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diff --git a/final/main.tex b/final/main.tex index 395f7ce..1ecfbca 100644 --- a/final/main.tex +++ b/final/main.tex @@ -147,19 +147,26 @@ $\hat{x}$ and whose revenue is a constant approximation to $\Rev(\hat{x},F)$?} \paragraph{Single-item case.} To make things more concrete, let us first look at the single-item case, which -has already been studied extensively and is well-understood. In this setting, $\hat{x}$ is a real number and the -function $\Rev(\hat{x},F)$ defined above is obtained by maximizing the revenue curve $R(\hat{x})$ subject to the allocation constraint, and turns out -to be a very useful object to understand the revenue maximizing multiple-agent -single-item auction (see \citep{hartline}, Chapter 3). +has already been studied extensively and is well understood. In this setting, +$\hat{x}$ is a real number and the function $\Rev(\hat{x},F)$ defined above is +obtained by maximizing the revenue curve $R(x)$ subject to the allocation +constraint $x\leq \hat{x}$, and turns out to be a very useful object to +understand the revenue maximizing multiple-agent single-item auction (see +\citep{hartline}, Chapter 3). In particular, if the type of the agent (her value) is drawn from a regular -distribution, the optimal mechanism which serves the agent with ex-ante -allocation probability $\hat{x}$ has revenue $\Rev(\hat{x},F)$, given by solving \begin{align*} -\max_{p} p(1-F(p)) \\ -\text{subject to }& 1 - F(p) \leq \hat{x} +distribution, the revenue curve $R(x)$ is equal to the posted price revenue +curve $P(x) = xF^{-1}(1-x)$ and the optimal mechanism which serves the agent +with ex-ante allocation probability at most $\hat{x}$ has revenue $\Rev(\hat{x},F)$, +given by solving +\begin{align*} + \begin{split} + \max_{x} & \;xF^{-1}(1-x) \\ +\text{subject to }& \; x \leq \hat{x} + \end{split} \end{align*} - -which is a concave function. +which is a convex optimization program since $P(x) = xF^{-1}(1-x)$ is concave +for regular distributions. \paragraph{Multiple-item case.} The multiple-agent multiple-item setting is not fully understood yet, but the revenue function $\Rev(\hat{x},F)$ defined above still |