My wife and I like to play Pokémon Go and today was Moltres raid day. In order to prepare for it, she talked to her coworkers and was able to get a group of them to meet up and go take on Moltres all together. I really enjoyed the time playing. We met up, walked around the mall and stopped at all the Pokémon gyms. While we had enough to defeat the raids on our own, at most gyms other people hopped in with us and talked to us about whether or not they were able to catch this phoenix like bird (see below). Even though the game may be beyond its maximum popularity point, there were still many trainers out today trying to get their hands on the legendary Pokémon.

Therefore, I thought I’d spend some time looking over the reward system in the game that keeps people playing. In particular, I wanted to look at how often players are rewarded for the time they put into the game compared to how much time has already been put in. That is, if I just started I expect that I would be rewarded more often than someone who has been playing a long time, so how exactly does this work?

As you catch Pokémon you gain experience which then allows you to level up. As you level up, you can unlock different features that make catching or finding Pokémon easier. As such, I’m going to work under the impression that people playing Pokémon are particularly happy whenever they level up. I should point out that people do get rewarded by catching new types of Pokémon, walking around, visiting new places, meeting people or various other things that occur while playing, but I will focus on the rewards provided by leveling.

Now, how do you get new Pokémon or level up. Well, you play the game. As you visit Poké stops you can spin a disk and get items, you can use your balls to catch Pokémon you run across, or you can participate in gym and raid battles. As you reach a determined amount of experience you get to level up. Now the amount required to level increases every time you level. To get from level 1 to 2, you only need 1,000 experience; however you need 5,000,000 experience to get from level 39 to 40. A graph is given below and a complete list of experience per level is available on the silph road website. From the graph, it appears that assuming an exponential growth for required experience would be warranted.

Instead of creating a list for every level, we will try to provide a continuous function to measure reward per effort. If we let TE be the total experience and L be the level, then we get the following equation by using an exponential regression,

While the amount of experience gained per hour is dependent on the game play style of the person playing, we will work with the assumption that a player can earn 50,000 experience per hour. We therefore get that your level can be determined as a function of time as

Now, we can determine your reward per hour of play as a function of time by looking the reward per hour as the change of your level with respect to the change in time. That is, we find the derivative of L with respect to t and get

Therefore, we see that your gain per hour is inversely proportional to amount of time you have spent playing. For example, if you have played the game for 1 hour, then you are gaining levels at a rate of 30% of a level per hour. However, if you have been playing for 100 hours, you are gaining .3% of a level per hour of play.

From this, we see that the game is set up to reward players frequently in the beginning. This gives them the a feeling of accomplishment as they play the game and gets the to continue playing. As the player continues to play; however, the frequency that rewards are supplied are decreased, causing them to have to work longer to get the same reward.

In my experience it feels like the game is meant to pull you in and get you excited. As you play, you need the rewards less frequently to continue to want to play the game, so the game gives them less frequently. Since there is an endpoint available, you want to continue to work until you get there. However, this process takes longer as you go and the finish line never seems to come, which keeps you playing. I should point out that many people have made it to the finish line of level 40, it just seems to never come for me as I am only at level 34.

This really does seem like an effective system to me, and I have heard of people trying to incorporate such a system into the classroom to get students to study more. I haven’t actually been able to find a way to do this myself, but, if you have tried, let me know how you did this and the results you got.

## 3 thoughts on “Pokémon Go leveling.”