We study interim strategy-proof (ISP) Mechanisms with interdependent values: It requires that truth-telling is an interim dominant strategy for each agent, i.e., conditional on an agent’s own private information, the truth-telling maximizes her interim expected payoff for all possible strategies the other agents could use. We provide full characterizations of ISP mechanisms in two classical settings: single unit auctions and binary collective decision-makings. Our results highlight the tension between informational externalities and strategic externalities when designing ISP mechanisms.

A social choice function (SCF) is robustly implementable in rationalizable strategies (RoRat-implementable) if every rationalizable outcome on every type space agrees
with the SCF. We propose it as a way of obtaining more permissive results than robust implementation in interim equilibria (RoEq-implementation), which requires
every equilibrium outcome on every type space to agree with the SCF. We first establish that RoRat-implementation is equivalent to weak rationalizable implementation, which is defined directly on the space of payoff types. Applying this equivalence, we identify
weak robust monotonicity (weak RM) as a necessary and almost sufficient condition for RoRat-implementation. We also provide simpler incentive and preference-reversal conditions that jointly imply weak RM but are equivalent to the latter if alternatives are finite. We argue that weak RM is strictly weaker than strict robust monotonicity, which is necessary and almost sufficient for RoEq-implementation. Thus, within general
mechanisms, RoRat-implementation is more permissive than RoEq-implementation. This gap stems from the strong requirement that players have correct conjectures in equilibrium. We apply our results to single crossing aggregator environments and also provide a comprehensive discussion on the implications of RoRat-implementation.

A single unit of a good is sold to one of a group risk-neutral bidders whose privately-known values are drawn independently from an identical distribution. The seller only has limited information about the value distribution and believes that the value distribution is designed by Nature adversarially to minimize revenue. In addition, the seller knows that bidders play undominated strategies. For the two-bidder case, we construct a strong maxmin solution, consisting of a mechanism, a value distribution, and an equilibrium in undominated strategies, such that neither the seller nor Nature
can move revenue in their respective preferred directions, even if the deviator can select
the new equilibrium in undominated strategies. The mechanism and value distribution solve a family of maxmin mechanism design and minmax information design problems, regardless of how an equilibrium in undominated strategies is selected. The maxmin
mechanism is a second-price auction with a random reserve price. For arbitrary number of bidders, a second-price auction with a random reserve price remains a maxmin mechanism among a subclass of dominant-strategy mechanisms.

We study the design of optimal mechanisms when the designer is uncertain both about the information held by the agents and also about which equilibrium will be played. The guarantee of a mechanism is the minimum of the designer’s welfare across all information structures and equilibria. The potential of an information structure is the maximum welfare across all mechanisms and equilibria. We formulate a pair of linear programs that upper bound the maximum guarantee across all mechanisms and lower bound the minimum potential across all information structures. In applications to public goods, bilateral trade, and optimal auctions, we use the bounding programs to characterize guarantee-maximizing mechanisms and potential-minimizing information structures and show that the max guarantee is equal to the min potential.