We revisit the connection between bargaining and equilibrium in exchange economies, and study its algorithmic implications. We consider bargaining outcomes to be allocations that cannot be blocked (i.e., profitably re-traded) by coalitions of small size and show that these allocations must be approximate Walrasian equilibria. Our results imply that deciding whether an allocation is approximately Walrasian can be done in polynomial time, even in economies for which finding an equilibrium is known to be computationally
hard.

Our work contributes to the literature on the Edgeworth conjecture, which seeks to  understand the scope of the price-taking assumption. We also introduce a new algorithmic question: given an allocation, do there exist prices for which the allocation is an approximate Walrasian equilibrium? We propose a polynomial time algorithm for this question, even for instances where hardness results for finding a Walrasian equilibrium are known.

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