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In a general interdependent preference environment, we characterize when two payoﬀ types can be distinguished by their rationalizable strategic choices without any prior knowledge of their beliefs and higher order beliefs. We show that two types are strategically distinguishable if and only if they satisfy a separability condition. The separability condition for each agent essentially requires that there is not too much interdependence in preferences across agents. A social choice function — mapping payoﬀ type proﬁles to outcomes — can be robustly virtually implemented if there exists a mechanism such that every equilibrium on every type space achieves an outcome arbitrarily close to the social choice function: this deﬁnition is equivalent to requiring virtual implementation in iterated deletion of strategies that are strictly dominated for all beliefs. The social choice function is robustly measurable if strategically indistinguishable types receive the same allocation. We show that ex post incentive compatibility and robust measurability are necessary and suﬀicient for robust virtual implementation.
Bergemann, Dirk and Morris, Stephen, "Strategic Distinguishability and Robust Virtual Implementation" (2007). Cowles Foundation Discussion Papers. 1904.