In this lecture, I will show different examples of different models and how asking a different question or a wrong question might actually get you to the wrong recommendation or prescription. This is why in prescriptive analytics it's very important to understand how the actions actually affect the goal that we're trying to maximize. So what we've covered so far is what are goals, what are actions, and what are models, okay? And we've seen how several examples in which changing the model, but using the same data, impacts the recommended action. We've seen that sometimes we recommend the lower price, let's say, to maximize the quantity sold. Sometimes we recommend the higher price to maximize the revenue sold. And sometimes we recommend an even higher price to maximize the profits generated, mostly because the cost range reduced. And it is very, very important to ask the right question. What are we trying to maximize, and what our consumers are doing in this market? Now let's see a few examples. The first example I would like to discuss is the competition between two different firms. So, the question is did we assume anything in the previous analysis that would not apply to a case of competitive firms and when there is a competing product that can take market share from our product? And the answer is, yes, we did. We assumed that if we are changing the price, the consumers behave according to that demand model that we've described from the data, but we didn't say if other companies might also reciprocate with a lower price, or maybe a higher price. But in reality, if we lower the price, wouldn't the competitor also lower the price? And then the question, what do we need to do in response? If the competitor lowers the price a little bit below us, and all of the consumers go and buy that competitor's product, should we lower the price even a little bit below that consumer? If we do that, that competitor will also lower their price, and we will lower the price, and then when we lower the price, we will get a price war]. And actually the profit will be eroded and maybe the profit will be zero. What I've just described, which is my company is responding to the competitor's company, and the competitor's company is responding to my price change, this is called strategic interaction. And this is what the field of game theory typically handles and tries to solve. Another problem I'm going to introduce now, and in the next lecture I'm going to show you some details about it. So let's take a look at how Online Advertising works on the web. Basically these are the websites that you all browse, too. These are all of the online ads that you see, the pop-ups, the ads that chase you and follow you after you put a product in the shopping cart. And what companies can do when they show you these ads, they need to make a choice, do they show the ads to you, or do they show the ads to a different consumer? And also what they can do is, they can say, I wanna show one consumer the same ad but on multiple websites. If this consumer visits Yahoo.com and eBay.com and Amazon.com and CNN.com, I'm gonna show the same ad to the same consumer, and these ads are gonna chase this consumer. Now because it is very, very hard to tell if those ads actually impacted the consumer's decision to buy a product, what companies do is they do something called attribution. At the end of the process, they try to measure what was the impact of each ad on the consumer's decision to purchase. One simple way to do that is to say, did the consumer click the ad? If consumers click the ads more, probably the ad have more impact. And what companies are trying to measure is something called click-through rate, which is, basically, if I showed this ad to 100 consumers, how many of them are gonna actually click the ad? And this ratio of the number of consumers that clicked to the number of consumers that have seen the ads is actually the click-through rate of the ad. So let's see an example. What you see in this graph is basically how advertisers analyze data online. What they say is, they say, I'm gonna show these ads to the same consumer on different Channels or different websites. On the x-axis you'll see is the number of Channels that showed an ad to the same consumer. So some consumers have seen 0 ads, just didn't see any ads. Some consumers have seen 1 ad, some consumers have seen ads on 2 websites, some consumers have seen ads on 3 websites. And on the left hand side, we see the click-through rate. What is the probability or the percentage of consumers that actually clicked on the ad? So if you see no ads, you don't click on ads. That's great, that's perfect. And it will appear from this analysis that if you show more and more ads to the consumers on different website,s they click more and more ads. The probability of clicking goes higher, which would imply that actually those ads become more and more impactful the more ads you show. In the next lecture, what I will show you is how you can analyze this data, and try to understand does this model make sense? Does showing more and more ads to a consumer actually increase the probability of them clicking the ads?