Dropping 50 Gold Quests

Starting from the list of quests and the fact that for each there are 9 different Two Class Victory/Dominance Quests and 9 different Class Victory Quests and considering the Watch and Learn quest to be worth 100 Gold, the quest pool looks like this:

• 15: 40 Gold (Destroy Them All, Only the Mighty, The Meek Shall Inherit, Spell Master, Beat Down, 3 Victories!, and Two Class Victory x 9)
• 9: 50 Gold (Class Victory x 9)
• 10: 60 Gold (Everybody! Get in here! and Two Class Dominance x 9)
• 2: 100 Gold (Total Dominance and Watch and Learn)

Since we are limited to 3 quests with 4 possible denominations, actually 5 denominations since there can be no quest as well, we can use combinations with repetition to find all possible quest states:

Of these 35 states we can't reach the 35^th, where there are 3×100 Gold quests, and there isn't much we can drop when we have no quests, so we're really interested in 33 states.

http://hearthstone.gamepedia.com/Quest#Gaining_daily_quests outlines 2 restrictions placed upon the quests that may be obtained from this pool:

1. Players will never be assigned a duplicate of a quest they are already on
2. A player who Drops a quest will receive a different, random quest in its place

We can use leverage this information to find the transitions between the 33 states which will maximize daily gold. I've written a spreadsheet to show the weighted average of transitions. I'll post it at the end of this answer, but fundamentally you'll be comparing the "Value" (the Gold that could be obtained from immediately completing all quests), "Pre Roll Value" (the weighted average of the Value obtained by dropping the quest worth the least Gold before completing any quests), and "Post Roll Value" (the weighted average of the Value obtained by completing the quest worth the most Gold then dropping the quest worth the least Gold.)

• If the "Value" is the highest, complete the highest value quest, then recheck the spreadsheet with the updated state
• If the "Pre Roll Value" is the highest, drop the lowest value quest, and recheck the spreadsheet with the updated state, if the "Value" is still lower than that state's "Pre Roll Value" don't complete a quest today unless you have 3 quests, in which case complete the most valuable one
• If the "Post Roll Value" is the highest, If it is possible to complete the highest value quest without completing a lower value quest do so, if that's not possible drop the blocking quest and then complete the highest value quest

[Click this image for a high resolution view]

EDIT:

Halfwar has informed me that there may be a cool down on completed quests. If we define this of quests number as COOLDOWN, this means that we must preserve COOLDOWN quests to obtain our "Pre Roll Value", and COOLDOWN - 1 quests to obtain our "Post Roll Value".

There are 2087 possible combinations so that's obviously too large to do by hand, and unfortunately too large to post here as a table.

I've written some code to generate these combinations: http://ideone.com/6jJa1u Sadly I haven't found a cloud compiler that will allow for enough time to generate the whole file. But feel free to use the code to generate locally. I'm working on finding a way to post the generated spreadsheet anonymously.

Several thank yous to answers that helped me find a better way to do this on http://www.stackoverflow.com:

1. I got the Technical Release 2 combination code from this answer: https://stackoverflow.com/a/35215540/2642059
2. The string::operator* was suggested as an answer to this question: https://stackoverflow.com/q/35506712/2642059
3. And I sadly decided to output by comma separated value in response to the answers to this question: https://stackoverflow.com/q/35747864/2642059

Looking at the list of quests and assuming each quest would appear with equal likeliness and also assuming the [class #1] or [class #2] Victory and [class #1] or [class #2] Dominance quests would appear at an equal rate to each other, this gives approximately:

• 7 quests that give 40 gold (counting all versions of [class #1] or [class #2] Victory as 1 quest)

• 1 quest that gives 50 gold

• 2 quests that give 60 gold (again, counting all versions of [class #1] or [class #2] Dominance as 1 quest)

• 1 quest that gives 100 gold

and

• 1 quest that gives a card pack.

This can probably be made more precise by figuring exact counts for the class victory and dominance quests, but at a glance, it seems that dropping a 50 gold quest would more likely result in a 40 gold quest than anything else.

---

Although, this doesn't necessarily mean dropping them is bad, as it only takes one "50->100" to compensate for five "50->40"s.

Given apparent odds of 7:4 for worse or better quests, over about 5 quests it looks like it should be...just about even.

This gets even more complicated if you start to take into account the value of a potential card pack reward.

In the long run, it doesn't seem like it would make a big difference. If you're patient enough to reroll the same quest tomorrow if you get a 40 out of it, by all means, reroll it today and wait.

Taking a simpler mathematical approach, we can just determine the expected value of a new quest and decide based on that value.

For this calculation, it's important to take into account the quests' cooldown of 4 days. That is, once a quest has been completed, it can only reappear 4 days later at the earliest. Quests completed in the last 3 days as well as quests currently in the quest log are thus unavailable as the new quest (this includes the one we're rerolling).

As far as I know, all quests appear with the same probability, which is 1 divided by the number of quests available if the quest is neither currently in the quest log nor on cooldown, 0 otherwise. The expected value of the new quest is thus the sum of all the rewards of all available quests divided by their number. There is little research on this (most of which has statistically insignificant sample sizes), so this assumption might be wrong.

This is a list of all currently available daily quests:

• 40 gold: 9 win 2 with [class] or [class]; 9 play 30 [class] cards; 6 do various stuff = 24 quests
• 50 gold: 9 win 3 with [class]; 15 play [x] [keyword] cards = 24 quests
• 60 gold: 9 win 5 with [class] or [class]; 9 play 50 [class] cards; 1 Tavern Brawl = 19 quests
• 80 gold: 1 play a friend = 1 quest
• 100 gold: 1 win 7 games, 1 play 75 Battlecry Minions, 1 play 75 Murlocs = 3 quests
• 1 pack (calculated as 100 gold): 1 spectate a friend = 1 quest
• Total: 72 quests

Now the Tavern Brawl quest only appears when Tavern Brawl is up, so from Wednesday to Monday (by rerolling at the right time, it's only truly unavailable on Tuesday), so when determining the available quests, take that into account.

So the best case, assuming one 50 gold quest in your quest log and 5 40 gold quests on cooldown, results in the following: 6 quests unavailable, 66 available. Total value: 19*40+23*50+19*60+1*80+4*100 = 3530, which gives us an expected value of nearly 53.5 gold for rerolling the quest.

Worst case, you have the 5 best quests on cooldown (all 100s and the 80) on a Tuesday, leaving you with 7 unavailable (65 available) and an expected value of just over 49 gold.

As you can see, even in the worst case (all 4 100s, the 80 and Tavern Brawl unavailable), you only lose 1 gold in expected value by rerolling the 50.

Here are the break even points depending on the number of quests on cooldown (completed in the last 3 days or currently in your quest log, not counting the one you're rerolling), assuming of course you only have 50s+ in your quest log:

• 0: Your expected value is necessarily 52.5 gold -> always reroll
• 1: Worst case, your expected value is 51.9 gold -> always reroll
• 2: Worst case, your expected value is 51.2 gold -> always reroll
• 3: Worst case, your expected value is 50.4 gold -> always reroll
• 4: You only lose gold in the worst case (all 4 100s on cooldown), even 3 100s and the 80 on cooldown breaks even at exactly 50 gold expected value -> reroll unless all 4 100s are on cooldown; no gain nor loss from rerolling if 3 100s and the 80 are on cooldown
• 5: Do not reroll if the total value of your quests on cooldown exceeds 430 (e.g. the 80, 3 100s and one 60). If your quests on cooldown total less than 430, always reroll. If they total exactly 430, no gain or loss, so whatever you feel like.
• 6: (only on Tuesdays if the Tavern Brawl quest is not naturally on cooldown) Do not reroll if the total value of your quests on cooldown (not including the Tavern Brawl quest) exceeds 420 (e.g. the 80, 3 100s and one 50). If your quests on cooldown total less than 420, always reroll. If they total exactly 420, no gain or loss, so whatever you feel like.

If it is a Tuesday or you're rerolling at the wrong time on a Monday or Wednesday (Tavern Brawl not currently available), only one case changes: If 4 quests are naturally on cooldown, you no longer break even if they are 3 100s and the 80.

Notably, this means that you always reroll if a maximum of one 100 quest is on cooldown. Even in the worst case (Tuesday, the 80 and 3 non-Tavern Brawl 60s on cooldown in addition to the one 100), you break even with this restriction.
While Tavern Brawl is available or the Tavern Brawl quest is naturally on cooldown, you always reroll if a maximum of 2 100 quests are on cooldown.
If 3 100s are on cooldown, but the 80 is not, you also always reroll.
If all 4 100s are on cooldown, never reroll a 50.

If a pack is worth less than 100 gold to you, adjust the value of the Spectate quest accordingly. You'll have to redo the calculations in that case.