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# Pokémon-Catching em all.

Yesterday, in Pokémon Go leveling., we talked about the rate at which players were able to level up while playing the game and the reward system this introduced.  I focused on that system because most games have an experience system built into them that allows the game to reward players for spending time in game.  However, many games also have a collection based reward system.  In fact, this is the main reward system of Pokémon as they even say “Gotta catch em all.”  Therefore, I wanted to look at the rate of rewards given when looking at collecting new Pokémon.

In order to construct our model of catching Pokémon, we will make a few assumptions that may not be exactly correct.  However, we should still get a good idea of what to expect in this or similar situations.

To begin with, how many Pokémon  are there?  According to the Silph Road, there are currently 376 Pokémon available in Pokémon Go.  Some of these are only available through raiding, special events, or in certain regions worldwide, so you wouldn’t be able to get all of these just by catching Pokémon.  For our purposes; however, we will assume that all of these available Pokémon can be caught “in the wild.”

Next, we need to determine how likely it is you would catch a new Pokémon.  If each Pokémon was equally likely to catch, then the probability of catching a new Pokémon would be (376-n)/376 where n is the number of Pokémon you have previously caught.  We should note that not all Pokémon are equally likely to be found and you cannot catch a fraction of a Pokémon.  However, in the interest of simplifying our work, we will work with these assumptions.  In this case, we note that if the probability of catching a new Pokémon is 1/10, we will proceed as if we catch 1/10th of a new Pokémon.  We therefore get

Using the equation we came up with above, we then get

If we now assume that you can catch 50 Pokémon per hour, we get that

We note in this case that the rate at which we are catching Pokémon is in fact exponentially decaying, so the frequency with which we get new Pokémon decreases rapidly. Below, we have the graph of both n and dn/dt.

As a comparison to last time, we note that a player can finish leveling in the game in approximately 400 hours of game play.  Using this model, we will never actually get that last Pokémon, but we will be at all but one Pokémon after approximately 45 hours of game play.

Checking the model against my own experience, I would say that the findings don’t match up well. In particular, I have played much more than 45 hours and I am only at 337 of the 376 Pokémon. The biggest problem with the model is that not all Pokémon are equally likely to show up. Pidgies and Rattatas are all over the place, but you can search for quite a while before you see a Lapras. We could adjust the model, using a probability of ((376-n)/376)2. However, the process of setting up the model would be extremely similar, so I leave it to the reader to decide what adjustments should be made based on the game they are interested in looking at.

As always I hope you’ve enjoyed the blog. If you haven’t played Pokémon, I would suggest that it’s worth trying. You can check out Pokémon Go in the App Store, or check out the new Let’s go Pikachu that will be released in November.

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