CS364B: Frontiers in Mechanism Design Lecture #8: MIR and MIDR Mechanisms

Last lecture concluded with a 1 2 -approximation algorithm for computing a welfare-maximizing allocation for bidders submodular valuations (scenario #6); the exercises give another such algorithm. How can we extend the allocation rules induced by these algorithms into DSIC mechanisms? This question turns out to be considerably harder than it looks. We won’t even offer any DSIC approximation mechanisms for scenario this lecture. Instead, this lecture and the next consider some simpler scenarios and develop new tools that yield good DSIC approximation mechanisms for them. We’ll then circle back to bidders with submodular valuations next week, wielding our expanded design toolbox. Why is the question so difficult? After all, we design some good DSIC approximation mechanisms last quarter, albeit in a single-parameter setting. A key step was to characterize the relevant design space — the allocation rules that are implementable, meaning the rules that can be coupled with a suitable payment rule to yield a DSIC mechanism. We proved Myerson’s Lemma (CS364A, Lecture #3), which includes the fact that the implementable allocation rules are precisely the monotone ones, meaning bidding higher (per unit of stuff) can only net a bidder more stuff, holding others’ bids fixed. Myerson’s Lemma reduces the problem of designing DSIC approximation mechanisms to a problem we could get our head around, namely designing an approximation algorithm that induces a monotone allocation rule. In CS364A (Lecture #4) we used the knapsack problem as our case study: the natural greedy algorithms yield monotone allocation rules; the standard fully-polynomial time approximation schemes do not, but can be tweaked to be monotone. For the knapsack problem,