Learning to Maximize Gains From Trade in Small Markets

TL;DR

The paper investigates learning DSIC, ex-post budget-balanced double-auctions for maximizing gains from trade in small markets. It first proves a general impossibility: for arbitrarily correlated valuations across sellers and buyers, learning a near-optimal simple mechanism from finite samples is impossible. It then exploits the independence structure to derive a tight, low-dimensional characterization of GFT-optimal simple mechanisms using monotone, tight function pairs, and provides a polynomial-time algorithm to compute them for finitely supported product distributions. Extending to continuous product distributions, the authors show that an empirical-sample-based approach yields near-optimal GFT with polynomial sample complexity, via standard $oldsymbol{ ext{ε}}$-sample guarantees. Overall, the work delineates when learning to maximize GFT in small markets is feasible and provides explicit mechanism-design procedures under independence that are provably near-optimal.

Abstract

We study the problem of designing a two-sided market (double auction) to maximize the gains from trade (social welfare) under the constraints of (dominant-strategy) incentive compatibility and budget-balance. Our goal is to do so for an unknown distribution from which we are given a polynomial number of samples. Our first result is a general impossibility for the case of correlated distributions of values even between just one seller and two buyers, in contrast to the case of one seller and one buyer (bilateral trade) where this is possible. Our second result is an efficient learning algorithm for one seller and two buyers in the case of independent distributions which is based on a novel algorithm for computing optimal mechanisms for finitely supported and explicitly given independent distributions. Both results rely heavily on characterizations of (dominant-strategy) incentive compatible mechanisms that are strongly budget-balanced.

Figures (10)

• Figure 1: Example of a pair $(f_1, f_2)$ of compatible functions. Such a pair satisfies that there is no point that is strictly above $f_2(v_1)$ and strictly on the right of $f_1(v_2)$.
• Figure 2: The pair of compatible functions that is associated with the GFT-maximizing simple mechanism for the uniform distribution over $[0,1]^3$.
• Figure 3: Optimal pair of functions for the independent distributions $v_s\sim U[0,1]$, $v_1\sim U[0,1]$, $v_2\sim U[0,\frac{1}{2}]$. Note that the buyer with lower value sometimes trades, e.g., for values $(v_s,v_1,v_2)= (0, 0.4,0.3)$.
• Figure 4: Optimal pair for $v_s\sim U[0,1]$, $v_1\sim U[0,\frac{1}{2}],\frac{1}{4}U[\frac{1}{2}, \frac{3}{4}],\frac{3}{4}$, $v_2\sim U[0,\frac{1}{4}],\frac{3}{4}U[\frac{1}{2}, 1],\frac{1}{4}$. Note that the functions are rather complicated.
• Figure 5: An example of the Compatibility Restriction function $r_2^{f_1}$, the compatibility restriction of $f_1$ on $f_2$.

Theorems & Definitions (95)

• Definition 1.1
• Definition 1.2
• Theorem 1
• Lemma 1.3
• Definition 1.4
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 2.1
• Definition 2.2