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In settings of incomplete information, we put forward a very conservative ---indeed, purely set-theoretic--- model of the knowledge that the players may have about the types of their opponents. Yet, we prove that such knowledge can be successfully and robustly leveraged by means of a solution concept relying on very weak assumptions: in essence, via extensive-form mechanisms under “mutual” knowledge of rationality. We demonstrate the potential of our approach in auctions of a single good by 1. considering a new revenue benchmark, always lying between the highest and second-highest valuation, 2. proving that no classical mechanism can even slightly approximate it in any robust way, and 3. providing a new mechanism that perfectly and robustly achieves it, with the extra property that the good will always be sold out at the end of the auction. Our impossibility result for robustly implementing our revenue benchmark applies not only to implementation in dominant strategies, but also to any implementation ``at equilibrium", as well as to implementation in undominated strategies.

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