A CHARACTERIZATION OF OSP MECHANISMS Introduction In this section, we delve intothe characterization of OSP(Ordinal Scoring Problem)mechanisms for binaryallocation problems. Weassume that the agents havefinite domains. The goal is tounderstand the structuralproperties and mechanismsgoverning OSP. We'll discusskey observations, definitions,and insights that lead to abetter understanding of thesemechanisms. Basic Properties of theOSP-Graph - OSP mechanisms deal with binary allocationproblems, where outcomes are binary (0or 1). - We observe that the OSP-graph has edgeswith different weightsbased on the agents'preferences: - Positive-weight edges when an agent prefers 0and 1 (w(a, b) = +ai). - Negative-weight edges when an agent prefers 1and 0 (w(a, b) = ai). - These edges have strict inequalities due to thenature of agent preferences. Types and Structure of theImplementation Tree - We introduce the concepts of "0-always,""1-always," and "unclear"types for agents. - An agent's type is considered "sel-always"if they consistently prefereither 0 (sel = 0) or 1 (sel= 1).
- Types that don't fall into either category aretermed "unclear." Structure of AdmissibleQueries - Admissible queries in OSPmechanisms are characterized by two conditions: 1. Homogeneous parts:Either all agents in apart consistently prefer0 or 1. 2. Agent revealability:When certain conditions are met, anagent's type becomes"revealable." Structure of Negative-Weight Cycles - Negative-weight cycles in OSP mechanisms have aspecial structure definedby four profiles. - These profiles involve preferences of agents for0 and 1. - The absence of certain edges in the OSP-graphis a key characteristic ofthese cycles. Equivalence to WeakInterleaving - We show that, without loss of generality, we canfocus on OSP mechanisms where agents are asked eithertop or bottom queries,except when agent typesbecome revealable. - The notion of "extremal mechanisms" and "weakinterleaving" is introduced. - We prove that an OSP mechanism exists if andonly if an extremalmechanism with weakinterleaving exists.
Two-Way Greedy Algorithms - We emphasize that the characterization appliesto both costs andvaluations. - Monotone and antimonotone functions,in terms of type, aredefined. - We discuss the connection between OSP mechanisms andadaptive priority algorithms. - New upper bounds on the approximation guarantee of OSPmechanisms are presented, independentof domain size or otherassumptions. This understanding of OSPmechanisms and theirstructural properties isessential for designing,analyzing, and optimizing allocation mechanisms invarious scenarios.