Across many billion-dollar industries, organizations pool proprietary data for mutual benefit without selling it. A striking feature of these arrangements is that participation is rarely open: access is gated on contribution, e.g., credit bureaus codify this as explicit "Principles of Reciprocity" — a subscriber receives the level of data it contributes and is expected to contribute all it has; fraud consortia describe themselves as "give-to-get". This motivates reciprocity as a first-class design goal: each participant should receive value from the exchange at least equal to the value its own data contributes to others, where contributions are measured by a standard credit-sharing rule such as the Shapley value. Reciprocity alone, however, is vacuous: the empty exchange in which no one shares anything is trivially reciprocal, yet useless. We therefore need a guarantee that the exchange is also efficient — that participants do not leave mutually beneficial trades on the table. We capture this through stability: no group of participants should be able to break away and arrange a private exchange among themselves that every member strictly prefers. Stability, too, is trivial in isolation — the exchange in which everyone shares everything is perfectly stable, since no breakaway group can do better — but it is generally not reciprocal. The two requirements thus pull in opposite directions and satisfying them together is far from obvious. Our main result shows that it is always possible: a data exchange that is simultaneously reciprocal and stable exists for an extremely broad class of participant utilities (any monotone, continuous valuation of received data) and for any credit-sharing rule meeting mild, standard conditions —  thereby, covering several settings in which such exchanges arise in practice.

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