Serial dictatorship

An allocation procedure using a fixed priority order: each participant in turn selects one remaining object, which is permanently assigned to them. For participants whose utilities depend only on their own assigned objects, strict rankings give a unique best available choice and the resulting allocation is Pareto optimal. If a proposed Pareto improvement changes an assignment, its earliest changed participant loses their unique best available object. With ties or allocation externalities, that argument requires additional assumptions.

Assign distinct rooms along a hallway to participants. A participant's utility is their own room value plus a positive bonus if their designated friend receives a physically adjacent room; participants without friends receive only their room value. When friendships form disjoint pairs, consecutive choosing slots allow each pair to settle its own adjacency before later participants act. Consecutive slots in serial dictatorship do not themselves require adjacent allocated rooms.

Suppose each participant has at most one mutual friend, each friendship pair chooses consecutively, room values lie in $[0,1]$ and are distinct for each participant, and each friendship bonus exceeds $1$. Then every subgame perfect equilibrium outcome of serial dictatorship is Pareto optimal.

For a friend pair, if any remaining adjacent room pair exists, the leader chooses their uniquely best room having an available neighbor; the follower chooses their uniquely best available neighbor. Adjacency dominates every possible difference in own-room value for both friends. If no adjacent room pair remains, neither friend can obtain a bonus and each chooses their unique best available room. A participant without a friend also chooses their unique best available room.

For the proof, compare a purported Pareto improvement block by block, where a block is a single participant without a friend or one consecutive friendship pair. Earlier fixed blocks leave the same rooms available. A block receiving adjacency cannot lose it in a weak improvement. Its leader must retain the uniquely best eligible room, and its follower must retain the uniquely best neighboring room. Without available adjacency, both rooms are fixed by strict own-room rankings. Mathematical induction fixes every assignment, excluding a strict improvement.