bilibili

Simple mathematical models are almost always too simple to model complex phenomena in the real world, and the one I want to discuss in this post is not an exception. However, I believe it is a good first order approximation. I know no empirical tests to support my belief, it is just the fit between the model assumptions and the reality, subjectively judged by myself to be a good fit. Let me begin with the model first and then talk about its potential real life application. Suppose you have an unknown probability distribution over the unit interval , and you draw i.i.d. samples . Each draw costs a fixed amount , and your goal is to maximize the expectation of by choosing the best . The solution depends on the unknown distribution of course. Model Assumptions Let us talk about the use case of this model. You may want to buy a consumer product from the market, and you'd like to do some inspection before buying one. This is especially relevant for technological gadgets. Inspecting each item costs you time/effort, and you need to stop at some point and buy the best item you inspected so far. Let's think about the fitness between model assumptions and real life. The model assumes there is no learning during the sampling process. You choose the next item randomly. This may be somewhat violated in real life. One way to increase the model faithfulness is to use a dynamic distribution that shifts to the higher values as one samples. This may cause the optimum to be lower than the case where a fixed distribution is used. The other model assumption is that the items will stay available and you don't fear some of them getting out-of-stock before you complete your exploration. This sounds reasonable for many cases. The inspection cost is assumed to be constant, which is a nice assumption since all items are goods of the same kind. And maybe the most unrealistic assumption is that you can judge an item's value exactly after the inspection process. An item's value is the utilit