Competitive equilibrium can fail near equal budgets for three additive agents

In a Fisher market with indivisible goods, each agent has a budget of artificial money and a competitive equilibrium is an allocation of all the goods with prices at which every agent gets a most preferred bundle within budget. With exactly equal budgets one good and two agents already have no equilibrium. Babaioff, Nisan and Talgam-Cohen proved that two agents with additive preferences over any number of goods have a competitive equilibrium whenever their budgets are sufficiently close to equal but not equal, and asked whether this extends to more than two agents (their Open Problem 1). It does not. We give three agents with strict additive valuations over five goods such that for every 0<δ<1/17 no competitive equilibrium exists at the budgets (1-3δ,1+2δ,1+δ), nor at any budget vector near them. So every neighborhood of equal budgets contains a set of positive measure on which no equilibrium exists. Adding goods of small value, as in Segal-Halevi's Lemma 5, extends this to every number of goods m≥5. This answers the case of three additive agents and m≥5 goods that Segal-Halevi left open: the budgets with no competitive equilibrium can have positive measure. The proof gives, for each allocation of the goods, an exact Farkas (linear-programming duality) certificate that no prices support it; an independent exact linear-programming computation confirms it. This work has not yet been reviewed by experts. The proofs, the computer checks and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.