{ "original_four_by_four_left_preferences": { "A": [ "Y", "X", "Z", "W" ], "B": [ "X", "Z", "W", "Y" ], "C": [ "Z", "W", "Y", "X" ], "D": [ "W", "Y", "X", "Z" ] }, "original_four_by_four_right_preferences": { "X": [ "A", "C", "D", "B" ], "Y": [ "B", "D", "A", "C" ], "Z": [ "D", "B", "C", "A" ], "W": [ "C", "A", "B", "D" ] }, "left_proposing": { "matching": { "A": "Y", "B": "X", "C": "Z", "D": "W" }, "proposal_trace": [ { "proposer": "A", "receiver": "Y", "previous_hold": null, "action": "held", "holds_after": { "Y": "A" } }, { "proposer": "B", "receiver": "X", "previous_hold": null, "action": "held", "holds_after": { "X": "B", "Y": "A" } }, { "proposer": "C", "receiver": "Z", "previous_hold": null, "action": "held", "holds_after": { "X": "B", "Y": "A", "Z": "C" } }, { "proposer": "D", "receiver": "W", "previous_hold": null, "action": "held", "holds_after": { "W": "D", "X": "B", "Y": "A", "Z": "C" } } ] }, "right_proposing": { "matching_in_left_to_right_orientation": { "C": "W", "A": "X", "B": "Y", "D": "Z" }, "proposal_trace": [ { "proposer": "W", "receiver": "C", "previous_hold": null, "action": "held", "holds_after": { "C": "W" } }, { "proposer": "X", "receiver": "A", "previous_hold": null, "action": "held", "holds_after": { "A": "X", "C": "W" } }, { "proposer": "Y", "receiver": "B", "previous_hold": null, "action": "held", "holds_after": { "A": "X", "B": "Y", "C": "W" } }, { "proposer": "Z", "receiver": "D", "previous_hold": null, "action": "held", "holds_after": { "A": "X", "B": "Y", "C": "W", "D": "Z" } } ] }, "all24_bijections_and_blocking_pairs": [ { "matching": { "A": "W", "B": "X", "C": "Y", "D": "Z" }, "blocking_pairs": [ [ "A", "X" ], [ "A", "Y" ], [ "C", "W" ], [ "D", "X" ], [ "D", "Y" ] ] }, { "matching": { "A": "W", "B": "X", "C": "Z", "D": "Y" }, "blocking_pairs": [ [ "A", "X" ] ] }, { "matching": { "A": "W", "B": "Y", "C": "X", "D": "Z" }, "blocking_pairs": [ [ "A", "X" ], [ "C", "W" ] ] }, { "matching": { "A": "W", "B": "Y", "C": "Z", "D": "X" }, "blocking_pairs": [ [ "A", "X" ], [ "B", "Z" ] ] }, { "matching": { "A": "W", "B": "Z", "C": "X", "D": "Y" }, "blocking_pairs": [ [ "A", "X" ], [ "C", "W" ] ] }, { "matching": { "A": "W", "B": "Z", "C": "Y", "D": "X" }, "blocking_pairs": [ [ "A", "X" ], [ "A", "Y" ], [ "C", "W" ], [ "D", "Y" ] ] }, { "matching": { "A": "X", "B": "W", "C": "Y", "D": "Z" }, "blocking_pairs": [ [ "A", "Y" ], [ "C", "W" ], [ "D", "Y" ] ] }, { "matching": { "A": "X", "B": "W", "C": "Z", "D": "Y" }, "blocking_pairs": [ [ "B", "Z" ] ] }, { "matching": { "A": "X", "B": "Y", "C": "W", "D": "Z" }, "blocking_pairs": [] }, { "matching": { "A": "X", "B": "Y", "C": "Z", "D": "W" }, "blocking_pairs": [ [ "B", "W" ], [ "B", "Z" ] ] }, { "matching": { "A": "X", "B": "Z", "C": "W", "D": "Y" }, "blocking_pairs": [] }, { "matching": { "A": "X", "B": "Z", "C": "Y", "D": "W" }, "blocking_pairs": [ [ "A", "Y" ], [ "C", "W" ] ] }, { "matching": { "A": "Y", "B": "W", "C": "X", "D": "Z" }, "blocking_pairs": [ [ "C", "W" ], [ "D", "Y" ] ] }, { "matching": { "A": "Y", "B": "W", "C": "Z", "D": "X" }, "blocking_pairs": [ [ "B", "Z" ], [ "D", "Y" ] ] }, { "matching": { "A": "Y", "B": "X", "C": "W", "D": "Z" }, "blocking_pairs": [ [ "D", "X" ], [ "D", "Y" ] ] }, { "matching": { "A": "Y", "B": "X", "C": "Z", "D": "W" }, "blocking_pairs": [] }, { "matching": { "A": "Y", "B": "Z", "C": "W", "D": "X" }, "blocking_pairs": [ [ "D", "Y" ] ] }, { "matching": { "A": "Y", "B": "Z", "C": "X", "D": "W" }, "blocking_pairs": [ [ "C", "W" ] ] }, { "matching": { "A": "Z", "B": "W", "C": "X", "D": "Y" }, "blocking_pairs": [ [ "A", "X" ], [ "B", "Z" ], [ "C", "W" ], [ "C", "Z" ] ] }, { "matching": { "A": "Z", "B": "W", "C": "Y", "D": "X" }, "blocking_pairs": [ [ "A", "X" ], [ "A", "Y" ], [ "B", "Z" ], [ "C", "W" ], [ "C", "Z" ], [ "D", "Y" ] ] }, { "matching": { "A": "Z", "B": "X", "C": "W", "D": "Y" }, "blocking_pairs": [ [ "A", "X" ], [ "C", "Z" ] ] }, { "matching": { "A": "Z", "B": "X", "C": "Y", "D": "W" }, "blocking_pairs": [ [ "A", "X" ], [ "A", "Y" ], [ "C", "W" ], [ "C", "Z" ] ] }, { "matching": { "A": "Z", "B": "Y", "C": "W", "D": "X" }, "blocking_pairs": [ [ "A", "X" ], [ "B", "Z" ], [ "C", "Z" ] ] }, { "matching": { "A": "Z", "B": "Y", "C": "X", "D": "W" }, "blocking_pairs": [ [ "A", "X" ], [ "B", "W" ], [ "B", "Z" ], [ "C", "W" ], [ "C", "Z" ] ] } ], "stable_matchings": [ { "A": "X", "B": "Y", "C": "W", "D": "Z" }, { "A": "X", "B": "Z", "C": "W", "D": "Y" }, { "A": "Y", "B": "X", "C": "Z", "D": "W" } ], "stable_count": 3, "proposer_order24_checks_same_left_result": 24, "all16_two_by_two_preference_profiles_both_directions": 16, "additional_authored_replacement_and_rejection_witness": { "left_preferences": { "A": [ "Y", "X", "Z", "W" ], "B": [ "Y", "X", "Z", "W" ], "C": [ "Y", "Z", "W", "X" ], "D": [ "W", "Y", "X", "Z" ] }, "right_preferences": { "X": [ "A", "C", "D", "B" ], "Y": [ "B", "D", "A", "C" ], "Z": [ "D", "B", "C", "A" ], "W": [ "C", "A", "B", "D" ] }, "matching": { "B": "Y", "D": "W", "A": "X", "C": "Z" }, "proposal_trace": [ { "proposer": "A", "receiver": "Y", "previous_hold": null, "action": "held", "holds_after": { "Y": "A" } }, { "proposer": "B", "receiver": "Y", "previous_hold": "A", "action": "replaces_previous_hold", "holds_after": { "Y": "B" } }, { "proposer": "C", "receiver": "Y", "previous_hold": "B", "action": "rejected", "holds_after": { "Y": "B" } }, { "proposer": "D", "receiver": "W", "previous_hold": null, "action": "held", "holds_after": { "W": "D", "Y": "B" } }, { "proposer": "A", "receiver": "X", "previous_hold": null, "action": "held", "holds_after": { "W": "D", "X": "A", "Y": "B" } }, { "proposer": "C", "receiver": "Z", "previous_hold": null, "action": "held", "holds_after": { "W": "D", "X": "A", "Y": "B", "Z": "C" } } ] }, "invalid_preferences_rejected": [ "incomplete", "duplicate", "unequal", "unknown" ], "scope": "strict complete equal fictional groups, original4x4 not renamed sourceExample1; stability is no jointly preferred blocking pair, not minimum cost, fairness, consent or real allocation; validation is local only" }