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| """Streamlit entrypoint""" | |
| import base64 | |
| import time | |
| import numpy as np | |
| import streamlit as st | |
| import sympy | |
| from helpers.thompson_sampling import ThompsonSampler | |
| eta, a, p, D, profit, var_cost, fixed_cost = sympy.symbols("eta a p D Profit varcost fixedcost") | |
| np.random.seed(42) | |
| st.set_page_config( | |
| page_title="๐ธ Dynamic Pricing ๐ธ", | |
| page_icon="๐ธ", | |
| layout="centered", | |
| initial_sidebar_state="auto", | |
| menu_items={ | |
| 'Get help': None, | |
| 'Report a bug': None, | |
| 'About': "https://www.ml6.eu/", | |
| } | |
| ) | |
| st.title("๐ธ Dynamic Pricing ๐ธ") | |
| st.subheader("Setting optimal prices with Bayesian stats ๐") | |
| # (0) Intro | |
| st.header("Why care about dynamic pricing? ๐ญ") | |
| st.markdown("""Dynamic pricing aims to actively adapt product prices based on insights about | |
| customer behaviour. \n | |
| This pricing strategy has proven exceptionally effective in a wide range of industries: from | |
| e-commerce (e.g., eBay) to airlines (e.g., Delta Airlines) and from retail (e.g., Walmart) to | |
| utilities (e.g., Tampa Electric) as it allows to **adjust prices to changes in demand patterns**.""") | |
| st.markdown("""It is hardly surprising that recent times of extraordinary uncertainty and volatility | |
| caused a surge in adoption of dynamic pricing strategies with an [estimated 21% of e-commerce | |
| businesses reportedly already using dynamic pricing](https://www.statista.com/statistics/1174557/dynamic-pricing-ecommerce-companies-worldwide/) | |
| and an additional 15% planning to adopt thestrategy in the upcoming year.""") | |
| st.markdown("""To find a big success story, we should look no further than Amazon who (on average) | |
| change their products' prices once every 10 minutes. They attribute roughly [25% of their e-commerce | |
| profits](https://dzone.com/articles/big-data-analytics-delivering-business-value-at-am) | |
| to their pricing strategy.""") | |
| st.markdown("""In what follows we'll discuss the working principles & challenges in the field of | |
| dynamic pricing, followed by an interactive demo and some final considerations.""") | |
| # (1) Basics | |
| st.header("Back to basics ๐") | |
| st.markdown("The beginning is usually a good place to start so we'll kick things off there.") | |
| st.markdown("""The one crucial piece information we need in order to find the optimal price is | |
| **how demand behaves over different price points**. \nIf we can make a decent guess of what we | |
| can expect demand to be for a wide range of prices, we can figure out which price optimizes our | |
| target (i.e., revenue, profit, ...).""") | |
| st.markdown("""For the keen economists amongst you, this is beginning to sound a lot like a | |
| **demand curve**.""") | |
| st.markdown("""Estimating a demand curve, sounds easy enough right? \nLet's assume we have | |
| demand with constant price elasticity; so a certain percent change in price will cause a | |
| constant percent change in demand, independent of the price level. In economics, this is often used | |
| as a proxy for demand curves in the wild.""") | |
| st.markdown("So our demand data looks something like this:") | |
| st.image("assets/images/ideal_case_demand.png") | |
| st.markdown("""Alright now we can get out our trusted regression toolbox and fit a nice curve | |
| through the data because we know that our constant-elasticity demand function has this form:""") | |
| st.latex(sympy.latex(sympy.Eq(sympy.Function(D)(p), a*p**(-eta), evaluate=False))) | |
| st.write("with shape parameter a and price elasticity ฮท") | |
| st.image("assets/images/ideal_case_demand_fitted.png") | |
| st.markdown("""Now that we have a reasonable estimate of our demand function, we can derive our | |
| expected profit at different price points because we know the following holds:""") | |
| st.latex(f"{profit} = {p}*{sympy.Function(D)(p)} - [{var_cost}*{sympy.Function(D)(p)} + {fixed_cost}]") | |
| st.image("assets/images/ideal_case_profit_curve.png") | |
| st.markdown("""Finally we can dust off our good old high-school math book and find the | |
| price which we expect will optimize profit which was ultimately the goal of all this.""") | |
| st.image("assets/images/ideal_case_optimal_profit.png") | |
| st.markdown("""Voilร there you have it: we should price this product at 4.24 and we can expect | |
| a bottom-line profit of 7.34""") | |
| st.markdown("So can we kick back & relax now? \nWell, there are a few issues with what we just did.") | |
| # (2) Dynamic demand curves | |
| st.header("The demands they are a-changin' ๐ธ") | |
| st.markdown("""We arrive at our first bit of bad news: unfortunately, you can't just estimate a | |
| demand curve once and be done with it. \nWhy? Because demand is influenced by many factors (e.g., | |
| market trends, competitor actions, human behavior, etc.) that tend to change a lot over time.""") | |
| st.write("Below you can see an (exaggerated) example of what we're talking about:") | |
| with open("assets/images/dynamic_demand.gif", "rb") as file_: | |
| contents = file_.read() | |
| data_url = base64.b64encode(contents).decode("utf-8") | |
| st.markdown( | |
| f'<img src="data:image/gif;base64,{data_url}" alt="dynamic demand">', | |
| unsafe_allow_html=True, | |
| ) | |
| st.markdown("""Now, you may think we can solve this issue by periodically re-estimating the demand | |
| curve. \nAnd you would be very right! But also very wrong as this leads us nicely to the | |
| next issue.""") | |
| # (3) Constrained data | |
| st.header("Where are we getting this data anyways? ๐ค") | |
| st.markdown("""So far, we have assumed that we get (and keep getting) data on demand levels at | |
| different price points. \n | |
| Not only is this assumption **unrealistic**, it is also very **undesirable**""") | |
| st.markdown("""Why? Because getting demand data on a wide spectrum of price points implies that | |
| we are spending a significant amount of time setting prices that are either too high or too low! \n | |
| Which is ironically exactly the opposite of what we set out to achieve.""") | |
| st.markdown("In practice, our demand observations will rather look something like this:") | |
| st.image("assets/images/realistic_demand.png") | |
| st.markdown("""As we can see, we have tried three price points in the past (โฌ7.5, โฌ10 and โฌ11) and | |
| collected demand data.""") | |
| st.markdown("""On a side note: keep in mind that we still assume the same latent demand curve and | |
| optimal price point of โฌ4.24 \n | |
| So (for the sake of the example) we have been massively overpricing our product in the past.""") | |
| st.image("assets/images/realistic_demand_latent_curve.png") | |
| st.markdown("""This limited data brings along a major challenge in estimating the demand curve | |
| though. \n | |
| Intuitively, it makes sense that we can make a reasonable estimate of expected demand at โฌ8 or โฌ9, | |
| given the observed demand at โฌ7.5 and โฌ10. \nBut can we extrapolate further to โฌ2 or โฌ20 with the | |
| same reasonable confidence? Probably not.""") | |
| st.markdown("""This is a nice example of a very well-known problem in statistics called the | |
| **\"exploration-exploitation trade-off\"** \n | |
| ๐ **Exploration**: We want to explore the demand for a diverse enough range of price points | |
| so that we can accurately estimate our demand curve. \n | |
| ๐ **Exploitation**: We want to exploit all the knowledge we have gained through exploring and | |
| actually do what we set out to do: set our price at an optimal level.""") | |
| # (4) Thompson sampling explanation | |
| st.header("Enter: Thompson Sampling ๐") | |
| st.markdown("""As we mentioned, this is a well-known problem in statistics. So luckily for us, | |
| there is a pretty neat solution in the form of **Thompson sampling**!""") | |
| st.markdown("""Basically instead of estimating one demand function based on the data available to | |
| us, we will estimate a probability distribution of demand functions or simply put, for every | |
| possible demand function that fits our functional form (i.e. constant elasticity) | |
| we will estimate the probability that it is the correct one, given our data.""") | |
| st.markdown("""Or mathematically speaking, we will place a prior distribution on the parameters | |
| that define our demand function and update these priors to posterior distributions via Bayes rule, | |
| thus obtaining a posterior distribution for our demand function""") | |
| st.markdown("""Thompson sampling then entails just sampling a demand function out of this | |
| distribution, calculating the optimal price given this demand function, observing demand for this | |
| new price point and using this information to refine our demand function estimates.""") | |
| st.image("assets/images/flywheel_1.png") | |
| st.markdown("""So: \n | |
| ๐ When we are **less certain** of our estimates, we will sample more diverse demand functions, | |
| which means that we will also explore more diverse price points. Thus, we will **explore**. \n | |
| ๐ When we are **more certain** of our estimates, we will sample a demand function close to | |
| the real one & set a price close to the optimal price more often. Thus, we will **exploit**.""") | |
| st.markdown("""With that said, we'll take another look at our constrained data and see whether | |
| Thompson sampling gets us any closer to the optimal price of โฌ4.24""") | |
| st.image("assets/images/realistic_demand_latent_curve.png") | |
| st.markdown("""Let's start working our mathemagic: \n | |
| We'll start off by placing semi-informed priors on the parameters that make up our | |
| demand function.""") | |
| st.latex(f"{sympy.latex(a)} \sim N(ฮผ=0,ฯ=2)") | |
| st.latex(f"{sympy.latex(eta)} \sim N(ฮผ=0.5,ฯ=0.5)") | |
| st.latex("sd \sim Exp(\lambda=1)") | |
| st.latex(f"{sympy.latex(D)}|P=p \sim N(ฮผ={sympy.latex(a*p**(-eta))},ฯ=sd)") | |
| st.markdown("""These priors are semi-informed because we have the prior knowledge that | |
| price elasticity is most likely between 0 and 1. As for the other parameters, we have little | |
| knowledge about them so we can place a pretty uninformative prior.""") | |
| st.markdown("If that made sense to you, great. If it didn't, don't worry about it") | |
| st.markdown("""Now that are priors are taken care of, we can update these beliefs by incorporating | |
| the data at the โฌ7.5, โฌ10 and โฌ11 price levels we have available to us.""") | |
| st.markdown("The resulting demand & profit curve distributions look a little something like this:") | |
| st.image(["assets/images/posterior_demand.png", "assets/images/posterior_profit.png"]) | |
| st.markdown("""It's time to sample one demand curve out of this posterior distribution. \n | |
| The lucky curve is:""") | |
| st.image("assets/images/posterior_demand_sample.png") | |
| st.markdown("This results in the following expected profit curve") | |
| st.image("assets/images/posterior_profit_sample.png") | |
| st.markdown("""And eventually we arrive at a new price: โฌ5.25! Which is indeed considerably closer | |
| to the actual optimal price of โฌ4.24""") | |
| st.markdown("""Now that we have our first updated price point, why stop there? Let's simulate 10 | |
| demand data points at this price point from out latent demand curve and check whether Thompson | |
| sampling will edge us even closer to that optimal โฌ4.24 point.""") | |
| st.image("assets/images/updated_prices_demand.png") | |
| st.markdown("""We know the drill by now. \n | |
| Let's recalculate our posteriors with this extra information.""") | |
| st.image(["assets/images/posterior_demand_2.png", "assets/images/posterior_profit_2.png"]) | |
| st.markdown("""We immediately notice that the demand (and profit) posteriors are much less spread | |
| apart this time around which implies that we are more confident in our predictions.""") | |
| st.markdown("Now, we can sample just one curve from the distribution.") | |
| st.image(["assets/images/posterior_demand_sample_2.png", "assets/images/posterior_profit_sample_2.png"]) | |
| st.markdown("""And finally we arrive at a price point of โฌ4.04 which is eerily close to | |
| the actual optimum of โฌ4.24""") | |
| # (5) Thompson sampling demo | |
| st.header("Demo time ๐ฎ") | |
| st.markdown("Now that we have covered the theory, you can go ahead and try it our for yourself!") | |
| st.markdown("""You will notice a few things: \n | |
| ๐ As you increase price elasticity, the demand becomes more sensitive to price changes and thus the | |
| profit-optimizing price becomes lower (& vice versa). \n | |
| ๐ As you decrease price elasticity, our demand observations at โฌ7.5, โฌ10 and โฌ11 become | |
| increasingly larger and increasingly more variable (as their variance is a constant fraction of the | |
| absolute value). This causes our demand posterior to become increasingly wider and thus Thompson | |
| sampling will lead to more exploration. | |
| """) | |
| thompson_sampler = ThompsonSampler() | |
| demo_button = st.checkbox( | |
| label='Ready for the Demo? ๐คฏ', | |
| help="Starts interactive Thompson sampling demo" | |
| ) | |
| elasticity = st.slider( | |
| "Adjust latent elasticity", | |
| key="latent_elasticity", | |
| min_value=0.05, | |
| max_value=0.95, | |
| value=0.25, | |
| step=0.05, | |
| ) | |
| while demo_button: | |
| thompson_sampler.run() | |
| time.sleep(1) | |
| # (6) Extra topics | |
| st.header("Some final remarks") | |
| st.markdown("""Because we have purposefully kept the example above quite simple, you may still be | |
| wondering what happens when added complexities show up. \n | |
| Let's discuss some of those concerns FAQ-style:""") | |
| st.subheader("๐ Isn't this constant-elasticity model a bit too simple to work in practice?") | |
| st.markdown("Brief answer: usually yes it is.") | |
| st.markdown("""Luckily, more flexible methods exist. \n | |
| We would recommend to use Gaussian Processes. We won't go into how these work here but the main idea | |
| is that it doesn't impose a restrictive functional form onto the demand function but rather lets | |
| the data speak for itself.""") | |
| with open("assets/images/gaussian_process.gif", "rb") as file_: | |
| contents = file_.read() | |
| data_url = base64.b64encode(contents).decode("utf-8") | |
| st.markdown( | |
| f'<img src="data:image/gif;base64,{data_url}" alt="gaussian process">', | |
| unsafe_allow_html=True, | |
| ) | |
| st.markdown("""If you do want to learn more, we recommend these links: | |
| [1](https://distill.pub/2019/visual-exploration-gaussian-processes/), | |
| [2](https://thegradient.pub/gaussian-process-not-quite-for-dummies/), | |
| [3](https://sidravi1.github.io/blog/2018/05/15/latent-gp-and-binomial-likelihood)""") | |
| st.subheader("""๐ Price optimization is much more complex than just optimizing a simple profit function?""") | |
| st.markdown("""It sure is. In reality, there are many added complexities that come into play, such | |
| as inventory/capacity constraints, complex cost structures, ...""") | |
| st.markdown("""The nice thing about our setup is that it consists of three components that you can | |
| change pretty much independently from each other. \n | |
| This means that you can make the price optimization pillar arbitrarily custom/complex. As long as | |
| it takes in a demand function and spits out a price.""") | |
| st.image("assets/images/flywheel_2.png") | |
| st.markdown("You can tune the other two steps as much as you like too.") | |
| st.image("assets/images/flywheel_3.png") | |
| st.subheader("๐ Changing prices has a huge impact. How can I mitigate this during experimentation?") | |
| st.markdown("There are a few things we can do to minimize risk:") | |
| st.markdown("""๐ **A/B testing**: You can do a gradual roll-out of the new pricing system where a | |
| small (but increasing) percentage of your transactions are based on this new system. This allows you | |
| to start small & track/grow the impact over time.""") | |
| st.markdown("""๐ **Limit products**: Similarly to A/B testing, you can also segment on the | |
| product-level. For instance, you can start gradually rolling out dynamic pricing for one product | |
| type and extend this over time.""") | |
| st.markdown("""๐ **Bound price range**: Theoretically, Thompson sampling in its purest form can | |
| lead to any arbitrary price point (albeit with an increasingly low probability). In order to limit | |
| the risk here, you can simply place a upper/lower bound on the price range you are comfortable | |
| experimenting in.""") | |
| st.markdown("""On top of all this, Bayesian methods (by design) explicitly quantify uncertainty. | |
| This allows you to have a very concrete view on the variance of our demand estimates""") | |
| st.subheader("๐ What if I have multiple products that can cannibalize each other?") | |
| st.markdown("Here it really depends") | |
| st.markdown("""๐ **If you have a handful of products**, we can simply reformulate our objective while | |
| keeping our methods analogous. \n | |
| Instead of tuning one price to optimize profit for the demand function of one product, we tune N | |
| prices to optimize profit for the joint demand function of N products. This joint demand function | |
| can then account for correlations in demand within products.""") | |
| st.markdown("""๐ **If you have hundreds, thousands or more products**, we're sure you can imagine that | |
| the procedure described above becomes increasingly infeasible. \n | |
| A practical alternative is to group substitutable products into "baskets" and define the "price of | |
| the basket" as the average price of all products in the basket. \n | |
| If we assume that the products in baskets are subtitutable but the products in different baskets are | |
| not, we can optimize basket prices indepedently from one another. \n | |
| Finally, if we also assume that cannibalization remains constant if the ratio of prices remains | |
| constant, we can calculate individual product prices as a fixed ratio of its basket price. \n""") | |
| st.markdown("""For example, if a "burger basket" consists of a hamburger (โฌ1) and a cheeseburger | |
| (โฌ3), then the "burger price" is ((โฌ1 + โฌ3) / 2 =) โฌ2. So a hamburger costs 50% of the burger price | |
| and a cheeseburger costs 150% of the burger price. \n | |
| If we change the burger's price to โฌ3, a hamburger will cost (50% * โฌ3 =) โฌ1.5 and a cheeseburger | |
| will cost (150% * โฌ3 =) โฌ4.5 because we assume that the cannibalization effect between hamburgers & | |
| cheeseburgers is the same when hamburgers cost โฌ1 & cheeseburgers cost โฌ3 and when hamburgers cost | |
| โฌ1.5 & cheeseburgers cost โฌ4.5""") | |
| st.image("assets/images/cannibalization.png") | |
| st.subheader("๐ Is dynamic pricing even relevant for slow-selling products?") | |
| st.markdown("""The boring answer is that it depends. It depends on how dynamic the market is, the | |
| quality of the prior information, ...""") | |
| st.markdown("""But obviously this isn't very helpful. \nIn general, we notice that you can already | |
| get quite far with limited data, especially if you have an accurate prior belief on how the demand | |
| likely behaves.""") | |
| st.markdown("""For reference, in our simple example where we showed a Thompson sampling update, we | |
| were already able to gain a lot of confidence in our estimates with just 10 extra demand | |
| observations.""") |