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On Pondering a possible Second-Order Strategy for Event Chest decisions

Sir Derf

Artisan
Since I'm drawing on numbers from an event that is not yet on the Live servers, I'll post this here for now. Moved out of Beta, since I discussed with number from an event that had been in Beta/

The scenario:

Your goal is to maximize Daily Prizes.
You are down to your final 100 SK.
You are being offered to choose between the 3 crappiest chests for Dailies, 48, 23 and 58.

Is it mathematically better to take the 48 chest or the 23 chest?
 
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Sir Derf

Artisan
Don't know where this is going. Going to post a few musings as I go along...


Taking the 48 chest this round means you can't take the 60 chest if it is offered next round.
Taking the 23 chest this round means you can take the 60 chest if it is offered next round.
What are those odds?

How many different chest offerings are there?

9!/(6! * 3!) = 9*8*7/3*2*1 = 3*4*7 = 84

How many of those offerings would contain the 60 chest?

8!/(6! * 2!) = 8*7/2*1 = 4*7 = 28

There is a 28/84 = 33.3% chance that you will be offered the 60 chest next round.


Or, put another way, there is a 33.3% chance that any particular chest will be part of a round.
 

Sir Derf

Artisan
I think this are the full decision trees...

Starting with 100 SK, you either pick the 48 chest or the 23 chest...

  • 48 chest, 13% of Daily, 52 leftover SK.
    • 28/84, 33.3% chance of 33 chest, 12.7% chance of Daily, 19 leftover SK
    • 21/84, 25.0% chance of 29 chest, 10.1% chance of Daily, 23 leftover SK
      • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 1 leftover SK
      • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 0 leftover SK
      • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
    • 15/84, 17.9% chance of 22 chest, 6.0% chance of Daily, 30 leftover SK
      • 28/84, 33.3% chance of 29 chest, 10.1% chance of Daily, 1 leftover SK
      • 21/84, 25.0% chance of 22 chest, 6.0% chance of Daily, 8 leftover SK
      • 15/84, 17.9% chance of 23 chest, 5.9% chance of Daily, 7 Leftover SK
      • 20/84, 23.8% chance of no chest, 0.0% chance of Daily, unusable SK
    • 10/84, 11.9% chance of 23 chest, 5.9% chance of Daily, 29 leftover SK
      • 28/84, 33.3% chance of 29 chest, 10.1% chance of Daily, 1 leftover SK
      • 21/84, 25.0% chance of 22 chest, 6.0% chance of Daily, 8 leftover SK
      • 15/84, 17.9% chance of 23 chest, 5.9% chance of Daily, 7 Leftover SK
      • 20/84, 23.8% chance of no chest, 0.0% chance of Daily, unusable SK
    • 10/84, 11.9% chance of no chest, 0.0% chance of Daily, unusable SK

  • 23 Chest, you have 5.9% chance of Daily, 77 leftover SK
    • 28/84, 33.3% chance of 60 chest, 23.2% chance of Daily, 17 leftover SK
    • 21/84, 25.0% chance of 33 chest, 12.7% chance of Daily, 44 leftover SK
      • 28/84, 33.3% chance of 33 chest, 12.7% chance of Daily, 11 leftover SK
      • 21/84, 25.0% chance of 29 chest, 9.9% chance of Daily, 15 leftover SK
      • 15/84, 17.9% chance of 22 chest, 6.0% chance of Daily, 22 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 0 leftover SK
        • 56/84, 66.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 10/84, 11.9% chance of 23 chest, 5.9% chance of Daily, 21 leftover SK
      • 10/84, 11.9% chance of no chest, 0.0% chance of Daily, unusable SK
    • 15/84, 17.9% chance of 75 chest, 27.3% chance of Daily, 2 leftover SK
    • 10/84, 11.9% chance of 29 chest, 9.9% chance of Daily, 48 leftover SK
      • 28/84, 33.3% chance of 33 chest, 12.7% chance of Daily, 15 leftover SK
      • 21/84, 25.0% chance of 29 chest, 9.9% chance of Daily, 19 leftover SK
      • 15/84, 17.9% chance of 22 chest, 6.0% chance of Daily, 26 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 4 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 3 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 10/84, 11.9% chance of 48 chest, 13.0% chance of Daily, 0 leftover SK
      • 6/84, 7.1% chance of 23 chest, 5.9% chance of Daily, 25 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 3 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 2 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 4/84, 4.8% chance of no chest, 0.0% chance of Daily, unusable SK
    • 6/84, 7.1% chance of 22 chest, 6.0% chance of Daily, 55 leftover SK
      • 28/84, 33.3% chance of 33 chest, 12.7% chance of Daily, 22 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 0 leftover SK
        • 56/84 66.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 21/84, 25.0% chance of 29 chest, 9.9% chance of Daily, 26 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 4 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 3 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 15/84, 17.9% chance of 22 chest, 6.0% chance of Daily, 33 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 11 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 10 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 10/84, 11.9% chance of 48 chest, 13.0% chance of Daily, 7 leftover SK
      • 6/84, 7.1% chance of 23 chest, 5.9% chance of Daily, 32 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 10 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 9 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 4/84, 4.8% chance of no chest, 0.0% chance of Daily, unusable SK
    • 3/84, 3.5% chance of 48 chest, 13.0% chance of Daily, 29 leftover SK
      • 28/84, 33.3% chance of 29 chest, 9.9% chance of Daily, 0 leftover SK
      • 21/84, 25.0% chance of 22 chest, 6.0% chance of Daily, 7 leftover SK
      • 15/84, 17.9% chance of 23 chest, 5.9% chance of Daily, 6 leftover SK
      • 20/84, 23.8% chance of no chest, 0.0% chance of Daily, unusable SK
    • 1/84, 1.2% chance of 23 chest, 5.9% chance of Daily, 54 leftover SK
      • 28/84, 33.3% chance of 33 chest, 12.7% chance of Daily, 21 leftover SK
      • 21/84, 25.0% chance of 29 chest, 9.9% chance of Daily, 25 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 3 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 2 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 15/84, 17.9% chance of 22 chest, 6.0% chance of Daily, 32 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 10 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 19 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 10/84, 11.9% chance of 48 chest, 13.0% chance of Daily, 6 leftover SK
      • 6/84, 7.1% chance of 23 chest, 5.9% chance of Daily, 31 leftover SK
        • 28/84, 33.3% chance of 22 chest, 6.0% chance of Daily, 9 leftover SK
        • 21/84, 25.0% chance of 23 chest, 5.9% chance of Daily, 8 leftover SK
        • 35/84, 41.7% chance of no chest, 0.0% chance of Daily, unusable SK
      • 4/84, 4.8% chance of no chest, 0.0% chance of Daily, unusable SK
 

Sir Derf

Artisan
And, I made a mistake in the above. In my above evaluation, you should not be deciding to pick the 23 chest, there is always a better option. Crap.
 

Sir Derf

Artisan
Wait, we're at end of game, if your only option is 23, you take it at that point.

My head hurts... I need to step away for a bit...
 

Sir Derf

Artisan
Let me try and make this smaller...

The scenario:

Your goal is to maximize Daily Prizes.
You are down to your final 55 SK.
You are being offered to choose between the 3 crappiest chests for Dailies, 48, 23 and 58.

This is the option of a single 48 vs. a single 23 with a 33% chance of a 22.

48 is 13% chance of Daily.

23 is 5% chance of Daily + 12% chance of 30 SK, which is roughly 5.93% chance total. 22 is 6% chance of Daily. 33% chance of being offered a 22 chest is therefore an added 2% chance of Daily. This only increases the total chance from 5.93% to 7.93%; not enough to make it a better strategy compared to the 13% chance from the 48 chest.

Turning that around, the unlocked future gain necessary to make choosing a 23 chest over a 48 chest would have to be greater than 8%, to overcome the 13% of the 48.


Edit - 22+23 != 55
 
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Sir Derf

Artisan
next...

The scenario:

Your goal is to maximize Daily Prizes.
You are down to your final 5648 SK.
You are being offered to choose between the 3 crappiest chests for Dailies, 48, 23 and 58.

This is the option of a single 48 vs. a single 23 with a 33% chance of a 22 and a 25% chance of a 23.

48 is 13% chance of Daily.

23 is 5% chance of Daily + 12% chance of 30 SK, which is roughly 5.93% chance total. Add to that a 33% chance at 5.92% and a 25% chance at 6%, for an additional 3.47%, or a total 9.40%. Better, but still not greater than 13%

Edit - 23+23 != 56
 
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Sir Derf

Artisan
Your goal is to maximize Daily Prizes.
You are down to your final 6252 SK.
You are being offered to choose between the 3 crappiest chests for Dailies, 48, 23 and 58.

This is the option of a single 48 vs. a single 23 with a 33% chance of a 29 and a 25% chance of a 22 and a 17.8% chance of a 23.

48 is 13% chance of Daily.

23 is 5% chance of Daily + 12% chance of 30 SK, which is roughly 5.93% chance total. Add to that a 33% chance at 10.1%, a 25% chance at 6%, and a 17.8% chance of 5.93%, for an additional 5.92%, or a total 11.85%. Better still, but still not greater than 13%.

Edit 23+29 != 62
 

Sir Derf

Artisan
Your goal is to maximize Daily Prizes.
You are down to your final 56 SK.
You are being offered to choose between the 3 crappiest chests for Dailies, 48, 23 and 58.

This is the option of a single 48 vs. a single 23 with a 33% chance of a 33 and a 25% chance of a 29, a 17.8% chance of a 22 and a 11.9% chance of a 23.

48 is 13% chance of Daily.

23 is 5% chance of Daily + 12% chance of 30 SK, which is roughly 5.93% chance total. Add to that a 33% chance at 12.7%, a 25% chance at 10.1%, a 17.8% chance of 6%, and a 11.9% chance of 5.93%, for an additional 8.53%, or a total 14.46%. Eureka.



Let me do a rough double-check, and ignore the attempts to factor in the bonus SK...

48 is still a 13% chance.

The original 23 is a 5% chance, leaving 33 SK.

33% chance of 33 chest offering 11% chance = 3.663%
25% chance of 29 chest offering 8% chance = 2.000%
17.8% chance of 22 chest offering 6% chance = 1.068%
11.9% chance of 23 chest offering 5% chance = 0.595%

5% + (3.663% + 2.000% + 1.068% + 0.595%) = 12.326%, very nearly 13%, and that is without factoring in the results of bonus SK.
 

Sir Derf

Artisan
So, proof of concept.

At the end of the game, when SK levels are low, there are circumstances where it can be better to choose the lower-cost lower-odds chest over the higher-cost higher odds chest for the opportunity to get an extra low-cost chest.
 

Sir Derf

Artisan
So, two cautions not to take this farther than you should.

1) This happens because we are at the end of the game, with low SK levels, which limits continuing options. At early points in the game, when SK levels are high, the added weight does not come in to play. When you have enough SK that whatever you choose you will be able to open many further chests, this effect gets smaller and smaller.

2) Just because you are at then end of the game, that does not mean you should always take the lowest-cost chest. 48 ad 52 SK left both still favored the 48 chest; it wasn't until we had 56 SK left, opening the future options for 4 different chests, that it became better to take the 23 chest over the 48 chest.
 

Sir Derf

Artisan
The above numbers are off. I used chest information from "The Forbidden Ruins" event, but worked out chest options as if this was a 2-out-of-9 random chest Event, rather than a one-1-one-2-one-3 chest event. So, while the numbers are off, the concept is still sound.
 

FieryArien

Necromancer
The above numbers are off. I used chest information from "The Forbidden Ruins" event, but worked out chest options as if this was a 2-out-of-9 random chest Event, rather than a one-1-one-2-one-3 chest event. So, while the numbers are off, the concept is still sound.
And to add to the complexity, you might want to consider the special situation that happens often near the right end: if less than 3 points are missing to finish the path, you’ve got a selection of 1s and 2s or even only 1s!! :p (I really miss an evil smiley here, admins, give us one, please!)
 

Sir Derf

Artisan
The realization I also had in another thread is that the mathematical analysis of best chest is not being worked out correctly for an Elvenar Event.

The current computations are based on the relative odds for a single round, or for a game with an infinite supply of SK. Elvenar Events are a series of multiple rounds ultimately constrained by finite SK.

As a goal for Dialies, on a head-to-head basis, 48 chest is better than 23 chest. On equal footings, in the long run, on average, take the 48 chest over the 23 chest. But Elvenar events are finite, and it appears there is a second-level, higher-order mathematically-supported strategy that can be employed.

As such, the opportunity cost of costlier chests need to be taken into account; that is, the potential benefit of taking a "worse odds" lower cost chest now, saving SK towards opening additional chests at the end of the event.

A proper mathematical assessment needs to be done separately for every set of three offered chests, assessing all three chests as if they cost the value of the most expensive chest of the set, and including guaranteed savings of SK as part of the rewards. For example, when presented with the triple-loser 48, 23 and 58, the 48 should be based on costing 58 but paying off 10 SK + 13% Daily, the 23 on costing 58 but paying off 35 SK, 5% Daily and 12% 30 SK, and the 58 on costing 58 but paying off 12% Daily and 13% 80 SK.
 

Sir Derf

Artisan
As a proud practitioner of the art of Semi-Pointless Math (tm), I here place a call to those who practice the Darker Arts, Useful Math, for an actual mathematical computation of the chest odds as described above, and a head-to-head comparison of this strategy versus the earlier strategy (versus 'always take highest direct payoff' vs. 'always take the most expensive' vs. 'always take the least expensive')