Online Supplement to “Resource Allocation via Message Passing”
نویسندگان
چکیده
Proof. Let L be a Lipschitz constant that applies to all utility functions. Suppose each message in the set V is Lipschitz continuous with Lipschitz constant L. Consider the message from an activity a to a resource r ∈ R(a). Define X a\r , ∏r∈R(a)\r Xr to be the space of consumption bundles for activity a, excluding resource r. Without loss of generality, assume that (FV )a→r(xar) ≥ (FV )a→r(xar). Then, for some z′ ∈ X a\r, (FV )a→r(xar)− (FV )a→r(xar) = ua(xar, z′) + ∑
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