Preference Manipulation Leads To The Uniform Rule

In the division problem with single-peaked preferences, it is well known that the uniform rule is robust to strategic manipulation. Under efficiency and symmetry, it is also the unique strategy-proof rule (Sprumont, 1991; Ching, 1994). We conversely analyze the consequences of strategic manipulation for some families of rules satisfying natural requirements. Given a rule, we interpret its associated direct revelation game as a manipulation game, and we characterize its equilibrium allocations. Our results are unequivocal: for any rule, at each preference profile there is a unique (strong) Nash equilibrium allocation and it is the uniform allocation for the profile. In other words, attempts to manipulate lead to the uniform rule.