Napkin Math on dCity Citizen Mining

Historically, dCity assets have been an interesting hedge against HIVE volatility. With HIVEUSD going up, people would sell off their dCity stuff to lock in USD profits. With HIVEUSD going down, they would buy up the stuff that became more attractive in USD terms (which would drive SIMHIVE and even HIVE-denominated card pricing up). Given the recent crypto crash, no wonder the interest in the game has been picking up recently.

With @ecoinstant as the most prominent dCity resource, a lot of focus is on his favourite pastime of citizen mining. I refuse to accept the "starter city" label because it is based on a false premise of low-cost being associated with beginners and high-cost with experts. These two attributes might corellate mildly but the core of gaming is rich people throwing money around for fun and sharp minds investing their time in picking it up, isn't it?

The Citizen Farming Villages are a time-intensive expert play aimed at a game's marginal area. They neither teach beginners much (although they can help with some of the very basics) nor provide great starting point for building a TOP400 balanced city (Ranking Rewards are where the money is).

If you want to start the game treating at as a gamified investment fund (that's what dCity really is), you should just deposit at Hive Engine (using HIVE token, you can save a lot on fees whenever BeeSwap has some liquidity - other cryptos can be deposited at TribalDEX) and grab a handful of SIM. That alone currently entitles you to 40%-ish APR. Then you can mint a dozen or two random cards and play around (preferably in the Simulation Tab) to balance the population with your jobs. Once you understand how things work, the market is there for you to buy up what you need to reach the city size that matches your budget. Your APR can then go up or down compared to holding SIM depending on your performance.

Having run through the disclaimer section, let's look at the actual math.

Citizen Mining 101

When you have enough popularity, you get a chance to mine a citizen (Homeless/Immigrant/Student). Every two hours, you roll a dice - the better your popularity, the better your chance (capped at 30%). You cannot get more than one citizen per day, though. That is the 101.

Now the fineprint. Recently, the value of mining skyrocketed. Not only is the base population more expensive after the popularity nerf, the addition of Student card raised the stakes as it is much more valuable than the other two.

The logical consequence was a nerf to citizen mining probability. Now you need way more popularity for the same mining chance, leading to mining villages growing in size (and costs). As a side effect, the chances are no longer bracketed the way they used to be (the chance was always a multiple of 3%).

I admit, I have no idea if they are rounded to units (multiples of 1%, if you wish) or fractions are used. If you want to be safe, keep just enough popularity to make the next bracket. Their size varies with the war tax (presidential only, no lobbying part), anywhere between 50 for 0% War tax and 550 for the max of 10% War tax (including fractions if the President happens to set it at 4.7862%). I only deal with rounded tick percentages for simplicity.

Citizen a Day Keeps Poverty Away

Eco's recent article shows how binomial distribution knowledge allows you to calculate your Daily Percentage (chance you get your citizen within a given day) from the Tick Percentage (chance on any given dice roll) displayed in the game. The math is right but only works for Day One.

We all assumed the rolls are made every tick and only then successful rolls are checked against the history of the day, discarding duplicates. However, practice showed that there are not as many citizens mined as expected and some of us started reevaluating.

Since noone ever observed two citizens being mined on an account within 24 hours, the original model of days where you can mine one citizen at 22:30 and another at 02:30 of the next day has been discarded. Apparently, mining a citizen puts you on a 24-hour cooldown. Then you can start mining until you hit and enter a new cooldown.

The tough part is that with the days no longer independent of each other, easy math no longer applies. Your Daily Percentage can be twelve (maybe just eleven if mining at 22:30 does not influence the next day's 22:30 tick) different numbers depending on what time (if any) was the previous day's citizen mined.

The Shortcut

I thought a simulation was needed (well, it is still the best way to get an answer) but the switch from 30% Tick Percentage starter cities to smaller numbers led me to spot an approximation that is quite useful if mining is a rare event.

Instead of dealing with twelve Daily Percentages, let's just ask ourselves if a citizen was mined on the previous day or not. If it was not, our new day's chance is C (for Clean sheet) and it can be calculated from the Tick Percentage (T) as shown by the previousl linked article:

C = 1 - (1-T)^12

If it was mined, however, the next day's probability (D for Dirty) is blurred. Now the shortcut. Looking at a day far from now, we know that the probability of mining a citizen within that day (let's call it P) is the same for the object day and the day before. Pretending we knew D, we could weight the object-day-percentages

P = P * D + (1-P) * C

P = C / (1+C-D)

Let's be lazy and say that on average, the citizen is mined right in the middle of the previous day. The rarer mining, the fairer guess we have. Obviously, if Tick Percentage were 100%, it is always mined the first opportunity, never in the middle, but whatever. For 2% or 4% Tick Percentage, this guesstimate is great. Now can approximate D with the 6-attempt version:

D = 1 - (1-T)^6

If T is close to zero, we can further approximate:

C = 12 * T
D = 6 * T

And therefore:

D = C/2
P = C/(1+C/2) = 2C/(2+C)

Since we still have enough space on our napkin to tabulate a couple of Ts and spreadsheets are doable on phones these days, let's compute Ps for the full-sized formulas. Remember, D is still underestimated as an average hit happens more often towards the beginning of the interval. For higher T, the estimate is less accurate (not all the way to T=100% but surely for the in-game cap of T=30%)

Did you ever mine a citizen exactly 24 hours after the previous one? Less than 24 hours? Please share.