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Heirlooms Math
http://wwww.one-ring.co.uk/viewtopic.php?f=5&t=32975		 Page 1 of 1
Author:  Ring_of_Gyges [ Wed May 03, 2017 8:57 pm ]
Post subject:  Heirlooms Math
Suppose you're playing the new "Heirlooms of Ages Past" scenario and you want to know which of the six objectives contains the heirloom, any of them could, but they aren't all equally likely to.

The rule is that when a model checks a site you roll a d6, on a 1-5 the heirloom isn't there, on a 6 it is. If you check five times unsuccessfully the heirloom is automatically at site #6.

So what are the odds of it being any particular place? Well, it has a 1/6 chance of being in the first place someone looks (~17%). But it only has a 5/36 chance of being in the second place. Why? To be in the second place it has to both be found on the first search (5/6) and be found on the second search (1/6), multiplied together that's 5/36 (~14%). The odds of being found on the third search are 5^2/6^3 or ~12% and so on.

So what do the odds come out to? Before anyone searches the odds of the heirloom ending up being at each site are as follows:
1st: ~17%
2nd: ~14%
3rd: ~12%
4th: ~10%
5th: ~8%
6th: ~40%

As you can see, the 6th objective searched will contain the heirloom dramatically more often than any other search (although in a majority of games it won't contain the heirloom)!

As the game goes on, the preference for being the final searcher grows ever stronger. Imagine you've been playing and the first three searches have come up failures, there are three objectives left, but the final objective has an almost 70% chance of being the winner.

How to use this in games? Well, you aren't forced to search. If you're the only side to have models near an objective, you can just leave your troops out of contact and wait. Suppose you and I both get control of three objectives each. If I rush to explore them I have a 42% chance to find the heirloom. If you explore yours after I do, you have a 58% chance to find the heirloom. Not a bad advantage given that both of us control exactly the same number of sites.

Likewise if it has gotten down towards the end of the game and only two objectives are left, whoever searches first is at a real disadvantage. If we're down to two, the first searcher has a 17% chance of finding the heirloom compared to the second searcher ending up with it 83% of the time.

Obviously it is best to control everything, be the only searcher, and kill anything that gets near the heirloom, but it isn't just up to you. Keep the math in mind and you can manipulate when and where the heirloom is found.
Author:  Angmarred [ Wed May 03, 2017 9:47 pm ]
Post subject:  Re: Heirlooms Math
That math works when looking at the scenario overall but in practice it doesn't quite play out that way. Once the first package is opened, if it doesn't have the heirloom, the second package had a 1/6 chance, higher than the 14% you state. So the real strategy is to not give your opponent the chance to open the last package. Basically don't open one unless you control two.
Author:  Ring_of_Gyges [ Wed May 03, 2017 10:31 pm ]
Post subject:  Re: Heirlooms Math
Angmarred wrote:
That math works when looking at the scenario overall but in practice it doesn't quite play out that way. Once the first package is opened, if it doesn't have the heirloom, the second package had a 1/6 chance, higher than the 14% you state. So the real strategy is to not give your opponent the chance to open the last package. Basically don't open one unless you control two.

Perhaps I was unclear, in 14% of all games the heirloom will be in the second place searched. Is that more clear?

Yes the heirloom will be in the second place 1/6 of the time *if it isn't in the first place*. But since it will sometimes be in the first place it will be in the second place less than 1/6 of the time.
Author:  Tar-Minastir [ Thu May 04, 2017 4:14 pm ]
Post subject:  Re: Heirlooms Math
Yes, that is a great analysis. I’ve played the scenario only once, but was warned going in to be prepared for the heirloom to be the final objective. It creates a lot of interesting strategy since you want to grab one objective with the plan to hold it for last. Then it seems you want to search the others as quickly as possible to force the heirloom into your hands. The opponent is doing the same. Of course there’s always the wild card where the objective could end up somewhere else and you have to be prepared for that too. It all seems like a fun puzzle.

I just absolutely hate the deployment in that scenario. It seems to take the focus away from the purpose of the scenario in order to gratuitously add complexity.
Author:  Valadorn [ Mon May 08, 2017 10:50 pm ]
Post subject:  Re: Heirlooms Math
Very very interesting post. I will keep it in mind. In case you have a very small and slow army (like dwarves) it is best to stay and defend an objective until your opponent searches everything else, then get it.
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