We explore brokerage between traders in an online learning framework. At any
round $t$, two traders meet to exchange an asset, provided the exchange is
mutually beneficial. The broker proposes a trading price, and each trader tries
to sell their asset or buy the asset from the other party, depending on whether
the price is higher or lower than their private valuations. A trade happens if
one trader is willing to sell and the other is willing to buy at the proposed
price. Previous work provided guidance to a broker aiming at enhancing traders'
total earnings by maximizing the gain from trade, defined as the sum of the
traders' net utilities after each interaction. In contrast, we investigate how
the broker should behave to maximize the trading volume, i.e., the total number
of trades. We model the traders' valuations as an i.i.d. process with an
unknown distribution. If the traders' valuations are revealed after each
interaction (full-feedback), and the traders' valuations cumulative
distribution function (cdf) is continuous, we provide an algorithm achieving
logarithmic regret and show its optimality up to constant factors. If only
their willingness to sell or buy at the proposed price is revealed after each
interaction ($2$-bit feedback), we provide an algorithm achieving
poly-logarithmic regret when the traders' valuations cdf is Lipschitz and show
that this rate is near-optimal. We complement our results by analyzing the
implications of dropping the regularity assumptions on the unknown traders'
valuations cdf. If we drop the continuous cdf assumption, the regret rate
degrades to $\Theta(\sqrt{T})$ in the full-feedback case, where $T$ is the time
horizon. If we drop the Lipschitz cdf assumption, learning becomes impossible
in the $2$-bit feedback case.

通过在在线学习框架中的交易者之间进行经纪，我们研究了经纪的关键问题。我们探索了经纪机构如何最大限度地提高交易量，并分析了对于交易者来说最优的交易策略和遗憾率。

在线学习中的交易量最大化

Trading Volume Maximization with Online Learning

We study the price impact of order book events - limit orders, market orders
and cancelations - using the NYSE TAQ data for 50 U.S. stocks. We show that,
over short time intervals, price changes are mainly driven by the order flow
imbalance, defined as the imbalance between supply and demand at the best bid
and ask prices. Our study reveals a linear relation between order flow
imbalance and price changes, with a slope inversely proportional to the market
depth. These results are shown to be robust to seasonality effects, and stable
across time scales and across stocks. We argue that this linear price impact
model, together with a scaling argument, implies the empirically observed
"square-root" relation between price changes and trading volume. However, the
relation between price changes and trade volume is found to be noisy and less
robust than the one based on order flow imbalance.

本研究使用 NYSE TAQ 数据研究了限价单、市价单和撤单等订单簿事件对价格的影响，发现在短时间内，价格变化主要由供求平衡度驱动，其呈现出线性关系且斜率与市场深度成反比，给出了价格变化和交易量之间的平方根关系及其影响因素。