Rigidity in Mechanism Design and its Applications

We introduce the notion of rigidity in auction design and use it to analyze some fundamental aspects of mechanism design. We focus on single-item auctions where the values of the bidders are drawn from some (possibly correlated) distribution \(\mathcal F\). Let \(f\) be the allocation function of an optimal mechanism for \(\mathcal F\). Informally, \(S\) is (linearly) rigid in \(\mathcal F\) if for every mechanism \(M'\) with an allocation function \(f'\) where \(f\) and \(f'\) agree on the allocation of at most \(x\)-fraction of the instances of \(S\), the expected revenue of \(M'\) is at most an \(x\) fraction of the optimal revenue. We use rigidity to explain the singular success of Cremer and McLean's auction. Recall that the revenue of Cremer and McLean's auction is the optimal welfare if the distribution obeys a certain ``full rank'' condition, but no analogous constructions are known if this condition does not hold. Note that the Kolmogorov complexity of the allocation function of Cremer and McLean's auction is logarithmic, whereas we use rigidity to show that for some distributions that do not obey the full rank condition, the Kolmogorov complexity of the allocation function of every mechanism that provides a constant approximation is almost linear. We further investigate rigidity assuming different notions of individual rationality. Assuming ex-post individual rationality, if there is a rigid set, the structure of the optimal mechanism is simple: the player with the highest value ``usually'' wins the item and contributes most of the revenue. In contrast, assuming interim individual rationality, there are distributions with a rigid set \(S\) where the optimal mechanism has no obvious allocation pattern (i.e., its Kolmogorov complexity is high). Our results help explain why we have little hope of developing good, simple and generic approximation mechanisms in the interim individual rationality world.