buyer-heavy exchange market. Since $b_k=(1-s_k)/3$, we concludeThe proof accounts for sellers arriving very early, where a fixed-time concentration argument alone would not be uniform.