The Simplicity of Optimal Dynamic Mechanisms

Abstract

A fundamental economic question is that of designing revenue-maximizing mechanisms in dynamic environments. This paper considers a simple yet compelling market model to tackle this question, where forward-looking buyers arrive at the market over discrete time periods, and a monopolistic seller is endowed with a limited supply of a single good. In the case of i.i.d. and regular valuations for the buyers, Board and Skrzypacz (2016) characterized the optimal mechanism and proved the optimality of posted prices in the continuous-time limit. Our main result considers the limit case of a continuum of buyers, establishing that for arbitrary independent buyers' valuations, posted prices and capacity rationing can implement the optimal anonymous mechanism. Our result departs from the literature in three ways: It does not make any regularity assumptions, it considers the case of general, not necessarily i.i.d., arrivals, and finally, not only posted prices but also capacity rationing takes part in the optimal mechanism. Additionally, if supply is unlimited, we show that the rationing effect vanishes, and the optimal mechanism can be implemented using posted prices only, à la Board (2008).

Figures (3)

• Figure 1: The pictures illustrate the examples where we describe when rationing is useful from a seller's perspective. In the two pictures on the left we compare the situation in a market with discrete buyers of mass one to a market with many non-atomic buyers of the same mass. While in the large market in picture 1 rationing is of no help, this changes with atomic buyers as in picture 2, see also \ref{['ex:rationing_usefull_discrete']}. A similar situation can be observed when considering a large market with finite inventory, here of $\frac{3}{2}$. While without rationing the prices marked in red in picture 3 are optimal, with rationing the seller can do better as depicted in picture 4, see also \ref{['ex:rationing_usefull_inventory']}.
• Figure 2: The picture has the goal to visualize \ref{['eq:subdiff']}. On the left we see a schematic picture, where the blue curve is the maximum over several curves $g_i$. At each point the subdifferential of each such $g_i$ is a subset of the subdifferential of the maximum. This can be seen at $v=\tfrac{1}{2}$, where the derivative is not defined and subdifferential of the blue curve (marked in blue) contains the subdifferential of the red curve (marked in red). On the right the same situation is depicted but for a concrete game. In period 1 a good is sold for a fixed price of $\frac{1}{2}$. In period 2, the same good is sold for a price of $0$ but only assigned with probability $\frac{1}{2}$. For $\delta_t=1$ for all $t \in [T]$, this results in the utility curves $U_1$ and $U_2$, where $U_1$ is not differentiable at $v=\frac{1}{2}$. The derivative of $U_2$ which is $\frac{1}{2}$ is in the subdifferential of $U_1$ (which is all tangents with slope in $[\frac{1}{2},1]$ and marked in blue). Since $r_1(\frac{1}{2})=0$, \ref{['eq:subdiff']} reduces to $\frac{1}{2} \in [\frac{1}{2},1]$.
• Figure 3: The marked interval of time slots is $\bar{T}$, the dots depict the discount factors and the region marks the shape of region $I$.

Theorems & Definitions (22)

• Theorem 1
• Theorem 2
• Example 1
• Example 2
• Example 3
• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3