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The Mathematics Behind Badges and Maximizing Rent

A lot of people seem to be confused as to why conventional strategy is to buy your first badge at ~40 parcels and go hog-wild after 150. This post explains the mathematics behind purchasing badges to maximize rent over time.

Before I begin, there are a few caveats I’d like to bring to the table:

  • This post is about maximizing rent per second over time. If your goal is to buy badges for reasons other than making the most money, this post isn’t for you.
  • If you plan to vacation/visit somewhere you’ll likely never visit again, then I highly recommend saving your AB to purchase Town/City and State badges along the way. Atlas Earth is a long-term app and badges are often hard to come by.
  • I won’t consider rarity. The end of the post contains a quick method to determine when to purchase your first badge based on rarity.

Note: For non-USA, check for notes such as this one. There are some differences between the USA and non-USA versions of Atlas Earth.

Calculating Expected Value

As you probably already know, there are four rarities: common, rare, epic, and legendary. The odds and rent produced are as follows:

  • Common: 50%, $0.0000000011 per second
  • Rare: 30%, $0.0000000016 per second
  • Epic: 15%, $0.0000000022 per second
  • Legendary: 5%, $0.0000000044 per second

What you may not know is that we can calculate the average value of a random parcel by doing a little bit of math. The formula is ∑ P(Xi)∗Xi. What does this gobbly goop mean? I’ll break it down.

∑ (or sigma) is used for summation. We’re going to be doing some adding!

P(Xi)∗Xi means “Probability of some value Xi multiplied by some value Xi.”. In this case, the probabilities are rarity probabilities and values are the rent per second.

Putting these two things together, we multiply the probability (in decimal form) of each rarity by its value. Then, we sum all these totals together. This gives us what is called the “expected value”.

It looks like this:

($0.0000000011 * 0.5) + ($0.0000000016 * 0.3) + (0.0000000022 * 0.15) + ($0.0000000044 * 0.05)

When we add these together, this gives us an expected value of $0.00000000158 per second, just a bit below “Rare”.

Why is this useful? It gives us an idea of how much rent we can expect to see every time we purchase a parcel. It also makes the math a LOT easier by using a consistent number rather than re-calculating based on your current rarity distribution.

For the purposes of this post, I’ll use 0.158 as the average distribution for parcels. Almost all the math involved in this post uses ratios, so it’s not necessary to use $0.00000000158 per second when 0.158 works just as well. If you want to do the math with rent per second, replace 0.158 with $0.00000000158. The results are exactly the same.

How do I find badges?

See badge hunting resources at https://atlasearthguides.com/resources/.

Also, for those interested in viewing an example of a planned badge run during a trip, select https://www.reddit.com/r/AtlasEarthOfficial/comments/yqw302/game_plan_for_day_after_thanksgiving_badge_hunt/. All the badges were collected on the run except for the experimental county badge.

Buying Your First Badge

Let’s start with the first question: why is 40 the number everyone seems to recommend for your first parcel? After all, 5% is quite a bit of an increase for your first badge! Let’s take a look at total rent increase using expected value 0.158.

When you started this app, Atlas Reality handed you your first parcel for free. Regardless of what rarity it was, we’ll assume that the rarity you got has expected value 0.158. Your first parcel produces 0.158 units over a period of time.

At this point we have a choice: do we buy another parcel or do you buy the first badge? Bear in mind that badges cost 200 Atlas Bucks (AB), while parcels only cost 100 AB. let’s see what happens in each scenario.

If we buy the badge, we will spend 200 AB to earn a 5% increase in rent. To compare this to parcels, we can divide the % increase and AB cost by 2 to calculate the overall rent % increase per 100 AB. In this case, 1 badge will produce 2.5% rent increase per 100 AB.

If we buy the parcel, this will give us another 0.158 units over a period of time. What is the % rent increase? We can calculate that by dividing the rent units for our new parcel by the current units for all parcels owned and multiplying the whole thing by 100 to convert decimal to %. In this case, it is (0.158 [new units] / 0.158 [current units owned]) * 100 [to convert decimal to %], or 100. Buying a parcel gives a 100% increase in rent.

Which is the better choice: 2.5% increase in rent or 100% increase? 100% of course! So, we buy a parcel. Now we have two parcels that produce (0.158 * 2) or 0.316 units over a period of time.

We’re left with the same choice again: Do we go with the first badge or parcel? The % increase for the first badge is the same no matter how many parcels we own. It will remain fixed at 2.5% per 100 AB. For the parcel, we can use the same formula as before to calculate % increase. (0.158/0.316) * 100 = 50%. Parcel crushes badge. We buy another parcel. Now we have three parcels and no badges.

What about your fourth, fifth, and sixth badges? If we do the math, this produces 33%, 25%, and 20%, respectively. Some of you math geeks might notice a pattern here. It’s just decreasing fractions with the same numerator! For those who don’t see it, the pattern is, 1/1, 1/2, 1/3, 1/4, and 1/5, etc. Each time we buy a parcel, the % added to overall rent decreases, but remains above 2.5%.

This begs the question: when does adding a parcel go below 2.5%? The answer is 42. Allow me to explain why.

In this example, when we own 40 parcels, we have 40 * 0.158, or 6.32 units over a period of time. If we buy the 41st parcel, we will add (0.158/6.32) * 100, or…well look at that! It’s exactly 2.5%! Logically, that means buying the 42nd parcel would be less than 2.5%.

This begs another question: why do we say “buy at 40″ and not ‘buy at 41”? From a practical standpoint, almost nobody will have a perfectly even distribution at 40 parcels. Also, 40 is an easy-to-remember number.

At this point, we now have 40 parcels and 1 badge. Our strategy is to buy 40 parcels, then by 1 badge.

Buying Additional badges up to 150 parcels

Now that you’ve bought your first badge, allow me to address a common question: does each additional badge add 5% to overall rent? No, it does not. Rent increases from badges are dependent on the passport level. Below are the requirements for each passport level:

  • Level 0: 0 badges required, no increase
  • Level 1: 1 badge required, 5% increase
  • Level 2: 11 badges required, 10% increase
  • Level 3: 31 badges required, 15% increase
  • Level 4: 61 badges required, 20% increase
  • Level 5: 101 badges required, 25% increase

Do we get anything for having, say, 6 badges? No, except that we only need 5 more badges to reach passport level 2. It is only when you buy badge #11 that you see an increase from 5% to 10%. In other words, you have to have 11 badges to move up.

For the remainder of this exercise, we will assume that you have the means to buy 101 badges. However, this may not be feasible for everyone. Your tolerance for passport level will vary depending on where/how often you travel.

Recall that in our earlier scenario, we attained 40 parcels and 1 badge. Now we must decide if we should jump to 11 badges or buy more parcels.

For badges, the math gets a bit more complicated. Remember: we are calculating average rent increase per 100 AB. There are two things we must account for: overall rent increase from 5% to 10% (hint: it’s not 5%) and average rent increase for 1 badge.

To calculate rent increase from 5% to 10%, we must first figure out how much rent we produce at the 5% and 10% marks. If we assume 6.32 as the base units (this is for 40 parcels), then at 5% increase, we should have 6.32 * 1.05, or 6.636 units. For 10% increase, this is 6.32 * 1.1, or 6.952 units. If we take the difference between these answers, we get 6.952 – 6.636, or 0.316 units. In other words, moving from 5% to 10% in passport level adds 0,316 units of rent to our overall total. If we divide this by the 5% level unit total and multiply by 100, this gives us the % rent increase from 5% to 10%. (0.316 / 6.636) * 100, or ~4.762%. Thus, moving from passport level 1 to passport level 2 will produce ~4.762% increase in rent.

Since it requires 10 badges to move from passport level 1 to 2, we’ll need to divide that into % rent increase. 4.76% / 10 = ~0.476% rent increase per badge. However, each badge costs 200 AB each. We need to divide this by 2 again so that we can compare badges to parcels. ~0.476% / 2 = ~0.2381%. We have our final answer: Each 100 AB invested in badges will increase overall rent by ~0.2381%, assuming that we achieve passport level 2 when we’re finished.

This is a good time to bring up boost levels. Atlas Reality posted the official boost level ranges at https://atlasreality.helpshift.com/hc/en/3-atlas-earth/faq/39-why-do-ad-boosts-change-and-how-do-i-see-my-current-boost-rate/. Although the rent outcomes are slightly off, and the whole thing assumes you boost 20 hours per day, it gives you an idea of the massive wall between 150 and 1,500 parcels. Yes, if you buy 151 parcels, you will decrease your overall rent by a lot. So, don’t do that! Let’s inch up to 150 parcels and see if we should buy parcels or badges.

In our scenario, we’re still at 40 parcels and 1 badge. We saw what happens when we add 10 badges. Each 100 AB in badges adds ~0.2381% in rent, assuming that we buy all 10 badges. What happens if we added parcels all the way up to 150? Let’s take a look at 149 to 150.

At 149 parcels, our base unit is 0.158 * 149, or 23.542 units over a period of time. If we add 1 more, we can use the same method as before to calculate overall % increase in rent. (0.158/23.542) * 100 = ~0.6711%. This is higher than badges. Thus, we conclude that we should buy up to 150 parcels.

In case you were wondering why I did not include 5% from passport level 1, the math would look like this:

(0.158 * 1.05/23.542 * 1.05) * 100 = ~0.6711% In other words, passport level doesn’t make a difference in terms of overall rent % increase for parcels.

In our scenario, we have 150 parcels and 1 badge. Our strategy is to buy:

  • 40 parcels
  • 1 badge
  • 110 parcels

Note: For non-USA, the math is a bit different but the result is still the same. Parcels is better than getting 10 additional badges.

Buying Additional badges beyond 150 parcels

See my Reddit post at https://www.reddit.com/r/AtlasEarthOfficial/comments/1argbe5/new_badge_optimization_strategy_based_on_64hour/.

Note: For non-USA, see https://www.reddit.com/r/AtlasEarthOfficial/comments/1bffr2i/canadian_badge_optimization_strategy_based_on/. This post was originally for Canada but has expanded into other territories.

Side Note: The math becomes a bit different when considering a user who doesn’t boost. I won’t explore this within this post because not boosting is anathema. However, for someone who boosts 24/7/365, the end result is the same: badges over parcels beyond 150 parcels.

A word on accidentally buying a parcel beyond 150

If you happen to accidentally buy a parcel that puts you at the bottom of the next booster level, your best bet is to either maximize your current booster level by purchasing parcels or selling one of your parcels.

Quick Calculation for First Badge

If you want to do the math, assign the following values to your rarity distribution:

  • Common: 0.11
  • Rare: 0.16
  • Epic: 0.22
  • Legendary: 0.44

If your total exceeds 6.32, then buy your first badge.

If you happen to be mayor, if your total exceeds 5.688, then buy your town badge. Since mayor income gives 10% of the badge cost to the mayor, you can multiply 6.32 by 0.9 to get 5.688 (or ~36 parcels) to find the threshold for your first badge as mayor.


05/10/2024 – Changed “Canada/Canadian” to “non-USA” where applicable.

03/15/2024 – Updated page with Canadian notes and section “Buying Additional badges beyond 150 parcels” content now points to my Reddit post on optimal strategy based on 64 hours SRB. Included is a separate post for Canadians.

01/30/2024 – Updated section “How do I find badges?” to point to the badge hunting resources.






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