We propose a complete ranking of strategy-proof mechanisms in terms of the contingent reasoning they require agents to engage in to recognize their dominant strategy. Our rankings are consistent with the coarser ones implied by the solution concepts of (strong) obvious strategy-proofness (Li, 2017b, Pycia and Troyan, 2023b). The added flexibility of our approach allows a designer to balance a mechanism’s simplicity with other objectives. Our measure characterizes the Ausubel (2004) auction as the simplest way to implement the VCG outcome in multi-unit allocation problems with transfers, and provides novel rankings of mechanisms that implement stable outcomes in matching problems. Finally, we characterize minimally complex mechanisms for a range of settings, and formalize the intuition that some mechanisms are as simple as if they were (strongly) obviously strategy-proof. We explain how this extension can be valuable for high-stakes applications such as the FCC incentive auction. 

We show that achieving dominant strategy incentive compatibility often requires a designer to choose a mechanism of severely limited transparency. However, experiments and theoretical arguments suggest that, sometimes, increasing transparency can improve reliability of a mechanism's predictions---even if it breaks the dominant strategy property. To help resolve this dilemma, we define as-if dominant strategy mechanisms: (i) Each agent has at least one undominated strategy that becomes dominant if the others were restricted to behave as if the mechanism was static, and (ii) all combinations of such strategies are ex-post equilibria. To behavioral agents who neglect that others may condition their behavior in sophisticated ways, the incentives of these mechanisms resemble those of a dominant strategy one.  Our framework rationalizes the auction format chosen by prominent online platforms---such as eBay. It also provides a unified explanation for experimental evidence in various settings.  Further, mechanisms satisfying a refinement of as-if dominance are also weak dominance solvable.  

An auction equilibrium is weakly credible if no deviation from the auction rules is such that (i) the auctioneer always profits from it—sometimes strictly, and (ii) no bidder alone can detect it. No static multiunit auction has an efficient and weakly credible equilibrium in which losers pay nothing. No auction for more than two units has an efficient and weakly credible equilibrium in dominant strategies. An adapted version of the ascending auction proposed by Ausubel (2004) has an efficient and weakly credible ex-post equilibrium, regardless of how many units are for sale.

We study exploding offers by considering the strategic interaction between a low-tier firm and a set of workers within a large job market. Each worker has a private value for the firm and may receive offers from preferred top-tier firms according to an exogenous stochastic process. We show that more risk averse workers receive offers with shorter deadlines, and that there is a level of risk aversion beyond which a worker only receives offers that expire as soon as possible---independently of what all other firms are doing. If workers are sufficiently risk averse, the workers' expected welfare is maximized if and only if exploding offers are banned. Finally, any minimal offer length that does not ban exploding offers may lead to workers falling through the cracks. All results are robust to a range of sequentially-rational strategies for the workers. Our predictions match existing evidence and have implications for policies regulating exploding offers.