Buy-Many Mechanisms: What Are They and Why Should You Care?

Multi-item mechanisms can be very complex, offering many different bundles to the buyer that could even be randomized. Such complexity is thought to be necessary as the revenue gaps between randomized and deterministic mechanisms, or deterministic and simple mechanisms are huge even for additive valuations. Furthermore, the optimal revenue displays strange properties such as noncontinuity: changing valuations by tiny multiplicative amounts can change the optimal revenue by an arbitrarily large multiplicative factor. Our work shows that these strange properties do not apply to most natural situations as they require that the mechanism overcharges the buyer for a bundle while selling individual items at much lower prices. In such cases, the buyer would prefer to break his order into smaller pieces paying a much lower price overall. We advocate studying a new revenue benchmark, namely the optimal revenue achievable by "buy-many" mechanisms, that is much better behaved. In a buy-many mechanism, the buyer is allowed to interact with the mechanism multiple times instead of just once. We show that the optimal buy-many revenue for any n item setting is at most O(log n) times the revenue achievable by item pricing. Furthermore, a mechanism of finite menu-size (n/epsilon) 2(O(n)) suffices to achieve (1 + epsilon)-approximation to the optimal buy-many revenue. Both these results are tight in a very strong sense, as any family of mechanisms with description complexity sub-doubly-exponential in n cannot achieve better than logarithmic approximation in revenue. In contrast, for buy-one mechanisms, no simple mechanism of finite menu-size can get a finite-approximation in revenue. Moreover, the revenue of buy-one mechanisms can be extremely sensitive to multiplicative changes in values, while as we show optimal buy-many mechanisms satisfy revenue continuity.