buying_to_bundle_asymptotic_surrogate_optimality

Theorem 4.6 of Buying to Bundle: Optimal Sourcing from Monopolistic Sellers (asymptotic optimality of the surrogate-profit mechanism in large markets). Buyers value seller $i$'s item at $\mu_i+\sigma Z_i$, where $Z$ is a strictly regular, sub-exponential $SE(\gamma^2,\xi)$ random variable with mean zero and unit variance, and the private qualities $\mu_i$ are i.i.d. with positive density on $[\mu_L,\mu_H]$. Let $\bar{x}$ be an incentive-compatible allocation rule (measurable, monotone non-increasing into $[0,1]$) maximizing the expected surrogate profit $\varpi(x)=N\,E[x(\mu)(\mu-\varphi(\mu))]$, where $\varphi$ is the Myerson virtual cost — by Theorem 4.4 of the paper the maximizer is a threshold rule. If $\varpi(\bar{x})>0$, then the profit-maximizing IC mechanism improves on $\bar{x}$ by at most a factor $1+O(N^{-1/3})$: there exist $C>0$ and $N_0$ such that for every market size $N\ge N_0$, the expected profit satisfies $\Pi(\bar{x})>0$ and every IC allocation rule $x$ has

Hence $\Pi(x^*)/\Pi(\bar{x})\le 1+O(N^{-1/3})$ for any profit-maximizing rule $x^*$: bundling via the simple surrogate threshold mechanism is asymptotically profit-optimal.