Planning With Mathematics
- Monday 22 January 2018
- 7 min. read (1507 words)
- Trial And Error
- Compound Interest
- Diminishing Returns
- Minimum Spanning Tree
- A* search
How many times have I over scoped a project to throw away most of the work? How many times have I focused on a goal only to find out I need to pivot? Creative people want to build first, and ask questions later. Here is how a little math can make our planning more reliable. I only use basic math like addition, subtraction, and multiplication.
To determine if a goal is worth it or not, assign a cost and benefit number to it. If the benefits are 2 times more than the costs, it is worth doing. Most people have experience estimating tasks on an hourly scale, so breaking big goals into tasks that can be done in 1 day will lead to accurate estimations.
- 1 point = per 40 hour work week
- 2 points = per 55 hour work week
- 3 points = per 70 hour work week
- 1 point = we have domain expertise in the goal
- 1 point = if a competitor implemented a similar goal, they probably have research to back it up
- 1 point = positive customer reviews for similar products or positive feedback on competitor's social media or forum
- 1 point = per week of salary that is paid from the expected number of paying users
- 1 point = per week saved from building the tool
Say we are working on a feature for our product that will take 1 month to complete. We have 3 employees working on it. The total cost is 3, or 1 point per person.
One person on the team has domain experience in the feature, +1 point. A competitor has implemented a similar feature, +1 point. The competitor's Amazon page has many reviews praising the feature and YouTube reviews also praise this feature, +1 point. We estimate the feature will pay for 3 months worth of salary for those 3 people, +3 points. In total, the feature has 6 benefit points.
Trial And Error
Is y = 28, and x = 10?
- y + 2 = 3x
- 28 + 2 = 3(10)
- 30 = 30
What if we got 60 = 10 on line 3? Either the original y = 28 or x = 10 was wrong, we did the math incorrectly, or the equation is wrong. In other words, we question the entire premise.
Likewise, if a plan exceeding the costs, do not keep working on it for hours, question the entire premise. Were the goals too ambitious? Were the initial conditions off? Spend time gathering more information, read a book, go Wiki, check out competitors, watch YouTube tutorials, and find out what went wrong. Set aside one day a week to question the entire premise of the plan.
Think of learning a skill as an investment. Even a little effort can pay off over time. We may start off dedicating 1 hour a week to learning a skill, but each week we get more productive than the previous week. Our productivity is increased, because we are getting better at using the tool for that skill.
In practice, it can be hard to figure out what goals or skills to invest in. Investments can crash, as shown by the stock market. We can lose knowledge too. If we don't consistently practice what we learned, we forget it.
Invest $100 at 10% interest
- 100 + 100*0.1 = 110
- 110 + 110*0.1 = 121
- 111 + 111*0.1 = 133
- 133 + 133*0.1 = 146
- 133 + 133*0.1 = 161
Have you ever done an activity with a friend to discover they seem to have a natural talent for it? They figured out a general solution that can work in many situations. Like compound interest, putting in a fixed amount of time into learning a skill will keep paying off over time. For example, we would expect a basketball player to learn another physical sport, like tennis, faster than they would to an unrelated skill. Their skills transfer over quickly to related domains.
After a certain point, investments stop growing quickly, and it is less enticing to invest. From personal experience, when it comes to learning a skill, I slow down around 2 years. A simple way to detect when investments stop growing is to keep track of it, and observe it for long term trends. When learning skills, people typically keep a to do list, compare that from week to week to see the growth trends.
In Dota 2, the average rating of a player is 3000 MMR. Reaching 4200 MMR, puts the player in the top 10%. We want from being better than 50% of player to better than 90%, a 40% increase. It's not hard to imagine going from 3000 to 4200 in two years. After that, there is only 10% to climb, which could take another 2 years. That extra 2 years could be invested into gaining 40% in another skill.
Avoid plans that depend on two unrelated events. The odds of unrelated events are multiplied, so the odds to get two heads in two coin flips is 1/2 x 1/2 = 1/4. Jake dropped out of high school, and loves video games.
Which is more likely?
- Jake is unemployed
- Jake is unemployed and plays video games
Option 1 is more likely. The two events in option 2 are unrelated, but the prior information may bias us to pick it. Let us pretend the odds of a high school drop out being unemployed is 1/5 or 20%. If the odds of boys being gamers is 3/4, the odds for option 2 is 3/4 * 1/5 = 3/20 or 15%.
Prefer plans that depend on either events, but neither event can happen at the same time in one attempt, because those probabilities are added. The chance of getting a tail or head in one coin flip is the probability of heads, plus the probability of tails, or 1/2 + 1/2 = 1. If we need people to invest in our start up, and know 1/33 people have a chance to become a millionaire, then making friends with 33 people now might give us access to 1 millionaire in the future.
Minimum Spanning Tree
Note, I'm not advocating we draw the graphs shown below. The key lesson is to try what we are familiar with, then slowly work towards the unfamiliar.
One problem with entertainment: games, movies, TV shows, comics, etc. is that there is a lot of top tier content, but we don't have time to explore them. Let's say we have 6 critically acclaimed games to go through. We can graph them by hours it takes to complete them.
Let's say game A grabs our attention the most, and we must play it. Game E is a complex strategy game that might take us 20 hours to play. Instead, we can play game B, which has some strategy elements, then game D which has a bit more strategy, and so by the time we play game E the learning curve is much lower. Going from A to B to D to E cost us 13 hours, while going directly to E cost us 20 hours. We saved 7 hours.
In reality, entertainment is very competitive, so the costs are very similar. In the below image, going from A to E to D cost us 5 points, and going from A to B to D cost us 4 points. A one point difference might as well be a rounding error. This causes analysis paralysis where we are unable to make a choice. Just pick one.
The A* search guarantees we find the shortest path given two conditions:
- we know where our goal is
- we can underestimate how far we are from our current location to the goal.
If we are lost in a forest, but know we must head north, then it is better to start walking north at first. Similarly, if we underestimate how far we are from the goal, we can never pass it.
It is common knowledge that we break problems into smaller steps, so this concept isn't too mind blowing. What isn't common knowledge is that we must constantly measure how far we are from the goal at each step. If we break a task into 5 subtasks and estimate it can be done in a week. A* says we need to update our estimate after finishing each subtask. This way we might discover the last task is unnecessary or too expensive to complete.