Computer Science and Game Theory 5 JAN, 2012

Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit

Bundling Customers: How to Exploit Trust Among Customers to Maximize Seller Profit

We consider an auction of identical digital goods to customers whose valuations are drawn independently from known distributions. Myerson’s classic result identifies the truthful mechanism that maximizes the seller’s expected profit. Under the assumption that in small groups customers can learn each others’ valuations, we show how Myerson’s result can be improved to yield a higher payoff to the seller using a mechanism that offers groups of customers to buy bundles of items.

💡 Research Summary

The paper studies the sale of identical digital goods—goods that can be reproduced at zero marginal cost—to a population of buyers whose private valuations are drawn independently from known probability distributions. In the classic single‑buyer setting, Myerson’s (1981) optimal auction theory shows that the seller can maximize expected profit by offering each buyer a take‑it‑or‑leave‑it price derived from the buyer’s virtual value, φ(v)=v−(1−F(v))/f(v). This mechanism is dominant‑strategy incentive compatible (DSIC) and individually rational (IR), and it is provably optimal when buyers act independently and have no information about each other’s valuations.

The authors relax the independence assumption by allowing buyers to form small groups (typically of size two to four) in which members can fully observe each other’s valuations before making a purchase decision. This “trust network” reflects realistic scenarios such as families sharing streaming subscriptions, corporate teams buying software licenses, or friends pooling resources for digital content. The central insight is that, when a group can coordinate, the seller can extract more surplus by selling a bundle of items to the whole group at a single price that depends on the aggregate valuation of the group rather than on each individual’s valuation.

Mechanism design.

  1. Group formation. The seller announces a set of admissible group sizes (e.g., k=2,3,4) and a pricing rule for each size. Buyers either self‑select a group or are randomly matched by the platform.
  2. Value revelation within the group. Because members trust each other, each buyer truthfully reports his private value vi to the other members. The mechanism must ensure that no individual can gain by misreporting, even though the report is shared.
  3. Aggregate virtual value. Let V_G = Σ_{i∈G} vi be the total value of group G. The distribution of V_G, denoted F_G, is the convolution of the individual distributions. The seller computes the group virtual value φ_G(V_G)=V_G−(1−F_G(V_G))/f_G(V_G).
  4. Bundle price. The seller offers the bundle to group G only if φ_G(V_G)≥0. The price p_G is set at the threshold V_G* where φ_G(V_G*)=0, i.e., p_G = V_G* − ∫_0^{V_G*} (1−F_G(t))/f_G(t) dt. This is the exact analogue of Myerson’s optimal price, now applied to the sum of valuations.
  5. Purchase decision. If V_G ≥ V_G*, the group collectively buys the bundle; each member receives one unit of the digital good and pays p_G/k. The utility for member i is ui = vi − p_G/k, which is non‑negative by construction, satisfying IR.

Why profit increases.
Because the bundle price is based on the sum of valuations, a high‑valued member can “subsidize” a low‑valued member, allowing the seller to charge a price that would be unattainable in a purely individual setting. The expected revenue from a group of size k is E