Real-Time Bidding (RTB) has become a relevant paradigm in display advertising. It mimics stock exchanges and utilizes computer algorithms to buy and sell ads in real-time automatically. Imagine that you have to participate in N ≫ 1 of those online ad auctions with a limited bidding budget. The task is to create such a bidding strategy that you can win some of them, and that the placed ads generate at least Nc clicks. That should be done by spending as little money as possible. In the following, we will look at a possible solution to this problem.

实时出价(RTB)已成为展示广告的一种相关范例。 它模仿证券交易所，并利用计算机算法自动实时地买卖广告。 想象一下，您必须在有限的出价预算下参加N≫ 1项在线广告拍卖。 我们的任务是制定一种出价策略，以使您可以赢得其中的一些，并且所放置的广告至少产生Nc次点击。 这应该通过花尽可能少的钱来完成。 在下文中，我们将研究该问题的可能解决方案。

1.实时出价生态系统 (1. Real-Time Bidding ecosystem)

A brief description of the RTB ecosystem is given in the figure above. When a user visits an ad-supported site each ad placement will trigger an auction. Bid requests will be sent via the ad exchange to the different bidding agents. Upon receiving a bid request, every bidding agent calculates a bid that is sent together with an ad to the Ad exchange. Finally, the winner’s ad will be shown to the visitor along with the regular content of the website. The whole process should be completed within a fraction of the second. A more detailed introduction to RTB could be found in [1,2].

上图中对RTB生态系统进行了简要说明。 当用户访问广告支持的网站时，每个广告展示位置都会触发拍卖。 出价请求将通过广告交易平台发送给其他出价代理。 收到出价请求后，每个出价代理都会计算与广告一起发送到Ad Exchange的出价。 最后，优胜者的广告将与网站的常规内容一起显示给访问者。 整个过程应该在一秒钟之内完成。 在[1,2]中可以找到有关RTB的更详细介绍。

2.问题描述 (2. Problem description)

For simplicity, we will consider that we only have to create a strategy for a particular ad (for example, white sneakers from a particular brand) but the approach can be easily generalized to multiple ads, each one of them having a different budget and target. The ad exchange generates a large number of bid requests which are processed by many bidding agents, each one of them having the opportunity to make a bid. The user and publisher data contained in every bid request could be used to predict the probability distribution function of the winning bid price s, and the probability that the user will click on the displayed ad. For every auction n ∈ {1,…N} they will be denoted as:

为简单起见，我们认为我们只需要为特定广告创建策略(例如，特定品牌的白色运动鞋)，但是这种方法可以轻松地推广到多个广告，每个广告都有不同的预算和目标。 广告交换产生大量的出价请求，这些出价请求由许多投标代理处理，每个投标代理都有机会进行投标。 每个投标请求中包含的用户和发布者数据可用于预测中标价格s的概率分布函数，以及用户单击显示的广告的概率。 对于每次拍卖n∈{1，…N}，它们将表示为：

For every auction n, we will place a bid price xn. The probability to win is then given by:

对于每次拍卖n，我们都会下一个竞标价格xn 。 然后，获胜的概率为：

The integral from 0 to xn takes into account all cases where the winning bid price generated by taking into account all other participants except us is smaller than our bid price xn. Because of the probabilistic nature of our assumptions, we can not guarantee which auction we are going to win or if a user will click on the displayed ad. To describe these random events we will use the following Bernoulli random variables:

从0到xn的积分考虑了所有情况，其中通过考虑除我们以外的所有其他参与者而产生的中标价格小于我们的出价xn 。 由于我们的假设具有概率性质，因此我们无法保证我们将赢得哪场拍卖，也无法保证用户是否会点击所展示的广告。 为了描述这些随机事件，我们将使用以下伯努利随机变量 ：

where Cn describes the user ad click events (click: Cn=1, no click: Cn=0) and Wn|xn — the event of winning the n-th auction by placing the bid price xn (win: Wn|xn=1, loss: Wn|xn=0). The probability for each one of these events to occur is given by:

其中Cn描述用户广告点击事件(点击： Cn = 1 ，没有点击： Cn = 0 )，并且Wn | xn-通过放置出价xn赢得第n次拍卖的事件(获胜： Wn | xn = 1 ，损失： Wn | xn = 0) 。 这些事件中每一个发生的概率由下式给出：

The total number user clicks on our ad obtained by placing the bids {xn|n = 1, 2 . . . N} is given by:

用户通过放置出价{xn | n = 1，2所获得的广告点击次数。 。 。 N}由下式给出：

This is a random variable, as well. For simplicity, we will look only at its expected value:

这也是一个随机变量。 为简单起见，我们将仅查看其预期值：

The amount of money spent on the auctions that we have won can be described by the following random variable:

我们赢得的竞标花费金额可以通过以下随机变量来描述：

As in the equation for the total number of click events, we will look only at the expected value of this variable:

就像点击事件总数的方程式一样，我们将仅查看此变量的期望值：

The problem of placing N bids x1, . . . xN such that the expected number of user clicks