How much gold should be spent to optimally run this Tavern?

Question

The Waterdeep: Dragon Heist adventure gives the players the option to run a tavern. The "Tavern Keeping Expenses" sidebar (p. 41) specifies the regular running expenses and how to determine profit or loss. 60 gp is spent every tenday, and a d100 + 10 is rolled against the "Running a Business" table in the DMG (p. 129).

Rolling a 50, for example, means that players earned 60 gp for that period, and so matched the money spent in maintenance, earning no profits, but incurring no losses.

Combining this information, the tavern's profit or loss (based on the total of the roll) is as follows:

• 20 or less: Loss of 90 gp.
• 21-30: Loss of 60 gp.
• 31-40: Loss of 30 gp.
• 41-60: No profit, no losses.
• 61-80: Profits = 1d6 × 5 gp.
• 81-90: Profits = 2d8 × 5 gp.
• 91 or more: Profits = 3d10 × 5 gp.

The sidebar also specifies that players can spend gold on marketing, adding 1 to the roll for each 1 gp they spend. What is the optimal amount of gold to spend in marketing to optimize the profits earned? Or is it not even worth to run the tavern?

Answer 1

Spend 30gp/tenday

Using this anydice program, I've calculated the expected profit based on how much money you spend on marketing. I've defined a function for calculating the profit or loss of a business as per the DMG's rules, and then run it in a loop to compare the result of differing marketing spends when used to run Trollskull Manor.

function: profit ROLL:n MAINT:n {
  if ROLL <= 20 { result: -(MAINT * 15) / 10 }
  if ROLL <= 30 { result: -MAINT }
  if ROLL <= 40 { result: -MAINT / 2 }
  if ROLL <= 60 { result: 0 }
  if ROLL <= 80 { result: 1d6 * 5 }
  if ROLL <= 90 { result: 2d8 * 5 }
  if ROLL >= 91 { result: 3d10 * 5 }
}

output [profit 1d100+10 60] named "Trollskull: No Marketing"

loop X over {1..80} {
  output [profit 1d100+10+X 60] - X named "Trollskull: [X]gp Marketing"
}

We're only going as high as 80gp on marketing spend, since at 80gp we have already guaranteed the result will be the best possible, and any extra money we throw at it will go to waste. The awkward -(MAINT * 15) / 10 is there because anydice can only do integers, so can't tell it directly to multiply by 1.5. I have also assumed (as in this answer) that the profit roll described by the rules in the DMG is a single amount for an entire operating period (30 days in a normal downtime activity, 10 days in the special case of the Trollskull Manor's rules) rather than being daily profits. Here's a table of results, using the table and summary view:

The results shown in anydice are a bit dense and impenetrable because it's checking for eighty different possible marketing spends, but by looking at the summary of mean results for each variation, we can see that the expected return on investment rises with diminishing returns for each gp spent on marketing to a peak of 19.25gp profit on a 30gp marketing investment; after that, expected profit declines. (The scale on the table is bizarre because this view also has minimum and maximum tables, so values are being graphed in relation to a minimum of -99 and a maximum of 150 off past the edge of my screencap.)

_{I note, with some trepidation, that this answer is entirely different to the analysis I previously offered in a comment on this answer to another question. However, very importantly, the analysis in this post was done just now and so I am absolutely convinced of its correctness, whereas I don't remember exactly what I did for that previous comment and probably mucked it up somehow.}

Answer 2

30gp per tenday

Using google sheets:

Calculating the averages of each outcome

• 20- Loss of 90g.
• 21-30 Loss of 60g.
• 31-40 Loss of 30g.
• 41-60 No profit, no losses.
• 61-80 Profits 17.5gp.
• 81-90 Profits 45gp.
• 91+ Profits 82.5gp.

For a given marketing budget calculate the odds of each category

The odds of rolling 5 to 10 (inclusive) on a d100 is (1 + (10 - 5)) / 100 = 6/100

If budget would be enough to increase the minimum number then we need to take that into account, plus the base +10:

=max(0, 1 + Max_Number - Min_Number - MAX(0, Budget - Min_Number + 1 + 10))/100

Sum together all possible profits

Now that we have the odds for each category, we can multiply by the average profit to find out the expected profit:

Eg for +0 budget, the odds of 91+ is 20%, so 20% of 82.5gp is 16.5gp. Repeat for every category.

Plot all budgets from 0 to 90

The max is at 30gp marketing budget, yielding an average profit of 19.25gp. 30gp has a 20% chance for no profit, 20% chance for 1d6×5gp, 10% chance for 2d8×5gp, and 50% chance for 3d10×5gp.

FWIW I calculated 29gp yields 19.125g, 31gp yields 19.075, and 30.5gp would yield 19.1625gp profit, so not quite as good.

Answer 3

The Thlot Plickens... highcough!

The first thing is to think directly on the profit rather than the original investment of the 60gp. That original investment is a given. Even if you get this back by incurring no losses, to sustain this investment in the tavern over time, you will need to keep reinvesting the 60gp each time. It becomes a rolling re-investment. So, for now forget about this.

The aim is to get a maximum profit every tenday, which is 3d10x5gp, or 82.5gp on average.

Next...

The average roll of a d100 is 50.5, just like any dice roll, to work out the average you add 0.5 to half of the maximum value of the roll, e.g. for 1d8 the average is 4.5, for 1d4 it's 2.5, etc.

The average roll of 1d100 + 10 is 60.5 (50.5 + 10)

To get maximum profit you need to roll 91 or higher.

So, 91 - 60.5 = 30.5! But, you cannot roll .5 on a die.

You need to go either higher or lower; you can either invest 30gp, or 31gp on marketing.

Investing 30gp on marketing means that you will mostly break even or make a profit, but you will lose your investment on 1 occasion every 100 throws on average.

Investing 31gp on marketing means that you will always come even and not lose your investment, but you will be marginally over-investing.

The profit for investing 30gp is indeed marginally greater than that of 31gp.

However, you can off-set this by alternating your marketing investments; this will give you the maximum profit because in effect your average investment will be 30.5 gp. You will lose your investment every 200 throws on average, yet you will make the highest profit.

Invest 30gp one tenday; invest 31gp the next tenday; and keep alternating!

To sum up:

Have a 60gp as on-going investment; and keep 60gp in the kitty for that odd occasion where you roll a 1 on a tenday in which you had invested 30gp.

Keep alternating your investment in marketing between 30gp and 31gp every tenday. Your average will produce maximum profits and you will get on average 82.5gp back for your regular investment.

It is definitely worth investing in the Tavern!

Bottoms up!

Note: special thanks to @pwi for pointing some helpful things out and helping me re-visit this problem.

Answer 4

Per Carcer's outstanding work, the maximum average profit you will see (using 30 gp per tenday on promoting the business) is 19.25 gp per tenday. It costs 1,250 gp and 12 days just to get the business started, so at that rate, in addition to the 12 days to renovate, you're looking at 1250/19.25 = ~65 tendays of running the business just to break even.

So to answer your last question, unless you plan to spend approximately more than 65 tendays of downtime to run the business, it's not worth it.

Edit: However, there are other benefits beyond gold to think about. In chapter 3, one of the reasons the characters are perceived as unlikely perpetrators is because the Manor is considered their place of business.

“Even though the characters are suspects by virtue of their proximity to the crime scene, it hardly seems plausible that they would unleash destructive magic so close to their place of business in broad daylight.”